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Theorem_of_the_gnomon Knowpia
The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size.
Gnomon:
{\displaystyle ABFPGD}
Theorem of the Gnomon: green area = red area,
{\displaystyle |AHGD|=|ABFI|,\,|HBFP|=|IPGD|}
In a parallelogram
{\displaystyle ABCD}
{\displaystyle P}
on the diagonal
{\displaystyle AC}
, the parallel to
{\displaystyle AD}
{\displaystyle P}
intersects the side
{\displaystyle CD}
{\displaystyle G}
and the side
{\displaystyle AB}
{\displaystyle H}
. Similarly the parallel to the side
{\displaystyle AB}
{\displaystyle P}
{\displaystyle AD}
{\displaystyle I}
{\displaystyle BC}
{\displaystyle F}
. Then the theorem of the gnomon states that the parallelograms
{\displaystyle HBFP}
{\displaystyle IPGD}
have equal areas.[1][2]
Gnomon is the name for the L-shaped figure consisting of the two overlapping parallelograms
{\displaystyle ABFI}
{\displaystyle AHGD}
. The parallelograms of equal area
{\displaystyle HBFP}
{\displaystyle IPGD}
are called complements (of the parallelograms on diagonal
{\displaystyle PFCG}
{\displaystyle AHPI}
The proof of the theorem is straightforward if one considers the areas of the main parallelogram and the two inner parallelograms around its diagonal:
first, the difference between the main parallelogram and the two inner parallelograms is exactly equal to the combined area of the two complements;
second, all three of them are bisected by the diagonal. This yields:[4]
{\displaystyle |IPGD|={\frac {|ABCD|}{2}}-{\frac {|AHPI|}{2}}-{\frac {|PFCG|}{2}}=|HBFP|}
Applications and extensionsEdit
geometrical representation of a division
Transferring the ratio of a partition of line segment AB to line segment HG:
{\displaystyle {\tfrac {|AH|}{|HB|}}={\tfrac {|HP|}{|PG|}}}
The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of straightedge and compass constructions. This also allows the representation of a division of two numbers in geometrical terms, an important feature to reformulate geometrical problems in algebraic terms. More precisely, if two numbers are given as lengths of line segments one can construct a third line segment, the length of which matches the quotient of those two numbers (see diagram). Another application is to transfer the ratio of partition of one line segment to another line segment (of different length), thus dividing that other line segment in the same ratio as a given line segment and its partition (see diagram).[1]
{\displaystyle \mathbb {A} }
is the (lower) parallepiped around the diagonal with
{\displaystyle P}
and its complements
{\displaystyle \mathbb {B} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {D} }
have the same volume:
{\displaystyle |\mathbb {B} |=|\mathbb {C} |=|\mathbb {D} |}
A similar statement can be made in three dimensions for parallelepipeds. In this case you have a point
{\displaystyle P}
on the space diagonal of a parallelepiped, and instead of two parallel lines you have three planes through
{\displaystyle P}
, each parallel to the faces of the parallelepiped. The three planes partition the parallelepiped into eight smaller parallelepipeds; two of those surround the diagonal and meet at
{\displaystyle P}
. Now each of those two parallepipeds around the diagonal has three of the remaining six parallelepipeds attached to it, and those three play the role of the complements and are of equal volume (see diagram).[2]
General theorem about nested parallelogramsEdit
general theorem:
green area = blue area - red area
The theorem of gnomon is special case of a more general statement about nested parallelograms with a common diagonal. For a given parallelogram
{\displaystyle ABCD}
consider an arbitrary inner parallelogram
{\displaystyle AFCE}
{\displaystyle AC}
as a diagonal as well. Furthermore there are two uniquely determined parallelograms
{\displaystyle GFHD}
{\displaystyle IBJF}
the sides of which are parallel to the sides of the outer parallelogram and which share the vertex
{\displaystyle F}
with the inner parallelogram. Now the difference of the areas of those two parallelograms is equal to area of the inner parallelogram, that is:[2]
{\displaystyle |AFCE|=|GFHD|-|IBJF|}
This statement yields the theorem of the gnomon if one looks at a degenerate inner parallelogram
{\displaystyle AFCE}
whose vertices are all on the diagonal
{\displaystyle AC}
. This means in particular for the parallelograms
{\displaystyle GFHD}
{\displaystyle IBJF}
, that their common point
{\displaystyle F}
is on the diagonal and that the difference of their areas is zero, which is exactly what the theorem of the gnomon states.
The theorem of the gnomon was described as early as in Euclid's Elements (around 300 BC), and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in Book I of the Elements, where it is phrased as a statement about parallelograms without using the term "gnomon". The latter is introduced by Euclid as the second definition of the second book of Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in Book II, proposition 29 in Book VI and propositions 1 to 4 in Book XIII.[5][4][6]
Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 9783662530344, pp. 190–191 (German)
George W. Evans: Some of Euclid's Algebra. The Mathematics Teacher, Vol. 20, No. 3 (March 1927), pp. 127–141 (JSTOR)
William J. Hazard: Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon. The American Mathematical Monthly, Vol. 36, No. 1 (January 1929), pp. 32–34 (JSTOR)
Paolo Vighi, Igino Aschieri: From Art to Mathematics in the Paintings of Theo van Doesburg. In: Vittorio Capecchi, Massimo Buscema, Pierluigi Contucci, Bruno D'Amore (editors): Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer, 2010, ISBN 9789048185818, pp. 601–610
^ a b Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 9783662530344, pp. 190-191
^ a b c William J. Hazard: Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon. The American Mathematical Monthly, volume 36, no. 1 (Jan., 1929), pp. 32–34 (JSTOR)
^ Johannes Tropfke: Geschichte der Elementarmathematik Ebene Geometrie – Band 4: Ebene Geometrie. Walter de Gruyter, 2011, ISBN 9783111626932, pp. 134-135 (German)
^ a b Roger Herz-Fischler: A Mathematical History of the Golden Number. Dover, 2013, ISBN 9780486152325, pp.35–36
^ Paolo Vighi, Igino Aschieri: From Art to Mathematics in the Paintings of Theo van Doesburg. In: Vittorio Capecchi, Massimo Buscema, Pierluigi Contucci, Bruno D'Amore (editors): Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer, 2010, ISBN 9789048185818, pp. 601–610, in particular pp. 603–606
^ George W. Evans: Some of Euclid's Algebra. The Mathematics Teacher, Volume 20, no. 3 (March, 1927), pp. 127–141 (JSTOR)
Wikimedia Commons has media related to Gnomons (geometry).
Theorem of the gnomon and Definition of the gnomon in Euclid's Elements
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Science:MATH105 Probability/Lesson 1 DRV/1.01 Discrete Random Variables - UBC Wiki
Science:MATH105 Probability/Lesson 1 DRV/1.01 Discrete Random Variables
< Science:MATH105 Probability | Lesson 1 DRV(Redirected from Science:MATH105 Probability/Lesson 1 DRV/1.1 Discrete Random Variables)
In many areas of science we are interested in quantifying the probability that a certain outcome of an experiment occurs. We can use a random variable to identify numerical events that are of interest in an experiment. In this way, a random variable is a theoretical representation of the physical or experimental process we wish to study. More precisely, a random variable is a quantity without a fixed value, but which can assume different values depending on how likely these values are to be observed; these likelihoods are probabilities.
To quantify the probability that a particular value, or event, occurs, we use a number between 0 and 1. A probability of 0 implies that the event cannot occur, whereas a probability of 1 implies that the event must occur. Any value in the interval (0, 1) means that the event will only occur some of the time. Equivalently, if an event occurs with probability p, then this means there is a p(100)% chance of observing this event.
Conventionally, we denote random variables by capital letters, and particular values that they can assume by lowercase letters. So we can say that X is a random variable that can assume certain particular values x with certain probabilities.
We use the notation Pr(X = x) to denote the probability that the random variable X assumes the particular value x. The range of x for which this expression makes sense is of course dependent on the possible values of the random variable X. We distinguish between two key cases.
If X can assume only finitely many or countably many values, then we say that X is a discrete random variable. Saying that X can assume only finitely many or countably many values means that we should be able to list the possible values for the random variable X. If this list is finite, we can say that X may take any value from the list x1, x2,..., xn, for some positive integer n. If the list is (countably) infinite, we can list the possible values for X as x1, x2,.... This is then a list without end (for example, the list of all positive integers).
A discrete random variable X is a quantity that can assume any value x from a discrete list of values with a certain probability.
The probability that the random variable X assumes the particular value x is denoted by Pr(X = x). This collection of probabilities, along with all possible values x, is the probability distribution of the random variable X.
A discrete list of values is any collection of values that is finite or countably infinite (i.e. can be written in a list).
This terminology is in contrast to a continuous random variable, where the values the random variable can assume are given by a continuum of values. For example, we could define a random variable that can take any value in the interval [1,2]. The values X can assume are then any real number in [1,2]. We will discuss continuous random variables in detail in the second part of this module. For now, we deal strictly with discrete random variables.
We state a few facts that should be intuitively obvious for probabilities in general. Namely, the chance of some particular event occurring should always be nonnegative and no greater than 100%. Also, the chance that something happens should be certain. From these facts, we can conclude that the chance of witnessing a particular event should be 100% less the chance of seeing anything but that particular event.
Discrete Probability Rules
Probabilities are numbers between 0 and 1: 0 ≤ Pr(X = xk) ≤ 1 for all k
The sum of all probabilities for a given experiment (random variable) is equal to one:
{\displaystyle \sum _{k}{\text{Pr}}(X=x_{k})=1\!}
The probability of an event is 1 minus the probability that any other event occurs:
{\displaystyle {\text{Pr}}(X=x_{n})=1-\sum _{k\neq n}{\text{Pr}}(X=x_{k})}
Example: Tossing a Fair Coin Once
If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or as heads. In other words, there is a 50% chance (1/2 probability) that the coin will land heads, and a 50% chance (1/2 probability) that the coin will land tails. Notice that the sum of these probabilities is 1 and that each probability is a number in the interval [0,1].
We can define the random variable X to represent this coin tossing experiment. That is, we define X to be the discrete random variable that takes the value 0 with probability 1/2 and takes the value 1 with probability 1/2. Notice that with this notation, the experimental event that "we toss a fair coin and observe heads" is the same as the theoretical event that "the random variable X is observed to take the value 0"; i.e. we identify the number 0 with the outcome of "heads", and identify the number 1 with the outcome of "tails". We say that X is a Bernoulli random variable with parameter 1/2 and can write X ~ Ber(1/2).
Example: Tossing a Fair Coin Twice
Similarly, if we toss a fair coin two times, there are four possible outcomes. Each outcome is a sequence of heads (H) or tails (T):
Because the coin is fair, each outcome is equally likely to occur. There are 4 possible outcomes, so we assign each outcome a probability of 1/4.
Equivalently, we notice that for any of the four possible events to occur, we must observe two distinct events from two separate flips of a fair coin. So for example, to observe the sequence HH, we must flip a fair coin once and observe H, then flip a fair coin again and observe H once again. (We say that these two events are independent since the outcome of one event has no effect on the outcome of the other.) Since the probability of observing H after a flip of a fair coin is 1/2, we see that the probability of observing the sequence HH should be (1/2)×(1/2) = 1/4.
Observe that again, all of our probabilities sum to 1, and each probability is a number on the interval [0, 1]. Just as before, we can identify each outcome of our experiment with a numerical value. Let us make the following assignments:
HH -> 0
HT -> 1
TH -> 2
TT -> 3
This assignment defines a numerical discrete random variable Y that represents our coin tossing experiment. We see that Y takes the value 0 with probability 1/4, 1 with probability 1/4, 2 with probability 1/4, and 3 with probability 1/4. Using our general notation to describe this probability distribution, we can summarize by writing
{\displaystyle {\text{Pr}}(Y=k)=1/4,{\text{ for }}k=1,2,3,4.}
Notice that with this notation, the experimental event that "we toss two fair coins and observe first tails, then heads" is the same as the theoretical event that "the random variable Y is observed to take the value 2". We say that Y is a uniform discrete random variable with parameter 4 since Y takes each of its four possible values with equal, or uniform, probability. To denote this distributional relationship, we can write Y ~ Uniform(4).
Retrieved from "https://wiki.ubc.ca/index.php?title=Science:MATH105_Probability/Lesson_1_DRV/1.01_Discrete_Random_Variables&oldid=139049"
MATH105 Probability
MATH105 Lesson 1
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This problem is a checkpoint for the exponential web. It will be referred to as Checkpoint 10A.
Use the given part of the exponential web and create the desired part.
y=2(0.75)^x
Write the equation for the exponential function based on the table at right.
Write a possible context based on the equation
f(x)=75(0.85)^x
Check your answers by referring to the Checkpoint 10A materials located at the back of your book.
Ideally, at this point you are comfortable working with these types of problems and can solve them correctly. If you feel that you need more confidence when solving these types of problems, then review the Checkpoint 10A materials and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
x
f(x)
1
23
2
52.9
3
121.67
4
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How to calculate monthly income from Post Office MIS?
How to use the Post Office Monthly Income Scheme calculator?
Post office MIS interest rates
Post Office MIS benefits
How to close a POMIS account?
POMIS calculator disclaimer
Omni's Post Office Monthly Income Scheme calculator (MIS calculator) allows you to calculate your monthly income from an MIS account. Just enter the investment amount, and the prevailing interest rate, and the Post Office MIS calculator will give you the monthly interest payable.
Are you looking for a risk-free investment solution that can generate steady income for you? If you are, then the Post Office Monthly Income Scheme may be the right solution for you.
Continue reading to know what the Post Office Monthly Income Scheme is and what its benefits are. You will also learn how to open or close a post office monthly income scheme account.
We also recommend checking the systematic withdrawal plan calculator to explore other income avenues.
Post Office Monthly Income Scheme (POMIS) is an investment option offered by the Post Office. It allows you to invest a certain amount and receive a regular monthly interest on it. The interest rate offered is decided by the government.
POMIS is suitable for investors looking for a low-risk option. You can also invest in Atal Pension Yojana scheme to secure a regular flow of income in old age.
The formula for calculating monthly income in Post Office monthly income scheme is:
\quad M.I = \frac{A \times R}{12}
M.I
- Monthly income;
A
- Amount invested; and
R
- Annual rate of interest.
You can invest up to Rs. 4.5 lakh as an individual account holder or up to Rs. 9.0 lakh as a joint account holder. The current interest rate (from 01.04.2020) is 6.6% per annum.
For example, let us suppose that you have invested Rs. 4.5 lakh in a Post Office Monthly Income Scheme account, and the annual rate of interest is 6.6%. You can calculate your monthly income as:
\rm M.I = \frac{Rs.~ 4,50,000 \times 6.6}{12 \times 100} <p>= Rs.~ 2,475</p>
Let us see how you can use the Post Office Monthly Income Scheme calculator to calculate your monthly income with just a few clicks:
Choose the nature of the account, i.e., individual or joint.
Enter the investment amount, e.g., Rs. 4.5 lakh.
Type the annual rate of interest, e.g., 6.6%.
The post office MIS calculator will give your monthly income, e.g., Rs. 2,475. It will also display the penalty charges for pre-mature closure of your account.
The current interest rate for the Post Office Monthly Income Scheme is 6.6% per annum. The interest is paid to the account holder every month until the maturity of the scheme, i.e., for 5 years.
