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https://physics.stackexchange.com/questions/509315/why-is-there-no-constant-factor-in-the-equation-e-mc2 | # Why is there no constant factor in the equation $E=mc^2$? [duplicate]
As I understand, the metric system starts with an arbitrary weight, and calls it $$1\, \mathrm{kg}$$, then the volume of this weight in water gives the meter. And then the energy is defined based on the meter, second and kilogram.
My question is: given that the relation between meter, kg and joule relies on the density of water (or did so originally), how can we possibly tie them again with $$E=mc^2$$ by adding the speed of light? The density of water seems completely uncorrelated to the speed of light.
Intuitively I would have added an extra constant to Einstein's equation.
• Also what is the constant you propose adding? That might help make the question clearer. – BioPhysicist Oct 21 '19 at 6:04
• Dear all, I feel all the answers arent addressing my issue. I'm not asking wether units are consistent in the sense that they boil down to the same sub units but rather that we have units that were defined initially around the density of water (and then redefined but it still matches with water) and that in E=mc2 we are matching mass & energy around the unrelated speed of light. – Manu de Hanoi Oct 22 '19 at 1:14
• @ManudeHanoi I don't quite understand your confusion. The equation e=MC2 is no different in principle to the equation for KE =1/2MV2- in both equations the physical quantity of energy is the product of a mass and a velocity squared. The choice of units is irrelevant. – Marco Ocram Oct 22 '19 at 8:50
The assumptions on the SI units are not correct.
The kilogram isn't 'defined by an arbitrary weight', it's defined using Planck's constant since 2018:
The kilogram is defined by setting the Planck constant h exactly to 6.62607015×10−34 J⋅s (J = kg⋅m2⋅s−2), given the definitions of the metre and the second
Hence, the kilogram is now essentially defined in terms of the second and the metre. Though it did used to be defined by a platinum-iridium cylinder (and before that a pure platinum cylinder), whose mass was (as close as possible) to the absolute weight of a volume of pure water equal to the cube of the hundredth part of the metre, at 4 Celsius.
The metre is not a derived unit in SI, it's one of the base units, defined in 1983 as:
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.
I'm not too sure where the water based definition you mention comes from, but if you can elaborate, I can see if I can see where it fits into the story for you.
You are correct that the Joule is based on the kilogram, metre and second (since $$J = kg \cdot m^2/s^2$$). And if you did a dimensional analysis on the Joule, you would find it based on the above combination of the units you mention, it doesn't necessarily mean whatever energy you are measuring (in this case some particle, i presume) is correlated to that cylinder of platinum or any other thing used to make a definition.
Put simply, a unit is just shorthand for saying "is equal to this many of a quantity". For example we can define our own unit, the Dave, as equal to 63 times the average mass of a male platypus. I weigh 1 Dave (yeh, I need to lose some mass...). It doesn't mean that me and the average male platypus are in anyway correlated.
To answer, the 'extra constant' bit, what I think you're asking is:
Why is there no constant of proportionality used in $$E=mc^2$$ to relate the Joule to kilogram, metre, and second, since their definitions seem unrelated?
Feel free to correct me if I misunderstood. Constants of proportionality (and other constants) are used, as a way to get a certain unit out of an equation. $$E=mc^2$$ is just an equation, it doesn't require any constant until you choose a system of units to work in. Let's try adding the constant you mentioned, we will call it, k, so now the equation is $$E=kmc^2$$. For getting an answer in Joules, from kilograms, metres and seconds, k would be 1 (it is how the Joule is defined). If you wanted an answer in milliJoules ($$1/1000$$ Joules) from kilograms, metres and seconds, k would now have to be 1000, to balance the units. If you wanted an answer in milliJoules, from grams ($$1/1000$$ Kilograms), metres and seconds, k would be back to being 1, since the alterations made to both sides balance.
A good real world example of this, is natural units, where c=1 (whereas, in SI it's 299792458 $$m/s^2$$, but here the value and units are adjusted to output an energy in electronVolts, instead of Joules), and Energy and mass are measured in eV, but the equation is still $$E=mc^2$$. All the equation says, regardless of any constants, is it takes this much energy to create this much mass from the vacuum.
Hope that helps, feel free to ask more if I was unclear or didn't cover some of your questions - either units or platypus related.
Sources:
SI Unit Definitions
Platypus Mass
• if you define 1 Joule as equal to platypus times mass. and then comes e=mc2 that can be used to define 1 joule as the energy contained in a mass of 1kg/c2 you end up with having 2 redundant definitions of energy implying that 1 platypus =1/c2 . My question is how could platypus(=density of water) and inverse square of the speed of light be related – Manu de Hanoi Oct 22 '19 at 1:32
• @ManudeHanoi The point that everybody is telling you is that the numeric value of $c^2$, in the units you are using, depends on the units you are using. If you change what "platypus" means, that indirectly changes what number $c^2$ is equal to, in platypus units. – knzhou Oct 22 '19 at 5:02
• @ManudeHanoi This is like being amazed that the height and width of a wooden square are always the same in any units, $h = w$, even though you could have measured them in, say, platypus lengths. What do platypuses have to do with the sides of a wooden square? Nothing, but still $h = w$. – knzhou Oct 22 '19 at 5:04
• @knzhou, if so, can you prove it ? Can you prove that 1 joule =1kg(1m/s)^2 is equivalent to 1 joule = energy of (1/c^2) kg ? – Manu de Hanoi Oct 22 '19 at 8:14
• @ManudeHanoi For the proof, see any special relativity textbook. You are correct that it is a nontrivial fact that there is no numeric coefficient in $E = mc^2$. But this has nothing to do with the unit system: if there were a coefficient, it would be the same in every reasonable unit system. – knzhou Oct 22 '19 at 16:13
Well, $$E=mc^2$$ doesn't mean that a Joule is equal to a kilogram times (the speed of light) squared. Actually, if a Joule is defined to be $$\operatorname{Joule} =\operatorname{Newton}\cdot \operatorname{meter} = (\operatorname{kg}\cdot \frac{\operatorname{meter}}{\operatorname{second}^2})\cdot \operatorname{meter}=\operatorname{kg}\cdot \left( \frac{\operatorname{meter}}{\operatorname{second}} \right) ^2$$ Then by plugging 1 kilogram to Einstein's equation, you get that rouphly
$$E = 1\operatorname{kg}\cdot (3\cdot 10^8 \frac{\operatorname{meter}}{\operatorname{second}})^2 = 9 \cdot 10^{16} \operatorname{kg}\cdot \left( \frac{\operatorname{meter}}{\operatorname{second}} \right) ^2=9\cdot10^{16}\operatorname{Joule}$$ Meaning the energy you need to create from nothing a kilogram is that much of energy.
Constants come in handy when we specifically wanted to create a relation that when you plug in some units you will get other units.
For example, if Newtonian gravity says that $$F\propto \frac{m_1 m_2}{r^2}$$ That means that on the left side we have the units $$\operatorname{kg}\cdot \frac{\operatorname{meter}}{\operatorname{second}^2}$$ and on the right we have $$\frac{\operatorname{kg}^2}{\operatorname{meter}^2}$$. If we want the units to match we need a constant with the units of $$\frac{\operatorname{meter}^3}{\operatorname{kg}\cdot \operatorname{second}^2}$$ and give it a value that really gives the force between two masses of a kg one meter apart (in Newtons). In $$E=mc^2$$ you don't need such a constant to balance the units, this formula is derived no matter which units you use.
This concept if constants in physical relations as a way to work with a specific unit system is actually what makes the equations of electromagnetism differ depending on which units you use. In the SI system (meters, kilograms, seconds), one likes to work with the "Coulomb" unit for electric charge, but then needs a constant in Coulomb's law. In the cgs system of units (centimeter, grams, seconds), one defines the electric charge to be such that you don't need a constant. I hope I didn't confuse you with the last paragraph. You are welcome to ask more questions :)
• Can you reconcile the fact that : 1) 1 joule is equal to the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of the force's motion through a distance of one metre AND that: 1)1 joule is the energy in 1/c2 kg ? Yes the units match but I see no correlation between the speed of light and the acceleration of 1 newton over 1 meter – Manu de Hanoi Oct 22 '19 at 1:59
• I think what the OP is really asking is why is it $E=mc^2$ and not, say, $E=\frac{1}{2}mc^2$, which would be similar to Newtonian kinetic energy which has the same kinds of variables on both sides of the equation. – Mark H Oct 22 '19 at 3:13
This is completely wrong. The international system of units SI
Time: $$1$$ second equals:
The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
Length: $$1$$ meter equals:
The distance travelled by light in vacuum in $$(1/299792458) \times$$ second.
Mass: $$1$$ kilogram equals:
The kilogram is defined by setting the Planck constant h exactly to $$6.62607015×10^{−34}\,\mathrm{J\cdot s}\;(\mathrm J = \,\mathrm{kg⋅m^2⋅s^{−2}}$$), given the definitions of the metre and the second.
So we are in the twenty-first century, and the definition of the kilogram depends on the Planck constant, it ain't simple, and it takes care of densities and such.
In addition, the m in $$E=mc^2$$ is not constant anyway, because it is velocity dependent. It is called the relativistic mass and is no longer used in physics, as it is useful to calculate the inertial mass for velocity nearing the speed of light, not the mass of the particle and the weight we find in our scales. One uses the invariant mass given by the length of the four vector:
$$\sqrt{P\cdot P}=\sqrt{E^2-(pc)^2}=m_0c^2.$$
The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference.
• dear Anna, the constants used to define the units were choosen so the side of a cube of 1000 kg of water still measures 1meter. Using arbitrary constants like "The duration of 9192631770 periods of the radiation" instead of the historical arbitrary choice of water doesnt change my question – Manu de Hanoi Oct 22 '19 at 1:37
• @ManudeHanoi Of course it changes it. The new definitions do not depend on density, this was the correction for the desnity,getting a stable to external influences definition. Worth reading the history of SIhttps://usma.org/si-unit-definitions – anna v Oct 22 '19 at 3:49
• it doesnt matter Anna, you've replaced the arbitrary constant of density of water with the arbitrary constant of "9192631770" and "1/299792458" in either case, you end up with 2 definitions of the joule, one based directly on the above constants (1joule=1kg x (1m/s)^2) and one based on the speed of light (1joule = energy of (1/c2) kg). So in order to reconcile the 2 definitions you'd have to somehow prove that they are the same mathematically. And I can't do that, can you ? – Manu de Hanoi Oct 22 '19 at 8:08
• @ManudeHanoi No, these definitions are not mathematically the same! You can not establish natural unit-of-measure-independent correspondence between mass and energy using the first definition. Second definition is not actually a mathematical definition, it says "energy of $1/9 * 10^{-16}$ kg" - but how much is it? Would it be enough to lift some object, or heat some water? It's more like not a definition of Joule, but a statement, that if object of mass $m$ decays into massless objects, an energy $m*c^2$ would be released. – lesnik Oct 22 '19 at 9:10
• Think a bit: there used to be an independent unit, the foot. Obviously it is connected with the human foot. Whose foot? my foot, your foot? An averaged foot?en.wikipedia.org/wiki/Foot_(unit)#Historical_origin . It is obvous that the older defintions have an error, dependent on the width of the average value of a male foot : thats why it was different from city to city. To request that the old foot measure can be connected with the new SI meter unit through a fomula is obviously not possible. The new SI definitions try to use theory constants whose measurement errors are very small. – anna v Oct 22 '19 at 12:01
Short answer: there are no additional factors in this formula because unit of measure for energy is consistent with units of measure for mass, time and distance.
Suppose you decided to create your own system of units and let's forget about relativity theory so far.
You can choose arbitrary unit for mass. You can choose arbitrary unit for time. You can choose arbitrary unit for distance.
But after that you can't arbitrarily choose unit of measure for speed. You'd better use (unit of distance)/(unit of time). Otherwise formulas in your system of units will have some weird constant factors in it.
And you can't use arbitrary choose unit of measure for energy. You'd better use (unit of mass) * (unit of speed)^2. For the same reason.
And in this designed from scratch system of units the classic formula for energy $$E = m * v^2/2$$ will hold. No additional factors appear in this formula, because you have chosen units of measure for speed and energy corresponding to (arbitrarily selected) units for mass, time and distance.
Similarly, numeric values of mass and the speed of light will be different in your system of units, but the equation $$E = m * c^2$$ will hold.
UPDATE (answer to a question in comment).
So, if formulas $$E = m * v^2/2$$ and $$E = m * c^2$$ are correct with one system of units of measure, these formulas would be correct with all other systems of units of measure if the unit of measure of energy "corresponds" to units of measure of time, distance and mass. This is just a pure math. From your question I guess that this is clear. Let me know if it is not. Please note, that these two formulas have very different physical meaning, but when switching from one units of measure to another the math is the same.
The classic formula $$E = m * v^2/2$$ is correct with all units of measure. It does not matter if the unit of distance was defined as some fraction of distance from pole to equator of some planet or as a distance between ears of some species living on it.
The second formula ($$E = m * c^2$$) can be formulated in units-of-measure-independent form. It says: there is some universal speed (speed of light), and the total energy of any object is twice as much as you would get if you use the classic formula of kinetic energy to calculate the kinetic energy of that object moving with our universal speed.
Why "twice as much" - it's a different question. It's relativity theory. But as long as there are no weird constant factors depending on units of measure in the formula $$E = m * v^2/2$$ there would be no such factors in $$E = m * c^2$$.
• Can you reconcile the fact that : 1) 1 joule is equal to the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of the force's motion through a distance of one metre AND that: 1)1 joule is the energy in 1/c2 kg ? Yes the units match but I see no correlation between the speed of light and the acceleration of 1 newton over 1 meter – Manu de Hanoi Oct 22 '19 at 2:00
• @ManudeHanoi I tried to answer your question and updated my original post - the answer did not fit into comment. – lesnik Oct 22 '19 at 7:48
• I have no problem with and never mentionned the kinetic energy definition, I dont know why your edit mentions it. My problem is that we have defined the joules using 2 definitions, one that involves the density of water and one that involves the speed of light. And these 2 constants are unrelted – Manu de Hanoi Oct 22 '19 at 7:57
• @ManudeHanoi Note, that definition of Newton is that this is the force which is required to accelerate 1 kg with 1 $(meter/sec)^2$. It also involves kilogram, so the first of your definitions of Joule (1 Newton * 1 meter) also involves density of water! – lesnik Oct 22 '19 at 8:08
• @ManudeHanoi So, density of water is involved in both definitions, so there are no problems with it? Ok. And only one of these formulas mentions speed of light. This second definition ($E=mc^2$) brings some new physical meaning, it establishes some natural correspondence between mass and energy. It tells that there is a natural unit-of-measure-independent way to measure energy in kilograms or mass in Joules or electron-vaults or whatever. Why the factor between mass and energy is exactly $c^2$ - well, this is subject of relativity theory. Absence of UOM-dependent constants in it - pure math. – lesnik Oct 22 '19 at 8:29
Historically, the kilogram was defined in terms of the metre and not the other way around. Originally, the metre was defined as "one ten-millionth of the shortest distance from the North Pole to the equator passing through Paris." Based on this definition of the metre, the kilogram was defined as the mass of one litre of water. So the kilogram was not just some arbitrary mass.
Of course, these definitions are historically where they came from. Today, these unit are defined in more sophisticated ways, as explained in some of the other answers provided here.
About the lack of constants in the famous equation $$E=m c^2$$. Remember, this equation is not proposed as a definition of units and should not be interpreted that way. Instead, it gives the relationship between mass and energy. It shows that one is proportional to the other. The role of the $$c^2$$ is to match the dimensions. In that sense, a numerical dimensionless constant factor would be meaningless. In fact, arbitrary numerical factors would appear when one assigns various different units of measure to the different quantities in the equation.
That does not mean that one cannot use the equation for the definition of units, by relating a new unit of measure to existing units. In terms of the SI units, the Joule is related to the kilogram and one metre-per-second for velocity. One could in principle use this equation to define a new unit of energy associated with one kilogram.
• we do define the joule based on water and paris position (with the proxy of wavelengths) but we could define the same joule as the energy in 1/c2 kg (E=mc2). How come the 2 definitions amount to the same joule ? – Manu de Hanoi Oct 22 '19 at 8:20
• The current system of units has been defined to be consistent with previous definitions. Remember that we do not use the water and the Paris position anymore, because they don't provide the same accuracy and precision that we can get with the current definitions. – flippiefanus Oct 22 '19 at 9:15
• I understand that, but the constants used to define the units are just as arbitrary as paris and density of water as explained in OP. Be it paris position or 184154131101 the wavelength of khsdfgkjbasdfuh – Manu de Hanoi Oct 22 '19 at 9:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 35, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9529881477355957, "perplexity": 408.6404198492481}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703506640.22/warc/CC-MAIN-20210116104719-20210116134719-00657.warc.gz"} |
https://johnhcochrane.blogspot.in/2015/04/ | ## Monday, April 27, 2015
### Unit roots in English and Pictures
After my unit roots redux post, a few people have asked for a nontechnical explanation of what this is all about.
Suppose there is an unexpected movement in any of the data we look at -- inflation, unemployment, GDP, prices, etc. Now, how does this "shock" affect our best estimate of where this variable will be in the future? The graph shows three possibilities.
## Friday, April 24, 2015
### Unit roots, redux
Arnold Kling's askblog and Roger Farmer have a little exchange on GDP and unit roots. My two cents here.
I did a lot of work on this topic a long time ago, in How Big is the Random Walk in GNP? (the first one) Permanent and Transitory Components of GNP and Stock Prices” (The last, and I think best one) "Multivariate estimates" with Argia Sbordone, and "A critique of the application of unit root tests", particularly appropriate to Roger's battery of tests.
The conclusions, which I still think hold up today:
Log GDP has both random walk and stationary components. Consumption is a pretty good indicator of the random walk component. This is also what the standard stochastic growth model predicts: a random walk technology shock induces a random walk component in output but there are transitory dynamics around that value.
## Wednesday, April 22, 2015
### The right to herd
Just when you thought financial regulation couldn't get more expansive and incoherent, our Justice Department comes in to defend morons' right to herd.
As explained in the Wall Street Journal at least, Mr. Navinder Singh Sarao is now under arrest, fighting extradition to the US, and his business ruined, for "spoofing" during the flash crash.
What is that? The Journal's beautiful graph at left explains.
The obvious question: Who are these traders who respond to spoofing orders by placing their own orders? Why is it a crucial goal of law and public policy to prevent Mr. Sarao from plucking their pockets? Is "herding trader" or "momentum trader" or "badly programmed high-speed trading program" or just simple "moron in the market" now a protected minority?
Why is Mr. Sarao being prosecuted and not all the people who wrote badly programmed algorithms that were so easily spoofed? If this caused the flash crash (how, not explained in the article) are they not equally at fault?
I don't mean by this a defense of the crazy stuff going on in high speed trading. As explained here, I think one second batch auctions are a much better market structure. But the whole high speed trading thing is largely a response to SEC regulations in the first place, the order routing regulation, discrete tick size regulation, and strict time precedence regulation. A fact which will probably not enter at Mr. Sarao's trial (he doesn't seem to have billions for a settlement) and will give him little comfort in jail.
And maybe, just maybe, there is something more coherent here than the Journal lets on. I'll keep reading hoping to find it and welcome comments who can.
A larger thought. We still really want to rely on regulators to spot all the problems of finance and keep us safe from more crashes?
Update: Craig Pirrong excellent commentary here via a good FT alphaville post. Great quote:
The complaint alleges that Sarao employed the layering strategy about 250 days, meaning that he caused 250 out of the last one flash crashes. [my emphasis] I can see the defense strategy. When the government expert is on the stand, the defense will go through every day. “You claim Sarao used layering on this day, correct?” “Yes.” “There was no Flash Crash on that day, was there?” “No.” Repeating this 250 times will make the causal connection between his trading and Flash Clash seem very problematic, at best.
Update 2: Reading various commentaries that I can't find to cite any more, I realize that "front running" more than "herding" is the protected class. You "spoof" by putting in a bunch of orders just outside the current spread. The algorithms that respond to that think this behavior means some big orders coming, so try to front run those by buying. They cross the spread to take the small order you put on the other side. Or so the story goes. In any case, viewed as spoofers vs. front-runners it's harder still to have sympathy for the latter.
Update 3: Good Bloomberg View coverage from Matt Levine and John Arnold, the source of the above front-running observation.
## Monday, April 20, 2015
### Consumption-based model and value premium
The consumption based model is not as bad as you think. (This is a problem set for my online PhD class, and I thought the result would be interesting to blog readers.)
I use 4th quarter to 4th quarter nondurable + services consumption, and corresponding annual returns on 10 portfolios sorted on book to market and the three Fama-French factors. (Ken French's website)
The graph is average excess returns plotted against the covariance of excess returns with consumption growth. (The graph is a distillation of Jagannathan and Wang's paper, who get any credit for this observation. The lines are OLS cross-sectional regressions with and without a free intercept.)
## Friday, April 17, 2015
### Macro Handbook 2
Last week I attended the first half of the conference on the Handbook of Macroeconomics Volume 2, organized by John Taylor and Harald Uhlig, held at Hoover. The conference program and most of the papers are here. The second half will be in Chicago April 23-25, program here
Overall, this Handbook is shaping up as a very useful resource. Really good summary and review papers are a natural way in to long literatures. Bad summary and review papers are long and boring. The conference produced the first kind. Most of the papers are rough first drafts, so make a note to come back when they're finished. A few highlights (with apologies to authors I've left out; I can't review them all here.)
## Thursday, April 16, 2015
### Banking at the IRS
A while ago in two blog posts here and here I suggested many ways other than currency to get a zero interest rate if the government tries to lower rates below zero. Buy gift cards, subway cards, stamps; prepay bills, rent, mortgage and especially taxes -- the IRS will happily take your money now and you can credit it against future tax payments; have your bank make out a big certified check in your name, and sit on it, don't cash incoming checks. Start a company that takes money and invests in all these things (as well as currency).
Chris and Miles Kimball have an interesting essay exploring these ideas "However low interest rates might go, the IRS will never act like a bank." Their central point: sure that's how things work now. But with substantial negative interest rates, all of these contracts can change. It's technically possible in each case for people and businesses to charge pre-payment penalties amounting to a negative nominal rate.
Reply: Sure, in principle. Nominal claims can all be dated, and positive or negative interest charged between all dates.
But this did not happen in the US and does not happen in other countries for positive inflation and high nominal rates, despite symmetric incentives, and at rates much higher than the contemplated 3-5% or so negative rates. Yes, with large nominal rates there is pressure to pay faster, inventory cash-management to reduce people's holdings of depreciating nominal claims, but this pervasive indexation of nominal payments did not break out. The IRS did not offer interest for early payment.
More deeply, what they're describing is a tiny step away from perfect price indexing. If all nominal payments are perfectly indexed to the nominal interest rate, accrued daily, then it's a tiny change to index all prices themselves to the CPI, accrued daily. If "how much you owe me," say to rent a house, is legally, contractually, and mechanically determined as a value times e^rt, and changes day by day, then e^(pi t) is just as easy.
So, price stickiness itself would (should!) disappear under this scenario.
Price stickiness has always been a bit of a puzzle for economists. As the Kimballs speculate how easy it is to index payments to negative interest rates, so economists speculate how easy it is to index payments to inflation. Yet it seems not to happen.
So this point of view strikes me as a bit of a catch-22 for its advocates, who generally are of the frame of mind that prices and nominal contracts are sticky and that’s why negative nominal rates are a good idea to "stimulate demand" in the first place. If we can have negative nominal rates and change all these legal and contractual zero-rate promises to allow it, then prices won't be sticky any more! Conversely, I should be cheering, as it amounts to a broad push to unstick prices. That has long seemed to me the natural policy response to the view that sticky prices are the root of all our troubles. It would allow negative rates, but eliminate their need as well.
Alas, the world seems remarkably resistant to time-indexing all payments.
## Wednesday, April 15, 2015
### Gdefault needs not Grexit
The little grumpy cartoon usually represents me pounding my coffee down in agreement as the WSJ exposes some idiocy. Last week, alas, I spilled my grumpy coffee in disagreement with a little part of its otherwise excellent "The case for letting Greece go."
Thursday marks another deadline in Greece’s struggle to avoid default, as a €450 million payment to the International Monetary Fund comes due. Athens says it will meet this obligation, but sooner or later Prime Minister Alexis Tsipras and his government will miss a payment to someone if it doesn’t agree with creditors on a new bailout. An exit from the euro would then be a real possibility.
Please can we stop passing along this canard -- that Greece defaulting on some of its bonds means that Greece must must change currencies. Greece no more needs to leave the euro zone than it needs to leave the meter zone and recalibrate all its rulers, or than it needs to leave the UTC+2 zone and reset all its clocks to Athens time. When large companies default, they do not need to leave the dollar zone. When cities and even US states default they do not need to leave the dollar zone. A common currency means that sovereigns default just like large financial companies. (Yes, a bit of humor in the last one.)
## Tuesday, April 14, 2015
### Blanchard on Countours of Policy
Olivier Blanchard, (IMF research director) has a thoughtful blog post, Contours of Macroeconomic Policy in the Future. In part it's background for the IMF's upcoming conference with the charming title Rethinking Macro Policy III: Progress or Confusion?” (You can guess my choice.)
Olivier cleanly poses some questions which in his view are likely to be the focus of policy-world debate for the next few years. Looking for policy-oriented thesis topics? It's a one-stop shop.
Whether these should be the questions is another matter. (Mostly no, in my view.)
As a blogger, I can't resist a few pithy answers. But please note, I'm mostly having fun, and the questions and essay are much more serious.
## Thursday, April 2, 2015
### The sources of stock market fluctuations
How much do dividend-growth vs. discount-rate shocks account for stock price variations?
An under-appreciated point occurred to me while preparing for my Coursera class and to comment on Daniel Greewald, Martin Lettau and Sydney Ludvigsson's nice paper "Origin of Stock Market Fluctuations" at the last NBER EFG meeting
The answer is, it depends the horizon and the measure. 100% of the variance of price dividend ratios corresponds to expected return (discount rate) shocks, and none to dividend growth (cash flow) shocks. 50% of the variance of one-year returns corresponds to cashflow shocks. And 100% of long-run price variation corresponds to from cashflow shocks, not expected return shocks. These facts all coexist
I think there is some confusion on the point. If nothing else, this makes for a good problem set question.
The last point is easiest to see just with a plot. Prices and dividends are cointegrated. Prices correspond to dividends and expected returns. Dividends have a unit root, but expected returns are stationary. Over the long run prices will not deviate far from dividends. So 100% of long-enough run price variation must come from dividend variation, not expected returns.
Ok, a little more carefully, with equations.
A quick review:
The most basic VAR for asset returns is $\Delta d_{t+1} = b_d \times dp_{t}+\varepsilon_{t+1}^{d}$ $dp_{t+1} = \phi \times dp_{t} +\varepsilon_{t+1}^{dp}$ Using only dividend yields dp, dividend growth is basically unforecastable $$b_d \approx 0$$ and $$\phi\approx0.94$$ and the shocks are conveniently uncorrelated. The behavior of returns follows from the identity, that you need more dividends or a higher price to get a return, $r_{t+1}\approx-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}%$ (This is the Campbell-Shiller return approximation, with $$\rho \approx 0.96$$.) Thus, the implied regression of returns on dividend yields, $r_{t+1} = b_r \times dp_{t}+\varepsilon_{t+1}^{r}$ has $$b_r = (1-\rho\phi)+0 = 1-0.96\times0.94 = 0.1$$ and a shock negatively correlated with dividend yield shocks and positively correlated with dividend growth shocks.
The impulse response function for this VAR naturally suggests "cashflow" (dividend) and "expected return" shocks, (d/p). (Sorry for recycling old points, but not everyone may know this.)
Three propositions:
• The variance of p/d is 100% risk premiums, 0% cashflow shocks
Iterate forward the return identity, to get $dp_{t} =\sum_{t=1}^{\infty}\rho^{j-1}r_{t+j}-\sum_{t=1}^{\infty}\rho ^{j-1}\Delta d_{t+j}$ multiply by $$dp_t$$ and take expectations (all variables are demeaned) $\sigma^{2}\left( \log\frac{P_{t}}{D_{t}}\right) =\sigma^{2}\left( dp_{t}\right) =\sum_{t=1}^{\infty}\rho^{j-1}cov(dp_{t},r_{t+j})-\sum _{t=1}^{\infty}\rho^{j-1}cov(dp_{t},\Delta d_{t+j}),$ But $$b_d \approx 0$$, so the dividend growth terms are all zero, and 100% of the variance of price-dividend ratios corresponds to time-varying expected returns. (I know this will bore people familiar with it and befuddle those who are not. "Discount rates" has a bit more leisurely review and citations)
But
• The variance of returns is 50% due to risk premiums, 50% due to cashflows.
$r_{t+1}=-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}%$ $\varepsilon_{t+1}^{r} =-\rho\varepsilon_{t+1}^{dp}+\varepsilon_{t+1}^{d}$ $\sigma^{2}\left( \varepsilon_{t+1}^{r}\right) =\rho^{2}\sigma^{2}\left( \varepsilon_{t+1}^{dp}\right) +\sigma^{2}\left( \varepsilon_{t+1}^{d}\right)$ The variance of the two shocks comes out very close to a 50/50 decomposition at an annual horizon. It's a lot more expected return at a daily horizon, and less at longer horizons. Here I use the fact that dividend growth and dividend yield shocks are basically uncorrelated.
Why are returns and p/d so different? Current cash flow shocks affect returns. But a shock to dividends, when prices rise at the same time, does not affect the dividend price ratio. (This is the essence of the Campbell-Ammer return decomposition.)
The third proposition is less familiar:
• The long-run variance of stock market values (and returns) is 100% due to cash flow shocks and none to expected return or discount rate shocks.
Here's why: $\Delta p_{t+1} =-dp_{t+1}+dp_{t}+\Delta d_{t+1}$ $p_{t+k}-p_{t} =-dp_{t+k}+dp_{t}+\sum_{j=1}^{k}\Delta d_{t+j}$ so as k gets big, ${var} (p_{t+k}-p_{t}) \rightarrow 2 {var}(dp_t) + k {var}(\Delta d_{t})$ The first term approaches a constant, but the second term keeps growing. As above the central fact is that P and D are cointegrated while expected returns are stationary.
This is related to a point made by Fama and French in their Equity Premium paper. Long run average returns are driven by long run dividend growth plus the average value of the dividend yield. The difference in valuation -- higher prices for given set of dividends -- can affect returns in a sample, as higher prices for a given set of dividends boost returns. But that mechanism can't last. (Avdis and Wachter have a nice recent paper formalizing this point.) It's related to a similar point made often by Bob Shiller: Long run investors should buy stocks for the dividends.
A little more generality as this is the new bit.
$p_{t+k}-p_t = dp_{t+k}-dp_t + \sum_{j=1}^{k}\Delta d _{t+j}$ $p_{t+k}-p_t = (\phi^{k}-1)dp_t + \sum_{j=1}^{k}\phi^{k-j} \varepsilon^{dp}_{t+j} + \sum_{j=1}^{k} \varepsilon^d _{t+j}$ $var(p_{t+k}-p_t) = \frac{(1-\phi^{k})^2}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + \frac{(1-\phi^{2k})}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + k\sigma^2(\varepsilon^d)$ $var(p_{t+k}-p_t) = 2\frac{(1-\phi^{k})}{1-\phi^2} var(\varepsilon^{dp}_{t+1}) + k var(\varepsilon^d_{t+j})$ So you can see the last bit takes over. It doesn't take over as fast as you might think. Here's a graph using sample values,
At a one year horizon, it's just about 50/50. The dividend shocks eventually take over, at rate 1/k. But at 50 years, it's still about 80/20.
Exercise for the interested reader/finance professor looking for problem set questions: Do the same thing for long horizon returns, $$r_{t+1}+r_{t+2}+...+r_{t+k}$$ using $$r_{t+1} = -\rho dp_{t+1} + dp_t + \Delta d_ {t+1}$$ It's not so pretty, but you can get a closed form expression here too, and again dividend shocks take over in the long run.
Be forewarned, the long run return has all sorts of pathological properties. But nobody holds assets forever, without eating some of the dividends.
Disclaimer: Notice I have tried to say "associated with" or "correspond to" and not "caused by" here! This is just about facts. The facts have just as easy a "behavioral" interpretation about fads and bubbles in prices as they do a "rationalist" interpretation. Exercise 2: Write the "behavioralist" and then "rationalist" introduction / interpretation of these facts. Hint: they reverse cause and effect about prices and expected returns, and whether people in the market have rational expectations about expected returns. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3424068093299866, "perplexity": 3397.381512590394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948513784.2/warc/CC-MAIN-20171211164220-20171211184220-00264.warc.gz"} |
https://www.physicsforums.com/threads/lie-dragging-a-vector-around-the-ellipse.156898/ | # Lie dragging a vector around the ellipse
1. Feb 18, 2007
### wandering.the.cosmos
I've been trying to understand exactly how the Lie derivative parallel transports a vector, by working out an explicit example: Lie dragging $\partial/\partial x$ at (a,0) on the x-y plane anticlockwise along the ellipse
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
I choose to parametrize the ellipse using
$$x=a\cos[\theta]$$
$$y=b\sin[\theta]$$
so that we have the tangent vector along the ellipse to be
$$\frac{d}{d\theta} = -\frac{a}{b} y \frac{\partial}{\partial x} + \frac{b}{a} x \frac{\partial}{\partial y}$$
Next I compute the Lie brackets
$$\left[\frac{d}{d\theta},\frac{\partial}{\partial x}\right] = \frac{b}{a} \frac{\partial}{\partial y}$$
$$\left[\frac{d}{d\theta},\frac{\partial}{\partial y}\right] = -\frac{a}{b} \frac{\partial}{\partial x}$$
From this I conclude that parallel transporting $\partial/\partial x$ from (a,0) through an angle $\theta$ along the ellipse yields
$$\exp\left[\theta \pounds_{d/d\theta}\right] \frac{\partial}{\partial x} = \sum_{n=0}^\infty \left( \frac{b}{a}\frac{(-1)^n}{(2n+1)!} \theta^{2n+1} \frac{\partial}{\partial y} + \frac{(-1)^n}{(2n)!} \theta^{2n} \frac{\partial}{\partial x} \right) \\ = \frac{b}{a} \sin[\theta] \frac{\partial}{\partial y} + \cos[\theta] \frac{\partial}{\partial x} \\ = \frac{y}{a} \frac{\partial}{\partial y} + \frac{x}{a} \frac{\partial}{\partial x}$$
It seems this implies that the parallel transported $\partial/\partial x$ gives a vector that is inclined at an angle $\phi$ such that
$$\tan[\phi] = \frac{y}{x}$$
But the derivative dy/dx of the ellipse at say the point x = y is $dy/dx = -b^2/a^2$, and so the "outward" normal to the ellipse has a gradient of $a^2/b^2$. Now, can we expect the $\partial/\partial x$ to remain perpendicular to the ellipse as we lie drag it on its circumference? Why or why not? It does not seem to be the case, as this computation shows. If so, how should we understand what parallel transportation on an ellipse means? (Of course, I could have made an error somewhere, and would appreciate any corrections.)
Last edited: Feb 18, 2007
2. Feb 18, 2007
### Doodle Bob
I can't really speak to your computations which at first glance look kosher.
But I can say that one would *not* expect the image of d/dx under the elliptical flow to remain perpendicular to the ellipse at all points. The flow of a vector field will preserve the angles of vectors (or even orthogonality) only if it preserves (up to a nonzero function) the Riemannian metric - which in this case is the Euclidean inner product. i.e. the Lie derivative of the metric with respect to the given vector field should be a multiple of that metric. I don't think that a vector field which is tangent along a noncircular ellipse will satisfy that -- or least the one that you have chosen.
Furthermore, don't confuse parallel translation along a curve with transportation by the flow of a vector field tangent to that curve. Parallel transport *will* by definition preserve angles, the dynamic flow (or rather a dynamic flow -- there are infinitely many after all because there are infinitely many ways to extend a vector field along a curve to a neighborhood abotu that curve) tangent to the curve will not necessarily.
3. Feb 18, 2007
### Doodle Bob
Regarding parallel translation...
Since we're dealing with a flat 2-dimensional Riem. manifold here (The Eucldiean plane), calculating the parallel transport of a vector around the ellipse is easy.
In your case, we have d/dx at the point (a,0). It is of unit length and perpendicular to the tangent vector of the ellipse there and pointing "outside" the filled ellipse. The parallel transport of this vector about the curve will have the very same properties at each given point, i.e. unit length and normal to the curve and pointing outside of the filled ellipse (its relative orientation with respect to the tangent vector will not change). Thus, it will be the outward pointing normal vector field of the ellipse.
Things change, of course, if the metric is not flat (say, hyperbolic 2-space), then we get nontrivial holonomy so that, when we parallel-transport a vector all the way around a closed curve, we often end up with a different vector at the place of origin. In the case of hyperbolci 2-space, the difference between the original vector and its wrap-around parallel-transport is related to the total area enclosed by the curve.
Hope that helps.
4. Feb 18, 2007
### pmb_phy
Note: Lie dragging a vector takes place on a one of the curves which forms a congruence which fills the manifold so keep in mind that the ellipse is part of this congruence.
Pete
5. Feb 19, 2007
### wandering.the.cosmos
I made a sign error with my Lie brackets -- both of them should read
$$\left[\frac{d}{d\theta},\frac{\partial}{\partial x}\right] = -\frac{b}{a} \frac{\partial}{\partial y}$$
$$\left[\frac{d}{d\theta},\frac{\partial}{\partial y}\right] = \frac{a}{b} \frac{\partial}{\partial x}$$
But that'd mean that
$$\exp\left[\theta \pounds_{d/d\theta}\right] \frac{\partial}{\partial x} = -\frac{y}{a} \frac{\partial}{\partial y} + \frac{x}{a} \frac{\partial}{\partial x}$$
And hence it seems that $\partial/\partial x$ would start dipping in the negative y direction as it's being dragged along the ellipse? If so this is quite far from my expectation that Lie dragging gives us a way of parallel transporting w/o using a metric.
Last edited: Feb 19, 2007
6. Feb 19, 2007
### Doodle Bob
That would be because the two processes are very different. The Lie process for example depends *completely* on how you extend the tangent vector of the curve away from the curve. You chose to extend it by creating the family of parallel ellipses. But you could have also chosen a family of conic sections with some other property, say, sharing a directrix. This would have resulted in a different family of curves hence a different vector field to "Lie drag" along and hence a different Lie transportation than the one you ended up with.
As I stated before, some vector fields do result in isometries, but you have to choose them wisely. As noted by Pete, your ellipse vector field will generate non-isometric transformations on the plane. I believe in this case they would have to foliate to circles and lines.
Similar Discussions: Lie dragging a vector around the ellipse | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9060774445533752, "perplexity": 407.3187518278573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825497.18/warc/CC-MAIN-20171023001732-20171023021732-00263.warc.gz"} |
https://math.stackexchange.com/questions/956684/what-is-probability-that-a-sample-covers-all-classes | # What is probability that a sample covers all classes?
Suppose:
1. there are $n$ distinct classes in a population (i.e. each member of the population is a member of exactly one class),
2. (For simplicity) the classes are all equally sized and the probability of selecting (with replacement) a member from any one class is $1/n$.
3. $x$ samples are taken (with replacement) where $x \ge n$.
What is the probability $p(x,n)$ that each class is represented by at least one of the samples?
What I've tried: special case $$p(n,n) = \frac{n!}{n^n}.$$ e.g. $$p(6,6) = \frac{6!}{6^6} = \frac{6}{6} \times \frac{5}{6} \times ... \times \frac{2}{6} \times \frac{1}{6}.$$
The 6 distinct classes could be the faces 1,2, .. , 6 on a fair die. $p(6,6)$ is the probability that when the die is rolled 6 times, no faces come up twice. Any face is OK for the first roll, then 5 faces, then 4, ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8608617782592773, "perplexity": 195.20858921886907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803737.78/warc/CC-MAIN-20210126202017-20210126232017-00314.warc.gz"} |
http://www.michaelshell.org/tex/mciteplus/ | # The Mciteplus Package Homepage
## Overview
The mciteplus LaTeX package is an enhanced reimplementation of the mcite package, which was written by Thorsten Ohl, which provides support for the grouping of multiple citations together as is often done in physics journals. An extensive set of features provide for other applications such as reference sublisting.
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https://astronomy.stackexchange.com/questions/28327/if-a-giant-object-would-approach-the-earth-would-earth-start-to-turn-faster-or | If a giant object would approach the Earth, would Earth start to turn faster or slower while the object starts to become nearer and nearer?
I am asking myself this question: if a giant celestial object would approach the Earth, would Earth start to turn faster (then day duration would be diminished) or slower (day duration increased) while the object starts to become nearer and nearer? Thank you!
• I believe it depend on which direction and speed it is coming toward the earth. Of it is a fly by then it depend otherwise, it is going to slow down if its is headed straight for the center of the earth – Arvin Kushwaha Nov 11 '18 at 2:27
• hi @ipmp . Welcome new user. The tidal effects would be absolutely tiny. There are plenty of other effects as well (relativistic! quantum! etc) But this question is simply about simple Newtonian physics of bodies. It is completely inappropriate to confuse the various domains. OP is just asking about how a point mass angular momentum is affected by being moved about. – Fattie Nov 11 '18 at 7:36
• It can be both depending if the approach is kind of prograde or or retrograde as well as depending on the actual position of the passing body. Earth has already gravitational "handles" as it is oblate and everything will depend on the relative position of the giant object. If it is massive enough it can even made its own handles and eventually things might get stripped apart. – Alchimista Dec 24 '18 at 12:24
Could be, you'd be better to ask this on the Physics site, by the way.
No, the spinning of a perfectly round object is not affected at all, it just keeps spinning identically.
Let's say ..
say another Sun appeared, just like that.
Yes, the path that Earth is taking would go crazy. We might fly off, head towards this new Sun, starting making a figure of 8, or some crazy path!
But the spinning as such of the Earth would be totally unchanged.
The "way we point" ("North up" so to speak) would be completely unchanged, and the speed we turn at (ie, 24 hours a turn) would be unchanged.
This might seem surprising but, well, you can learn all about it in Newtonian physics!
Once again, no change to the spin as such of the Earth. the "direction we point" (ie the axis we spin around) and the time taken is totally unchanged if (for some bizarre reason!) we are "moved around in the heavens".
Note that there would be some absolutely tiny effects due to things like relativity (have a read!), the water on our planet sloshing around, and after all our Moon could be moved, bashed, etc. These effects on the speed we are spinning would be extremely small over the short period of time you're considering, in the "sci fi" scenario you outline, as seen in the diagram.
• Can you support "But the spinning as such of the Earth would be totally unchanged" with some science based explanation? Also, tidal distortion is a reality of planetary rotation, and to just say "The tidal effects would be absolutely tiny" and "this question is simply about simple Newtonian physics" and then give a wrong answer is unhelpful and unscientific. – uhoh Nov 11 '18 at 10:12
• The tidal effects would be absolutely tiny. If you can prove otherwise, go for it. This question is about basic Newtonian physics. It is foolish to answer in a different domain. It's not clever or pedantic, it's just incorrect to answer in the wrong domain. – Fattie Nov 11 '18 at 10:15
• tidal effects on Earth alone may be tiny because it is "very round" - but what about tidal effects on the Earth-Moon combo, which is much less symmetric? I suppose the moon orbit could be changed, even drastically, even be "stolen". In the long run this would influence the earth rotation because predictions how Earth rotation changes by 2.5 ms per century needs to be adjusted. Then again, all this is still either negligible or going in unpredictable directions - not to mention that the influx on the Earth orbit (as demonstrated in the highly accurate image) might be a lot more drastic – Hagen von Eitzen Nov 11 '18 at 14:30
• per Wikipedia The solar tidal force is 46% as large as the lunar. Moon: 1.1 × 10−7 g, Sun: 0.52 × 10−7 g, so here you are just saying wrong stuff that no amount of bold, italics or "!!!!!!!!! Gosh!" can right. However, I sure would like to know how you were able to produce the bold italics! – uhoh Nov 11 '18 at 17:10
• @uhoh italics is one asterisk either side of the expression, bold needs two asterisks, bold italics is three :-) – Chappo Hasn't Forgotten Monica Dec 12 '18 at 9:16
If a giant object would approach the Earth, would Earth start to turn faster or slower while the object starts to become nearer and nearer?
tl;dr: Probably will slow it down a small amount, but may induce a wobble that make a simple answer impossible without further details.
The problem is complicated. I'll leave a short answer.
The question is carefully written in (at least two) ways. It is agnostic to the size of the effect, instead asking only about its sign. So arguments that the effects are too tiny to matter are moot. It also does not specify the nature, distance, or duration of the approach. That makes the question quite difficult to answer definitively.
induced tidal distortion
The most straight-forward effect we know about is slowing due to tidal forces. The Theory subsection of Wikipedia's article Tidal acceleration outlines this nicely.
Because the Earth is slightly stretchy, the Moon's or another close body's gravity induces a tidal bulge. Wikipedia's Earth tide gives several components of a thorough decomposition, but the top two are about 38 cm due to the Moon and 17 cm due to the Sun. These exist on both the near and far side of each body.
Because the Earth responds slowly, these have a "lag". The lumps exist offset from the line between the Earth and the other body, and this allows for a torque. In the case of the Earth-Moon system that torque then tends to slow the Earth's rotation.
If the body is double in mass, the induced bulge's mass will be roughly linearly proportional and since the torque proportional to both it will be quadruple, proportional to $$m^2$$. As shown in that subsection it is also proportional to \$r^{-6}. So just for example, a planet about the size of the Earth (~80 Moons) and half the distance of the moon would have a torque 80 x 80 x 64 times larger.
However, it's hard to imagine a scenario where that could last for a thousand years to have a huge effect on Earth's rotation speed. This would be small, but definitely negative.
interaction with equatorial bulge
The body could pass perpendicular to the Earth's equator as well, and produce a really huge torque on Earth's equatorial bulge (expressed often as the $$J_2$$ term). That bulge is more like 30 kilometers, compared to the 30 centimeters of the induced tidal bulge by the Moon.
A perpendicular pass like that would induce a substantial wobble in the Earth's rotation during the time of passage, but I don't think it is easy to say with certainty the sign of the effect on the rotation rate with such complex motion and no further details.
• :) Quantative hurts. So the example is an object the size of Earth, which arrives from outer space and passes us - we'll say as close as you like , 10,000 km separation. Let's say it's near us (within a million miles) for 10 minutes. No wait, let's say a day .. no, let's say a month. How much is the day slowed by tidal effects? – Fattie Nov 13 '18 at 1:39
• @Fattie that's a detailed calculation requiring a detailed description of the particular conditions. The beauty of the question is that it does not ask for the magnitude, only the sign. – uhoh Nov 13 '18 at 1:42
• Heh! :) I just gave you exact figures, that match the story description in the question. – Fattie Nov 13 '18 at 1:43
• @Fattie you can't ask a new question in comments. You are not new to SE and so you already know what comments are for, and what they are not for. – uhoh Nov 13 '18 at 1:46
• BTW it is three asterisks front and back for this :) – Fattie Nov 13 '18 at 3:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6981985569000244, "perplexity": 863.061310587522}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487620971.25/warc/CC-MAIN-20210615084235-20210615114235-00291.warc.gz"} |
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http://www.aanda.org/articles/aa/abs/2005/48/aa3592-05/aa3592-05.html | Free Access
Issue A&A Volume 444, Number 3, December IV 2005 767 - 775 Extragalactic astronomy https://doi.org/10.1051/0004-6361:20053592
A&A 444, 767-775 (2005)
DOI: 10.1051/0004-6361:20053592
## COMPTEL observations of the -ray blazars 3C 454.3 and CTA 102 during the CGRO mission
S. Zhang1, 2, W. Collmar2 and V. Schönfelder2
1 Laboratory for Particle Astrophysics, Institute of High Energy Physics, PO Box 918-3, Beijing 100049, PR China
e-mail: szhang@mail.ihep.ac.cn
2 Max-Planck-Institut für extraterrestrische Physik, Postfach 1613, 85741 Garching, Germany
(Received 8 June 2005 / Accepted 5 August 2005 )
Abstract
We have investigated the MeV behaviour of the -ray blazars 3C 454.3 and CTA 102 by analyzing all COMPTEL observations of this sky region during the complete CGRO mission. Both sources are detected by COMPTEL at the upper COMPTEL energies, although their flux estimates may be uncertain by possible minor contributions of nearby unidentified EGRET -ray sources. While CTA 102 was only detected at energies above 10 MeV during the early mission, 3C 454.3 is most significantly detected in the COMPTEL 3-10 MeV band in the sum of all data. Time-resolved analyses indicate a weak (near COMPTEL threshold) but likely steady 3-10 MeV emission over years, being independent of the observed time variability at energies above 100 MeV as observed by EGRET. This energy-dependent variability behaviour suggests different emission mechanisms at work in the two bands. Putting the COMPTEL fluxes in multifrequency perspective (radio to -rays) reveals for both sources the typical two-hump blazar spectrum, with a low-energy maximum around the IR and a high-energy maximum at MeV energies. The latter one dominates the energy output across the whole electro-magnetic spectrum. The results of our analyses are discussed in the framework of current blazar modeling.
Key words: rays: observations -- galaxies: active -- galaxies: quasars: individual: 3C 454.3, CTA 102 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8941606283187866, "perplexity": 5964.108840441644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501173866.98/warc/CC-MAIN-20170219104613-00301-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://thiscondensedlife.wordpress.com/tag/graphene/ | # Tag Archives: Graphene
## Timing
Rather than being linear, the historical progression of topics in physics sometimes takes a tortuous route. There are two Annual Reviews of Condensed Matter Physics articles, one by P. Nozieres and one by M. Dresselhaus, that describe how widespread interest on certain subjects in the study of condensed matter were affected by timing.
In the article by Dresselhaus, she notes that HP Boehm and co-workers had actually isolated monolayer graphene back in 1962 (pdf!, and in German). On the theoretical front, P. Nozieres says in his article:
But neither I nor any of these famous people ever suspected what was hiding behind that linear dispersion. Fifty years later, graphene became a frontier of physics with far-reaching quantum effects.
Dresselhaus also mentions that carbon nanotubes were observed in 1952 in Russia followed by another reported discovery in the 1970s by M. Endo. These reports occurred well before its rediscovery in 1991 by Iijima that sparked a wealth of studies. The controversy over the discovery of nanotubes actually seems to date back even further, perhaps even to 1889 (pdf)!
In the field of topological insulators, again there seems to have been an oversight from the greater condensed matter physics community. As early as 1985, in the Soviet journal JETP, B.A. Volhov and O.A. Pankratov discussed the possibility of Dirac electrons at the surface between a normal band-gap semiconductor and an “inverted” band-gap semiconductor (pdf). Startlingly, the authors suggest CdHgTe and PbSnSe as materials in which to investigate the possibility. A HgTe/(Hg,Cd)Te quantum well hosted the first definitive observation of the quantum spin hall effect, while the Pb$_{1-x}$Sn$_x$Se system was later found to be a topological crystalline insulator.
One can probably find many more examples of historical inattention if one were to do a thorough study. One also wonders what other kinds of gems are hidden within the vastness of the scientific literature. P. Nozieres notes that perhaps the timing of these discoveries has something to do with why these initial discoveries went relatively unnoticed:
When a problem is not ripe you simply do not see it.
I don’t know how one quantifies “ripeness”, but he seems to be suggesting that the perceived importance of scientific works are correlated in some way to the scientific zeitgeist. In this vein, it is amusing to think about what would have happened had one discovered, say, topological insulators in Newton’s time. In all likelihood, no one would have paid the slightest attention.
## What Happens in 2D Stays in 2D.
There was a recent paper published in Nature Nanotechnology demonstrating that single-layer NbSe$_2$ exhibits a charge density wave transition at 145K and superconductivity at 2K. Bulk NbSe$_2$ has a CDW transition at ~34K and a superconducting transition at ~7.5K. The authors speculate (plausibly) that the enhanced CDW transition temperature occurs because of an increase in electron-phonon coupling due to the reduction in screening. An important detail is that the authors used a sapphire substrate for the experiments.
This paper is among a general trend of papers that examine the physics of solids in the 2D limit in single-layer form or at the interface between two solids. This frontier was opened up by the discovery of graphene and also by the discovery of superconductivity and ferromagnetism in the 2D electron gas at the LAO/STO interface. The nature of these transitions at the LAO/STO interface is a prominent area of research in condensed matter physics. Part of the reason for this interest stems from researchers having been ingrained with the Mermin-Wagner theorem. I have written before about the limitations of such theorems.
Nevertheless, it has now been found that the transition temperatures of materials can be significantly enhanced in single layer form. Besides the NbSe$_2$ case, it was found that the CDW transition temperature in single-layer TiSe$_2$ was also enhanced by about 40K in monolayer form. Probably most spectacularly, it was reported that single-layer FeSe on an STO substrate exhibited superconductivity at temperatures higher than 100K (bulk FeSe only exhibits superconductivity at 8K). It should be mentioned that in bulk form the aforementioned materials are all quasi-2D and layered.
The phase transitions in these compounds obviously raise some fundamental questions about the nature of solids in 2D. One would expect, naively, for the transition temperature to be suppressed in reduced dimensions due to enhanced fluctuations. Obviously, this is not experimentally observed, and there must therefore be a boost from another parameter, such as the electron-phonon coupling in the NbSe$_2$ case, that must be taken into account.
I find this trend towards studying 2D compounds a particularly interesting avenue in the current condensed matter physics climate for a few reasons: (1) whether or not these phase transitions make sense within the Kosterlitz-Thouless paradigm (which works well to explain transitions in 2D superfluid and superconducting films) still needs to be investigated, (2) the need for adequate probes to study interfacial and monolayer compounds will necessarily lead to new experimental techniques and (3) qualitatively different phenomena can occur in the 2D limit that do not necessarily occur in their 3D counterparts (the quantum hall effect being a prime example).
Sometimes trends in condensed matter physics can lead to intellectual atrophy — I think that this one may lead to some fundamental and major discoveries in the years to come on the theoretical, experimental and perhaps even on the technological fronts.
Update: The day after I wrote this post, I also came upon an article demonstrating evidence for a ferroelectric phase transition in thin Strontium Titanate (STO), a material known to exhibit no ferroelectric phase transition in bulk form at all.
## Do “Theorems” in Condensed Matter Physics Limit the Imagination?
There are many so-called “theorems” in physics. The most famously quoted in the field of condensed matter are the ones associated with the names of Goldstone, Mermin-Wagner, and McMillan.
If you aren’t familiar with these often (mis)quoted theorems, then let me (mis)quote them for you:
1) Goldstone: For each continuous symmetry a phase of matter breaks, there is an associated collective excitation that is gapless for long wavelengths, usually referred to as a Nambu-Goldstone mode.
2) Mermin-Wagner: Continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. (From Wikipedia)
3) McMillan (PDF link!): Electron-phonon induced superconductivity cannot have a higher Tc than approximately 40K.
All these three theorems in condensed matter physics have been violated to a certain extent. My gut feeling, though, is that these theorems can have the adverse consequence of limiting one’s imagination. As an experimental physicist, I can see the value in such theorems, but I don’t think that it is constructive to believe them outright. The number of times that nature has proven that she is much more creative and elusive than our human minds should tell us that we should use these theorems as guidance but to always be wary of such ideas.
For instance, had one believed the Mermin-Wagner theorem outright, would someone have thought the existence of graphene possible? In a solid, which breaks translational symmetry in three directions and rotational symmetry in three directions, why are there only three acoustic phonons? McMillan’s formula still holds true for electron-phonon coupled superconductors (marginal case being MgB2 which has a Tc~40K), though a startling discovery recently may even shatter this claim. However, placed in its historical context (it was stated before the discovery of high-temperature superconductors), one wonders whether McMillan’s formula disheartened some experimentalists from pursuing the goal of a higher transition temperature superconductor.
My message: One may use the theorems as guidance, but they are really there to be broken. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 7, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7912139296531677, "perplexity": 1234.8495944422627}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989812.47/warc/CC-MAIN-20210515035645-20210515065645-00560.warc.gz"} |
http://dianken.mew.ru/library/art-and-aesthetics-after-adorno-by-j-m-bernstein-2014-08-18 | By J. M. Bernstein, Claudia Brodsky, Anthony J. Cascardi, among others
Theodor Adorno's Aesthetic concept (1970) bargains probably the most robust and finished reviews of artwork and of the self-discipline of aesthetics ever written. The paintings deals a deeply serious engagement with the heritage and philosophy of aesthetics and with the traditions of eu artwork during the heart of the twentieth century. it really is coupled with formidable claims approximately what aesthetic concept needs to be. however the cultural horizon of Adorno's Aesthetic conception was once the area of excessive modernism, and lots more and plenty has occurred due to the fact then either in thought and in perform. Adorno's robust imaginative and prescient of aesthetics demands reconsideration during this mild. needs to his paintings be defended, up-to-date, resisted, or just left in the back of? This quantity gathers new essays by way of best philosophers, critics, and theorists writing within the wake of Adorno on the way to tackle those questions. They carry in universal a deep recognize for the facility of Adorno's aesthetic critique and a priority for the way forward for aesthetic concept according to fresh advancements in aesthetics and its contexts.
Best Art books
n-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians (Memoirs of the American Mathematical Society)
The vital subject matter of this paper is the variational research of homeomorphisms $h: {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y}$ among given domain names ${\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n$. The authors search for the extremal mappings within the Sobolev house ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which reduce the strength vital \${\mathscr E}_h=\int_{{\mathbb X}} \,|\!
Model Driven Architecture – Foundations and Applications: First European Conference, ECMDA-FA 2005, Nuremberg, Germany, November 7-10, 2005. Proceedings
This booklet constitutes the refereed lawsuits of the 1st eu convention, Workshops on version pushed structure - Foundations and functions, ECMDA-FA 2005, held in Nuremberg, Germany in November 2005. The 24 revised complete papers offered, nine papers from the purposes tune and 15 from the principles song, have been rigorously reviewed and chosen from eighty two submissions.
Additional resources for Art and Aesthetics After Adorno by J. M. Bernstein (2014-08-18)
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Rated 4.76 of 5 – based on 17 votes | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25931525230407715, "perplexity": 7138.332445632528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160568.87/warc/CC-MAIN-20180924145620-20180924170020-00185.warc.gz"} |
http://mathhelpforum.com/trigonometry/49435-how-d-they-get-cos-out-here.html | # Math Help - How'd they get cos out here?
1. ## How'd they get cos out here?
Hi folks,
Firstly this is the page I'm looking at to try and learn some trig,
SparkNotes: SAT Subject Test: Math Level 1: Trigonometric Identities
The question goes as follows
$(\cos\theta \times \tan\theta) / (\sin\theta -\cos\theta^2)$
In their working, they re write this as follows
$\frac{\cos\theta \times \tan\theta}{\sin\theta} - \cos\theta^2$
My question is - How did they get the $\cos\theta^2$ out from under the fraction? Shouldn't it be under there instead of out seperately?
Why isn't it
$\frac{\cos\theta \times \tan\theta}{\sin\theta - \cos\theta^2}$
Thanks folks.