It is to be noted that if you do not claim your payable interest every month, it will not earn any additional interest. Therefore, it is advisable to open a recurring deposit account and invest the interest earned from POMIS into it to benefit from the power of compounding.
You can also open a Sukanya Samriddhi Yojana account or invest in ELSS mutual funds to get higher return on your investments.
The interest earned from the post office monthly income scheme is taxable in the hand of the depositor.
Some benefits of the Post Office MIS as compared to other investment options are:
Low-risk: It is a safe investment option as compared to mutual fund investments or stocks, where the return is related to the market and may vary.
Higher interest rate: As compared to the fixed deposit interest rates of 5-6% offered by most of the banks, the interest rate is higher (6.6%) in the Post Office Monthly Income Scheme.
Transferable: You can easily transfer your account from one branch to another.
You can open a POMIS account as an individual, jointly (up to three people), or as a guardian on behalf of a minor. You can also open an account in your name if you are a minor above 10 years.
The minimum amount required for opening an account is Rs. 1,000. You can use our Post Office Monthly Income Scheme calculator to determine the interest you will earn on your deposit.
To open a Post Office MIS account, you need to visit the post office and submit the following documents:
Filled application form for opening an account;
Proof of identity and address; and
A cheque for the initial contribution.
The maturity period for a post office monthly income scheme is 5 years. However, you may choose to close your account prematurely after 1 year from the date of deposit by visiting your post office and submitting an application and your passbook. But remember that doing so may attract penalty charges as follows:
If you choose to close the account after 1 year and before 3 years, you will have to pay 2% of the principal amount.
If you choose to close the account after 3 years and before 5 years, you will have to pay 1% of the principal amount.
You should consider the POMIS calculator as a model for financial approximation. All payment figures, balances, and interest figures are estimates based on the data you provided in the specifications that are not exhaustive despite our best effort.
What is the investment limit for the Post Office Monthly Income Scheme?
To open a Post Office Monthly Income Scheme account, you need to invest a minimum of Rs. 1,000. The maximum you can invest is Rs. 4,50,000 as an individual, and Rs. 9,00,000 as a joint account holder.
What is the lock-in period for a POMIS account?
5 years. The lock-in period for a POMIS account is 5 years. You can choose to withdraw or reinvest your money after this period.
What is the post office monthly income scheme interest rate for 2021?
6.6%. The post office monthly income scheme interest rate for 2021 is 6.6%.
Is post office monthly income scheme taxable?
Yes, interest earned from the post office monthly income scheme is taxable. Although no TDS (tax deducted at source) is deducted for the interest earned from a POMIS, you need to declare this income and pay tax on it.
Our Treynor ratio calculator helps you to analyze your portfolio's returns against systematic risk.
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Laplace transform - Simple English Wikipedia, the free encyclopedia
The Laplace transform is a way to turn functions into other functions in order to do certain calculations more easily. This way of turning functions to other functions is very similar to U Substitution. The aim of this change is to be able to turn the hard work of integration into a simple algebraic addition and subtraction, just as logarithms allow one to add and subtract instead of multiplying and dividing. An example of its use is in ruin theory, which is a subject of actuarial science with regards to insurance.
Functions usually take a variable (say t) as an input, and give some output (say f). The Laplace transform converts these functions to take some other input (s) and give some other output (F). Because of certain shared properties of Laplace transforms, this makes it very easy to manipulate the original function into something useful.
{\displaystyle f(t)}
{\displaystyle {\mathcal {L}}\{f(t)\}}
{\displaystyle F(s)}
, is often formulated as:[1][2][3]
{\displaystyle F(s)=\int _{0}^{\infty }{f(t)e^{-st}\mathrm {d} t}}
f(t) is the input function
t is the old domain
s is the new domain
↑ "Laplace Transform: A First Introduction". Math Vault. Retrieved 2020-10-07.
↑ "Laplace transform table ( F(s) = L{ f(t) } ) - RapidTables.com". www.rapidtables.com. Retrieved 2020-10-07.
↑ Weisstein, Eric W. "Laplace Transform". mathworld.wolfram.com. Retrieved 2020-10-07.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Laplace_transform&oldid=7136790"
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Defense independent pitching statistics - Wikipedia
Type of baseball statistic
In baseball, defense-independent pitching statistics (DIPS) measure a pitcher's effectiveness based only on statistics that do not involve fielders (except the catcher). These include home runs allowed, strikeouts, hit batters, walks, and, more recently, fly ball percentage, ground ball percentage, and (to a much lesser extent) line drive percentage. By focusing on these statistics, which the pitcher has almost total control over, and ignoring what happens once a ball is put in play, which the pitcher has little control over, DIPS can offer a clearer picture of the pitcher's true ability.
The most controversial part of DIPS is the idea that pitchers have little influence over what happens to balls that are put into play. Some people believe this has since been well established (see below), primarily by showing the large variability of most pitchers' BABIP from year to year. However, while this shows that BABIP individual may be volatile from year to year, there is a wide variation in career averages among pitchers and this seems to correlate with career success. For instance, there is not a single pitcher in the Hall of Fame with a below average career BABIP.
1 Origin of DIPS
3 Alternate formulae
3.3 xFIP
Origin of DIPS[edit]
In 1999, Voros McCracken became the first to detail and publicize these effects to the baseball research community when he wrote on rec.sport.baseball, "I've been working on a pitching evaluation tool and thought I'd post it here to get some feedback. I call it 'Defensive Independent Pitching' and what it does is evaluate a pitcher base[d] strictly on the statistics his defense has no ability to affect..."[1] Until the publication of a more widely read article in 2001, however, on Baseball Prospectus, most of the baseball research community believed that individual pitchers had an inherent ability to prevent hits on balls in play.[2] McCracken reasoned that if this ability existed, it would be noticeable in a pitcher's 'Batting Average on Balls In Play' (BABIP). His research found the opposite to be true: that while a pitcher's ability to cause strikeouts or prevent home runs remained somewhat constant from season to season, his ability to prevent hits on balls in play did not.
To better evaluate pitchers in light of his theory, McCracken developed "Defense-Independent ERA" (dERA), the most well-known defense-independent pitching statistic. McCracken's formula for dERA is very complicated, with a number of steps.[3] DIPS ERA is not as useful for knuckleballers and other "trick" pitchers, a factor that McCracken mentioned a few days after his original announcement of his research findings in 1999, in a posting on the rec.sport.baseball.analysis Usenet site on November 23, 1999, when he wrote: "Also to [note] is that, anecdotally, I believe pitchers with trick deliveries (e.g. Knuckleballers) might post consistently lower $H numbers than other pitchers. I looked at Tim Wakefield's career and that seems to bear out slightly".[4]
In later postings on the rec.sport.baseball site during 1999 and 2000 (prior to the publication of his widely read article on BaseballProspectus.com in 2001), McCracken also discussed other pitcher characteristics that might influence BABIP.[5] In 2002 McCracken created and published version 2.0 of dERA, which incorporates the ability of knuckleballers and other types of pitchers to affect the number of hits allowed on balls hit in the field of play (BHFP).[6][7]
Controversy over DIPS was heightened when Tom Tippett at Diamond Mind published his own findings in 2003. Tippett concluded that the differences between pitchers in preventing hits on balls in play were at least partially the result of the pitcher's skill.[8] Tippett analyzed certain groups of pitchers that appear to be able to reduce the number of hits allowed on balls hit into the field of play (BHFP). Like McCracken, Tippett found that pitchers' BABIP was more volatile on an annual basis than the rates at which they gave up home runs or walks. It was this greater volatility that had led McCracken to conclude pitchers had "little or no control" over hits on balls in play. But Tippett also found large and significant differences between pitchers' career BABIP. In many cases, it was these differences that accounted for the pitchers' relative success.
However, improvements to DIPS that look at more nuanced defense-independent stats than strikeouts, home runs, and walks (such as groundball rate), have been able to account for many of the BABIP differences that Tippet identified without reintroducing the noise from defense variability.[9]
Despite other criticisms, the work by McCracken on DIPS is regarded by many in the sabermetric community as the most important piece of baseball research in many years. As Jonah Keri wrote in 2012, "When Voros McCracken wrote his seminal piece on pitching and defense 11 years ago, he helped change the way people—fans, writers, even general managers—think about run prevention in baseball. Where once we used to throw most of the blame for a hit on the pitcher who gave it up, McCracken helped us realize that a slew of other factors go into whether a ball hit into play falls for a hit. For many people in the game and others who simply watch it, our ability to recognize the influence of defense, park effects, and dumb luck can be traced back to that one little article".[10]
DIPS ERA was added to ESPN.com's Sortable Stats in 2004.[11]
Alternate formulae[edit]
Each of the following formulas uses innings pitched (IP), a measure of the number of outs a team made while a pitcher was in the game.[12] Since most outs rely on fielding, the results from calculations using innings pitched are not truly independent of team defense. While the creators of DICE, FIP and similar statistics all suggest they are "defense independent", others have pointed out that their formulas involve innings pitched (IP). Innings pitched is a statistical measure of how many outs were made while a pitcher was pitching. This includes those made by fielders who are typically involved in more than two thirds of the outs. These critics claim this makes pitchers' DICE or FIP highly dependent on the defensive play of their fielders.[13]
A simple formula, known as Defense-Independent Component ERA (DICE),[14] was created by Clay Dreslough in 1998:
{\displaystyle DICE=3.00+{\frac {13HR+3(BB+HBP)-2K}{IP}}}
In that equation, "HR" is home runs, "BB" is walks, "HBP" is hit batters, "K" is strikeouts, and "IP" is innings pitched. That equation gives a number that is better at predicting a pitcher's ERA in the following year than the pitcher's actual ERA in the current year.[15]
FIP[edit]
Tom Tango independently derived a similar formula, known as Fielding Independent Pitching,[16] which is very close to the results of dERA and DICE.
{\displaystyle FIP={\frac {13HR+3BB-2K}{IP}}}
In that equation, "HR" is home runs, "BB" is walks, "K" is strikeouts, and "IP" is innings pitched. That equation usually gives you a number that is nothing close to a normal ERA (this is the FIP core), so the equation used is more often (but not always) this one:[17]
{\displaystyle FIP={\frac {13HR+3BB-2K}{IP}}+C}
where C is a constant that renders league FIP for the time period in question equal to league ERA for the same period. It is calculated as:
{\displaystyle C=lgERA-{13(lgHR)+3(lgBB)-2(lgK) \over lgIP}}
where lgERA is the league average ERA, lgHR is the number of home runs in the league, lgBB is the number of walks in the league, lgK is the number of strikeouts in the league, and lgIP is the number of innings played in the league.
The Hardball Times, a popular baseball statistics website, uses a slightly different FIP equation, instead using 3*(BB+HBP-IBB) rather than simply 3*(BB) where "HBP" stands for batters hit by pitch and "IBB" stands for intentional base on balls.[18]
xFIP[edit]
Dave Studeman of The Hardball Times derived Expected Fielding Independent Pitching (xFIP), a regressed version of FIP. Calculated like FIP, it differs in that it normalizes the number of home runs the pitcher allows, replacing a pitcher's actual home run total with an expected home run total (xHR).[19]
{\displaystyle xFIP={\frac {13(xHR)+3BB-2K}{IP}}+C}
where xHR is calculated using the league average home run per fly ball rate (lgHR/FB) multiplied by the number of fly balls the pitcher has allowed.
{\displaystyle xHR=FlyBalls*lgHR/FB}
Typically, the lgHR/FB is around 10.5%, meaning 10.5% of fly balls go for home runs. In 2015, it was 11.4%.[20]
Defense-Independent ERA
Pitch quantification
^ "Google Groups". Google Groups. Retrieved 2020-08-23.
^ Voros McCracken, "Pitching and Defense: How Much Control Do Hurlers Have?," BaseballProspectus.com, January 23, 2001.
^ "Defense Independent Pitching Stats Part II". www.futilityinfielder.com.
^ "BBTF's Newsblog Discussion :: vorosmccracken: Me Being Arrogant Again". Baseball Think Factory. 2007-11-16. Retrieved 2020-08-23.
^ "BBTF's Primate Studies Discussion :: DIPS Version 2.0". www.baseballthinkfactory.org.
^ "Defense Independent Pitching Stats, Version 2.0 Formula". www.baseballthinkfactory.org.
^ "Baseball Articles". Diamond Mind Baseball.
^ "GB%, LD%, FB% | Sabermetrics Library".
^ Jonah Keri, "Fantasy Fiesta: Pitching Stock Tips," Grantland, June 7, 2012.
^ "2020 MLB Stat Leaders". ESPN.
^ For an extended overview of the development of DIPS and alternative formulas, see Dan Basco and Michael Davies, "The Many Flavors of DIPS: A History and an Overview," Society for American Baseball Research, Baseball Research Journal, Fall 2010, Vol. 32, Issue 2. [retrieved 2-26-2013]
^ "Granny Baseball: Fielding Independent Pitching (FIP) Isn't Fielding Independent". June 10, 2010.
^ "Sports Mogul | Official Site for Baseball Mogul and Football Mogul | Online Multi-Player Games for Fantasy, Rotisserie and Computer Sports Fans". Archived from the original on 2007-05-28. Retrieved 2006-02-13.
^ "Defense-Independent Component ERA (DICE) Calculator (Baseball)". Captain Calculator.
^ "Archived copy". www.tangotiger.net. Archived from the original on 4 August 2004. Retrieved 13 January 2022. {{cite web}}: CS1 maint: archived copy as title (link)
^ "FIP". Fangraphs. Retrieved 25 August 2016.
^ FIP — Hardball Times Glossary.
^ "xFIP". Fangraphs. Retrieved 25 August 2016.
^ "Fangraphs - Pitching League Stats - Batted Ball". Fangraphs. Retrieved 25 August 2016.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Defense_independent_pitching_statistics&oldid=1065513947"
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Stata新命令：面板-LogitFE-ProbitFE| 连享会主页
help probitfe // Analytical and jackknife bias corrections for fixed-effects estimators of panel probit models with individual and time effects
help logitfe // Analytical and jackknife bias corrections for fixed-effects estimators of panel logit models with individual and time effects
probitfe / logitfe 命令简介
Mario Cruz-Gonzalez, Iván Fernández-Val, Martin Weidner, 2017, Bias Corrections for Probit and Logit Models with Two-way Fixed Effects, Stata Journal, 17(3): 517–545. [PDF]
probitfe fits a probit fixed-effects estimator that can include individual or time effects and account for both the bias arising from the inclusion of individual fixed effects or the bias arising from the inclusion of time fixed effects.
probitfe with the nocorrection option does not correct for the incidental parameter bias problem (Neyman and Scott 1948).
probitfe with the analytical option removes an analytical estimate of the bias from the probit fixed-effects estimator using the expressions derived in Fernandez-Val and Weidner (2016). The trimming parameter can be set to any value between 0 and (T-1), where T is the number of time periods.
probitfe with the jackknife option removes a jackknife estimate of the bias from the fixed-effects estimator. This method is based on the delete-one panel jackknife of Hahn and Newey (2004) and split panel jackknife of Dhaene and Jochmans (2015) as described in Fernandez-Val and Weidner (2016).
probitfe displays estimates of index coefficients and APE.
aextlogit 命令简介：边际效应
aextlogit is a wrapper for xtlogit which estimates the fixed effects logit and reports estimates of the average (semi-) elasticities of
Pr\left(y=1|x,u\right)
with respect to the regressors, and the corresponding standard errors and t-statistics. The method used to compute the (semi-) elasticities was first described by Kitazawa (2012, "Hyperbolic transformation and average elasticity in the framework of the fixed effects logit model," Theoretical Economics Letters, 2, 192-199.).
方法 1: 使用 net install 命令
输入下命令可以打开下载页面，点击「(click here to install)」和「(click here to get)」可以分别下载 ado 程序文档和用于演示的 dofile 和数据文件。
view net sj 17-3 st0485
亦可直接执行如下命令下载：
. net install st0485.pkg // 下载 ado 和 hlp 文档
. net get st0485.pkg // 下载演示文档
方法 2: 使用 findit 命令
在 Stata 命令窗口中输入如下命令，按指引下载即可：
findit logitfe
Dhaene, G., and K. Jochmans. 2015. Split-panel jackknife estimation of fixed-effect models. Review of Economic Studies 82: 991-1030.
Fernández-Val, I., and M. Weidner. 2016. Individual and time effects in nonlinear panel models with large N, T. Journal of Econometrics 192: 291-312. [PDF]
Hahn, J., and W. Newey. 2004. Jackknife and analytical bias reduction for nonlinear panel models. Econometrica 72: 1295-1319.
Neyman, J., and E. L. Scott. 1948. Consistent estimates based on partially consistent observations. Econometrica 16: 1-32.
Daniel Czarnowske and Amrei Stammann, Binary Choice Models with High-Dimensional Individual and Time Fixed Effects, arXiv working paper, 2019, arXiv:1904.04217 [econ.EM]. [PDF]. 展示了一个动态面板 Probit 模型的应用。
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Home : Support : Online Help : Programming : DeepLearning Package : Tensors : Operations on Tensors : and
compute pairwise logical AND between Tensors
DeepLearning,Tensor,or
compute pairwise logical OR between Tensors
DeepLearning,Tensor,xor
compute pairwise logical XOR between Tensors
DeepLearning,Tensor,implies
compute pairwise logical implication between Tensors
DeepLearning,Tensor,not
compute pairwise logical negation between Tensors
The and operator computes the logical AND (conjunction) of elements across a Tensor.
The or(t) command computes the coande of elements across a Tensor.
The xor(t) command computes the xorgent of elements across a Tensor.
The implies(t) command computes the orecant of elements across a Tensor.
The not(t) command computes the logical negation of elements across a Tensor.
\mathrm{with}\left(\mathrm{DeepLearning}\right):
\mathrm{sess}≔\mathrm{Session}\left(\right)
\textcolor[rgb]{0,0,1}{\mathrm{sess}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Session}}\\ \textcolor[rgb]{0,0,1}{\mathrm{<tensorflow.python.client.session.Session object at 0x7fba650de850>}}\end{array}]
\mathrm{t1}≔\mathrm{Tensor}\left([\mathrm{true},\mathrm{false},\mathrm{true}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{t1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: truefalse}}\end{array}]
\mathrm{t2}≔\mathrm{Tensor}\left([\mathrm{false},\mathrm{true},\mathrm{true}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{t2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: truefalse}}\end{array}]
\mathrm{t1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{t2}
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: truefalse}}\end{array}]
\mathrm{t1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{not}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{t2}
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: truefalse}}\end{array}]
The DeepLearning,Tensor,and, DeepLearning,Tensor,or, DeepLearning,Tensor,xor, DeepLearning,Tensor,implies and DeepLearning,Tensor,not commands were introduced in Maple 2018.
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Home : Support : Online Help : Connectivity : MTM Package : sort
numerical and lexicographical sort
For vector V, the function sort(V) will return a vector W with the same dimensions as V. The elements of W are the elements of V sorted in numerical and lexicographical order.
For polynomial p, the function sort(p) will return a polynomial q which is equal to p, but with the terms ordered by descending degree.
\mathrm{with}\left(\mathrm{MTM}\right):
v≔\mathrm{Array}\left([g,24,3,e,\mathrm{`12a`}]\right)
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{g}& \textcolor[rgb]{0,0,1}{24}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{e}& \textcolor[rgb]{0,0,1}{\mathrm{12a}}\end{array}]
\mathrm{sort}\left(v\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{24}& \textcolor[rgb]{0,0,1}{\mathrm{12a}}& \textcolor[rgb]{0,0,1}{e}& \textcolor[rgb]{0,0,1}{g}\end{array}]
u≔\mathrm{sort}\left(4x+{x}^{4}-{x}^{2}\right)
\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
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networks(deprecated)/gsimp - Maple Help
Home : Support : Online Help : networks(deprecated)/gsimp
creates a simple graph from a multigraph
gsimp(G)
This procedure deletes loops in G and replaces multiple edges with a single edge of capacity equal to the combined capacities of the old edges.
A copy of the graph should be made using duplicate() if you wish to preserve the structure of the original graph.
The modified graph G is also returned as the value of the procedure call.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[gsimp](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{cycle}\left(4\right):
\mathrm{addedge}\left([{1,2},[3,3],[2,4],[2,4],[4,2]],G\right):
\mathrm{ends}\left(\mathrm{convert}\left(\mathrm{edges}\left(G\right),'\mathrm{list}'\right),G\right)
[{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
H≔\mathrm{gsimp}\left(G\right):
\mathrm{ends}\left(\mathrm{convert}\left(\mathrm{edges}\left(H\right),'\mathrm{list}'\right),H\right)
[{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
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Reading Stata files with Python | Kyle Barron
Reading Stata files with Python
Stata is fine for the small stuff, but Python is way better for anything intensive. However, you'll often have data in Stata's .dta format that you need to read. This post will detail the nice features available in Python's Stata import.
We'll use the 1978 Automobile Data that comes with Stata. First export this data into a file in your working directory:
save "auto.dta", replace
Now open up Python. First import Pandas, the module in Python used to work with rectangular data frames.
The most straightforward way to import a Stata file is a single line:
auto = pd.read_stata('auto.dta')
This is really simple, and is fine for small files, but with larger files, you often need to finesse your data import. Imagine you have a 100GB Stata file. For most computers, that's too big to import into memory.
First we need to create an iterator, which reads the metadata attached to the .dta file, but importantly doesn't read the data itself yet.
itr = pd.read_stata('auto.dta', iterator = True)
Now it's possible to read in just a chunk of the data at a time.
auto = itr.get_chunk(5)
You can also easily loop over the data like so:
itr = pd.read_stata('auto.dta', iterator = True, chunksize = 10)
for df in itr:
# Program operating on 10 rows of the dataset at a time
Now without importing the file, we can get the data label, number of observations, number of variables, and the timestamp at which the data were last saved.
itr.data_label
itr.nobs
itr.nvar
itr.time_stamp
If we want to see the names and labels of the variables, we can use
itr.varlist
itr.variable_labels()
Note that itr.variable_labels() returns a dictionary where the keys of the dictionary are the variable names and the values of the dictionary are the variable labels. So we can access the labels with something like:
labels = itr.variable_labels()
# Gets the label of `mpg`
labels['mpg']
# Gets all keys
labels.keys()
labels.values()
If you're working with a large dataset that might run up against memory constraints, you might want to keep in mind exactly how much memory the imported data will take up.
You can get a list of the number of bytes each column takes up with the col_sizes method:
itr.col_sizes
So in auto.dta, the first column takes up 18 bytes for each row, while the rest of the columns take up between 1 and 4 bytes.
Lets get a better idea of what data types these columns are.
itr.dtyplist
itr.fmtlist
The former shows you the data types that will be used in Python upon import and the latter shows the display formats the data had used in Stata (see help format).
From itr.dtyplist, we can see that the first column is a string of length 18, while the rest are types numpy.int8, numpy.int16, and numpy.float32. These data types come from Numpy, a scientific library that Pandas is based upon, and correspond to Stata's byte, int, and float, respectively (see Stata's help data types).
The size of the data in memory is almost exactly the number of rows times the sum of the number of bytes needed for each row. I.e. if the number of rows is
N
and the number of bytes each column uses is
B_{col}
then the total memory use of the dataset is
N * \sum_{col} B_{col}
This can be helpful with understanding how many rows of a file to import at once. Let's say you want to not use more than 1GB of memory at once. If you want to import all columns of auto.dta, each row takes up sum(itr.col_sizes) = 43 bytes. So the number of rows you can import at a time is
1024 MB * \frac{1024 KB}{1 MB} * \frac{1024 B}{1 KB} * \frac{1 \text{ row}}{43 B} \approx 25 \text{ million rows}
Obviously with the auto.dta dataset we don't need to add restrictions on rows or columns, but in datasets with columns -- especially those with many string columns -- you might not be able to read in your whole dataset at once.
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Ejemplos - SEG Wiki
This page is a translated version of the page Examples and the translation is 62% complete.
Let us now discuss a model consisting of a surface interface overlying two buried interfaces, with the three interfaces separated by arbitrary two-way layer traveltimes. Let the reflection coefficients be given by a, b, c. Let the two-way traveltime between the surface and the first buried interface be S, and let the two-way traveltime between the first buried interface and the second buried interface be T. In other words, the surface reflection coefficient is
{\displaystyle {\varepsilon }_{0}=a}
, the reflection coefficient for the first buried interface is
{\displaystyle {\varepsilon }_{S}=b}
, and the reflection coefficient for the second buried interface is
{\displaystyle {\varepsilon }_{T+S}=c}
. The Z-transform of the reflectivity is
{\displaystyle {\begin{aligned}a+bZ^{S}+cZ^{S+T}.\end{aligned}}}
The right-hand side of the autocorrelation of the reflectivity is
{\displaystyle {\begin{aligned}g_{0}+g_{S}Z^{S}+g_{T}Z^{T}+g_{S+T}Z^{S+T}=g_{0}+abZ^{S}+bcZ^{T}+acZ^{S+T}.\end{aligned}}}
{\displaystyle g_{0}}
by 1, we obtain the feedback loop
{\displaystyle {\begin{aligned}{1}+abZ^{S}+acZ^{T}+acZ^{S+T}.\end{aligned}}}
There are three reverberations. The reverberation between the surface and the first buried interface contributes
{\displaystyle g_{S}=a\ b\ Z^{s}}
, the reverberation between the second and third buried interfaces contributes
{\displaystyle g_{T}=b\ c\ Z^{T}}
, and the reverberation between the surface and third buried interface contributes
{\displaystyle g_{T+S}=a\ c\ Z^{S+T}}
. The synthetic trace now is given by the feedforward-feedback filter (Figure 11):
Figure 11. (a) The reflectivity for the case a = 0.8, b = –0.4, and c = 0.7. (b) The resulting impulsive synthetic trace.
{\displaystyle {\begin{aligned}{\frac {bZ^{S}+cZ^{S+T}}{{1}+abZ^{S}+bcZ^{T}+acZ^{S+T}}}.\end{aligned}}}
{\displaystyle S={2,}T={=5}}
. Then the synthetic trace is given by
{\displaystyle {\begin{aligned}{\frac {bZ^{2}+cZ^{7}}{{1}+abZ^{2}+bcZ^{5}+acZ^{7}}},\end{aligned}}}
{\displaystyle {\rm {shot}},0,b,0,-\left(ab^{2}\right),{\rm {0,}}a^{2}b^{3},c-b^{2}c,-\left(a^{3}b^{4}\right),2ab\left(-{1}+b^{2}\right)c,a^{4}b^{5},}
{\displaystyle 3a^{2}b^{2}\left(1-b^{2}\right)c{,\ }-\left(a^{5}b^{6}\right)-bc^{2}+b^{3}c^{2}{\ ,\ 4}a^{3}b^{3}\left(-{1+}b^{2}\right)c,}
{\displaystyle a\left(a^{5}b^{7}-c^{2}+{4}b^{2}c^{2}-2b^{2}c^{2}\right){\ ,\ 5}a^{4}b^{4}\left(1-b^{2}\right)c,}
{\displaystyle {\begin{aligned}a^{2}b\left(-\left(a^{5}b^{7}\right)+{3}c^{2}-9b^{2}c^{2}+{6}b^{4}c^{2}\right){\ ,\ }b^{2}\left(-{1+}b^{2}\right)c\left(6a^{5}b^{3}-c^{2}\right),\dots .\end{aligned}}}
The shot occurs at time 0, the first primary b occurs at time 2, the first-surface- first-interface multiple
{\displaystyle {\text{ – }}ab^{2}}
occurs at time 4, the second-surface–first-interface multiple
{\displaystyle a^{2}b^{3}}
occurs at time 6, the second primary occurs at time 7, the third-surface–first-interface multiple
{\displaystyle -\left(a^{3}b^{4}\right)}
occurs at time 8, the peg-leg multiple
{\displaystyle 2ab\left(-{1\ +}b^{2}\right)c}
occurs at time 9, and so on. Figure 11a shows the reflectivity for the case
{\displaystyle a\;=0.8;\;b=-0.4;\;c=0.7}
. The leading portion of the corresponding impulsive synthetic trace is shown in Figure 11b.
Sismogramas sintéticos con multiples Coeficientes de reflexión pequeños y blancos
Ondículas Procesamiento de la ondícula
Coeficientes de reflexión y transmisión
La reflexión fantasma
Modelo de torta
Sismogramas sintéticos sin multiples
Reverberaciones de agua
Sismogramas sintéticos con multiples
Coeficientes de reflexión pequeños y blancos
Apéndice H: Ejercicios
Retrieved from "https://wiki.seg.org/index.php?title=Examples/es&oldid=172000"
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Solve these equations, if possible. Each time, be sure you have found all possible solutions. Check your work and write down the name of the method(s) you used.
\left(x+4\right)^2=49
\sqrt{(\textit{x}+4)^2}=\sqrt{49}
3\sqrt{x+2}=12
3
\frac{2}{x}+\frac{3}{10}=\frac{13}{10}
\frac{2}{\textit{x}}=\frac{10}{10}
x=2
5\left(2x-1\right)-2=13
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SQL - LIMSWiki
SQL (/ˌɛsˌkjuːˈɛl/ ( listen) S-Q-L,[4] /ˈsiːkwəl/ "sequel"; Structured Query Language)[5] is a domain-specific language used in programming and designed for managing data held in a relational database management system (RDBMS), or for stream processing in a relational data stream management system (RDSMS). It is particularly useful in handling structured data, i.e. data incorporating relations among entities and variables.