2. Originally Posted by Peleus
Hi folks,
Firstly this is the page I'm looking at to try and learn some trig,
SparkNotes: SAT Subject Test: Math Level 1: Trigonometric Identities
The question goes as follows
$(\cos\theta \times \tan\theta) / (\sin\theta -\cos\theta^2)$
In their working, they re write this as follows
$\frac{\cos\theta \times \tan\theta}{\sin\theta} - \cos\theta^2$ Mr F says: This is wrong.
My question is - How did they get the $\cos\theta^2$ out from under the fraction? Shouldn't it be under there instead of out seperately?
Why isn't it
$\frac{\cos\theta \times \tan\theta}{\sin\theta - \cos\theta^2}$ Mr F says: This is right.
Thanks folks.
.. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9107150435447693, "perplexity": 2842.6395111080074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999653645/warc/CC-MAIN-20140305060733-00057-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/107133-integral-print.html | # Integral
• October 10th 2009, 02:18 AM
Dranalion
Integral
Hey,
I am looking to antiderive the following:
$\int (3-sin{3x})(cos{3x})dx$
Any help is much appreciated,
Dranalion
• October 10th 2009, 03:42 AM
galactus
$\int (3-sin(3x))cos(3x)dx$
Just make the sub $u=sin(3x), \;\ \frac{du}{3}=cos(3x)dx$
Now, it should be easier.
$\frac{1}{3}\int (3-u)du$
• October 10th 2009, 03:50 AM
Dranalion
Using this, does this give the answer:
= sin(3x) - 1/6(sin^2(3x)) + c ?
I am just checking to see if I followed it correctly.
• October 10th 2009, 04:17 AM
mr fantastic
Quote:
Originally Posted by Dranalion
Using this, does this give the answer:
= sin(3x) - 1/6(sin^2(3x)) + c ?
I am just checking to see if I followed it correctly.
When you differentiate your answer do you get the integrand?
• October 10th 2009, 04:19 AM
Defunkt
Quote:
Originally Posted by Dranalion
Using this, does this give the answer:
= sin(3x) - 1/6(sin^2(3x)) + c ?
I am just checking to see if I followed it correctly.
$\frac{1}{3} \int(3-u)du = \frac{1}{3} (3u - \frac{u^2}{2}) + C = u - \frac{u^2}{6} + C = sin(3x) - \frac{sin^2(3x)}{6} + C$ so yes | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7899285554885864, "perplexity": 3199.4033365703426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661675.84/warc/CC-MAIN-20150417045741-00273-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.tutorialspoint.com/a-c-programming-language-puzzle | # A C Programming Language Puzzle?
CServer Side ProgrammingProgramming
#### C in Depth: The Complete C Programming Guide for Beginners
45 Lectures 4.5 hours
#### Practical C++: Learn C++ Basics Step by Step
Most Popular
50 Lectures 4.5 hours
#### Master C and Embedded C Programming- Learn as you go
66 Lectures 5.5 hours
Here we will see one C programming language puzzle question. Suppose we have two numbers 48 and 96. We have to add the first number after the second one. So final result will be like 9648. But we cannot use any logical, arithmetic, string related operations, also cannot use any pre-defined functions. So how can we do that?
This is easy. We can do by using Token Pasting operator(##) in C. The Token Pasting operator is a preprocessor operator. It sends commands to compiler to add or concatenate two tokens into one string. We use this operator at the macro definition.
## Example
Live Demo
#include<stdio.h>
#define MERGE(x, y) y##x
main() {
printf("%d", MERGE(48, 96));
}
## Output
9648
Updated on 30-Jul-2019 22:30:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6312024593353271, "perplexity": 6075.6814282486985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.22/warc/CC-MAIN-20220930181143-20220930211143-00015.warc.gz"} |
http://deggles.csoft.net/index.php?month=10&year=2015 | ## U-19 Qualifiers and ICC News; Associate and Affiliate Cricket Podcast Russell Degnan
The podcast returns with a little cricket and a lot of news. Andrew Nixon (@andrewnixon79) joins Russell Degnan (@idlesummers) to discuss the under-19 World Cup qualifiers (0:23), the Ireland (1:50) and Afghanistan (7:20) tours of Zimbabwe, the Africa T20 Cup (9:58), and the South American championship in Chile (13:11). There are lots of news topics to digest, including player eligibility issue for Suriname at WCL6 (17:17). From the ICC meeting there were mooted changes to the world cup qualification process (20:46) , the womens world cup format (24:23), the formalisation of changes to funding for associates and affiliates (26:06), a renewed look at the Olympics (29:19), administrative problems in Nepal (34:18) and the end of WCL6? (37:52). There is more news from the Americas with the ICC team and the Warne-Tendulkar All-Star series (39:38), and globally: associate involvement in the WBBL (52:05) amongst some previews for WCL and Hong Kongs tour of the UAE.
Direct Download Running Time 55min. Music from Martin Solveig, "Big in Japan"
The associate and affiliate cricket podcast is an attempt to expand coverage of associate tournaments by obtaining local knowledge of the relevant nations. If you have or intend to go to a tournament at associate level - mens womens, ICC, unaffiliated - then please get in touch in the comments or by email.
Idle Summers 29th October, 2015 08:25:16 [#] [0 comments]
## Batting form and the "hot hand" Russell Degnan
There has been an interesting paper in circulation recently that deals with the idea of a "hot hand", which in cricket terms wed refer to as "good form". It is primarily interesting because for several decades, the idea of streaks being anything other than random luck has been derided. Attempts to measure it in cricket have been few and far between, but there was little to suggest a batsman was more likely to score runs having just done so - or indeed, that they werent largely replicable with a random number generator.
The paper in question upends a key piece of prior research because of a rather simple, but slightly counter-intuitive piece of statistics. The various explanationstend towards the counter-intuitive end of the scale but Ill try to explain.
The technique being used is very elegant: take an action that occurs roughly 50% of the time and measure the number of successes that follow a previous success. If they are completely independent, the subsequent attempt will continue to have a probability of 50%. If the successes are clustered around other successes, that number ought to be higher.
The quirk, is that because strings of attempts are being measured, the average probability found in those strings will not actually be 50%. This is the counter-intuitive part, but is relatively easily seen on a simple graph:
Here is shown strings of eight attempts broken down into the number of times a measure took place (that is, the previous attempt saw a success). The number of instances of measurements breaks down as a binomial distribution centred on 3.5: 2 of zero (ooooooox and oooooooo) and seven (xxxxxxxx and xxxxxxxo), 14 of one and six, and so on as follows:
AttemptsOpportunities usedInstancesPercentage
002
114147.14
2844221.43
32107035.71
42807050.00
52104264.29
6841478.57
714292.86
The distribution of opportunities to take a measurement is similar, but because it takes more successful attempts to generate higher opportunities, it is shifted slightly across, and centred around 4 (or n/2). The breakdown also demonstrates the key to the problem: if four opportunities are to be had, the attempts will be distributed in such a way that the average success rate is 50%. But the only way to generate 7 opportunities is to have succeeded in each of the first 7 measurements. The percentage will be either 6/7 (83.33%) or 7/7 (100%). And as a consequence, the average of multiple strings of measurements ought to sit not at 50% (the middle of the opportunity distribution), but at the centre of the instances of measurement distribution (plus a term for the two extras) - around 45% for strings of length 8.
All very fascinating, particularly as it implies that previous studies showed a "hot hand" after all. But what does it say about cricket? The short answer is that this is a very elegant way of measuring form: find the median score for a batsman, if they surpass it, then test their subsequent score.
For Tendulkar, who played so many innings that the expected percentage is close to 50, his test "form" saw a 53.1% success rate in innings where hed surpassed the median (excluding not outs below the median). In ODIs however (counting only matches where he opened) the figure drops to 50.5%.
That is only a single data point, and some batsmen are likely to be more prone to runs of form than others, but it also points to an issue. In ODI cricket, where multi-lateral series exist, a batsman tends to shuffle opposition quite quickly, and therefore face a reasonable variety of bowling strength from match to match. In test cricket, the subsequent innings is less likely to be independent from the first, without being held in identical conditions - the second innings being on a wearing pitch. Apparent runs of form may just be a string of matches against poor opposition.
Conversely, ODI cricket may be less prone to form, being a format that requires a higher amount of risk-taking, and therefore more luck. Hence a discrepancy between test and ODI matches is feasible. Comparing all innings adds in time gaps when a player might fall out of form (and vice versa), and a proper study ought to remove them. The relative sparsity of innings means that when a player is really in form, it would be hard to distinguish between that and luck with any method. Most likely the effect is small - perhaps three or four runs on a batting average, but probably half that.
Hence measuring the effect, if any, of form remains difficult. On selection matters, - the only avenue where form might matter - there is a lot to be said for judging a player on technique, temperament and overall career trajectory and ignoring runs of form. Everything else is largely academic, albeit an interesting question.
Idle Summers 16th October, 2015 00:45:10 [#] [0 comments]
## Control is not always everything, but often, ratings 10th October Russell Degnan
A somewhat belated recap somewhat belatedly marking the start of the new season
5th TestEnglandvAustralia
Pre-rating1107.01204.3
Form+39.4-51.2
Expected MarginEngland by 1 run
Actual MarginAustralia by an innings and 46 runs
Post-rating1098.81214.4
Series rating1153.71162.5
It would be easy to point to the opening of this test match and say that Rogers and Warner batted as they ought to have throughout, laying a foundation foa big total, and allowing their bowlers to attack. But for the most part, Rogers and Warner tried (and largely succeeded) to make that foundation. The issue was that their inevitable failures (all failures are) resulted in cataclysms through the middle order. It isnt clear whether Smith is a number three, as he is only making starts against the old ball. But as with the openers, he mostly did the job. A team should expect to be two down in the first hour relatively often, partcularly in England - it isnt acceptable to turn that into all out by tea.
The lessons of the last three Ashes series in England (if not every series in England, ever) remain unheeded. At home, Australia can largely avoid collapses because the pitches are flat. In England, there are times when the play is wth the bowlers and the side that lost (the series but also individual tests) in 2005, 2009 and 2013, was the one that failed to arrest the collapse, and wait for better conditions.
Australias bowling showed similar flaws (as indeed they have in previous series), failing to keep control of the run-scoring when times were bad, and underestimating the value of putting the ball in the right spot consistently when times were good. Siddle, in this respect, ought to have played more matches. The allure of a 2013/14 Johnson and a 2015 World Cup Starc running rampant was too strong to exclude either; and the reputation Hazlewood has attained for being McGrath-like seems to have afflicted the judgement of his actual accuracy.
In the end, the series rating shows this to be close, when actually, Englands victory was near inevitable given the fragility of Australias batting. A evidenced in this match, England have their own worries in this department, but they have a young side with talent, so they can expect to improve. Australia have gone from an old side to half a side within days of the Ashes ending. That is a problem for another day though.
2nd/3rd TestsSri LankavIndia
Pre-rating1013.81083.9
Form-16.1-14.5
Expected MarginSri Lanka by 15 runs
Actual MarginIndia by 278 runs
India by 117 runs
Post-rating980.81109.5
Series rating938.31159.6
If you took Angelo Mathews out of this Sri Lankan side and theyd be a rabble. Chandimal provided runs in the first test, but in the following two, India managed consistent contributions across the board - with bat and ball - while Sri Lanka had two tons from Mathews, 14 wickets from Prasad and not a whole lot else. They didnt lose by a huge amount, but a failure to pass 306 in four innings nor keep India below 274 meant that they were well behind.
The result pushes Sri Lanka beow New Zealand and lifts India above England, with a reasonable chance to move into third or better across the forthcoming series. Sri Lanka, meanwhile, need to stave off the West Indies, who they are slowly drifting towards. With so few contributions of note, and the loss now of Sangakarra and Jayawardene as well, it may be some time before they again challenge for the top few places in the rankings.
I-Cup MatchNetherlandsvScotland
Pre-rating164.7418.9
Form-45.3-12.6
Expected MarginScotland by 77 runs
Actual MarginNetherlands by 44 runs
Post-rating174.8408.0
The sort of low-scoring match that is exciting to watch unfold, and ought to be covered better. There were only three fifties in the match, and two came on the closing morning when Scotland looked briefly to have put themselves back in contention, after the top-order collapsed in both innings. The unlikely bowling of Borren capped the victory for the Dutch, after Rippon again provided the partnership breaker.
Netherlands play so few first class fixtures it wasnt until they won that you realised just how long theyd been poor. Their rating reflects the seven year gap between wins, and their form-line shows how far they have to regain any sort of respectability. If they can keep van der Merwe in the side though, coupled with the relatively solid and committed lineup that is playing well, they will move back to where they belong. Challenging for a victory in the I-Cup is unlikely - though not impossible, as neither Ireland nor Afghanistan look strong - but this victory did seriously dent the chances of the Scottish.
2 TestsPakistanvEngland
Pre-rating1115.21098.8
Form+15.7+13.1
Expected MarginPakistan by 8 runs
Nominally an incredibly close series on neutral territory, in reality, Pakistan plays in the UAE as well as, or even better than they did at home. They are also facing a side that doesnt play well in the UAE, is young, and notwithstanding the home win, recently drew series against lesser rank sides, in more familiar conditions. Compounding that, while Yasir Shah is a loss for Pakistan, his opposite spinners are prone to loose deliveries, and in Mooen Alis case, will need to open on top of the bowling workload, having been shifted out of number eight for Adil Rashid.
It is questionable Wood offers a lot extra, on top of Broad and Anderson, with Stokes and two spinners. If Englands spinners cant take the wickets required, then theyll need to the extra batting moreso than the extra bowler. The benefit of three all-rounders being lost when the tail still starts at nine.
If Pakistan play nearly as well as they did against Australia theyll win this comfortably. They did subsequently lose to New Zealand, but that seems to be a growing trend, and in any case, was an unusual match in thr wake of Hughes death in Australia. Assuming their ageless batsmen dont suddenly do so, England will be in for a difficult short tour.
3 TestsSr LankavWest Indies
Pre-rating980.8867.0
Form-40.8-10.9
Expected MarginSri Lanka by 107 runs
A series that exemplifies the need for the ICC discussions on context and bilaterals. Optimally, not having bilaterals series, and absorbing this contest (and it is a close contest) into a championship would make it less of an afterthought for fans and players alike. Sri Lankas plan will be the same as always: try and score enugh runs for Herath to grind the opposition into dust. The West Indies bowling lineup is as good as it has been in some time (Taylor, Roach, Holder, Gabriel) but they are sporadically world class, and the batting looks shaky. If they can make this close it will count as a success.
Rankings at 10th October 2015
1.South Africa1267.9
2.Australia1214.4
3.Pakistan1115.2
4.India1109.5
5.England1098.8
6.New Zealand1000.5
7.Sri Lanka980.8
8.West Indies867.0
9.Bangladesh613.3
12.Zimbabwe559.8
10.Ireland611.1
11.Afghanistan586.2
13.Scotland408.0
14.Namibia369.1
15.Kenya276.4
16.U.A.E.248.5
17.Papua New Guinea219.4
18.Netherlands174.8
19.Hong Kong150.1
20.Canada147.9
Shaded teams have played fewer than 2 games per season. Non-test team ratings are not comparable to test ratings as they don`t play each other.
Idle Summers 14th October, 2015 22:57:29 [#] [0 comments] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31369373202323914, "perplexity": 3303.9177249856675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703495901.0/warc/CC-MAIN-20210115134101-20210115164101-00452.warc.gz"} |
https://www.wisdomjobs.com/e-university/php-tutorial-223/choosing-banner-ads-430.html | Another use for random numbers is choosing from banner ads. Suppose you've signed up three sponsors for your Web site. Each has a single banner you promise to display on an equal proportion of hits to your site. To accomplish this, generate a random number and match each number to a particular banner. It is used a switch statement on a call to mt_rand. In a situation like this, you don't need to worry too much about using good seeds. You simply want a reasonable distribution of the three choices. Someone guessing which banner will display at midnight poses no security risk.
Generating a Session Identifier
<?
//Seed the generator
mt_srand(doubleval(microtime()) * 100000000);
//choose banner
switch(mt_rand(1,3))
{
case 1:
$bannerURL =$bannerImage = "leon.jpg";
break;
case 2:
$bannerImage = "php_lang.jpg"; break; default:$bannerImage = "phptr.jpg";
}
//display banner
print("<A HREF="$bannerURL">"); print("<IMG SRC= "$bannerImage" ");
print("WIDTH="400" HEIGHT="148" BORDER="0"
>");
print("</A>");
?> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18175920844078064, "perplexity": 3613.0791009138907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217951.76/warc/CC-MAIN-20180821034002-20180821054002-00136.warc.gz"} |
http://link.springer.com/article/10.1023/A:1014275232713 | , Volume 5, Issue 1, pp 1-9
# Miracles and the limits of medical knowledge
Rent the article at a discount
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* Final gross prices may vary according to local VAT.
## Abstract
In considering whether medical miracles occur, thelimits of epistemology bring us to confront ourmetaphysical worldview of medicine and nature ingeneral. This raises epistemological questions of ahigher order. David Hume's understanding of miraclesas violations of the laws of nature assumes thatnature is completely regular, whereas doctrines suchas C. S. Peirce's tychism'' hold that there is anelement of absolute chance in the workings of theuniverse. Process philosophy gives yet another viewof the working of nature. Physicians have noepistemological grounds for declaring any cure to bemiraculous. Miracles are theological (orphilosophical) entities, and not medical entities. All physicians can do is to determine whether or nota cure is scientifically inexplicable according to thecurrent epistemological standards of medical science. As these standards change, what is currentlyunexplainable may become explainable. However, we canalso come to realize that our current explanations arein fact unsatisfactory. Our justifications ofknowledge claims about miracles will depend on ourviews about determinism and indeterminism. If theuniverse is not a deterministic one, we should be opento the possibility of encountering what appear to usas sui generis events. These would not beviolations of immutable laws of nature, butmanifestations of the true workings of nature, andcertainly causes for wonder. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5164061784744263, "perplexity": 9431.4943088911}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510268660.14/warc/CC-MAIN-20140728011748-00064-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://eccc.weizmann.ac.il/eccc-reports/1995/TR95-021/ | Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > DETAIL:
### Paper:
TR95-021 | 20th April 1995 00:00
#### On Randomized Versus Deterministic Computation
TR95-021
Authors: Marek Karpinski, Rutger Verbeek
Publication: 20th April 1995 16:12
Keywords:
Abstract:
In contrast to deterministic or nondeterministic computation, it is
a fundamental open problem in randomized computation how to separate
different randomized time classes (at this point we do not even know
how to separate linear randomized time from ${\mathcal O}(n^{\log n})$
randomized time) or how to compare them relative to corresponding
deterministic time classes. In other words we are far from
understanding the power of {\em random coin tosses} in the
computation, and the possible ways of simulating them
deterministically.
In this paper we study the relative power of linear and polynomial
randomized time compared with exponential deterministic time.
Surprisingly, we are able to construct an oracle $A$ such that
exponential time (with or without the oracle $A$) is simulated by
linear time Las~Vegas algorithms using the oracle $A$.
ISSN 1433-8092 | Imprint | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8751042485237122, "perplexity": 3081.7835955850983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487623596.16/warc/CC-MAIN-20210616093937-20210616123937-00268.warc.gz"} |
http://mathematica.stackexchange.com/questions/38294/nsolve-with-numerical-function | NSolve with numerical function
I would like to solve numerically an equation which involves a numerical function constructed by fitting some data:
data := {{0, 1}, {1, 1}, {.2, 3}, {.4, 5}, {.6, 2}};
try[x_?NumberQ, y_?NumberQ] := (auy = y;
solution = FindFit[data, A (aux - auy) + B Exp[aux - auy], {A, B}, aux];
myFit[aux_?NumberQ] = (A (aux - auy) + B Exp[aux - auy]) /.
solution; Return[myFit[x]])
NSolve[y - try[1, y] == 0, y]
I tried with this code but it does not function (various errors). Does anyone know how to help me?
-
Did you even try to test call your try function? What happens when you type try[1, y] ? does not return anything? This is really strange that no body seems to do any testing of anything any more. – Nasser Dec 5 '13 at 11:55
I tried try[1,2] and it works. try[1,y] has not to return an output since it need numerical parameters to work. What I want is a way to solve the problem I described, i.e. using NSolve to solve the equation which involve a function numerically determined by a fit. My code obviously does not work, that's why I asked help. Does anyone knows how to solve my issue? I hope I clarified my post! – user9994 Dec 5 '13 at 12:44
I modified a little the example code above, now it should be more clear. – user9994 Dec 5 '13 at 13:40
Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! – belisarius Dec 5 '13 at 15:34
data = {{0, 1}, {1, 1}, {.2, 3}, {.4, 5}, {.6, 2}};
try[x_?NumericQ, y_?NumericQ, data_] := (A (x - y) + B Exp[x - y]) /.
FindFit[data, A (xx - y) + B Exp[xx - y], {A, B}, xx]
FindRoot[y - try[1, y, data] == 0, {y, 1}]
(*
{y -> 1.78209}
*)
-
ok, this helps a lot: FindRoot is what I need. But I have to scal a big range of "y", right? Maybe with a Do[ ] statement. is there someway to optimize it?. Thank you for your help belisarius! – user9994 Dec 5 '13 at 15:17
Sorry, I don't understand your sentence But I have to scal a big range of "y", right?. Can you explain it further for me, please? – belisarius Dec 5 '13 at 15:37
I am really sorry, I had to write "scan" and not "scal". What I meant is that I have to search for solution in a range of ys to find more than one solution. – user9994 Dec 5 '13 at 16:34
@user9994 FindRoot should take care of that. Just try to give it a good initial guess – belisarius Dec 5 '13 at 16:45
But if I suspect the existence of different solutions I have to repeat the FindRoot with different intial guesses right? – user9994 Dec 5 '13 at 17:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24776944518089294, "perplexity": 2228.2538973386763}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507447819.35/warc/CC-MAIN-20141017005727-00231-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://www.jmlr.org/papers/v17/collier16a.html | ## Minimax Rates in Permutation Estimation for Feature Matching
Olivier Collier, Arnak S. Dalalyan; 17(6):1−31, 2016.
### Abstract
The problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation. We address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators. To this end, the minimax rate of separation is investigated and its expression is obtained as a function of the sample size, noise level and dimension of the features. We consider the cases of homoscedastic and heteroscedastic noise and establish, in each case, tight upper bounds on the separation distance of several estimators. These upper bounds are shown to be unimprovable both in the homoscedastic and heteroscedastic settings. Interestingly, these bounds demonstrate that a phase transition occurs when the dimension $d$ of the features is of the order of the logarithm of the number of features $n$. For $d=O(\log n)$, the rate is dimension free and equals $\sigma (\log n)^{1/2}$, where $\sigma$ is the noise level. In contrast, when $d$ is larger than $c\log n$ for some constant $c>0$, the minimax rate increases with $d$ and is of the order of $\sigma(d\log n)^{1/4}$. We also discuss the computational aspects of the estimators and provide empirical evidence of their consistency on synthetic data. Finally, we show that our results extend to more general matching criteria.
[abs][pdf][bib] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9655516147613525, "perplexity": 116.93332831784628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188717.24/warc/CC-MAIN-20170322212948-00082-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://jupyter.brynmawr.edu/services/public/dblank/CS110%20Intro%20to%20Computing/2017-Spring/Notes/Mathematical%20Functions.ipynb | ## Jupyter at Bryn Mawr College
Public notebooks: /services/public/dblank / CS110 Intro to Computing / 2017-Spring / Notes
# 1. Mathematical Functions¶
Functions that return true or false (boolean values).
## 1.1 isPrime¶
In [1]:
boolean isPrime(int number) {
// if number is evenly divided by any number less than number
// (except 1) then it is prime
// otherwise, not prime
for (int i = 2; i < number; i++) {
if ((number % i) == 0) {
return false;
}
}
return true;
}
void setup() {
println("isPrime(100): " + isPrime(100));
println("isPrime(101): " + isPrime(101));
println("isPrime(107): " + isPrime(107));
}
Sketch #1:
## 1.2 GCD¶
Greatest common divisor: the largest positive integer that divides all of the numbers without a remainder. For example, the GCD of 8 and 12 is 4.
Euclid's Algorithm (after https://en.wikipedia.org/wiki/Greatest_common_divisor):
1. Find the GCD of a and b
2. If a or b is zero, then return the other
3. if a > b, set a to a - b
4. if b > a, set b to b - a
5. go to step 2
Example:
1. Let a = 8, and b = 12
2. Neither is zero
3. a = 4 (12 - 8), b = 12
4. a = 4, b = 8 (12 - 4)
5. a = 4, b = 4 (8 - 4)
6. a = 0 (4 - 4), b = 4
6. return 4
In [5]:
int gcd(int a, int b) {
if (a == 0) // base case
return b;
else if (b == 0) // base case
return a;
else if (a > b)
return gcd(a - b, b);
else //
return gcd(a, b - a);
}
void setup() {
println("gcd(8, 12): " + gcd(8, 12));
println("gcd(3, 2): " + gcd(3, 2));
}
Sketch #4:
## 1.3 isLeapyear¶
In [6]:
boolean isLeapyear(int year) {
// if a year is evenly divided by 100
// but it is not evenly divided by 400
// it is a leapyear
// else if a year is evenly divided by 4, it is a leap year
// otherwise, not a leapyear
return true;
}
void setup() {
println("isLeapyear(2000): " + isLeapyear(2000) + " (should be true)");
println("isLeapyear(1904): " + isLeapyear(1904) + " (should be true)");
println("isLeapyear(1900): " + isLeapyear(1900) + " (should be false)");
println("isLeapyear(1901): " + isLeapyear(1901) + " (should be false)");
}
Sketch #5: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5036958456039429, "perplexity": 4662.941715922705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058589.72/warc/CC-MAIN-20210928002254-20210928032254-00048.warc.gz"} |
http://mathhelpforum.com/pre-calculus/170604-distance-between-point-line.html | # Math Help - distance between a point and a line.
1. ## distance between a point and a line.
Show that the distance from a point (xZERO,yZERO) to a line Ax+By+C=0 is given by
|AxZERO+ByZERO+C| / sqrt(A^2+B^2)
Why absolute value? I get that hes putting the points into the slope of the other one to get maybe a parallel line. but thats all I get. why the sqrt of (a^2+b^2) ?
2. Hello, frankinaround!
$\text}Show that the distance from a point }(x_o,y_o)\text{ to a line }Ax+By+C\:=\:0$
$\text{is given by: }\;d \;=\;\dfrac{|Ax_o +By_o+C|}{\sqrt{A^2+B^2}}$
Why absolute value? . Because distance is a positive measure.
I get that hes putting the points into the slope of the other one . What?
to get maybe a parallel line, but thats all I get.
Why the sqrt of (A^2+B^2) ? . Why not?
You're expected to derive that formula, aren't you?
Well, do it . . . and you'll see why.
3. Originally Posted by Soroban
Hello, frankinaround!
You're expected to derive that formula, aren't you?
Well, do it . . . and you'll see why.
That's a bit harsh, the OP obviously does not know how to...
Hint, the slope of the shortest line segment going through $\displaystyle (x_0, y_0)$ and touching $\displaystyle Ax + By + C = 0$ will be perpendicular to the slope of $\displaystyle Ax + By + C = 0$.
4. um... so im trying to do this. I can see ax/-b + c/-b = y and that the slop is a/-b, so then bx0/a should be the perpendicular slope. then I think I would need to take the point of intersection and use the distance formula to get to the next part of this question. but Im not sure how, because how do I know the point of intersection ? I have almost 2 equations. y=ax/-b+c/-b and bx/a + ? = y. I can TRY to substitute with y, so like bx/a + ? = ax/-b + c/-b. But it seems strange. what do you think?
5. the equation of a line with slope $\frac{B}{A}$ (perpendicular to the given line) through the point $(x_0, y_0)$ is $y= \frac{B}{A}(x- x_0)+ y_0$. Where does that intersect $y= -\frac{A}{B}x- \frac{C}{A}$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9217534065246582, "perplexity": 718.7722899326351}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860118807.54/warc/CC-MAIN-20160428161518-00138-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs11804-011-1035-8 | , Volume 10, Issue 1, pp 7-16
Date: 02 Apr 2011
# Effect of bottom undulation on the waves generated due to rolling of a plate
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## Abstract
In the present paper, the effect of a small bottom undulation of the sea bed in the form of periodic bed form on the surface waves generated due to a rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of non uniform finite depth is investigated. A simplified perturbation technique involving a non dimensional parameter characterizing the smallness of the bottom deformation is applied to reduce the given boundary value problem to two independent boundary value problems upto first order. The first boundary value problem corresponds to the problem of water wave generation due to rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of uniform finite depth. This is a well known problem whose solution is available in the literature. From the second boundary value problem, the first order correction to the wave amplitude at infinity is evaluated in terms of the shape function characterizing the bottom undulation, by employing Green’s integral theorem. For a patch of sinusoidal ripples at the sea bottom, the first order correction to the wave amplitude at infinity for both the configuration of the barrier is then evaluated numerically and illustrated graphically for various values of the wave number. It is observed that resonant interaction of the wave generated, with the sinusoidal bottom undulation occurs when the ratio of twice the wavelength of the sinusoidal ripple to the wave length of waves generated, approaches unity. Also it is found that the resonance increases as the length of the barrier increases.
Foundation item: Supported by DST through the Research Project No. SR/SY/MS: 521/08.
Puspendu Rakshit was born in 1979. He is presently a PhD candidate at Jadavpur University Kolkata, India. His current research interest includes theory of water waves and integral equations.
Sudeshna Banerjea was born in 1963. She presently works in the Department of Mathematics, at Jadavpur University, Kolkata, India. Her current research interest includes theory of water waves and integral equations. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9343881011009216, "perplexity": 632.8731181911544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119648438.26/warc/CC-MAIN-20141024030048-00072-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://scholars.ncu.edu.tw/zh/publications/radial-variations-of-outward-and-inward-alfv%C3%A9nic-fluctuations-bas | # Radial Variations of Outward and Inward Alfvénic Fluctuations Based on Ulysses Observations
L. Yang, L. C. Lee, J. P. Li, Q. Y. Luo, C. L. Kuo, J. K. Shi, D. J. Wu
1 引文 斯高帕斯（Scopus）
## 摘要
Ulysses magnetic and plasma data are used to study hourly scale Alfvénic fluctuations in the solar polar wind. The calculated energy ratio R2VA(cal) of inward to outward Alfvén waves is obtained from the observed Walén slope through an analytical expression, and the observed R2VA(obs) is based on a direct decomposition of original Alfvénic fluctuations into outward- and inward-propagating Alfvén waves. The radial variation of R2VA(cal) shows a monotonically increasing trend with heliocentric distance r, implying the increasing local generation or contribution of inward Alfvén waves. The contribution is also shown by the radial increase in the occurrence of dominant inward fluctuations. We further pointed out a higher occurrence (∼80% of a day in average) of dominant outward Alfvénic fluctuations in the solar wind than previously estimated. Since R2VA(cal) is more accurate than R2VA(obs) in the measurement of the energy ratio for dominant outward fluctuations, the values of R2VA(cal) in our results are likely more realistic in the solar wind than those previously estimated as well as R2VA(obs) in our results. The duration ratio RT of dominant inward to all Alfvénic fluctuations increases monotonically with r, and is about two or more times that from Voyager 2 observations at r ≥ 4 au. These results reveal new qualitative and quantitative features of Alfvénic fluctuations therein compared with previous studies and put constraints on modeling the variation of solar wind fluctuations.
原文 ???core.languages.en_GB??? 177 Astrophysical Journal 850 2 https://doi.org/10.3847/1538-4357/aa9615 已出版 - 1 12月 2017 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9428049921989441, "perplexity": 3246.7899301414964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500813.58/warc/CC-MAIN-20230208123621-20230208153621-00362.warc.gz"} |
http://authors.library.caltech.edu/1312/ | CaltechAUTHORS
A Caltech Library Service
# Induced-charge electro-osmosis
Squires, Todd M. and Bazant, Martin Z. (2004) Induced-charge electro-osmosis. Journal of Fluid Mechanics, 509 . pp. 217-252. ISSN 0022-1120. http://resolver.caltech.edu/CaltechAUTHORS:SQUjfm04
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## Abstract
We describe the general phenomenon of ‘induced-charge electro-osmosis’ (ICEO) – the nonlinear electro-osmotic slip that occurs when an applied field acts on the ionic charge it induces around a polarizable surface. Motivated by a simple physical picture, we calculate ICEO flows around conducting cylinders in steady (DC), oscillatory (AC), and suddenly applied electric fields. This picture, and these systems, represent perhaps the clearest example of nonlinear electrokinetic phenomena. We complement and verify this physically motivated approach using a matched asymptotic expansion to the electrokinetic equations in the thin-double-layer and low-potential limits. ICEO slip velocities vary as $u_s \,{\propto}\,E_0^2 L$, where $E_0$ is the field strength and $L$ is a geometric length scale, and are set up on a time scale $\tau_c \,{=}\,\lambda_D L/D$, where $\lambda_D$ is the screening length and $D$ is the ionic diffusion constant. We propose and analyse ICEO microfluidic pumps and mixers that operate without moving parts under low applied potentials. Similar flows around metallic colloids with fixed total charge have been described in the Russian literature (largely unnoticed in the West). ICEO flows around conductors with fixed potential, on the other hand, have no colloidal analogue and offer further possibilities for microfluidic applications.
Item Type: Article "Reprinted with the permission of Cambridge University Press." Received May 5 2003; Revised February 13 2004; Published Online 07 Jun 2004 The authors would like to thank E´cole Supe´rieure de Physique et Chimie Industrielles (Laboratoire de Physico-chimie Th´eorique) for hospitality and partial support, and the referees for extensive comments and Russian references. This research was supported in part by the US Army through the Institute for Soldier Nanotechnologies, under Contract DAAD-19-02-0002 with the US Army Research Office (M. Z.B.), and by the NSF Mathematical Sciences Postdoctoral Fellowship and Lee A. Dubridge Prize Postdoctoral Fellowship (T. M. S.). CaltechAUTHORS:SQUjfm04 http://resolver.caltech.edu/CaltechAUTHORS:SQUjfm04 http://dx.doi.org/10.1017/S0022112004009309 No commercial reproduction, distribution, display or performance rights in this work are provided. 1312 CaltechAUTHORS Archive Administrator 10 Jan 2006 26 Dec 2012 08:43
Repository Staff Only: item control page | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5636194348335266, "perplexity": 4255.613980830516}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989507.42/warc/CC-MAIN-20150728002309-00091-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/33710-tangent-e-x-through-origin.html | # Thread: Tangent to e^x through origin
1. ## Tangent to e^x through origin
Find the value of A such as that the tangent to y=e^x is a line through origin when x=a
This is what I've done so far,
Derivative of e^x is e^x, so the slope is e^x
(y-e^a)=e^a(x-a)
0-e^a=e^a(0-a)
-e^a=-a(e^a)
That's what I get in the end, how can I solve for a? Did I make an error? This corresponds to a question asking a similar question for lnx, where a would be the value e, so I suppose there has to be some connection....
2. ## Here is what you do
Originally Posted by theowne
Find the value of A such as that the tangent to y=e^x is a line through origin when x=a
This is what I've done so far,
Derivative of e^x is e^x, so the slope is e^x
(y-e^a)=e^a(x-a)
0-e^a=e^a(0-a)
-e^a=-a(e^a)
That's what I get in the end, how can I solve for a? Did I make an error? This corresponds to a question asking a similar question for lnx, where a would be the value e, so I suppose there has to be some connection....
$-e^{a}=-a\cdot{e^{a}}$... $a=1$?...o you made a mistake I think you should have $y-0=e^{a}(x-0)$...so $y=e^{a}x$....a=0...you just forgot that the slope for tangent line must be the same point as is shown in the equation for a tangent line $y-f(x_1)=f'(x_1)(x-x_1)$
3. That's what I figured, but what connection exists between that and the value e? The question seems to imply a relation between the answer here, and e..
4. ## The seeming connection would be
Originally Posted by theowne
That's what I figured, but what connection exists between that and the value e? The question seems to imply a relation between the answer here, and e..
$f(x)=\ln(x)$ both give the equation $y=x$? since $y-e=\ln(e)(x-e)$...so $y=\ln(e)x-\ln(e)e+e=x$...andy maybe that $\ln(e)=1,e^{0}=1$?
5. I think my earlier form was correct....(which results in a=1)..I plugged it into graphmatica and I can see the relationship more clearly now...for ln(x), it's when x=e, for e^x, it's when y=e. Thanks.....
6. ## Hmm
Originally Posted by theowne
I think my earlier form was correct....(which results in a=1)..I plugged it into graphmatica and I can see the relationship more clearly now...for ln(x), it's when x=e, for e^x, it's when y=e. Thanks.....
Maybe I am misunderstanding your question then...but based on you think what you just said...maybe you are supposed to realize that since $\ln(x)$ and $e^{x}$ are inverse functions their domain and ranges are switched | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9280014038085938, "perplexity": 1080.416092643256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824931.84/warc/CC-MAIN-20171022003552-20171022023552-00183.warc.gz"} |
http://mathhelpforum.com/algebra/213930-arithematic-progression-logarithm.html | 1. ## Arithematic Progression -Logarithm
If $log_k x, log_m x , log_nx$ are in A.P then prove that $n^2 = (kn)^{log_km}$
If the terms are in A.P then : $2 log_mx = log_k x + log_nx = \frac{2}{log_xm} = \frac{1}{log_xk}+ \frac{1}{log_xn}$
= $\frac{2}{log_xm}= \frac{log_xn+log_xk}{log_xklog_xn}$ can we solve this way or not..............please guide
2. ## Re: Arithematic Progression -Logarithm
$2\log_{m}x = \log_{k}x + \log_{n}x.$
Start by using the change of base formula so that all logs are to base $k.$
Cancel the $\log_{k}x$ throughout, cross multiply and that gets you a $2\log_{k}n$ (which becomes) $\log_{k}n^2$ on the LHS.
Try finishing from there.
3. ## Re: Arithematic Progression -Logarithm
thanks a lot....i got it..regards,sachin | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9136804938316345, "perplexity": 3351.559408753751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121453.27/warc/CC-MAIN-20170423031201-00581-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/168076-question-uniform-convergence-print.html | # A Question on Uniform Convergence
• January 11th 2011, 12:09 PM
garunas
A Question on Uniform Convergence
Give an example of a uniformly convergent sequence $(f_{n})_{n\in\mathbb N}$ of continuously differentiable functions such that $(f\prime_{n})_{n\in\mathbb N}$ is not uniformly continuous.
Just preparing for my exams in a few weeks. Need to know the answer to this one to have as an example. Just in case!
• January 11th 2011, 12:26 PM
FernandoRevilla
Choose:
$f_n(x)=\dfrac{\sin nx}{n}$
Fernando Revilla
• January 11th 2011, 12:28 PM
garunas
Many many thanks. Simple when you know how isn't it??
• January 11th 2011, 12:32 PM
FernandoRevilla
Quote:
Originally Posted by garunas
Many many thanks. Simple when you know how isn't it??
No wonder. It is a typical counter example. :)
Fernando Revilla
• January 11th 2011, 12:39 PM
emakarov
Quote:
Originally Posted by garunas
Give an example of a uniformly convergent sequence $(f_{n})_{n\in\mathbb N}$ of continuously differentiable functions such that $(f\prime_{n})_{n\in\mathbb N}$ is not uniformly continuous.
Sorry, what exactly is not uniformly continuous? Should $f'_n$ be not uniformly continuous for every $n$?
• January 11th 2011, 12:46 PM
FernandoRevilla
Quote:
Originally Posted by emakarov
Sorry, what exactly is not uniformly continuous? Should $f'_n$ be not uniformly continuous for every $n$?
I supposed that is a typo and he meant " ... $(f'_n)$ not uniformly convergent".
Fernando Revilla
• January 11th 2011, 01:30 PM
garunas
Yeah sorry I am a bit new to Latex | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9101979732513428, "perplexity": 1116.8735228921119}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982294158.7/warc/CC-MAIN-20160823195814-00177-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/pulsar-rapidly-rotating-neutron-star-emits-radio-beam-way-lighthouse-emits-light-beam-rece-q2693073 | A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of T = 0.033 s that is increasing at the rate of 1.26 x 10^-5 sec/year. (a) What is the pulsar�s angular acceleration? (b) If the angular acceleration is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054. Assuming constant angular acceleration, find the initial T. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9319376945495605, "perplexity": 471.40348955034085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396224.52/warc/CC-MAIN-20160624154956-00020-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://www.koreascience.or.kr/search.page?keywords=supercritical+fluid+extraction | • Title, Summary, Keyword: supercritical fluid extraction
### Quality and characteristics of ginseng seed oil treated using different extraction methods
• Lee, Myung-Hee;Kim, Sung-Soo;Cho, Chang-Won;Choi, Sang-Yoon;In, Gyo;Kim, Kyung-Tack
• Journal of Ginseng Research
• /
• v.37 no.4
• /
• pp.468-474
• /
• 2013
• Ginseng seed oil was prepared using compressed, solvent, and supercritical fluid extraction methods of ginseng seeds, and the extraction yield, color, phenolic compounds, fatty acid contents, and phytosterol contents of the ginseng seed oil were analyzed. Yields were different depending on the roasting pretreatment and extraction method. Among the extraction methods, the yield of ginseng seed oil from supercritical fluid extraction under the conditions of 500 bar and $65^{\circ}C$ was the highest, at 17.48%. Color was not different based on the extraction method, but the b-value increased as the roasting time for compression extraction was increased. The b-values of ginseng seed oil following supercritical fluid extraction were 3.54 to 15.6 and those following compression extraction after roasting treatment at $200^{\circ}C$ for 30 min, were 20.49, which was the highest value. The result of the phenolic compounds composition showed the presence of gentisic acid, vanillic acid, ferulic acid, and cinnamic acid in the ginseng seed oil. No differences were detected in phenolic acid levels in ginseng seed oil extracted by compression extraction or solvent extraction, but vanillic acid tended to decrease as extraction pressure and temperature were increased for seed oil extracted by a supercritical fluid extraction method. The fatty acid composition of ginseng seed oil was not different based on the extraction method, and unsaturated fatty acids were >90% of all fatty acids, among which, oleic acid was the highest at 80%. Phytosterol analysis showed that ${\beta}$-sitosterol and stigmasterol were detected. The phytosterol content of ginseng seed oil following supercritical fluid extraction was 100.4 to 135.5 mg/100 g, and the phytosterol content following compression extraction and solvent extraction was 71.8 to 80.9 mg/100 g.
### The Current Status of Supercritical Fluid Extraction Technology and Industrial Applications (초임계유체 추출 기술 및 상업화 현황)
• Ju Young-Woon;Lee Moon Young;Woo Moon Jea;Byun Sang Yo
• KSBB Journal
• /
• v.20 no.5
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• pp.329-337
• /
• 2005
• Because of their unique properties, supercritical fluids have been known as one of the most promising materials for the future technology. Supercritical fluid technologies have been widely applied to various operations such as extraction, impregnation, nano-particle generation, oxidation, reaction etc. Industrial applications, especially their successful usage of supercritical fluid, have been reviewed. A special case for the first successful industrial application of supercritical $CO_2$ extraction in Korea was reviewed. Its unique characteristics of enriched antioxidant, $'\grmma-tocopherol'$ enabled this industrial application in Korea in spite of its low market price. Also its size and operation conditions were known as world records.
### Extraction of Docosahexaenoic Acid (DHA) from Lyophilized Thraustochitrium sp.
• CHO, JOONG-HOON;GWI-SUK HEO;JI-WON YANG
• Journal of Microbiology and Biotechnology
• /
• v.6 no.5
• /
• pp.358-360
• /
• 1996
• Solvent extraction, soxhlet method, and supercritical fluid extracion were considered, respectively, as the method of choice for the recovery of DHA from lyophilized Thraustochitrium sp., and the results of corresponding extraction were compared. Supercritical fluid extraction seems to be the most appropriate process with respect to time, process simplicity, and extractant intoxicity.
### Antimicrobial Activity of Extracted by Supercritical Fluid from Origanum vulgare, Cinnamomum cassia, Chamaecyparis obtusa and Scutellariae baicalensis (오레가노, 육계, 편백 및 황금의 초임계 유체 추출물의 항균 활성)
• Kim, Woo-Jin;Cho, Jun-Young;Choi, Chang-Suk;Yoon, Gee-Sun;Lee, Won-Kyu;Ryu, Yeon-Woo
• KSBB Journal
• /
• v.23 no.2
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• pp.147-152
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• 2008
• The variety of functional plants has an attention for new natural food preservation and natural antiseptic development. The extracts from functional plants with various methods (ethanol extraction, hot water extraction and supercritical fluid extraction) tested antimicrobial activity against 10 strains including the pathogenic and food poisoning bacteria, the yeast and fungi. The antimicrobial activities of supercritical fluid extracts were shown higher than ethanol extract and hot water extract when tested with disc-diffusion method and minimum inhibitory concentration (MIC). Antimicrobial activity of supercritical fluid extract was two times higher than ethanol extract in Cinnaonomum cassia. In addition, the supercritical fluid extractions of Chamaecyparis obtuas and the C. cassia showed the higher yield than Origanum vulgare and Scutellariae baicalensis. The supercritical fluid extract of C. cassia showed an antimicrobial activity against all strains tested. The supercritical fluid extract of S. baicalensis showed strong antimicrobial activity on Listeria monocytogenes. Supercritical fluid extraction of O. vulgare and C. obtuas showed strong antimicrobial activity on Salmonella typhimuriium. In MIC test, C. obtuas was shown the best natural material for the preparation of natural antimicrobial agent by supercritical fluid extraction. In conclusion, these results suggest that supercritical fluid extraction technique was effective to obtain functional ingredient with higher antimicrobial activity in the development of new antimicrobial reagent from natural materials.
### Supercritical Fluid Extraction of Physiologically Active materials from Agaricus blazei Fruiting Bodies (Agaricus blazei 자실체로부터 초임계 유체를 이용한 생리활성물질 추출공정)
• 최정우
• KSBB Journal
• /
• v.15 no.4
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• pp.408-410
• /
• 2000
• The supercritical fluid extraction(SFE) technique was applied for the isolation and purification of nonpolar physiologically acitve mateials from Agaricus blazei fruiting bodies. the qualitative analysis of extract was accomplished by gas chromatography-mass spectrometer(GC-MS) and extract was determined as linoleic acid(cis-9 cis-12-octadecadienoic acid) In order to obtain the optimum operating conditions of supercritical fluid extraction process the various temperatures and pressures were applied for process operation. From the comparison of exraction efficiencies $50^{\circ}C and 200 kg_f/cm^2$ were determined as optimum conditions.
### Effect of Solvents as Subcritical and Supercritical Fluid on Decomposition and Extraction of Used Automotive Tire (아임계와 초임계유체로써 폐타이어 분해와 추출에 미치는 용매의 영향)
• Kang, W.S.;Na, D.Y.;Kim, I.S.;Han, S.B.;Park, P.W.
• Elastomers and Composites
• /
• v.34 no.3
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• pp.239-246
• /
• 1999
• Side wall samples from a used automotive tire were subjected to subcritical and supercritical decomposition and extraction with three solvents, water, 28% ammonia solution and ammonia. For 6mm cube samples the rate of supercritical extraction with water followed a first-order kinetics with an activation energy of 140 kJ/mol. Solvent power of 28% ammonia so lotion at supercritical condition was found to be higher than supercritical water at initial extraction as pressure decreased. These phenomena were considered to be an effect of ammonia involved in water.
### Removal of toxic compounds from Acer tegmentosum using supercritical fluid extraction (초임계유체 추출을 이용한 산겨릅나무로부터 독성성분들의 제거)
• Pyo, Dongjin;Jin, Jungeun
• Analytical Science and Technology
• /
• v.21 no.5
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• pp.392-396
• /
• 2008
• Acer tegmentosum is a tree used to treat various liver diseases in Korea. There have been some concern regarding the safety of Acer tegmentosum due to some toxic chemical compounds in its stems. Supercritical fluid extraction (SFE) was employed to develop a removing method of toxic compounds from Acer tegmentosum. The toxic compounds were effectively extracted with ethanol modified supercritical fluid $CO_2$. The optimum condition of SFE was 100 bar of pressure, $40^{\circ}C$ of extraction temperature, 3 mL/min of $CO_2$ flow rate, 0.2 mL/min of modifier (ethanol) flow rate.
### Efficient Extraction of Bioethanol from Freshwater Cyanobacteria Using Supercritical Fluid Pretreatment
• Pyo, Dongjin;Kim, Taemin;Yoo, Jisun
• Bulletin of the Korean Chemical Society
• /
• v.34 no.2
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• pp.379-383
• /
• 2013
• For the production of ethanol from freshwater cyanobacteria, a new pretreatment method using supercritical fluid was introduced. In this study, it was found that the supercritical fluid could penetrate inside the cell wall and help to liberate starch from cyanobacterial cells which resulted in the increase of the efficiency of ethanol production. For Microcystis aeruginosa, supercritical fluid pretreatment increased the amount of ethanol produced from cyanobacteria from 1.53 g/L to 2.66 g/L. For Anabaena variabilis, the amount of ethanol was increased from 1.25 g/L to 2.28 g/L. With use of supercritical fluid pretreatment, the efficiency of the process to obtain higher ethanol yields from freshwater cyanobacteria was improved upto 80%. The optimum temperature and pressure conditions for supercritical fluid pretreatment were determined as the temperature of $40^{\circ}C$ and the pressure of 120 atm. This study demonstrates the feasibility of using supercritical fluid pretreatment for ethanol production using freshwater cyanobacteria.
### Comparison of Essential Oil Composition Extracted from Agastache rugosa by Steam Distillation and Supercritical Fluid Extraction (수증기 증류법과 초임계유체 추출법으로 분리한 배초향의 정유성분 조성 비교)
• 김근수;김삼곤;김용하;김영회;이종철
• Journal of the Korean Society of Tobacco Science
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• v.23 no.1
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• pp.65-70
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• 2001
• In order to compare the extraction patterns of main components from the raw material between the extraction methods, the aerial parts (dried stem, leaves, and flowers) of Agastache rugosa were extracted by SDE simultaneous steam distillation & extraction) and SFE (supercritical fluid extraction). Volatile components of essential oil and extract were identified by GC and GC-MSD. The contents of essential oil extracted by SDE were 0.49% in aerial part of Agastache rugosa on dry basis. Major components were methyl chavicol(27.2%), isomenthone(24.6%), hexadecanoic acid(13.0%). menthone (5.5%) among 32 kinds of components confirmed in essential oil. On the other hand, the contents of SFE extracts revealed 3.21% on dry basis, 6 times higher than those of SDE. Major components were isomenthone(15.3%), hexadecanoic acid(13.7%), methyl chavicol(12.6%), benzoic acid(3.8%) among 33 kinds of components identified in extract.
### Detection of estrogenic hormone 17β-estradiol in soil samples by a recombinant yeast bioassay and supercritical fluid extraction
• Shim, Jae-Han;Kim, Mi-Ra;Topp, Edward;Choi, Jeong-Heui;Mamun, Iqbal Rouf
• Korean Journal of Environmental Agriculture
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• v.27 no.4
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• pp.447-455
• /
• 2008
• Recombinant yeast estrogenicity (YES) assay was used as a bioanalytical tool in order to screen $17{\beta}$-estradiol in the soil samples collected from different sites of South Korea. Solvent extraction and supercritical fluid extraction (SFE) methods were compared for the extraction of the estradiol from the soils. Most high detection of the estradiol based on YES assay was observed in the soils extracted with methanol. Different types of estrogenic hormones including $17{\beta}$-estradiol were suggested to be possibly exiting in the soils, since the methanol extracts of the soils showed an estrogenic activity that was not observed in the hexane extracts of the soil. SFE extracts showed estrogenic activity in some of the samples but methanol extract showed best activity. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6302857398986816, "perplexity": 20678.992384767378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740838.3/warc/CC-MAIN-20200815094903-20200815124903-00225.warc.gz"} |
https://www.physicsforums.com/threads/confused-charles-law.145885/ | Confused, charles law
1. Nov 29, 2006
MRCHEM
why is the answer different if you didnt convert T into kelvin considering that
they have the same temp in actual process
2. Nov 29, 2006
HallsofIvy
Staff Emeritus
The same temperature? Yes, but Kelvin and Celcius use different numbers to represent that temperature! Using different numbers in a formula typically gives different answers.
Oh, I see what you mean! Charles Law says that $V_1/T_1= V_2/T_2$ so $V_1/V_2= T_1/T_2$ and you are asking why the differences don't "cancel". They would IF Kelvin temperature were simply a multiple of Celcius Temperature. If K= aC, then $K_1/K_2= (aC_1)/(aC_2)= C_1/C_2$. But they are not: Kelvin temperature is Celcius temperature minus 273.15. That is K= C-a. $K_1/K_2= (C_1- a)/(C_2-a)$ and we can't cancel.
3. Nov 29, 2006
dextercioby
And keep it then in Celsius, Fahrenheit, Reaumur, Beufort, Mercali, Richter or other scales ? Maybe it's probably Charles law was valid only when both the volume and the temperature are expressed in SI units, with T the absolute thermodynamical temperature measured in Kelvin.
Daniel. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.779824435710907, "perplexity": 3052.1943722881024}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171632.91/warc/CC-MAIN-20170219104611-00199-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://aimpl.org/highlowforcing/3/ | $\newcommand{\Cat}{{\rm Cat}}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\freestar}{ \framebox[7pt]{\star} }$
## 3. Problems in the Overlap of High and Low Forcing
1. #### Problem 3.1.
Is there a model for $\text{TP}_{\kappa^{++}}$, with $\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset?
• #### Problem 3.2.
Is there a model for $\text{TP}_{\aleph_2} + \text{TP}_{\aleph_3}$ using side conditions?
• #### Problem 3.3.
Is there a model where both $I[\aleph_2]$ and $I[\aleph_3]$ are trivial? In other words, is $\text{AP}_{\aleph_2} + \text{AP}_{\aleph_3}$ consistent?
• #### Problem 3.4.
Is the forcing axiom for proper posets of size $\aleph_1$ with $\aleph_2$-many dense sets consistent? If so, what if we replace the size-$\aleph_2$ requirement with the $\aleph_2$-chain condition?
• There is a coloring $F:[\aleph_2]^2 \rightarrow {0,1}$ of pairs from $\aleph_2$ in $2$ colors so that the poset of finite approximations to a $0$- or $1$-homogeneous set has the $\aleph_2$-chain condition. (We mean either the poset $\mathbb P$ of finite functions $f:\omega \rightarrow F^{-1}(0)$ or the poset of $f:\omega \rightarrow F^{-1}(0)$, ordered by inclusion)
#### Problem 3.5.
Can this forcing be made proper using side conditions? Does this preserve the $\aleph_2$-chain condition?
If we do obtain properness, then the fprcing axiom for $\aleph_2$-many dense sets proper posets of size $\aleph_1$—and for proper posets with the $\aleph_2$-chain condition—is inconsistent.
• #### Problem 3.6.
[David Aspero] Suppose $\aleph_\omega$ is a strong limit. Is there a poset ${\mathbb P}$ of size $<\aleph_\omega$, which adds a club guessing sequence for $S^{\omega_2}_{\omega_1}$, preserving $\omega_1$, $\omega_2$, and $\omega_3$.
If yes than by PCF theory, in ZFC it would be provable that $2^{\aleph_\omega}<\aleph_{\omega_3}$ when $\aleph_\omega$ is a strong limit.
Cite this as: AimPL: High and Low forcing, available at http://aimpl.org/highlowforcing. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9573376178741455, "perplexity": 914.3976056935571}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527865.32/warc/CC-MAIN-20190419161226-20190419183226-00156.warc.gz"} |
https://pureportal.coventry.ac.uk/en/publications/evaluation-of-near-field-electromagnetic-shielding-effectiveness- | # Evaluation of near-field electromagnetic shielding effectiveness at low frequencies
Yessica Arellano, Andrew Hunt, Olivier Haas
Research output: Contribution to journalArticlepeer-review
5 Citations (Scopus)
## Abstract
Magnetic induction tomography (MIT) is a novel technology for flow measurement offering significant promise in the measurement of multiphase flows containing low-conductivity fluids such as saline water. Such measurements rely on optimal effective shielding to avoid external field interference and extraneous capacitive coupling that can lead to false readings and overestimations of the eddy current-induced fields. The performance of various attenuation materials in the low megahertz frequency spectrum is presented and compared with outcomes from a numerical computational method. The results demonstrate that the shielding mechanism that prevails at low frequencies is that of reflection. Consequently, hard shields such as metals show superior wave attenuation performance for MIT systems operating below 13 MHz. For higher frequencies, the absorption effect on the incident wave path within soft electromagnetic shields presents enhanced shielding properties. This paper also explores the limitations of traditional testing geometry for shielding effectiveness and proposes an alternative approach to near-field, free-space measurement for MIT sensors. The proposed semi-enclosed approach shows enhanced shielding effectiveness measurements compared with the traditional transversal barrier method. The proposed method was used to assess the electromagnetic shielding effectiveness of ferromagnetic and various metallic materials.