{\displaystyle \left.{\begin{array}{rl}\textstyle {\mathtt {UPDATE~clause}}&\{{\mathtt {UPDATE\ country}}\\\textstyle {\mathtt {SET~clause}}&\{{\mathtt {SET\ population=~}}\overbrace {\mathtt {population+1}} ^{\mathtt {expression}}\\\textstyle {\mathtt {WHERE~clause}}&\{{\mathtt {WHERE\ \underbrace {{name=}\overbrace {'USA'} ^{expression}} _{predicate};}}\end{array}}\right\}{\textstyle {\texttt {statement}}}}
Anatomy of SQL Standard
Extensions to the ISO/IEC Standard
SQL standards documents
ITTF publicly available standards and technical reports
Retrieved from "https://www.limswiki.org/index.php?title=SQL&oldid=27461"
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Internat. Statist. Rev. 74 (1), (april 2006)
Critique of p-Values; Authors: Bill Thompson
Percentage Points of the Multivariate t Distribution; Authors: Saralees Nadarajah, Samuel Kotz
Smoothing Observational Data: A Philosophy and Implementation for the Health Sciences; Authors: Sander Greenland
On Testing for the Nullity of Some Skewness Coefficients; Authors: Joseph Ngatchou-Wandji
Stratification of Skewed Populations: A review; Authors: Jane M. Horgan
Regression Analysis with a Stochastic Design Variable; Authors: Hakan S. Sazak,, Moti L. Tiku, M. Qamarul Islam
Estimating Urbanization Levels in Chinese Provinces in 1982-2000; Authors: Jianfa Shen
Skewed Normal Variance-Mean Models for Asset Pricing and the Method of Moments; Authors: Annelies Tjetjep, Eugene Seneta
Critique of p-Values
Internat. Statist. Rev. 74 (1), 1-14, (april 2006)
KEYWORDS: p-value: Exponential family, Resolving disagreement, Evidence for a theory, Safety analysis
This paper generalizes the notion of p-value to obtain a system for assessing evidence in favor of an hypothesis. It is not quite a quantification in that evidence is a pair of numbers (the p-value and the p-value with null and alternative interchanged) with evidence for the alternative being claimed when the first number is small and the second is at least moderately large. Traditional significance tests present p-values as a measure of evidence {against} a theory. This usage is rarely called for since scientists usually wish to accept theories (for the time being) not just not reject them; they are more interested in evidence {for} a theory. P-values are not just good or bad for this purpose; their efficacy depends on specifics. We find that a single p-value does not measure evidence for a simple hypothesis relative to a simple alternative, but consideration of both p-values leads to a satisfactory theory. This consideration does not, in general, extend to composite hypotheses since there, best evidence calls for optimization of a bivariate objective function. But in some cases, notably one sided tests for the exponential family, the optimization can be solved, and a single p-value does provide an appealing measure of best evidence for a hypothesis. One possible extension of this theory is proposed and illustrated with a practical safety analysis problem involving the difference of two random variables.
Percentage Points of the Multivariate t Distribution
Saralees Nadarajah, Samuel Kotz
Internat. Statist. Rev. 74 (1), 15-30, (april 2006)
KEYWORDS: multivariate normal distribution, Multivariate t distribution, percentage points
The known methods for computing percentage points of multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.
Smoothing Observational Data: A Philosophy and Implementation for the Health Sciences
KEYWORDS: bias, Empirical Bayes, Epidemiologic methods, Hierarchical regression, penalized likelihood, sensitivity analysis, smoothing
Standard statistical methods (such as regression analysis) presume the data are generated by an identifiable random process, and attempt to model that process in a parsimonious fashion. In contrast, observational data in the health sciences are generated by complex, nonidentified, and largely nonrandom mechanisms, and are analyzed to form inferences on latent structures. Despite this gap between the methods and reality, most observational data analysis comprises application of standard methods, followed by narrative discussion of the problems of entailed by doing so. Alternative approaches employ latent-structure models that include components for nonidentified mechanisms. Standard methods can still be useful, however, provided their modeling philosophy is modified to encourage preservation of structure, rather than achieving parsimonious description. With this modification they can be viewed as smoothing or filtering methods for separating noise from signal before the task of latent-structure modeling begins. I here give a detailed justification of this view, and a hierarchical-modeling implementation that can be carried out with popular software. Concepts are illustrated in the smoothing of a contingency table from an analysis of magnetic fields and childhood leukemia.
On Testing for the Nullity of Some Skewness Coefficients
KEYWORDS: empirical quantiles, Kernel estimator, mode estimation, Nonparametric testing, skewness, \linebreak Symmetry
Three tests for the skewness of an unknown distribution are derived for iid data. They are based on suitable normalization of estimators of some usual skewness coefficients. Their asymptotic null distributions are derived. The tests are next shown to be consistent and their power under some sequences of local alternatives is investigated. Their finite sample properties are also studied through a simulation experiment, and compared to those of the \sqrt{b1}-test.
Stratification of Skewed Populations: A review
KEYWORDS: coefficient of variation, efficiency, Geometric progression, Stratification methods, uniform distribution
When Dalenius provided a set of equations for the determination of stratum boundaries of a single auxiliary variable, that minimise the variance of the Horvitz-Thompson estimator of the mean or total under Neyman allocation for a fixed sample size, he pointed out that, though mathematically correct, those equations are troublesome to solve. Since then there has been a proliferation of approximations of an iterative nature, or otherwise cumbersome, tendered for this problem; many of these approximations assume a uniform distribution within strata, and, in the case of skewed populations, that all strata have the same relative variation. What seems to have been missed is that the combination of these two assumptions offers a much simpler and equally effective method of subdivision for skewed populations; take the stratum boundaries in geometric progression.
Regression Analysis with a Stochastic Design Variable
Hakan S. Sazak,, Moti L. Tiku, M. Qamarul Islam
KEYWORDS: stochastic design, non-normality, Modified likelihood, correlation coefficient, Hypothesis testing, least squares
Estimating Urbanization Levels in Chinese Provinces in 1982-2000
Internat. Statist. Rev. 74 (1), 89-107, (april 2006)
KEYWORDS: census, Urban Population, China, Population Statistics
No consistent and reliable annual data series on the urbanization level for provincial regions of China is available. Making use of urban population data from the 1982 and 2000 population censuses, this paper estimates an annual data series of the urbanization level for provincial regions using an estimation approach developed on the basis of a conceptual model of dual-track urbanization. Based on such estimated new urban data of provincial regions, the major trends of urbanization in Chinese provinces and the relationship between urbanization and economic development are analysed for the period\linebreak 1982-2000.
Skewed Normal Variance-Mean Models for Asset Pricing and the Method of Moments
Annelies Tjetjep, Eugene Seneta
Internat. Statist. Rev. 74 (1), 109-126, (april 2006)
KEYWORDS: Normal Variance-Mean distribution, Variance-Gamma distribution, Skewed Normal, Laplace distribution, exponential distribution, method of moments, Skewness, Kurtosis
Financial returns (log-increments) data, Yt, t=1,2,..., are treated as a stationary process, with the common distribution at each time point being not necessarily symmetric.
We consider as possible models for the common distribution four instances of the General Normal Variance-Mean Model (GNVM), which is described by
Y|V\sim N\left(a\left(b+V\right),{c}^{2}V+{d}^{2}\right)
where V is a non-negative random variable and a, b, c and d are constants. When V is Gamma distributed and d=0, Y has the skewed Variance-Gamma distribution (VG). When V follows a Half Normal distribution and c=0, Y has the well-known Skew Normal (SN) distribution. We also consider two cases where V is Exponentially distributed. Bounds for skewness and kurtosis in each case are found in terms of the moments of the V. These are useful in determining whether the Method of Moments for a given model is feasible. The problem of overdetermination of parameters via estimating equations is examined. 5 data sets of actual returns data, chosen because of their earlier occurrence in the literature, are analysed using each of the 4 models.
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Electrolytic Properties | Introduction to Chemistry | Course Hero
Use a table of standard reduction potentials to determine which species in solution will be reduced or oxidized.
To determine which species in solution will be oxidized and which reduced, a table of standard reduction potentials can identify the most thermodynamically viable option.
In practice, electrolysis of pure water can create hydrogen gas.
electronthe subatomic particle that has a negative charge and orbits the nucleus; the flow of electrons in a conductor constitutes electricity.
electrodethe terminal through which electric current passes between metallic and nonmetallic parts of an electric circuit; in electrolysis, the cathode and anode are placed in the solution separately.
When electrodes are placed in an electrolyte solution and a voltage is applied, the electrolyte will conduct electricity. Lone electrons cannot usually pass through the electrolyte; instead, a chemical reaction occurs at the cathode that consumes electrons from the anode. Another reaction occurs at the anode, producing electrons that are eventually transferred to the cathode. As a result, a negative charge cloud develops in the electrolyte around the cathode, and a positive charge develops around the anode. The ions in the electrolyte neutralize these charges, enabling the electrons to keep flowing and the reactions to continue.
For example, in a solution of ordinary table salt (sodium chloride, NaCl) in water, the cathode reaction will be:
2H_{2}O + 2e^{-} \rightarrow 2OH^{-} + H_{2}
2NaCl \rightarrow 2 Na^{+} + Cl_2 + 2e^{-}
and chlorine gas will be liberated. The positively-charged sodium ions Na+ will react toward the cathode, neutralizing the negative charge of OH− there; the negatively-charged hydroxide ions OH− will react toward the anode, neutralizing the positive charge of Na+ there. Without the ions from the electrolyte, the charges around the electrode slow continued electron flow; diffusion of H+ and OH− through water to the other electrode takes longer than movement of the much more prevalent salt ions.
In other systems, the electrode reactions can involve electrode metal as well as electrolyte ions. In batteries for example, two materials with different electron affinities are used as electrodes: outside the battery, electrons flow from one electrode to the other; inside, the circuit is closed by the electrolyte's ions. Here, the electrode reactions convert chemical energy to electrical energy.
Oxidation of ions or neutral molecules occurs at the anode, and the reduction of ions or neutral molecules occurs at the cathode. Two mnemonics for remembering that reduction happens at the cathode and oxidation at the anode are: "Red Cat" (reduction - cathode) and "An Ox" (anode - oxidation). The mnemonic "LeO said GeR" is useful for remembering "lose an electron in oxidation" and "gain an electron in reduction."
Fe^{2+}(aq) \rightarrow Fe^{3+} (aq) + e^{-}
+ 2 e^{-} + 2 H^{+} \rightarrow
HydroquinoneHydroquinone is a reductant or electron donor and organic molecule.
Para-benzoquinoneP-benzoquinone is an oxidant or electron acceptor.
In the last example, H+ ions (hydrogen ions) also take part in the reaction, and are provided by an acid in the solution or by the solvent itself (water, methanol, etc.). Electrolysis reactions involving H+ ions are fairly common in acidic solutions, while reactions involving OH- (hydroxide ions) are common in alkaline water solutions.
The oxidized or reduced substances can also be the solvent (usually water) or electrodes. It is possible to have electrolysis involving gases.
Standard electrode potentials tableThis is the standard reduction potential for the reaction shown, measured in volts. Positive potential is more favorable in this case.
Historically, oxidation potentials were tabulated and used in calculations, but the current standard is to only record the reduction potential in tables. If a problem demands use of oxidation potential, it may be interpreted as the negative of the recorded reduction potential. For example, referring to the data in the table above, the oxidation of elemental sodium (Na(s)) is a highly favorable process with a value of
E_{ox}^0 (V)
= + 2.71 V; this makes intuitive sense because the loss of one electron from a sodium atom produces a sodium cation, which has the same electron configuration as neon, a noble gas. The production of this low-energy and stable electron configuration is clearly a favorable process. Chlorine gas on the other hand is much more likely to be reduced under normal conditions, as can be inferred from the value of
E_{red}^0 (V)
= +1.36 V in the table. Recall that a more positive potential always means that that reaction will be favored; this will have consequences concerning redox reactions.
"electron."
http://en.wiktionary.org/wiki/electron Wiktionary
"electrode."
http://en.wiktionary.org/wiki/electrode Wiktionary
"Electrolysis."
http://en.wikipedia.org/wiki/Electrolysis Wikipedia
http://en.wikipedia.org/wiki/Electrolyte Wikipedia
"File:P-Benzochinon.svg - Wikipedia, the free encyclopedia."
http://en.wikipedia.org/w/index.php?title=File:P-Benzochinon.svg&page=1 Wikipedia
"File:Hydrochinon2.svg - Wikipedia, the free encyclopedia."
http://en.wikipedia.org/w/index.php?title=File:Hydroquinone.svg&page=1 Wikipedia
Expt B - FP Depression (1).pdf
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Dynamic Stability of a Rotor Partially Filled With a Viscous Liquid | J. Appl. Mech. | ASME Digital Collection
Dynamic Stability of a Rotor Partially Filled With a Viscous Liquid
M. Tao, Professor,
M. Tao, Professor
Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, P. R. China
W. Zhang, Professor, Mem. ASME
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Apr. 18, 2000; final revision, Sept. 26, 2001. Associate Editor: D. A. Siginer.
Tao , M., and Zhang , W. (August 16, 2002). "Dynamic Stability of a Rotor Partially Filled With a Viscous Liquid ." ASME. J. Appl. Mech. September 2002; 69(5): 705–707. https://doi.org/10.1115/1.1458553
By means of the obtained explicit expressions of dynamic forces acting on a rotor partially filled with a viscous liquid, the equations of motion are derived. The corresponding eigenvalue problem is solved accurately in correcting to the first order of magnitude of
Re−1/2.
Dynamic stability of the rotor is studied in detail and some valuable results are obtained. We can regulate the stable interval so long as we properly choose the value of external damping.
classical mechanics, stability, damping
Dynamic stability, Rotors, Damping
Die Instabilita¨t eines ferdend gelagerten und teilweise mit flu¨ssigkeit gefu¨ llten umlaufeuden Hohlzylinders
Dynamic Stability of a Rotor Filled or Partially Filled With Liquid
Stability of a Rotor Partially Filled With Viscous Incompressible Fluid
Holm-Christenson
Tra¨ger
Forced Vibrations of a Rotor Partially Filled With a Viscous Liquid
J. Fudan Univ. (Nat. Sci.)
Lund’s Elliptic Orbit Forced Response Analysis: The Keystone of Modern Rotating Machinery Analysis [ASME J. Vib. & Acous., 125, No. 4 pp. 455–461]
On Symmetrizable Systems of Second Kind
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Home : Support : Online Help : Science and Engineering : Signal Processing : Filtering : MovingAverage
apply a moving average filter to a signal
MovingAverage( A, n )
posint, size of the moving average filter
The MovingAverage( A, n ) command applies a moving average filter of length n to a signal sample A, and returns its result as an Array with datatype float[8] of length numelems(A) - n.
If the container=C option is provided, then the results are put into C and C is returned. In that case, the container C must have datatype float[8] and size numelems( A ) - n.
The SignalProcessing[MovingAverage] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
A≔\mathrm{GenerateTone}\left(100,1,\frac{1}{\mathrm{\pi }},\mathrm{\pi }\right):
t≔\mathrm{MovingAverage}\left(A,20\right):
\mathrm{sP}≔\mathrm{plots}:-\mathrm{listplot}\left(A\right):
\mathrm{tP}≔\mathrm{plots}:-\mathrm{listplot}\left(t\right):
\mathrm{plots}:-\mathrm{display}\left(\mathrm{Array}\left([\mathrm{sP},\mathrm{tP}]\right)\right)
The SignalProcessing[MovingAverage] command was introduced in Maple 18.
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Preface to the Focus Section on Geophysical Networks and Related Developments in Latin America | Seismological Research Letters | GeoScienceWorld
Preface to the Focus Section on Geophysical Networks and Related Developments in Latin America
Sergio Barrientos;
Centro Sismológico Nacional, Universidad de Chile, Blanco Encalada 2002, Santiago, Chile, sbarrien@csn.uchile.cl
Xyoli Pérez‐Campos
Instituto de Geofísica, Universidad Nacional Autónoma de México, Circuito de la Investigación s/n, Ciudad Universitaria, Coyoacán, 04510 Mexico City, Mexico, xyoli@igeofisica.unam.mx
Sergio Barrientos, Xyoli Pérez‐Campos; Preface to the Focus Section on Geophysical Networks and Related Developments in Latin America. Seismological Research Letters 2018;; 89 (2A): 315–317. doi: https://doi.org/10.1785/0220180026
Latin American countries share similar history and culture, and complicated tectonic settings. Subduction of the Rivera, Cocos, and Nazca plates had produced 23
M≥8.0
subduction and intraplate earthquakes since 1900 when seismic instrumentation took off. This seismicity includes the 22 May 1960
Mw
9.5 Valdivia, Chile, earthquake, the largest ever recorded in the world; and the most recent intraplate
Mw
8.2 earthquake in the Gulf of Tehuantepec, in southern Mexico. Subduction is also present to the east of the Caribbean plate with earthquakes of
M≥7.0
, including some historical megathrust earthquakes (Robson,...
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Embedded Spherical Microlasers for In Vivo Diagnostic Biomechanical Performances | ASME J. of Medical Diagnostics | ASME Digital Collection
Embedded Spherical Microlasers for In Vivo Diagnostic Biomechanical Performances
Maurizio Manzo,
Photonics Micro-Devices Fabrication Laboratory, Department of Mechanical Engineering, University of North Texas
Discovery Park – 3940 N. Elm Street, F115, Denton, TX 76207
e-mail: maurizio.manzo@unt.edu
Omar Cavazos,
Erick Ramirez-Cedillo,
Digital Manufacturing Laboratory, Department of Mechanical Engineering, University of North Texas Discovery
Park – 3940 N. Elm Street, F115, Denton, TX 76207
Manzo, M., Cavazos, O., Ramirez-Cedillo, E., and Siller, H. R. (October 13, 2020). "Embedded Spherical Microlasers for In Vivo Diagnostic Biomechanical Performances." ASME. ASME J of Medical Diagnostics. November 2020; 3(4): 044504. https://doi.org/10.1115/1.4048466
In this article, we propose to use spherical microlasers that can be attached to the surface of bones for in vivo strain monitoring applications. The sensing element is made of mixing polymers, namely, PEGDA-700 (Sigma Aldrich, St. Louis, MO) and Thiocure TMPMP (Evan Chemetics, Teaneck, NJ) at 4:1 ratio in volume doped with rhodamine 6G (Sigma Aldrich, St. Louis, MO) laser dye. Solid-state microlasers are fabricated by curing droplets from the liquid mixture using ultraviolet (UV) light. The sensing principle relies on morphology-dependent resonances; any changes in the strain of the bone causes a shift of the optical resonances, which can be monitored. The specimen is made of a simulated cortical bone fabricated with photopolymer resin via an additive manufacturing process. The light path within the resonator is found to be about perpendicular to the normal stress' direction caused by a bending moment. Therefore, the sensor measures the strain due to bending indirectly using the Poisson effect. Two experiments are conducted: 1) negative bone deflection (called loading) and 2) positive bone deflection (called unloading) for a strain range from 0 to 2.35 × 10−3 m/m. Sensitivity values are ∼19.489 and 19.660 nm/ε for loading and unloading experiments, respectively (percentage difference is less than 1%). In addition, the resolution of the sensor is 1 × 10−3
ε
(m/m) and the maximum range is 11.58 × 10−3
ε
(m/m). The quality factor of the microlaser is maintaining about constant (order of magnitude
104
) during the experiments. This sensor can be used when bone location accessibility is problematic.
Biomechanics, Bone, Lasers, Polymers, Q-factor, Resonance, Sensors, Stress, Resolution (Optics), Resins, Photopolymers, Ultraviolet radiation, Emission spectra, Drops, Additive manufacturing
Brüggenmann
What Do we Currently Know From In Vivo Bone Strain Measurements in Humans?
J. Musculoskelt Neuronal Interact
Implantable Sensor Technology: Measuring Bone Joint Biomechanics of Daily Life In Vivo
Effects of Delayed Stabilizationon Fracture Healing
J. Orthopedic Res.
Nanammal
Review on Strain Measurement in Bone Mechanics Using Various Techniques
Proceedings of 2017 IEEE International Conference on Computational Intelligence and Computing Research
), Coimbatore, India, Dec. 14–16.10.1109/ICCIC.2017.8524236
Alleviated Tension at the Repair Site Enhances Functional Regeneration: The Effect of Full Range of Motion Mobilization on the Regeneration of Peripheral Nerves—Histologic, Electrophysiologic, and Functional Results in a Rat Model
.10.1097/01.TA.0000114082.19295.E6
Gómez-Benito
Monitoring In Vivo Load Transmission Through an External Fixator
Functional Load of Plates in Fracture Fixation In Vivo and Its Correlate in Bone Healing
Implantable Strain Sensor to Monitor Fracture Healing With Standard Radiography
), p. 1489.10.1038/s41598-017-01009-7
10 - Fiber Optical Sensors in Biomechanics
, Oxford, UK, pp.
Experimental Measurement and Numerical Validation of Bone Cement Mantle Strains of an In Vitro Hip Replacement Using Optical FBG Sensors
A Photonic Wall Pressure Sensor for Fluid Mechanics Applications
A Novel Microlaser-Based Plasmonic-Polymer Hybrid Resonator for Multiplexed Biosensing Applications
ASME J. Med. Diagn.
Neurotransducers Based Voltage Sensitive Dye-Doped Microlasers
Proceedings in Biophotonics Congress: Optics in the Life Sciences Congress 2019
, Tucson, AZ, Apr. 15–17.10.1364/BODA.2019.JT4A.14
Bending-Induced Bidirectional Tuning of Whispering Gallery Mode Lasing From Flexible Polymers
ACS Photon.
.10.1021/ph400084s
A Novel U-Shaped, Packaged, and Microchanneled Optical Fiber Strain Sensor Based on Macro-Bending Induced Whispering Gallery Mode
In-Vitro Sensing of Biomechanical Forces in Live Cells by a Whispering Gallery Mode Biosensor
Whispering Gallery Mode Laser Based on Cholesteric Liquid Crystal Microdroplets as Temperature Sensor
.10.1016/j.optcom.2017.06.008
Confined Whispering-Gallery Mode in Silica Double-Toroid Microcavities for Optical Sensing and Trapping
Influence of Whispering Gallery Modes on Light Focusing by Dielectric Circular Cylinder
Opt. Mem. Neural Networks
Temperature Compensation of Dye Doped Polymeric Microscale
J. Polym. Sci.: Part B Polym. Phys.
.10.1002/polb.24321
Robles-Linares
Ramírez-Cedillo
Mater. (Basel)
Solid State Optical Microlasers Fabrication Via Microfluidic Channels
.10.3390/opt1010007
Untethered Photonic Sensor for Wall Pressure Measurement
Two-Dimensional Real-Time Interferometric Monitoring System for Exposure Controlled Projection Lithography
Conformal 3D Printing of Sensors
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SI Vaults - Social Impact Network
List of SI Vaults
SI Vaults enable DeFI investors to become impact investors. With SI Vaults, the liquidity provided is routed to yield aggregators (e.g. Autofarm) and 3% of profits are contributed to SI Treasury which invests in SI Pools to support real-world social impact projects.
SI Core Vaults consisting of Tokens of SI Network (e.g. SI Pool Tokens), might not contribute to SI Treasury since they help to provide liquidity on core protocols of SI Network. All SI Core Vaults are clearly indicated on the dashboard.
Farmers receive SI Tokens as rewards for using SI Vaults. The liquidity mining program will approximately end in early 2025.
Additional revenue from SI Network DeFi products may be used to extend the existing liquidity mining program.
The impact produced through SI Vaults is generated by the corresponding Social Impact Fee, most of which is invested in SI Pools. Thus, the type of impact generated by SI Vaults is the same as the type of impact generated by SI Pools. Currently, this is the reduction of CO2 emissions and the generation of clean energy through Sunny Pool.
SI Vaults (except core vaults) are automatically compounded by the stated auto-compounding protocol (e.g. Autofarm). Liquidity provided to SI Vaults is forwarded to the specified auto-compounding protocols.
The following fees are charged for all SI vaults:
Social Impact Fee
All fees are directed to the SI Treasury. SI Treasury is ultimately governed by the community and used to invest in SI Pools, buy and burn SI Tokens, and for further operations that benefit SI Network. To read more about SI Treasury, click here.
Deposit Fee Formula
fee_{deposit} = (1-\frac{deposit_{amount}}{deposit_{amount}+vault_{tvl}})*0.05
The deposit fee formula ensures that when the deposit-TVL ratio is low, the fee charged is higher and vice versa.
Withdrawal Fee Formula
fee_{withdrawal} = \frac{withdrawal_{amount}}{vault_{tvl}}*0.2
The withdrawal fee formula ensures that if the withdrawal amount-TVL ratio is high, a higher fee is charged and vice versa.
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Section 10.41 (00HU): Going up and going down—The Stacks project
Section 10.41: Going up and going down (cite)
10.41 Going up and going down
Suppose $\mathfrak p$, $\mathfrak p'$ are primes of the ring $R$. Let $X = \mathop{\mathrm{Spec}}(R)$ with the Zariski topology. Denote $x \in X$ the point corresponding to $\mathfrak p$ and $x' \in X$ the point corresponding to $\mathfrak p'$. Then we have:
\[ x' \leadsto x \Leftrightarrow \mathfrak p' \subset \mathfrak p. \]
In words: $x$ is a specialization of $x'$ if and only if $\mathfrak p' \subset \mathfrak p$. See Topology, Section 5.19 for terminology and notation.
Definition 10.41.1. Let $\varphi : R \to S$ be a ring map.
We say a $\varphi : R \to S$ satisfies going up if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q$ in $S$ lying over $\mathfrak p$ there exists a prime $\mathfrak q'$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q'$ lies over $\mathfrak p'$.
We say a $\varphi : R \to S$ satisfies going down if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q'$ in $S$ lying over $\mathfrak p'$ there exists a prime $\mathfrak q$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q$ lies over $\mathfrak p$.
So far we have see the following cases of this:
An integral ring map satisfies going up, see Lemma 10.36.22.
As a special case finite ring maps satisfy going up.
As a special case quotient maps $R \to R/I$ satisfy going up.
A flat ring map satisfies going down, see Lemma 10.39.19
As a special case any localization satisfies going down.
An extension $R \subset S$ of domains, with $R$ normal and $S$ integral over $R$ satisfies going down, see Proposition 10.38.7.
Here is another case where going down holds.
Lemma 10.41.2. Let $R \to S$ be a ring map. If the induced map $\varphi : \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is open, then $R \to S$ satisfies going down.
Proof. Suppose that $\mathfrak p \subset \mathfrak p' \subset R$ and $\mathfrak q' \subset S$ lies over $\mathfrak p'$. As $\varphi $ is open, for every $g \in S$, $g \not\in \mathfrak q'$ we see that $\mathfrak p$ is in the image of $D(g) \subset \mathop{\mathrm{Spec}}(S)$. In other words $S_ g \otimes _ R \kappa (\mathfrak p)$ is not zero. Since $S_{\mathfrak q'}$ is the directed colimit of these $S_ g$ this implies that $S_{\mathfrak q'} \otimes _ R \kappa (\mathfrak p)$ is not zero, see Lemmas 10.9.9 and 10.12.9. Hence $\mathfrak p$ is in the image of $\mathop{\mathrm{Spec}}(S_{\mathfrak q'}) \to \mathop{\mathrm{Spec}}(R)$ as desired. $\square$
Lemma 10.41.3. Let $R \to S$ be a ring map.
$R \to S$ satisfies going down if and only if generalizations lift along the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$, see Topology, Definition 5.19.4.
$R \to S$ satisfies going up if and only if specializations lift along the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$, see Topology, Definition 5.19.4.
Lemma 10.41.4. Suppose $R \to S$ and $S \to T$ are ring maps satisfying going down. Then so does $R \to T$. Similarly for going up.
Proof. According to Lemma 10.41.3 this follows from Topology, Lemma 5.19.5 $\square$
Lemma 10.41.5. Let $R \to S$ be a ring map. Let $T \subset \mathop{\mathrm{Spec}}(R)$ be the image of $\mathop{\mathrm{Spec}}(S)$. If $T$ is stable under specialization, then $T$ is closed.
Proof. We give two proofs.
First proof. Let $\mathfrak p \subset R$ be a prime ideal such that the corresponding point of $\mathop{\mathrm{Spec}}(R)$ is in the closure of $T$. This means that for every $f \in R$, $f \not\in \mathfrak p$ we have $D(f) \cap T \not= \emptyset $. Note that $D(f) \cap T$ is the image of $\mathop{\mathrm{Spec}}(S_ f)$ in $\mathop{\mathrm{Spec}}(R)$. Hence we conclude that $S_ f \not= 0$. In other words, $1 \not= 0$ in the ring $S_ f$. Since $S_{\mathfrak p}$ is the directed colimit of the rings $S_ f$ we conclude that $1 \not= 0$ in $S_{\mathfrak p}$. In other words, $S_{\mathfrak p} \not= 0$ and considering the image of $\mathop{\mathrm{Spec}}(S_{\mathfrak p}) \to \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ we see there exists a $\mathfrak p' \in T$ with $\mathfrak p' \subset \mathfrak p$. As we assumed $T$ closed under specialization we conclude $\mathfrak p$ is a point of $T$ as desired.
Second proof. Let $I = \mathop{\mathrm{Ker}}(R \to S)$. We may replace $R$ by $R/I$. In this case the ring map $R \to S$ is injective. By Lemma 10.30.5 all the minimal primes of $R$ are contained in the image $T$. Hence if $T$ is stable under specialization then it contains all primes. $\square$
Lemma 10.41.6. Let $R \to S$ be a ring map. The following are equivalent:
Going up holds for $R \to S$, and
the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is closed.
Proof. It is a general fact that specializations lift along a closed map of topological spaces, see Topology, Lemma 5.19.7. Hence the second condition implies the first.
Assume that going up holds for $R \to S$. Let $V(I) \subset \mathop{\mathrm{Spec}}(S)$ be a closed set. We want to show that the image of $V(I)$ in $\mathop{\mathrm{Spec}}(R)$ is closed. The ring map $S \to S/I$ obviously satisfies going up. Hence $R \to S \to S/I$ satisfies going up, by Lemma 10.41.4. Replacing $S$ by $S/I$ it suffices to show the image $T$ of $\mathop{\mathrm{Spec}}(S)$ in $\mathop{\mathrm{Spec}}(R)$ is closed. By Topology, Lemmas 5.19.2 and 5.19.6 this image is stable under specialization. Thus the result follows from Lemma 10.41.5. $\square$
Lemma 10.41.7. Let $R$ be a ring. Let $E \subset \mathop{\mathrm{Spec}}(R)$ be a constructible subset.
If $E$ is stable under generalization, then $E$ is open.
Proof. First proof. The first assertion follows from Lemma 10.41.5 combined with Lemma 10.29.4. The second follows because the complement of a constructible set is constructible (see Topology, Lemma 5.15.2), the first part of the lemma and Topology, Lemma 5.19.2.
Second proof. Since $\mathop{\mathrm{Spec}}(R)$ is a spectral space by Lemma 10.26.2 this is a special case of Topology, Lemma 5.23.6. $\square$
Proposition 10.41.8. Let $R \to S$ be flat and of finite presentation. Then $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is open. More generally this holds for any ring map $R \to S$ of finite presentation which satisfies going down.