Original language English 8481434 121 - 128 8 IEEE Sensors Journal 19 1 4 Oct 2018 https://doi.org/10.1109/JSEN.2018.2873909 Published - 1 Jan 2019
### Bibliographical note
© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
## Keywords
• Eddy currents
• electromagnetic compatibility
• electromagnetic induction
• electromagnetic shielding
• tomography
## ASJC Scopus subject areas
• Instrumentation
• Electrical and Electronic Engineering | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8033386468887329, "perplexity": 1443.272278823165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988850.21/warc/CC-MAIN-20210508061546-20210508091546-00280.warc.gz"} |
http://crypto.stackexchange.com/users/592/samuel-neves?tab=activity&sort=posts | Samuel Neves
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Jun13 answered Can we reduce Diffie-Hellman problem to “Discrete-log inversion” problem? Jun8 answered RSA public key recovery from signatures Jun5 answered Why does anyone use elliptic curves for a CSPRNG? May23 answered Logjam on Elliptic Curves? May7 answered Encoding a message to a point of curve y^2=x^3+7 and Bitcoin Core May6 answered How is HMAC(message,key) more secure than Hash(key1+message+key2) May4 answered Factoring two RSA moduli $N_i=p_i\cdot q_i$ knowing that $p_2=p_1+2$? Mar28 answered What aspect of elliptic curve encryption paradigms makes them especially susceptible to quantum based attack algorithms? Nov19 answered Do data-dependent rotations have any advantage over fixed rotations? Oct29 answered Understanding Twist Security with respect to short Weierstrass curves Aug18 answered SRP-6 vulnerabilities when N is small Jun19 answered Parallelized Pollard's Rho algorithm for ECDLP + Jacobian coordinates Jun19 answered Fast hashing into elliptic curve Jun15 answered curve25519 weak points for contributory behaviour May6 answered Is knowing the private key of RSA equivalent to the factorization of $N$? Feb9 answered Safe elliptic curve point addition using projective coordinates: How do I tell if the points are the same? Jan19 answered Why is this authentication procedure using Rabin crypto not useful? Jan13 answered Time complexity to solve Discrete log problem Dec26 answered Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? Dec23 answered Why do these Python XTEA implementations require different deltas? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.675001859664917, "perplexity": 12759.308435357127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097038.44/warc/CC-MAIN-20150627031817-00202-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://math.stackexchange.com/users/5676/peter-taylor?tab=activity&sort=revisions | # Peter Taylor
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# 90 Revisions
Nov30 revised What is the minimum number of locks on the cabinet that would satisfy these conditions? Add missing "fi" - I suspect this was copy-pasted from a PDF which used a ligature Nov7 revised Probabilty to win in die rolling game Formatting and linguistic style Nov4 revised is this operating procedure an Abelian Group? added 307 characters in body Nov4 revised is this operating procedure an Abelian Group? added 78 characters in body Nov2 revised Generator of a subgroup of a cyclic group Title and minor tidying Nov1 revised Characteristic roots-recursion. LaTeX, form into sentences, punctuate Oct18 revised Real analysis Proving a sequence lim inf |an|=0 LaTeX Oct14 revised Solutions to $x^3+y^3+z^3 = x^2 + y^2 + z^2=x+y+z=0$ All-Mathjax titles prevent people from right-clicking to open in new tab/window Oct11 revised odds / percent is the game flawed or this seem accurate Formatting, no need for signature Oct9 revised What is the difference between the terms 'equation' and 'algorithm'? This admits a definitive answer and hence is not a "soft question" Oct8 revised Which complete graphs are (edge) magic? added 1 characters in body Oct8 revised Which complete graphs are (edge) magic? added 46 characters in body Sep18 revised Proof for solar declination angle? added 764 characters in body Sep17 revised Proof for solar declination angle? deleted 40 characters in body Sep17 revised Proof for solar declination angle? Grammar, LaTeX Sep17 revised History of the theory of equations: John Colson Remove the link to the wrong paper to avoid confusing more people Sep16 revised Kaprekar's constant-related problem Add graph Sep16 revised Kaprekar's constant-related problem edited body Sep15 revised Find all $n\in\mathbb{Z}^+$ satisfying : $n\mid(2^{\varphi(n)}+3^{\varphi(n)}+\cdots +n^{\varphi(n)}).$ Fix a misplaced } Sep13 revised Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number Don't need the Stewart/Sierpinski characterisation for these cases - that was a hangover from previous thinking | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7627665996551514, "perplexity": 5816.072318636889}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163053669/warc/CC-MAIN-20131204131733-00031-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.mapleprimes.com/questions/95060-How-To-Generate-All-Vectors-Of-Length | # Question:How to generate all vectors of length m with entries are 0's or 1's and entries sum to n then sum over these entries?
## Question:How to generate all vectors of length m with entries are 0's or 1's and entries sum to n then sum over these entries?
Maple
Hi,
$\mathop {\sum {...\sum }}\limits_{z_r \in {Q_{2}^{*}}}$ such that
$Q_{2}^{*}=((z_1,...,z_r):z_{j}=0,1,\sum_{j=1}^{r}z_{j}=\ell)$, where $\ell = 0, 1, . . . , r$.
For example. If r=3, $\ell=1$; then we are looking for triples with elements 0's or 1's and sum to 1 i.e. $Q_{2}^{*}={(0,0,1),(0,1,0),(1,0,0)}$. So we would like to sum over theses $z_r$'s. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9417150020599365, "perplexity": 398.27003021034574}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300533.72/warc/CC-MAIN-20220117091246-20220117121246-00661.warc.gz"} |
https://brilliant.org/problems/an-algebra-problem-by-rashi-kumar/ | # A geometry problem by Rashi Kumar
Geometry Level pending
If the length of a rectangle is increased by 10% and the area is unchanged then the corresponding breadth must be decreased by............%
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.849597692489624, "perplexity": 1562.0150859477849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948587496.62/warc/CC-MAIN-20171216084601-20171216110601-00079.warc.gz"} |
https://www.lessonplanet.com/teachers/rr-rachel-runs-in-the-rain | # Rr-- Rachel Runs in the Rain
In this printing practice learning exercise, students study both upper and lower case letter r by tracing the words "Rachel runs in the rain" four times. Students then color the picture. This is Zaner-Bloser style.
Concepts
Resource Details | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.837443470954895, "perplexity": 5788.901105450579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425737.60/warc/CC-MAIN-20170726002333-20170726022333-00449.warc.gz"} |
https://stat.ethz.ch/pipermail/r-devel/2008-August/050542.html | # [Rd] Outdated dependencies and install.packages()
Martin Maechler maechler at stat.math.ethz.ch
Thu Aug 21 09:39:13 CEST 2008
>>>>> "PD" == Peter Dalgaard <P.Dalgaard at biostat.ku.dk>
>>>>> on Wed, 20 Aug 2008 17:06:36 +0200 writes:
PD> Prof Brian Ripley wrote:
>> On Wed, 20 Aug 2008, Peter Dalgaard wrote:
>>
>>> This looks like a buglet:
>>>
>>>> install.packages("lme4", depend=TRUE, lib="~/Rlibrary/")
>>> --- Please select a CRAN mirror for use in this session ---
>>> [...snip...]
>>> Error: package 'Matrix' 0.999375-4 was found, but >= 0.999375.11 is
>>> required by 'lme4'
>>> Execution halted
>>> ** Removing '/home/bs/pd/Rlibrary/lme4'
>>> ** Restoring previous '/home/bs/pd/Rlibrary/lme4'
>>>
>>> and the DESCRIPTION file for lme 4 does have
>>>
>>> Depends: methods, R(≥ 2.7.0), Matrix(≥ 0.999375-11), lattice
>>> <http://cran.r-project.org/web/packages/lattice/index.html>
>>>
>>>
>>> Looks like we're either not checking the version requirement of
>>> dependencies, or something is making 4 > 11... (from my reading of the
>>> code, it is the former, but I could have missed something).
>>
>> We don't check version requirements of dependencies in
>> install.packages(). That's a minefield related to having multiple
>> versions of a package on a system and which gets loaded/namespace
>> loaded, so you can easily have the needed version but pick up a
>> different one.
>>
>> issues) you may or may not need the dependencies to install the
>> package. It's also a problem that you can update dependencies to
>> incompatible ones, so the versions available at installation are only
>> part of the story.
>>
>> I don't see why the message reports .11 not -11, but that may be
>> related to "package_version" classes.
>>
>>> We should at least document the behaviour.
>>
>> I am not sure where you think it should be documented and is not.
>> But for install.packages():
>>
>> dependencies: logical indicating to also install uninstalled packages
>> on which these packages depend/suggest/import (and so on
>> recursively). Not used if 'repos = NULL'. Can also be a
>> character vector, a subset of 'c("Depends", "Imports",
>> "Suggests")'.
>>
>> 'uninstalled' seems pretty clear to me.
>>
PD> It isn't. Matrix 0.999375-11 was 'uninstalled'. The above could just
PD> mean that if you installed it, it wouldn't be installed again.
PD> Something like "Notice that version dependencies are not checked." would
PD> help. (Assuming that people read the help page.)
Or even more:
"
Notice that version dependencies are not checked such that
outdated versions of installed packages are \emph{not} updated.
\code{install.packages()}.
"
Note that --- thanks to Kate Mullen's nice help.request() proposal ---
one of my top "TODOs" for R-devel is
to finish a generalization of the current
update.packages()
function (that I have not quite finished)
which will allow to only look for updates of specified set of
packages rather than all packages in a given lib.loc
{something that should *not* be triggered by a help.request()
for a user (as ours) who has ~ 2000 R packages installed}.
Martin | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5354468822479248, "perplexity": 13365.840657382541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00035-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://eprint.iacr.org/2019/095 | ### Variants of the AES Key Schedule for Better Truncated Differential Bounds
Patrick Derbez, Pierre-Alain Fouque, Jérémy Jean, and Baptiste Lambin
##### Abstract
Differential attacks are one of the main ways to attack block ciphers. Hence, we need to evaluate the security of a given block cipher against these attacks. One way to do so is to determine the minimal number of active S-boxes, and use this number along with the maximal differential probability of the S-box to determine the minimal probability of any differential characteristic. Thus, if one wants to build a new block cipher, one should try to maximize the minimal number of active S-boxes. On the other hand, the related-key security model is now quite important, hence, we also need to study the security of block ciphers in this model. In this work, we search how one could design a key schedule to maximize the number of active S-boxes in the related-key model. However, we also want this key schedule to be efficient, and therefore choose to only consider permutations. Our target is AES, and along with a few generic results about the best reachable bounds, we found a permutation to replace the original key schedule that reaches a minimal number of active S-boxes of 20 over 6 rounds, while no differential characteristic with a probability larger than $2^{-128}$ exists. We also describe an algorithm which helped us to show that there is no permutation that can reach 18 or more active S-boxes in 5 rounds. Finally, we give several pairs $(P_s, P_k)$, replacing respectively the ShiftRows operation and the key schedule of the AES, reaching a minimum of 21 active S-boxes over 6 rounds, while again, there is no differential characteristic with a probability larger than $2^{-128}$.
Available format(s)
Category
Secret-key cryptography
Publication info
Published elsewhere. MINOR revision.SAC 2018
DOI
10.1007/978-3-030-10970-7_2
Contact author(s)
baptiste lambin @ irisa fr
History
Short URL
https://ia.cr/2019/095
CC BY
BibTeX
@misc{cryptoeprint:2019/095,
author = {Patrick Derbez and Pierre-Alain Fouque and Jérémy Jean and Baptiste Lambin},
title = {Variants of the AES Key Schedule for Better Truncated Differential Bounds},
howpublished = {Cryptology ePrint Archive, Paper 2019/095},
year = {2019},
doi = {10.1007/978-3-030-10970-7_2},
note = {\url{https://eprint.iacr.org/2019/095}},
url = {https://eprint.iacr.org/2019/095}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3872942626476288, "perplexity": 1422.7899778030937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499768.15/warc/CC-MAIN-20230129211612-20230130001612-00527.warc.gz"} |
https://www.physicsforums.com/threads/thermo-derivation.283004/ | # Thermo Derivation
1. Jan 5, 2009
### olechka722
Hello,
I am trying to derive a formula for entropy. I have:
dS= Cv/T dT + R/(V-b) dV
and want to get:
S= Cv*ln(T) + R*ln(V-b) + constant.
Math rules seem to say i cant just integrate this up even though it looks obvious since i have two different d's on the right hand side. Maybe something using Maxwell relations? Not sure.
Thank you for any help!
olechka
2. Jan 5, 2009
### Mapes
The first term on the right side contains only a constant, a function of T, and dT. The second term contains only constants, a function of V, and dV. It's OK to integrate the terms separately in this case. You can verify this by differentiating the second expression.
3. Jan 5, 2009
### Renge Ishyo
You can just integrate it up I believe. The derivatives in this instance are actually partial derivates if I am not mistaken (the evidence is that Cv is the heat capacity at constant V...which implies that the second term is the change due to volume at constant T). So the total change in S is the sum of the integration of the two partial derivatives.
4. Jan 5, 2009
### Mapes
Not quite. It's true that the terms represent partial derivatives; that is, we're looking at
$$dS=\left(\frac{\partial S}{\partial T}\right)_V dT+\left(\frac{\partial S}{\partial V}\right)_T dV$$
However, this observation isn't sufficient to conclude that the integral of the right hand side equals the sum of the integrals of the individual terms. We must also require that $C_V(T,V)=T\left(\frac{\partial S(T,V)}{\partial T}\right)_V$, which is generally a function of both T and V, is idealized as a constant, as I stated above. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9728992581367493, "perplexity": 507.32270437767613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510749.55/warc/CC-MAIN-20181016113811-20181016135311-00283.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/185642-proof-first-date-event-happens-stopping-time-print.html | # Proof of ''The first date for which an event happens is a stopping time.''
Suppose ${X(t,\omega)}$ is a stochastic process on a filtered probability space $(\Omega,\mathbb{F},P)$, and let $A$ be any Borel set.
Then, I want to show that the first date for which $X(t,\omega)\in A$ is a stopping time. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8174018263816833, "perplexity": 122.10742789319787}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187717.19/warc/CC-MAIN-20170322212947-00445-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/critical-field-for-a-superconductor-why.163903/ | # Critical field for a superconductor - why?
1. Apr 3, 2007
### postscript07
Cool a cylinder made from a Type I superconducting material below the superconducting transition temperature.
Apply a magnetic field parallel to the cylinder. The cylinder expels the flux.. up until the field reaches a critical value.
Why does a critical value exist? That is, why doesn't the cylinder expel magnetic flux independent of the field strength?
Many thanks.
2. Apr 3, 2007
### marcusl
In short, it is a question of energy balance. The superconducting state has lower energy due to "condensation" of electrons into Cooper pairs. In the absence of a strong field, it is energetically favorable to be in the superconducting state.
The expelled flux, however, exerts a pressure inwards and this represents an unfavorable energy situation. As the field strength rises, the energy penalty to exclude flux exceeds the energy savings of condensation, and the material goes normal.
3. Apr 4, 2007
### postscript07
Thank you marcusl, that is very helpful.
Does a critical field exist for a zero-resistance (not superconducting) cylinder? Hall (1st ed Manchester) p264 suggests this is the case. Why?
4. Apr 4, 2007
### marcusl
Since a non-SC zero resistance material doesn't exist, this is a theoretical discussion. There is no critical field so far as I know.
Do you know the difference in behavior between the Meissner effect (SC) and a perfect conductor? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9592244029045105, "perplexity": 2137.1878851706592}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867417.71/warc/CC-MAIN-20180526112331-20180526132331-00635.warc.gz"} |
http://mathsvideos.net/2015/01/ | # Golden Ratio Proof
Prove that:
$\frac { a }{ b } =\frac { a+b }{ a } =\frac { 1+\sqrt { 5 } }{ 2 }$
First say that:
$p=\frac { a }{ b }$
$bp=a$
So:
$\frac { a }{ b } =\frac { a+b }{ a } \\ \\ p=\frac { bp+b }{ bp } \\ \\ p=\frac { b\left( p+1 \right) }{ bp } \\ \\ p=\frac { p+1 }{ p } \\ \\ { p }^{ 2 }=p+1\\ \\ { p }^{ 2 }-p=1\\ \\ { \left( p-\frac { 1 }{ 2 } \right) }^{ 2 }-{ \left( \frac { 1 }{ 2 } \right) }^{ 2 }=1\\ \\ { \left( p-\frac { 1 }{ 2 } \right) }^{ 2 }=\frac { 4 }{ 4 } +\frac { 1 }{ 4 } \\ \\ { \left( p-\frac { 1 }{ 2 } \right) }^{ 2 }=\frac { 5 }{ 4 } \\ \\ p-\frac { 1 }{ 2 } =\frac { \sqrt { 5 } }{ \sqrt { 4 } } \\ \\ p-\frac { 1 }{ 2 } =\frac { \sqrt { 5 } }{ 2 } \\ \\ p=\frac { 1 }{ 2 } +\frac { \sqrt { 5 } }{ 2 } \\ \\ p=\frac { 1+\sqrt { 5 } }{ 2 }$
Now:
$\frac { a+b }{ a } \\ \\ =\frac { \left( 1+\sqrt { 5 } \right) +2 }{ \left( 1+\sqrt { 5 } \right) } \\ \\ =\frac { \left( 1+\sqrt { 5 } \right) }{ \left( 1+\sqrt { 5 } \right) } +\frac { 2 }{ \left( 1+\sqrt { 5 } \right) } \cdot \frac { \left( 1-\sqrt { 5 } \right) }{ \left( 1-\sqrt { 5 } \right) } \\ \\ =1+\frac { \left( 2-2\sqrt { 5 } \right) }{ -4 } \\ \\ =\frac { -4 }{ -4 } +\frac { \left( 2-2\sqrt { 5 } \right) }{ -4 } \\ \\ =\frac { -4+\left( 2-2\sqrt { 5 } \right) }{ -4 } \\ \\ =\frac { -4+2-2\sqrt { 5 } }{ -4 } \\ \\ =\frac { -2-2\sqrt { 5 } }{ -4 } \\ \\ =\frac { -2\left( 1+\sqrt { 5 } \right) }{ -2\cdot 2 } \\ \\ =\frac { 1+\sqrt { 5 } }{ 2 }$
# Proving That Two Different Variance Formulas Are Equal To One Another
Prove that:
$\frac { \Sigma { \left( x-\overline { x } \right) }^{ 2 } }{ n } =\frac { \Sigma { x }^{ 2 } }{ n } -{ \left( \frac { \Sigma x }{ n } \right) }^{ 2 }$
Well you first identify these truths:
$x=\left\{ a,\quad b,\quad c \right\} \\ \\ \therefore \quad \overline { x } =\frac { \Sigma x }{ n } =\frac { a+b+c }{ n } ,\quad \\ \\ \therefore \quad n=3\\ \\ \therefore \quad n\overline { x } =a+b+c\\ \\ \therefore \quad \frac { \Sigma { x }^{ 2 } }{ n } =\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ n } .\\ \\ \therefore \quad \Sigma { x }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }$
And next:
The standard deviation formula looks like this:
$\sigma =\sqrt { \frac { \Sigma { \left( x-\overline { x } \right) }^{ 2 } }{ n } } =\sqrt { \frac { \Sigma { x }^{ 2 } }{ n } -{ \left( \frac { \Sigma x }{ n } \right) }^{ 2 } }$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981181621551514, "perplexity": 7442.500425233811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741151.56/warc/CC-MAIN-20181112215517-20181113001517-00399.warc.gz"} |
https://testbook.com/blog/signals-and-systems-for-gate-ec-quiz-1/ | • Share
# Signals and Systems for GATE EC Quiz 1
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Here is Signals and Systems for GATE EC Quiz 1 to help you prepare for your upcoming GATE exam. The GATE EC paper has several subjects, each one as important as the last. However, one of the most important subjects in GATE EC is Signals and Systems. The subject is vast, but practice makes tackling it easy.
This quiz contains important questions which match the pattern of the GATE exam. Check your preparation level in every chapter of Signals and Systems for GATE. Simply take the quiz and comparing your ranks. Learn about control system, Feedback principle and more.
Signals and Systems for GATE EC Quiz 1
Que. 1
A real valued signal $$\rm x(t)$$ limited to frequency band $$\rm \left| f \right| < \frac{\omega }{2}$$ is passed through an LTI system whose frequency response is
$$\rm H\left( f \right) = \;\left\{ {\begin{array}{*{20}{c}} {{e^{ – j4\pi f}},}&\rm {\left| f \right| < \frac{\omega }{2}}\\ {0,}& \rm {\left| f \right| > 0} \end{array}} \right.$$
The output of the system is
1.
$$\rm x(t+4)$$
2.
$$\rm x(t-4)$$
3.
$$\rm x(t+2)$$
4.
$$\rm x(t-2)$$
Que. 2
The unilateral Laplace transform of $$f\left( t \right) = \frac{1}{{{s^2} + s + 1}}$$. Which one of the following is unilateral Laplace transform of $$g(t)=tf(t)$$?
1.
$$\frac{{ – s}}{{{{\left( {{s^2} + s + 1} \right)}^2}}}$$
2.
$$\frac{-({2s + 1})}{{{{\left( {{s^2} + s + 1} \right)}^2}}}$$
3.
$$\frac{{ s}}{{{{\left( {{s^2} + s + 1} \right)}^2}}}$$
4.
$$\frac{({2s + 1})}{{{{\left( {{s^2} + s + 1} \right)}^2}}}$$
Que. 3
A stable LTI system has transfer function $$H\left( s \right) = \frac{1}{{{s^2} + s – 6}}$$. To make this system causal and stable it needs to be cascaded with another LTI system having transfer function $$\rm{H_1(s)}$$. Correct choice for $$\rm{H_1(s)}$$ is
1.
$$\rm{s+3}$$
2.
$$\rm{s-2}$$
3.
$$\rm{s-6}$$
4.
$$\rm{s+1}$$
Que. 4
The LTI causal system is given as $$\rm y\left[ n \right] = \alpha y\left[ {n – 1} \right] + \beta x\left[ n \right]$$
If impulse response is given as $$\rm \mathop \sum \limits_{n = 0}^\infty h\left[ n \right] = 2,$$ the relationship between $$\rm \alpha$$ and $$\rm \beta$$ is:
1.
$$\rm \;\alpha = 1 – \frac{\beta }{2}$$
2.
$$\rm \alpha = 1 + \frac{\beta }{2}$$
3.
$$\rm \alpha = 2\beta$$
4.
$$\rm \;\alpha = – 2\beta$$
Que. 5
Which of the following statements is $$\rm{\textbf{not true}}$$ for continuous time causal and stable LTI system.
1.
All poles must lie on left side of jω – axis
2.
Zeros of the system can lie anywhere in s – plane
3.
All poles must lie within |s|=1
4.
All the roots of the characteristics equation must be located on left side of jω axis
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3 years ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8700371980667114, "perplexity": 2481.412343721685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998923.96/warc/CC-MAIN-20190619063711-20190619085711-00424.warc.gz"} |
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https://discourse.julialang.org/t/unstable-julia-language-server-was-quarto-vscode-and-unicode-characters/90317 | # Unstable Julia Language Server? (was Quarto VSCode and unicode characters?)
I’m writing a Quarto document as part of a series of tutorials I’m writing, and will announce in a few weeks.
My Quarto document explains the basic ideas of differential calculus using nonstandard analysis. I’ve got a couple places where it would be useful to stick in some math characters… like a \theta specifically within the Julia code blocks… it doesn’t seem like there is any kind of tab-completion for unicode either in the text blocks or the code blocks in Quarto.
Anyone know any suggested method for entering unicode in Quarto code blocks or text blocks? Is there a convenient vscode unicode extension?
EDIT: apparently it was due to the fact that the Julia Language Server had crashed! so now I am able to use \theta to insert a theta… Maybe I need to figure out if I need to file a bug against the language server. I often get a message when I first start vscode that the language server has crashed N times in a row…
EDIT2: It worked once, then crashed again… I’ll look into it. Anyone else experiencing instability in the Julia Language Server?
Ok, here’s some interesting results:
This is the content of my Project.toml in the project directory. There are a bunch of bare null characters in the file which is borking the Language Server. I see that this is happening off and on in many of my projects! Anyone have any idea what’s up?
1 Like | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8004044890403748, "perplexity": 1567.9832224621018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711077.50/warc/CC-MAIN-20221206092907-20221206122907-00669.warc.gz"} |
http://math.stackexchange.com/questions/210132/invertible-measurable-and-measure-preserving | # invertible, measurable and measure preserving [closed]
$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by
$T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$
and
$T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$
In class we said this $T$ is
a) invertivle
b) measurable
c) measure preserving.
My last Analysis and Stochasitc class is quite a time ago and so I do not see that easy how $T$ fulfills a)-c)....
I hope someone can tell my why this is true, or give me some hints how I can find that out myseld :)
Best, Luca
-
## closed as off-topic by Jonas, Shailesh, Leucippus, zz20s, Daniel W. FarlowJun 21 at 2:50
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jonas, Shailesh, Leucippus, zz20s, Daniel W. Farlow
If this question can be reworded to fit the rules in the help center, please edit the question.
To check it's invertible, you could simply find the inverse. It's a piecewise linear map so it shouldn't be too bad. You know it's measurable because it's piece-wise continuous. You should check the measure-preserving property by drawing what happens to points under $T$. – Christopher A. Wong Oct 9 '12 at 23:37
youre taking the square, smooshing it down and stretching it out. then cut off the excess and put it on top. (draw a picture). clearly invertible (just undo what you did). preserves measure (doesnt change areas, should be clear, what happens to the area of a little rectangle under $T^{-1}$?).
Hey! First thanks for the Answers! My $T^{-1}(x,y) = (\frac{x}{2},2y)$ for $0 \leq y < \frac{1}{2}$ And $= (\frac{x+1}{2},2y-1)$ for $\frac{1}{2} \leq y < 1$. – Luca Oct 10 '12 at 0:40
I pushed enter too early ;) Hey! First thanks for the Answers! The baker trafo describes taking a rectangle, strech it, fold it and turn it a quater. Hence if I name the two halfs $A$ and $B$, then $A$ has $\frac{1}{2}$ of the area and $T(A)$ sends it again to $\frac{1}{2}$ of the area. And then $T(A)$ intersects $A$ and $B$ in a square of area $\frac{1}{2}$. Well so informally, if we take the measure of $A$, and go the streching/folding.. back, than we still have the same area for $A$? How do I write this more formally? – Luca Oct 10 '12 at 0:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6959730386734009, "perplexity": 801.5868798466041}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395346.72/warc/CC-MAIN-20160624154955-00155-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.ima.umn.edu/convergence-multigrid-algorithm-Navier-Stokes-equations | Campuses:
# The convergence on the multigrid algorithm for Navier-Stokes equations
Author:
Chen Zhangxin and Li Kaitai
February
1987
Document:
Number:
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https://math.stackexchange.com/questions/868720/integration-by-substitution-why-do-we-change-the-limits | # Integration by substitution, why do we change the limits?
I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make?
Source of Quotation: Calculus: Early Transcendentals, 7th Edition, James Stewart
• Just because you changed variable. When $x=1$ with $u=1+9x/4$, what is $u$ ? – Claude Leibovici Jul 16 '14 at 7:18
Because the function has changed. Let's do an example:
$$\int_{-1}^1 x\,dx =0$$
because the integrand is odd and the interval is symmetric (you can also check directly).
Let's do a simple $u=x+1$ so that $du=dx$ then the right way we have:
$$\int_0^2 u-1\,du = {1\over 2}(u-1)^2\bigg|_0^2={1\over 2}-{1\over 2}=0$$
but if we fail to change the limits:
$$\int_{-1}^1(u-1)\,du = {1\over 2}(u-1)^2\bigg|_{-1}^1=-2$$
The underlying reason is that integration comes from Riemann sums, the function values depend on the interval of integration. When you change the interval, the heights of the rectangles you use in the definition change (remember the heights are the function values) so that you end up adding up different things if you don't change the function to compensate.
No, we don't change the limits. Who said that. They stay the same. Only their representation changes.
$$L = \int\limits^{x=4}_{x=1}\sqrt{1 + \dfrac{9}{4}x}dx$$
Now we substitute $u=1+\dfrac{9}{4}x$:
$$L = \int\limits^{x=4}_{x=1} \sqrt{u} dx$$
See? We didn't change anything. We only substituted $u$ for the $x$ expression in the integrand. But now, it contains mixed variables and not solvable as is. We need to substitute $u$ expressions for other $x$ expressions too.
$$L = \int\limits^{x=4}_{x=1} \sqrt{u} dx = \int\limits^{\dfrac{4}{9}(u-1)=4}_{\dfrac{4}{9}(u-1)=1} \sqrt{u} d\left( \dfrac{4}{9}(u-1) \right)$$
Now, all the variables are in $u$, and the integral is solvable.
$$\dfrac{4}{9}(u-1)=4 \implies u=10, \quad \dfrac{4}{9}(u-1)=1 \implies \dfrac{13}{4} \\ d\left( \dfrac{4}{9}(u-1) \right) = \dfrac{4}{9}du$$
We rewrite the integral expression:
$$L = \int\limits^{\dfrac{4}{9}(u-1)=4}_{\dfrac{4}{9}(u-1)=1} \sqrt{u} d\left( \dfrac{4}{9}(u-1) \right) = \dfrac{4}{9} \int\limits^{u=10}_{u=\dfrac{13}{4}} \sqrt{u} du$$
Have we ever changed the limits in any step? No! They are the exact same limits. There only represented by $u$ instead of $x$ in the final expression.
• I get your point (and I agree that this is more or less the proper way to think about substitution in integrals), but it is, at the very least, really confusing to say that the limits don't change. I would word it differently. – David Z Jul 16 '14 at 17:29
• I'm with @DavidZ, the limits change to $u$ limits, which while representing the same region for $x$ limits, are different. – Adam Hughes Jul 16 '14 at 23:01
By the substitution $x=\varphi(t)$ we have
$$\int_a^b f(x)dx=\int_{\varphi^{-1}(a)}^{\varphi^{-1}(b)}f(\varphi(t))\varphi'(t)dt$$
You asked two questions: “Why do we change the limits of integration here?” and “What difference does it make?” I wonder whether you are also asking “why do we substitute $u=1+\frac9 4 x$ ?” so I’ll cover that briefly too.
Why do we change the limits of integration?
The limits of integration are not actually being changed – just expressed in the language of the new variable $u$. Ironically, this is to stop the value of the answer getting changed!
If you were somehow able to perform the integral in your question directly (without substituting in $u=1+ \frac94 x$), you would naturally then evaluate the answer from $x=1$ to $x=4$. So when someone writes $$L=∫_1^4√(1+9/4 x) dx$$ it's assumed (by them and you) that $4$ and $1$ are $x$-values. That is, they really meant $$L=∫_{x=1}^{x=4}√(1+\frac94x)dx$$
(Make sense? If you performed $∫_1^42xdx$ you'd evaluate the answer, $x^2$, by substituting in $4$ and $1$ for $x$, giving $4^2-1^2=15$. i.e. you just evaluated the answer at $x=4$ and $x=1$, but it was so obvious that $4$ and $1$ referred to $x$-values, that no-one needed to write $∫_{x=1}^{x=4}2xdx$ even though that's what the question really meant. It’s important because if the question was, say, $∫_{y=1}^{y=4}2xdx$ then the answer would probably be different. After you got your general answer of $x^2$, you couldn’t then just substitute $4^2-1^2$; you would need to find out what $x$ is worth at $y=4$ and $y=1$.)
So the question could be more precisely written $$L=∫_{x=1}^{x=4}√(1+\frac94x)dx$$ By the way, we are going to substitute $u=1+\frac94 x$ because it looks like it’s going to be so much easier to integrate: $$L=∫_{x=1}^{x=4}√u dx$$ But we have a problem: integrating $u$ with respect to $x$, we may as well just be back with the original question. So instead we figure out what $dx$ means in the world of $u$: $$\frac{du}{dx}=\frac{d}{dx}(1+\frac94x)=\frac94⇒du=\frac94dx$$ or $$dx=\frac49du$$ So $$L=∫_{x=1}^{x=4}√u⋅\frac49du=\frac49∫_{x=1}^{x=4}u^{\frac12}du=\frac49⋅\frac32 u^{\frac32}\Bigg|_{x=1}^{x=4}=\frac23u^{\frac32}\Bigg|_{x=1}^{x=4}$$ Oh dear. Problem. All very well figuring out what $dx$ meant in the world of $u$, and all very well substituting $u$ for $1+\frac94x$, but we also need to know what $u$ is worth at $x=4$ and $x=1$ or we can’t evaluate the integral. That is why we "change" or "substitute for" the limits of integration: $$u=1+\frac94x⇒$$ $$at\, x=4,\,u=1+\frac94×4=10$$ $$at\, x=1,\,u=1+\frac94×1=3.25$$ Now we can evaluate the integral because the following two statements mean exactly the same thing: $$\frac23u^{\frac32}\Bigg|_{x=1}^{x=4} \,and \, \frac23u^{\frac32}\Bigg|_{u=3.25}^{u=10}$$ the difference being that we can work with the second, but not with the first. Answer: $$L=\bigg(\frac23×10^{\frac32}\bigg)-\bigg(\frac23×3.25^{\frac32}\bigg)=21.08-3.91=17.17 (2 d.p.)$$
What difference does it make?
If we hadn’t changed the limits of integration (i.e. recognised that $4$ and $1$ meant $x=4$ and $x=1$ not $u=4$ and $u=1$) then we’d get a different answer by mistake: $$L=\frac23u^{\frac32}\Bigg|_1^4=\bigg(\frac23×4^{\frac32}\bigg)-\bigg(\frac23×1^{\frac32}\bigg)=16/3-2/3=14/3\,\,i.e.\,4.33 \,(2 d.p.)$$ because we wrongly assumed that 4 and 1 referred to u rather than to x… By the way, it also makes sense to change the limits earlier i.e. at the same time as the other substitutions (although doing it earlier won’t affect your answer.)
The original limits is for variable $x$, and the new limits is for the new variable $u$. If you can get a primitive function $$\frac{8}{27}(1+\frac{9}{4}x)^\frac{3}{2}$$ by observation, it is unnecessary. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9482994079589844, "perplexity": 254.55482724942644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370493818.32/warc/CC-MAIN-20200329045008-20200329075008-00001.warc.gz"} |
https://publications.aap.org/pediatrics/article-abstract/110/3/471/64224/Perinatal-Screening-for-Group-B-Streptococci-Cost?redirectedFrom=fulltext | Objective. To evaluate the costs and benefits of a group B streptococci screening strategy using a new, rapid polymerase chain reaction test in a hypothetical cohort of expectant mothers in the United States.
Methods. Design. Cost-benefit analysis using the human capital method. We developed a decision model to analyze the costs and benefits of a hypothetical group B streptococci screening strategy using a new, rapid polymerase chain reaction test as compared with the currently recommended group B streptococci screening guidelines—prenatal culture performed at 35 to 37 weeks or risk-factor-based strategy with subsequent intrapartum treatment of the expectant mothers with antibiotics to prevent early-onset group B streptococcal infections in their infants.
Participants. A hypothetical cohort of pregnant women and their newborns.
Interventions. Screening strategies for group B streptococci using the new polymerase chain reaction technique, the 35- to 37-week culture, or maternal risk factors.
Outcome Measures. Infant infections averted, infant deaths, infant disabilities, costs, and societal benefits of healthy infants.
Results. A screening strategy using the new polymerase chain reaction test generates a net benefit of $7 per birth when compared with the maternal risk-factor strategy. For every 1 million births, 80 700 more women would receive antibiotics, 884 fewer infants would become infected with early-onset group B streptococci, and 23 infants would be saved from death or disability. The polymerase chain reaction-based strategy generates a net benefit of$6 per birth when compared with the 35- to 37-week prenatal culture strategy and results in fewer maternal courses of antibiotics (64 080 per million births), fewer perinatal infections with early-onset group B streptococci (218/million), and a reduction in 6 infant deaths and severe infant disability per million births. The benefits hold over a wide range of assumptions regarding key factors in the analysis.
Conclusions. Although additional clinical trials are needed to establish the accuracy of this new polymerase chain reaction test, initial studies suggest that strategies using this test will be superior to the other 2 strategies. Using the rapid polymerase chain reaction test becomes less attractive as the cost of the test increases. The test’s greatest strengths lie in its ability to identify women and infants at risk at the time of labor, thereby decreasing the number of false-positives and false-negatives seen with the other 2 strategies and allowing for more accurate and effective intrapartum prophylaxis. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18115223944187164, "perplexity": 4119.984864777081}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662521152.22/warc/CC-MAIN-20220518052503-20220518082503-00280.warc.gz"} |
https://brilliant.org/discussions/thread/chandrashaker-limit/ | ×
# Chandrashaker limit
find me near Chandrashaker limit M = 1.49*M☼
Body made up of 'n' particales, so
for spherical symmetric object we write 4/3 ΠR^3 = n(4/3 Πr^3) where R- radius of object; & r- radius of particle of object made from; R = r * (n^1/3) now, mass of object = M = nmass of particles
M = n*m
ζ = M/R = n*m / r *(n^1/3)
= (m/r)*(n^2/3)
= constant
taking m is the mass of neutron = 1.67*E^-27
& r = E^-15
(m/r)^-1 = 0.59*E^12
Now for Black Hole escape velocity is given by, c = (2GM/R)^1/2
so M/R = 0.6304E^27 M/R = (n^2/3)(m/r) n = [(M/R)(r/m)]^3/2 n = 0.733E^58
M = 1.2241*E^31 kg
M☼= 8.169*E^30 kg
M/ M☼ = 1.49
M = 1.49*M☼
3 years, 11 months ago
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\boxed{123} $$\boxed{123}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9934537410736084, "perplexity": 24980.24850628457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948544124.40/warc/CC-MAIN-20171214124830-20171214144830-00515.warc.gz"} |
https://jmlr.org/papers/v20/15-504.html | ## Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion
Lijun Zhang, Tianbao Yang, Rong Jin, Zhi-Hua Zhou; 20(97):1−22, 2019.
### Abstract
In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix. Although multiple works have been devoted to analyzing the recovery error of full-rank matrix completion, their error bounds are usually additive, making it impossible to obtain the perfect recovery case and more generally difficult to leverage the skewed distribution of eigenvalues. Our analysis is built upon the optimality condition of the regularized formulation and existing guarantees for low-rank matrix completion. To the best of our knowledge, this is the first relative bound that has been proved for the regularized formulation of matrix completion.
[abs][pdf][bib]
© JMLR 2019. (edit, beta) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.89106684923172, "perplexity": 546.4284274092963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704824728.92/warc/CC-MAIN-20210127121330-20210127151330-00614.warc.gz"} |
http://tug.org/pipermail/macostex-archives/2014-September/052980.html | # [OS X TeX] bug with the cyrillic fonts
V.Yu. Shavrukov v.yu.shavrukov at gmail.com
Wed Sep 24 04:36:57 CEST 2014
dear Juan,
On Sep 24, 2014, at 03:09, juan tolosa <juantolo at me.com> wrote:
> I am afraid I will show my ignorance, but I don't know what T2A means.
The T2A font encoding is one of the supported ways to associate characters (such as "letter ы") to slots in a TeX font table. In particular, an encoding determines a character set which may — as with T2A — or may not (T1) include Cyrillic.
> The same applies to "UT8 input encoding,"
Same thing but for plain text files. It defines which bit sequences stand for which characters.
> which sounds pretty imposing. I use cyrillic for Russian. What I use is surely outdated, since I have been using it for more years than I care to remember---ever since I started doing translation of math papers for the AMS, which was a long time ago. What I do to typeset in Russian is, follow the indications given for the \cyracc AMS family: one uses the {\cyr ...} environment and, within it, one uses the standard Russian transliteration, except for things like p1 for the soft sign, and p2 for the hard sign, which are not obvious.
>
> If you can help me out with concrete examples of what T2A is, and how one can use it, I would be very grateful. A sample file, with the necessary font declarations included would probably be the best, given my ignorance in the subject.
Here is a minimal example in LaTeX — yours was in plain TeX, to which it ought to be adaptable, but I am not sure how. Success depends on the presence of all the appropriate files for the cmr font family, but this should be automatic on most of this century installations.
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\documentclass{amsart}
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{\cyr Как бы {\bfseries цыплята} нас по осени {\itshape не {\bfseries посчитали}}.}\par
You don't have to use the ø-less T2A if you don't want to.
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> Thank your for your comment about что касается упражнения, you are absolutely right, my fault. Only I need to make a small correction to your Grammar comment: after the verb касатъся, one uses Genitive, not Accusative (which I should have done, but didn't). The Accusative for упражнение is still упражнение (as a neuter noun).
I stand corrected. Genitive is right. Don't know what I was thinking.
as ever,
Volodya | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.938873291015625, "perplexity": 3877.2311197644476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866926.96/warc/CC-MAIN-20180624083011-20180624103011-00232.warc.gz"} |
https://www.bankofcanada.ca/2008/09/working-paper-2008-31/ | # Aggregate and Welfare Effects of Redistribution of Wealth Under Inflation and Price-Level Targeting
Available as: PDF
Since the work of Doepke and Schneider (2006a) and Meh and Terajima (2008), we know that inflation causes major redistribution of wealth – between households and the government, between nationals and foreigners, and between households within the same country. Two types of monetary policy, inflation targeting (IT) and price level targeting (PT), have very different implications for the price level path subsequent to a price-level shock, and consequently, have different redistributional properties which is what we explore in this paper. For Canada, we show that the magnitude of redistributions of an unexpected 1% price-level increase under IT is about three times larger than under PT. Households' and foreigners' wealth losses from a price level increase is matched by the gains of the government. Even though this redistribution is zero-sum, we observe positive effects on GDP due to the wealth loss, the lower value of the debt and its associated fiscal adjustment, and the non-linear effects on work effort of the redistribution of wealth across households. Finally, the direction of the change in the weighted welfare of households depends on the fiscal policy.
### Published In:
Journal of Monetary Economics (0304-3932)
Sept. 2010. Vol. 57, Iss. 6, pp. 637-652
JEL Code(s): D, D3, D31, E, E2, E21, E3, E31, E4, E44, E5, E52, E6, E63 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9078394174575806, "perplexity": 4085.5679824506224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988725.79/warc/CC-MAIN-20210506023918-20210506053918-00591.warc.gz"} |
http://mathhelpforum.com/calculus/59954-convergence-infinite-series-print.html | # convergence of an infinite series
Decide when $\sum _{n=1} ^{\infty} \frac{n^{n}}{a^{n} n!}$ converges when the ratio test fails.
The ratio test fails when $a=\pm e$. At $a=e$, use the comparison test to compare the series with $\sum n^{-1/2}$, using Stirling's formula. At $a=-e$, use the alternating series test. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9983483552932739, "perplexity": 176.64834623626678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049281869.96/warc/CC-MAIN-20160524002121-00121-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://blogcompsci.wordpress.com/2015/01/20/problem-045/ | # Project Euler
## Triangular, pentagonal, and hexagonal
### Problem 45
Triangle, pentagonal, and hexagonal numbers are generated by the following formulas:
Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, … Pentagonal Pn=n(3n−1)/2 1, 5, 12, 22, 35, … Hexagonal Hn=n(2n−1) 1, 6, 15, 28, 45, …
It can be verified that T285 = P165 = H143 = 40755.
Find the next triangle number that is also pentagonal and hexagonal.
Following from last problem, we can use similar strategy to check if a number is a triangular, pentagonal, and hexagonal by verifying the following two formulas from here:
$P(s, n) = \frac{n^2(s-2) - n(s-4)}{2}$
$n = \frac{\sqrt{8 (s-2)x + (s-4)^2} + (s-4)}{2(s-2)}$
where P(s, n) = x and s is equal to the number of sides and n is the nth s-gonal number.
Another key observation from above is that every hexagonal number is also a triangular number (you can verify this using the formula above).
$P(6, n) = P(3, 2n-1)$
Therefore, we can minimize our check by finding the next hexagonal number (starting with n = 144) and verify if it is also a pentagonal number. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 3, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7913721799850464, "perplexity": 873.0813642808421}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823839.40/warc/CC-MAIN-20171020063725-20171020083725-00727.warc.gz"} |
http://nodus.ligo.caltech.edu:8080/40m/?id=13366 | 40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
40m Log, Page 74 of 339 Not logged in
ID Date Author Type Category Subject
13377 Thu Oct 12 07:56:33 2017 SteveHowToCamerasETMX GigE side view at 50 deg of IR scattering
Telescope front lens to wall distance 25 cm, GigE camera lenght 6 cm and cat6 cable 2cm
Atm3, Existing short camera can has 16cm lenght to lexan guard on viewport. Available 2" od periscope tube lenght is 8cm. The one in use 16 cm long.
Note: we can fabricate a lite cover with tube that would accomodate longer telescope.
Can we calibrate the AR coated M5018-SW and compare it's performance agains the 2" periscope
Look at the Edmond Optics 3" od camera lens with AR
This lower priced 1" apeture Navitar lens can be an option too.
Atm1, Now I can see dust. This is much better. The focus is not right yet.
Atm2, Chamber viewport wiped and image refocused. Actually I was focusing on the dust.
Quote: I calculated a better lens solution for the ETMX side view with the simple python script that's attached. The camera is still not as close to the viewport as we would like, and now the front lens is almost all the up to the end of the tube. With a little more playing around there maybe a better way, especially if we expand the repertoire of focal lengths. Using Steve's wonderful camera fixture I put the beam spot in focus. I turned the camera sideways for better use of the field of view, and now the beam spot actually fills the center area of the beam, to the point where we probably don't want more magnification or else we start losing the tails of the Gaussian. We'll take a serious of images tomorrow, and will have an estimate of the scatter loss by the end of tomorrow.
Attachment 1: Image__2017-10-11__15-29-52_15k400g.png
Attachment 2: Image__2017-10-12__15-50-18wipedRefocud2.png
Attachment 3: camCan16cm.jpg
13376 Thu Oct 12 01:50:11 2017 gautamUpdateLSCAS55Q Dark Noise
I worked on the daughter board a little more in the evening. I have somehow managed to make the dark noise ~25% worse [Attachment #1].
• Earlier in the day, I had switched out both on-board AD797s for OP27. The latter has ~3x the input voltage noise, and LISO modeling suggests that this is the dominant contribution to the output voltage noise.
• There are some differences in the actual components with which the board is stuffed, and the schematic.
• After updating the LISO model, I expect to get an output voltage noise of ~50nV/rtHz. But I measured ~2x this value (measured with LO input of demod board driven, RFPD input terminated).
• While I had the board out, I replaced most of the installed thick-film resistors with thin film ones. For good measure, I also changed the AD829s.
After making all these changes, I re-installed the card in the eurocrate and repeated the measurement. The Q channel noise was close to the expected value (~50nV/rtHz), but the I channel is twice as noisy. I will continue this investigation tomorrow.
Attachment 1: AS55_dark.png
13375 Thu Oct 12 01:03:49 2017 johannesHowToCamerasETMX GigE side view
I calculated a better lens solution for the ETMX side view with the simple python script that's attached. The camera is still not as close to the viewport as we would like, and now the front lens is almost all the up to the end of the tube. With a little more playing around there maybe a better way, especially if we expand the repertoire of focal lengths. Using Steve's wonderful camera fixture I put the beam spot in focus. I turned the camera sideways for better use of the field of view, and now the beam spot actually fills the center area of the beam, to the point where we probably don't want more magnification or else we start losing the tails of the Gaussian.
We'll take a serious of images tomorrow, and will have an estimate of the scatter loss by the end of tomorrow.
Attachment 1: IMG_20171011_164549698.jpg
Attachment 2: Image__2017-10-11__16-52-01.png
Attachment 3: GigE_lens_position_helper.py.zip
13374 Wed Oct 11 19:31:32 2017 gautamUpdateLSCAS55Q Dark Noise
I tried replacing the AD797s on the daughter board with OP27s, and saw no significant improvement in the electronics noise of the demod board. Note that according to LISO, in this configuration, the voltage noise of the Op27 is expected to dominate the total noise of the daughter board. Measurement condition was that the RFPD input was terminated, but the LO input was still being driven (SR785 input range is -50dBVpk for all traces, and the input ranging was set to "UpOnly"). Need to do a more systematic investigation to figure out where this excess noise is coming from. I will upload photos of the board later.
Quote: This supports the hypothesis that something is wonky on the daughter board, because the purple trace should only be the quad sum of the orange and green traces. I will pull it out and have a look.
Attachment 1: AS55Q_darkNoises.pdf
13373 Wed Oct 11 17:59:45 2017 ranaUpdateElectronicsSR560 noise level
these are not the SR785 settings that you're looking for
Quote: Gautam and Steve, All 3 show the same noise level ~80 nV / rt Hz at 1 kHz as shown. Batteries ordered to be replaced in the top 2 We'll do more measurement to see how can we get to 4 nV / rt Hz specification level.
To get low noise measurements on the SR785, you have to have the input range set to -50 dB, not +20 dB. Its not within the powers of commercial electronics ADCs to give you a 10 nV noise floor with +10 V input signals. The SR560 has an input referred noise of 5 nV/rHz, so the output noise should be 5e-9 x 500 = 2.5 uV/rHz. Your picture shows it giving 1 uV RMS, so you also need to use the PSD units.
13372 Wed Oct 11 14:42:03 2017 gautamUpdateLSCAS55Q Dark Noise
I keep adding traces to this plot, here is the most complete one I have now. Looks like the input noise to the D040179 (measured at "Q out" SMA jack of D990511 with RFPD input terminated) is ~10nV/rtHz. This supports the hypothesis that something is wonky on the daughter board, because the purple trace should only be the quad sum of the orange and green traces. I will pull it out and have a look.
Some other follow-ups on the questions raised at the meeting:
1. Doesn't look like I've implemented thin film resistors on the input of the coil driver boards. De-whitening boards have the critical signal path resistors (judged as the ones with largest contribution as per LISO model) changed to thin film. Pictures are here.
2. I think I didn't make a full elog of my demod board efficiency investigations, but from my notes and Attachment #4 of elog 12972, I calculated the gain in the signal path as the ratio of Vpp_out / Vpp_in.
Quote: I measured the output voltage noise of the Q output of the AS55 Demod Board with the PSL shutter closed, using the SR785 (see Attachment #1). The measured noise is consistent with the expected number of ~120nV/rtHz around 100Hz. I had measured the gain of this board from RFPD input to Q output to be ~5.1: so if the PD dark noise is 16nV/rtHz, this would be amplified to ~80nV/rtHz. Still a discrepancy of ~50%. I didn't measure the noise with the PD input terminated. Added the noise of the demod board output with the RFPD input terminated. The level of ~100nV/rtHz seems consistent with the actual PD dark noise being ~80nV/rtHz, as their quadrature sum is around 130nV/rtHz. Need to dig up the schematics for the demod board + daughter board, and check against LISO, to see if this is consistent with what is expected. Also - I think I was using the wrong value of the DC power on the AS55 photodiode for shot noise calculations - 13mW was for REFL55, not AS55. I did a crude measurement of the power by sticking the Ophir power meter (filter removed) in front of the AS55 PD with the Michelson flashing around, and noticed the maximum value registered was ~1.2mW. So in the DRMI lock, there would be ~2.4mW, which is 10x lower than the value I was assuming. I've made the correction in the NB code, for the next time the plot is generated. A more rigorous measurement would involve sticking the Ophir in front of the AS110 PD during a DRMI lock. The light from the AS port is split by a 50-50 BS to the AS55 and AS110 PDs (so measuring at AS110 is a reasonable proxy for power at AS55), and the AS110 signals are not used for triggering in the DRMI lock, so this is feasible.
Attachment 1: AS55Q_darkNoises.pdf
13371 Wed Oct 11 10:29:43 2017 SteveUpdateElectronicsSR560 noise level
Gautam and Steve,
All 3 show the same noise level ~80 nV / rt Hz at 1 kHz as shown. Batteries ordered to be replaced in the top 2
We'll do more measurement to see how can we get to 4 nV / rt Hz specification level.
Attachment 1: sr560.jpg
Attachment 2: sr560noise.jpg
13370 Tue Oct 10 22:04:06 2017 ranaUpdateLSCAS55Q Dark Noise
how about calibrate the DC channels so that you can just get the acutal power levels from the trend data?
13369 Mon Oct 9 22:18:34 2017 gautamUpdateLSCAS55Q Dark Noise
I measured the output voltage noise of the Q output of the AS55 Demod Board with the PSL shutter closed, using the SR785 (see Attachment #1). The measured noise is consistent with the expected number of ~120nV/rtHz around 100Hz. I had measured the gain of this board from RFPD input to Q output to be ~5.1: so if the PD dark noise is 16nV/rtHz, this would be amplified to ~80nV/rtHz. Still a discrepancy of ~50%. I didn't measure the noise with the PD input terminated. Added the noise of the demod board output with the RFPD input terminated. The level of ~100nV/rtHz seems consistent with the actual PD dark noise being ~80nV/rtHz, as their quadrature sum is around 130nV/rtHz. Need to dig up the schematics for the demod board + daughter board, and check against LISO, to see if this is consistent with what is expected.
Also - I think I was using the wrong value of the DC power on the AS55 photodiode for shot noise calculations - 13mW was for REFL55, not AS55. I did a crude measurement of the power by sticking the Ophir power meter (filter removed) in front of the AS55 PD with the Michelson flashing around, and noticed the maximum value registered was ~1.2mW. So in the DRMI lock, there would be ~2.4mW, which is 10x lower than the value I was assuming. I've made the correction in the NB code, for the next time the plot is generated. A more rigorous measurement would involve sticking the Ophir in front of the AS110 PD during a DRMI lock. The light from the AS port is split by a 50-50 BS to the AS55 and AS110 PDs (so measuring at AS110 is a reasonable proxy for power at AS55), and the AS110 signals are not used for triggering in the DRMI lock, so this is feasible.
Attachment 1: AS55Q_dark.pdf
13368 Mon Oct 9 11:55:01 2017 KojiUpdateLSCDRMI Nosie Budget v3.0
My last characterization of the AS55 PD was on Feb 2013. ELOG 8100
There I said the dark noise at the PD output was 16nV/rtHz. I don't have the measurement of the Voltage noise at the output of the demod board.
Note that the PD can only be limited by shot noise when the DC current is larger then 4mA.
13367 Mon Oct 9 01:29:26 2017 gautamUpdateLSCDRMI Nosie Budget v3.0
## Summary:
I spent this weekend doing a more careful investigation of the DRMI noise. I think I have some new information/insights. Attachment #1 is the noise budget (png attached because pdf takes forever to upload, probably some ImageMagick problem. The last attachment is a tarball of the PDF). Long elog, so here are the Highlights:
1. Coil de-whitening does result in small improvement in noise in the 60-200Hz band.
2. Above 200Hz, we seem to be limited by "Dark" noise. More on this below.
3. The coupling from SRCL->MICH is the other limiting noise in the 60-200Hz band now.
## Sensing Matrix Measurement:
• I rotated the AS55 demod phase from -42 degrees to -82 degrees, the idea being to get more of the MICH error signal in AS55_Q.
• Consequently, the MICH servo gain has been lowered from -0.035 to -0.021. Settings have been updated in the snap file used by the locking script.
• Seems to have worked.
• Attachment #2 is the measured sensing elements.
• One major source of uncertainty in these sensing element numbers is the actuator gains for PRM, SRM and BS. The coil driver electronics for the latter two have been modified recently, and for them, I am using numbers from this elog scaled by the expected factor as a result of removing the x3 gain in the de-whitening boards for SRM and BS.
## MICH OLTF
• Measurement was done in lock using the usual IN1/IN2 method.
• Model made by loading the FOTON filters + assumed models for the BS pendulum and AA/AI filters in Matlab, and fitting to an overall gain + delay.
• Attachment #3 shows the agreement between measurement and model.
• The model was exported and used to invert in-loop signals to their out-of-loop counterparts in the noise budget.
## DAC Noise
• I had claimed that turning on the coil de-whitening did not improve the MICH noise.
• This was not exactly true - I had only compared MICH noise with the BS de-whitening turned ON/OFF, while the ITM de-whitening was always on.
• Turns out that there is in fact a small improvement - see Attachment #4 (DTT crashes everytime I try to print a pdf, so png screenshot will have to do for now).
• I have also changed the way in which DAC noise is plotted in the Noise Budget code:
• I used to directly convert the measured voltage noise (multiplied by appropriate scalar to account for quadrature sum of 4 coils each in 3 optics) to displacement noise using the sensing measurement cts/m values.
• Now I convert the measured voltage noise first to current noise (knowing the series resistance), then to force noise (using the number 0.016 N/A per coil), then to displacement noise (assuming a mirror mas of 250g).
• Quadrature sum is again taken for 4 coils on 3 optics.
• I've also added the option to plot the DAC noise with the de-whitening filter TF applied (taking care that the maximum of filtered DAC noise / coil driver electronics noise is used at each frequency).
• So the major source of uncertainty in the calculated DAC noise is the assumed actuator gain of 0.016 N/A.
The DAC noise is not limiting us anywhere when the coil de-whitening is switched on.
## I think this is the major find.
• The dark noise spectrum is measured with:
• the PSL shutter closed
• the AS55 I and Q analog whitening filters (and corresponding digital de-whitening filters) engaged, to mimic the operating conditions under which the in-lock error signal is acquired.
• Comparing the blue and black traces, it is clear that turning on the analog whitening is having some effect on the dark noise.
• However, the analog whitening filters should suppress the ADC noise by ~30dB @ 100Hz - so assuming 1uV/rtHz, this would be ~30nV/rtHz @100Hz.
• But the measured noise seems to be ~5x higher, with 4*10^-4 cts/rtHz translating to roughly 120nV/rtHz.
• The photodiode dark noise is only 15nV/rtHz according to the wiki. Where is this measured?
So I don't understand the measured Dark Noise level, and it is limiting us at frequencies > 200Hz. Some busted electronics in the input signal chain? Or can the LSC demod daughter board gain of ~5 explain the observed noise?
## Shot noise
• The DC power on AS55 photodiode was measured to be ~13mW with the SRM misaligned.
• This corresponds to ~100cts peak amplitude on the ASDC channel (derived from AS55 photodiode).
• In the DRMI lock, the ASDC level is ~200cts.
• I used these numbers, and equation 2.17 in Tobin's thesis, to calculate this curve.
Edit 1730 9 Oct: I had missed out the factor of 5 gain in the demod board in calculating the shot noise curve. Attachment #7 shows the corrected shot noise level. Explicitly:
$n_{\mathrm{shot}} [m/\sqrt{\mathrm{Hz}}] = \alpha \sqrt{2 h \nu \bar{P} (\frac{1}{2} - \frac{1}{4}\mathrm{cos}2\theta)}$, where $\alpha [m/W] = (\mathcal{M}_{\mathrm{MICH}} [V/m] / 5 [V/V] / 420 [V/A] / 0.7 [A/W])^{-1}$is to convert shot noise in W to displacement units.
## This is the other find.
• While chatting with Gabriele, he suggested measuring the SRCL->MICH and PRCL->MICH cross couplings.
• I injected a signal in SRCL servo EXC channel, and adjusted amplitude till coherence in MICH_IN1 was good.
• The actual TF measured was MICH_IN1 / SRCL_IN1 (so units of cts/ct).
• My multiplying the in-lock PRCL and SRCL IN1 signals by these coupling coefficients (assumed flat in frequency for now, note that measurement was only made between 100Hz and 1kHz), I get the trace labelled "AUX coupling" in Attachment #1 (this is the quadrature sum for SRCL and PRCL couplings).
• Also repeated for PRCL -> MICH coupling in the same way.
• Measurements of these TFs and coherence are shown in Attachment #5 (again png screenshot because of DTT).
• However, there is no significant coherence in MICH/SRCL or MICH/PRCL in this frequency range.
This seems to be limiting us from saturating the dark noise once the coil de-whitening is engaged. But lack of coherence means the mechanism is not re-injection of SRCL/PRCL sensing noise? Need to think about what this means / how we can mitigate it.
## OL A2L coupling
• I didn't measure these
• These couplings would have changed because I modified the Oplev loop shapes to allow engaging of coil de-whitening filters.
• But anyways, their effect will only be below 100Hz because I made the roll-offs steeper.
## Still to measure (but not likely to be limiting us anywhere in the current state):
• Laser intensity noise -> MICH coupling (using AOM).
• Laser frequency noise -> MICH coupling (using CM board IN2).
• Oscillator noise (amplitude + phase) -> MICH coupling (using AM/FM input of Marconi).
I've also made several changes to the NB code - will push to git once I finish cleaning stuff up, but it is now much faster to make these plots and see what's what.
Attachment 1: DRMI_NB.png
Attachment 2: DRMI1f_Oct8.pdf
Attachment 3: MICH_OLTF_model.pdf
Attachment 4: MICH_noises.png
Attachment 5: AUX_couplings.png
Attachment 6: C1NB_disp_40m_MICH_NB_2017-10-08.pdf.tar.gz
Attachment 7: MICH_NB_corrected.png
13366 Fri Oct 6 17:08:09 2017 SteveUpdateALSX End table beam traps corrected
There are no more double sided tape on this table.