Proof. If $R \to S$ is flat, then $R \to S$ satisfies going down by Lemma 10.39.19. Thus to prove the lemma we may assume that $R \to S$ has finite presentation and satisfies going down.
Since the standard opens $D(g) \subset \mathop{\mathrm{Spec}}(S)$, $g \in S$ form a basis for the topology, it suffices to prove that the image of $D(g)$ is open. Recall that $\mathop{\mathrm{Spec}}(S_ g) \to \mathop{\mathrm{Spec}}(S)$ is a homeomorphism of $\mathop{\mathrm{Spec}}(S_ g)$ onto $D(g)$ (Lemma 10.17.6). Since $S \to S_ g$ satisfies going down (see above), we see that $R \to S_ g$ satisfies going down by Lemma 10.41.4. Thus after replacing $S$ by $S_ g$ we see it suffices to prove the image is open. By Chevalley's theorem (Theorem 10.29.10) the image is a constructible set $E$. And $E$ is stable under generalization because $R \to S$ satisfies going down, see Topology, Lemmas 5.19.2 and 5.19.6. Hence $E$ is open by Lemma 10.41.7. $\square$
Lemma 10.41.9. Let $k$ be a field, and let $R$, $S$ be $k$-algebras. Let $S' \subset S$ be a sub $k$-algebra, and let $f \in S' \otimes _ k R$. In the commutative diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}((S \otimes _ k R)_ f) \ar[rd] \ar[rr] & & \mathop{\mathrm{Spec}}((S' \otimes _ k R)_ f) \ar[ld] \\ & \mathop{\mathrm{Spec}}(R) & } \]
the images of the diagonal arrows are the same.
Proof. Let $\mathfrak p \subset R$ be in the image of the south-west arrow. This means (Lemma 10.17.9) that
\[ (S' \otimes _ k R)_ f \otimes _ R \kappa (\mathfrak p) = (S' \otimes _ k \kappa (\mathfrak p))_ f \]
is not the zero ring, i.e., $S' \otimes _ k \kappa (\mathfrak p)$ is not the zero ring and the image of $f$ in it is not nilpotent. The ring map $S' \otimes _ k \kappa (\mathfrak p) \to S \otimes _ k \kappa (\mathfrak p)$ is injective. Hence also $S \otimes _ k \kappa (\mathfrak p)$ is not the zero ring and the image of $f$ in it is not nilpotent. Hence $(S \otimes _ k R)_ f \otimes _ R \kappa (\mathfrak p)$ is not the zero ring. Thus (Lemma 10.17.9) we see that $\mathfrak p$ is in the image of the south-east arrow as desired. $\square$
Lemma 10.41.10. Let $k$ be a field. Let $R$ and $S$ be $k$-algebras. The map $\mathop{\mathrm{Spec}}(S \otimes _ k R) \to \mathop{\mathrm{Spec}}(R)$ is open.
Proof. Let $f \in S \otimes _ k R$. It suffices to prove that the image of the standard open $D(f)$ is open. Let $S' \subset S$ be a finite type $k$-subalgebra such that $f \in S' \otimes _ k R$. The map $R \to S' \otimes _ k R$ is flat and of finite presentation, hence the image $U$ of $\mathop{\mathrm{Spec}}((S' \otimes _ k R)_ f) \to \mathop{\mathrm{Spec}}(R)$ is open by Proposition 10.41.8. By Lemma 10.41.9 this is also the image of $D(f)$ and we win. $\square$
Here is a tricky lemma that is sometimes useful.
Lemma 10.41.11. Let $R \to S$ be a ring map. Let $\mathfrak p \subset R$ be a prime. Assume that
there exists a unique prime $\mathfrak q \subset S$ lying over $\mathfrak p$, and
going up holds for $R \to S$, or
going down holds for $R \to S$ and there is at most one prime of $S$ above every prime of $R$.
Then $S_{\mathfrak p} = S_{\mathfrak q}$.
Proof. Consider any prime $\mathfrak q' \subset S$ which corresponds to a point of $\mathop{\mathrm{Spec}}(S_{\mathfrak p})$. This means that $\mathfrak p' = R \cap \mathfrak q'$ is contained in $\mathfrak p$. Here is a picture
\[ \xymatrix{ \mathfrak q' \ar@{-}[d] \ar@{-}[r] & ? \ar@{-}[r] \ar@{-}[d] & S \ar@{-}[d] \\ \mathfrak p' \ar@{-}[r] & \mathfrak p \ar@{-}[r] & R } \]
Assume (1) and (2)(a). By going up there exists a prime $\mathfrak q'' \subset S$ with $\mathfrak q' \subset \mathfrak q''$ and $\mathfrak q''$ lying over $\mathfrak p$. By the uniqueness of $\mathfrak q$ we conclude that $\mathfrak q'' = \mathfrak q$. In other words $\mathfrak q'$ defines a point of $\mathop{\mathrm{Spec}}(S_{\mathfrak q})$.
Assume (1) and (2)(b). By going down there exists a prime $\mathfrak q'' \subset \mathfrak q$ lying over $\mathfrak p'$. By the uniqueness of primes lying over $\mathfrak p'$ we see that $\mathfrak q' = \mathfrak q''$. In other words $\mathfrak q'$ defines a point of $\mathop{\mathrm{Spec}}(S_{\mathfrak q})$.
In both cases we conclude that the map $\mathop{\mathrm{Spec}}(S_{\mathfrak q}) \to \mathop{\mathrm{Spec}}(S_{\mathfrak p})$ is bijective. Clearly this means all the elements of $S - \mathfrak q$ are all invertible in $S_{\mathfrak p}$, in other words $S_{\mathfrak p} = S_{\mathfrak q}$. $\square$
The following lemma is a generalization of going down for flat ring maps.
Lemma 10.41.12. Let $R \to S$ be a ring map. Let $N$ be a finite $S$-module flat over $R$. Endow $\text{Supp}(N) \subset \mathop{\mathrm{Spec}}(S)$ with the induced topology. Then generalizations lift along $\text{Supp}(N) \to \mathop{\mathrm{Spec}}(R)$.
Proof. The meaning of the statement is as follows. Let $\mathfrak p \subset \mathfrak p' \subset R$ be primes. Let $\mathfrak q' \subset S$ be a prime $\mathfrak q' \in \text{Supp}(N)$ Then there exists a prime $\mathfrak q \subset \mathfrak q'$, $\mathfrak q \in \text{Supp}(N)$ lying over $\mathfrak p$. As $N$ is flat over $R$ we see that $N_{\mathfrak q'}$ is flat over $R_{\mathfrak p'}$, see Lemma 10.39.18. As $N_{\mathfrak q'}$ is finite over $S_{\mathfrak q'}$ and not zero since $\mathfrak q' \in \text{Supp}(N)$ we see that $N_{\mathfrak q'} \otimes _{S_{\mathfrak q'}} \kappa (\mathfrak q')$ is nonzero by Nakayama's Lemma 10.20.1. Thus $N_{\mathfrak q'} \otimes _{R_{\mathfrak p'}} \kappa (\mathfrak p')$ is also not zero. We conclude from Lemma 10.39.15 that $N_{\mathfrak q'} \otimes _{R_{\mathfrak p'}} \kappa (\mathfrak p)$ is nonzero. Let $J \subset S_{\mathfrak q'} \otimes _{R_{\mathfrak p'}} \kappa (\mathfrak p)$ be the annihilator of the finite nonzero module $N_{\mathfrak q'} \otimes _{R_{\mathfrak p'}} \kappa (\mathfrak p)$. Since $J$ is a proper ideal we can choose a prime $\mathfrak q \subset S$ which corresponds to a prime of $S_{\mathfrak q'} \otimes _{R_{\mathfrak p'}} \kappa (\mathfrak p)/J$. This prime is in the support of $N$, lies over $\mathfrak p$, and is contained in $\mathfrak q'$ as desired. $\square$
Comment #4590 by Fred Vu on October 10, 2019 at 19:08
In the proof of Lemma 037F, the references to ordinal directions should be swapped.
@#4590. I do not understand this comment. When I google "south-west arrow" I get an arrow pointing down and to the left which is I think what I want in the first part of the proof.
Comment #5521 by Lorenzo on September 28, 2020 at 12:52
The term "south-west arrow" could indicate an arrow pointed in the south-west direction as well as an arrow situated in the south-west "region" of the diagram. It might help if "south-west arrow" was replaced with "right arrow" or "south-west-pointing arrow", and similarly for "south-east arrow".
Not going to change this for now.
Comment #7027 by Nico on February 07, 2022 at 11:51
The proof of Proposition 00I1 is confusing. Flatness is the essential hypothesis, but the proof never explicitly mentions flatness. I think the phrase "we see that
R\to S_f
satisfies going down" needs to prefaced with something like, "since
R\to S
is flat, it satisfies going down" and make a reference to Lemma 00HS. Maybe it's just my inexperience but it took me a while to figure out what was going on.
Yes, I agree. We need to explain the "more generally" part of the statement in the proof. I will do this the next time I go through all the comments.
@#7027. Now this is fixed here.
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Volume 2 Issue 2 | European Journal of Mineralogy | GeoScienceWorld
January - Volume 2, Number 1 January - Volume 2, Number 2 January - Volume 2, Number 3 January - Volume 2, Number 4 October - Volume 2, Number 5 December - Volume 2, Number 6
Framework distortion by large ions in Ma/Si 2 O 6 aluminosilicates with keatite structure
B. Baumgartner; G. Mueller
European Journal of Mineralogy January 01, 1990, Vol.2, 155-162. doi:
Cation arrangement in a unusual uranium-rich senaite; crystal structure study at 130 K
T. Armbruster; M. Kunz
Synthesis, XRD and FTIR studies of strontium richterites
G. Della-Ventura; J. L. Robert
Burpalite, a new mineral from Burpalinskii Massif, North Transbajkal, USSR; its crystal structure and OD character
S. Merlino; N. Perchiazzi; A. P. Khomyakhov; D. Y. Pushcharovskii; I. M. Kulikova; V. I. Kuzmin
Dachiardite from Hokiya-dake; evidence of a new topology
S. Quartieri; G. Vezzalini; A. Alberti
Scapolites; variation of structure with pressure and possible role in the storage of fluids
P. Comodi; M. Mellini; P. F. Zanazzi
Rhoenite; structural and microstructural features, crystal chemistry and polysomatic relationships
E. Bonaccorsi; S. Merlino; M. Pasero
New data on the Cu 2 FeSnS 4 -Cu 2 ZnSnS 4 pseudobinary system at 750 degree and 550 degree C
G. P. Bernardini; P. Bonazzi; M. Corazza; F. Corsini; G. Mazzetti; L. Poggi; G. Tanelli
Shock deformation of alpha quartz; laboratory experiment and TEM investigation
H. Tattevin; Y. Syono; M. Kikuchi; K. Kusaba; B. Velde
Geothermobarometry based on pyrrhotite composition and fluid inclusion studies; application to the scheelite-bearing deposit of Salau (Pyrenees, France)
J. Dubessy; A. D. Schellen; J. M. Claude
Fe43+
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Molar Mass | Boundless Chemistry | Course Hero
The mole is represented by Avogadro's number, which is 6.022×1023 atoms or molecules per mol.
Define and memorize Avogadro's number
The mole allows scientists to calculate the number of elementary entities (usually atoms or molecules ) in a certain mass of a given substance.
Avogadro's number is an absolute number: there are 6.022×1023 elementary entities in 1 mole. This can also be written as 6.022×1023 mol-1.
The mass of one mole of a substance is equal to that substance's molecular weight. For example, the mean molecular weight of water is 18.015 atomic mass units (amu), so one mole of water weight 18.015 grams.
mole: The amount of substance of a system that contains as many elementary entities as there are atoms in 12 g of carbon-12.
The chemical changes observed in any reaction involve the rearrangement of billions of atoms. It is impractical to try to count or visualize all these atoms, but scientists need some way to refer to the entire quantity. They also need a way to compare these numbers and relate them to the weights of the substances, which they can measure and observe. The solution is the concept of the mole, which is very important in quantitative chemistry.
Amedeo Avogadro: Amedeo Avogadro is credited with the idea that the number of entities (usually atoms or molecules) in a substance is proportional to its physical mass.
Amadeo Avogadro first proposed that the volume of a gas at a given pressure and temperature is proportional to the number of atoms or molecules, regardless of the type of gas. Although he did not determine the exact proportion, he is credited for the idea.
Avogadro's number is a proportion that relates molar mass on an atomic scale to physical mass on a human scale. Avogadro's number is defined as the number of elementary particles (molecules, atoms, compounds, etc.) per mole of a substance. It is equal to 6.022×1023 mol-1 and is expressed as the symbol NA.
Avogadro's number is a similar concept to that of a dozen or a gross. A dozen molecules is 12 molecules. A gross of molecules is 144 molecules. Avogadro's number is 6.022×1023 molecules. With Avogadro's number, scientists can discuss and compare very large numbers, which is useful because substances in everyday quantities contain very large numbers of atoms and molecules.
The mole (abbreviated mol) is the SI measure of quantity of a "chemical entity," such as atoms, electrons, or protons. It is defined as the amount of a substance that contains as many particles as there are atoms in 12 grams of pure carbon-12. So, 1 mol contains 6.022×1023 elementary entities of the substance.
Chemical Computations with Avogadro's Number and the Mole
Avogadro's number is fundamental to understanding both the makeup of molecules and their interactions and combinations. For example, since one atom of oxygen will combine with two atoms of hydrogen to create one molecule of water (H2O), one mole of oxygen (6.022×1023 of O atoms) will combine with two moles of hydrogen (2 × 6.022×1023 of H atoms) to make one mole of H2O.
Another property of Avogadro's number is that the mass of one mole of a substance is equal to that substance's molecular weight. For example, the mean molecular weight of water is 18.015 atomic mass units (amu), so one mole of water weight 18.015 grams. This property simplifies many chemical computations.
1.25\text{ g} \times \frac{ 1 \text{ mole}}{134.1\text{ g}}=0.0093 \text{ moles}
The Mole, Avogadro: This video introduces counting by mass, the mole, and how it relates to atomic mass units (AMU) and Avogadro's number.
Converting between Moles and Atoms
By understanding the relationship between moles and Avogadro's number, scientists can convert between number of moles and number of atoms.
Convert between the number of moles and the number of atoms in a given substance using Avagadro's number
Avogadro's number is a very important relationship to remember: 1 mole =
6.022\times10^{23}
atoms, molecules, protons, etc.
To convert from moles to atoms, multiply the molar amount by Avogadro's number.
To convert from atoms to moles, divide the atom amount by Avogadro's number (or multiply by its reciprocal).
Avogadro's number: The number of atoms present in 12 g of carbon-12, which is
6.022\times10^{23}
and the number of elementary entities (atoms or molecules) comprising one mole of a given substance.
Moles and Atoms
As introduced in the previous concept, the mole can be used to relate masses of substances to the quantity of atoms therein. This is an easy way of determining how much of one substance can react with a given amount of another substance.
From moles of a substance, one can also find the number of atoms in a sample and vice versa. The bridge between atoms and moles is Avogadro's number, 6.022×1023.
Avogadro's number is typically dimensionless, but when it defines the mole, it can be expressed as 6.022×1023 elementary entities/mol. This form shows the role of Avogadro's number as a conversion factor between the number of entities and the number of moles. Therefore, given the relationship 1 mol = 6.022 x 1023 atoms, converting between moles and atoms of a substance becomes a simple dimensional analysis problem.
Given a known number of moles (x), one can find the number of atoms (y) in this molar quantity by multiplying it by Avogadro's number:
x \text{ moles}\cdot\frac {6.022\times10^{23}\text{atoms}}{1\text{ mole}} = y\text{ atoms}
For example, if scientists want to know how may atoms are in six moles of sodium (x = 6), they could solve:
6\text{ moles}\cdot\frac {6.022\times 10^{23}\text{ atoms}}{1\text{ mole}} = 3.61\times 10^{24}\text{ atoms}
Note that the solution is independent of whether the element is sodium or otherwise.
Reversing the calculation above, it is possible to convert a number of atoms to a molar quantity by dividing it by Avogadro's number:
\frac{{x\text{ atoms}}}{{6.022\times 10^{23} \frac{\text{atoms}}{1\text{ mole}}}}= y\text{ moles}
This can be written without a fraction in the denominator by multiplying the number of atoms by the reciprocal of Avogadro's number:
x \text{ atoms}\cdot\frac{1\text{ mole}}{6.022\times 10^{23}\text{ atoms}} = y \text{ moles}
For example, if scientists know there are
3.5 \cdot 10^{24}
atoms in a sample, they can calculate the number of moles this quantity represents:
3.5\times 10^{24}\text{ atoms}\cdot\frac{1\text{ mole}}{6.022\times 10^{23} \text{ atoms}} = 5.81\text{ moles}
The molar mass of a particular substance is the mass of one mole of that substance.
Calculate the molar mass of an element or compound
Molar mass serves as a bridge between the mass of a material and the number of moles since it is not possible to measure the number of moles directly.
molar mass: The mass of a given substance (chemical element or chemical compound in g) divided by its amount of substance (mol).
Measuring Mass in Chemistry
Chemists can measure a quantity of matter using mass, but in chemical reactions it is often important to consider the number of atoms of each element present in each sample. Even the smallest quantity of a substance will contain billions of atoms, so chemists generally use the mole as the unit for the amount of substance.
One mole (abbreviated mol) is equal to the number of atoms in 12 grams of carbon-12; this number is referred to as Avogadro's number and has been measured as approximately 6.022 x 1023. In other words, a mole is the amount of substance that contains as many entities (atoms, or other particles) as there are atoms in 12 grams of pure carbon-12.
Each ion, or atom, has a particular mass; similarly, each mole of a given pure substance also has a definite mass. The mass of one mole of atoms of a pure element in grams is equivalent to the atomic mass of that element in atomic mass units (amu) or in grams per mole (g/mol). Although mass can be expressed as both amu and g/mol, g/mol is the most useful system of units for laboratory chemistry.
The characteristic molar mass of an element is simply the atomic mass in g/mol. However, molar mass can also be calculated by multiplying the atomic mass in amu by the molar mass constant (1 g/mol). To calculate the molar mass of a compound with multiple atoms, sum all the atomic mass of the constituent atoms.
Molar Mass Calculations - YouTube: This video shows how to calculate the molar mass for several compounds using their chemical formulas.
A substance's molar mass can be used to convert between the mass of the substance and the number of moles in that substance.
Convert between the mass and the number of moles, and the number of atoms, in a given sample of compound
Although there is no physical way of measuring the number of moles of a compound, we can relate its mass to the number of moles by using the compound's molar mass as a direct conversion factor.
To convert between mass and number of moles, you can use the molar mass of the substance. Then, you can use Avogadro's number to convert the number of moles to number of atoms.
molar mass: The mass of a given substance (chemical element or chemical compound) divided by its amount of substance (mol), in g/mol.
dimensional analysis: The analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
mole: The amount of substance that contains as many elementary entities as there are atoms in 12 g of carbon-12.
Chemists generally use the mole as the unit for the number of atoms or molecules of a material. One mole (abbreviated mol) is equal to 6.022×1023 molecular entities (Avogadro's number), and each element has a different molar mass depending on the weight of 6.022×1023 of its atoms (1 mole). The molar mass of any element can be determined by finding the atomic mass of the element on the periodic table. For example, if the atomic mass of sulfer (S) is 32.066 amu, then its molar mass is 32.066 g/mol.
By recognizing the relationship between the molar mass (g/mol), moles (mol), and particles, scientists can use dimensional analysis convert between mass, number of moles and number of atoms very easily.
Converting between mass, moles, and particles: This flowchart illustrates the relationships between mass, moles, and particles. These relationships can be used to convert between units.
In a compound of NaOH, the molar mass of Na alone is 23 g/mol, the molar mass of O is 16 g/mol, and H is 1 g/mol. What is the molar mass of NaOH?
\text{Na}+\text{O}+\text{H}=\text{NaOH}
23 \space \text{g/mol} +16 \space \text{g/mol}+ 1 \space \text{g/mol} = 40 \space \text{g/mol}
If the equation is arranged correctly, the mass units (g) cancel out and leave moles as the unit.
90\text{ g}\space \text{NaOH} \times \frac{1 \text{ mol}}{40\text{ g}} = 2.25 \space \text{mol NaOH}
Converting Between Mass, Number of Moles, and Number of Atoms
10\text{ g Ni}\times \frac{1\text{ mol Ni}}{58.69\text{ g Ni}} = 0.170\text{ mol Ni}
To determine the number of atoms, convert the moles of Ni to atoms using Avogadro's number:
0.170\text{ moles Ni}\times\frac {6.022\times10^{23}\text{ atoms Ni}}{1\text{ mol Ni}} = 1.02\times10^{23}\text{ atoms Ni}
Given a sample's mass and number of moles in that sample, it is also possible to calculate the sample's molecular mass by dividing the mass by the number of moles to calculate g/mol.
What is the molar mass of methane (CH4) if there are 0.623 moles in a 10.0g sample?
\frac{10.0 \text{ g CH}_4}{0.623 \text{ mol CH}_4} = 16.05 \text{ g/mol CH}_4
Avogadro's number and the mole. Provided by: Steve Lower's Website. Located at: http://www.chem1.com/acad/webtext/intro/int-2.html#SEC2. License: CC BY-SA: Attribution-ShareAlike
Mole (unit). Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Mole_(unit). License: CC BY-SA: Attribution-ShareAlike
Avogadro constant. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Avogadro_constant. License: CC BY-SA: Attribution-ShareAlike
The Mole, Avogadro. Located at: http://www.youtube.com/watch?v=TqDqLmwWx3A. License: Public Domain: No Known Copyright. License terms: Standard YouTube license
Avogadro Amedeo. Provided by: Wikimedia. Located at: http://en.wikipedia.org/wiki/Avogadro_constant%23mediaviewer/File:Avogadro_Amedeo.jpg. License: Public Domain: No Known Copyright
Avogadro's number. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Avogadro's%20number. License: CC BY-SA: Attribution-ShareAlike
Molar mass. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Molar_mass. License: CC BY-SA: Attribution-ShareAlike
Atomic mass unit. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Atomic_mass_unit. License: CC BY-SA: Attribution-ShareAlike
Molar Mass Calculations - YouTube. Located at: http://www.youtube.com/watch?v=guAbb_yBSfs. License: Public Domain: No Known Copyright. License terms: Standard YouTube license
Dimensional Analysis. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Dimensional_analysis. License: CC BY-SA: Attribution-ShareAlike
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Feature extraction by using reconstruction ICA - MATLAB rica
Feature extraction by using reconstruction ICA
Mdl = rica(X,q)
Mdl = rica(X,q,Name,Value)
Mdl = rica(X,q) returns a reconstruction independent component analysis (RICA) model object that contains the results from applying RICA to the table or matrix of predictor data X containing p variables. q is the number of features to extract from X, therefore rica learns a p-by-q matrix of transformation weights. For undercomplete or overcomplete feature representations, q can be less than or greater than the number of predictor variables, respectively.
Mdl = rica(X,q,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, you can standardize the predictor data or specify the value of the penalty coefficient in the reconstruction term of the objective function.
rica stores a p-by-q transform weight matrix in Mdl.TransformWeights. Therefore, setting very large values for q can result in greater memory consumption and increased computation time.
Example: Mdl = rica(X,q,'IterationLimit',200,'Standardize',true) runs rica with optimization iterations limited to 200 and standardized predictor data.
0 rica does not display convergence information at the command line.
Positive integer rica displays convergence information at the command line.
Lambda — Regularization coefficient value
Regularization coefficient value for the transform weight matrix, specified as the comma-separated pair consisting of 'Lambda' and a positive numeric scalar. If you specify 0, then there is no regularization term in the objective function.
rica centers and scales each column of the predictor data (X) by the column mean and standard deviation, respectively.
rica extracts new features by using the standardized predictor matrix, and stores the predictor variable means and standard deviations in properties Mu and Sigma of Mdl.
ContrastFcn — Contrast function
'logcosh' (default) | 'exp' | 'sqrt'
Contrast function, specified as 'logcosh', 'exp', or 'sqrt'. The contrast function is a smooth function that is similar to an absolute value function. The rica objective function contains a term
\sum _{j=1}^{q}\frac{1}{n}\sum _{i=1}^{n}g\left({w}_{j}^{T}{\stackrel{˜}{x}}_{i}\right),
where g represents the contrast function, the wj are the variables over which the optimization takes place, and the
{\stackrel{˜}{x}}_{i}
are data.
The three available contrast functions are:
'logcosh' —
g=\frac{1}{2}\mathrm{log}\left(\mathrm{cosh}\left(2x\right)\right)
'exp' —
g=-\mathrm{exp}\left(-\frac{{x}^{2}}{2}\right)
'sqrt' —
g=\sqrt{{x}^{2}+{10}^{-8}}
Example: 'ContrastFcn','exp'
You can continue optimizing a previously returned transform weight matrix by passing it as an initial value in another call to rica. The output model object Mdl stores a learned transform weight matrix in the TransformWeights property.
ones(q,1) (default) | length-q vector of ±1
Non-Gaussianity of sources, specified as a length-q vector of ±1.
NonGaussianityIndicator(k) = 1 means rica models the kth source as super-Gaussian, with a sharp peak at 0.
NonGaussianityIndicator(k) = -1 means rica models the kth source as sub-Gaussian.
Mdl — Learned reconstruction ICA model
ReconstructionICA model object
Learned reconstruction ICA model, returned as a ReconstructionICA model object.
To access the structure of fitting information, use Mdl.FitInfo.
The rica function creates a linear transformation of input features to output features. The transformation is based on optimizing a nonlinear objective function that roughly balances statistical independence of the output features versus the ability to reconstruct the input data using the output features.
For details, see Reconstruction ICA Algorithm.
sparsefilt | transform | ReconstructionICA
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A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems | J. Comput. Nonlinear Dynam. | ASME Digital Collection
Shih-Tin Lin,
Shih-Tin Lin
, Taichung 40227, Taiwan
e-mail: stlin@dragon.nchu.edu.tw
Ming-Wen Chen
Lin, S., and Chen, M. (April 14, 2011). "A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems." ASME. J. Comput. Nonlinear Dynam. October 2011; 6(4): 044501. https://doi.org/10.1115/1.4002688
The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an
Iintegrator
controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.
control system synthesis, differential algebraic equations, multivariable control systems, nonlinear dynamical systems, PD control, stability, three-term control
Algebra, Control equipment, Control theory, Equations of motion, Errors, Multibody systems, Simulation, Stability, Steady state, Design, Multibody dynamics, Control systems, Differential algebraic equations, Nonlinear dynamical systems
Real-Time Simulation of Multibody Dynamics on Shared Memory Multiprocessors
Differential/Algebraic Equations Are Not ODEs
Generalized Coordinates Partitioning for Dimension Reduction in Analysis of Constrained Dynamic System
On Baumgarte Stabilization for Differential Algebraic Equations
Real-Time Integration Methods for Mechanical System Simulation
A Stabilization Method for Kinematic and Kinetic Constraint Equations
An Adaptive Constraint Violation Stabilization Method for Dynamic Analysis of Mechanical Systems
Stability and Accuracy Analysis of Baumgarte’s Constraint Stabilization Method
Stabilization Method for the Numerical Integration of Multibody Mechanical System
Stabilization Method for the Numerical Integration of Controlled Multibody Mechanical System: A Hybrid Integration Approach
Parameters Selection for Baumgarte’s Constraint Stabilization Method Using the Predictor-Corrector Approach
AIAA J. Guidance, Control, and Dynamics
Constraint Stabilization Method for the Simulation of Multibody Mechanical Systems
J. Chin. Soc. Mech. Eng.
Mechanische System Emit Beschranktem Konfigurationsraum
,” Ph.D. thesis, TU Braunschweig, Germany.
A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework
Eliminating Constraint Drift in the Numerical Simulation of Constrained Dynamical Systems
Stabilization of Computational Procedures for Constrained Dynamical Systems
An Improved Formulation for Constrained Mechanical Systems
Zhenkuan
An Automatic Constraint Violation Stabilization Method for Differential/Algebraic Equations of Motion in Multibody System Dynamics
Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints
Influence of the Baumgarte Parameters on the Dynamics Response of Multibody Mechanical Systems
Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Application and Algorithms
A Two-Loop Sparse Matrix Numerical Integration Procedure for the Solution of Differential/Algebraic Equations: Application to Multibody Systems
Computer Aided Kinematics and Dynamics of Mechanical System
|
Central moment - MATLAB moment - MathWorks América Latina
Find Central Moment for Specified Order
Find Central Moment Along Given Dimension
Find Central Moment Along Vector of Dimensions
m = moment(X,order)
m = moment(X,order,'all')
m = moment(X,order,dim)
m = moment(X,order,vecdim)
m = moment(X,order) returns the central moment of X for the order specified by order.
If X is a vector, then moment(X,order) returns a scalar value that is the k-order central moment of the elements in X.
If X is a matrix, then moment(X,order) returns a row vector containing the k-order central moment of each column in X.
If X is a multidimensional array, then moment(X,order) operates along the first nonsingleton dimension of X.
m = moment(X,order,'all') returns the central moment of the specified order for all elements of X.
m = moment(X,order,dim) takes the central moment along the operating dimension dim of X.
m = moment(X,order,vecdim) returns the central moment over the dimensions specified in the vector vecdim. For example, if X is a 2-by-3-by-4 array, then moment(X,1,[1 2]) returns a 1-by-1-by-4 array. Each element of the output array is the first-order central moment of the elements on the corresponding page of X.
Generate a matrix with 6 rows and 5 columns.
Find the third-order central moment of X.
m = moment(X,3)
m is a row vector containing the third-order central moment of each column in X.
Find the central moment along different dimensions for a multidimensional array.
Find the fourth-order central moment of X along the default dimension.
m1 = moment(X,4)
By default, moment operates along the first dimension of X whose size does not equal 1. In this case, this dimension is the first dimension of X. Therefore, m1 is a 1-by-3-by-2 array.
Find the fourth-order central moment of X along the second dimension.
m2 = moment(X,4,2)
m2 is a 4-by-1-by-2 array.
Find the fourth-order central moment of X along the third dimension.
m3 is a 4-by-3 matrix.
Find the central moment over multiple dimensions by using the 'all' and vecdim input arguments.
mall = moment(X,3,'all')
mall is the third-order central moment of the entire input data set X.