Attachment 1: c1.jpg
Attachment 2: c2.jpg
Attachment 3: c3.jpg
Attachment 4: c4.jpg
13365 Fri Oct 6 12:56:40 2017 gautamSummaryLSCRTCDS NN
[gabriele, gautam]
Gabriele had prepared a C code implementation of his NN for MICH/PRCL state estimation. He had been trying to get it going on some of the machines in WB, but was unsuccessful. The version of RCG he was trying to compile and run the code on was rather dated so we decided to give it a whirl on our new RCG3.4 here at the 40m. Just noting down stuff we tried here:
• Code has been installed at /opt/rtcds/userapps/release/cds/c1/src/nn.This is to facilitate compilation by the RCG.
• There is also a simulink block diagram (x3tst.mdl) in there which we copied and pasted into c1pem. Changed the appropriate paths in the C-Code block to point to the location in the previous bullet point.
• c1pem was chosen for this test as it runs at 2k which is what the network is designed for.
• Since we were running the test on c1sus and expected the machine to crash, I shutdown all the watchdogs for the test.
• Code compiled without any problems (i.e. rtcds make c1pem and rtcds install c1pem executed successfully). There were some warning messages related to C-Code blocks but these are not new, they show up in all models in which we have custom C-code blocks.
• Unfortunately, the whole c1sus FE crashed when we tried rtcds restart c1pem.
• We tried a couple of more iterations, and attempted to monitor dmesg during the restart process to see if there were any clues as to why this wasn't working, but got nothing useful.
All models have been reverted to their state prior to this test, and everything on the CDS_OVERVIEW MEDM screen is green now.
Attachment 1: post_NN_test.png
13364 Fri Oct 6 12:46:17 2017 gautamUpdateCDS40m files backup situation
Looks to have worked this time around.
controls@fb1:~ 0sudo dd if=/dev/sda of=/dev/sdc bs=64K conv=noerror,sync 33554416+0 records in 33554416+0 records out 2199022206976 bytes (2.2 TB) copied, 55910.3 s, 39.3 MB/s You have new mail in /var/mail/controls I was able to mount all the partitions on the cloned disk. Will now try booting from this disk on the spare machine I am testing in the office area now. That'd be a "real" test of if this backup is useful in the event of a disk failure. Quote: The 4TB HGST drives have arrived. I've started the FB1 dd backup process. Should take a day or so. 13363 Fri Oct 6 00:25:45 2017 ranaUpdateLSCFS725 for Marconi stabilization Steve, can you please connect this fan to the rack power and remove this extra power supply? Quote: Re-arranged the DC bench supply on the shelf in the PSL enclosure, whose only purpose seems to be to supply 12V to a fan attached to the rear of the PSL NPRO controller. Seems to be a waste of space! The fan was momentarily disconnected but has since been reconnected and is spinning again. 13362 Thu Oct 5 18:40:27 2017 gautamUpdateLSCFS725 for Marconi stabilization [steve, gautam] 1. We installed the FS725 on the shelf inside the PSL enclosure - see Attachment #1. 2. We ran a long BNC cable (labelled "GPS 1pps" on both ends) from 1X7 to the PSL enclosure - this was to pipe the 1PPS signal from the GPS timing unit (EndRun Technologies Tempus LX) rear panel (50 ohm output according to the datasheet) to the 1PPS input of the FS725 (high impedance). See Attachments #2. Note that the 1pps output was already tee'd on the rear panel. One port of the tee was unused (this now goes to the FS725) while the other was going to the 1PPS input of the Master Timing Sequencer (D050239), so I decided that there was no need to tee the 1pps input of the FS725 with a 50ohm terminator. In a few minutes, the Rb standard indicated that it was locked to its internal reference, and also to the external 1pps input (see Attachment #1). 3. We ran a long BNC cable (labelled "Rb 10MHz" on both ends) from the 10MHz output of the FS725 (50 ohm output impedance), in the PSL enclosure to the rear BNC "FREQ_STD IN/OUT" BNC connector of the Marconi (1kohm input impedance). Changed the frequency reference setting on the Marconi to "External Direct". The FS725 datasheet recommends terminating the load with a 50ohm inline terminator, I have not yet done this (see Attachment #3). Is it appropriate to use a Balun (FTB-1-1) here? This would avoid ground loops between the Marconi and the FS725, and also make the load seen by the FS725 50ohms 4. Found that there was an unused long cable from the PSL enclosure to the 1X2 electronics rack. We re-purposed this to drive the AOM driver via the DAC output in 1Y2. The cable is labelled "AOM driver" on both ends. This was to facilitate measurement of the coupling of laser intensity noise to AS55_Q in a DRMI lock. 5. Removed 2 long cables between 1X7 and 1X2 that weren't connected to anything. 6. Re-arranged the DC bench supply on the shelf in the PSL enclosure, whose only purpose seems to be to supply 12V to a fan attached to the rear of the PSL NPRO controller. Seems to be a waste of space! The fan was momentarily disconnected but has since been reconnected and is spinning again. 7. Removed a couple of unused power cables from the mess on the shelf in the PSL enclosure. Also removed an unused Sony Video Squential Switcher YS-S6 from the PSL enclosure. Quote: I've located the Stanford Research FS725 Rb reference unit. The question is where to put it. This afternoon Steve and I put it inside the little electronics rack next to 1X3, but in hindsight, this probably isn't such a great place for a timing reference as there are a bunch of Sorensen power supplies in there (and presumably the accompanying harmonics from these switching supplies). The unit itself was repaired in 2015, and powering it on, it locked to the internal reference within a few minutes as prescribed in the manual. Attachment 1: IMG_7617.JPG Attachment 2: IMG_7619.JPG Attachment 3: IMG_7618.JPG 13361 Thu Oct 5 13:58:26 2017 gautamUpdateCDS40m files backup situation The 4TB HGST drives have arrived. I've started the FB1 dd backup process. Should take a day or so. Quote: Edit: unmounting /frames won't help, since dd makes a bit for bit copy of the drive being cloned. So we need a drive with size that is >= that of the drive we are trying to clone. On FB1, this is /dev/sda, which has a size of 2TB. The HGST drive we got has an advertised size of 2TB, but looks like actually only 1.8TB is available. So I think we need to order a 4TB drive. 13360 Thu Oct 5 11:46:15 2017 gautamUpdateCDSslow machine bootfest MC Autolocker was umnhappy because c1iool0 was unresponsive and hence it couldn't write to the "C1:IOO-MC_AUTOLOCK_BEAT" channel. I keyed the crate and IMC locked almost immediately. I'm moving this channel into the RTCDS model as we did for the IFO_STATE EPICS channel so that the autolocker isn't dependant on c1iool0 (which was the whole point of migrating the IFO-STATE variable anyways). I also commented out all of these channels in /cvs/cds/caltech/target/c1iool0/autolocker.db so that there aren't duplicate channels. Quote: Eurocrate key turning reboots for c1susaux, c1auxex,c1auxey, c1iscaux, c1iscaux2 and c1aux. Usual precautions were taken for ITMX. Did burtrestore for c1iscaux andc1iscaux2 in order to restore the LSC PD whitening gains. Un-related to this work: input pointing into PMC was tweaked as the PMC_REFL spot was pretty bright. 13359 Thu Oct 5 02:14:51 2017 gautamUpdateLSCMore DRMI coupling measurements - setup In the end I decided to access the available spare DAC channels via the C1ASS model - for this purpose, I added a namespace block "TEST" in the C1ASS simulink model, which is a SISO block. Inside is just a single CDS filter module. My idea is to use the EXC of this filter module to inject excitations for measuring various couplings. Rather than have a simple testpoint, we also have the option of adding in some filter shapes in the filter module which could possibly allow a more direct read-off of some coupling TF. Recompiling the model went smooth - there was a crash earlier in the day which required me to hard-reboot c1lsc (and also restart all models on c1sus and c1ioo but no reboots necessary for those machines). Note that to get the newly added channels to show up in the channel lists in DTT/AWGGUI etc, you need to ssh into fb1 and restart the daqd processes via sudo systemctl restart daqd_*. If I remember right, it used to be enough to do telnet fb 8088 followed by shutdown. This is no longer sufficient. It took me a while to get the DRMI locking going again. The model restarts earlier in the evening had changed a bunch of EPICS channel settings (and out config scripts don't catch all of these settings). In particular, I forgot to re-enable the x3 digital gain for the ITMs, BS and SRM (necessitated by removing an analog x3 gain on the de-whitening boards). I was hesitant to spend time re-adjusting all damping / oplev loop gains because if we change the series resistor on the coil driver board, we will have to do this again. I also didn't want this arbitrary FM to be enabled in the SDF safe.snap. But maybe it's worth doing it anyways - if nothing it'll be good practise. Once I hunted down all the setting diffs and tweaked alignment, the DRMI locks were pretty robust. I had hoped to make some of these TF measurements tonight. But I realized I needed to look up a bunch of stuff in manuals/datasheets, and think about these measurements a little. I wasn't sure if the DW/AI board could drive a signal over 40m of BNC cabling so I added an SR560 (DC coupled, gain=1, low noise mode, 50ohm output used) to buffer the output. The Marconi's external modulation input is high impedance (100k) but for the AOM driver we want 50ohm. For the Marconi, the external input accepts 1Vrms max, while for the AOM driver, we want to drive a signal between 0V and 1V at most. The general measurement setup is schematically shown in Fig 1. Questions to address: • What happens if we apply a negative voltage to the input of the AOM driver? What is the damage threshold? Do we have to worry about SR560 offset level? • Is there a way to dynamically adjust the offset in DTT such that we can have different amplitude signals at different frequencies (usually done by specifying an envelope in DTT) but still satisfy the requirement that the entire signal lie between 0-1V? • For the Laser Intensity noise -> MICH coupling TF measurement, I guess we can use the AOM to inject an excitation, and measure the ratio of the response in MC_TRANS and in MICH_IN1. Then we multiply the in-loop MC_TRANS spectrum by the magnitude of this TF to get the Laser Intensity Noise contribution to MICH. • The Laser Frequency Noise coupling should be negligible in MICH - but the measurement principle should be the same. Drive the AO input of the Mode Cleaner Servo board from the DAC, look at ratio of response in MICH_IN1 and MC_F. Multiply the DRMI in-lock MC_F spectrum by this TF. • The oscillator noise seems more tricky to me (also Finesse modeling suggests this may be the most significant of the 3 couplings described in this elog, though I may just be computing the coupling in Finesse wrongly) • I don't understand all the External Modulation options specified in the manual. • DC? AC? FM? PM? AM? Need to figure out what is the right settings to use. • I'm not sure how independent the various modulations will be - i.e. if I select PM, how much AM is induced as a result of me driving the EXT MOD input? • What is the right level of excitation drive? I tried this a bunch of times tonight - set the PM range to 0.1rad (for the full scale 1Vrms sine wave input), but with an excitation of just a few counts, already saw non-lineaer coupling in MICH_IN1 which probably means I'm driving this too hard. • This measurement needs a bit more algebra. We have an estimate of the Marconi phase noise from Rana (is this the right one to use?). But the "Transfer Function" we'd measure is cts in MICH_IN1 in response to counts to Marconi via the signal chain in Attachment #1. So we'd need to know (and divide out) the AI/DW board TF, and the Marconi's TF, which the datasheet suggests has a lower 3dB frequency of 100Hz (assuming SR560 and cable can be treated as flat). • A simpler test may be to just hook up the Marconi to the Rb standard, and the Rb to 1pps from GPS, and look for a change in the MICH noise. Am I missing something? Attachment 1: CB4709D0-3FA7-43E3-BC25-3CF4164E6C6A.jpeg 13357 Wed Oct 4 17:38:25 2017 gautamUpdateLSCFS725 for Marconi stabilization I've located the Stanford Research FS725 Rb reference unit. The question is where to put it. This afternoon Steve and I put it inside the little electronics rack next to 1X3, but in hindsight, this probably isn't such a great place for a timing reference as there are a bunch of Sorensen power supplies in there (and presumably the accompanying harmonics from these switching supplies). The unit itself was repaired in 2015, and powering it on, it locked to the internal reference within a few minutes as prescribed in the manual. 13356 Wed Oct 4 17:18:15 2017 gautamUpdateCDSFree DAC channels in c1lsc There are at least 5 free DAC channels (4 if you discount the one channel from these that I am hijacking) available in the 1Y2 electronics rack. Jamie's nice wiring diagram shows the topology - the actual DAC card sits in 1Y3 inside the c1lsc expansion chassis (while the c1lsc frontend itself is in 1X4). The output of the DAC goes via SCSI to an interface box (D080303) and then to some dewhitening/AI boards (D000316). There are a total of 16 DAC channels available, out of which 8 are used for the TTs, 2 are used for the DAFI model, and one is labeleld "From c1ioo 1X2" (I don't know what this one is for). So I'm going to use some of these channels for measuring the coupling of oscillator noise and intensity noise to MICH in the DRMI lock. The de-whitening/AI board seems to be old - it has 2x 800Hz Butterworth LPFs and no notch for the clock frequency, but maybe this doesn't matter for the tests I have in mind. The AI board available on 1X2 is more modern but routing the DAC channels from 1Y2 to it is going to be some work. I'm going to add my testpoint to c1daf given that it seems to be the least critical model on c1lsc. EDIT: testpoints added to c1daf don't show up in the list of available channels - there was some issue with this model while we were getting the new RTCDS going. So I'm moving my temporary testpoint to c1cal instead. 13355 Tue Oct 3 19:39:10 2017 gautamUpdateCDSslow machine bootfest Eurocrate key turning reboots for c1susaux, c1auxex,c1auxey, c1iscaux, c1iscaux2 and c1aux. Usual precautions were taken for ITMX. Did burtrestore for c1iscaux andc1iscaux2 in order to restore the LSC PD whitening gains. Un-related to this work: input pointing into PMC was tweaked as the PMC_REFL spot was pretty bright. 13354 Tue Oct 3 01:58:32 2017 johannesHowToCamerasCCD calibration Disclaimer: Wrong calibration factors! See https://nodus.ligo.caltech.edu:8081/40m/13391 The factors were indeed enormously off. The correct table reads: Camera IP Calibration Factor CF 192.168.113.152 85.8 pW*s 192.168.113.153 78.3 pW*s I did subtract a 'dark' frame from the images, though not in the sense of your point 1, just an exposure of identical duration with the laser turned off. This was mostly to reduce the effect of residual light, but given similar initial conditions would somewhat compensate for the offset that pre-existing charge and electronics noise put on the pixel values. The white field is of course a difference story. I wonder how close we can get to a white field by putting a thin piece of paper in front of the camera without lenses and illuminate it from the other side. A problem is of course the coherence if we use a laser source... Or we scrap any sort of screen/paper and illuminate directly with a strongly divergent beam? Then there wouldn't be a specular pattern. I'm not sure I understand your point about the 1.5V/A. Just to make sure we're talking about the same thing I made a crude drawing: The PD sees plenty of light at all times, and the 1.5V/uW came from a comparative measurement PD<-->Ophir (which took the place of the CCD) while adjusting the power deflected with the AOM, so it doesn't have immediate connection to the conversion gain of silicon in this case. I can't remember the gain setting of the PD, but I believe it was 0dB, 20dB at most. Attachment 1: gige_calibration.pdf 13353 Tue Oct 3 01:32:39 2017 gautamUpdateLSCLaser intensity noise coupling to MICH (simulated) GV Oct 6: This coupling is probably not correct - Finesse outputs TF magnitude in units of W/W, and not W/RIN Since I was foiled (by lack of DAC) in my attempt to measure the coupling of laser intensity noise to MICH in the DRMI (no arms) configuration, I decided to try understanding the effect with a simulation. For this purpose, I used my DRMI Finesse model - this had mirror positions tuned for locking and photodiode demod phases tuned to give a sensing matrix model that wasn't too far from an actual measurement (within factor of a few). So the model seems okay for a first pass at estimating this coupling. Measuring transfer functions in Finesse is straightforward - use the fsig command to modulate some quantity (in this case the input beam intensity), and use the pd2 detector to demodulate the effect of this modulation at the port of interest (in this case AS55_Q). **Note that to apply a modulation to an input beam (i.e. Laser) in Finesse, the keyword for the "type" argument given to fsig is "amp" and not "amplitude" as the manual would had me believe. In fact, there seem to be quite a few such caveats. The best way to figure this out is to go to the pykat installation directory, find the file components.py, and look for the fsig_name for the component of interest. It is also indicated in the same file, via the canFsig argument, if that property of the component can be modulated for transfer function measurements. Attachment #1 shows the result of such a sweep. To estimate what the actual contribution to the displacement noise is, I used the DQ-ed MC transmission (recorded at 1024Hz) from the DRMI lock, computed the ASD using scipy.signal.welch, divided by the nominal MC transmission of ~15,000 counts to convert to RIN/rtHz. The RIN was then multiplied by the above calculated coupling function, and divided by the sensing matrix element for AS55_Q (in units of W/m) to give the curve shown in Attachment #2. If we believe the simulation, then Laser Intensity Noise shouldn't be the limiting noise between 10Hz-1kHz. I will of course measure the actual coupling and see how it lines up with Attachment #1 - would be a nice additional validation of the Finesse model. I will also try using the Finesse model to estimate some other coupling transfer functions (e.g. Laser Frequency Noise, Oscillator Noise). Quote: The absence of evidence is not evidence of absence. Attachment 1: MICH_intensityNoiseCoupling.pdf Attachment 2: MICH_intensityNoiseASD.pdf 13352 Mon Oct 2 23:16:05 2017 gautamHowToCamerasCCD calibration Going through some astronomy CCD calibration resources ([1]-[3]), I gather that there are in general 3 distinct types of correction that are applied: 1. Dark frames --- this would be what we get with a "zero duration" capture, some documents further subdivide this into various categories like thermal noise in the CCD / readout electronics, poissonian offsets on individual pixels etc. 2. Bias frames --- this effect is attributed to the charge applied to the CCD array prior to the readout. 3. Flat-field calibration --- this effect accounts for the non-uniform responsivity of individual pixels on the CCDs. The flat-field calibration seems to be the most complicated - the idea is to use a source of known radiance, and capture an image of this known radiance with the CCD. Then assuming we know the source radiance well enough, we can use some math to back out what the actual response function of individual pixels are. Then, for an actual image, we would divide by this response-map to get the actual image. There are a number of assumptions that go into this, such as: • We know the source radiance perfectly (I guess we are assuming that the white paper is a Lambertian scatterer so we know its BRDF, and hence the radiance, perfectly, although the work that Jigyas and Amani did this summer suggest that white paper isn't really a Lambertian scatterer). • There is only one wavelength incident on the CCD. • We can neglect the effects of dust on the telescope/CCD array itself, which would obviously modify the responsivity of the CCD, and is presumably not stationary. Best we can do is try and keep the setup as clean as possible during installation. I am not sure what error is incurred by ignoring 2 and 3 in the list at the beginning of this elog, perhaps this won't affect our ability to estimate the scattered power from the test-masses to within a factor of 2. But it may be worth it to do these additional calibration steps. I also wonder what the uncertainty in the 1.5V/A number for the photodiode is (i.e. how much do we trust the Ophir power meter at low power levels?). The datasheet for the PDA100A says the transimpedance gain at 60dB gain is 1.5 MV/A (into high impedance load), and the Si responsivity at 1064nm is ~0.25A/W, so naively I would expect 0.375 V/uW which is ~factor of 4 lower. Is there a reason to trust one method over the other? Also, are the calibration factor units correct? Jigyasa reported something like 0.5nW s / ct in her report. Camera IP Calibration Factor CF 192.168.113.152 8.58 W*s 192.168.113.153 7.83 W*s The incident power can be calculated as Pin =CF*Total(Counts-DarkCounts)/ExposureTime. References: [1] http://www.astrophoto.net/calibration.php [2] https://www.eso.org/~ohainaut/ccd/ [3] http://www.astro.ufl.edu/~lee/ast325/handouts/ccd.pdf 13351 Mon Oct 2 19:03:49 2017 gautamUpdateCDS[Solved] c1ioo DC errors This did the trick - I simply ran sudo systemctl restart daqd_* on FB1, and now all the CDS overview lights are green again. I thought I had done this already, but I realize that I was supposed to restart the daqd processes on FB1 (which is where they are running) and not on c1ioo . Thanks Jamie for the speedy resolution! Quote: Quote: This time the model came back up but I saw a "0x2000" error in the GDS overview MEDM screen. Since there are no DACs installed in the c1ioo expansion chassis, I thought perhaps the problem had to do with the fact that there was a "DAC_0" block in the c1x03 simulink diagram - so I deleted this block, recompiled c1x03, and for good measure, restarted all (three) models on c1ioo. Now, however, I get the same 0x2000 error on both the c1x03 and c1als GDS overview MEDM screens (see Attachment #1). From page 21 of T1100625, DAQ status "0x2000" means that the channel list is out of sync between the front end and the daqd. This usually happens when you add channels to the model and don't restart the daqd processes, which sounds like it might be applicable here. It looks like open-mx is loaded fine (via "rtcds lsmod"), even though the systemd unit is complaining. I think this is because the open-mx service is old style and is not intended for module loading/unloading with the new style systemd stuff. Attachment 1: CDSoverview.png 13350 Mon Oct 2 18:50:55 2017 jamieUpdateCDSc1ioo DC errors Quote: This time the model came back up but I saw a "0x2000" error in the GDS overview MEDM screen. Since there are no DACs installed in the c1ioo expansion chassis, I thought perhaps the problem had to do with the fact that there was a "DAC_0" block in the c1x03 simulink diagram - so I deleted this block, recompiled c1x03, and for good measure, restarted all (three) models on c1ioo. Now, however, I get the same 0x2000 error on both the c1x03 and c1als GDS overview MEDM screens (see Attachment #1). From page 21 of T1100625, DAQ status "0x2000" means that the channel list is out of sync between the front end and the daqd. This usually happens when you add channels to the model and don't restart the daqd processes, which sounds like it might be applicable here. It looks like open-mx is loaded fine (via "rtcds lsmod"), even though the systemd unit is complaining. I think this is because the open-mx service is old style and is not intended for module loading/unloading with the new style systemd stuff. 13349 Mon Oct 2 18:08:10 2017 gautamUpdateCDSc1ioo DC errors I was trying to set up a DAC channel to interface with the AOM driver on the PSL table. • It would have been most convenient to use channels from c1ioo given proximity to the PSL table. • Looking at the 1X2 rack, it looked like there were indeed some spare DAC channels available. • So I thought I'd run a test by adding some TPs to the c1als model (because it seems to have the most head room in terms of CPU time used). • I added the DAC_0 block from CDS_PARTS library to c1als model (after confirming that the same part existed in the IOP model, c1x03). • Model recompiled fine (I ran rtcds make c1als and rtcds install c1als on c1ioo). • However, I got a bunch of errors when I tried to restart the model with rtcds restart c1als. The model itself never came up. • Looking at dmesg, I saw stuff like [4072817.132040] c1als: Failed to allocate DAC channel. [4072817.132040] c1als: DAC local 0 global 16 channel 4 is already allocated. [4072817.132040] c1als: Failed to allocate DAC channel. [4072817.132040] c1als: DAC local 0 global 16 channel 5 is already allocated. [4072817.132040] c1als: Failed to allocate DAC channel. [4072817.132040] c1als: DAC local 0 global 16 channel 6 is already allocated. [4072817.132040] c1als: Failed to allocate DAC channel. [4072817.132040] c1als: DAC local 0 global 16 channel 7 is already allocated. [4073325.317369] c1als: Setting stop_working_threads to 1 • Looking more closely at the log messages, it seemed like rtcds could not find any DAC cards on c1ioo. • I went back to 1X2 and looked inside the expansion chassis. I could only find two ADC cards and 1 BIO card installed. The SCSI cable labelled ("DAC 0") running from the rear of the expansion chassis to the 1U SCSI->40pin IDE breakout chassis wasn't actually connected to anything inside the expansion chassis. • I then undid my changes (i.e. deleted all parts I added in the simulink diagram), and recompiled c1als. • This time the model came back up but I saw a "0x2000" error in the GDS overview MEDM screen. • Since there are no DACs installed in the c1ioo expansion chassis, I thought perhaps the problem had to do with the fact that there was a "DAC_0" block in the c1x03 simulink diagram - so I deleted this block, recompiled c1x03, and for good measure, restarted all (three) models on c1ioo. • Now, however, I get the same 0x2000 error on both the c1x03 and c1als GDS overview MEDM screens (see Attachment #1). • An elog search revealed that perhaps this error is related to DAQ channels being specified without recording rates (e.g. 16384, 2048 etc). There were a few DAQ channels inside c1als which didn't have recording rates specified, so I added the rates, and restarted the models, but the errors persist. • According to the RCG runtime diagnostics document, T1100625 (which admittedly is for RCG v 2.7 while we are running v3.4), this error has to do with a mismatch between the DAQ config files read by the RTS and the DAQD system, but I'm not sure how to debug this further. • I also suspect there is something wrong with the mx processes: controls@c1ioo:~ 130 sudo systemctl status mx ● open-mx.service - LSB: starts Open-MX driver Loaded: loaded (/etc/init.d/open-mx) Active: failed (Result: exit-code) since Tue 2017-10-03 00:27:32 UTC; 34min ago Process: 29572 ExecStop=/etc/init.d/open-mx stop (code=exited, status=1/FAILURE) Process: 32507 ExecStart=/etc/init.d/open-mx start (code=exited, status=1/FAILURE)
Oct 03 00:27:32 c1ioo systemd[1]: Starting LSB: starts Open-MX driver... Oct 03 00:27:32 c1ioo open-mx[32507]: Loading Open-MX driver (with ifnames=eth1 ) Oct 03 00:27:32 c1ioo open-mx[32507]: insmod: ERROR: could not insert module /opt/3.2.88-csp/open-mx-1.5.4/modules/3.2.88-csp/open-mx.ko: File exists Oct 03 00:27:32 c1ioo systemd[1]: open-mx.service: control process exited, code=exited status=1 Oct 03 00:27:32 c1ioo systemd[1]: Failed to start LSB: starts Open-MX driver. Oct 03 00:27:32 c1ioo systemd[1]: Unit open-mx.service entered failed state.
• Not sure if this is related to the DC error though.
Attachment 1: c1ioo_CDS_errors.png
13348 Mon Oct 2 12:44:45 2017 johannesUpdateCamerasBasler 120gm calibration
Disclaimer: Wrong calibration factors! See https://nodus.ligo.caltech.edu:8081/40m/13391
The two acA640-120gm Basler GigE-Cams have been calibrated. I used the collimated output of a fiber that carried the auxiliary laser light from the PSL table. With a non-polarizing beam splitter some of the light was picked off onto a PD, and I modified the RF amplitude of the AOM drive signal to vary the power coming out of the fiber. The fiber output was directed at a white paper, which was placed 1.06m from the front of the lens tube assembly, which is where the focal plane is. Using the Pylon Viewer App I made sure that the entirety of the beam spot was imaged onto the CCD. Since the camera sensor is 1/4" across, I removed the camera from the lens tube and instead placed the Ophir power meter head at the position of the sensor and measured the power reported versus PD voltage, which turned out to be 1.5 V/uW.
The camera was put back in place and I used the Pypylon package Gautam had stumbled upon to sweep the exposure time from 100us to 10ms at different light power settings including no laser light at all for background subtraction, and rather than keeping the full bitmap data for O(100s) of images I recorded only the quantities
1. Pixel Max
2. Pixel Sum
3. Pixel Mean
4. Pixel Standard Deviation
5. Pixel Median
I performed this procedure for both the 152 and 153 cameras and plotted the pixel sum and the pixel max vs the exposure time. All the exposures were taken at a gain setting of 100, which is the smallest possible setting (out of 100-600). To obtain the calibration factor I use the input power Pin=75nW in the 'safe' region 1ms to 10ms where the pixel sum looks smooth and the CCD is reportedly not saturated.
Camera IP Calibration Factor CF 192.168.113.152 8.58 W*s 192.168.113.153 7.83 W*s
The incident power can be calculated as Pin =CF*Total(Counts-DarkCounts)/ExposureTime.
Attachment 1: calib_20170930_152.pdf
Attachment 2: calib_20170930_153.pdf
13347 Fri Sep 29 18:36:25 2017 gautamUpdateLSCDAC noise measurement (again)
BS connections and damping restored.
Quote:
### I will restore these tomorrow.
13346 Fri Sep 29 11:16:52 2017 SteveUpdateALSY End table corrected
The first Faraday isolater rejected beam path from the NPRO is fixed.
Attachment 1: ETMYf1.jpg
13345 Fri Sep 29 11:07:16 2017 gautamUpdateCDS40m files backup situation
The FB1 dd backup process seems to have finished too - but I got the following message:
dd: error writing ‘/dev/sdc’: No space left on device 30523666+0 records in 30523665+0 records out 2000398934016 bytes (2.0 TB) copied, 50865.1 s, 39.3 MB/s
Running lsblk shows the following:
controls@fb1:~ 32$lsblk NAME MAJ:MIN RM SIZE RO TYPE MOUNTPOINT sdb 8:16 0 23.5T 0 disk └─sdb1 8:17 0 23.5T 0 part /frames sda 8:0 0 2T 0 disk ├─sda1 8:1 0 476M 0 part /boot ├─sda2 8:2 0 18.6G 0 part /var ├─sda3 8:3 0 8.4G 0 part [SWAP] └─sda4 8:4 0 2T 0 part / sdc 8:32 0 1.8T 0 disk ├─sdc1 8:33 0 476M 0 part ├─sdc2 8:34 0 18.6G 0 part ├─sdc3 8:35 0 8.4G 0 part └─sdc4 8:36 0 1.8T 0 part While I am able to mount /dev/sdc1, I can't mount /dev/sdc4, for which I get the error message controls@fb1:~ 0$ sudo mount /dev/sdc4 /mnt/HGSTbackup/ mount: wrong fs type, bad option, bad superblock on /dev/sdc4, missing codepage or helper program, or other error
In some cases useful info is found in syslog - try dmesg | tail or so.
Looking at dmesg, it looks like this error is related to the fact that we are trying to clone a 2TB disk onto a 1.8TB disk - it complains about block size exceeding device size.
## So if we either have to get a larger disk (4TB?) to do the dd backup, or do the backing up some other way (e.g. unmount /frames RAID, delete everything in /frames, and then do dd, as Jamie suggested). If I understand correctly, unmounting /frames RAID will require that we stop all the daqd processes for the duration of the dd backup
Quote: sudo dd if=/dev/sda of=/dev/sdb bs=64K conv=noerror,sync in a tmux session on nodus (as I did for chiara and FB1, latter backup is still running).
## Edit: unmounting /frames won't help, since dd makes a bit for bit copy of the drive being cloned. So we need a drive with size that is >= that of the drive we are trying to clone. On FB1, this is /dev/sda, which has a size of 2TB. The HGST drive we got has an advertised size of 2TB, but looks like actually only 1.8TB is available. So I think we need to order a 4TB drive.
13344 Fri Sep 29 09:43:52 2017 jamieHowToCDSpyawg
Quote: I've modified the __init.py__ file located at /ligo/apps/linux-x86_64/cdsutils-480/lib/python2.7/site-packages/cdsutils/__init__.py so that you can now simply import pyawg from cdsutils. On the control room workstations, iPython is set up such that cdsutils is automatically imported as "cds". Now this import also includes the pyawg stuff. So to use some pyawg function, you would just do (for example): exc=cds.awg.ArbitraryLoop(excChan,excit,rate=fs) One could also explicitly do the import if cdsutils isn't automatically imported: from cdsutils import awg pyawg-away! Linking this useful instructional elog from Chris here: https://nodus.ligo.caltech.edu:8081/Cryo_Lab/1748
? Why aren't you able to just import 'awg' directly? You shouldn't have to import it through cdsutils. Something must be funny with the config.
13343 Thu Sep 28 23:50:04 2017 gautamUpdateLSCDAC noise measurement (again)
### I will restore these tomorrow.
As Rana pointed out to me, one important fact to keep in mind w.r.t. DAC noise is that it can be non-linear. So the RMS of the DAC noise in a higher frequency band (say 60-100Hz) can be affected by the RMS of the requested DAC signal in some lower frequency band (say 10-20Hz). One test to see if this hypothesis can explain the difference @100Hz between the ITMX channels and BS channels I observed a couple of days ago is to see if the noise around 100Hz becomes lower when I enable a 20-40Hz bandstop in the digital signal chain.
13342 Thu Sep 28 23:47:38 2017 gautamUpdateCDS40m files backup situation
The nodus backup too is now complete - however, I am unable to mount the backup disk anywhere. I tried on a couple of different machines (optimus, chiara and pianosa), but always get the same error:
mount: unknown filesystem type 'LVM2_member'
The disk itself is being recognized, and I can see the partitions when I run lsblk, but I can't get the disk to actually mount.
Doing a web-search, I came across a few blog posts that look like the problem can be resolved using the vgchange utility - but I am not sure what exactly this does so I am holding off on trying.
To clarify, I performed the cloning by running
sudo dd if=/dev/sda of=/dev/sdb bs=64K conv=noerror,sync
in a tmux session on nodus (as I did for chiara and FB1, latter backup is still running).
13341 Thu Sep 28 23:32:38 2017 gautamHowToCDSpyawg
I've modified the __init.py__ file located at /ligo/apps/linux-x86_64/cdsutils-480/lib/python2.7/site-packages/cdsutils/__init__.py so that you can now simply import pyawg from cdsutils. On the control room workstations, iPython is set up such that cdsutils is automatically imported as "cds". Now this import also includes the pyawg stuff. So to use some pyawg function, you would just do (for example):
exc=cds.awg.ArbitraryLoop(excChan,excit,rate=fs)
One could also explicitly do the import if cdsutils isn't automatically imported:
from cdsutils import awg
pyawg-away!
Linking this useful instructional elog from Chris here: https://nodus.ligo.caltech.edu:8081/Cryo_Lab/1748
13340 Thu Sep 28 11:13:32 2017 jamieUpdateCDS40m files backup situation
Quote: After consulting with Jamie, we reached the conclusion that the reason why the root of FB1 is so huge is because of the way the RAID for /frames is setup. Based on my googling, I couldn't find a way to exclude the nfs stuff while doing a backup using dd, which isn't all that surprising because dd is supposed to make an exact replica of the disk being cloned, including any empty space. So we don't have that flexibility with dd. The advantage of using dd is that if it works, we have a plug-and-play clone of the boot disk and root filesystem which we can use in the event of a hard-disk failure. One option would be to stop all the daqd processes, unmount /frames, and then do a dd backup of the true boot disk and root filesystem. Another option would be to use rsync to do the backup - this way we can selectively copy the files we want and ignore the nfs stuff. I suspect this is what we will have to do for the second layer of backup we have planned, which will be run as a daily cron job. But I don't think this approach will give us a plug-and-play replacement disk in the event of a disk failure. Third option is to use one of the 2TB HGST drives, and just do a dd backup - some of this will be /frames, but that's okay I guess.
This is not quite right. First of all, /frames is not NFS. It's a mount of a local filesystem that happens to be on a RAID. Second, the frames RAID is mounted at /frames. If you do a dd of the underlying block device (in this case /dev/sda*, you're not going to copy anything that's mounted on top of it.
What i was saying about /frames is that I believe there is data in the underlying directory /frames that the frames RAID is mounted on top of. In order to not get that in the copy of /dev/sda4 you would need to unmount the frames RAID from /frames, and delete everything from the /frames directory. This would not harm the frames RAID at all.
But it doesn't really matter because the backup disk has space to cover the whole thing so just don't worry about it. Just dd /dev/sda to the backup disk and you'll just be copying the root filesystem, which is what we want.
13339 Thu Sep 28 10:33:46 2017 gautamUpdateCDS40m files backup situation
After consulting with Jamie, we reached the conclusion that the reason why the root of FB1 is so huge is because of the way the RAID for /frames is setup. Based on my googling, I couldn't find a way to exclude the nfs stuff while doing a backup using dd, which isn't all that surprising because dd is supposed to make an exact replica of the disk being cloned, including any empty space. So we don't have that flexibility with dd. The advantage of using dd is that if it works, we have a plug-and-play clone of the boot disk and root filesystem which we can use in the event of a hard-disk failure.
1. One option would be to stop all the daqd processes, unmount /frames, and then do a dd backup of the true boot disk and root filesystem.
2. Another option would be to use rsync to do the backup - this way we can selectively copy the files we want and ignore the nfs stuff. I suspect this is what we will have to do for the second layer of backup we have planned, which will be run as a daily cron job. But I don't think this approach will give us a plug-and-play replacement disk in the event of a disk failure.
3. Third option is to use one of the 2TB HGST drives, and just do a dd backup - some of this will be /frames, but that's okay I guess.
I am trying option 3 now. dd however does requrie that the destination drive size be >= source drive size - I'm not sure if this is true for the HGST drives. lsblk suggests that the drive size is 1.8TB, while the boot disk, /dev/sda, is 2TB. Let's see if it works.
Backup of chiara is done. I checked that I could mount the external drive at /mnt and access the files. We should still do a check of trying to boot from the LaCie backup disk, need another computer for that.
nodus backup is still not complete according to the console - there is no progress indicator so we just have to wait I guess.
Quote:
### Backups of the root filesystems of chiara and nodus are underway right now. I am backing them up to the 1 TB LaCie external hard drives we recently acquired.
We also wanted to do a backup of the root of FB1 - but I'm not sure if dd will work with the external hard drive, because I think it requires the backup disk size (for us, 1TB) to be >= origin disk size (which on FB1, according to df -h, is 2TB). Unsure why the root filesystem of FB is so big, I'm checking with Jamie what we expect it to be. Anyways we have also acquired 2TB HGST SATA drives, which I will use if the LaCie disks aren't an option.
13338 Thu Sep 28 06:35:44 2017 ChrisUpdateLSCDAC noise measurement (again)
Since you're monitoring two channels simultaneously, you could try subtracting them, as an alternative to carving out bandstops.
1. Drive both channels with the same time series
2. Tweak the filter module gains if needed to equalize the analog outputs
3. Apply SR785 user math to subtract the two channels (or A-B mode of a single input, if you're not using that already?)
4. Measure the residual, which should be the incoherent part containing DAC + coil driver noise
Subtraction can conceal certain annoying effects (like numerical noise or level crossing glitches) that remain coherent for two identical outputs. It might be worth experimenting with a differential offset or sinusoid, to try to break up that kind of coherence if it exists.
13337 Wed Sep 27 23:44:45 2017 gautamUpdateALSProposed PM measurement setup
Attachment #1 is a sketch of the proposed setup to measure the PM response of the EX NPRO. Previously, this measurement was done via PLL. In this approach, we will need to calibrate the DFD output into units of phase, in order to calibrate the transfer function measurement into rad/V. The idea is to repeat the same measurement technique used for the AM - take ~50 1 average measurements with the AG4395, and look at the statistics.
Some more notes:
• Delay line box is passive, just contains a length of cable.
• IQ Demodulation is done using an aLIGO 1U chassis unit, with the actual demod board electronics being D0902745
• The RF beatnote amplitude out of the IR beat PD is ~ -8dBm.
• The ZHL-3A amplifiers have gain of 24dB, so the amplified beat should be ~16dBm
• At the LSC rack, the amplified beat is split into two - one path goes to the LO input of D0902745 (so at most 13dBm), the other goes through the delay line.
• On the demod board, the LO signal is amplified with a AP1053, rated at 10dB gain, max output of 26dBm, so the signal levels should be fine for us, even though the schematic says the nominal LO level is 10dBm - moreover, I've ignored cable losses, insertion losses etc so we should be well within spec.
• The mixer is PE4140. The datasheet quotes LO levels of 17dBm for all the "nominal" tests, we should be within a couple of dBm of this number.
• There is no maximum value specified for the RF input signal level to the mixer on the datasheet, but I expect it to be <10dBm.
• We should park the beatnote around 30MHz as this should be well within the operational ranges for the various components in the signal chain.
Attachment 1: IMG_7609.JPG
13336 Wed Sep 27 22:25:21 2017 gautamUpdateLSCDAC noise measurement (again)
Attachment #1: Summary of results of measurements made on Friday. There is a lot in this plot, here is a breakdown:
• I drove the excitation points of the coil output filter banks with raw time-series data downloaded during a DRMI lock with pyawg. Today during the meeting, Rana pointed out that we could just acquire median (as opposed to mean since the former is more immune to glitches during the averaging process) spectra during a lock, and then do the ifft in python to generate time series data for pyawg. Another advantage of doing it this way is that we don't need to store a large (~200MB in my case) file of 16k data for numerous channels. But since I already had this file, I decided against changing the methodology for this round of tests. Time series plots of the signals do not show any large glitches.
• The SR785 was used in dual channel mode to acquire spectra from 2 coil driver outputs simultaneously, in the interest of saving time. Input range was set to -32dbVpk, AC coupled, which was the smallest value that worked for the given signal profile. Spectra were taken from DC-200Hz, with 801 points, 25 averages. The DB15 output of the coil driver board was connected to a DB15 breakout board, and then a BNC->Pomona mini-grabber adapter was used to connect to the SR785 input. The newly acquired linear power supplies for the GPIB box mean that spurious 60Hz harmonics were not present.
• Initially, I had planned to enable various bandstops from 20Hz-200Hz, to get a more complete profile of the noise. But in the end, I only used two elliptic bandstops (6th order, 60dB stopband attenuation): 60-90Hz, for which data is plotted in red and 90-200Hz, for which data is plotted in green
• I've used the same noise model as I used here, plotted in dashed grey (summed with SR785 noise at the above input range, with input terminated via 50ohm terminator) - but had to tweak the parameters to get the curve to line up with the measurement. It looks like there is considerable variation between DAC channels, and certainly between the ITMX channels and the BS channels as groups.
• I took the measurement in two conditions - with the coil de-whitening off (left column) and coil de-whitening on (right column). Note that the input to the excitation was acquired at the IN1 of the relevant filter bank, and since the de-whitening happens downstream of this, we don't have to do anything special.
• In the right column, I have also plotted the LISO modelled noise, which was shown to match well with the measured curve, admittedly only for one channel (for the coil driver alone, so I am not taking into account the noise of the de-whitening board - I will fix this once I dig up that data).
Some remarks:
1. According to this measurement, the de-whitening filters are the same on the ITMX channels and BS channels. So I don't understand the difference in the right column for BS and ITMX channels.
2. While there is considerable variation between channels and also between ITMX and BS, there is certainly >6dB of reduction in the DAC noise when the de-whitening is engaged. However, no improvement was seen in the MICH error signal spectrum between 60-300Hz. So we have to continue to investigate other noises that can explain the noise in that band.
3. Also, the realized improvement in DAC noise by turning on the coil de-whitening seems marginal - the low pass has gain of ~-80dB at 100Hz, but we seem to be hitting some sort of electronics noise in all channels at the level of ~100nV/rtHz (assuming the actual DAC noise doesn't degrade significantly when the digital simulated de-whitening filter is engaged).
4. It remains to do the test for the ITMY channels.
5. It would be useful to visualize the incoherent sum of all these channels - this is what should go into the MICH displacement NB. To be added.
6. I'm currently loading pyawg from my user directory. Need to figure out a place to put this and add it to \$PATH.
Data + code for this plot will be attached later.
Attachment 1: coilNoises.pdf
13335 Wed Sep 27 00:20:19 2017 gautamUpdateALSMore AM sweeps
Attachment #1: Result of AM sweeps with EX laser crystal at nominal operating temperature ~ 31.75 C.
Attachment #2: Tarball of data for Attachment #1.
Attachment #3: Result of AM sweeps with EX laser crystal at higher operating temperature ~ 40.95 C.
Attachment #4: Tarball of data for Attachment #2.
Remarks:
• Confirmed that PDA 55 is in the "0dB" setting - the actual dial is unmarked, and has 5 states. I guessed that the left-most one is 0dB, and checked that if I twiddled the dial by one state to the right, the DC level on the scope increased by 10dB as advertized. Didn't check all the states.
• DC level is ~2.3V on a high-impedance scope. So it will be ~1.15V to a 50ohm load, which is what the DC block is. The inverse of this value is used to calibrate the vertical axis of the TF measurement to RIN/V.
• Input R (split RF source signal) attenuation: 20dB. Input A (PDA55 output) attenuation: 0dB.
• Main problem is still network hangups when trying to do many sweeps.
• Seems to persist even when I connect the GPIB box to one of the network switches - so don't think we can blame the WiFi.
• Need to explore possibility of speedup - takes >2hours to run ~50scans!
To-do:
• Overlay median and uncertainty plots for the two temp. settings. There is a visible diference in both the locations and depths/heights of various notches/peaks in the AM profile.
• Repeat test with a fast focusing lens to focus the beam more tightly on the PD active area to confirm that the measured AM is indeed due to the PZT drive and not from beam-jitter (presently, spot diameter is ~0.5x active area diameter, to eye).
• Get the PM data.
• Depending on what the PM data looks like, do a more fine-grained scan around some promising AM notches / PM peaks.
Attachment 1: TFAG4395A_26-09-2017_202344_FourSquare.pdf
Attachment 2: lowTemp.tgz
Attachment 3: TFAG4395A_26-09-2017_231630_FourSquare.pdf
Attachment 4: highTemp.tgz
13334 Tue Sep 26 22:11:08 2017 johannesUpdateCameraspost-vent camera capture comparison
I configured the remaining GigE-Camera to work on the 40m network. We currently have 3 operational Basler cameras:
The 120gm's have been assigned the IPs 192.168.113.152 (was already configured) and 192.168.113.153 (freshly configured) and have been labeled accordingly. Note that it was not necessary to connect the out-of-the-box camera directly to a dedicated ethernet adapter whose IP was set manually to 169.254.0.XXX as pointed out in earlier posts - a few seconds after connecting the camera to the control room switch (with PoE adapter to power it) the camera showed up in the configuration software tool which is launched via
/opt/rtcds/caltech/c1/scripts/GigE/pylon5/bin/./IpConfigurator
and can be assigned a corrected, static IP.
We have a plethora of 2" tubes for the lens assembly, but not a great variety of focal lengths for 2" lenses. Present with the camera gear were two f=250 mm and one f=150 mm 2" lenses with a NIR broadband AR coating
To determine the lens positions relativ to the sensor I assumed that the camera we're setting up looks at its test mass from a distance of 1m. Using the two available focal lengths we can look for solutions which have reasonable lens separations <~10cm and suitable magnification. We primarily want to image the central mirror area onto a 1/4" sized sensor, which can be achieved with a magnification of ~1/8.
I chose a lens separation of 6cm, which gives a theoretical magnification of -.12 and a sensor-lens 2 distance of 7.95 cm. I placed the lenses accordingly in the tubes and checked the focusing with Gautam's help:
It's pretty close to what we would expect. We will do the calibration using the auxiliary laser on the PSL table. For this I temporarily routed a fiber from the PSL enclosure to the SP table. Since the main cable hole is sort of cramped it's going in through a gap near the ceiling instead.
Attachment 1: lens_distance.pdf
13333 Tue Sep 26 19:10:13 2017 gautamUpdateALSFiber ALS setup neatened
[steve, gautam]
The Fiber ALS box has been installed on the existing shelf on the PSL table. We had to re-arrange some existing cabling to make this possible, but the end result seems okay (to me). The box lid was also re-installed.
Some stuff that still needs to be fixed:
1. Power supply to ZHL amplifiers - it is coming from a table-top DC supply currently, we should hook these up to the Sorensens.
2. We should probably extend the corrugated fiber protection tubing for the three fibers all the way up to the shelf.
Beat spectrum post changes to follow.
Quote:
Is it better to mount the box in the PSL under the existing shelf, or in a nearby PSL rack?
Quote: Further characterization needs to be done, but the results of this test are encouraging. If we are able to get this kind of out of loop ALS noise with the IR beat, perhaps we can avoid having to frequently fine-tune the green beat alignment on the PSL table. It would also be ideal to mount this whole 1U setup in an electronics rack instead of leaving it on the PSL table
Attachment 1: IMG_7605.JPG
13332 Tue Sep 26 15:55:20 2017 gautamUpdateCDS40m files backup situation
### Backups of the root filesystems of chiara and nodus are underway right now. I am backing them up to the 1 TB LaCie external hard drives we recently acquired.
I first initialized the drives by hooking them up to my computer and running the setup.app file. After this, plugging the drive into the respective machine and running lsblk, I was able to see the mount point of the external drive. To actually initialize the backup, I ran the following command from a tmux session called ddBackupLaCie:
sudo dd if=/dev/sda of=/dev/sdb bs=64K conv=noerror,sync
Here, /dev/sda is the disk with the root filesystem, and /dev/sdb is the external hard-drive. The installed version of dd is 8.13, and from version 8.21 onwards, there is a progress flag available, but I didn't want to go through the exercise of upgrading coreutils on multiple machines, so we just have to wait till the backup finishes.
We also wanted to do a backup of the root of FB1 - but I'm not sure if dd will work with the external hard drive, because I think it requires the backup disk size (for us, 1TB) to be >= origin disk size (which on FB1, according to df -h, is 2TB). Unsure why the root filesystem of FB is so big, I'm checking with Jamie what we expect it to be. Anyways we have also acquired 2TB HGST SATA drives, which I will use if the LaCie disks aren't an option.
13331 Tue Sep 26 13:40:45 2017 gautamUpdateCDSNDS2 server restarted on megatron
Gabriele reported problems with the nds2 server again. I restarted it again.
update: had to do it again at 1730 today - unclear why nds2 is so flaky. Log files don't suggest anything obvious to me...
Quote: I was unable to download data using nds2. Gabriele had reported similar problems a week ago but I hadn't followed up on this. I repeated steps 5-7 from elog 13161, and now it seems that I can get data from the nds2 servers again. Unclear why the nds2 server had to be restarted. I wonder if this is somehow related to the mysterious acromag EPICS server tmux session dropout.
13330 Mon Sep 25 17:56:33 2017 johannesUpdateComputer Scripts / Programstransmitted power during lossmap
I had to do a reboot + burt restore of c1psl today. It was unresponsive and I couldn't get the PMC to lock. I also had to slightly realign the PMC, and the IMC was too misaligned for the autolocker to catch lock. Adjusting it manually, it was predominantly MC1 PIT that was off. The YARM locked on a 10 mode and had to be aligned manually as well.
I left a script running on Donnatella that tilts ETMX and thus moves the beam on ITMX. I'm monitoring the transmitted power to evaluate sane thresholds for the demodulation offsets in a lossmap measurement. The script will return the IFO to normal after it is done and will take <2 hours to complete (no real clue, but there's no way it takes longer than that for ~50 datapoints).
13329 Sun Sep 24 20:47:15 2017 ranaUpdateComputer Scripts / ProgramsRF TF Uncertainties
I have made several changes to Craig's script for better pythonism. Its more robust with different libraries and syntaxes and makes a tarball by default (w/o a command line flag). These kinds of general util scripts will be going into a general use folder in the git.ligo.org/40m/ team area so that it can be used throughout the LSC.
I don't think we need/want a coherence calculation, so I have not included it. Usually, we use coherence to estimate the uncertainty, and here we are just plotting it directly from the dist of the sweeps so coherence seems superfluous.
Attachment 1: TFAG4395A_21-09-2017_115547_FourSquare.pdf
Attachment 2: TFAG4395A_21-09-2017_115547.tgz
13328 Fri Sep 22 18:12:27 2017 gautamUpdateLSCDAC noise measurement (again)
I've been working on setting up some scripts for measuring the DAC noise.
In all the DRMI noise budgets I've posted, the coil-driver noise contribution has been based on this measurement, which could be improved in a couple of ways:
• The measurement was made at the output of the AI board - we can make the measurement at the output of the coil driver board, which will be a closer reflection of the actual current noise at the OSEM coils.
• The measurement was made by driving the DAC with shaped random noise - but we can record the signal to the coils during a lock and make the noise measurement by using awg to drive the coil with this signal, with elliptic bandstops at appropriate frequencies to reveal the electronics noise.
1. The IN1 signals to the coils aren't DQ-ed, but ideally this is the signal we want to inject into the coil_EXC channel for this measurement - so I re-locked the DRMI a couple of nights ago and downloaded the coil IN1 channel data for ~5mins for the ITMs and BS.
2. AWGGUI supposedly has a feature that allows you to drive an EXC channel with an arbitrary signal - but I couldn't figure out how to get this working. I did find some examples of this kind of application using the Python awg packages, so I cobbled together some scripts that allows me to drive some channels and place elliptic bandstop filters as I required.
3. I wasted quite a bit of time trying to implement these signals in Python using available scipy functions, on account of me being a DSP n00b . When trying to design discrete-time filters, of course numerical precision errors become important. Initially I was trying to do everything in the "Transfer function (numerator-denominator)" basis, but as Rana pointed out, the way to go is using SOSs. Fortunately, this is a simple additional argument to the relevant python functions, after which elliptic bandstop filter design was trivial.
4. The actual test was done as follows:
• Save EPICS PIT/YAW offsets, set them to 0, disable Oplev servos, and then shut down optic watchdog once the optic is somewhat damped. This is to avoid the optics getting a large kick when disconnecting the DB15 connector from the coil driver board output.
• Disconnect above-mentioned DB15 connector from the appropriate coil driver board output.
• Turn off inputs to coils in filter module EPICs screens. Since the full signal (local damping servo output + Oplev servo output + LSC servo output) to the coil during a DRMI lock will be injected as an excitation, we don't need any other input.
• Use scripts (which I will upload to a git repo soon) to set up the appropriate excitation.
• To measure the spectrum, I used a DB15 breakout board with test-points soldered on and some mini-grabber-to-BNC adaptors, in order to interface the SR785 to the coil driver output. We can use the two input channels of the SR785 to simultaneously measure two coil driver board output channels to save some time.
• Take a measurement of the SR785 noise (at the appropriate "Input Range" setting) with inputs terminated to estimate the analyzer noise floor.
• Just for kicks, I made the measurement with the de-whitening both OFF/ON.
I only managed to get in measurements for the BS and ITMX today. ITMY to be measured later, and data/analysis to follow.
The ITMX and BS alignments have been restored after this work in case anyone else wants to work with the IFO.
Some slow machine reboots were required today - c1susaux was down, and later, the MC autolocker got stuck because of c1iool0 being unresponsive. I thought we had removed all dependency of the autolocker on c1iool0 when we moved the "IFO-STATE" EPICS variable to the c1ioo model, but clearly there is still some dependancy. To be investigated.
13327 Thu Sep 21 15:23:04 2017 gautamOmnistructureALSLong cable from LSC->IOO
[steve,gautam]
We laid out a 45m long BNC cable from the LSC rack to the IOO rack via overhead cable trays. There is ~5m excess length on either side, which have been coiled up and cable-tied for now. The ends are labelled "TO LSC RACK" and "TO IOO RACK" on the appropriate ends. This is to facilitate hooking up the output of the DFD for making a PM measurement of the AUX X laser. There is already a long cable that runs from the IOO rack to the X end.
ELOG V3.1.3- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6400650143623352, "perplexity": 3323.4949532877786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573145.32/warc/CC-MAIN-20220818003501-20220818033501-00138.warc.gz"} |
http://maikolsolis.com/2012/09/ | ## The Delta method: Main Result
Let $latex {T_{n}}&fg=000000$ an estimator of $latex {\theta}&fg=000000$, we want to estimate the parameter $latex {\phi(\theta)}&fg=000000$ where $latex {\phi}&fg=000000$ is a known function. It is natural to estimate $latex {\phi(\theta)}&fg=000000$ by $latex {\phi(T_{n})}&fg=000000$. Now, we can then ask: How the asymptotic properties of $latex {T_{n}}&fg=000000$ could be transfer to $latex {\phi(T_{n})}&fg=000000$?
## Weak Law of Large Numbers and Central Limit Theorem via the Levy’s continuity theorem
The Levy’s continuity theorem is a very important tool in the statistical machinery. For example, it will give us two simple proofs to two classical statistical problems: The Law of Large Numbers and the Central Limit Theorem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9834875464439392, "perplexity": 102.52615593248807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514576047.85/warc/CC-MAIN-20190923043830-20190923065830-00188.warc.gz"} |
https://tpfto.wordpress.com/2011/09/ | ## Parametric equations for regular and Reuleaux polygons
September 15, 2011
Not having any other place to note this down, I suppose this blog entry is as good as any.
From this m.SE answer, user Raskolnikov demonstrates a polar equation whose graph is a regular $n$-gon (slightly reformulated):
$r=\cos\frac{\pi}{n}\sec\left(\theta-\frac{\pi}{n}\left(2\left\lfloor\frac{n\theta}{2\pi}\right\rfloor+1\right)\right)$
Through similar considerations, one can derive parametric equations for Reuleaux polygons ($n$ of course being odd):
$x=2\cos\frac{\pi}{2n}\cos\left(\frac12\left(t+\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)\right)-\cos\left(\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)$
$y=2\cos\frac{\pi}{2n}\sin\left(\frac12\left(t+\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)\right)-\sin\left(\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)$
Here’s a side-by-side comparison of some odd-sided regular and Reuleaux polygons:
I’m not sure if there are simpler expressions for the parametric equations of the Reuleaux polygon, though, and I would be interested if you, the reader, might come up with something a bit simpler. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 5, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7509503960609436, "perplexity": 871.8287147423513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886112682.87/warc/CC-MAIN-20170822201124-20170822221124-00241.warc.gz"} |
http://quant.stackexchange.com/tags/monte-carlo/new | Tag Info
New answers tagged monte-carlo
2
I would definitely recommend Volopta as a reliable source of self-contained and commented financial engineering source codes (useful for prototyping/understanding but clearly not production code). I have for instance copy-pasted, the explicit PDE solver you are looking for (centred in space, backward in time) below (+ edited for clarity + improved ...
0
Monte Carlo simulation in the context of Quantitative Finance refers to a set of techniques to generate artificial time series of the stock price overtime, from which option prices can be derived. In this case, the Least-Squares Monte Carlo can be applied to any stock price stochastic process that lends itself to simulation. It is especially useful for ...
0
Assuming deterministic interest rates, the price of an American call option struck at $K$ and expiring at $T$ is given by $$V_0 = \text{sup}_{\tau \in \mathcal{T}[0,T]} \mathbb{E}_0^\mathbb{Q}\left[ e^{-r\tau} \max(S_{\tau}-K, 0) \right]$$ where $\mathcal{T}[0,T]$ denotes a family of stopping times with values in $[0,T]$ and where, under the risk-neutral ...
0
I would suggest to use : $$f(t,S^\theta_t)=\max(K-S^\theta_t,0)\exp(-\theta W_t\color{red}{\mathbf{-}}\frac{1}{2}\theta^2t)$$ where $dS^\theta_t=(r+\sigma\theta)S^\theta_t dt + \sigma S^\theta_t dW_t$
4
Typically when running a Monte Carlo simulation we might simulate an SDE similar to $$\dfrac{dS}{S} = \mu\:dt + \sigma \: dW(t)$$ by some appropriate method (e.g. Euler-Maruyama, Milstein, etc). We notice by dimensional analysis that if $t$ is in units of $\textrm{years}$ then $\mu \sim \textrm{years}^{-1}$ and $\sigma \sim \textrm{years}^{-1/2}$. ...
0
Overall you are not mistaken, although it is worth revisiting a few steps in your question. We assume $S$ follows the SDE $$\dfrac{dS}{S} = \mu\:dt+ \sigma\:dW^\mathbb{P}(t)$$ under the physical measure $\mathbb{P}$. If we change to the risk neutral measure $\mathbb{Q}$ (using Girsanov's theorem) then $\mu \to r$ and we have the following SDE $$\dfrac{dS}... 5 [Short answer] IMHO there is a fundamental problem with wanting to extract a sound implied volatility figure out of a deep ITM option's price. You should use out-of-the-money forward options (OTMF) instead: put options for strikes smaller than the forward price (left wing of the volatility surface) and call options otherwise (right wing of the volatility ... 0 There's nothing wrong with your formulation, in my opinion. If you model the rate z_30 with a fixed mean, then indeed the forward ZCB price is long vega. This means that the forward interest rate is short vega (i.e. the 30yr into 10yr forward rate goes down when vol goes up). This is self-consistent. In most textbooks, however, the forward interest ... 0 I just made things clearer hoping it would help. Let define \mathbb{Q}_\theta as$$\frac{d\mathbb{Q}_\theta}{d\mathbb{P}}|_{\mathcal{F}_t}=\exp(\theta W_t -\frac{1}{2}\theta^2 t)=Z^\theta_t$$By girsanov, if W is a brownian motion under \mathbb{P}, then W^\theta_t=W_t-\theta t is a brownian motion under \mathbb{Q}^\theta$$\begin{split} \mathbb{...
1
Applying Itô's lemma to the Black-Scholes SDE and integrating from $t$ to $t+\Delta t$ gives: $$S_{t+\Delta t} = S_t e^{(r-\frac{1}{2}\sigma^2)\Delta t + \sigma \sqrt{\Delta t}Z}$$ with $Z \sim N(0,1)$, showing that $S_{t+\Delta t}$ given $S_t$ is log-normally distributed. It is then straightforward to write, for any compact $\mathcal{A} = [a_1,a_2]$ ...
1
Note that $${{f}_{W(t)\left| W(s) \right.}}\left(x\left| y \right. \right)=\frac{{{f}_{ W(s),W(t)}}\left( x,y \right)}{{{f}_{ W(s)}}\left( y \right)}=\frac{1}{\sqrt{2\pi(t-s)}}\exp \left[-\frac{{{(x-y)}^{2}}}{2(t-s)} \right]$$ By application of Ito's lemma we have $$ln\,S_{t+\Delta t}=ln\,S_t+\left((\mu-\frac{1}{2}\sigma^2)\Delta t+\sigma(W_{t+\Delta t}-... 4 You are trying to price an option through Monte Carlo simulations. Here is how it should work, assuming the Black-Scholes diffusion framework. Under the Black-Scholes model's assumptions, the value of a risky asset S at the time t=T is a random variable which reads$$ S_T = S_0 e^{\left(\mu-\frac{\sigma^2}{2}\right)T + \sigma \sqrt{T} Z}\tag{1} with ...
Top 50 recent answers are included | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8852811455726624, "perplexity": 589.9506966240847}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783391634.7/warc/CC-MAIN-20160624154951-00025-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://learnyousomeml.com/nb/ul-ml-catalogue.html | # 08.00 Machine Learning Groupings, Again¶
We saw that we can divide machine learning models in four categories.
• Supervised Learning
• Classification
• Regression
• Unsupervised Learning
• Dimensionality Reduction
• Clustering
And although this grouping is quite outdated we will follow it for the time being. We did skim over classification and regression but more importantly we did see how we evaluate ML models and how we automate the evaluation. Now, one may be surprised that we already went through the majority of way of evaluating an ML model.
In unsupervised learning there are no good ways to know whether a model works well or not. Instead, faced with a problem against which we need to attempt unsupervised learning, we need to attempt to transform at least part of the problem into supervised learning and evaluate it with accuracy scores, f1 measures, and so on.
Often we do not have labels for the data we use unsupervised learning on, in such a case any form of model evaluation will be very difficult. Perhaps the easiest way to evaluate a model for a problem for which we have no labels at all and cannot get any labels, is to find a similar problem for which we have labels and train a model on that similar problem. Once we evaluate a good model on the similar problem we assume that it will work in a similar fashion on the original problem.
Most of the time with unsupervised learning we have or can get at least a handful of labels. We can then evaluate our unsupervised model on the handful of labels in a similar fashion as we would do for a supervised one. This can be sometimes called semi-supervised learning, and is our first indication that the list of ML groupings at the top is quite incomplete.
### Dimensionality Reduction¶
Will allow us to explore the data in lesser dimensions, whatever we call the dimension. Here we come back to the concept of distance. We did see different distances such as L-norms, including L1 and L2, but many other measures can be made to work as a distance: sum of coordinates and polynomial functions on top of the data are distances we have already used; other distances include measures such as cosine similarity or probabilities. Don't worry what these exactly are right now, we will explore some of the ways of measuring distance.
Independent of what we use for distance, we say that two points (samples) are similar if they are close to one another, and dissimilar if they are far apart.
One way to evaluate dimensionality reduction is to perform the reduction to a set of numbers we can visualize and then check by-the-eye that things close together are the ones we expect to be similar and things far apart are the ones we expect to be dissimilar.
When we have a handful of labels we can perform the dimensionality reduction and then train another model on top of the reduced data. The performance of the resulting model will be an indication of the performance of the dimensionality reduction technique.
### Clustering¶
Is similar to the models we have already seen when looking at supervised learning. Instead of changing the structure of the space where the data lives as is done with dimensionality reduction, clustering attempts to identify the structure in the data. The identification of this structure is again performed by means of distance in one form or another.
One builds clusters, structures which then allow new (unseen) data to be classified into one of these clusters. We argue that samples that are similar to each other are in a cluster together, and in a different cluster from samples that are dissimilar to them. Contrary to supervised learning one such cluster has no meaning, unless we give it one using a guesstimate, by-the-eye, evaluation. Without labels clusters are just groupings of data, yet seeing such groupings we may be able to guess labels for them.
If we have some labels then we can check what label is present for the majority of samples in a cluster and give the cluster a meaning. In this fashion we can evaluate a clustering model in a similar way to supervised learning models. We can then say that a good clustering would group together the samples according to their labels.
When one has no labels at all and cannot guesstimate them there remains possibility of evaluating the shape of the clusters. Clusters live in the space where the data lives. Evaluation techniques can geometrically check whether the clusters are round or dense. And the concept of round or dense can be changed slightly depending of how we measure distance. Still, geometric techniques for clustering evaluation are rather limited, and are an area of active research.
Let's import a handful of things that will help us with the next topic.
We will need a handful of mathematical concepts, although not necessarily equations, in order to advance from here. We will also have a quick look at a few mathematicians that allowed us to get where we are today.
In [1]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.style.use('seaborn-talk')
ul-godel.svg
Kurt Gödel's incompleteness theorems, from 1931, were one of the greatest turns in Mathematical thinking. The theorem's themselves only prove the impossibility of building a full set of mathematical axioms which would explain the axioms themselves. Despite the fact that the theorem's only build upon the axioms of arithmetic, the deepest ideas behind it proved to be revolutionary. Later works and interpretations of Gödel's theorems built the idea that mathematics cannot ever be proven by mathematics themselves, which placed applied mathematics on the same level as "pure" theoretical mathematics. This line of thinking also led to the rejection of the idea that there must exist a set of first principles for every phenomena.
## Linearity¶
For a long time science assumed the linearity of the world. This is another way to say that science believed that every phenomena can be distilled to its first principles and then those first principles will combine in a linear fashion to produce the more complex phenomena. That is a lot of talk that does not actually tell us what this linearity is. It often is the case that the exact definition of linearity is slightly different depending on the discipline one looks at. Instead we will attempt to define where linearity fails and work backward from there.
On the assumption that one would be capable of dividing the world into all its first principles then, by definition, none of the first principles could be correlated with any of the other first principles. Otherwise, if the first principles are indeed correlated, then one first principle is dependent on the other and hence not a first principle but phenomena caused by the other first principle.
Let us play with this concept and build two graphs: the capital letter B, and some white noise - a completely uniform random set of points - and make their histograms in each dimension.
In [2]:
B = np.array([
[.1, .9], [.20, .9],
[.23, .87],
[.1, .8], [.25, .8],
[.1, .7], [.25, .7],
[.23, .63],
[.1, .6], [.20, .6],
[.1, .5], [.20, .5],
[.1, .4], [.27, .4],
[.27, .37],
[.1, .3], [.30, .3],
[.27, .23],
[.1, .2], [.27, .2],
[.1, .1], [.20, .1],
])
noise = []
for i in range(70):
noise.append(B + np.random.random(B.shape)*.12)
B = np.vstack(noise)
white = np.random.random(B.shape)
fig, ax = plt.subplots(2, 5, figsize=(16, 9))
ax[1, 0].scatter(B[:, 0], B[:, 1], s=20, c='crimson')
ax[0, 0].hist(B[:, 0], color='crimson')
ax[0, 0].set_title('Letter B, corr %.06f' % np.corrcoef(B[:, 0], B[:, 1])[0, 1])
ax[1, 1].hist(B[:, 1], color='crimson', orientation='horizontal')
ax[1, 3].scatter(white[:, 0], white[:, 1], s=20, c='steelblue')
ax[0, 3].hist(white[:, 0], color='steelblue')
ax[0, 3].set_title('White Noise, corr %.06f' % np.corrcoef(white[:, 0], white[:, 1])[0, 1])
ax[1, 4].hist(white[:, 1], color='steelblue', orientation='horizontal')
for img in ax.flat:
img.axis('off')
In both graphs we only have points in $2$ dimensions, and we evaluated the correlation between the $2$ dimensions in each case. Both the capital letter B and the white noise have pretty much no correlation between the dimensions in which they exist.
If we accept the approximation that the correlations are indeed exactly zero, then we can argue that we have separated the world where the letter B and the white noise live into its first principles: the positions on the x-axis and the positions on the y-axis. And since the positions and correlations are pretty much the same the letter B and the white noise are exactly the same thing!
Obviously they are not the same thing. We can see the letter B because our human brain is very good at finding non-linear patters. When we draw a capital letter B we do not take notice of exactly where we place each part of the letter. Instead we take effort to make sure that the letter B has one straight side and round holes in it. The exact position of the parts of B does not matter, what matters is the position between the parts of B.
## Here be Dragons - where Linearity Breaks¶
The letter B has a pattern we can identify with our human brain which do not follow from the definition of the two-dimensional world built by positions of points with two coordinates. Finding such patters as the letter B in the example is a case of a non-linear problem.
How do we know that a problem is non-linear? Well, we do not. The only way to know that there is more to be seen than what is possible by splitting the entire world where the problem lives into its first principles, is to perform the splitting and attempting to solve the problem. If the problem cannot be solved by splitting it into first (uncorrelated) principles then it is a non-linear problem. With experience one learns to identify non-linear problems. Notably, the world has many more non-linear problems than linear ones.
Linearity of problems, even when just an approximation, allows for several nice properties. Linear Algebra and Univariate statistics describe linear problems well, and have several closed form solutions (solutions than can be evaluated straight away). Hence it is worth to first attempt a linear solution to a problem before assuming that it is non-linear.
That being said, assuming that everything in the world around us can be explained by splitting it into first principles - and that linear techniques work on every problem - is akin of hitting a square peg into a round hole. In the case above one would just assume that every letter B is just white noise and never find out what B may actually be.
### A Note on Statistics¶
Univariate Statistics, where each random variable (or each feature when thinking about date) is described by a distribution with a singe peak, are not the only form of statistics. There are statistics which account for the possibility of non-linear patterns in the data. Univariate statistics are just more common, since they include the assumption of a Gaussian distribution of the data. And the Central Limit Theorem we mentioned earlier is also a form of univariate statistics since it is a sum of several univariate distributions (distributions with a single peak). Wilcoxon statistics and Tsallis statistics are examples of statistics that account for non-linearities.
Whether the commonalty of univariate statistics is due to the fact that linear problems are easier is up to debate. One can argue that the only reason why linear problems are easier is because we have more tools to deal with them. If mathematics has had developed in a different way than it did, perhaps today we would argue that non-linear problems are easy and non-linear problems are hard. That said, we currently have many more tools to deal with linear problems, hence we often argue that for practical purposes linear problems are easy to solve and non-linear ones are hard. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6698766350746155, "perplexity": 541.7878842884085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335034.61/warc/CC-MAIN-20220927131111-20220927161111-00006.warc.gz"} |
https://tex.stackexchange.com/questions/416155/etoolbox-for-loop-over-range | # etoolbox for loop over range
Is there a way to run a command on a range of consecutive integers (say, 11, 12, 13, 14, 15), specifying only the first and last numbers (11 and 15)? Basically I'm looking for the general LaTeX equivalent of tikz's \foreach \x in {11,…,15}.
In the MWE below (using etoolbox), \forcsvlist\mycommand{11,12,13,14,15} correctly runs \mycommand on each of the four inputs, and \myrange{11}{15} indeed expands to 11,12,13,14,15. But putting them together \forcsvlist\mycommand{\myrange{11}{15}} doesn't work. I think it comes down to order of expansion, which is way over my head.
\documentclass{article}
\usepackage{amsmath, etoolbox}
\newcommand\mycommand[1]{\boxed{#1} }
\newcounter{mycounter}
\newcommand\myrange[2]{
\defcounter{mycounter}{#1}
\themycounter%
\whileboolexpr
{test {\ifnumless{\themycounter}{#2}}}
{\stepcounter{mycounter},\themycounter}
}
\begin{document}
\forcsvlist\mycommand{11,12,13,14,15}
does not equal
\forcsvlist\mycommand{\myrange{11}{15}}
\end{document}
I've looked at Loop Multi-Contingency using etoolbox, Print all elements of a working array created with etoolbox package, and Remove extra curly braces, but couldn't figure out how to apply them to this situation.
• You are defining and setting counters in your \myrangemacro -- this can't be expandable. There might be more issues, but I can't check right now – user31729 Feb 19 '18 at 20:45
• Similar (?) tex.stackexchange.com/a/360959/31729 – user31729 Feb 19 '18 at 20:59
It is just as easy to loop without any specific package code in this case.
\documentclass{article}
\usepackage{amsmath}
\newcommand\mycommand[1]{\boxed{#1} }
\makeatletter
\newcommand\zz[3]{%
#1{#2}%
\ifnum#2=\numexpr#3\relax\expandafter\@gobblefour\fi
\zz#1{\the\numexpr#2+1\relax}{#3}%
}
\makeatother
\begin{document}
\zz\mycommand{11}{15}
\end{document}
• you get nested \numexpr which numexperts usually remove by suitable \expandafter ;-) (this may also surprise macro \mycommand if the latter parses in some way its argument and chokes on ten nested \numexpr's) – user4686 Feb 20 '18 at 13:41
• @jfbu I wondered about adding more expandafter but thought it would obscure the main point, it's not actually wrong is it? – David Carlisle Feb 20 '18 at 13:48
• nothing wrong ;-) – user4686 Feb 20 '18 at 15:02
\documentclass{article}
\usepackage{xinttools}
\newcommand\mycommand[1]{\fbox{#1} }
\newcommand\forrange[4][1]%
{\xintFor*##1in{\xintSeq[#1]{#2}{#3}}\do{#4{##1}}}
\begin{document}
\forrange{11}{15}{\mycommand}
\forrange[4]{11}{27}{\mycommand}
\forrange[-1]{15}{11}{\mycommand}
\end{document}
Now I picked over that abstraction from another authoritative answer. For everyday use you can also use
\xintFor #1 in {99, 37, -53, 'zouzou'}\do{ whatever }
and replace #1 by ##1 if inside a macro definition.
Optionally providing also the step; the \int_step_function:nnnN macro takes as argument the starting point, the step, the final point and finally the one parameter macro to which the current value is passed as argument.
\documentclass{article}
\usepackage{xparse}
\newcommand\mycommand[1]{\fbox{#1} }
\ExplSyntaxOn
\NewDocumentCommand{\forrange}{mO{1}mm}
{
\int_step_function:nnnN { #1 } { #2 } { #3 } #4
}
\ExplSyntaxOff
\begin{document}
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https://www.eevblog.com/forum/programming/macro-functions-in-headers/?nowap;PHPSESSID=rept6bug4nn3eq0t6jtk0d9h36 | 0 Members and 1 Guest are viewing this topic.
#### dunkemhigh
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« on: December 02, 2021, 12:44:00 am »
Title doesn't really cover it properly, so here is what it is:
I have a C module which has a lot of functions which all basically do the same thing apart from using different variables. They are get/set functions for the config, reading and writing NVS. So the string functions might look like this (it is ESP-IDF):
Code: [Select]
void confSetURL(char *url){ nvs_set_string(hconfig, NVS_KEY, url); nvs_commit(hconfig);}
OK, so I am writing a macro that saves me typing that all out:
Code: [Select]
#define confSetStr(NAME, KEY) \void confSet##NAME(char *str) \{ \ nvs_set_str(hconfig, KEY, str); \ nvs_commit(hconfig); \}
and I can then do lots of line like:
Code: [Select]
confSetStr(URL, CFG_KEY_URL);
and call it:
Quote
Except, I have to manually add the real function prototype to the .h in order to achieve that, and I am now looking at not just typos but a non-correlation between header prototype and macro instantiation.
Any way around this?
(I have no doubt someone will say that this - the many nearly identical functions - is not the way to do this, but it's the getting callable prototypes into the header that I'm really interested in here.)
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##### Re: Macro functions in headers
« Reply #1 on: December 02, 2021, 12:54:06 am »
#### SiliconWizard
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##### Re: Macro functions in headers
« Reply #2 on: December 02, 2021, 01:12:42 am »
Several ways to deal with it. Two of them:
- Either define an additional macro to generate the prototypes. Would just be:
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#define confSetStr_Proto(NAME) void confSet##NAME(char *str)And then in the header file, you'd have to declare the prototypes like so:
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confSetStr_Proto(URL);
- Or you could define the functions themselves inside the header file. You just need to qualify them static. The added benefit is that they'll get inlined everywhere they'll be called when optimizations are enabled. The drawback is that: possibly excess code size if the functions in question are large, but in the case of very small functions like getters and setters, that's IMHO a good approach:
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#define confSetStr(NAME, KEY) \static void confSet##NAME(char *str) \{ \ nvs_set_str(hconfig, KEY, str); \ nvs_commit(hconfig); \}And then in the header file, you'd have to define the functions like so:
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confSetStr(URL, CFG_KEY_URL)(Note that you can't add a final semicolon as you did in your example for defining functions with a body. The semicolon right after the final brace of the function's body will normally give a syntax error.)
#### brucehoult
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##### Re: Macro functions in headers
« Reply #3 on: December 02, 2021, 01:12:55 am »
Put all the calls to confSetStr() in a .inc file (name doesn't matter, it's just a convention) and then #include it into both the header and the C file, with different definitions of the macro in each (and #undef it after)
As I do here:
https://github.com/brucehoult/trv
The following users thanked this post: newbrain
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #4 on: December 02, 2021, 08:21:50 am »
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #5 on: December 02, 2021, 08:31:50 am »
- Either define an additional macro to generate the prototypes. Would just be:
Code: [Select]
#define confSetStr_Proto(NAME) void confSet##NAME(char *str)And then in the header file, you'd have to declare the prototypes like so:
Code: [Select]
confSetStr_Proto(URL);
That has possibilities, yes. Hmmm.
Quote
- Or you could define the functions themselves inside the header file.
I am reluctant to do that because of potential coupling. The calling code would need to know of, and be calling, nvs_set_str(), and know of hconfig, etc.
Quote
The drawback is that: possibly excess code size if the functions in question are large
The getter is bigger, albeit not by a lot. But I don't think doing this (putting it in the header) would change things since this is all really just saving typing and source size - there would still be the duplicated functions with just differerent names and key variables.
#### Nominal Animal
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##### Re: Macro functions in headers
« Reply #6 on: December 02, 2021, 08:32:16 am »
Do you really need to define a set of functions to perform the functionality? Why not use
Code: [Select]
#define confSet(key, value) do { nvs_set_str(hconfig, NVS_##key, value); nvs_commit(hconfig); } while (0)instead, in only the header file? This way, there is only one "function" to call. Each confSet() is preprocessed into a pair of calls: one to nvs_set_str(), and the other to nvs_commit(). See how the NVS_ prefix is automagically added to the key? This limits the key namespace to constants and macros that begin with NVS_.
I'm pretty sure confSet(URL, value); is just as easy to use as confSetURL(value); is.
Personally, I'd do the following instead:
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#define confAdd(key, value) nvs_set_str(hconfig, NVS_##key, value)#define confCommit() nvs_commit(hconfig)so that in your code, you have one or more confAdd() lines, followed by a confCommit(). For example,
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confAdd(URL, CFG_KEY_URL); confAdd(FOO, CFG_KEY_FOO); confAdd(BAR, CFG_KEY_BAR); confCommit();instead. You then put say confAdd(URL, whatever); confAdd(FOO, foovalue); confAdd(BAR, barvalue); confCommit(); in the code.
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #7 on: December 02, 2021, 08:38:34 am »
Put all the calls to confSetStr() in a .inc file (name doesn't matter, it's just a convention) and then #include it into both the header and the C file, with different definitions of the macro in each (and #undef it after)
As I do here:
https://github.com/brucehoult/trv
That certainly has potential, thanks
#### Nominal Animal
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##### Re: Macro functions in headers
« Reply #8 on: December 02, 2021, 08:39:52 am »
The calling code would need to know of, and be calling, nvs_set_str(), and know of hconfig, etc.
To hide those, expose the common call in the header file,
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void confSet(int key, const char *val);You do need to expose the NVS_ constants somehow – but they don't need to have the same names, just the corresponding integer values. So, in addition to the above, the header file does need
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#define CONFSET_URL the_same_value_as_NVS_URL_hasand so on.
This way the "user" code can do confSet(CONFSET_URL, the_url); .
Then, in the internal implementation that does have access to hconfig etc, define
Code: [Select]
void confSet(int key, const char *val){ nvs_set_str(hconfig, key, str); nvs_commit(hconfig);}This one must have global linkage (not be static), so that it can be called from the "user" code.
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #9 on: December 02, 2021, 08:43:59 am »
Quote
Do you really need to define a set of functions to perform the functionality?
No, however it can be done opaquely is OK. But there are things internal to the config module that shouldn't be visible outside (hconfig, for instance, and the commit).
Edit: sorry, you overtook me as I was typing
Quote
You do need to expose the NVS_ constants somehow – but they don't need to have the same names, just the corresponding integer values.
Yes. Hmmm. Well this is another aspect which I haven't actually decided on yet, whether discrete functions for the data is sensible or should there be a single function and an ID key. On a small scale the discrete functions are nice because the prototype can be commented, whereas using an ID it is a bit trickier because it covers many different issues. However, if there were, say, 50 of the things it's wasting code space.
« Last Edit: December 02, 2021, 08:50:39 am by dunkemhigh »
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##### Re: Macro functions in headers
« Reply #10 on: December 02, 2021, 10:18:48 am »
maybe something like the last answer here?
I couldn't see how that was relevant, sorry.
put the functions in the header file so you don't need the prototypes and use __attribute__((weak)) so you only get one copy of each function
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #11 on: December 02, 2021, 10:48:08 am »
Ah, OK. As you've no doubt seen since, I was reluctant to put the function body in the header. Bu thanks for the clarification
#### SiliconWizard
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##### Re: Macro functions in headers
« Reply #12 on: December 02, 2021, 07:05:27 pm »
Quote
- Or you could define the functions themselves inside the header file.
I am reluctant to do that because of potential coupling. The calling code would need to know of, and be calling, nvs_set_str(), and know of hconfig, etc.
Unless your concern here is to "hide" this particular information from the caller, it's just a matter of including the required corresponding prototypes and external definitions, in another header file, inside this one header file.
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #13 on: December 02, 2021, 10:06:57 pm »
Sure, but it's like leaving live wires hanging from the roof - of course you're not going to grab them so they're not a problem. But we wouldn't be seen dead (sorry!) doing it.
#### Nominal Animal
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##### Re: Macro functions in headers
« Reply #14 on: December 02, 2021, 11:22:36 pm »
If you go by the numeric key route, here is one "trick" that might be useful.
Instead of declaring the keys only as preprocessor macros, you declare them as enums instead:
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enum { CONF_URL = 1, /*!< Main configuration URL string */#define CONF_URL CONF_URL CONF_FOO, /* !< Some feature named FOO */#define CONF_FOO CONF_FOO};In C, these (CONF_URL and CONF_FOO) are ints. In C++, you can use enum typename: integer-type { ... }; instead, so that you get a new type named typename that is of integer-type, say uint8_t for example.
Instead of maintaining these in that form, you can make a file in your preferred format with three fields per record: name-suffix, integer-value (or a marker for auto-numbering, like CONF_FOO is above), and comment-description, and then use a script to generate a pair of lines for each. If the header file then contains
Code: [Select]
enum {#include "generated-enums.inc"};with the script output saved to generated-enums.inc, you can include the generation in your Makefile-based build facilities very easily. (Just make generated-enums.inc a prerequisite of the target header file, with the rule running the generator script when necessary. If you ever add proper dependency tracking (make dep) so that all affected files get recompiled when any source file gets modified, Everything Just Works, this way.)
If you want, I can easily construct a small example, but do remember I use Linux. (So state your preferred language for the generator: bash, sh, awk, sed, perl, python. Since the generator is run on the host, it makes sense to use a scripting language instead of C, because when cross-compiling to a different architecture, you may not even have a C toolchain that can generate code to run on the host itself.)
#### Ian.M
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##### Re: Macro functions in headers
« Reply #15 on: December 02, 2021, 11:58:13 pm »
It should be possible to avoid the need for OS specific generator scripts by using X-macros: https://en.wikibooks.org/wiki/C_Programming/Preprocessor_directives_and_macros#X-Macros
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #16 on: December 03, 2021, 12:11:30 am »
Quote
Instead of declaring the keys only as preprocessor macros, you declare them as enums instead:
I always do key-type stuff as typedef enums.
Er, except for these because they're actually strings
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #17 on: December 03, 2021, 12:13:14 am »
Quote
It should be possible to avoid the need for OS specific generator scripts by using X-macros
Thanks. Looks like the thing that Bruce described - not heard the term xmacro before, but it's descriptive.
#### Nominal Animal
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##### Re: Macro functions in headers
« Reply #18 on: December 03, 2021, 01:00:00 am »
Quote
Instead of declaring the keys only as preprocessor macros, you declare them as enums instead:
I always do key-type stuff as typedef enums.
Er, except for these because they're actually strings
That's one use where I do use the above-kind of header-file generation: collecting a set of strings into a hash table (usually using DJB2-xor variant, so very, very fast) of minimal size; one that is mapped and verified at build time.
That way you could only expose int confSet(const char *key, const char *val); but still very efficiently parse the keys. On an ESP32, the run time difference between an integer key and a string key would then be absolutely neglible, and the only downside be the additional ROM/Flash use (for the key strings and associated little bit of data).
I like to do this for (embedded) configuration parsing. Then, the hash table strings are configuration keys, and the values are pointers to the data, and the value-parsing function and its parameters in an union, so that although each configuration value is parsed using a single function, there only needs to be one per parameter type, instead of one per parameter.
The nice thing is that everything is trivially verified to be okay at build time, and the interface is always easy to extend: there is no risk of accidentally changing the keys (as there is when enumerating them), because the old keys always stay as they are. Only the firmware hash table varies, but since it is only needed to accelerate the finding of the matching key, it's perfectly okay.
#### SiliconWizard
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##### Re: Macro functions in headers
« Reply #19 on: December 03, 2021, 01:42:15 am »
I would also do this with a single function. And yes the difference would be absolutely negligible. Unless you were going to call those functions extensively in time-constrained parts of the code, which I doubt here.
#### SiliconWizard
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##### Re: Macro functions in headers
« Reply #20 on: December 03, 2021, 01:48:17 am »
Sure, but it's like leaving live wires hanging from the roof - of course you're not going to grab them so they're not a problem. But we wouldn't be seen dead (sorry!) doing it.
Frankly, unless you're writing an API for others to use, or are expecting this piece of code to be maintained by many developers over the next decades, it won't make a difference.
Just because it would be exposing those helper functions to the caller doesn't mean you have to use them. Nothing is hanging. As much as I like hiding and encapsulating, you're probably overthinking it here.
That said, all in all I prefer Nominal Animal's approach for your need here. My proposal is more adapted to functions that do not have external dependencies, and that are possibly called very frequently - and most of all, that potentially all have a different argument list. In your case, it's always the same and only the name of the function differs, which makes Nominal's approach easier and a lot more consistent.
#### brucehoult
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##### Re: Macro functions in headers
« Reply #21 on: December 03, 2021, 02:00:09 am »
Quote
It should be possible to avoid the need for OS specific generator scripts by using X-macros
Thanks. Looks like the thing that Bruce described - not heard the term xmacro before, but it's descriptive.
Ha! Exactly the same, yes.
I didn't know it had a name. Or that someone had done it before (but it's unsurprising).
Preprocessor ab-uuuuuuse
I'm pretty certain the ugly hacky preprocessor is 90% of the reason C beat the Wirthian languages outside of the Unix environment.
80% of early C++ was getting rid of preprocessor hacks, one by one.
#### SiliconWizard
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##### Re: Macro functions in headers
« Reply #22 on: December 03, 2021, 02:48:04 am »
80% of early C++ was getting rid of preprocessor hacks, one by one.
And then replacing it with even uglier stuff.
Of course that's a bit tongue-in-cheek, but point is, many languages with a C root that have tried getting rid of the preprocessor have come up with a bunch of stuff that, when it eventually gets to the point of being as flexible as the original preprocessor, becomes horribly convoluted.
« Last Edit: December 03, 2021, 02:50:05 am by SiliconWizard »
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #23 on: December 03, 2021, 12:40:50 pm »
Quote
I like to do this for (embedded) configuration parsing. Then, the hash table strings are configuration keys, and the values are pointers to the data, and the value-parsing function and its parameters in an union, so that although each configuration value is parsed using a single function, there only needs to be one per parameter type, instead of one per parameter.
I might be drifting towards this kind of thing. But probably I think I will more likely try each approach and see how they turn out.
#### dunkemhigh
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##### Re: Macro functions in headers
« Reply #24 on: December 03, 2021, 12:44:32 pm »
Quote
Unless you were going to call those functions extensively in time-constrained parts of the code, which I doubt here
If there is any constraint it would be towards code size. They would typically be called at init, and if a time-sensitive section needed repetitive access it would cache a copy.
Smf | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3369591534137726, "perplexity": 6411.4973437875315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00595.warc.gz"} |
https://chemistry.stackexchange.com/questions/43122/desorption-instead-of-adsorption-from-solution-using-activated-charcoal | I was just trying to see the effect of concentration on adsorption in solutions, i.e. I was trying to verify the Freundlich isotherm. I used oxalic acid and a batch of homemade activated charcoal (using this site).
First I prepared solutions of different concentrations. Then I added the activated charcoal, stirred it for 5 minutes and filtered the solution. When I titrated the filtrate using KMnO4, after calculating the concentration, I found that the concentration of the solution had been increased!
I tried the same process with different concentrations of oxalic acid solution and the result was the same. Can anyone tell me where did I go wrong?
Note: I used the same batch of activated charcoal in all the experiments. (Even then, how did the concentration of the solution increase? Shouldn't it be the same or lower than the initial concentration?)
For titrating oxalic acid, I heated the solution to about 60 degrees C.( After filtering the solution) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8846035003662109, "perplexity": 912.4221599234276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000545.97/warc/CC-MAIN-20190626194744-20190626220744-00264.warc.gz"} |
https://cseducators.stackexchange.com/questions/7205/how-to-respond-to-the-two-handed-approach-to-swapping/7218 | # How to respond to the two-handed approach to swapping
I was introducing lists to someone young (in Scratch). Creating a list is boring, so I opted to introduce something that you can do with lists, but requires some thinking: swapping. I introduced it by putting out four pencils in a row, like this. I asked them to swap the second and fourth items in the array (in reality).
| | | |
They took both their hands, picked up the second and fourth pencils (each in one hand), and swapped them! That works in reality... but isn't right in computer science. So, I told them, try doing it with only one hand. Predictably, they couldn't. I introduced a temporary spot, where they could put one pencil. I was hoping they would do something like this:
Second Item -> Temp spot
Fourth Item -> Second spot
Temp spot -> Fourth spot
They figured it out. But they asked me something like this: "Why can't I use two hands?". I wasn't sure how to answer that. The obvious answer is... the computer only has one hand (this was Scratch, so multithreading didn't exist). But... I'm hesitant to say that for two reasons: (1) in Python, the "two-hands" approach works (a, b = b, a, see this Stack Overflow question) and (2) multithreading exists, and there are some (obscure) ways to swap two variables without a temporary variable. So, how should I answer the question "why can't I use two hands"?
Remember, they are just learning lists, so explaining something like multithreading would be too difficult.
What's a good way to answer that question, without confusing them?
• Comments are not for extended discussion; this conversation has been moved to chat.
– Ben I.
Jan 7 at 22:58
Don't overcomplicate things. Your first instinct, that the computer only has one hand, was the correct one.
This is a regular problem that new teachers have, and it sometimes takes a few years of frustration to burn out the idea that everything you say must be formally true. If we say something that is untrue, the reasoning goes, the students will remember it and use that wrong idea to badly misunderstand things later.
Let me give you a slightly different mental model of learning, and you'll find all of the support you need in the absolutely fantastic book "How People Learn II", which is a big volume that describes everything we really know from research about how learning actually works in the brain.
We (1) build mental models by (2) connecting new information (3) to things that we already know. Furthermore, (4) abstraction is the brain-process of finding the similarity in (5) similar mental models. (6) In order to build upon prior knowledge, (7) it must already be fairly well understood. If it's not, and you simply don't have the prerequisite knowledge, you are unlikely to build any mental model at all.
(That should make some sense; if it's not already fairly concrete within your mind, it's too nebulous to glom new ideas onto.)
So, with that as a model of learning, there are a few implications:
1. We need to help students build small, clear mental models. They can then use these in the future to build further, richer models. This lends itself very nicely to the KISS principle. Don't try to introduce too many variations to the idea at once. Leave the complicating for the next step, or you won't obtain any model at all.
2. Before there is already a fairly clear mental model, learning isn't really taking place, so little of what is said will be directly remembered. Corollary to this is that the danger of them being confused by some overly literal interpretation of what you said while explaining how to swap in Scratch once they encounter threading later down the line is completely negligible. They won't remember much of the explanation until they can build a mental model, so they won't remember those kinds of details. In addition, there are simply too many cognitive models that have to be built between swap in Scratch and multithreading for one to hamper the other.
Don't let perfect be the enemy of progress. Keep it simple and focused on the small concept they are learning right now, and be comfortable that that will support them best as they grow and develop and complexify ideas later.
• Let me add that to justify using the simple (sequential, explicit temp) model, ask the students to think about all the ways the two handed model could go wrong. Especially how the simple model doesn't have those defects. Jan 4 at 20:29
• It would also be fine to add the caveat that "right now in our example, the computer only has one hand." This would be less technically incorrect without distracting the student too much, I think. Jan 6 at 20:42
• @2rs2ts Sure! But realize that you do that basically for your own benefit. It won't help the student, and might, by admitting that more complex layers exist, hinder them in their current struggle. But I don't want to sound too negative; I play those sorts of word games every day, and could scarcely stop it if I tried. After all, it bothers me to say it wrong. I just have to be cautious about how much additional complexity I allow into a model that the student already isn't getting (ie they're already at the edge of their cognitive limits). In those moments, when in doubt, leave it out.
– Ben I.
Jan 6 at 22:32
• One name for the "small, clear mental model" you mentioned is Notional Machine, as someone else commented. This idea is why I often say that we have to teach people how computers actually work. I say: computers can only do 5 things - fetch, store, arithmetic, compare and jump. With that in mind, students are more likely to understand programming than if we start with something other than a Notional Machine. Jan 7 at 2:04
### Re-evaluate what you think is "right", because you aren't
but isn't right in computer science
This is simply incorrect. Your student can create two variables called LeftHand and RightHand. Copy value "red" from position 1 to LeftHand, copy value "green" from position 4 to RightHand, copy values back to positions 4 and 1 respectively. Job done.
Is this optimal? No - but you didn't ask for optimal, you asked for a solution which works. Your student gave you a working solution, and you've rejected it because of your own preconceptions of how many variables they "should" use, or how many steps it "should" take. Neither of those preconceptions are correct. There's no time or RAM pressures here, so the student is perfectly free to do it their way. (Edit following your comments elsewhere: Nor do they have to consider concurrency where someone else could move the pencils in between.)
And not only are they free to do it this way - if they find it easier to understand then they should. My background is in safety-related software. Coding standards in safety-related software prioritise readability, maintainability and ease of understanding by other coders, ahead of pure speed, because the most intractable bugs almost always come from people trying to be too clever. If you keep the concepts simple, you don't make mistakes and your code is more likely to work first time.
In short, you are committing a cardinal mistake in software engineering, which is premature optimisation (which as every software engineer should know is the root of all evil). Your student is not making that mistake, and that currently makes them a better software engineer than you!
Certainly after they've got a two-handed solution, you can have them do a one-handed version with a single slot. You can then show them that it takes fewer steps and one less variable. The answer to "why can't we...?" is simply "you can if you want, but look, it's more work". Most people would rather do less work, so it's a win all round.
• Comments are not for extended discussion; this conversation has been moved to chat.
– Ben I.
Jan 7 at 22:58
• I've added an answer that shows your LeftHand, RightHand version as actually better — for one extant architecture — than the OP's : cseducators.stackexchange.com/a/7223/8837
– Rusi
Jan 9 at 5:55
• The gcc compiler happily turns the "standard one temp variable" in C into a 2 registers "both hands" versions. Probably because it is more efficient (read-write, cache...) Jan 13 at 21:27
• @Michelbillaud That's really worth an answer! Hard data that the less efficient notionally is the more efficient actually. Just (1) input program (2) output cut down to relevant parts (3) gcc flags if any (4) gcc version just in case
– Rusi
Jan 18 at 4:09
• Well compiling this with gcc -03 -S void swap(int *a, int *b) { int c = *a; *a = *b; *b = c; } produces this assembler code movl (%rdi), %eax ; movl (%rsi), %edx ; movl %edx, (%rdi) ; movl %eax, (%rsi) ; ret Jan 21 at 9:30
Your problem is that you tried to let your student "program" without establishing a language. Thus they were free to write their own "language" in which you can do two things at once because you have two hands. If you'd asked them to rotate 3 objects, they would have picked up the third pencil in their mouth or so.
You could have asked them "suppose you're a crane driver" then, because they obviously have only one crane, they would have solved the problem "correctly".
May this also be a lesson to you to think very hard about everything you ask your students. Don't improvise. Or just learn for next time.
• I do like your crane idea, and in hindsight, I should have done that... however, how do you come up with these ideas for alternate ways to phrase problems to get students to think about it in a certain way or come to a specific answer? Jan 4 at 23:27
• I actually ran into a very similar problem, but it was about communcating processes. After a student uses the two-handed solution, I now tell them, "you have only one communication channel, so put your left hand in your pocket, and use your right hand". So I had to learn the hard way, like you. Jan 5 at 1:41
• @cocomac -- apart from the obvious canned answer of "practice will do it" that, although true, doesn't help you up front :-), mostly just by going the opposite direction. To think about it first, like "I don't want them to use both hands, so I need to come up with some setup that makes it one hand only, so I need an interface between two-handed people and one-handed manipulator, let me see". Jan 5 at 11:23
• @computercarguy And I'm sure there are people who can juggle with one hand, so they could still do the swap without a temporary variable. Analogies are always imprecise. Jan 5 at 20:05
• @VictorEijkhout, juggling usually means you can handle more than one object with one hand. I've juggled for over 35 years and can do 3 balls in one hand, for example. So yes, the analogy is imprecise, but that doesn't mean it can't still work. I said there was an issue, not a problem. I should have been more precise that it wasn't a critical reason not to use the analogy. This issue with the analogy can easily be avoided by specifying the language, as you say, by not allowing double-picking. We just need to know how to specify the language to avoid these issues, that's all. Jan 5 at 20:23
This question reminds me of when a student failed to understand why a third variable was needed to exchange the value of two variables (i.e. to swap A and B, C=A, A=B, B=C). I managed to illustrate this to them by placing an item (mobile phone) in each of their hands, and asking them to swap the items between the hands. On attempting to do this, they immediately realised they needed a third location (the table) to achieve this.
You may find your list-swapping activity works better if you place the pencils in students' hands. This will help reinforce the idea that each pencil needs a location (hand) to be stored in at all times. You'll obviously need more than one student to make a list that's larger than two pencils, and pandemic-related distancing may be an obstacle.
I'd suggest trying this activity with two or three students sat in a row, and using an odd number of pencils. This will ensure that there is only one hand available for moving pencils about at any time, without appearing to be an arbitrary rule that you've introduced.
• Neat! Hand not the mover (verb) but the place (noun) 👍
– Rusi
Jan 7 at 17:43
They took both their hands, picked up the second and fourth pencils (each in one hand), and swapped them! That works in reality... but isn't right in computer science.
It depends on your paradigm. You are teaching the swap algorithm you know, but haven't explained to the students (or at least in the question) what things are. Here is a perfectly valid way to look at the operation, using two temp variables, 'Left Hand' and 'Right Hand':
Second Item -> Left Hand
Fourth Item -> Right Hand
Left Hand -> Fourth spot
Right Hand -> Second spot
In fact many students will do it this way (with two temp variables) when they attempt to do swapping for the first time, I'm pretty sure I did. Some languages have swap operations built-in so you don't require a temporary variable.
In this particular case I think it would make more sense to say that you can only move one pencil at a time rather than you have only one hand. After all I could pick up the pencil in spot two and place it in slot four while picking up the pencil there at the same time with one hand.
The analogy breaks down in other ways, for instance normally you make a copy of the item in the temp variable and it exists in two places until you overwrite it in the array. I think using things like colored blocks here and making copies into variables would be a better way to visualize what is actually happening too...
In thinking about this, and especially the answer of Graham, I think that what we really do in the standard three step process is a two handed swap without thinking of it. So, it is a failure of the metaphor, actually. So, here is a reframing of the standard process:
Pick up one stick with the left hand (i.e. temp is associated with the left hand). THEN, move the other stick with the right hand (the assignment). THEN put the stick in the left hand into the proper slot.
Or:
Second Item -> Left Hand Fourth Item -> Second spot // using Right Hand Left Hand -> Fourth spot
All you need to explain is that (with a single processor) only one thing can be done at a time and so the swap-with-two-hands has to be sequenced in some way rather than happening simultaneously.
This is to avoid issues in the more general case where you want to swap(m,n) where m and n could be the same. Both hands can't pick up the same stick simultaneously if it has to be sequenced. The bright kids will want to complain that you can't pick up a stick from an empty slot, but the small variation isn't too hard to explain if you think of it beforehand. The really astute ones will notice that assignment is copying, so "picking up one stick" in the program doesn't really leave that slot empty, as it does with physical objects.
(ASIDE: there is a way to avoid that physical/virtual disconnect by using a reference (rather than copy) metaphor of variables, but it is likely not where you want to go with youngsters unless you do it ubiquitously. In essence, a variable is a reference to a value not a box that "holds" the value.)
It also avoids discussions of race conditions with multiple processors.
• Why does "single processor" keep getting brought up? How would you sequence operations by 2 cores to get a swap done correctly? The relevant thing here is a scalar CPU, as opposed to modern superscalar CPU cores that literally can execute two independent load instructions in the same clock cycle. (e.g. AMD since K8, Intel since Sandybridge: realworldtech.com/sandy-bridge/7). Intel since Ice Lake can even execute two store-data uops in the same clock cycle, although commit from store-buffer to L1d cache can only commit two stores in 1 cycle if they're to the same line. Jan 6 at 16:19
• As you say, a normal swap is 2 loads then 2 stores (assuming an optimizing compiler for an AoT compiled language for a register machine), whether we name one or both temporaries. (Or in Python, neither). For (much) more advanced students of CPU architecture, this might be a good illustration of hazards like write-after-read when finding parallelism between operations: you have to finish picking up an object (loading into regs) before a store of the other object writes that location. Jan 6 at 16:27
• @PeterCordes "Why does "single processor" keep getting brought up?" Not because using multiple processors or multiple threads would enable a different solution, but because if you are explaining to a student why they have to solve the problem with one hand then the analogy is justified by the fact that their program can "only do one thing at a time". Jan 6 at 17:10
• @kaya3: Thanks, you're right, I was thinking in terms of finding instruction-level parallelism within instructions for a sequential model of execution, not the need for there to be a sequence in the first place. I think I'd find it less weird to say the code or program is single-threaded, rather than to say there's only one processor to execute it on. As you rightly pointed out, the issue here is the execution model of the programming language, not the CPU details. Jan 6 at 18:17
• @PeterCordes Because if you ask a centipede how it knows which leg to move next, it gets in a hopeless tangle. (That's just a metaphor, of course.) Jan 7 at 2:19
I do the swap example pretending to have two largish (2-hands-to-lift) boxes side-by-side on a desk (with only enough room for two boxes). After we lift the first we have to look for a place to put it, which feels like creating the temp variable.
But I don't think swap is an especially good example with lists. It's easier to explain swapping just any two variables, x and y. Lists are about using indexes. In Scratch we don't get as much fun with loops, but can do 2 step find-the-index-first stuff like "turn the first 9 into a 6" or "put 12's before and after the first 3".
As far as "computer rules" I like a lighter touch. Swap is one of many clever tricks which happen to work, for now. Motivate the problem with how switching x and y starting with "set x to y" permanently erases x. Then explain how that 3-step dance is actually the standard swap trick. We could have had a built-in swap command but everyone was all "no, everyone knows the trick and it's just one extra step".
Esp. in Scratch you're hoping they like it and move onto a real language. At that point you're going to explain how "Scratch rules" like pre-making variables aren't computer rules at all. But we learned lots of concepts which mostly transfer over with minor changes. In the end: switching values of two variables is a thing; everyone calls it a swap; and it's good to know the trick just-in-case there's no built-in way.
• well... to be fair you do need to "pre-make" variables in most languages... Jan 5 at 20:43
• @somebody My understanding is the Scratch language has no declarations -- variables are created manually through the GUI. I assume a Scratch user learning a text-based language would be surprised at something like int x=4. But I've only seen Scratch a few times, is there a better example? Jan 5 at 23:27
• well. while that's true i'd argue that scratch variables are all global, and global variable declarations in other languages are little different from that of scratch - you still "create variables manually". to look at it another way, in scratch, not just variables but everything is created "through the GUI", no? (equivalently, scratch has no (official) textual syntax at all). but either way the point is, imo "pre-making variables" (and scratch in general) is little different from the so-called "computer rules" Jan 5 at 23:47
• tl;dr imo how variables are created in scratch are mostly a side effect of being a graphical (?) language rather than it being fundamentally different from other lanugages Jan 5 at 23:48
Analogies & metaphors are imperfect, so let's share the expectation that they are just illustrative.
Rules of what is possible (and not) are very important, and we can compare the rules of pencils moved with hands and what computers can do — but we need to define both, in part so the rules for the metaphor can be compared with the those of a computer.
These rules can be reasonably terse and simple to state.
The computer can copy a value from one storage location to another storage location. (Computers generally copy data rather than moving it.)
Your hand can simulate action of making a copy, and whether the hand itself represent another storage location that can be utilized while doing something else is a matter of definition and explanation of rules.
Since we're talking about information rather than physical items, I might prefer a metaphor of a spreadsheet: swapping two cells in a spreadsheet generally requires another free cell; a pencil and piece of paper with some boxes holding symbols might also work, you need to stipulate whether human memory is outside of the metaphor (i.e. not be used for storage).
Programming languages support constructs & expressions that require additional implicit storage, and language implementations introduced temporaries for these purposes: Python's a,b=b,a is such a construct, while any language's general purpose expressions is another: you can't really compute a*b+c*d without holding the result of one of the multiplications somewhere (on paper, in your head, in storage...), while doing the other.
• "a * b + c * d" : lovely! This insight is at the heart of functional programming : elide useless names. The more technical name for that is combinatory logic. (The proper name "Haskell" is the name of its second founder)
– Rusi
Jan 6 at 3:45
• @Rusi Are there any Functional Machines? Computers that execute your high level idea of the calculation and not some lower level translation? I would like to see such a machine. I would also like to get to work using a map instead of a car, but you can't have everything. (Where would you put it?) Brains really do seem to implement a Neural Network, but computers not so much. Some day perhaps. Jan 7 at 2:11
• @Scott: Lisp machine, Scheme88. In fact Data flow architectures were actually made for improving efficiency of von Neumann languages eg SISAL. In fact among theoretical models, lambda calculus is more realiable than Turing machines because the latter has an infinite sub part -- the tape. How the F does an engineer make an infinite object?! Nothing infinite about the lambda calculus. Evolution...... is slow
– Rusi
Jan 7 at 2:51
• @ScottRowe All that FP does is generalize "a * b + c * d" from numbers to arbitrary memory data structures. Heck even python Java C# etc -- the whole current mainstream gang -- will allow you to handle complex data structures, manage ing the memory with gc etc. Only difference is the modern crop mess up memory semantics. FPLs don't. Java, C#, JS are identical modulo syntax and minor details.
– Rusi
Jan 7 at 3:23
• Of course IF you're prepared to go outside the mainstream there are heroic exceptions. Eg Rust which strives for C-level efficiency, memory-footprint. Plus almost haskellish purity. And is even highly portable (Compare to C's autoconf!!!). IF... Are you?? (In 1987 I pioneered and initiated C in edu. It was as new and uncharted then as it's passe now. Today I would use rust for that space (if I were younger... If...))"
– Rusi
Jan 7 at 3:42
Ben's answer is great, but allow me to suggest that you don't need to "respond" to the two-handed approach at all: just say when you're explaining the problem in the first place that they can only use one hand and they can only hold one item at a time.
Students are less likely to need a justification this way, because you haven't changed the rules to forbid a solution after they've come up with it themselves. If a justification is required, the question is not "why can't I use two hands?", but rather "why is it useful to know how to do this with one hand?". And the answer to that is that the solution using one hand can be translated directly into code, whereas the solution with two hands can't be.
• Except you perfectly well can translate the solution with two hands into code. Not picking up both simultaneously, sure - at least on most systems - but one at a time works fine. Jan 5 at 13:36
• except you cannot just let it slide without explanation - it's extremely important to make sure the student is understanding it how you think they are. so you'd at the very least have to explain (and get the student to agree) that they are describing a 4 step process left = a; right = b; a = right; b = left rather than a 1 step process a, b = b, a. the problem with this is that it runs the same risk of the student asking why the process must be split into those steps. Jan 5 at 20:49
• @Graham By "the solution with two hands" I refer to the one in the OP where a student picks up both items with both hands at the same time. If you interpret hands as variables rather than as the computer's action of moving (or really, copying) a value from one variable to another, then there are other two-handed solutions, but "the" two-handed solution still doesn't work. As for "on some systems", we are talking about whatever language the students are learning, not the capabilities of all programming languages. Jan 5 at 22:51
• Maybe someday we will build computers that can do many things at once. Quantum, massively parallel, whatever. Then we will not be able to teach anyone how to program them, but we won't have to. Jan 7 at 2:29
• @ScottRowe I don't know what future you're imagining where nobody is a computer programmer, but I want no part of it - I like programming! Jan 7 at 6:03
Graham pointed out that the assumtions of the question are wrong. And Peter Cordes elaborated a bit in the comments.
Considering a target VLIW (Very Long Instruction Word) architecture can help sharpen this argument further.
The basic idea of VLIW is that a Very Long Instruction "horizontally" packs a number of instructions per single VL-instruction, all of which are executed in parallel in one machine cycle. The requirement is that the multiple instructions in a VL-instruction need to access disjoint memory/registers.
In terms of more popular architectures, VLIW is:
• Like pipelined superscalar architectures except that the compiler rather than the hardware does parallelization scheduling
• Like RISC in that the complexity is shifted to the compiler
So
x = 1
y = 2
can be packed as one VL-instruction
x = 1 // y = 2
Whereas
x = 1
y = x + 1
can't because x overlaps.
(Using the notation that multiple VL instructions are on multiple lines and "subinstructions" packed into a VL instruction are on one line "//" separated).
With that background we can write swap as
lh = x // rh = y
x = rh // y = lh
t = x
x = y
y = t
is interlocked at each stage and so is hopelessly sequential.
## tl;dr
The VLIW oriented swap is 33% faster — 2 vs 3 — than the standard imperative one. When it is naively sequentialized it's 25% slower — 4 vs 3.
## Note
The two-hand analogy is right in being more efficient than with one hand. However it's not so right in this that it seems to be able to do it in one single 2-handed swoop. Whereas (for VLIW at least) it's two 2-handed swoops. Our hands have enough muscle intelligence to criss-cross without colliding whereas data-buses dynamically hop-skipping over each other is.. uh... still futuristic sci-fi😂.
IOW analogies/metaphors may help but don't go all the way.
• Premature optimization is... Oh, you already heard that one. Could we venture to teach beginning students something without having to teach them everything? I had a great welding instructor, he could cover one topic or technique each class night. I still think of him as one of the best teachers I had, and still smile when I recall him whistling during class as he thought about going out with his wife afterwards. Jan 13 at 1:40
• 👍 @ScottRowe. In upholding the "Premature..." proverb you'll find me more extreme left (or is it right?) than anyone. The point of this post was.... O well... May the blessing of Fermat be on these comment columns...
– Rusi
Jan 13 at 2:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4709716737270355, "perplexity": 1259.7307118307344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571911.5/warc/CC-MAIN-20220813081639-20220813111639-00192.warc.gz"} |
https://www.cut-the-knot.org/m/Geometry/CirclesAtVertices.shtml | # Lines through Circles at Vertices of Equilateral Triangle
### Problem
Circles $(A),$ $(B),$ $(C)$ centered at the vertices of an equilateral triangle $ABC$ with side length $4$ all have radius $\sqrt{3}.$
Describe all straight lines that have common points with all three circles.
The purpose of this writeup is to establish the following result:
Circles $(A),$ $(B),$ $(C)$ centered at the vertices of an equilateral triangle $ABC$ with side length $4,$ all have radius $\sqrt{3}.$
The only straight lines that have common points with all three circles are their common tangents, of which there are three.
### Lemma
The midlines of $\Delta ABC$ are tangent to all three circles $(A),$ $(B),$ $(C).$
For a proof, draw the common tangent of circles $(B)$ and $(C)$ on the side of $(A)$ ($EF$ in the diagram.) Let $D$ be the point of tangency on $(B)$ and $E$ the intersection of that line with $AB.$ $\Delta BDE$ is right with $\angle DBE=30^{\circ}.$ Since $BD=\sqrt{3},$ it follows that $BE=2,$ implying that $E$ is the midpoint of $AB.$
Points $F$ and $G,$ and lines $EG,$ $FG$ are constructed similarly and, by symmetry, are shown to be the common tangents to all three circles.
### Solution
The proof is based on the assertion that a straight line that crosses the interior of $\Delta EG$ may at best have common points with only two of the three circles. This is because, such a line, may have common points with, say, $(A)$ only if it crosses the interior of side $EF.$ On the other hand, a straight line that pass through the interior of a triangle but not through a vertex crosses exactly two of the sides of the triangle. If a line passes through a vertex then it crosses the interior of the opposite side.
More transparently, a straight line that has no common points with $\Delta ABC$ may, at best, have common points with two of the circles. For example, consider a line that crosses $(A).$ If it is parallel to $EF$ it is certainly does not meet either of the remaining circles. Otherwise, it crosses line $EF$ just once outside the segment $EF$ in which case, too, it has common points with at most one of the remaining circles.
In conclusion, the only straight lines that have common points with all three circles are the midlines $EF$, $EG,$ and $FG.$
### Story
The following story could be found in the book 60-odd YEARS of MOSCOW MATHEMATICAL OLYMPIADS now available online.
At the 8-th Olympiad (1945). The organizers believed that the following Problem for grade 8 was relatively easy.
Vertices $A,$ $B,$ and $C$ of triangle $ABC$ are connected with points $A',$ $B',$ and $C'$ lying on the opposite sides, but not in the vertices, as below:
Prove that the midpoints of segments $AA',$ $BB',$ and $CC'$ do not lie on the same straight line.
Indeed, the midpoints $A_{1},$ $B_{1},$ and $C_{1}$ in question lie sides of the medial triangle of $\Delta ABC,$ one per side. Hence, the assertion required, since no line not passing through the vertices of a triangle can cross all the sides in their inner points.
The organizers believed the statement is obvious. However, a participant, Yulik Dobrushin from the 8-th grade (now the world-famous mathematician, Roland Lvovich Dobrushin, Dr.Sc.), reached this stage in the solution and added: "For a long time I have tried to prove that a straight line cannot cross all three sides of a triangle at their inner points but failed to do so. I am horrified to realize that I do not know what a straight line is!"
Dobrushin was crowned with the first prize for this frank recognition of his failure. The members of the organizing committee might have understood the meaning of Dobrushin's phrase better than its author himself. The point is that in modern geometry the answer to the question what is a straight line is given only by listing the line's properties among which the impossibility to cross all three sides of a triangle (or an equivalent property) is usually included.
We'll look into that next.
### Basics
II, 4. $A,$ $B,$ $C$ be three points that do not lie on a line and let $a$ be a line in the plane $ABC$ which does not meet any of the points $A,$ $B,$ $C.$ If the line $a$ passes through a point of the segment $AB,$ it also passes through a point of the segment $AC,$ or through a point of the segment $BC.$
Expressed intuitively, if a line enters the interior of a triangle, it also leaves it. The fact that both segments $AC$ and $BC$ are not intersected by the line $a$ can be proved. (see Supplement I,1.)
From Supplement I,1.
If the line $a$ met the segments $BC,$ $CA,$ $AB$ at points $D,$ $E,$ $F$ then these points wold be distinct. By Theorem 4 one of these points would lie between the other two
If, say, $D$ lay between $E$ and $F,$ then an application of Axiom II,4 to the triangle $AEF$ and the line $BC$ would show that this line would have to pass through a point on segment $AE$ or on segment $AF.$ In both cases a contradiction of Axiom II,3 or Axiom I,2 would result.
References
1. D. Hilbert, Foundations of Geometry, Open Court, 1999
### Acknowledgment
This problem has been posted by Leo Giugiuc (Romania) at the CutTheKnotMath facebook page. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6184404492378235, "perplexity": 230.94734393877044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742020.26/warc/CC-MAIN-20181114125234-20181114151234-00487.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs10856-009-3732-2 | Journal of Materials Science: Materials in Medicine
, Volume 20, Issue 8, pp 1611–1618
# The effect of phosphate content on the bioactivity of soda-lime-phosphosilicate glasses
• M. D. O’Donnell
• S. J. Watts
• R. G. Hill
• R. V. Law
Article
DOI: 10.1007/s10856-009-3732-2
O’Donnell, M.D., Watts, S.J., Hill, R.G. et al. J Mater Sci: Mater Med (2009) 20: 1611. doi:10.1007/s10856-009-3732-2
## Abstract
We report on the bioactivity of two series of glasses in the SiO2–Na2O–CaO–P2O5 system after immersion in simulated body fluid (SBF) after 21 days. The effect of P2O5 content was examined for compositions containing 0–9.25 mol.% phosphate. Both series of glasses degraded to basic pH, but the solutions tended towards to neutrality with increasing phosphate content; a result of the acidic phosphate buffering the effect of the alkali metal and alkaline earth ions on degradation. Bioactivity was assessed by the appearance of features in the X-ray diffraction (XRD) traces and Fourier transform infrared (FTIR) spectra consistent with crystalline hydroxyl-carbonate-apatite (HCAp): such as the appearance of the (002) Bragg reflection in XRD and splitting of the P–O stretching vibration around 550 cm−1 in the FTIR respectively. All glasses formed HCAp in SBF over the time periods studied and the time for formation of this crystalline phase occurred more rapidly in both series as the phosphate contents were increased. For P2O5 content >3 mol.% both series exhibited highly crystalline apatite by 16 h immersion in SBF. This indicates that in the compositions studied, phosphate content is more important for bioactivity than network connectivity (NC) of the silicate phase and compositions showing rapid apatite formation are presented, superior to 45S5 Bioglass® which was tested under identical conditions for comparison.
## Authors and Affiliations
• M. D. O’Donnell
• 1
• 2
• 3
• S. J. Watts
• 1
• R. G. Hill
• 1
• R. V. Law
• 2
1. 1.Department of MaterialsImperial College LondonLondonUK
2. 2.Department of ChemistryImperial College LondonLondonUK
3. 3.BioCeramic Therapeutics Ltd.LondonUK | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8685997128486633, "perplexity": 11077.57946625665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661449.76/warc/CC-MAIN-20160924173741-00158-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/586272/mathematical-definition-of-exponent | # Mathematical definition of exponent [duplicate]
What does it mean by a^b in real number system? How is it defined mathematically?
It is clear in case of exponent being an integer.
i.e., a real number a is multiplied b times where b belongs to Z
If b is a rational..say b=p/q, then a^b can be interpreted as a^(1/q) multiplied p times.
But how is it defined when b is irrational?
-
## marked as duplicate by T. Bongers, Daniel Fischer♦, apnorton, ncmathsadist, leoNov 30 '13 at 4:01
You use the density of rationals in $\Bbb R$.Find a sequence $x_n\to \sqrt 2$ and $x_n<\sqrt 2$ for every $n$. Same for a sequence $y_n\to \sqrt 2$ with $y_n>\sqrt 2$. Then ...? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9650431275367737, "perplexity": 598.3228341441762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737931434.67/warc/CC-MAIN-20151001221851-00013-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://rd.springer.com/article/10.1007/s00170-018-2246-9 | # Method for the geometric modeling and rapid prototyping of involute bevel gears
• Ricardo García-García
• Max A. González-Palacios
ORIGINAL ARTICLE
## Abstract
The growth of additive manufacturing technology allows the fabrication of complete functional devices with complex geometries with ease and low cost. This technology allows the fabrication of pieces that could not be made in the past with traditional manufacturing techniques, in this case, bevel gears with exact spherical involute (ESI). Focusing on this issue, this paper presents a method for the geometric design and fabrication of practical industrial-like ESI bevel gears with variable surface detail. The definition of the tooth profiles on back cones is proposed introducing a projection procedure for straight and spiral toothing. An in-house-developed software package was developed to test the method, and a pair of plastic examples were fabricated on a generic 3D printer. A comparison between the fabricated pinion and its STL model was conducted for validation purposes. Mean deviations of $$0.22$$ and $$0.15~mm$$ were obtained for the whole model and for the contact surfaces comparison, respectively. Thus, the fabrication of bevel gears using AMT is achieved, and can be implemented to a specific application using a particular additive manufacturing technique.
## Keywords
Bevel gear Geometric design Exact spherical involute Additive manufacturing
## Notes
### Acknowledgements
The first author acknowledges the support of CONACYT (Consejo Nacional de Ciencia y Tecnología), México. The second, acknowledges the support of the SNI (Sistema Nacional de Investigadores), México.
## References
1. 1.
Al-Daccak MJ, Angeles J, Gonzalez-Palacios MA (1994) Modeling of bevel gears using the exact spherical involute. J Mech Des Trans ASME 116(2):364–368
2. 2.
Berger U (2015) Aspects of accuracy and precision in the additive manufacturing of plastic gears. Virtual Phys Prototyp 10(2):49–57
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Blaszczak M (1999) Professional MFC with Visual C++ 6, 4th edn. Wrox PressGoogle Scholar
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Fuentes-Aznar A, Gonzalez-Perez I (2016) Mathematical definition and computerized modeling of spherical involute and octoidal bevel gears generated by crown gear. Mech Mach Theory 106:94– 114
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Gao W, Zhang Y, Ramanujan D, Ramani K, Chen Y, Williams CB, Wang CC, Shin YC, Zhang S, Zavattieri PD (2015) The status, challenges, and future of additive manufacturing in engineering. Comput Aided Des 69:65–89
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González-Palacios M (2012) Advanced engineering platform for industrial development. J Appl Res Technol 10(3):309–326Google Scholar
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González-Palacios M A (1995) An algorithm for the synthesis of bevel gears. In: Unicopli E (ed) Ninth World Congress on the Theory of Machines and Mechanisms IFToMM. Politecnico Di Milano, Italy, pp 570–574Google Scholar
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Litvin FL, Rahman P, Goldrich RN (1982) Mathematical models for the synthesis and optimization of spiral bevel gear tooth surfaces. Contractor Report 3553, NASAGoogle Scholar
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Park NG, Lee HW (2011) The spherical involute bevel gear: its geometry, kinematic behavior and standardization. J Mech Sci Technol 25(4):1023–1034
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Shreiner D, Sellers G, Kessenich J, Licea-Kane B (2013) OpenGL programming guide: the official guide to learning OpenGL, version 4.3. Addison-WesleyGoogle Scholar
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Stock Drive Products / Sterling Instrument (2010) Elements of metric gear technology. Technical section. Catalog D805. http://sdp-si.com/D805/D805_PDFS/sections/technical.pdf
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Traeger MF, Krieger YS, Lueth TC (2013) Automated construction of gear racks, spur gears and helical gears using matlab & stl files for rapid manufacturing. In: 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics: mechatronics for human wellbeing, AIM 2013. IEEE, pp 1603–1608Google Scholar
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Tsai Y, Chin P (1987) Surface geometry of straight and spiral bevel gears. J Mech Transmiss Autom Des 109(4):443–449
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Uicker JJ, Pennock GR, Shigley JE (2017) Theory of machines and mechanisms, 5th edn. Oxford University Press, New York, pp 436–439Google Scholar
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Wang J, Kong L, Liu B, Hu X, Yu X, Kong W (2014) The mathematical model of spiral bevel gears—a review. Strojniški vestnik-J Mech Eng 60(2):93–105
© Springer-Verlag London Ltd., part of Springer Nature 2018
## Authors and Affiliations
• Ricardo García-García
• 1
• Max A. González-Palacios
• 1
1. 1.División de Ingenierías Campus Irapuato-SalamancaUniversidad de GuanajuatoSalamancaMéxico | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7402699589729309, "perplexity": 21937.651940298438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744750.80/warc/CC-MAIN-20181118221818-20181119003818-00113.warc.gz"} |
https://www.illustrativemathematics.org/content-standards/5/NBT/A/3/tasks/1803 | Update all PDFs
# Placing Thousandths on the Number Line.
Alignments to Content Standards: 5.NBT.A.3
Label all of the tick marks on the number line. Plot and label each of the following numbers on the number line.
\begin{align} &0.100\\ &0.010\\ &0.072\\ &0.038 \end{align}
Which of these numbers is greatest? Which is least? How can you tell by looking at the number line?
## IM Commentary
Though this task primarily deals with comparing decimal numbers on a number line, it also requires students to draw upon what they know about the base ten system. They must recognize that the distance from 0 to 0.1 is partitioned into ten equal pieces and that one tenth of one tenth is 0.01. Once they realize that each increment goes up by 0.01, they can fill in the framework to place other numbers. One of the key instructional points in this task is that our number line orders numbers least to greatest. Therefore, numbers farther to the right are greater than those to the left. This understanding will hold true for future explorations of rational numbers as well. A good precursor task for this one would be 5.NBT Comparing Decimals on the Number Line. Side note: It would be good to have versions of this for thousandths that are not between 0 and 1.
## Solution
0.010 is least because it is the farthest to the left on the number line and 0.100 is greatest because it is the farthest to the right on the number line. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4297228753566742, "perplexity": 583.5986598713642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593004.92/warc/CC-MAIN-20180722022235-20180722042235-00469.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/40576-help-intergral-domain.html | # Thread: Help on Intergral Domain
1. ## Help on Intergral Domain
Consider the integral domain $\displaystyle Z(\sqrt10)=${$\displaystyle a+b\sqrt10|a,b \in Z$}
(a) Using the fact that the norm $\displaystyle N(a+b\sqrt10)=a^2 - 10b^2$ is multaplicative, describle the units of $\displaystyle Z(\sqrt10)$.
(b) Show that all four numbers: $\displaystyle 2,3,4+\sqrt10, 4-\sqrt10$ are irreducible. Are any of these numbers prime?
(c) Deduce that Z(\sqrt10) is not unique factorization domain.
My lecturer go though this topic too fast, so I have difficulty on above question. Please help me.
2. Originally Posted by kleenex
Consider the integral domain $\displaystyle Z(\sqrt10)=${$\displaystyle a+b\sqrt10|a,b \in Z$}
(a) Using the fact that the norm $\displaystyle N(a+b\sqrt10)=a^2 - 10b^2$ is multaplicative, describle the units of $\displaystyle Z(\sqrt10)$.
Put an absolute value on it, then $\displaystyle N(a+b\sqrt{10}) = |a^2 - 10b^2|$ will satisfy $\displaystyle N(\alpha \beta) = N(\alpha)N(\beta)$.
If $\displaystyle u$ is a unit then there is $\displaystyle v$ so that $\displaystyle uv=1$ but then $\displaystyle N(uv) = N(1) = 1$ thus $\displaystyle N(u)N(v) = 1$ which means $\displaystyle N(u)=1$. Thus, the units must have norm equal to $\displaystyle 1$. Conversely, if $\displaystyle N(a+b\sqrt{10}) = 1$ then $\displaystyle \tfrac{a}{a^2 - 10b^2} - \tfrac{b}{a^2 - 10b^2} \sqrt{10}$ (which is in $\displaystyle \mathbb{Z}(\sqrt{10})$) is inverse of $\displaystyle a+b\sqrt{10}$. To find the units we need to solve $\displaystyle |a^2 - 10b^2| = 1 \implies a^2 - 10b^2 = \pm 1$. This equation is Pellian and got infinitely many solution. I assume all you need to do is find a test to determine if a number is a unit or not, that is how to do it.
(b) Show that all four numbers: $\displaystyle 2,3,4+\sqrt10, 4-\sqrt10$ are irreducible. Are any of these numbers prime?
A non-zero non-unit element $\displaystyle \alpha$ is called irreducible iff $\displaystyle \alpha = \beta \gamma$ is an improper trivial factorization, i.e. one of them, say $\displaystyle \beta$, is associate to $\displaystyle \alpha$ and the other one $\displaystyle \gamma$ is a unit. Hence it cannot be reduced further. A non-zero non-unit element $\displaystyle \alpha$ is called prime iff $\displaystyle \alpha | \beta \gamma \implies \alpha | \beta \mbox{ or }\alpha | \gamma$. A simple consequence is that prime elements are irreducible, while not necessarily irreducible elements are prime.
(c) Deduce that Z(\sqrt10) is not unique factorization domain.
In a unique factorization domain the irreducible elements are also prime. Thus, if you can find an irreducible non-prime element (which is probably from the above list) then it would mean this is not a unique factorization domain. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9644820094108582, "perplexity": 242.32809018996682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583671342.16/warc/CC-MAIN-20190119160425-20190119182425-00064.warc.gz"} |
https://en.wikipedia.org/wiki/Negative_frequency | Negative frequency
The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way: a signed value of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions (a.k.a. cycles) per second (hertz) or radian/second (where 1 cycle corresponds to 2π radians).
Sinusoids
Let ω be a nonnegative parameter with units of radians/sec. Then the angular function (angle vs. time) ωt + θ, has slope −ω, which is called a negative frequency. But when the function is used as the argument of a cosine operator, the result is indistinguishable from cos(ωtθ). Similarly, sin(−ωt + θ) is indistinguishable from sin(ωtθ + π). Thus any sinusoids can be represented in terms of positive frequencies. The sign of the underlying phase slope is ambiguous.
A negative frequency causes the sin function (violet) to lead the cos (red) by 1/4 cycle.
The vector (cos t, sin t) rotates counter-clockwise at 1 radian/second, and completes a circle every 2π seconds. The vector (cos −t, sin −t) rotates in the other direction (not shown).
The ambiguity is resolved when the cosine and sine operators can be observed simultaneously, because cos(ωt + θ) leads sin(ωt + θ) by 1/4 cycle (= π/2 radians) when ω > 0, and lags by 1/4 cycle when ω < 0. Similarly, a vector, (cos t, sin t), rotates counter-clockwise at 1 radian/sec, and completes a circle every 2π seconds, and the vector (cos −t, sin −t) rotates in the other direction.
The sign of ω is also preserved in the complex-valued function:
${\displaystyle e^{i\omega t}=\underbrace {\cos(\omega t)} _{R(t)}+i\cdot \underbrace {\sin(\omega t)} _{I(t)},}$ [1]
(Eq.1)
since R(t) and I(t) can be separately extracted and compared. Although ${\displaystyle e^{i\omega t}}$ clearly contains more information than either of its components, a common interpretation is that it is a simpler function, because:
${\displaystyle \cos(\omega t)={\begin{matrix}{\frac {1}{2}}\end{matrix}}\left(e^{i\omega t}+e^{-i\omega t}\right),}$
(Eq.2)
which gives rise to the interpretation that cos(ωt) comprises both positive and negative frequencies. But the sum is actually a cancellation that contains less, not more, information. Any measure that indicates both frequencies includes a false positive, because ω can have only one sign.[3] The Fourier transform, for instance, merely tells us that cos(ωt) correlates equally well with both cos(ωt) + i sin(ωt) and with cos(ωt) − i sin(ωt).
Applications
Perhaps the most well-known application of negative frequency is the calculation:
${\displaystyle X(\omega )=\int _{a}^{b}x(t)\cdot e^{-i\omega t}dt,}$
which is a measure of the amount of frequency ω in the function x(t) over the interval (a, b). When evaluated as a continuous function of ω for the theoretical interval (−∞, ∞), it is known as the Fourier transform of x(t). A brief explanation is that the product of two complex sinusoids is also a complex sinusoid whose frequency is the sum of the original frequencies. So when ω is positive, ${\displaystyle e^{-i\omega t}}$ causes all the frequencies of x(t) to be reduced by amount ω. Whatever part of x(t) that was at frequency ω is changed to frequency zero, which is just a constant whose amplitude level is a measure of the strength of the original ω content. And whatever part of x(t) that was at frequency zero is changed to a sinusoid at frequency −ω. Similarly, all other frequencies are changed to non-zero values. As the interval (a, b) increases, the contribution of the constant term grows in proportion. But the contributions of the sinusoidal terms only oscillate around zero. So X(ω) improves as a relative measure of the amount of frequency ω in the function x(t).
The Fourier transform of ${\displaystyle e^{i\omega t}}$ produces a non-zero response only at frequency ω. The transform of ${\displaystyle \cos(\omega t)}$ has responses at both ω and −ω, as anticipated by Eq.2.
Sampling of positive and negative frequencies and aliasing
This figure depicts two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points. They are thus aliases of each other when sampled at the rate (fs) indicated by the grid lines. The gold-colored function depicts a positive frequency, because its real part (the cos function) leads its imaginary part by 1/4 of one cycle. The cyan function depicts a negative frequency, because its real part lags the imaginary part.
Notes
1. ^ The equivalence is called Euler's formula
2. ^ See Euler's formula § Relationship to trigonometry and Phasor § Addition for examples of calculations simplified by the complex representation.
3. ^ Conversely, any measure that indicates only one frequency has made an assumption, perhaps based on collateral information.
References
• Positive and Negative Frequencies
• Lyons, Richard G. (Nov 11, 2010). Chapt 8.4. Understanding Digital Signal Processing (3rd ed.). Prentice Hall. 944 pgs. ISBN 0137027419. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.967449426651001, "perplexity": 1441.133754864033}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583656577.40/warc/CC-MAIN-20190116011131-20190116033131-00538.warc.gz"} |
http://neon.nervanasys.com/generated/neon.layers.recurrent.GRU.html | # neon.layers.recurrent.GRU¶
class neon.layers.recurrent.GRU(output_size, init, init_inner=None, activation=None, gate_activation=None, reset_cells=False, name=None)[source]
Implementation of the Gated Recurrent Unit based on [Cho2014].
• It uses two gates: reset gate (r) and update gate (z)
• The update gate (z) decides how much the activation is updated
• The reset gate (r) decides how much to reset (when r = 0) from the previous activation
• Activation (h_t) is a linear interpolation (by z) between the previous activation (h_t-1) and the new candidate activation ( h_can )
• r and z are computed the same way, using different weights
• gate activation function and unit activation function are usually different
• gate activation is usually logistic
• unit activation is usually tanh
• consider there are 3 gates: r, z, h_can
Parameters: output_size (int) – Number of hidden/output units init (Initializer) – Function for initializing the model’s input to hidden weights. By default, this initializer will also be used for recurrent parameters unless init_inner is also specified. Biases will always be initialized to zero. init_inner (Initializer, optional) – Function for initializing the model’s recurrent parameters. If absent, will default to using same initializer provided to init. activation (Transform) – Activation function for the input modulation. gate_activation (Transform) – Activation function for the gates. reset_cells (bool) – default to be False to make the layer stateful, set to True to be stateless. name (str, optional) – name to refer to this layer as.
x
Tensor – Input data tensor (input_size, sequence_length * batch_size)
W_input
Tensor – Weights on the input units (out size * 3, input size)
W_recur
Tensor – Weights on the recursive inputs (out size * 3, out size)
b
Tensor – Biases (out size * 3 , 1)
References
• Learning phrase representations using rnn encoder-decoder for statistical machine translation [Cho2014]
• Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling [Chung2014]
[Cho2014] (1, 2) http://arxiv.org/abs/1406.1078
__init__(output_size, init, init_inner=None, activation=None, gate_activation=None, reset_cells=False, name=None)[source]
Methods
__init__(output_size, init[, init_inner, …]) accumulates(f) Higher order decorator function that enables accumulation functionality for that function. allocate([shared_outputs]) Allocate output buffer to store activations from fprop. allocate_deltas(global_deltas) bprop(deltas[, alpha, beta]) Backpropagation of errors, output delta for previous layer, and calculate the update on model params. configure(in_obj) Set shape based parameters of this layer given an input tuple, int or input layer. final_state() Return final state for sequence to sequence models fprop(inputs[, inference, init_state]) Apply the forward pass transformation to the input data. gen_class(pdict) get_description([get_weights, keep_states]) Get layer parameters. get_final_hidden_error() Return hidden delta after bprop and adjusting for bprop from decoder to encoder in sequence to sequence models. get_is_mklop() is_mklop true means this op is on mkl backend get_param_attrs() get_params() Get layer parameters, gradients, and states for optimization. get_params_serialize([keep_states]) get_terminal() Used for recursively getting final nodes from layer containers. init_buffers(inputs) Initialize buffers for recurrent internal units and outputs. init_params(shape) Initialize params for GRU including weights and biases. load_weights(pdict[, load_states]) Load weights. nested_str([level]) Utility function for displaying layer info with a given indentation level. recursive_gen(pdict, key) helper method to check whether the definition serialize() Get state parameters for this layer. set_acc_on(acc_on) Set the acc_on flag according to bool argument. set_batch_size(N) Set minibatch size. set_deltas(delta_buffers) Use pre-allocated (by layer containers) list of buffers for backpropagated error. set_is_mklop() set_next(layer) Set next_layer to provided layer. set_not_mklop() set_params(pdict) Set layer parameters (weights). set_seq_len(S) Set sequence length. set_states(pdict)
accumulates(f)
Higher order decorator function that enables accumulation functionality for that function. Object that use this decorator are required to have an acc_param attribute. This attribute tuple declares the names for existing temp parameter and real parameter buffers. The temp parameter buffer copies the value of the parameter buffer before f is called, and after f is called the temp and normal buffers are summed. This decorator could be used to wrap any function that may want to accumulate parameters instead of overwriting.
allocate(shared_outputs=None)[source]
Allocate output buffer to store activations from fprop.
Parameters: shared_outputs (Tensor, optional) – pre-allocated tensor for activations to be computed into
allocate_deltas(global_deltas)
be = None
bprop(deltas, alpha=1.0, beta=0.0)[source]
Backpropagation of errors, output delta for previous layer, and calculate the update on
model params.
Parameters: deltas (Tensor) – error tensors for each time step of unrolling alpha (float, optional) – scale to apply to input for activation gradient bprop. Defaults to 1.0 beta (float, optional) – scale to apply to output activation gradient bprop. Defaults to 0.0
dW_input
dW_recur
db
Returns: Backpropagated errors for each time step of model unrolling Tensor
classnm
Returns the class name.
configure(in_obj)
Set shape based parameters of this layer given an input tuple, int or input layer.
Parameters: in_obj (int, tuple, Layer, Tensor or dataset) – object that provides shape information for layer shape of output data (tuple)
final_state()
Return final state for sequence to sequence models
fprop(inputs, inference=False, init_state=None)[source]
Apply the forward pass transformation to the input data. The input data is a list of
inputs with an element for each time step of model unrolling.
Parameters: inputs (Tensor) – input data as 3D tensors, then converted into a list of 2D tensors. The dimension is (input_size, sequence_length * batch_size) inference (bool, optional) – Set to true if you are running inference (only care about forward propagation without associated backward propagation). Default is False. GRU output for each model time step Tensor
gen_class(pdict)
get_description(get_weights=False, keep_states=True)
Get layer parameters. All parameters are needed for optimization, but only weights are serialized.
Parameters: get_weights (bool, optional) – Control whether all parameters are returned or just weights for serialization. keep_states (bool, optional) – Control whether all parameters are returned or just weights for serialization.
get_final_hidden_error()
Return hidden delta after bprop and adjusting for bprop from decoder to encoder in sequence to sequence models.
get_is_mklop()
is_mklop true means this op is on mkl backend and may require convert when from non-mkl op
get_param_attrs()
get_params()
Get layer parameters, gradients, and states for optimization.
get_params_serialize(keep_states=True)
get_terminal()
Used for recursively getting final nodes from layer containers.
init_buffers(inputs)
Initialize buffers for recurrent internal units and outputs. Buffers are initialized as 2D tensors with second dimension being steps * batch_size. The second dimension is ordered as [s1b1, s1b2, …, s1bn, s2b1, s2b2, …, s2bn, …] A list of views are created on the buffer for easy manipulation of data related to a certain time step
Parameters: inputs (Tensor) – input data as 2D tensor. The dimension is (input_size, sequence_length * batch_size)
init_params(shape)[source]
Initialize params for GRU including weights and biases. The weight matrix and bias matrix are concatenated from the weights for inputs and weights for recurrent inputs and bias. The shape of the weights are (number of inputs + number of outputs +1 ) by (number of outputs * 3)
Parameters: shape (Tuple) – contains number of outputs and number of inputs
load_weights(pdict, load_states=True)
Parameters: pdict – load_states – (Default value = True)
modulenm
Returns the full module path.
nested_str(level=0)
Utility function for displaying layer info with a given indentation level.
Parameters: level (int, optional) – indentation level layer info at the given indentation level str
recursive_gen(pdict, key)
helper method to check whether the definition dictionary is defining a NervanaObject child, if so it will instantiate that object and replace the dictionary element with an instance of that object
serialize()
Get state parameters for this layer.
Returns: whatever data this model wants to receive in order to restore state varies
set_acc_on(acc_on)
Set the acc_on flag according to bool argument. If set to true, the layer will accumulate some (preset) parameters on calls to functions that are decorated with the accumulates decorator. In order to use this feature, accumulate_updates=True must have been passed to the layer’s allocate function
This currently only works for a few hard coded parameters in select layers
Parameters: acc_on (bool) – Value to set the acc_on flag to.
set_batch_size(N)
Set minibatch size.
Parameters: N (int) – minibatch size
set_deltas(delta_buffers)
Use pre-allocated (by layer containers) list of buffers for backpropagated error. Only set deltas for layers that own their own deltas Only allocate space if layer owns its own deltas (e.g., bias and activation work in-place, so do not own their deltas).
Parameters: delta_buffers (list) – list of pre-allocated tensors (provided by layer container)
set_is_mklop()
set_next(layer)
Set next_layer to provided layer.
Parameters: layer (layer) – Next layer
set_not_mklop()
set_params(pdict)
Set layer parameters (weights). Allocate space for other parameters but do not initialize them.
Parameters: pdict (dict, ndarray) – dictionary or ndarray with layer parameters [support for ndarray is DEPRECATED and will be removed]
set_seq_len(S)
Set sequence length.
Parameters: S (int) – sequence length
set_states(pdict) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1646992713212967, "perplexity": 10022.770728639449}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794868248.78/warc/CC-MAIN-20180527111631-20180527131631-00332.warc.gz"} |
https://impa.br/sobre/memoria/reunioes-cientificas/nonlinear-pdes-impa/ | # Nonlinear PDE`s @ IMPA
IMPA, Rio de Janeiro, June 20 – 24, 2016
The field of nonlinear partial differential equations has experienced a striking evolution over the last fifteen years in Brazil. The number of PhD’s and students has increased creating a critical mass. In this process it is fundamental to have a scientific interchange with researchers leaders in the field. The aim of the workshop is to continuing this process by allowing the community to attend to lectures of recent and important progress in the field delivered by renowned specialists. The workshop will bring together leading experts in nonlinear dispersive equations, nonlinear hyperbolic systems of conservation laws, nonlinear elliptic equations and free-boundary problems. Graduate students and young PhD’s are encouraged to participate of this meeting.
Picture
Videos
## Conference Venue
IMPA is located in the South Zone of Rio de Janeiro, near the Tijuca Forest and the Botanical Gardens, 20 minutes from major hotels of Leblon and Ipanema. You can find our address and some practical information about how to get to the institution below.
Estrada Dona Castorina, 110 Jardim Botânico Zip Code 22460-320 Rio de Janeiro, RJ – Brazil Phone number: (55 21) 2529-5000 Fax: (55 21) 2512-4115 E-mail: eventosimpa.br
How to get to IMPA:
From the airport
On leaving the airport we suggest that you take a taxi. There are several companies that have booths located inside the airport facilities, on the way out. A ride to South Zone costs around R$120 to R$150 and takes about 50 minutes. There are also special buses leaving from Terminals 1 and 2, whose routes you can find here.
Buses
There are three bus lines running near the Institute:
• 409 – Saens Peña – Horto (via Lapa)
• 416 – Saens Peña – Jardim Botânico (Horto) (via Túnel Rebouças)
You can check which buses line is the best for you by clicking here.
The passenger must pay as they enter, using the front door to get on the bus and the rear door to get off the bus. The bus fare is R$3,80 and the maximum change given is for a R$ 20,00 bill.
Should you prefer to use a taxi, we suggest the following companies:
MAP of the area
## Registration
Registration fee
Category Price Researchers and Professors (Ph.D’s) R$200.00 Students (Master and Ph.D) R$ 80.00 IMPA´s Students R\$ 50.00
## Organizing and Scientific Committee
Hermano Frid (IMPA)
## Conference Topics
The main themes in the workshop Nonlinear PDE’s @ IMPA will be:
– Nonlinear Dispersive Equations.
– Nonlinear Hyperbolic systems of Conservation Laws.
– Nonlinear Elliptic Equations.
– Free-Boundary Problems
The meeting will also include related topics such as:
– Equations of Fluid Mechanics.
– Kinetic Equations.
– Optimal Transport
– Variational Problems.
– Homogenization.
Program
## Plenary Speakers
Luis Caffarelli (University of Texas at Austin, USA)
Some non local versions of the Monge Ampere equation
Thierry Cazenave (Univ. Paris 6, France)
Regularity issues in semilinear Schrödinger and heat equations
Scattering of small solutions of the cubic NLS with short range potential
Kenneth H. Karlsen (University of Oslo)
Stochastic Conservation Laws
Ronghua Pan (Georgia Tech)
Global regularity v.s. finite time blowup for compressible Euler equations
Jean-Claude Saut (Univ. Paris Sud, France )
Full dispersion water waves models
Denis Serre (Ecole Normale Superieure Lyon, France)
The singularity of a perfect gas at the vacuum boundary
## 30-Minute Talks
Ricardo Alonso (PUC)
Short introduction to one dimensional dissipative Boltzmann equation
Miguel Angel Alejo (Universidade Federal de Santa Catarina)
Stability of Breathers and Numerical Results
Jaime Angulo (USP)
Stability of Peak Solutions for Nls Equations with Point Interactions
Stefan Berres (Universidad Católica de Temuco, Chile)
Identification of shock profile solutions for bidisperse suspensions
Raimund Bürger (Universidad de Concepción, Chile)
A random sampling approach for a family of Templeclass systems of conservation laws
Emanuel Carneiro (IMPA)
On the regularity of maximal operators on BV spaces
Marcelo M. Cavalcanti (State University of Maringá)
Unilateral Problem for the Wave Equation With Spatial-Time Degenerate Nonlinear Damping: Well-Posedness And Exponential Stability
Gyula Csato (Universidad de Concepción, Chile)
About the Hardy-Sobolev, Moser-Trudinger and isoperimetric inequalities with densities
Generalized Zakharov-Kuznetsov equation in anisotropic weighted Sobolev spaces
Olivier Kneuss (Universidade Federal do Rio de Janeiro)
Nelson’s conjecture revisited
Yachun Li (Shanghai Jiao Tong University, China)
Global Classical Solutions to the Cauchy Problem of Conservation Laws with Degenerate Diffusion
Diego Moreira (UFC)
Hopf-Oleinik Lemma and Applications
Well-posedness for the Novikov-Veselov equation
Stochastic transport equation in bounded domains
Scattering for a 3D Coupled Nonlinear Schrodinger System
Stabilization of a Boussinesq system with generalized damping
Edgard Pimentel (UFSCar)
A priori Sobolev regularity for fully nonlinear parabolic equations
Moritz Reintjes (IMPA)
Is Spacetime Locally Inertial for General Relativistic Shock Wave Solutions?
Jean Carlos da Silva (UFRJ)
On The Almost Periodic Homogenization of Non-Linear Scalar Conservations Laws
Boyan Sirakov (PUC)
A Priori Bounds for Elliptic Inequalities via Regularity Estimates
Erlend Briseid Storrosten (University of Oslo)
Error Estimates for Finite Difference Approximations of Degenerate Convection-Diffusion Equations
## Short Communications
Eduardo Arbieto Alarcon (Universidade Federal de Goiás)
On the Cauchy Problem Associated to the Brinkman Flow in Rn+
Pablo Braz e Silva (Universidade Federal de Pernambuco)
Some asymptotic supnorm estimates for convection-diffusion equations and systems
Gerardo Camacho-de La Rosa (University of Mexico)
Existence and Stability of the Travellingwave Solutions for a Semilinear Wave Equation
Roberto Capistrano-Filho (Universidade Federal de Pernambuco)
Neumann Boundary Controllability of the Korteweg-De Vries Equation
Hardy-Sobolev Type Inequalties with Monomial Weights
Márcio Cavalcante de Melo (Universidade Federal de Alagoas)
The Initial-Boundary-Value Problem For Some Quadratic Nonlinear Schrodinger Equations On The Half-Line
Liliana Esquivel (UNAM Campus Morelia)
Neumann problem for nonlinear Schrödinger equation with the Riesz fractional derivative operator
Marco A. M. Guaraco (IMPA)
Geometric min-max methods and the space of solutions to the Allen-Cahn equation on closed manifolds
Isnaldo Isaac Barbosa (Universidade Federal de Alagoas)
On the Cauchy Problem for Nonlinear Interactions type Schrödinger
Uriel Kaufmann (Universidad Nacional de Cordoba)
Positive Solutions for Singular Problems Involving the P-Laplacian
Bifurcation and Stability of Steady States of Parabolic Problems Under Logistic Flux Boundary Conditions
Wanderley Nunes do Nascimento (IMEEC – UNICAMP)
Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation
Tiago Picon (FFCLRP/USP)
Global existence of small data solutions to the semilinear fractional wave equation
Juliana Pimentel (Universidade Federal do ABC)
Infinite time blow-up for the Chafee-Infante equation
Marcelo Rempel Ebert (Universidade de São Paulo)
Some optimal decay estimates for damped semilinear evolution equations
## Poster Session
Amanda A. Feltrin Nunes (Universidade Federal de São Carlos)
Multiple Positive Solutions for a Non-Local Quasilinear Problem from Population Genetics
Marcelo Fernandes de Almeida (Federal University of Sergipe)
Uniform Global Well-Posedness of the Navier-Stokes-Corioles in a new Critical Space
Fernando A. Gallego (Universidade Federal do Rio de Janeiro)
Neumann Boundary Controllability of the Gear-Grimshaw System with Critical Size Restrictions on the Spacial Domain
Carlos M. Guzmán (Universidade Federal de Minas Gerais)
On the inhomogeneous nonlinear Schrödinger equation
Alex Hernandez Ardila (IME-USP)
Stability of ground states for logarithmic Schrödinger equation with a δ‘-interaction
Lucas Fabiano Lima (Universidade Federal de São Carlos)
Regularity Theory for a Mean-Field Game Model of Wealth and Capital Accumulation
Cecilia Saavedra Fresia (Universidad Nacional de Tucumán)
EDP’s no Lineales en el Sistema Vascular
César K. Soares dos Santos
Pointwise regularity for a parabolic equation with log-term singularity
## Call For Abstracts
Besides the main lectures, there will be 30 minutes lectures, 20 minutes short communications and posters. We would like to invite the interested to submit abstracts for these presentations. The abstract should be no longer than 2 pages in PDF or PS format. Full consideration will be given to abstracts which are submitted before April 15th, 2016. The accepted presentations will be communicated by April 30th, 2016.
Be sure your abstract includes the title, and all author names and addresses, indicating who is going to present.
It is strongly recommended that the following template be used to produce the pdf file:
Please send the abstract by e-mail to: nlpde@impa.br
Back
List
## Suggested Accommodations
Rio de Janeiro has a large number of hotels. Lately, for the main congresses, we have been using Hotel Acapulco, Everest Rio Hotel, Everest Park Hotel, Hotel Debret, Hotel Vermont, among others. To take advantage of IMPA´s special rate, please contact Ms. Michele Leite at MMX Eventos – micheleleite@mmxcongressos.com – phone: + 55 21 2210-1732 or visit their homepage at: www.mmxcongressos.com
For other type of accommodation like hostels and bed and breakfast, we suggest the following:
Pouso Verde Bed & Breakfast (very close to IMPA and strongly recommended)
Rua Caminhoá, 14 – Jardim Botânico
Tel: + 55 21 2529-2942 , + 55 21 99971-0511
e-mail: pousoverde@globo.com
www.pousoverde.com
Hostels
Botafogo:
Meiai Hostel:
R. Guilhermina Guinle, 127 – Botafogo
Tel: 55(21) 3495-4481
www.meiai.com.br
Copacabana:
Travessa Angrense, 25 A – Copacabana (entre Santa Clara e Raimundo Correa)
Tel: 55(21)2256-6951 / Fax: 55(21) 2286-5652
Humaitá:
Lagoa Guest House
Rua do Humaitá, 392 – Humaitá
Tel: 55(21) 3518-9000
E-mail: contato@lagoaguesthouse.com
www.lagoaguesthouse.com
Ipanema:
Che Lagarto Ipanema
Rua Paul Redfern 48 – Ipanema
Tel: 55(21) 2512-8076
www.chelagarto.com
Leblon:
Leblon Spot (associado à Hostelling International)
Rua Dias Ferreira, 626 – Leblon
Tel: 55(21) 2137-0090
E-mail: contact@leblonspot.com
www.leblonspot.com
Please check availability and prices directly with them
# Institutional
Flats
Hotels
Others
## Visa Information
The Brazilian visa system operates under a strict reciprocity policy. That means that all citizens from countries which require visas from Brazilians will have to obtain visas to attend the conference. Beware that to attend a scientific conference in Brazil a tourist visa suffices. Source: Consulate General of Brazil
Foreign participants should get in touch with the nearest Brazilian Consulate (click on the map) to find out whether a visa is required.
Here you find the list of nationalities that are exempt from this requirement, see also the official Ministry of Tourism page, that includes the whole of the European Union and Mercosur. Those who do need visa should note that a tourist visa suffices for the conference attendance; for more information about Brazil, visit the official Brazilian Institute of Tourism website. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22982257604599, "perplexity": 10017.34812954705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686169.5/warc/CC-MAIN-20170920033426-20170920053426-00253.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=2661561 | MathSciNet bibliographic data MR2661561 32A38 (30B50 42B10 46E10) Abanin, A. V.; Khoi, Le Hai Dual of the function algebra $A^{-\infty}(D)$$A^{-\infty}(D)$ and representation of functions in Dirichlet series. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3623–3635. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9852445721626282, "perplexity": 4457.2817354391445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425381.3/warc/CC-MAIN-20170725202416-20170725222416-00662.warc.gz"} |
http://math.stackexchange.com/questions/119176/showing-uniqueness-of-solution-to-ivp-under-certain-conditions | # Showing uniqueness of solution to IVP under certain conditions
Consider the IVP $\mathbf{\dot x} = f(\mathbf{x},t)$ where $\mathbf{x}(0) = 0$, $f$ is continuous in some neighborhood of $(x,t) = (0,0)$ and $|f(x,t)-f(y,t)| \leq \frac{|x-y|}{t^\alpha}$. I would like to show the uniquenesss of the solution if $\alpha\in (0,1)$.
It seems like I would like to show that $f$ is locally Lipschitz in $x$ when $t = 0$ if $\alpha\in(0,1)$, (since it is clearly locally Lipschitz for all $t > 0$) but I haven't been able to tackle that by elementary methods (or any other, really). Another idea might be to bound the sequence of Picard iterates, but that seems very messy. A third idea would be to appeal to general facts about moduli of continuity. The fact that a solution is unique when the integral $\int_0^x\frac{dx}{\omega(x)}$ diverges for the modulus $\omega$ seems almost useful, except that in this case my modulus is a function of both $x$ and $t$. Am I in the right ballpark with these approaches, or is there something very obvious that I'm missing?
Also, this is homework, so I would be grateful for a small hint rather than a full solution. Thanks in advance!
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9843271970748901, "perplexity": 92.9261899294166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824756.90/warc/CC-MAIN-20160723071024-00050-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/224896-linear-operator-question.html | 1. ## Linear Operator Question
I'm working on a practice final exam, and I'm stuck on a problem:
Let $\displaystyle T: \mathcal{P}_2 \to \mathcal{P}_2$ be a linear operator defined by $\displaystyle T(f) = f' + f''$. ($\displaystyle \mathcal{P}_2$ is the set of all polynomials of order $\displaystyle \leq 2$).
(Note that $\displaystyle T$ is corresponding to a matrix)
(1) Find the matrix $\displaystyle A$.
For this, I'm not sure if I need to use a basis as a reference or not. Is the matrix representation of $\displaystyle T$ supposed to operate on coordinates of $\displaystyle f$? Could I not just assume the standard basis for this, $\displaystyle B = \{1,x,x^2\}$, and use that to find $\displaystyle A$?
I imagine I'd be solving the problem $\displaystyle A[a \ b \ c]^T = [0 \ 2a \ (2a + b)]^T$ since the general polynomial of order two has the coordinates $\displaystyle [a \ b \ c]^T$ in $\displaystyle B$ and, after being operated on by $\displaystyle T$, has coordinates $\displaystyle [0 \ 2a \ (2a + b)]^T$.
Can you tell me if I'm on the right track?
(2) Is $\displaystyle A$ diagonalizable?
If I continue with the method from (1), then $\displaystyle A$ is NOT diagonalizable, but, of course, I'm not certain that my answer in (1) is correct or not.
(3) Find a basis, $\displaystyle B$ of $\displaystyle \mathcal{P}_2$ such that $\displaystyle [T]_B$ is a diagonal matrix.
In this context, is $\displaystyle [T]_B$ just my $\displaystyle A$ from part (1) just with a different basis? I figured that it would be, but, if that were the case, couldn't I just work backwards from (3) to (1) and everything is obvious?
2. ## Re: Linear Operator Question
Originally Posted by Aryth
I'm working on a practice final exam, and I'm stuck on a problem:
Let $\displaystyle T: \mathcal{P}_2 \to \mathcal{P}_2$ be a linear operator defined by $\displaystyle T(f) = f' + f''$. ($\displaystyle \mathcal{P}_2$ is the set of all polynomials of order $\displaystyle \leq 2$).
(Note that $\displaystyle T$ is corresponding to a matrix)
(1) Find the matrix $\displaystyle A$.
For this, I'm not sure if I need to use a basis as a reference or not. Is the matrix representation of $\displaystyle T$ supposed to operate on coordinates of $\displaystyle f$? Could I not just assume the standard basis for this, $\displaystyle B = \{1,x,x^2\}$, and use that to find $\displaystyle A$?
I imagine I'd be solving the problem $\displaystyle A[a \ b \ c]^T = [0 \ 2a \ (2a + b)]^T$ since the general polynomial of order two has the coordinates $\displaystyle [a \ b \ c]^T$ in $\displaystyle B$ and, after being operated on by $\displaystyle T$, has coordinates $\displaystyle [0 \ 2a \ (2a + b)]^T$.
Can you tell me if I'm on the right track?
(2) Is $\displaystyle A$ diagonalizable?
If I continue with the method from (1), then $\displaystyle A$ is NOT diagonalizable, but, of course, I'm not certain that my answer in (1) is correct or not.
(3) Find a basis, $\displaystyle B$ of $\displaystyle \mathcal{P}_2$ such that $\displaystyle [T]_B$ is a diagonal matrix.
In this context, is $\displaystyle [T]_B$ just my $\displaystyle A$ from part (1) just with a different basis? I figured that it would be, but, if that were the case, couldn't I just work backwards from (3) to (1) and everything is obvious?
I just read a pretty good treatment of this. You do use the basis you mentioned above and you note
T{1 0 0} = {0 0 0}, T{0 1 0} = {1 0 0}, T{0 0 1} = {2 2 0}
in other words T(1) = 0, T(x) = 1, T(x^2) = 2+2x
0 1 2 0 0 2 0 0 0
It's pretty clearly not diagonizable.
3) is above my head atm but you might try the Fourier basis, I saw that done in another article I just read trying to come up to speed on this.
3. ## Re: Linear Operator Question
Originally Posted by romsek
I just read a pretty good treatment of this. You do use the basis you mentioned above and you note
T{1 0 0} = {0 0 0}, T{0 1 0} = {1 0 0}, T{0 0 1} = {2 2 0}
in other words T(1) = 0, T(x) = 1, T(x^2) = 2+2x
0 1 2 0 0 2 0 0 0
It's pretty clearly not diagonizable.
3) is above my head atm but you might try the Fourier basis, I saw that done in another article I just read trying to come up to speed on this.
Yeah, I'm glad I was on the right track. For (3), I have a few bases I can try. If those don't work, I'll certainly give the Fourier basis a try. Thanks for your help.
4. ## Re: Linear Operator Question
Originally Posted by Aryth
Yeah, I'm glad I was on the right track. For (3), I have a few bases I can try. If those don't work, I'll certainly give the Fourier basis a try. Thanks for your help.
a little toying with this convinced me Fourier is not the way to go. It looks like minimal polynomials are going to play a role in this.
This may be of interest.
5. ## Re: Linear Operator Question
I'm not sure why you talk about "trying" a particular basis. A linear transformation, from vector space U to vector V, can only be written as a matrix in given bases for U and V. Are you not told what basis to use? Yes, "1", "x", and "$\displaystyle x^2$". Then T(1)= 0, T(x)= 0+ 1, and [tex]T(x^2)= 2x so, yes, the matrix, in that (ordered) basis is $\displaystyle \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 2 \end{bmatrix}$.
Now, a 3 by 3 matrix is invertible if and only if it has three independent eigenvectors. So, yes, find the "characteristic polynomial", set it equal to 0, and solve for the eigenvalues. If there are three distinct eigenvalues ,there definitely are three independent eigenvectors and the matrix is diagonalizable. If there are only one or two distinct eigenvalues there still may be three independent eigenvalues, but you would have to check that.
(3) kind of gives the answer to (2) away- it wouldn't make sense to ask for a basis in which A is diagonal if A were NOT "diagonalizable". That basis is the basis consisting of the eigenvectors of A.
6. ## Re: Linear Operator Question
Halls point is pretty obvious if you think about it. If T were diagonalizable you could recover f from f' + f''. This is obviously not possible since you lose any information on the 0th order coefficients when you differentiate. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9059679508209229, "perplexity": 285.18296641579934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648177.88/warc/CC-MAIN-20180323024544-20180323044544-00387.warc.gz"} |
https://brilliant.org/problems/solve-it-38/ | # clue - starting two numbers of the answer is 14
Calculus Level pending
The digits of a three digit number form a G.P. If 400 is subtracted from it then we get another three digit number whose digits form an arithmetic series what is the sum of these two numbers?
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8906724452972412, "perplexity": 456.207973693014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424645.77/warc/CC-MAIN-20170724002355-20170724022355-00703.warc.gz"} |
https://help.syncfusion.com/js/calculate/overview | # Overview in JQuery Calculate widget
18 Apr 20171 minute to read
Essential Calculate enables to parse and evaluate the formulas. These formulas are equations which performs calculation on values. It allows the user to add extensive calculation support for business objects.
The range of calculations include simple algebraic expressions such as `(1.2^3-1)/8`, to formulas using intrinsic functions like `4 * sqrt(exp(8.4))`, to formulas relying on variables that are defined through controls on a form such as `cos([textBox1] * pi()/180)`, to spreadsheet-like formulas such as `Sum(A2:B14)`.
### Key Features
Important features of Essential Calculate are listed below,
• Parses and evaluates excel based formulas and basic equations.
• It comes with a function library of more than 200 entries of formulas.
• Named ranges, dynamic references, array formulas and formula dependencies are supported.
• Supports cross sheet references (can work with multiple sheets).
• Essential Calculate does not depend upon Microsoft Excel and thus enables you to perform calculations independent of Excel.
• Supports custom functions with n-number of optional arguments.
• Culture-sensitive decimal separator and argument separator .
• Calculate can work without any sheets or references.
• It also supports CalcQuick or Indexer method for calculation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.957382321357727, "perplexity": 5060.401516520511}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154214.63/warc/CC-MAIN-20210801154943-20210801184943-00312.warc.gz"} |
http://esrf.eu/home/news/spotlight/content-news/spotlight/spotlight225.html | # New iron hydrides synthesised under pressure
12-05-2015
Many calculations have predicted exceptional properties of hydrides at high pressure, namely a high hydrogen storage capacity and a possible conventional high Tc superconductivity. Yet, these properties remain to be tested. Scientists working at the ESRF have observed an increase of hydrogen content in iron under pressure, with the formation of two novel hydrides, FeH~2 and FeH3. In particular, the structure of FeH~2 is remarkable with alternate planes of Fe and of atomic H.
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Most transition metals form hydrides in contact with H2, usually with a H:metal ratio close to 1. However, this chemistry is predicted to change under pressure. Numerous first-principles calculations have predicted that novel forms of polyhydrides should exist at high pressure, such as LiH2, LiH6 [1], FeH3 and FeH4 [2]. A rough rule is emerging, i.e. the hydrogen content in hydrides should increase significantly with pressure. So far, this rule and the existence of such unusual stoichiometries have yet to be confirmed experimentally.
In this study, we have discovered that under pressure the hydrogen content in iron dramatically increases in discontinuous steps: heating FeH in the presence of hydrogen leads to the formation of FeH~2 at 67 GPa and of FeH3 at 86 GPa.
Iron samples (~2 µm thick) were loaded in diamond anvil cells with hydrogen as a pressure medium, and YAG-laser heated using the online setup at the ID27 beamline. dhcp-FeH forms under pressure at 3.5 GPa and no changes in the X-ray diffraction (XRD) pattern were observed when purely compressing the sample up to 136 GPa. Laser-heating, however, allowed us to induce a chemical reaction, identified by the disappearance of the dhcp-FeH XRD peaks and the appearance of new XRD peaks (Figure 1). Rietveld refinement of the integrated new diffraction patterns give, at 67 GPa, FeH~2 with a tetragonal unit cell (space group I4/mmm), and at 86 GPa, FeH3 with a simple cubic unit cell (space group Pm-3m) corresponding to the structure predicted in Ref. [2] at 300 GPa. The structural positional parameters of the H atoms have been determined by first-principles density functional calculations. The new phase diagram for the Fe-H system is shown in Figure 2.
Figure 1. X-ray diffraction pattern for FeH, FeH2 and FeH3 (right: image plates, left: integrated patterns).
This discovery is a textbook case for the expected increase of the hydrogen stoichiometry in hydrides with pressure. The investigation of the electronic properties of these novel iron compounds could reveal unusual magnetism, dimensionality, proton zero point energy, and correlations. Apart from these fundamental issues, the observed stability of novel iron hydrides under high pressure is of significant importance for planetary interior modelling since iron and hydrogen are two of their main constituents [3]. In particular, hydrogen is considered as a possible light element in the Earth’s core, which is mainly composed of iron.
Figure 2. Phase diagram of the Fe-H system showing the different structures.
Principal publication and authors
New Iron Hydrides under Pressure, C.M. Pépin (a), A. Dewaele (a), P. Loubeyre (a) and M. Mezouar (b), Phys. Rev. Letter 113, 265504 (2014).
(a) Commissariat à l’Énergie Atomique, Direction des Applications Militaires Île de France Arpajon (France)
(b) ESRF
References
[1] E. Zurek, R. Hoffmann, N. Ashcroft, A. Oganov, and A. Lyakhov, Proc. Natl. Acad. Sci. U.S.A. 106, 17640 (2009).
[2] Z. Bazhanova, A. Oganov, and O. Gianola, Phys. Usp. 55, 489 (2012).
[3] D. Stevenson, Nature (London) 268, 1300 (1977).
Top image: Sketch of a diamond anvil cell and microscope image of the sample studied. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.843957781791687, "perplexity": 4480.779051785206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376467.86/warc/CC-MAIN-20210307105633-20210307135633-00320.warc.gz"} |
https://0cch.com/archives/2014/04/ | ### 一个查找指定栈回溯符号并执行命令的Windbg脚本
>a<${$arg0} 5 ntdll* ~n就表示在前5个线程里寻找栈回溯有关ntdll的线程，然后执行~n命令挂起线程。
### Windows 8.1中获得系统版本信息的方法
With the release of Windows 8.1, the behavior of the GetVersionEx API has changed in the value it will return for the operating system version. The value returned by the GetVersionEx function now depends on how the application is manifested.
Applications not manifested for Windows 8.1 will return the Windows 8 OS version value (6.2). Once an application is manifested for a given operating system version, GetVersionEx will always return the version that the application is manifested for in future releases.
===============想睡觉的分割线==================== | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.15640024840831757, "perplexity": 2422.40342195132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187820927.48/warc/CC-MAIN-20171017052945-20171017072945-00599.warc.gz"} |
http://ramiro18dabbufhdu.soup.io/ | You are at the newest post.
## January022015
### 8 Romantic Green Coffee Tone And Colon Thin Holidays
There's nothing simpler than putting a espresso pod into the maker and simply pushing a button. You can forget about grinding beans and then attempting to measure out the ideal quantity. Espresso pods are a lot like a tea bag, but filled with precisely the correct quantity of coffee rather. Every cup you brew will usually style new and perfect. There's nothing you can do to screw issues up!
Almost any equipment utilized to pop popcorn can be used to roast coffee beans. The scorching air poppers are great, however, you may want to roast outdoors or in your garage as they blow the chaff out and can be messy. There are a quantity of manufacturers of home coffee roaster, such as a number of that use a gasoline grill and rotisserie.
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Now there are two thoughts about green bean 900. Some people say it is good while some say it gives aspect results. So, the only way to discover the reality is to find facts. It might be talked about that green bean 900 is considered as 1 of the most effective ingredients. And the pure type is safe. But, now there are a great deal of phony products with fake components. So, this could be the purpose why the component is getting lots of criticism.
Before we begin keep in mind that beans are at their peak of fresh flavor in between 1 working day and two weeks after they are roasted. The initial twenty-4 hours right after the coffee comes out of the roaster is regarded as the resting time period. Throughout this time espresso are as well new to style their very best and require a small time to rest. If you are a house roaster, please maintain this in thoughts when roasting Green Coffee Tone Pills. If you brew coffee during this resting period, as well much carbon dioxide is still contained in the bean. When drinking water is added the beans will increase to the top preventing the espresso from being extracted fully into the drinking water. Numerous purchasers will have little specialty retailers that roast their beans fresh every working day, make sure you maintain the above info in thoughts when buying freshly roasted espresso.
One of the very best places to buy Green Coffee Tone Pills wholesale is on-line rather than buying them from nearby specialty retailers. A number of distributors will let you buy bulk Green Coffee Tone Pills for a fraction of the price you'd pay at the grocery shop for beans that have already been roasted.
Coffee trees develop berries and the fruit of the plant is picked by hand and then processed by machines that strip off the fruit flesh. The raw seeds undergo a type of drying process, after which what arrive out are our coffee beans. They aren't what you'll see in a bag of coffee beans from the supermarket, however. Green Coffee Tone Pills are much smaller sized than the roasted variety and don't have extremely great flavor.
There are several advantages to taking Pure Green Coffee Tone Review capsules containing 50%twenty five Cholorogenic acid. Chlorogenic acid is an amazing natural compound which not only delivers about fast excess weight loss, but it also helps to boost metabolism, is a powerful antioxidant, controls blood stress and strengthens the immune system.
So the month-to-month shipping and delivery or espresso club is an outstanding present and gives you numerous choices to choose from. Ideally you can use these choices to make a good gift and to assist mankind too.
For coffee enthusiasts, there is nothing that compares with the smell of a freshly brewed cup. Even the scent of the floor espresso prior to being brewed can make your senses tingle. But the reality of the make a difference is that these beans do not arrive by that scent naturally. Green espresso beans actually scent kind of bland. It is the roasting of the beans that gives them the aroma that so many people crave. Now envision if you could get that fantastic scent of roasting beans in your home. Really you can by obtaining 1 of the numerous great espresso roasting devices on the market.
## January082014
### How Significantly Caffeine Is There In A Cup Of Coffee
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One more gain of green coffee tone tone pills from complete nutrition beans is that they can be substantially more cost-effective to order as an unroasted bean. They are also often ordered from smaller grower spouse and children operations in the establishing world. Ninety p.c or extra of coffee is grown by modest farmers in developing nations. When it is bought and will come less than the Certified Fair Trade, this is promoted by the entire world organisation for fair trade and carries a label to establish manufacturers that have paid out growers a appropriate price tag. This can be to the value of five occasions additional than what other companies pay to growers.
The leaves of the coffee plant are huge and are shiny, dim environmentally friendly. The espresso plant helps make a charming dwelling plant. The espresso plant can develop to be three ft tall.
Analysis is however getting conducted, but it is speculated that coffee can help protect the body towards liver sickness and numerous kinds of most cancers as perfectly. It also seems to lessen the possibility of obtaining type two diabetes, and serves to reduced blood sugar degrees.
With in excess of 28 decades of medical expertise, alternative well being expert and celeb nutritionist is one of the world's major gurus on super meals, organic medicine, purely natural remedies and normal wellness.
Whether or not you contact this your "Slender Coffee" or your "Slimming Coffee" I get the emotion that if you are currently a espresso drinker then creating the alternative to try this "straightforward pounds reduction" discovery will be right up your ally.
I am not always a pattern follower, as traits can promptly fade. Subsequent a terrible development can outcome in a closet total of high-squandered mother denims and leg warmers. I do believe that the green movement must not be treated as a trend. In truth, dwelling a eco-friendly way should be attempted by each and every American loved ones. By going environmentally friendly, a household can exercise social responsibility and make small but significant changes in their everyday life.
Uncover the ideas of the health practitioner. If you have issues together with high-blood pres-sure, stomach concerns o-r sugar troubles ahead of you make the most of the green beans extract. When you do get complications and in particular confusion and faintness get your physicians watch moreover be incredibly mindful about regardless of whether you need to actually deal with an car o-r travel wherever the light headedness might cause you to fall. I understand that is popular awareness none the significantly less you might be shocked how many ladies and adult men have so little. LOL!
Typically all coffee roasters are delicate to your residence voltage. Roast on a circuit that isn't really becoming utilized at the identical time you are roasting. Do not get confused. Home roasters are not the similar as business roasters they should be permitted to amazing down in involving batches or you could established off the thermal protection that will demand the roaster to be reset by the maker. You never want that variety of trouble.
## December302013
### What Goes On To Coffee All Through The Roasted Strategy
Green Coffee Tone Pills
Simple body weight decline is hurriedly used to merchandise we see on late evening television promising over weight males and women of all ages the rapid fix. Typically this variety of simple weight decline adverts get the variety of an infomercial at times lasting as very long as a late night time film.
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I am not normally a craze follower, as traits can quickly fade. Next a bad pattern can end result in a closet whole of higher-wasted mom jeans and leg warmers. I do believe that the eco-friendly motion ought to not be treated as a development. In point, dwelling a inexperienced way should really be tried by every American relatives. By likely environmentally friendly, a relatives can observe social accountability and make modest but meaningful alterations in their every day lives.
If you are in the commercial/industrial stop of the decorating/structure business enterprise, what mood maximizing colors do the job very best in a reception location, an open up region with a lot of staff members and the place of work of the CEO?
Residence roasted is once more than and in excess of repeatedly getting popular with all creation of digital drum roasters that definitely support de-strain the technique. Are likely to be a lot of some people who consist of identified sorts of really roasted beans using distinctive warmth pop corn producers.
The pancreas generates glucagon and insulin, which are 2 very important hormones for our system to operate ordinarily. The research that have been built all-around chlorogenic acids that can be observed in the Green Coffee Tone Pills beam, proved the truth that people today can have their sugar metabolism affected by it in a optimistic fashion and this will materialize in the write-up-eating period.
In buy to drop fat there is no alternative to proscribing your calorie intake so that you take in less energy than your body is employing. The overall body thus has to make up the big difference by burning body fat. And anything you can do to reduce your calorie consumption, or maximize the charge you burn extra fat (this sort of as physical exercise) will therefore support pace up the course of action of shedding body weight.
Natural is a label popular in our grocery retailers and anything we are most familiar with. The main big difference between a basic Natural and organic and USDA licensed organic and natural eco-labels is that United States Section of Agriculture labels organic and natural goods dependent on their certification requirements.
### Does The Laxative Diet Plan Get The Job Done
green coffee tone reviews
Exactly where do individuals espresso anti-oxidants arrive from - and are they maximum in green coffee tone reviews beans or beans that have been roasted? According to new research, the organic anti-oxidants in green coffee beans identified as chlorogenic acids are destroyed when coffee is heated to large temperatures. But during the roasting course of action they're changed with much more strong anti-oxidants known as maillard response goods. The anti-oxidants generated by the maillard response when coffee is roasted can make up the vast majority of the espresso anti-oxidants Individuals consume when they sip a cup of coffee.
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A twelve-7 days examination of the efficiency of green beans extract was just currently achieved. Via the investigate, each individual a person of these who employed the extract dropped about sixteen.6% of their typical body fats together with 11% of their overall physique unwanted fat, all with no changes inside of their food diets or exercise sessions. Additionally, those people who tried it drop on common seventeen lbs . just about every without the need of section results.
Who realized that getting espresso can be so complicated? Though it could choose time to investigate and educate by yourself about eco-labels and their meanings, you will be superior off realizing the distinction a espresso acquire can make.
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Nevertheless, the course of action of getting the espresso beans picked from their cherries, dried, processed, delivered, packed and Ultimately placed on the shelf for the buyer is a extended one particular, and each working day that passes is a further working day your espresso has aged and missing it is really flavour.
Overview: Coffee bean belongs to rubiaceae household. It originates in tropical zone of Africa with 2000 yrs planting record. By 525 BC, Arabs had now started to plant coffee beans and right after 15 century it started to be planted thoroughly. Then soon after 18 century, this plant commonly distributed in Europe, Asia, Africa and Latin The usa, turning out to be the head of 3 beverage crops in the world.
Ahead of now the coffee beans are commonly roasting, decaf coffee is really initially getting that will help you go through decaffeination. Straight and indirect approach, bodily activity ingesting h2o tactic, CO2/O2 remedy and triglceride software could be the unique approaches of generating decaf coffee.
### Unfavorable Outcomes Of Espresso
green Coffee tone pills from complete nutrition
The positive aspects of coffee have been acknowledged for generations. In simple fact, green Coffee tone pills from complete nutrition bean extract has been bought in wellbeing foodstuff outlets and huge-box chains for several years, marketed as a nutritional complement, a effective antioxidant, an electricity booster, an anti-hypertensive health supplement, and a stimulant for nutritious circulation.
The only kinds who should not drink espresso are individuals who are expecting, are struggling from heart illness or children who will turn into restless with the extra electrical power thanks to the caffeine.
To be regarded as Organic and natural or Certified Organic and natural espresso, farmers expanding the item should really encourage use of renewable assets, and follow water and soil conservation. Espresso is grown without having pesticides or artificial fertilizers, and there is no use of genetic engineering.
This is what is recognized as a "super foods", and is packed whole of anti-oxidants. Anti-oxidants, as you almost certainly know, are wonderful for supporting general overall health, but they also attack excess fat cells, way too, and that means you ought to be able to reduce a fair quantity of fat. When you burn up extra fat in this fashion, you end up with extra "energy" to burn, and that can lead into more activity on your element, burning extra energy!
Residence roasted is when more than and in excess of having common with all development of electronic drum roasters that assist de-strain the course of action. Are usually many some folks who include things like observed methods of really roasted beans employing exceptional hot-air area corn producers.
As soon as you've got begun roasting, you can keep keep track of of the progress by listening. At a specified level, as sugars caramelize and dampness evaporates from the beans, they will make a cracking noise. At this issue, if you favor a light-weight roast, you can contemplate your beans completed. Darker roasts have to have for a longer period, although, and some favor to cook dinner the beans as considerably as the second "crack." At this place, the beans may even burst aside, leaving smaller items. In addition to these sound cues, you can also choose how considerably together your beans are, and when you look at them finished, by their scent.
Is there a big difference in between how kids, teenagers, grownups and the aged respond to shade? If there is, what colors and forms of colors (pastel vs. bright) are greatest for which age group and why?
The all-natural reward of the Green Coffee 800 is the potent Anti-oxidants present in the Green Coffee are valuable for human system for numerous good reasons.
### Green Coffee Extract Bodyweight Decline
green coffee tone pills from complete nutrition
Blooms Svelte, contains Coffea canephora. This is truly an extract of green coffee tone pills from complete nutrition bean and several persons who are struggling from weight problems have efficiently managed it by applying this extract.
No. But what I locate is that way too much saturation of a colour can have a adverse impact (like yellow and nausea) Most of the time it is delicate and you don't recognize it is the space colour or the computer monitor color or or or... Intuitively your entire body will get treatment of you, that blue home will produce tranquil as lengthy as you require tranquil designed.
Seek out out the coffee and tea that have traveled the minimum distance to get to you and also aim at supporting regional, impartial farms, cafs, and roasters.
Recent study has proven that coffee is also wealthy with antioxidants. In truth espresso is the range a person source of antioxidants among the individuals in the state, with black tea becoming next. However, black tea truly consists of a significantly larger sized total of antioxidants, and as its attractiveness catches on, it may well quickly go into the amount one particular posture.
When you examine the Garcinia Cambogia opinions, you will in all probability choose for this products, if only mainly because it does not artificially elevate your metabolic fee. This tends to make it greater for people with heart difficulties or other health issues, but of system you really should often speak to your physician prior to starting any food plan.
Having said that, the procedure of owning the coffee beans picked from their cherries, dried, processed, transported, packed and Eventually placed on the shelf for the purchaser is a extended just one, and each day that passes is another working day your espresso has aged and shed it is really flavour.
Roasting espresso is in essence to regulate the flavour that arrives out of the beans. Environmentally friendly unroasted coffee beans are smaller sized and much heavier than a roasted espresso bean. Unroasted coffee bean can also previous for decades when stored effectively and carefully. Nonetheless, roasted coffee beans are equally essential by people today simply because the coffees that men and women consume are the consequence of coffee roasting the darker the coffee bean soon after roasting, the more robust the style. In actuality, folks are so into coffee roasting that coffee roasting specialty outlets emerged. These coffee roasters use only clean espresso beans roasted into different warmth levels to give their faithful patrons some excellent tasting coffee.
The normal profit of the Green Coffee 800 is the highly effective Anti-oxidants existing in the Green Coffee are useful for human system for a variety of explanations.
## December292013
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Blooms Svelte, incorporates Coffea canephora. This is really an extract of green coffee tone reviews bean and lots of folks who are struggling from excess weight concerns have effectively managed it by utilizing this extract.
The only types who shouldn't consume coffee are these who are pregnant, are suffering from heart condition or kids who will turn into restless with the excess energy thanks to the caffeine.
New innovations are staying designed routinely for the best solutions to command the human body of your fats. The researchers suggest the having problem could result in a signalling system, which prompts the body to take in more fats. Amidst the several promising experiments, Slimshots have made a breakthrough in the battle versus being overweight. With this, you can management metabolic swelling that potential customers to serious success in the weight problems-connected difficulties.
This is what is identified as a "super food", and is packed total of anti-oxidants. Anti-oxidants, as you likely know, are great for supporting in general overall health, but they also assault fat cells, too, and that implies you should really be ready to eliminate a reasonable amount of money of weight. When you melt away fats in this way, you end up with far more "power" to burn up, and that can lead into far more activity on your section, burning far more calories!
002 saw the start of the S350, the significant advance becoming the large pressure steam jet enabling cappuccino and other frothy drinks to be made available. 2007 saw the start of the most recent design, the C400, compact with a classy contemporary retro look with field main electricity efficiency. With the start of this equipment manufacturing was moved from Basingstoke to Shanghai, the filter pack output stays in Basingstoke.
However, the bold consequences of green coffee extract on body weight decline have astonished many. In a the latest clinical research, executed in India, participants misplaced about 10 % of their body bodyweight in 22 weeks, all with no alter to their each day diet plan or exercise regimen. They lose whopping seventeen lbs ., on average, just by adding the green coffee complement to their everyday regimen.
It is essential to start out with a table that is no cost of any pre-present paints, stains, or grime. (If these are existing, take away with a little bit of paint thinner and scrub with a heavy duty wire brush.) Carefully sand the desk completely smooth with medium grit sandpaper. Eliminate all excessive dust soon after sanding with a tack cloth.
Members ranged from 22 to forty six decades of age and ended up deemed clinically overweight. They ingested products containing either seven hundred milligrams or 1,050 milligrams of the extract about 50 % an hour prior to each and every meal, consuming loads of drinking water to counter the bitterness of the tablets. By the end of the research, lots of returned to their typical fat.
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The Nature's Ideal web-site description of Verdesse is "We are happy to present Verdesse, a solution built for folks pursuing a calorie managed body weight loss diet plan. Verdesse is a decaffeinated Green Coffee Tone Pills extract named Svetol containing high stages of energetic compounds named chlorogenic acids, and a person in distinct which is called 5-caffeoylquinic acid. The coffee extract in Verdesse is made from unroasted espresso beans, because the typical roasting procedure destroys many of the all-natural compounds.
A great basic principle would be to sustain a history with regards to your pursuits of using the supplements. A lot of folks confuse difficulties pursuing any period of time and the realities get misunderstood. A journal will undoubtedly enable in trying to keep your data files pretty exact. For any having system it really is a exceptional approach to maintain a log and history you are snacking information to assist you regulate your consuming plan to be able to get extraordinary innovations together with aid hold your details exact concerning any type of unwelcome side results of inexperienced beans extract which could probably arrive.
Research is however becoming done, but it is speculated that coffee can assist guard the overall body from liver ailment and lots of styles of cancer as very well. It also seems to lower the chance of receiving type two diabetes, and serves to reduce blood sugar amounts.
I know for-a confirmed reality that putting products to your eating plan routine or any type of alter in your life time together with touring may possibly have an impact in your tummy and bowel motions. I found out any individual experienced acquire-n a laxative and the aches handed they experienced great effects. I uncovered a different specific associated far more fibre for their dinners and their problems vanished they located accomplishment. That is just some detail for a single to look at.
No matter if you connect with this your "Slender Espresso" or your "Slimming Coffee" I get the sensation that if you are previously a espresso drinker then making the choice to attempt this "effortless body weight reduction" discovery will be proper up your ally.
The certain tannins in coffee very easily minimize the cariogenic imminent of foods. In just vitro experiments demonstrate that these polyphenolic worsens could interfere with glucosyltransferase work out of mutans streptococci, which could reduce cavity enducing plaque development.
Overview: Coffee bean belongs to rubiaceae household. It originates in tropical zone of Africa with 2000 years planting historical past. By 525 BC, Arabs experienced currently begun to plant coffee beans and after 15 century it commenced to be planted thoroughly. Then just after eighteen century, this plant widely dispersed in Europe, Asia, Africa and Latin The us, becoming the head of 3 beverage crops in the planet.
## December272013
### Green Coffee Bean Extract. Wonder Diet Capsule
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The Arabica tree is imagined to have originated from the northeastern part of Africa. The trees are now discovered all above the earth. Gourmand coffee beans that have been generated from the Arabica bushes are ordinarily taken from the bushes three moments every single year for approximately thirty years.
To me, supporting tiny farmers throughout the world is vital as they are the ones escalating and retaining espresso crops. In addition to pretty compensating a farmer's challenging work, it is critical to glimpse for licensed organic and natural labels for coffee that is grown devoid of destructive chemicals.
Exactly where do all those coffee anti-oxidants arrive from - and are they best in green coffee tone Reviews beans or beans that have been roasted? According to new study, the all-natural anti-oxidants in green coffee beans called chlorogenic acids are destroyed when coffee is heated to large temperatures. But during the roasting approach they're changed with a lot more strong antioxidants called maillard reaction solutions. The anti-oxidants developed by the maillard response when espresso is roasted makes up the majority of the coffee antioxidants People in america take in when they sip a cup of coffee.
Tea leaves and in particular coffee grounds make exceptional compost. Coffee's substantial nitrogen content has designed it a fertilizer of preference considering the fact that days of yore. Composting leaves and grounds aids retain organic and natural squander out of landfills, will make terrific soil, and keeps waste baskets dry. If you you should not have a heap to toss it on, just distribute coffee grounds on the top of your plants' soil.
It is really welcome news that coffee has antioxidants considering the fact that several Us residents eat adequate antioxidant-wealthy fruits and vegetables. On the other hand, it's far better to consume espresso in moderation and get at least some of your antioxidants from other resources. Much too a great deal caffeine contributes to sleeplessness, blood pressure elevations and heartburn. It also raises degrees of the strain hormone, cortisol. Take pleasure in a cup of freshly roasted espresso if you can tolerate the caffeine, but will not make it your only resource of anti-oxidants.
## December262013
### The Health Gains Of Green Coffee Beans Extract
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The fantasy that way too much coffee can guide to coronary heart condition has been disproved by a team of researchers immediately after conducting a lengthy examine. They surmised that the correlation that experienced previously existed in between coronary heart disorder and coffee drinkers can now be attributed to the truth that quite a few shoppers who drink huge amounts of coffee also are inclined to smoke and get very little training. Nevertheless, persons with large blood strain are nonetheless advised to minimize coffee ingestion. Coffee may perhaps also secure against Parkinson's disease, and maybe Alzheimer's condition.
Several coffee consumers have espresso styles on the kitchen area desk and elect to get beans and operate their pretty possess espresso. They will come across a full new excellent, flavor and odor, when a single grinds their particular beans. If you select to create your have espresso in-the benefit of your possess assets, you may perhaps make it in accordance with your individual style.
Are there any side consequences? Like coffee, green coffee tone tone pills from complete nutrition bean does have a modest quantity of caffeine in it. So if you happen to be really delicate it truly is very best to check out it early in the working day so you will not have sleeplessness difficulties. With any complement of system you need to constantly search for the approval of your health care marketing consultant.
Software: Coffee bean is the most essential substance to make beverages. Espresso beverage is drunk worldwide and welcomed many folks. Aside from, espresso bean is also a health care product. We can acquire caffeine and serve it as anesthetic, diuretic, cordial and cardio tonic.
Quite a few men and women may possibly will need quite espresso mills and that is some factor to just take into account when looking for the quite best espresso mill. Some mills are extremely loud when grinding the beans.
A French press espresso maker fully removes the need for filters at all (kind of) and necessitates no electricity to function (kind of). Invented someday during the late 19th century, French presses mix boiling drinking water with floor coffee allowing for a richer and darker taste to arise from the espresso beans. A plunger presses the espresso grounds to the base of the coffee with a metal strainer or cheesecloth, eliminating the grounds from the espresso as its currently being poured.
The Robusta espresso beans are generally developed in an ecosystem that is extra humid as effectively as hotter than the usual setting for the Arabica beans. The Robusta coffee beans are largely grown in Central The us, parts of South The us and the southeast part of Asia. The Robusta beans are usually blended with the beans from the Arabica trees. The mixing provides a extra sophisticated and richer flavor to gourmet flavored coffee.
Fans of dwelling brewed cups of Joe can also search espresso aisles at their regional grocery retailers for other very good finds. Natural and organic is a time period you will now see on coffee baggage, and it signifies espresso beans are grown with no genetic engineering or synthetic fertilizers.
Tocopherols in espresso possess e Vitamin. Tocopherols in coffee act significantly like area phenols. Tocopherols function as antioxidants and performs a aspect in the development of carbohydrates and lipids. Tocopherols more around guard cells versus hurt and ruin absolutely free radicals in the human system. Tocopherols existing in coffee have turn into crucial for ocular health and fitness, which associated with the functionality of the human eye, and the pores and skin. Tocopherols have the capability to inhibit the raising of gallbladder and kidney stones and could possibly give safety for you versus colon cancer.
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Men and women really cherished the freshly roasted espresso and its exotic flavor and aroma. Even if there have been any roasting stores, men and women most popular the taste of dwelling roasted espresso since they felt that the previous resulted in a cup of swill. Green Coffee Tone beans were saved by the people and roasted on a day-to-day basis as for each their demands.
Almost 70 a long time back, roasting coffee in the kitchen at property was as a great deal a section of the day-to-day regimen of persons as observing Tv is ours now-a-days. Most of the folks at that time could not find the money for to acquire roasted espresso so they relished roasting espresso at property with the help of solid iron stoves, above their fireplaces or out on their porches. A coffee roasting store at that time was deemed a luxurious enjoyed only by the city community dwellers.
New inventions are being made often for the perfect goods to handle the entire body of your fats. The scientists recommend the consuming disorder might set off a signalling procedure, which prompts the physique to consume additional fat. Amidst the many promising experiments, Slimshots have manufactured a breakthrough in the battle from obesity. With this, you can command metabolic irritation that sales opportunities to continual final results in the weight problems-similar difficulties.
Not only does certified good trade coffee and tea help ensure living wages and safe functioning circumstances for farmers, but TransFair and Rainforest Alliance both include things like demanding environmental benchmarks in their certification standards.
002 observed the launch of the S350, the major advance currently being the large stress steam jet enabling cappuccino and other frothy beverages to be made available. 2007 noticed the launch of the most recent model, the C400, compact with a trendy modern day retro glimpse with field top energy performance. With the start of this machine production was moved from Basingstoke to Shanghai, the filter pack production stays in Basingstoke.
Having said that, the approach of acquiring the espresso beans picked from their cherries, dried, processed, transported, packed and Last but not least put on the shelf for the buyer is a extended a person, and just about every day that passes is a different day your coffee has aged and dropped it is flavour.
Tocopherols in espresso have e Vitamin. Tocopherols in espresso act substantially like location phenols. Tocopherols functionality as anti-oxidants and plays a element in the development of carbohydrates and lipids. Tocopherols much more over guard cells in opposition to harm and wipe out absolutely free radicals in the human body. Tocopherols existing in espresso have become vital for ocular health and fitness, which affiliated with the perform of the human eye, and the pores and skin. Tocopherols have the potential to inhibit the rising of gallbladder and kidney stones and could possibly supply defense for you towards colon cancer.
Generally all coffee roasters are sensitive to your residence voltage. Roast on a circuit that isn't really staying made use of at the very same time you are roasting. You should not get puzzled. Dwelling roasters are not the same as industrial roasters they have to be allowed to amazing down in amongst batches or you could established off the thermal protection that will have to have the roaster to be reset by the company. You don't want that form of hassle.
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## December252013
### The Place To Obtain Green Coffee Beans
Obtaining a coffee maker that works by using a reusable filter like a gold plated mesh display screen or stainless metal filter can get rid of the need to have for wasteful paper filters. Though, according to some, the use of reusable steel filters alterations the taste of the espresso. Unbleached cotton filters are reusable and keep coffee tasting as fantastic (if not superior in my belief) than metallic filters without making paper filter waste. Simply clean and dry the filters immediately after each use and you are great to go.
Hen Welcoming also refers to espresso beans that are grown in places that supply great hen habitats. Plantations that follow Chook Welcoming requirements should preserve tree canopies, deliver various species of crops and protect purely natural habitats for birds.
But I am empowering myself to now stick to a new extra fair weight loss program. I will much more strictly adhere to the Mediterranean Diet program, that contains far more greens, fruit, olive oil and fish--and decrease my carbs. And at the exact same time, develop into a "health and fitness center rat," and try out to training at my club routinely occasions a 7 days, be it swimming, yoga, or on the cardiovascular devices.
There are a large selection of connoisseur flavored coffees which includes mint, raspberry, vanilla, hazelnut, chocolate, and many other tasty varieties. When you are picking flavored espresso beans it can be a good thought to purchase large high-quality coffee beans. They cost much more than the standard flavored espresso beans, but ordinarily you will observe the greater taste that is provided with the top quality beans.
Folks are recommended SlimShots to shallow tri-system. This indicates that you have to acquire three doses in per day foundation. The formulation is manufactured up of diverse natural ingredients that start out functioning inside 5 minutes. Each and every components commences with your morning breakfast till your very last meal. The quite first dose consumed in the morning involves Green Coffee Tone Review (read this article), mate, meadowsweet, cherry stalk and green tea. Having said that, the prescription for noon is a ideal mix of cider vinegar, apple pectin, and guar gum. And lastly, the important ingredients in the evening include pineapple, grape marc extracts, chromium and orange pores and skin.
The Robusta espresso beans are generally developed in an natural environment that is a lot more humid as very well as hotter than the usual surroundings for the Arabica beans. The Robusta coffee beans are typically grown in Central The united states, parts of South The usa and the southeast portion of Asia. The Robusta beans are commonly blended with the beans from the Arabica trees. The blending offers a much more complex and richer flavor to connoisseur flavored espresso.
Possessing a great mill is just a very good remedy to establish your individual individual excellent tea. Naturally, it may be hard-to select the mill, but with the aid of the Web, you may perhaps read through opinions to be ready to identify which is the better for you. There are various good mills out there, you only have to have to see them.
A whole lot of folks depend on having their early morning stroll as a way to begin their day. Some really feel like they are unable to development, without first having their sit down in other places. Below are a couple of strategies to guarantee it is the prime of place. ought to you require good quality coffee.
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So, I have henceforth supplied up on the fad diet plan capsule that is the Green Coffee Tone Free Trial tablet. I have also resigned to give up any "wonder heal" to eliminate fat as perfectly. The aged adage that if it seems much too great to be real, then it is too superior to be real, is almost certainly the soundest science in relation to this trend diet program fiasco but.
"Natural and organic fat burners have now been gaining a good deal of reputation simply because they are protected and cost-free of part results. A few of commonly the most well-liked forms currently being encouraged by training experts now have garcinia cambogia, inexperienced espresso obtain and strawberry ketones. All a few of these are proven fat busters and appetite suppressants" claims a SlimmersExpert.co.united kingdom spokesperson.
Though there is no rating process for coffee makers like EnergyStar or Purchaser Reviews Environmentally friendly Selections, there are solutions readily available to people who want to decrease there affect on the earth by reducing the usage of electricity and disposable items that can fill up our landfills. Paper espresso filters are a disposable products that triggers an environmental effects and is not eco-helpful.
With a ruler, evaluate the width and size of the table top. Get rid of one/2" all all over from this measurement, and mark with a pencil on the A two Z Essentials "Paisley Stripe" Print Scrapbook Paper. Reduce the paper out in sections to cover the entire tabletop with a one/two" border of un-papered surface close to the edge. Implement a generous quantity of Mod Podge or other decoupage medium to the back again of the paper. Include an further amount of money to the table itself. Carefully press the paper to the table. Smooth out any bubbles by utilizing a popsicle adhere or your fingers. Permit the paper dry right away.
Straightforward fat reduction is hurriedly applied to goods we see on late evening television promising over weight males and women of all ages the fast take care of. Usually this kind of simple fat decline advertisements choose the form of an infomercial sometimes lasting as extended as a late night time movie.
One more review reveals that the anti-oxidants contained in espresso are a lot more effective than individuals coming from fruits and vegetables. This is for the reason that not all of the nutrition listed here have completed significantly to safeguard the unique from a variety of conditions.
GCB MAX or Purely natural Espresso Vegetable MAX is normally the most well-liked of all eco-friendly espresso fat reduction health supplements as a outcome of substantial amount of chlorogenic p in it. It incorporates 800 mg of genuine environmentally friendly espresso get with at the extremely least fifty% chlorogenic acid. This is a health supplement that might warranty bodyweight loss without diet plan and physical exercise. Persons have now been obtaining fantastic outcomes with it. Mixing it with food plan and work out is confirmed to make absolutely sure a lot faster excess fat decline.
Like many of us, the new calendar year represents a modify in our lives that we could be trying to get or striving to. If weight decline is your major precedence then like many, you have been hunting for a position to commence. No matter whether it's your poor feeding on behaviors, workout plan, or complement program, we are all even now hunting for that "miracle pill" that will allow us to hold indulging in substantial caloric foodstuff that are not useful to our well being in any way.
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Could green coffee tone tone pills from complete nutrition bean extract be the long-awaited pounds decline tremendous tablet? New study implies that this could certainly be the scenario, delivering a ray of hope for all people who have struggled to drop the additional lbs .. The purely natural electricity of green coffee extract could permit individuals to shed fat properly and successfully, with no necessitating a transform in exercise or consuming habits.
No. But what I find is that as well significantly saturation of a shade can have a damaging effect (like yellow and nausea) Most of the time it is subtle and you you should not know it is the space shade or the computer system display screen shade or or or... Intuitively your human body will take treatment of you, that blue place will create tranquil as lengthy as you will need tranquil established.
To be viewed as Natural or Accredited Natural and organic espresso, farmers escalating the product or service need to motivate use of renewable assets, and follow water and soil conservation. Espresso is grown without the need of pesticides or artificial fertilizers, and there is no use of genetic engineering.
Who realized that obtaining espresso can be so complicated? When it might get time to investigation and teach your self about eco-labels and their meanings, you will be greater off knowing the change a coffee buy can make.
The other individuals, which includes Antony Wild, a author of Black Gold, have advised a pounds and individuals mixed up in espresso market place get significantly in surplus because of to Rhinehart's estimates.
Bravi, Behmor, or Gene Caf are without a doubt the best drum roasters. Burman Coffee Trader Organization has carefully tested each espresso roaster and offered excellent facts with regards to the roasters. On their leading website you will discover the execs and cons regarding the use of these espresso roasters. These roasters are a bit far more high priced.
If you are in the commercial/industrial conclusion of the decorating/style and design company, what mood improving hues do the job greatest in a reception spot, an open place with a lot of workforce and the office environment of the CEO?
Method: The very first technique is dry system which is effortless. Commonly we use electric powered dry oven to dry the beans, and then take out the shells and peels. Right after impurity filtering, we can get the beans we can promote in the industry. The 2nd a person is soaked procedure which has the measures like clean, ferment, dry and so on. This process now is adopted in most planting parts.
## December242013
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This undertaking generally consists of spray paints and glues. As a result, it is best done in a nicely ventilated outdoors workspace that is protected with newspaper.
When you examine the Garcinia Cambogia assessments, you will likely opt for this product, if only for the reason that it would not artificially increase your metabolic level. This tends to make it far better for men and women with coronary heart difficulties or other wellness troubles, but of program you should normally communicate to your medical doctor right before starting any diet.
"The benefits ended up astounding" explained Dr. Oz, the men and women of all ages who eaten green coffee tone extracts; Suggested Web site, beans missing an ordinary of 17 lbs just about every - ten.5% of their in general overall body bodyweight & sixteen% of their human body body fat! as noticed a the latest "Dr. Oz" televised method.
In most instances, there are experiments that demonstrate that these products will perform, but you should not assume to be equipped to sit again on the sofa, consume potato chips, and check out the pounds just fall off. The dietary supplements only do element of the get the job done, and you will have to do the rest to truly get rid of a good deal of fat.
Drinking of SlimShots has been shown to assist minimize calorie ingestion with no leading to any detrimental aspect outcomes. You do truly feel hungry but not immensely. Far more so, it slashes in your hungry by eight hrs. This raises a sensation of satiety, the extended-expression outcome of a food. In this way, you develop into hungry again when food stuff is eaten at the subsequent food. This urge for food suppressant assists a excess weight management plan that involves wholesome calorie mindful foods.
GCB MAX or Natural Coffee Vegetable MAX is typically the most preferred of all eco-friendly espresso bodyweight loss health supplements as a end result of higher amount of chlorogenic p in it. It incorporates 800 mg of true green espresso receive with at the incredibly least 50% chlorogenic acid. This is a dietary supplement that may possibly guarantee pounds decline without the need of eating plan and exercise. People have now been locating good effects with it. Mixing it with diet program and exercise is tested to make confident significantly quicker excess fat reduction.
Nonetheless, thankfully, in the late 60s and 70s, there commenced a back again to the basics' movement. Men and women resorted to the forgotten and dead industry of coffee roasting once more. Espresso is the 2nd most traded commodity immediately after oil on earth. The proliferation of espresso shops and properties has achieved these huge proportions that sure neighborhoods have in fact placed a ban on any new espresso cafes.
## December222013
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The fantasy that as well considerably coffee can lead to heart disease has been disproved by a group of scientists following conducting a lengthy research. They surmised that the correlation that experienced earlier existed amongst heart ailment and espresso drinkers can now be attributed to the fact that quite a few people who consume huge amounts of coffee also are likely to smoke and get small training. However, individuals with high blood tension are nonetheless encouraged to lessen coffee consumption. Coffee might also secure towards Parkinson's ailment, and maybe Alzheimer's condition.
Coffee should definitely be consumed in moderation, as described previously mentioned, to acquire the best gains. A honest volume of caffeine might improve psychological comprehension and consciousness. It's presently been recommended that espresso allows intellectual capacity and could lower the possibilities of the progress of Parkinson's disease and Alzheimer's ailment.. Due to the fact caffeine can basically raise the heart price and blood tension caffeine mightn't be rather wholesome for a handful of persons who have heart challenges. The caffeine in espresso may possibly also be contraindicated with particular medicines, such as stimulants or diuretics.
There are a lot of procedures prior to the espresso beans come to be the espresso that you drink. Initially, espresso seeds are planted in wealthy soil in get to improve a great good quality espresso plant. The seed will be taken care of for a long interval of time until it is totally developed and bears fruit. At the time the coffee plant bears fruit, the farmers wait until eventually the fruit is ripe before they start harvesting it. green coffee tone extracts (Going At this website) beans are then made. After all the coffee crops have been ripped of their fruits, the farmers will roast all the espresso beans. Just after they are all roasted in the ideal level, coffee beans are grinded into espresso powder. At the time pulverized, the resulting raw material will then be packed and is all set to be offered.
Coffees are priced from $1.eighty five to$three.00, espressos array in rate from $1.75 to$4.00, and ice blended coffee beverages commence at $4.50 and go up to$five.twenty five. Very hot chocolate or vanilla is readily available from $2.sixty five to$3.40. No sugar added drinks are also available.
Violet (which includes purple and indigo below) is about inspiration, intuition, spirituality. Younger little ones and artists like violet. It incorporates the power of purple with the qualities of blue, so it is a fantastic shade to have all over when you are seeking for that 'muse'. Medically it relates to troubles of the head, eyes, brow, pituitary.
The leaves of the espresso plant are massive and are glossy, darkish environmentally friendly. The espresso plant makes a pretty property plant. The espresso plant can develop to be a few toes tall.
Even though creating gourmand espresso the temperature that the beans ended up roasted at performs an critical position in the taste of the coffee. Europeans usually delight in darker types of coffee beans and Individuals seem to get pleasure from the lighter coloured beans. Connoisseur coffee beans will retain their freshness and flavor better when they are put in oxygen evidence bags. If you get pleasure from espresso there are various versions of gourmand flavored coffee that are obtainable.
## December132013
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To be considered Organic and natural or Accredited Organic and natural coffee, farmers developing the solution must motivate use of renewable methods, and practice drinking water and soil conservation. Coffee is grown devoid of pesticides or artificial fertilizers, and there is no use of genetic engineering.
Just how green coffee tone extracts bean extract contributes to pounds loss is not well recognized. Researchers speculate that it may well velocity up metabolic process, reduce absorption of unwanted fat and glucose in the intestine, lessen insulin levels, or all of the previously mentioned. Substances identified as polyphenols, which are current in a wide variety of plants like the espresso bean, have lengthy been suspected to stimulate unwanted fat decline and may be the active substances selling these venues. On the other hand, deducing their correct system of motion will involve far more analysis.
Virtually 70 several years in the past, roasting coffee in the kitchen area at household was as substantially a component of the everyday routine of individuals as observing Television is ours now-a-days. Most of the folks at that time could not find the money for to purchase roasted coffee so they relished roasting coffee at house with the help of solid iron stoves, over their fireplaces or out on their porches. A coffee roasting store at that time was viewed as a luxury enjoyed only by the city neighborhood dwellers.
The draw back of working with a solitary serve espresso machine is the wasteful use of single provide packs when a number of users, family members or places of work use them. Improved electrical power intake is also an problem, but only if other people use the machine. Single provide coffee makers should be just that-one provide if you want them to be electrical power successful as doable.
Overview: Coffee bean belongs to rubiaceae spouse and children. It originates in tropical zone of Africa with 2000 decades planting history. By 525 BC, Arabs experienced now begun to plant coffee beans and just after fifteen century it commenced to be planted extensively. Then immediately after 18 century, this plant broadly dispersed in Europe, Asia, Africa and Latin The usa, getting the head of 3 beverage crops in the planet.
## November292013
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Persons are approved SlimShots to shallow tri-formula. This indicates that you have to acquire three doses in for each day foundation. The formula is built up of distinct all-natural components that begin working within five minutes. Each individual formulation starts with your morning breakfast till your last meal. The quite to start with dose consumed in the early morning contains green coffee tone green tone pills from complete, mate, meadowsweet, cherry stalk and green tea. However, the prescription for midday is a excellent blend of cider vinegar, apple pectin, and guar gum. Last of all, the crucial ingredients in the night time incorporate pineapple, grape marc extracts, chromium and orange pores and skin.
If you are just commencing to conceive your personal label coffee program, Virtues will help you to offer, marketplace, and create your coffee product. Their dedicated and professional staff will acquire you all the way from the first strategy to the products stream. Based on what you want to offer you your prospects, Virtues can give you green coffee, fair-trade espresso, natural espresso, or whichever coffee profile you desire. You have to have just to ask, and Virtues will support you to define your application just the way your prospects will motivation it.
Having said that, luckily, in the late 60s and 70s, there started off a again to the basics' movement. People today resorted to the neglected and dead marketplace of espresso roasting once again. Coffee is the next most traded commodity following oil on earth. The proliferation of coffee stores and properties has attained such huge proportions that certain neighborhoods have actually put a ban on any new espresso cafes.
Dependent on how darkish you want your roast to be, the roasting procedure can just take from 6 to fifteen minutes or so. Also, the darker you roast your beans, the additional smoke you'll produce, so be certain you have satisfactory ventilation. The roasting approach creates chaff, which is the pores and skin of the beans staying drop. This can be messy if you're not organized.
In most conditions, there are reports that demonstrate that these products and solutions will function, but you shouldn't anticipate to be equipped to sit back again on the couch, eat potato chips, and observe the weight just drop off. The dietary supplements only do part of the perform, and you will have to do the rest to truly reduce a lot of excess weight.
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White is what I refer to as the 'Christ Light' it elevates any colour, so pastels are normally calming and take on the 'higher' attributes of the coloration it originates from, pink is nurturing and spiritual due to the fact it requires the passion and power of purple and clams it down to the nurturing features of pink. Exact with all the other colours. Black is what I refer to as a magnifier. Individuals that test to intimidate by carrying all black are actually just speaking that they need to have course! (Believe of ministers and clergymen on the lookout to God for route-have on black with a white collar) Black magnifies any bit of coloration that is with it, it absorbs all shade whereas white is reflecting all color.
Ingesting of SlimShots has been revealed to assist lower calorie ingestion without the need of leading to any destructive side effects. You do come to feel hungry but not immensely. A lot more so, it slashes in your hungry by eight several hours. This raises a emotion of satiety, the very long-phrase effect of a meal. In this way, you turn out to be hungry once more when foodstuff is eaten at the upcoming food. This hunger suppressant helps a body weight administration system that involves healthful calorie conscious foods.
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https://accessphysiotherapy.mhmedical.com/content.aspx?bookid=2460§ionid=193999253 | Skip to Main Content
×close section menu
## OBJECTIVES
By studying this chapter, you should be able to do the following:
1. Express work, power, and energy in standardized (SI) units and convert those units to others commonly used in exercise physiology.
2. Give a brief explanation of the procedure used to calculate work performed during step, cycle ergometer, and treadmill exercise.
3. Describe the concept behind the measurement of energy expenditure using (a) direct calorimetry and (b) indirect calorimetry.
4. Calculate the following expressions of energy expenditure when given the oxygen uptake in liters per minute: kcal · min−1, ml · kg−1 · min−1, METs, and kcal · kg−1 · hr−1.
5. Estimate energy expenditure during horizontal treadmill walking and running, and cycle ergometry.
6. Describe the procedure used to calculate net efficiency during steady-state exercise; distinguish efficiency from economy.
## OUTLINE
• Units of Measure 17
• Metric System 17
• SI Units 17
• Work and Power Defined 17
• Work 17
• Power 18
• Measurement of Work and Power 18
• Bench Step 18
• Cycle Ergometer 19
• Treadmill 20
• Measurement of Energy Expenditure 21
• Direct Calorimetry 21
• Indirect Calorimetry 22
• Common Expressions of Energy Expenditure 23
• Estimation of Energy Expenditure 24
• Calculation of Exercise Efficiency 25
• Factors That Influence Exercise Efficiency 26
• Running Economy 27
## KEY TERMS
cycle ergometer
direct calorimetry
ergometer
ergometry
indirect calorimetry
kilocalorie (kcal)
MET (metabolic equivalent)
net efficiency
open-circuit spirometry
percent grade
power
relative $V˙O2$
System International (SI) units work
## INTRODUCTION
How much energy do you expend when you run a mile? How fast can you run 100 m? How high can you jump? These questions deal with energy, speed, and explosive power—and so will you as you study exercise physiology. Throughout this text, we will discuss such terms as aerobic and anaerobic power, efficiency, work capacity, and energy expenditure. The purpose of this chapter is to introduce you to some of the most common pieces of equipment and measurements linked to these terms. It is very important to understand this information at the outset, as it is used throughout the text. However, the details associated with specific exercise tests for fitness and performance are covered in detail in Chaps. 15 and 20. Let’s begin with the most basic units of measurement.
## UNITS OF MEASURE
### Metric System
In the United States, the English system of measurement remains in common use. In contrast, the metric system, which is used in most countries, is the standard system of measurement for scientists and is used by almost ...
### Pop-up div Successfully Displayed
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https://www.bas.ac.uk/data/our-data/publication/estimates-of-structural-complexity-in-clonal-plant-morphology-comparisons-of/ | # Estimates of structural complexity in clonal plant morphology: comparisons of grazed and ungrazed Acaena magellanica rhizomes
The aim of this study is to examine the information given by various indices of rhizome morphology that describe grazed and ungrazed rhizome systems of Acaena magellanica (Rosaceae). Internode lengths, branching probabilities, and branching angles were estimated from grazed and ungrazed rhizomes in the field. These parameter values were then used in computer simulations of rhizome growth, and the structural complexity of the simulated rhizomes were described using size, topology, and fractal dimensions. Grazed rhizomes had shorter internodes, higher probabilities of branching, and more open branching angles than ungrazed rhizomes. This resulted in a more directional growth (herring-bone pattern) in the simulated ungrazed rhizomes, whereas the grazed rhizomes had a more space-filling growth pattern. Most indices, even though they are based on different mathematical and theoretical backgrounds, were highly correlated and thus equally good at describing the structural complexity exhibited by the rhizomes. However, indices have different relationships to theories about function, and we suggest that any study of structural complexity of branching systems should use several different indices of shape depending on the questions asked.Key words: Acaena magellanica, fractal dimension, grazing, growth simulation, topology.
### Details
Publication status:
Published
Author(s):
Authors: Moen, Jon, Ingvarsson, Pär K., Walton, David W.H. ORCID record for David W.H. Walton
Date:
1 January, 1999
Journal/Source:
Canadian Journal of Botany / 77
Page(s):
869-876
Digital Object Identifier (DOI):
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https://www.zzzdavid.tech/NAS/ | ## Overview
Neural Architecture Search, or NAS, can be seen as a subtask of AutoML. It tries to automate the designing process of neural network architectures. An architecture search space can be seen as a Directed Asyclic Graph (DAG), and each possible architecture is a subgraph. We denote the search space as $A$, the subgraph $a$, the searched neural network $N(a,w)$. The goals of NAS are often two-fold:
• Weight Optimization
• Architecture Optimization
The constraints are diverse: such as computation (FLOPS), speed (latency), or even energy consumption and memory usage.
## Surveys
### Neural Architecture Search: A Survey
NAS methods encompasses three sub-tasks: designing search space, select search strategy, and use proper estimation strategy.
#### Search Space
The search space defines what architectures are discoverable during search. Search space designs can be largely classified into global search space and cell-based search space.
• Global: chain-structured, chain-structured with skip, segment-based
• Cell-based
Global Search Space
##### Chain-Structured Search Space
The simplist form of search space. Parameters include: the number of layers, the type of layers, hyperparameters of the layers, e.g., number of filters, kernel sizes and strides for convolutional layer.
##### Chain-Structured with Skips
Obviously the idea comes from ResNet, DenseNet or Xception. The branching can be described with layer’s inputs as a function
Taking existing architectures for example, for chain-structured net, $g_i (L_{i-1}^{out}, ..., L_0^{out}) = L_{i-1}^{out}$. For ResNet, $g_i (L_{i-1}^{out}, ..., L_0^{out}) = L_{i-1}^{out} + L_j^{out}, (j < i-1)$. For DenseNet, $g_i (L_{i-1}^{out}, ..., L_0^{out}) = concat(L_{i-1}^{out},...,L_0^{out})$
##### Segment-based
Segment-based method is considered as a global search space design, but with another level of granularity: segment, or block.
Mnas Search Space
Taking Mnas as an example, the network layers are grouped into a many blocks. The skeleton constructed with blocks is predefined. Inside each block, there are a variable number of repeated layers. Blocks are not necessarily the the same, the layer amount and operations can be different.
#### Cell-based Structure
Cell-based architecture is a lot like the segment-based one. However, instead of searching the topology/hyperparameters of each block, cell-based approach defines several types of cells, then connects repeated cells in a pre-defined manner. In such ways, the search space is further downsized.
Illustration of the cell search space
(Zoph, 2018) defined two types of cell: normal cell or reduction cell. Normal cell preserves dimensionality while reduction cell reduces spatial dimensions.
The problem of cell-based structure is that the macro-architecture, i.e. the structure based on cells, are still manually designed. (Liu, 2018b) tried to overcome macro-architecture engineering by introducing hierarchical search space.
#### Search Strategy
##### Reinforcement Learning Strategy
Different RL approaches differ in how they represent the agent’s policy and how they optimize it.
• Architecture generation: agent’s action
• Search space: action space
• Evaluation: agent’s reward is based on an estimate of the performance.
Agent’s Policy:
• RNN policy: sequentially sample a string that encodes the architecture
Training method:
• Proximal Policy Optimization
• Q-learning
##### Neron-Evolutionary Strategy
Neuro-evolutionary methods differ in how they sample parents, udpate populations, and generate offsprings.
In the context of NAS, mutations are local operations, such as adding or removing a layer, altering the hyperparameters of a layer, adding skip connections.
(Liu, 2019b) proposed a continuous relaxation to enable direct gradient-based optimization: instead of fixing a single operation $o_i$ to be executed at a specific layer, the authors compute a convex combination from a set of operations ${o_1, ..., o_m}$. Given a layer input $x$, the layer output $y$ is computed as $y = \sum_{i=1}^m \alpha_i o_i(x)$, $(\alpha_i \geq 0, \sum_{i=1}^m \alpha_i = 1)$
#### Performance Estimation Strategy
Overview of different methods for speeding up performance estimation in NAS
• Lower Fidelity
The bias in the estimation is the gap between the full evaluation. This may not be problematic as only relative ranking is considered. But when the gap is large, the relative ranking may not be stable.
• Learning Curve Extrapolation or Surrogate Model
Using early learning curve of architecture information to extrapolate performance.
• Network Morphism
No inherent search space upper bound. However, strict netwrok morphisms can only make architectures larger, therefore leading to redundancy.
• One-Shot
Training only one supernet, so only validation is needed at evaluation stage. The supernet restricts the search space, and may be limited by GPU memory.
• ENAS learns a RNN controller that samples architectures from the search space and trains the one-shot model based on approximate gradients obtained through REINFORCE.
• DARTS jointly optimize all weights with a continuous relaxation of the search space.
• SNAS optimizes a distribution over the candidate operations. The authors employ the concrete distribution and reparameterization to relax the discrete distribution and make it differentiable.
• Proxyless NAS “binarizes” the weights, maksing out all but one edge per operation to overcome the necessity of keeping the entire supernet model in GPU memory.
It is currently not well understood which biases they introduce into the search if the sampling of distribution of architectures is optimized along with the one-shot model instead of fixing it. For example, an initial bias in exploring certain parts of the search space more than others might lead to the weights of the one-shot model being better adapted for these architectures, which in turn would reinforce the bias of the search to these parts in the search space. This might result in premature convergence of NAS or little correlation between the one-shot and true performance of an architecture.
#### Future Directions
1. Applying NAS to less explored domains, such as image restoration, semantic segmentation, transfer learning, machine translation, GAN, or sensor fusion.
2. Develop MAS methos for multi-task problems, and for multi-objective problems, such as latency and power optimization.
3. Universal NAS on different tasks. More general and flexible search space design.
4. Developing benchmarks.
While NAS has achieved impressive performance, so far it provides little insights into why specific architectures work well and how similar the architectures derived in indepen- dent runs would be. Identifying common motifs, providing an understanding why those motifs are important for high performance, and investigating if these motifs generalize over different problems would be desirable.
## Paper Summary
### Once-For-All: Train One Network and Specialize It For Efficient Deployment
#### Motivation
Designing a once-for-all network that flexibly supports different depths, widths, kernel sizes, and resolutions without retraining.
#### Contributions
• The model training stage and architecture search stage is decoupled.
• A progressive shrinking method to trian the supernet
• Authors: Shulin Zeng, Hanbo Sun, Yu Xing, Xuefei Ning, Yi Shan, Xiaoming Chen, Yu Wang, Huazhong Yang
• Venue: 2020 ASP-DAC
• Institution: Tsinghua University, and Institute of Computing Technology, Chinese Academy of Sciences
• Xplore: https://ieeexplore.ieee.org/document/9045179
#### Motivation
1. Designing a suitable search space for DNN accelerators is still challenging.
2. Evaluating latency across platforms with various architectures and compilers is hard because we don’t have prior knowlege of the hardware & compiler.
#### Contribution
1. A black-box search space tuning method for Instruction Set Architecture (ISA) accelerators.
2. A layer-adaptive latency correction method.
##### The Complete NAS Workflow
Entire NAS workflow
The magic happens in “Basenet Pool” and “Policy-aware Latency”, where the authors constructs a automatic search space design method. “Policy-aware Latency LUT” stands for an improved latency profiling method. Previous works use standalone block latency to build a LUT, ignoring inter-layer latency influences. Such influences are caused by compiler policies, such as “layer fusion” that combines layers to shorten latency. This new latency profiling method aims to take inter-layer latency dependencies into account without knowing the underlying compiler or hardware.
##### How to Optimize Search Space?
Search Space Optimization
SSBN: stands for Search Space Base Network. Based on a specific net from the Search Space, change at most one block at a time to get a SSBN. Because the original Search Space can be regarded as linear combination of SSBNs, this reduces complexity without harming degree of freedom.
Generating Basenet Pool: there are currently 3 types of mainstream hand-designed blocks: VGG, Res, and MobileNet-based. For each type the authors construct a Specified Network (SN), and obtain its SSBNs. All SSBNs together are the Basenet Pool.
Selecting Search Space: for each group of SSBNs, the authors design a cost function combining accuracy and latency. For each group of SSBNs, their costs are first calculated. After that, they are sorted in ascending w.r.t. the cost. A threshold defined by the user selects a range of
SSBNs that goes into the new search space. The new search space is much smaller.
##### How to Correct Block Latency?
The authors design two methods for latency correction. The first one uses overall network latency measurements to calculate the difference between standalone and actual block latency. The second one is a refinement of the first. It takes other block’s latency into account, thus more fine-grained.
• $L_{BNblock}(n,l)$ is the block latency we want to know.
• $n$ is the type of block. $l$ is the index of layer.
• $L_{SNblock}(l)$ is the standalone layer latency.
• $L_{SN}$ is the overal latency of the network.
• $L_{BN}(n,l)$ is the overall latency where the target block is substituted.
This one is difficult to understand.
$L$ is the number of layers (or blocks) in the network. The authors first create $L$ networks preserving the target block but the rest are all one particular type. Then within each of these “Specified Network”, our target block’s latency is measured with method 1. At last, the block latency measurements are averaged to get the estimated “candidate network (CN)” block latency. | {"extraction_info": {"found_math": true, "script_math_tex": 7, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6045535206794739, "perplexity": 4083.161554515497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499758.83/warc/CC-MAIN-20230129180008-20230129210008-00043.warc.gz"} |
https://biblio.ugent.be/publication/1200772 | Ghent University Academic Bibliography
Subharmonic branching at a reversible 1:1 resonance
Maria-Cristina Ciocci (2005) 11(13). p.1119-1135
abstract
We investigate the bifurcation of q-periodic points from a fixed point of a 2-parameter family of n-dimensional reversible diffeomorphisms (n >= 4). The focus is on the codimension 2 non-semisimple 1:1 resonancecase, that is, when the linearization at the fixed point has a pair of double eigenvalues on the unit circle, exp (+/- 2i pi p/q) (with gcd(p, q) = 1), with geometric multiplicity 1. Through a modified version of Lyapunov Schmidt reduction, the reduced bifurcation equations are obtained. These are then analysed using reversible normal form theory and (reversible) singularity theory. We obtain the existence of two branches of symmetric q-periodic orbits bifurcating from the fixed point. We also describe the corresponding bifurcation scenario for a family of reversible systems in the case of a two-fold resonance, cfr. [ 12,5].
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
reversible systems, subharmonic branching, reversible maps, 1:1 resonance, SYSTEMS, BIFURCATIONS, SYMMETRY
journal title
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
J. Differ. Equ. Appl.
volume
11
issue
13
pages
1119 - 1135
Web of Science type
Article
Web of Science id
000233870100002
JCR category
MATHEMATICS, APPLIED
JCR impact factor
0.615 (2005)
JCR rank
75/151 (2005)
JCR quartile
2 (2005)
ISSN
1023-6198
DOI
10.1080/10236190500331214
language
English
UGent publication?
no
classification
A1
id
1200772
handle
http://hdl.handle.net/1854/LU-1200772
date created
2011-03-31 15:40:29
date last changed
2016-12-19 15:37:59
```@article{1200772,
abstract = {We investigate the bifurcation of q-periodic points from a fixed point of a 2-parameter family of n-dimensional reversible diffeomorphisms (n {\textrangle}= 4). The focus is on the codimension 2 non-semisimple 1:1 resonancecase, that is, when the linearization at the fixed point has a pair of double eigenvalues on the unit circle, exp (+/- 2i pi p/q) (with gcd(p, q) = 1), with geometric multiplicity 1. Through a modified version of Lyapunov Schmidt reduction, the reduced bifurcation equations are obtained. These are then analysed using reversible normal form theory and (reversible) singularity theory. We obtain the existence of two branches of symmetric q-periodic orbits bifurcating from the fixed point. We also describe the corresponding bifurcation scenario for a family of reversible systems in the case of a two-fold resonance, cfr. [ 12,5].},
author = {Ciocci, Maria-Cristina},
issn = {1023-6198},
journal = {JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS},
keyword = {reversible systems,subharmonic branching,reversible maps,1:1 resonance,SYSTEMS,BIFURCATIONS,SYMMETRY},
language = {eng},
number = {13},
pages = {1119--1135},
title = {Subharmonic branching at a reversible 1:1 resonance},
url = {http://dx.doi.org/10.1080/10236190500331214},
volume = {11},
year = {2005},
}
```
Chicago
Ciocci, Maria-Cristina. 2005. “Subharmonic Branching at a Reversible 1:1 Resonance.” Journal of Difference Equations and Applications 11 (13): 1119–1135.
APA
Ciocci, M.-C. (2005). Subharmonic branching at a reversible 1:1 resonance. JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 11(13), 1119–1135.
Vancouver
1.
Ciocci M-C. Subharmonic branching at a reversible 1:1 resonance. JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS. 2005;11(13):1119–35.
MLA
Ciocci, Maria-Cristina. “Subharmonic Branching at a Reversible 1:1 Resonance.” JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 11.13 (2005): 1119–1135. Print. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8294590711593628, "perplexity": 6774.6800769348765}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794863662.15/warc/CC-MAIN-20180520170536-20180520190536-00030.warc.gz"} |
https://txt.binnyva.com/2007/05/extracting-daa-image-files/ | 25 May
I once got a DAA file from a P2P network. Initally I was at a lose about how to extract the data. Finally I found a method that works. First get the ‘poweriso’ utilty for Linux. You can get it at the PowerISO website
Then convert the DAA file to a regular ISO file using this command…
poweriso convert daa_image.daa -o iso_image.iso -ot iso
[tags]iso,daa,image,poweriso,command,linux[/tags]
## One thought on “Extracting DAA Image Files”
1. magicISO is a better tool for extracting any kind of image file… | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33232665061950684, "perplexity": 8604.814426594568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107912807.78/warc/CC-MAIN-20201031032847-20201031062847-00498.warc.gz"} |
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# How much time is needed to do well in IR?
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How much time is needed to do well in IR? [#permalink] 04 Jul 2013, 09:10
I am scoring 48-49 in quant and 40+ in verbal section on the GMAT Prep test. I want to schedule the exam but I haven't started studying for IR section. So my question is, how much time and effort is needed to do well on this section? I would be happy with 6 or more.
Thanks.
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Re: How much time is needed to do well in IR? [#permalink] 04 Jul 2013, 09:30
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Baracuda123 wrote:
I am scoring 48-49 in quant and 40+ in verbal section on the GMAT Prep test. I want to schedule the exam but I haven't started studying for IR section. So my question is, how much time and effort is needed to do well on this section? I would be happy with 6 or more.
Thanks.
Check this video.
The 150 Hours Rule of Thumb
"One of the most fundamental concepts to understand about your GMAT prep is how much time you'll need to devote to your studies. Prior to the recent test change, which saw the inclusion of the Integrated Reasoning section into the GMAT, we recommended 100-120 hours of prep time in order to achieve a 700+ score. With the new section added, however, and the extra skills study that goes along with it, we recommend devoting 120-150 hours of prep time to land that same sort of score."
From it follows that from total of 120-150 hours of preparation you should spend approximately 30 hours or more learning about integrated reasoning data sets and the new question types that accompany them.
Hope it helps.
_________________
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Joined: 09 Jul 2012
Posts: 60
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Kudos [?]: 13 [0], given: 29
Re: How much time is needed to do well in IR? [#permalink] 05 Jul 2013, 06:20
Bunuel wrote:
Baracuda123 wrote:
I am scoring 48-49 in quant and 40+ in verbal section on the GMAT Prep test. I want to schedule the exam but I haven't started studying for IR section. So my question is, how much time and effort is needed to do well on this section? I would be happy with 6 or more.
Thanks.
Check this video.
The 150 Hours Rule of Thumb
"One of the most fundamental concepts to understand about your GMAT prep is how much time you'll need to devote to your studies. Prior to the recent test change, which saw the inclusion of the Integrated Reasoning section into the GMAT, we recommended 100-120 hours of prep time in order to achieve a 700+ score. With the new section added, however, and the extra skills study that goes along with it, we recommend devoting 120-150 hours of prep time to land that same sort of score."
From it follows that from total of 12-150 hours of preparation you should spend approximately 30 hours or more learning about integrated reasoning data sets and the new question types that accompany them.
Hope it helps.
Gotcha, thanks!
Re: How much time is needed to do well in IR? [#permalink] 05 Jul 2013, 06:20
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Display posts from previous: Sort by | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5122868418693542, "perplexity": 2823.7843465506153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661095.66/warc/CC-MAIN-20150417045741-00030-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2013.33.4157 | # American Institute of Mathematical Sciences
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September 2013, 33(9): 4157-4171. doi: 10.3934/dcds.2013.33.4157
## Simple skew category algebras associated with minimal partially defined dynamical systems
1 University West, Department of Engineering Science, SE-46186 Trollhättan, Sweden 2 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
Received June 2012 Revised February 2013 Published March 2013
In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $Top^{op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e ∈ ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.
Citation: Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157
##### References:
[1] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $C^*$-dynamical systems, Proc. of Edinburgh Math. Soc. (2), 37 (1993), 119-124. doi: 10.1017/S0013091500018733. Google Scholar [2] B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras," Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006. Google Scholar [3] K. R. Davidson, "$C^*$-Algebras by Example," Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996. Google Scholar [4] E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras," Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence RI, 1967. Google Scholar [5] G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci., 16 (1980), 299-311. doi: 10.2977/prims/1195187509. Google Scholar [6] R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems, Canad. J. Math., 58 (2006), 39-63. doi: 10.4153/CJM-2006-003-x. Google Scholar [7] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras, Tokyo. J. Math., 13 (1990), 251-257. doi: 10.3836/tjm/1270132260. Google Scholar [8] A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras, Comm. Math. Phys., 81 (1981), 429-435. Google Scholar [9] G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem, Algebra Colloq., 13 (2006), 181-196. Google Scholar [10] P. Lundström, Separable groupoid rings, Comm. Algebra, 34 (2006), 3029-3041. doi: 10.1080/00927870600639906. Google Scholar [11] P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems, Internat. J. Math., 23 (2012), 1250040, 16 pp. doi: 10.1142/S0129167X12500401. Google Scholar [12] T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 929-957. doi: 10.2977/prims/1195180873. Google Scholar [13] T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 959-970. doi: 10.2977/prims/1195180874. Google Scholar [14] J. R. Munkres, "Topology," $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 2000. Google Scholar [15] F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2), 37 (1936), 116-229. doi: 10.2307/1968693. Google Scholar [16] F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2), 44 (1943), 716-808. Google Scholar [17] J. von Neumann, "Collected Works. Vol. III: Rings of Operators," Pergamon Press, New York-Oxford-London-Paris, 1961. Google Scholar [18] J. Öinert, Simple group graded rings and maximal commutativity, in "Operator Structures and Dynamical Systems," Contemporary Mathematics, 503, American Mathematical Society, Providence, RI, (2009), 159-175. doi: 10.1090/conm/503/09899. Google Scholar [19] J. Öinert and P. Lundström, Commutativity and ideals in category crossed products, Proc. Est. Acad. Sci., 59 (2010), 338-346. doi: 10.3176/proc.2010.4.13. Google Scholar [20] J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings, Comm. Algebra, 40 (2012), 1860-1871. doi: 10.1080/00927872.2011.559181. Google Scholar [21] J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings, Int. Electron. J. Algebra, 11 (2012), 46-63. Google Scholar [22] J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions, J. Algebra Appl., 12 (2013), 1250192, 16 pp. doi: 10.1142/S0219498812501927. Google Scholar [23] J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 287-302. doi: 10.4303/jglta/S070404. Google Scholar [24] J. Öinert and S. D. Silvestrov, On a correspondence between ideals and commutativity in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 216-220. Google Scholar [25] J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings, in "Generalized Lie Theory in Mathematics, Physics and Beyond" (eds. S. Silvestrov, E. Paal, V. Abramov and A. Stolin), Springer, Berlin, (2009), 281-296. doi: 10.1007/978-3-540-85332-9_24. Google Scholar [26] J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings, Acta Appl. Math., 108 (2009), 585-602. doi: 10.1007/s10440-009-9435-3. Google Scholar [27] J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, (). Google Scholar [28] A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras," Progress in Mathematics, 170, Birkhäuser Boston, Inc., Boston, MA, 1999. Google Scholar [29] G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups," London Mathematical Society Monographs, 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. Google Scholar [30] S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems, J. London Math. Soc. (2), 18 (1978), 534-538. doi: 10.1112/jlms/s2-18.3.534. Google Scholar [31] J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system, Houston J. Math., 18 (1992), 139-152. Google Scholar [32] J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260. Google Scholar [33] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products, Internat. J. Math., 18 (2007), 455-471. doi: 10.1142/S0129167X07004217. Google Scholar [34] C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$, in "Methods of Spectral Analysis in Mathematical Physics," Operator Theory: Advances and Applications, 186, Birkhäuser Verlag, (2009), 391-401. doi: 10.1007/978-3-7643-8755-6_19. Google Scholar [35] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products, Acta Appl. Math., 108 (2009), 547-559. doi: 10.1007/s10440-009-9506-5. Google Scholar [36] M. Takesaki, "Theory of Operator Algebras. II," Encyclopaedia of Mathematical Sciences, 125, Operator Algebras and Non-Commutative Geometry, 6, Springer-Verlag, Berlin, 2003. Google Scholar [37] J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics," World Scientific Advanced Series in Dynamical Systems, 3, World Scientific Publishing Co., Singapore, 1987. Google Scholar [38] J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras," Lecture Notes Series, 2, Seoul National University, Research Institute of Mathematics, Global Anal. Research Center, Seoul, 1992. Google Scholar [39] D. P. Williams, "Crossed Products of $C^*$-Algebras," Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. Google Scholar [40] G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9), 47 (1968), 101-239. Google Scholar
show all references
##### References:
[1] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $C^*$-dynamical systems, Proc. of Edinburgh Math. Soc. (2), 37 (1993), 119-124. doi: 10.1017/S0013091500018733. Google Scholar [2] B. Blackadar, "Operator Algebras. Theory of $C^*$-Algebras and von Neumann Algebras," Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006. Google Scholar [3] K. R. Davidson, "$C^*$-Algebras by Example," Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996. Google Scholar [4] E. G. Effros and F. Hahn, "Locally Compact Transformation Groups and $C^*$-Algebras," Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence RI, 1967. Google Scholar [5] G. A. Elliott, Some simple $C^*$-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci., 16 (1980), 299-311. doi: 10.2977/prims/1195187509. Google Scholar [6] R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems, Canad. J. Math., 58 (2006), 39-63. doi: 10.4153/CJM-2006-003-x. Google Scholar [7] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras, Tokyo. J. Math., 13 (1990), 251-257. doi: 10.3836/tjm/1270132260. Google Scholar [8] A. Kishimoto, Outer automorphisms and reduced crossed products of simple $C^*$-algebras, Comm. Math. Phys., 81 (1981), 429-435. Google Scholar [9] G. Liu and F. Li, On strongly groupoid graded rings and the corresponding Clifford theorem, Algebra Colloq., 13 (2006), 181-196. Google Scholar [10] P. Lundström, Separable groupoid rings, Comm. Algebra, 34 (2006), 3029-3041. doi: 10.1080/00927870600639906. Google Scholar [11] P. Lundström and J. Öinert, Skew category algebras associated with partially defined dynamical systems, Internat. J. Math., 23 (2012), 1250040, 16 pp. doi: 10.1142/S0129167X12500401. Google Scholar [12] T. Masuda, Groupoid dynamical systems and crossed product. I. The case of W*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 929-957. doi: 10.2977/prims/1195180873. Google Scholar [13] T. Masuda, Groupoid dynamical systems and crossed product. II. The case of C*-systems, Publ. Res. Inst. Math. Sci., 20 (1984), 959-970. doi: 10.2977/prims/1195180874. Google Scholar [14] J. R. Munkres, "Topology," $2^{nd}$ edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 2000. Google Scholar [15] F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2), 37 (1936), 116-229. doi: 10.2307/1968693. Google Scholar [16] F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2), 44 (1943), 716-808. Google Scholar [17] J. von Neumann, "Collected Works. Vol. III: Rings of Operators," Pergamon Press, New York-Oxford-London-Paris, 1961. Google Scholar [18] J. Öinert, Simple group graded rings and maximal commutativity, in "Operator Structures and Dynamical Systems," Contemporary Mathematics, 503, American Mathematical Society, Providence, RI, (2009), 159-175. doi: 10.1090/conm/503/09899. Google Scholar [19] J. Öinert and P. Lundström, Commutativity and ideals in category crossed products, Proc. Est. Acad. Sci., 59 (2010), 338-346. doi: 10.3176/proc.2010.4.13. Google Scholar [20] J. Öinert and P. Lundström, The ideal intersection property for groupoid graded rings, Comm. Algebra, 40 (2012), 1860-1871. doi: 10.1080/00927872.2011.559181. Google Scholar [21] J. Öinert and P. Lundström, Miyashita action in strongly groupoid graded rings, Int. Electron. J. Algebra, 11 (2012), 46-63. Google Scholar [22] J. Öinert, J. Richter and S. D. Silvestrov, Maximal commutative subalgebras and simplicity of Ore extensions, J. Algebra Appl., 12 (2013), 1250192, 16 pp. doi: 10.1142/S0219498812501927. Google Scholar [23] J. Öinert and S. D. Silvestrov, Commutativity and ideals in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 287-302. doi: 10.4303/jglta/S070404. Google Scholar [24] J. Öinert and S. D. Silvestrov, On a correspondence between ideals and commutativity in algebraic crossed products, J. Gen. Lie T. Appl., 2 (2008), 216-220. Google Scholar [25] J. Öinert and S. D. Silvestrov, Crossed product-like and pre-crystalline graded rings, in "Generalized Lie Theory in Mathematics, Physics and Beyond" (eds. S. Silvestrov, E. Paal, V. Abramov and A. Stolin), Springer, Berlin, (2009), 281-296. doi: 10.1007/978-3-540-85332-9_24. Google Scholar [26] J. Öinert, S. Silvestrov, T. Theohari-Apostolidi and H. Vavatsoulas, Commutativity and ideals in strongly graded rings, Acta Appl. Math., 108 (2009), 585-602. doi: 10.1007/s10440-009-9435-3. Google Scholar [27] J. Öinert, Simplicity of skew group rings of abelian groups,, to appear in Communications in Algebra, (). Google Scholar [28] A. L. T. Paterson, "Groupoids, Inverse Semigroups, and their Operator Algebras," Progress in Mathematics, 170, Birkhäuser Boston, Inc., Boston, MA, 1999. Google Scholar [29] G. K. Pedersen, "$C^*$-algebras and their Automorphism Groups," London Mathematical Society Monographs, 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. Google Scholar [30] S. C. Power, Simplicity of $C^*$-algebras of minimal dynamical systems, J. London Math. Soc. (2), 18 (1978), 534-538. doi: 10.1112/jlms/s2-18.3.534. Google Scholar [31] J. C. Quigg and J. S. Spielberg, Regularity and hyporegularity in $C^*$-dynamical system, Houston J. Math., 18 (1992), 139-152. Google Scholar [32] J. S. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260. Google Scholar [33] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems and commutants in crossed products, Internat. J. Math., 18 (2007), 455-471. doi: 10.1142/S0129167X07004217. Google Scholar [34] C. Svensson, S. Silvestrov and M. de Jeu, Connections between dynamical systems and crossed products of Banach algebras by $\mathbbZ$, in "Methods of Spectral Analysis in Mathematical Physics," Operator Theory: Advances and Applications, 186, Birkhäuser Verlag, (2009), 391-401. doi: 10.1007/978-3-7643-8755-6_19. Google Scholar [35] C. Svensson, S. Silvestrov and M. de Jeu, Dynamical systems associated with crossed products, Acta Appl. Math., 108 (2009), 547-559. doi: 10.1007/s10440-009-9506-5. Google Scholar [36] M. Takesaki, "Theory of Operator Algebras. II," Encyclopaedia of Mathematical Sciences, 125, Operator Algebras and Non-Commutative Geometry, 6, Springer-Verlag, Berlin, 2003. Google Scholar [37] J. Tomiyama, "Invitation to $C^*$-Algebras and Topological Dynamics," World Scientific Advanced Series in Dynamical Systems, 3, World Scientific Publishing Co., Singapore, 1987. Google Scholar [38] J. Tomiyama, "The Interplay Between Topological Dynamics and Theory of $C^*$-Algebras," Lecture Notes Series, 2, Seoul National University, Research Institute of Mathematics, Global Anal. Research Center, Seoul, 1992. Google Scholar [39] D. P. Williams, "Crossed Products of $C^*$-Algebras," Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. Google Scholar [40] G. Zeller-Meier, Produits croisés d'une $C^*$-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9), 47 (1968), 101-239. Google Scholar
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2019 Impact Factor: 1.338 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7605360150337219, "perplexity": 4589.99978934119}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487611089.19/warc/CC-MAIN-20210613222907-20210614012907-00336.warc.gz"} |
https://www.isa-afp.org/entries/Hoare_Time.html | # Hoare Logics for Time Bounds
Title: Hoare Logics for Time Bounds Authors: Maximilian P. L. Haslbeck and Tobias Nipkow Submission date: 2018-02-26 Abstract: We study three different Hoare logics for reasoning about time bounds of imperative programs and formalize them in Isabelle/HOL: a classical Hoare like logic due to Nielson, a logic with potentials due to Carbonneaux et al. and a separation logic following work by Atkey, Chaguérand and Pottier. These logics are formally shown to be sound and complete. Verification condition generators are developed and are shown sound and complete too. We also consider variants of the systems where we abstract from multiplicative constants in the running time bounds, thus supporting a big-O style of reasoning. Finally we compare the expressive power of the three systems. BibTeX: ```@article{Hoare_Time-AFP, author = {Maximilian P. L. Haslbeck and Tobias Nipkow}, title = {Hoare Logics for Time Bounds}, journal = {Archive of Formal Proofs}, month = feb, year = 2018, note = {\url{http://isa-afp.org/entries/Hoare_Time.html}, Formal proof development}, ISSN = {2150-914x}, }``` License: BSD License Depends on: Separation_Algebra | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9737182855606079, "perplexity": 2183.2951503502686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650188.31/warc/CC-MAIN-20180324093251-20180324113251-00304.warc.gz"} |
http://eprint.iacr.org/2003/195 | Cryptology ePrint Archive: Report 2003/195
Public Key Encryption with keyword Search
Dan Boneh and Giovanni Di Crescenzo and Rafail Ostrovsky and Giuseppe Persiano
Abstract: We study the problem of searching on data that is encrypted using a public key system. Consider user Bob who sends email to user Alice encrypted under Alice's public key. An email gateway wants to test whether the email contains the keyword urgent' so that it could route the email accordingly. Alice, on the other hand does not wish to give the gateway the ability to decrypt all her messages. We define and construct a mechanism that enables Alice to provide a key to the gateway that enables the gateway to test whether the word urgent' is a keyword in the email without learning anything else about the email. We refer to this mechanism as <I>Public Key Encryption with keyword Search</I>. As another example, consider a mail server that stores various messages publicly encrypted for Alice by others. Using our mechanism Alice can send the mail server a key that will enable the server to identify all messages containing some specific keyword, but learn nothing else. We define the concept of public key encryption with keyword search and give several constructions.
Category / Keywords: public-key cryptography /
Date: received 22 Sep 2003, last revised 24 Mar 2004
Contact author: dabo at cs stanford edu
Available format(s): PDF | BibTeX Citation
Short URL: ia.cr/2003/195
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https://publikationen.bibliothek.kit.edu/1000129679 | # Tests for multivariate normality—a critical review with emphasis on weighted $L^2$-statistics
Ebner, Bruno 1; Henze, Norbert 1
1 Institut für Stochastik (STOCH), Karlsruher Institut für Technologie (KIT)
##### Abstract:
This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted L$^{2}$-statistics. Since weighted L$^{2}$-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt.
Zugehörige Institution(en) am KIT Institut für Stochastik (STOCH) Publikationstyp Zeitschriftenaufsatz Publikationsmonat/-jahr 12.2020 Sprache Englisch Identifikator ISSN: 1133-0686, 1863-8260 KITopen-ID: 1000129679 Erschienen in TEST Verlag Springer Band 29 Heft 4 Seiten 845–892 Vorab online veröffentlicht am 01.12.2020 Nachgewiesen in DimensionsWeb of Science
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
KITopen Landing Page | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46829953789711, "perplexity": 13570.246288649454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00252.warc.gz"} |
https://www.quantumstudy.com/tag/work-energy-power/ | A neutron travelling with a velocity v and K.E. ‘E’ collides perfectly elastically head on with nucleus…
Q: A neutron travelling with a velocity v and K.E. ‘E’ collides perfectly elastically head on with nucleus of an atom of mass number A at rest. The fraction of total energy retained by the neutron is :
(a) $\displaystyle (\frac{A-1}{A+1})^2$
(b) $\displaystyle (\frac{A-1}{A})^2$
(c) $\displaystyle (\frac{A+1}{A-1})^2$
(d) $\displaystyle (\frac{A-1}{A})^2$
Ans: (a)
velocity of body after collision is
$\displaystyle v_1 = \frac{(m_1-m_2)u_1}{m_1 +m_2}$
$\displaystyle v_1 = \frac{(1-A)u_1}{1 + A}$
Fraction of K.E retained is
$\displaystyle \frac{1/2 m v_1^2}{1/2 m u_1^2} = (\frac{v_1}{u_1})^2$
$\displaystyle = (\frac{(1-A)}{1 + A})^2$
A bullet is fired at a target with a velocity of 80 m/s and penetrates 50 cm into it. If this bullet were fired into…
A bullet is fired at a target with a velocity of 80 m/s and penetrates 50 cm into it. If this bullet were fired into a target 25 cm thick with equal velocity, with what velocity would it emerge, supposing the resistance is uniform and same in both the cases?
(a) √80 m/s
(b) 40/√2 m/s
(c) 40 m/s
(d) 40√2 m/s
Ans: (d)
Sol:
Applying , $\displaystyle v^2 = u^2 + 2 a S$
$\displaystyle 0 = 80^2 + 2 a (0.5)$
a = – 6400 m⁄s2
Again , $\displaystyle v^2 = u^2 + 2 a S$
$\displaystyle v^2 = 80^2 + 2 (-6400) (0.25)$
$\displaystyle v^2 = 6400 - 3200 = 3200$
$\displaystyle v = \sqrt{3200} = 40 \sqrt{2} m/s$
A child on a swing is 1 metre above the ground at the lowest point and 2 metre above the ground…
Q: A child on a swing is 1 metre above the ground at the lowest point and 2 metre above the ground at the highest point. The horizontal speed of the child at the lowest point of the swing is (g = 10 m/s2 ) :
(a) 4.5 m/sec
(b) 6.4 m/sec
(c) 7.8 m/sec
(d) 10 m/sec
Ans: (a)
Sol: $\displaystyle v = \sqrt{2 g (h_2 - h_1)}$
$\displaystyle v = \sqrt{2 \times 10 (2 - 1)}$
$\displaystyle v = \sqrt{20} = 4.5 m/s$
A ball is allowed to fall down with initial speed v from a height of 10 m. It loses 50% kinetic energy…
Q: A ball is allowed to fall down with initial speed v from a height of 10 m. It loses 50% kinetic energy after striking the floor. It reaches to the same height after collision. What is the value of v ?
(a) 28 m/s
(b) 7 m/s
(c) 14 m/s
(d) it is never possible
Ans: (c)
Sol:
To reach same height of 10 m after collision,
Velocity After collision $\displaystyle = \sqrt{2 g h}$
$\displaystyle = \sqrt{2 \times 10 \times 10} = 10\sqrt{2} m/s$
As energy lost in collision is 50% , velocity becomes v⁄√2
∴ Velocity of ball on striking,
v = (10 √2) √2 = 20 m/s
Applying $\displaystyle v^2 = u^2 + 2 a S$
$\displaystyle 20^2 = u^2 + 2 \times 10 \times 10$
$\displaystyle u^2 = 20^2 - 200 = 200$
$\displaystyle u = \sqrt{200} = 10\sqrt{2}$
u = 14.14 m/s
A ball of mass ‘m’ moving with a horizontal velocity ‘v’ strikes the bob of mass ‘m’ of a pendulum at rest. ..
Q: A ball of mass ‘m’ moving with a horizontal velocity ‘v’ strikes the bob of mass ‘m’ of a pendulum at rest. During this collision, the ball sticks with the bob of the pendulum. The height to which the combined mass raises is (g = acceleration due to gravity)
(a) $\displaystyle \frac{v^2}{4 g}$
(b) $\displaystyle \frac{v^2}{8 g}$
(c) $\displaystyle \frac{v^2}{ g}$
(d) $\displaystyle \frac{v^2}{2 g}$
Ans: (b)
Sol: Applying conservation of momentum ,
m v = (m + m)v’
v’ = v/2
Applying conservation of Energy ,
$\displaystyle \frac{1}{2} (2 m) v'^2 = (2 m) g h$
$\displaystyle h = \frac{v'^2}{2 g}$
$\displaystyle h = (\frac{v}{2})^2 \times \frac{1}{2 g}$
$\displaystyle h = \frac{v^2}{8 g}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 62, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8627830743789673, "perplexity": 1386.4979313033004}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655882051.19/warc/CC-MAIN-20200703122347-20200703152347-00511.warc.gz"} |
http://tex.stackexchange.com/questions/58427/any-way-to-change-color-of-not | # Any way to change color of \not
I often use command \not just to cancel off symbols in my equations and i would need an advice on how to change the color of this command.
-
Have a look at the cancel package manual and the disclaimer in the end. – percusse Jun 3 '12 at 20:44
Related Question: Diagonal strikeout starting too low and ending too high. – Peter Grill Jun 4 '12 at 0:31
A very simple method would be to temporarily change the text color and then switch back to black. This can be done by a new command.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{xcolor}
\newcommand*{\cnot}[1][black]{\color{#1}\not\color{black}}
\begin{document}
$\cnot[red]\in$
\end{document}
The new command takes an arbitrary color as optional argument.
-
\newcommand*{\cnot}[1][black]{\mathrel{\color{#1}\not}} would avoid restating the current color. – egreg Jun 3 '12 at 20:55
Even more, the "current color" might not be black. – lockstep Jun 3 '12 at 21:00
do you need fontenc as part of the MWE? – cmhughes Jun 3 '12 at 21:55
@egreg: Hm, I always forget things like \mathrel. – Thorsten Donig Jun 5 '12 at 18:46
@ThorstenDonig A casual user might think that T1 and fontenc are essential for the solution. – egreg Jun 5 '12 at 19:21
show 3 more comments
As mentioned by percusse in the comments, it is better to use cancel package since \not can not strike out more than one variable (say (x+y)). You can define a macro to change the color of strike line as your need:
\documentclass{article}
\usepackage{xcolor}
\usepackage[thicklines]{cancel}
% \renewcommand*{\CancelColor}{\color{red}}%%if only one color is enough
\newcommand*{\mynot}[1]{\renewcommand{\CancelColor}{\color{#1}}\cancel}
\begin{document}
$\mynot{red}{\in}$
$(\mynot{green}{x+y})(x-y)$
$\mynot{black}{x+y}$
\mynot{blue}{This} is cancelled out. % \cancel works in text mode also.
\end{document}
-
The OP does not seem to ask about canceling complete expressions but about negating single relation symbols. For this purpose \not is quite sufficient. Furthermore the author of »cancel« explicitly discourages users from using the package (at the very end of its manual). – Thorsten Donig Jun 5 '12 at 18:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7095630168914795, "perplexity": 3799.7240040172906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163055855/warc/CC-MAIN-20131204131735-00029-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://wordassociations.net/en/words-associated-with/Metric | # Associations to the word «Metric»
## Wiktionary
METRIC, adjective. Of or relating to the metric system of measurement
METRIC, adjective. (music) of or relating to the meter of a piece of music.
METRIC, adjective. (mathematics) (physics) of or relating to distance
METRIC, noun. A measure for something; a means of deriving a quantitative measurement or approximation for otherwise qualitative phenomena (especially used in engineering)
METRIC, noun. (mathematics) A measurement of the "distance" between two points in some metric space: it is a real-valued function d(x,y) between points x and y satisfying the following properties: (1) "positive definiteness": $$d(x,y) \ge 0$$ and $$d(x,y) = 0 \mbox{ iff } x=y$$, (2) "symmetry": $$d(x,y) = d(y,x)$$, and (3) "triangle inequality": $$d(x,y) \le d(x,z) + d(z,y)$$.
METRIC, noun. Abbreviation of metric system.
METRIC, verb. (transitive) (aerospace, systems engineering) To measure or analyse statistical data concerning the quality or effectiveness of a process.
METRIC CARAT, noun. A unit of weight, 0.2 grams, used for gemstones, etc.
METRIC FEET, noun. Plural of metric foot
METRIC FOOT, noun. A unit of length (30 cm) used to sell wood in the United Kingdom.
METRIC LEVEL, noun. (music) The pulses in a metric structure.
METRIC LEVELS, noun. Plural of metric level
METRIC MILE, noun. A distance in international athletics (particularly track), the 1500 meter race, being about a mile (~1609 meters).
METRIC MILE, noun. (US) In high school track, 4 laps of the standard 400 meter track, 1600 meters.
METRIC MILES, noun. Plural of metric mile
METRIC OUNCE, noun. A metric approximation of the ounce; specifically:
METRIC OUNCE, noun. 25 grams.
METRIC OUNCE, noun. 30 grams.
METRIC OUNCES, noun. Plural of metric ounce
METRIC POUND, noun. Usually an informal term for half of a kilogram or 500 grams.
METRIC POUNDS, noun. Plural of metric pound
METRIC SHITLOAD, noun. (vulgar) (humorous) A very large number or amount.
METRIC SPACE, noun. (mathematics) Any space whose elements are points, between any two of which a non-negative real number can be defined as the distance between the points; an example is Euclidean space.
METRIC SPACES, noun. Plural of metric space
METRIC STRUCTURE, noun. (music) regular rhythmic components of a song, including its metre and tempo
METRIC STRUCTURES, noun. Plural of metric structure
METRIC SYSTEM, noun. The system of measurements developed in France in the 1790s and now used worldwide.
METRIC SYSTEM, noun. The modern version of that system, Systeme Internationale d'Unites (International System of Units), or SI system of measurements that is based on the base units of the meter/metre, the kilogram, the second, the ampere, the kelvin, the mole, and the candela.
METRIC SYSTEM, noun. Any variant of that system, that was not codified as SI, such as cgs
METRIC TON, noun. A tonne, a unit of mass equal to 1000 kilograms.
METRIC TONS, noun. Plural of metric ton
## Dictionary definition
METRIC, noun. A function of a topological space that gives, for any two points in the space, a value equal to the distance between them.
METRIC, noun. A decimal unit of measurement of the metric system (based on meters and kilograms and seconds); "convert all the measurements to metric units"; "it is easier to work in metric".
METRIC, noun. A system of related measures that facilitates the quantification of some particular characteristic.
METRIC, adjective. Based on the meter as a standard of measurement; "the metric system"; "metrical equivalents".
METRIC, adjective. The rhythmic arrangement of syllables.
## Wise words
Once a word has been allowed to escape, it cannot be recalled.
Horace | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8416747450828552, "perplexity": 7580.663779759336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145438.12/warc/CC-MAIN-20200221014826-20200221044826-00136.warc.gz"} |
https://toontownrewritten.fandom.com/wiki/Level_12_Cogs | ## FANDOM
2,288 Pages
Level 12 Cogs are the highest possible level of regular Cogs. They have a total of 200 health points. They can be found in boss battles and Cog buildings that have three or more floors, and they are the highest Cogs on the Corporate Ladder.
## Health
The health of Cogs ranging from level 1 to level 11 can be calculated using the equation $HP = (x + 1) * (x + 2)$ where x represents the Cog's level, and HP represents the cog's health points. Level 12 Cogs, however, are given an extra 10% boost. Therefore, instead of having 182 HP- as expected from the standard equation, they have 200 HP. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6664498448371887, "perplexity": 1770.3729885815353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525374.43/warc/CC-MAIN-20190717181736-20190717203736-00547.warc.gz"} |
http://gallabytes.com/2014/03/exporting-literate-haskell-with.html | # Exporting Literate Haskell With HSColour and Pandoc
Posted on March 14, 2014
In yesterday’s post I said I would detail the workflow from literate Haskell + markdown to postable html, so here we go:
Requirements: You need to have Haskell installed somehow. I generally recommend the Haskell Platform installed through your OS’s package manager (pacman, apt-get, homebrew…). At a minimum, you need ghc and cabal-install Next, you need hscolour and pandoc installed. If you already have them, great! If not, run these commands:
cabal install hscolour
cabal install pandoc
Once you have those installed, it’s relatively simple - just run this script:
There you go! Now you’ve highlighted the code bits, and left the rest to be formatted as markdown! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6018745303153992, "perplexity": 11518.218300788607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886123312.44/warc/CC-MAIN-20170823171414-20170823191414-00719.warc.gz"} |
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# Decoding the phase structure of QCD via particle production at high energy
## Abstract
Recent studies based on lattice Monte Carlo simulations of quantum chromodynamics (QCD)—the theory of strong interactions—have demonstrated that at high temperature there is a phase change from confined hadronic matter to a deconfined quark–gluon plasma in which quarks and gluons can travel distances that greatly exceed the size of hadrons. Here we show that the phase structure of such strongly interacting matter can be decoded by analysing particle production in high-energy nuclear collisions within the framework of statistical hadronization, which accounts for the thermal distribution of particle species. Our results represent a phenomenological determination of the location of the phase boundary of strongly interacting matter, and imply quark–hadron duality at this boundary.
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## Acknowledgements
K.R. acknowledges support by the Polish National Science Centre under Maestro grant DEC-2013/10/A/ST2/00106. This work is part of and supported by the DFG Collaborative Research Centre ‘SFB1225/ISOQUANT’.
### Author contributions
All authors contributed equally to the physics analysis and to writing the manuscript.
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Andronic, A., Braun-Munzinger, P., Redlich, K. et al. Decoding the phase structure of QCD via particle production at high energy. Nature 561, 321–330 (2018). https://doi.org/10.1038/s41586-018-0491-6
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### Keywords
• Ultra-relativistic Nuclear Collisions
• Relativistic Heavy Ion Collider (RHIC)
• Heavy Quark
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Journal of High Energy Physics (2022) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9646390676498413, "perplexity": 8453.47235342163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00544.warc.gz"} |
http://mathhelpforum.com/calculus/192865-equation-normal-line.html | # Math Help - Equation of the normal line?
1. ## Equation of the normal line?
Find the equation of the normal line to the graph of y=√ 8x at x=2.
The part that I'm stuck at is...how do you get the derivative of y=√ 8x? The book gives me 4 / √ 8x for the derivative. How do you get that?
I know how to do the rest, just not how to take the derivative of that type of function
Thanks so much
2. ## Re: Equation of the normal line?
Their notation is very messy, in my opinion.
$y=\sqrt{8x}=2\sqrt{2}\cdot x^{0.5}$
Now apply the power rule. If you prefer, you can go straight into the power rule, as we have $8^{0.5}x^{0.5}$ and the $8^{0.5}$ is just a constant. Either way, I'd avoid their notation if possible; I get an answer of $\sqrt{\frac{2}{x}}$ which is equivalent to theirs, although I've simplified slightly. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9641616344451904, "perplexity": 403.30119102767054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936461988.0/warc/CC-MAIN-20150226074101-00306-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://mathmodelsblog.wordpress.com/category/credibility/ | # Introduction to Buhlmann credibility
In this post, we continue our discussion in credibility theory. Suppose that for a particular insured (either an individual entity or a group of insureds), we have observed data $X_1,X_2, \cdots, X_n$ (the numbers of claims or loss amounts). We are interested in setting a rate to cover the claim experience $X_{n+1}$ from the next period. In two previous posts (Examples of Bayesian prediction in insurance, Examples of Bayesian prediction in insurance-continued), we discussed this estimation problem from a Bayesian perspective and presented two examples. In this post, we discuss the Buhlmann credibility model and work the same two examples using the Buhlmann method.
First, let’s further describe the setting of the problem. For a particular insured, the experience data corresponding to various exposure periods are assumed to be independent. Statistically speaking, conditional on a risk parameter $\Theta$, the claim numbers or loss amounts $X_1, \cdots, X_n,X_{n+1}$ are independent and identically distributed. Furthermore, the distribution of the risk characteristics in the population of insureds and potential insureds is represented by $\pi_{\Theta}(\theta)$. The experience (either claim numbers or loss amounts) of a particular insured with risk parameter $\Theta=\theta$ is modeled by the conditional distribution $f_{X \lvert \Theta}(x \lvert \theta)$ given $\Theta=\theta$.
The Buhlmann Credibility Estimator
Given the observations $X_1, \cdots, X_n$ in the prior exposure periods, the Buhlmann credibility estimate $C$ of the claim experience $X_{n+1}$ is
$\displaystyle C=Z \overline{X}+(1-Z)\mu$
where $Z$ is the credibility factor assigned to the observed experience data and $\mu$ is the unconditional mean $E[X]$ (the mean taken over all members of the risk parameter $\Theta$). The credibility factor $Z$ is of the form $\displaystyle Z=\frac{n}{n+K}$ where $n$ is a measure of the exposure size (it is the number of observation periods in our examples) and $\displaystyle K=\frac{E[Var[X \lvert \Theta]]}{Var[E[X \lvert \Theta]]}$. The parameter $K$ will be further explained below.
The Buhlmann credibility estimator $C$ is a linear function of the past data. Note that it is of the form:
$\displaystyle C=Z \overline{X}+(1-Z)\mu=w_0+\sum \limits_{i=1}^{n} w_i X_i$
where $w_0=(1-Z)\mu$ and $\displaystyle w_i=\frac{Z}{n}$ for $i=1, \cdots, n$.
Not only is the Buhlmann credibility estimator a linear estimator, it is the best linear estimator to the Bayesian predictive mean $E[X_{n+1} \lvert X_1, \cdots, X_n]$ and the hypothetical mean $E[X_{n+1} \lvert \Theta]$ in terms of minimizing squared error loss. In other words, the coefficients $w_i$ are obtained in such a way that the following expectations (loss functions) are minimized where the expectations are taken over all observations and/or $\Theta$ (see [1]):
$\displaystyle L_1=E\biggl( \biggl[E[X_{n+1} \lvert \Theta]-w_0-\sum \limits_{i=1}^{n} w_i X_i \biggr]^2 \biggr)$
$\displaystyle L_2=E\biggl( \biggl[E[X_{n+1} \lvert X_1, \cdots, X_n]-w_0-\sum \limits_{i=1}^{n} w_i X_i \biggr]^2 \biggr)$
The Buhlmann Method
As discussed above, the Buhlmann credibility factor $Z=\frac{n}{n+K}$ is chosen such that $C=Z \overline{X}+(1-Z) \mu$ is the best linear approximation to the Bayesian estimate of the next period’s claim experience. Now we focus on the calculation of the parameter $K$.
Conditional on the risk parameter $\Theta$, $E[X \lvert \Theta]$ is called the hypothetical mean and $Var[X \lvert \Theta]$ is called the process variance. Then $\mu=E[X]=E[E[X \lvert \Theta]]$ is the expected value of hypothetical means (the unconditional mean). The total variance of this random process is:
$\displaystyle Var[X]=E[Var[X \lvert \Theta]]+Var[E[X \lvert \Theta]]$
The first part of the total variance $E[Var[X \lvert \Theta]]$ is called the expected value of process variance (EPV) and the second part $Var[E[X \lvert \Theta]]$ is called the variance of the hypothetical means (VHM). The parameter $K$ in the Buhlmann method is simply the ratio $K=\frac{EPV}{VHM}$.
We can get an intuitive feel of this formula by considering the variability of the hypothetical means $E[X \lvert \Theta]$ across many values of the risk parameter $\Theta$. If the entire population of insureds (and potential insureds) is fairly homogeneous with respect to the risk parameter $\Theta$, then $VHM=Var[E[X \lvert \Theta]]$ does not vary a great deal and is relatively small in relation to $EPV=E[Var[X \lvert \Theta]]$. As a result, $K$ is large and $Z$ is closer to 0. This agrees with the notion that in a homogeneous population, the unconditional mean (the overall mean) is of more value as a predictor of the next period’s claim experience. On the other hand, if the population of insureds is heterogeneous with respect to the risk parameter $\Theta$, then the overall mean is of less value as a predictor of future experience and we should reply more on the experience of the particular insured. Again, the Buhlmann formula agrees with this notion. If $VHM=Var[E[X \lvert \Theta]]$ is large relative to $EPV=E[Var[X \lvert \Theta]]$, then $K$ is small and $Z$ is closer to 1.
Another attractive feature of the Buhlmann formula is that as more experience data accumulate (as $n \rightarrow \infty$), the credibility factor $Z$ approaches 1 (the experience data become more and more credible).
Example 1
In this random experiment, there are a big bowl (called B) and two boxes (Box 1 and Box 2). Bowl B consists of a large quantity of balls, 80% of which are white and 20% of which are red. In Box 1, 60% of the balls are labeled 0, 30% are labeled 1 and 10% are labeled 2. In Box 2, 15% of the balls are labeled 0, 35% are labeled 1 and 50% are labeled 2. In the experiment, a ball is selected at random from bowl B. The color of the selected ball from bowl B determines which box to use (if the ball is white, then use Box 1, if red, use Box 2). Then balls are drawn at random from the selected box (Box $i$) repeatedly with replacement and the values of the series of selected balls are recorded. The value of first selected ball is $X_1$, the value of the second selected ball is $X_2$ and so on.
Suppose that your friend performs this random experiment (you do not know whether he uses Box 1 or Box 2) and that his first selected ball is a 1 ($X_1=1$) and his second selected ball is a 2 ($X_2=2$). What is the predicted value $X_3$ of the third selected ball?
This example was solved in (Examples of Bayesian prediction in insurance) using the Bayesian approach. We now work this example in the Buhlmann approach.
The following restates the prior distribution of $\Theta$ and the conditional distribution of $X \lvert \Theta$. We denote “white ball from bowl B” by $\Theta=1$ and “red ball from bowl B” by $\Theta=2$.
$\pi_{\Theta}(1)=0.8$
$\pi_{\Theta}(2)=0.2$
$\displaystyle f_{X \lvert \Theta}(0 \lvert \Theta=1)=0.60$
$\displaystyle f_{X \lvert \Theta}(1 \lvert \Theta=1)=0.30$
$\displaystyle f_{X \lvert \Theta}(2 \lvert \Theta=1)=0.10$
$\displaystyle f_{X \lvert \Theta}(0 \lvert \Theta=2)=0.15$
$\displaystyle f_{X \lvert \Theta}(1 \lvert \Theta=2)=0.35$
$\displaystyle f_{X \lvert \Theta}(2 \lvert \Theta=2)=0.50$
The following computes the conditional means (hypothetical means) and conditional variances (process variances) and the other parameters of the Buhlmann method.
Hypothetical Means
$\displaystyle E[X \lvert \Theta=1]=0.60(0)+0.30(1)+0.10(2)=0.50$
$\displaystyle E[X \lvert \Theta=2]=0.15(0)+0.35(1)+0.50(2)=1.35$
$\displaystyle E[X^2 \lvert \Theta=1]=0.60(0)+0.30(1)+0.10(4)=0.70$
$\displaystyle E[X^2 \lvert \Theta=2]=0.15(0)+0.35(1)+0.50(4)=2.35$
Process Variances
$\displaystyle Var[X \lvert \Theta=1]=0.70-0.50^2=0.45$
$\displaystyle Var[X \lvert \Theta=2]=2.35-1.35^2=0.5275$
Expected Value of the Hypothetical Means
$\displaystyle \mu=E[X]=E[E[X \lvert \Theta]]=0.80(0.50)+0.20(1.35)=0.67$
Expected Value of the Process Variance
$\displaystyle EPV=E[Var[X \lvert \Theta]]=0.8(0.45)+0.20(0.5275)=0.4655$
Variance of the Hypothetical Means
$\displaystyle VHM=Var[E[X \lvert \Theta]]=0.80(0.50)^2+0.20(1.35)^2-0.67^2=0.1156$
Buhlmann Credibility Factor
$\displaystyle K=\frac{4655}{1156}$
$\displaystyle Z=\frac{2}{2+\frac{4655}{1156}}=\frac{2312}{6967}=0.33185$
Buhlmann Credibility Estimate
$\displaystyle C=\frac{2312}{6967} \frac{3}{2}+\frac{4655}{6967} (0.67)=\frac{6586.85}{6967}=0.9454356$
Note that the Bayesian estimate obtained in Examples of Bayesian prediction in insurance is 1.004237288. Under the Buhlmann model, the past claim experience of the insured in this example is assigned 33% weight in projecting the claim frequency in the next period.
Example 2
The number of claims $X$ generated by an insured in a potfolio of independent insurance policies has a Poisson distribution with parameter $\Theta$. In the portfolio of policies, the parameter $\Theta$ varies according to a gamma distribution with parameters $\alpha$ and $\beta$. We have the following conditional distributions of $X$ and distribution of the risk parameter $\Theta$.
$\displaystyle f_{X \lvert \Theta}(x \lvert \theta)=\frac{\theta^x e^{-\theta}}{x!}$ where $x=0,1,2, \cdots$
$\displaystyle \pi_{\Theta}(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}$ where $\Gamma(\cdot)$ is the gamma function.
Suppose that a particular insured in this portfolio has generated 0 and 3 claims in the first 2 policy periods. What is the Buhlmann estimate of the number of claims for this insured in period 3?
Since the conditional distribution of $X$ is Poisson, we have $E[X \lvert \Theta]=\Theta$ and $Var[X \lvert \Theta]=\Theta$. As a result, the $EPV$, $VHM$ and $K$ are:
$\displaystyle EPV=E[\Theta]=\frac{\alpha}{\beta}$
$\displaystyle VHM=Var[\Theta]=\frac{\alpha}{\beta^2}$
$\displaystyle K=\frac{EPV}{VHM}=\beta$
As a result, the credibility factor for a 2-period experience period is $Z=\frac{2}{2+\beta}$ and the Buhlmann estimate of the claim frequency in the next period is:
$\displaystyle C=\frac{2}{2+\beta} \thinspace \biggl(\frac{3}{2}\biggr)+\frac{\beta}{2+\beta} \thinspace \biggl(\frac{\alpha}{\beta}\biggr)$
To generalize the above results, suppose that we have observed $X_1=x_1, \cdots, X_n=x_n$ for this insured in the prior periods. Then the Buhlmann estimate for the claim frequency in the next period is:
$\displaystyle C=\frac{n}{n+\beta} \thinspace \biggl(\frac{\sum \limits_{i=1}^{n}x_i}{n}\biggr)+\frac{\beta}{n+\beta} \thinspace \biggl(\frac{\alpha}{\beta}\biggr)$
In this example, the Buhlmann estimate is exactly the same as the Bayesian estimate (Examples of Bayesian prediction in insurance-continued).
Reference
1. Klugman S. A., Panjer H. H., Willmot G. E., Loss Models, From Data To Decisions, Second Edition, 2004, John Wiley & Sons, Inc.
# Examples of Bayesian prediction in insurance-continued
This post is a continuation of the previous post Examples of Bayesian prediction in insurance. We present another example as an illustration of the methodology of Bayesian estimation. The example in this post, along with the example in the previous post, serve to motivate the concept of Bayeisan credibility and Buhlmann credibility theory. So these two posts are part of an introduction to credibility theory.
Suppose $X_1, \cdots, X_n,X_{n+1}$ are independent and identically distributed conditional on $\Theta=\theta$. We denote the density function of the common distribution of $X_j$ by $f_{X \lvert \Theta}(x \lvert \theta)$. We denote the prior distribution of the risk parameter $\Theta$ by $\pi_{\Theta}(\theta)$. The following shows the steps of the Bayesian estimate of the next observation $X_{n+1}$ given $X_1, \cdots, X_n$.
The Marginal Distribution
$\displaystyle f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)=\int \limits_{\theta} \biggl[\prod \limits_{i=1}^{n} f_{X \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta) d \theta$
The Posterior Distribution
$\displaystyle \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \frac{1}{f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)} \biggl[\prod \limits_{i=1}^{n} f_{X \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta)$
The Predictive Distribution
$\displaystyle f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{\theta} \biggl(f_{X \lvert \Theta}(x \lvert \theta)\biggr) \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n) d \theta$
The Bayesian Predictive Mean of the Next Period
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{x} x \thinspace f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n) dx$
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{\theta} E[X \lvert \theta] \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n) d \theta$
Example 2
The number of claims $X$ generated by an insured in a potfolio of independent insurance policies has a Poisson distribution with parameter $\Theta$. In the portfolio of policies, the parameter $\Theta$ varies according to a gamma distribution with parameters $\alpha$ and $\beta$. We have the following conditional distributions of $X$ and prior distribution of $\theta$.
$\displaystyle f_{X \lvert \Theta}(x \lvert \theta)=\frac{\theta^x e^{-\theta}}{x!}$ where $x=0,1,2, \cdots$
$\displaystyle \pi_{\Theta}(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}$ where $\Gamma(\cdot)$ is the gamma function.
Suppose that a particular insured in this portfolio has generated 0 and 3 claims in the first 2 policy periods. What is the Bayesian estimate of the number of claims for this insured in period 3?
Note that the conditional mean $E[X \lvert \Theta]=\Theta$. Thus the unconditional mean $E[X]=E[\Theta]=\frac{\alpha}{\beta}$.
Comment
Note that the unconditional distribution of $X$ is a negative binomial distribution. In a previous post (Compound negative binomial distribution), it was shown that if $N \sim Poisson(\Lambda)$ and $\Lambda \sim Gamma(\alpha,\beta)$, then the unconditional distribution of $X$ has the following probability function. We make use of this result in the Bayesian estimation problem in this post.
$\displaystyle P[N=n]=\frac{\Gamma(\alpha+n}{\Gamma(\alpha) \Gamma(n)} \biggl[\frac{\beta}{\beta+1}\biggr]^{\alpha} \biggl[\frac{1}{\beta+1}\biggr]^n$
The Marginal Distribution
$\displaystyle f_{X_1,X_2}(0,3)=\int_{0}^{\infty} e^{-\theta} \frac{\theta^3 e^{-\theta}}{3!} \frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta} d \theta$
$\displaystyle =\int_{0}^{\infty} \frac{\beta^{\alpha}}{3! \Gamma(\alpha)} \theta^{\alpha+3-1} e^{\beta+2} d \theta=\frac{\Gamma(\alpha+3)}{6 \Gamma(\alpha)} \frac{\beta^{\alpha}}{(\beta+2)^{\alpha+3}}$
The Posterior Distribution
$\displaystyle \pi_{\Theta \lvert X_1,X_2}(\theta \lvert 0,3)=\frac{1}{f_{X_1,X_2}(0,3)} e^{-\theta} \frac{\theta^3 e^{-\theta}}{3!} \frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} e^{-\beta \theta}$
$\displaystyle =K \thinspace \theta^{\alpha+3-1} e^{-(\beta+2) \theta}$
In the above expression $K$ is a constant making $\pi_{\Theta \lvert X_1,X_2}(\theta \lvert 0,3)$ a density function. Note that it has the form of a gamma distribution. Thus the posterior distribution must be:
$\displaystyle \pi_{\Theta \lvert X_1,X_2}(\theta \lvert 0,3)=\frac{(\beta+2)^{\alpha+1}}{\Gamma(\alpha+3)} \thinspace \theta^{\alpha+3-1} e^{-(\beta+2) \theta}$
Thus the posterior distribution of $\Theta$ is a gamma distribution with parameter $\alpha+3$ and $\beta+2$.
The Predictive Distribution
Note that the predictive distribution is simply the mixture of $Poisson(\Theta)$ with $Gamma(\alpha+3,\beta+2)$ as mixing weights. By the comment above, the predictive distribution is a negative binomial distribution with the following probability function:
$\displaystyle f_{X_3 \lvert X_1,X_2}(x \lvert 0,3)=\frac{\Gamma(\alpha+5)}{\Gamma(\alpha+3) \Gamma(2)} \biggl[\frac{\beta+2}{\beta+3}\biggr]^{\alpha+3} \biggl[\frac{1}{\beta+3}\biggr]^{2}$
The Bayesian Predictive Mean
$\displaystyle E[X_3 \lvert 0,3]=\frac{\alpha+3}{\beta+2}=\frac{2}{\beta+2} \biggl(\frac{3}{2}\biggr)+\frac{\beta}{\beta+2} \biggl(\frac{\alpha}{\beta}\biggr) \ \ \ \ \ \ \ \ \ \ (1)$
Note that $E[X \lvert \Theta]=\Theta$. Thus the Bayesian predictive mean in this example is simply the mean of the posterior distribution of $\Theta$, which is $E[\Theta \vert 0,3]=\frac{\alpha+3}{\beta+2}$.
Comment
Generalizing the example, suppose that in the first $n$ periods, the claim counts for the insured are $X_1=x_1, \cdots, X_n=x_n$. Then the posterior distribution of the parameter $\Theta$ is a gamma distribution.
$\biggl[\Theta \lvert X_1=x_1, \cdots, X_n=x_n\biggr] \sim Gamma(\alpha+\sum_{i=1}^{n} x_i,\beta+n)$
Then the predictive distribution of $X_{n+1}$ given the observations has a negative binomial distribution. More importantly, the Bayesian predictive mean is:
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]=\frac{\alpha+\sum_{i=1}^{n} x_i}{\beta+n}$
$\displaystyle =\frac{n}{\beta+n} \biggl(\frac{\sum \limits_{i=0}^{n} x_i}{n}\biggr)+\frac{\beta}{\beta+n} \biggl(\frac{\alpha}{\beta}\biggr)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
It is interesting that the Bayesian predictive mean of the $(n+1)^{th}$ period is a weighted average of the mean of the observed data ($\overline{X}$) and the unconditional mean $E[X]=\frac{\alpha}{\beta}$. Consequently, the above Bayesian estimate is a credibility estimate. The weight given to the observed data $Z=\frac{n}{\beta+n}$ is called the credibility factor. The estimate and the factor are called Bayesian credibility estimate and Bayesian credibility factor, respectively.
In general, the credibility estimate is an estimator of the following form:
$\displaystyle E=Z \thinspace \overline{X}+ (1-Z) \thinspace \mu_0$
where $\overline{X}$ is the mean of the observed data and $\mu_0$ is the mean based on other information. In our example here, $\mu_0$ is the unconditional mean. In practice, $\mu_0$ could be the mean based on the entire book of business, or a mean based on a different block of similar insurance policies. Another interpretation is that $\overline{X}$ is the mean of the recent experience data and $\mu_0$ is the mean of prior periods.
One more comment about the credibility factor $Z=\frac{n}{\beta+n}$ derived in this example. As $n \rightarrow \infty$, $Z \rightarrow 1$. This makes intuitive sense since this gives more weight to $\overline{X}$ as more and more data are accumulated.
# Examples of Bayesian prediction in insurance
We present two examples to illustrate the notion of Bayesian predictive distributions. The general insurance problem we aim to illustrate is that of using past claim experience data from an individual insured or a group of insureds to predict the future claim experience. Suppose we have $X_1,X_2, \cdots, X_n$ with each $X_i$ being the number of claims or an aggregate amount of claims in a prior period of observation. Given such results, what will be the number of claims during the next period, or what will be the aggregate claim amount in the next period? These two examples will motivate the notion of credibility, both Bayesian credibility theory and Buhlmann credibility theory. We present Example 1 in this post. Example 2 is presented in the next post (Examples of Bayesian prediction in insurance-continued).
Example 1
In this random experiment, there are a big bowl (called B) and two boxes (Box 1 and Box 2). Bowl B consists of a large quantity of balls, 80% of which are white and 20% of which are red. In Box 1, 60% of the balls are labeled 0, 30% are labeled 1 and 10% are labeled 2. In Box 2, 15% of the balls are labeled 0, 35% are labeled 1 and 50% are labeled 2. In the experiment, a ball is selected at random from bowl B. The color of the selected ball from bowl B determines which box to use (if the ball is white, then use Box 1, if red, use Box 2). Then balls are drawn at random from the selected box (Box $i$) repeatedly with replacement and the values of the series of selected balls are recorded. The value of first selected ball is $X_1$, the value of the second selected ball is $X_2$ and so on.
Suppose that your friend performs this random experiment (you do not know whether he uses Box 1 or Box 2) and that his first ball is a 1 ($X_1=1$) and his second ball is a 2 ($X_2=2$). What is the predicted value $X_3$ of the third selected ball?
Though it is straightforward to apply the Bayes’ theorem to this problem (the solution can be seen easily using a tree diagram) to obtain a numerical answer, we use this example to draw out the principle of Bayesian prediction. So it may appear that we are making a simple problem overly complicated. We are merely using this example to motivate the method of Bayesian estimation.
For convenience, we denote “draw a white ball from bowl B” by $\theta=1$ and “draw a red ball from bowl B” by $\theta=2$. Box 1 and Box 2 are conditional distributions. The Bowl B is a distribution for the parameter $\theta$. The distribution given in Bowl B is a probability distribution over the space of all parameter values (called a prior distribution). The prior distribution of $\theta$ and the conditional distributions of $X$ given $\theta$ are restated as follows:
$\pi_{\theta}(1)=0.8$
$\pi_{\theta}(2)=0.2$
$\displaystyle f_{X \lvert \Theta}(0 \lvert \theta=1)=0.60$
$\displaystyle f_{X \lvert \Theta}(1 \lvert \theta=1)=0.30$
$\displaystyle f_{X \lvert \Theta}(2 \lvert \theta=1)=0.10$
$\displaystyle f_{X \lvert \Theta}(0 \lvert \theta=2)=0.15$
$\displaystyle f_{X \lvert \Theta}(1 \lvert \theta=2)=0.35$
$\displaystyle f_{X \lvert \Theta}(2 \lvert \theta=2)=0.50$
The following shows the conditional means $E[X \lvert \theta]$ and the unconditional mean $E[X]$.
$\displaystyle E[X \lvert \theta=1]=0.6(0)+0.3(1)+0.1(2)=0.50$
$\displaystyle E[X \lvert \theta=2]=0.15(0)+0.35(1)+0.5(2)=1.35$
$\displaystyle E[X]=0.8(0.50)+0.2(1.35)=0.67$
If you know which particular box your friend is using ($\theta=1$ or $\theta=2$), then the estimate of the next ball should be $E[X \lvert \theta]$. But the value of $\theta$ is unkown to you. Another alternative for a predicted value is the unconditional mean $E[X]=0.67$. While the estimate $E[X]=0.67$ is easy to calculate, this estimate does not take the observed data ($X_1=1$ and $X_2=2$) into account and it certainly does not take the parameter $\theta$ into account. A third alternative is to incorporate the observed data into the estimate of the next ball. We now continue with the calculation of the Bayesian estimation.
Unconditional Distribution
$\displaystyle f_X(0)=0.6(0.8)+0.15(0.2)=0.51$
$\displaystyle f_X(1)=0.3(0.8)+0.35(0.2)=0.31$
$\displaystyle f_X(2)=0.1(0.8)+0.50(0.2)=0.18$
Marginal Probability
$\displaystyle f_{X_1,X_2}(1,2)=0.1(0.3)(0.8)+0.5(0.35)(0.2)=0.059$
Posterior Distribution of $\theta$
$\displaystyle \pi_{\Theta \lvert X_1,X_2}(1 \lvert 1,2)=\frac{0.1(0.3)(0.8)}{0.059}=\frac{24}{59}$
$\displaystyle \pi_{\Theta \lvert X_1,X_2}(2 \lvert 1,2)=\frac{0.5(0.35)(0.2)}{0.059}=\frac{35}{59}$
Predictive Distribution of $X$
$\displaystyle f_{X_3 \lvert X_1,X_2}(0 \lvert 1,2)=0.6 \frac{24}{59} + 0.15 \frac{35}{59}=\frac{19.65}{59}$
$\displaystyle f_{X_3 \lvert X_1,X_2}(1 \lvert 1,2)=0.3 \frac{24}{59} + 0.35 \frac{35}{59}=\frac{19.45}{59}$
$\displaystyle f_{X_3 \lvert X_1,X_2}(2 \lvert 1,2)=0.1 \frac{24}{59} + 0.50 \frac{35}{59}=\frac{19.90}{59}$
Here is another formulation of the predictive distribution of $X_3$. See the general methodology section below.
$\displaystyle f_{X_3 \lvert X_1,X_2}(0 \lvert 1,2)=\frac{0.6(0.1)(0.3)(0.8)+0.15(0.5)(0.35)(0.2)}{0.059}=\frac{19.65}{59}$
$\displaystyle f_{X_3 \lvert X_1,X_2}(1 \lvert 1,2)=\frac{0.3(0.1)(0.3)(0.8)+0.35(0.5)(0.35)(0.2)}{0.059}=\frac{19.45}{59}$
$\displaystyle f_{X_3 \lvert X_1,X_2}(2 \lvert 1,2)=\frac{0.1(0.1)(0.3)(0.8)+0.5(0.5)(0.35)(0.2)}{0.059}=\frac{19.90}{59}$
The posterior distribution $\pi_{\theta}(\cdot \lvert 1,2)$ is the conditional probability distribution of the parameter $\theta$ given the observed data $X_1=1$ and $X_2=2$. This is a result of applying the Bayes’ theorem. The predictive distribution $f_{X_3 \lvert X_1,X_2}(\cdot \lvert 1,2)$ is the conditional probability distribution of a new observation given the past observed data of $X_1=1$ and $X_2=2$. Since both of these distributions incorporate the past observations, the Bayesian estimate of the next observation is the mean of the predictive distribution.
$\displaystyle E[X_3 \lvert X_1=1,X_2=2]$
$\displaystyle =0 \thinspace f_{X_3 \lvert X_1,X_2}(0 \lvert 1,2)+1 \thinspace f_{X_3 \lvert X_1,X_2}(1 \lvert 1,2)+2 \thinspace f_{X_3 \lvert X_1,X_2}(2 \lvert 1,2)$
$\displaystyle =0 \frac{19.65}{59}+1 \frac{19.45}{59}+ 2 \frac{19.90}{59}$
$\displaystyle =\frac{59.25}{59}=1.0042372$
$\displaystyle E[X_3 \lvert X_1=1,X_2=2]$
$\displaystyle =E[X \lvert \theta=1] \medspace \pi_{\Theta \lvert X_1,X_2}(1 \lvert 1,2)+E[X \lvert \theta=2] \medspace \pi_{\Theta \lvert X_1,X_2}(2 \lvert 1,2)$
$\displaystyle =0.5 \frac{24}{59}+1.35 \frac{35}{59}=\frac{59.25}{59}$
Note that we compute the Bayesian estimate $E[X_3 \vert X_1,X_2]$ in two ways, one using the predictive distribution and the other using the posterior distribution of the parameter $\theta$. The Bayesian estimate is the mean of the hypothetical means $E[X \lvert \theta]$ with expectation taken over the entire posterior distribution $\pi_{\theta}(\cdot \lvert 1,2)$.
Discussion of General Methodology
We now use Example 1 to draw out general methodology. We first describe the discrete case and have the continuous case as a generalization.
Suppose we have a family of conditional density functions $f_{X \lvert \Theta}(x \lvert \theta)$. In Example 1, the bowl B is the distribution of the parameter $\theta$. Box 1 and Box 2 are the conditional distributions with density $f_{X \lvert \Theta}(x \lvert \theta)$. In an insurance application, the $\theta$ is a risk parameter and the conditional distribution $f_{X \lvert \Theta}(x \lvert \theta)$ is the claim experience in a given fixed period (conditional on $\Theta=\theta$).
Suppose that $X_1,X_2, \cdots, X_n,X_{n+1}$ (conditional on $\Theta=\theta$) are independent and identically distributed where the common density function is $f_{X \lvert \Theta}(x \lvert \theta)$. In our Example 1, once a box is selected (e.g. Box 1), then the repeated drawing of the balls are independent and identically distributed. In an insurance application, the $X_k$ are the claim experience from an insured (or a group of insureds) where the insured belongs to the risk class with parameter $\theta$.
We are interested in the conditional distribution of $X_{n+1}$ given $\Theta=\theta$ to predict $X_{n+1}$. In our example, $X_{n+1}$ is the value of the ball in the $(n+1)^{st}$ draw. In an insurance application, $X_{n+1}$ may be the claim experience of an insured (or a group of insureds) in the next policy period. We can use the unconditional mean $E[X]=E[E(X \lvert \Theta)]$ (the mean of the hypothetical means). This approach does not take the risk parameter of the insured into the equation. On the other hand, if we know the value of $\theta$, then we can use $f_{X \lvert \Theta}(x \lvert \theta)$. But the risk parameter is usually unknown. The natural alternative is to condition on the observed experience in the $n$ prior periods $X_1, \cdots, X_n$ rather than conditioning on the risk parameter $\theta$. Thus we derive the predictive distribution of $X_{n+1}$ given the observation $X_1, \cdots, X_n$. Given the observed experience data $X_1=x_1,X_2=x_2, \cdots, X_n=x_n$, the following is the derivation of the Bayesian predictive distribution. Note that the prior distribution of the parameter $\theta$ is $\pi_{\Theta}(\theta)$.
The Unconditional Distribution
$\displaystyle f_X(x)=\sum \limits_{\theta} f_{X \lvert \Theta}(x \lvert \theta) \ \pi_{\Theta}(\theta)$
The Marginal Distribution
$\displaystyle f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)=\sum \limits_{\theta} \biggl[\prod \limits_{i=1}^{n} f_{X_i \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta)$
The Posterior Distribution
$\displaystyle \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n)$
$\displaystyle = \ \ \ \ \ \ \ \ \ \ \frac{1}{f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)} \biggl[\prod \limits_{i=1}^{n} f_{X_i \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta)$
The Predictive Distribution
$\displaystyle f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \sum \limits_{\theta} f_{X \lvert \Theta}(x \lvert \theta) \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n)$
Another formulation is:
$\displaystyle f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \frac{1}{f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)} \sum \limits_{\theta} f_{X_{n+1} \lvert \Theta}(x \lvert \theta) \biggl[ \prod \limits_{j=1}^{n}f_{X_j \lvert \Theta}(x_j \lvert \theta)\biggr] \thinspace \pi_{\Theta}(\theta)$
The Bayesian Predictive Mean of the Next Period
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \sum \limits_{x} x \thinspace f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \sum \limits_{\theta} E[X \lvert \theta] \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n)$
We state the same results for the case that the claim experience $X$ is continuous.
The Unconditional Distribution
$\displaystyle f_{X}(x) = \int_{\theta} f_{X \lvert \Theta} (x \lvert \theta) \ \pi_{\Theta}(\theta) \ d \theta$
The Marginal Distribution
$\displaystyle f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)=\int \limits_{\theta} \biggl[\prod \limits_{i=1}^{n} f_{X \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta) d \theta$
The Posterior Distribution
$\displaystyle \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \frac{1}{f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)} \biggl[\prod \limits_{i=1}^{n} f_{X \lvert \Theta}(x_i \lvert \theta)\biggr] \pi_{\Theta}(\theta)$
The Predictive Distribution
$\displaystyle f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{\theta} f_{X \lvert \Theta}(x \lvert \theta) \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n) \ d \theta$
Another formulation is:
$\displaystyle f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n)$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \frac{1}{f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)} \int \limits_{\theta} f_{X_{n+1} \lvert \Theta}(x \lvert \theta) \biggl[ \prod \limits_{j=1}^{n}f_{X_j \lvert \Theta}(x_j \lvert \theta)\biggr] \thinspace \pi_{\Theta}(\theta) \ d \theta$
The Bayesian Predictive Mean of the Next Period
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{x} x \thinspace f_{X_{n+1} \lvert X_1, \cdots, X_n}(x \vert x_1, \cdots, x_n) dx$
$\displaystyle E[X_{n+1} \lvert X_1=x_1, \cdots, X_n=x_n]$
$\displaystyle =\ \ \ \ \ \ \ \ \ \ \int \limits_{\theta} E[X \lvert \theta] \thinspace \pi_{\Theta \lvert X_1, \cdots, X_n}(\theta \lvert x_1, \cdots, x_n) d \theta$
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