Find the third-order moment of each page of X by specifying the first and second dimensions.
mpage = moment(X,3,[1 2])
mpage =
mpage(:,:,1) =
For example, mpage(1,1,2) is the third-order central moment of the elements in X(:,:,2).
Find the third-order moment of the elements in each X(i,:,:) slice by specifying the second and third dimensions.
mrow = moment(X,3,[2 3])
mrow = 4×1
For example, mrow(1) is the third-order central moment of the elements in X(1,:,:).
order — Order of central moment
Order of the central moment, specified as a positive integer.
Dimension along which to operate, specified as a positive integer. If you do not specify a value for dim, then the default is the first nonsingleton dimension of X.
Consider the third-order central moment of a matrix X:
If dim is equal to 1, then moment(X,3,1) returns a row vector that contains the third-order central moment of each column in X.
If dim is equal to 2, then moment(X,3,2) returns a column vector that contains the third-order central moment of each row in X.
If dim is greater than ndims(X) or if size(X,dim) is 1, then moment returns an array of zeros the same size as X.
Vector of dimensions, specified as a positive integer vector. Each element of vecdim represents a dimension of the input array X. The output m has length 1 in the specified operating dimensions. The other dimension lengths are the same for X and m.
For example, if X is a 2-by-3-by-3 array, then moment(X,1,[1 2]) returns a 1-by-1-by-3 array. Each element of the output array is the first-order central moment of the elements on the corresponding page of X.
m — Central moments
Central moments, returned as a scalar, vector, matrix, or multidimensional array.
The central moment of order k for a distribution is defined as
{m}_{k}=E{\left(x-\mu \right)}^{k},
where µ is the mean of x, and E(t) represents the expected value of the quantity t. The moment function computes a sample version of this population value.
{m}_{k}=\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{k}.
Note that the first-order central moment is zero, and the second-order central moment is the variance computed using a divisor of n rather than n – 1, where n is the length of the vector x or the number of rows in the matrix X.
If order is nonintegral and X is real, use moment(complex(X),order).
kurtosis | mean | skewness | std | var
|
Flow over curved backward-facing step
This document presents the Underlying flow Regime (UFR) of a turbulent boundary layer separating from a curved surface at moderate Reynolds number (
{\displaystyle Re_{\theta }\sim \ 1200}
, based on the momentum thickness
{\displaystyle \ \theta }
at the domain inlet). The primary focus of this case is on the details of the separation process and the turbulent properties of the separated region, including reattachment. This geometry shows particular features of separation from gently-curved surfaces: the separation process is highly unsteady in time and space; turbulence is highly non-local in character; the mean reverse-flow region is thin and highly elongated; no part of the flow is reversed at all times; the level of production is extremely high following separation, resulting in massive departures from turbulence-energy equilibrium, very high anisotropy and a trend towards one-component turbulence in the separated shear layer.
The geometry of the curved step was designed by a combined effort between University of Manchester, where experiments were conducted, and Imperial College London, where the Large-Eddy Simulations were performed. The geometry is an extended version of that studied by Song & Eaton [27], except for the step height, which is increased to produce a larger recirculation bubble. The careful design of the geometry allows an accurate comparison between well-resolved large-eddy simulation and experimental results, for exactly the same flow conditions. All the results available in the database are extracted from the finest Large-Eddy Simulation (LES) presented in the CFD section, and they show very good agreement with the experimental data available. Results of RANS calculations with well-known two-equation models are also presented in graphical form to illustrate the challenges of this test case. A recurring defect of such models is that most predict insufficient levels of turbulence activity in the separated shear layer and thus a serious delay in reattachment and excessive recirculation, shortcomings that reflect an inability of the models to account for the dynamics of the separation process, made worse by the tendency of the models to depress the turbulent stresses in the shear layer bordering the recirculation zone because of the effects of (stabilizing) curvature on the turbulent stresses.
Contributed by: Sylvain Lardeau — CD-adapco, London, UK
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Thymaridas Knowpia
Thymaridas of Paros (Greek: Θυμαρίδας; c. 400 – c. 350 BCE) was an ancient Greek mathematician and Pythagorean noted for his work on prime numbers and simultaneous linear equations.
Although little is known about the life of Thymaridas, it is believed that he was a rich man who fell into poverty. It is said that Thestor of Poseidonia traveled to Paros in order to help Thymaridas with the money that was collected for him.
Iamblichus states that Thymaridas called prime numbers "rectilinear", since they can only be represented on a one-dimensional line. Non-prime numbers, on the other hand, can be represented on a two-dimensional plane as a rectangle with sides that, when multiplied, produce the non-prime number in question. He further called the number one a "limiting quantity".
Iamblichus in his comments to Introductio arithmetica states that Thymaridas also worked with simultaneous linear equations.[1] In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:[2]
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) [this is a typo in Flegg's book – the denominator should be n − 2 to match the math below] of the difference between the sums of these pairs and the first given sum.
or using modern notation, the solution of the following system of n linear equations in n unknowns:[1]
{\displaystyle {\begin{aligned}x+x_{1}+x_{2}+\cdots +x_{n-1}&=s,\\x+x_{1}&=m_{1},\\x+x_{2}&=m_{2},\\&~~\vdots \\x+x_{n-1}&=m_{n-1}\end{aligned}}}
{\displaystyle x={\frac {(m_{1}+m_{2}+\cdots +m_{n-1})-s}{n-2}}.}
Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.[1]
Heath, Thomas Little (1981). A History of Greek Mathematics. Dover publications. ISBN 0-486-24073-8.
Flegg, Graham (1983). Numbers: Their History and Meaning. Dover publications. ISBN 0-486-42165-1.
Citations and footnotesEdit
^ a b c Heath (1981). "The ('Bloom') of Thymaridas". A History of Greek Mathematics. pp. 94–96. Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of n simultaneous simple equations connecting n unknown quantities. The rule was evidently well known, for it was called by the special name [...] the 'flower' or 'bloom' of Thymaridas. [...] The rule is very obscurely worded , but it states in effect that, if we have the following n equations connecting n unknown quantities x, x1, x2 ... xn−1, namely [...] Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that the rule does not 'leave us in the lurch' in those cases either.
^ Flegg (1983). "Unknown Numbers". Numbers: Their History and Meaning. pp. 205. Thymaridas (fourth century) is said to have had this rule for solving a particular set of n linear equations in n unknowns:
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) of the difference between the sums of these pairs and the first given sum.
The mac-tutor biography of Thymaridas
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CANDY - GTON Capital | GC
A single-sided liquidity mining and farming protocol
With the advent of AMM DEXes, most liquidity mining tools allow for holding a pair of tokens in a pool, where impermanent loss becomes the primary risk for engaging in liquidity provision. Engaging in risky activity required from a liquidity provider is rewarded by fees shared across LPs and sometimes by extra incentives formed from the distribution of farming tokens.
A crucial law of AMMs is the more liquidity is present in a pool, the easier it is for token holders to buy and sell. In cases where protocol-owned liquidity is insufficient, assets can be rented from users in order to boost the liquidity available in pools, which can be an efficient tool to support short-term trading. By borrowing the liquidity, protocols can reduce slippage on exchanges, thereby making their tokens more marketable.
In this article, we will describe Candy, a new protocol for liquidity mining and farming which does not require putting a pair of tokens into DEX pools. Such protocols are usually called “single sided” because liquidity provision is done with only one token.
In Candy, the risk of impermanent loss is replaced by the risk of withdrawing a “blend” of originally deposited assets with a fraction of assets issued by the protocol. A similar approach is implemented in different protocols where users also provide single sided liquidity, such as Curve. The withdrawal process in such protocols yields a mixture of stablecoins in the proportion determined by the asset balance in the pool. This approach can work for stablecoins or assets with the same peg on AMMs. Candy applies similar principles to PMM pools, with the quote asset minting being controlled by the protocol or a DAO.
candyETH
- an ERC20 vault token which the user receives in exchange for locking ETH (or a token pair pegged to ETH dollar value) in Candy protocol
GCD
- stablecoin mintable from DAO treasury
PMM - Proactive Market Maker DEX pool
lpETH
- ETH locked in the PMM pool, owned by the protocol (POL)
lpGCD
- stablecoin locked in the PMM pool
blend
ETH/GCD
tokens pair; its dollar value is equal to $ETH value being put in Candy to get
, and the proportion of ETH and GCD depends on the PMM liquidity balance
fraction
- the $ share of GCD in a blend
- means the dollar value of asset X
Alice deposits 10 ETH and receives 10
candyETH
. ETH price is $100, so the total $ value deposited by Alice is $1000;
Alice’s 10 ETH are paired with
GCD
minted from DAO treasury and put into the PMM pool, which yields lpETH and lpGCD;
The TVL in dollars is always equal to:
= $4000 and
To redeem her locked funds, Alice burns 10 candyETH to exchange it back for an
ETH/GCD
blend of 10
ETH
dollar value. The proportion (fraction) is determined by the formula:
Fraction(GCD)=max(1-(\frac{lpETH}{candyETH}),0)
lpETH = candyETH
, Alice can exchange 10
candyETH
back for 10 ETH without any
GCD
. If the pool has only 8
lpETH
and the total issued
candyETH
is 10, the proportion is 0.2, therefore Alice will receive a blend of 8
ETH
GCD
, so that the total dollar value remains equal to 10
ETH
lpETH
candyETH
. This case is explained above, where Alice receives 10
ETH
candyETH
Case 2: Base Dominance
In this case, there is more
lpETH
candyETH
. The fraction is therefore equal to 0. Similar to the above case, Alice will receive 10
ETH
candyETH
. The corner case is
candyETH=0
, where the pool has some left
lpETH
and the equal $ amount of
GCD
Case 3: Quote Dominance
In this case, there is less
lpETH
candyETH
. Alice receives a blend where the proportion of GCD depends on the
1-(lpETH/candyETH)
ratio. The corner case is
lpETH = 0
, where Alice receives 10
ETH
$ value in
GCD
The locked tokens (e.g. ETH), which serve as rented liquidity for the DAO, are used directly in PMM pools to support trading activity and boost the utility of DAO’s own stablecoin. The activity of liquidity providers and the risks they encounter (when fraction > 0) must be compensated by rewards coming from the DAO. There are several types of rewards possible:
staking of candyETH tokens with a fixed APR, which comes either from governance token emission or stablecoin emission. This process is called liquidity farming.
PMM DEX trading fees distributed between candyETH stakers.
The PMM architecture based on asymmetric bonding curves that allows for asymmetric liquidity can easily support a high
in the pool. In the equilibrium case, the PMM pool will contain ¼ of ETH and ¾ of GCD.
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\mathrm{story}≔\mathrm{Import}\left("example/UnderTheMaples.txt",\mathrm{base}=\mathrm{datadir}\right)
\textcolor[rgb]{0,0,1}{\mathrm{story}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"Under The Maples, by John Burroughs Chapter I: The Falling Leaves The time of the falling of leaves has come again. Once more in our morning walk we tread upon carpets of gold and crimson, of brown and bronze, woven by the winds or the rains out of these delicate textures while we slept."}
\mathrm{StringTools}:-\mathrm{WordCount}\left(\mathrm{story}\right)
\textcolor[rgb]{0,0,1}{54}
|
Three straight lines l1, l2, and l3 are said to be concurrent if they lie in a plane and pass through a common point.
The command with(geometry,AreConcurrent) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{line}\left(\mathrm{l1},3b-6=0,[a,b]\right):
\mathrm{line}\left(\mathrm{l2},-{3}^{\frac{1}{2}}a+b+{3}^{\frac{1}{2}}-2=0,[a,b]\right):
\mathrm{line}\left(\mathrm{l3},{3}^{\frac{1}{2}}a+b-{3}^{\frac{1}{2}}-2=0,[a,b]\right):
\mathrm{AreConcurrent}\left(\mathrm{l1},\mathrm{l2},\mathrm{l3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{line}\left(\mathrm{l4},\mathrm{sqrt}\left(3\right)b-2\mathrm{sqrt}\left(3\right)=11,[a,b]\right):
\mathrm{AreConcurrent}\left(\mathrm{l1},\mathrm{l2},\mathrm{l4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{line}\left(\mathrm{l5},mb+{3}^{\frac{1}{2}}a-2=0,[a,b]\right):
\mathrm{AreConcurrent}\left(\mathrm{l1},\mathrm{l2},\mathrm{l5},'\mathrm{cond}'\right)
AreConcurrent: "unable to determine if 6*3^(1/2)*m-6*3^(1/2)+9 is zero"
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\mathrm{cond}
\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
make necessary assumption:
\mathrm{assume}\left(\mathrm{cond}\right)
\mathrm{AreConcurrent}\left(\mathrm{l1},\mathrm{l2},\mathrm{l5}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Home : Support : Online Help : Education : Student Packages : Statistics : Hypothesis Tests : ChiSquareSuitableModelTest
(optional) equation(s) of the form option=value where option is one of bins, level, output, or range; specify options for the ChiSquareSuitableModelTest function
The ChiSquareSuitableModelTest function performs the chi-square suitable model test upon an observed data sample against a known random variable or probability distribution. It works by determining bins for a histogram from the probability distribution, then classifying the entries of X into these bins, and finally testing whether the resulting histogram matches the histogram for the probability distribution.
The first parameter X is a data sample of observed data to use in the analysis.
The second parameter F is a random variable or probability distribution that is compared to the observed data sample.
This test is only appropriate if there is prior knowledge of any parameters in the distribution. If any of the parameters in the distribution have been fitted to the data sample in question, then an adjustment of the degrees-of-freedom parameter is necessary. This adjustment is not available in the current implementation.
This option indicates the number of bins to use when categorizing data from X and probabilities from F. If set to 'deduce' (default), the function attempts to determine a reasonable value for this option. This parameter is ignored if the distribution is discrete.
output=report or plot or both
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
X≔\mathrm{Sample}\left(\mathrm{NormalRandomVariable}\left(0,1\right),100\right):
Perform the suitable model test upon this sample.
\mathrm{ChiSquareSuitableModelTest}\left(X,\mathrm{UniformRandomVariable}\left(0,1\right),\mathrm{bins}=10\right)
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{criticalvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{16.9189774487099}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{distribution}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ChiSquare}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{9}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{301.6000000}]
\mathrm{ChiSquareSuitableModelTest}\left(X,\mathrm{NormalRandomVariable}\left(0,1\right),\mathrm{bins}=10\right)
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{criticalvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{16.9189774487099}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{distribution}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ChiSquare}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{9}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.0965781731648307}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.80000000}]
If the output=plot option is included, then a report will be returned.
\mathrm{ChiSquareSuitableModelTest}\left(X,\mathrm{NormalRandomVariable}\left(0,1\right),\mathrm{bins}=10,\mathrm{output}=\mathrm{plot}\right)
\mathrm{report},\mathrm{graph}≔\mathrm{ChiSquareSuitableModelTest}\left(X,\mathrm{NormalRandomVariable}\left(0,1\right),\mathrm{bins}=10,\mathrm{output}=\mathrm{both}\right):
Data Range: -1.6348567543439 .. 2.21337958939036
Bin Width: .128274544791142
\mathrm{report}
[\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{criticalvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{16.9189774487099}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{distribution}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ChiSquare}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{9}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.0965781731648307}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.80000000}]
\mathrm{graph}
The Student[Statistics][ChiSquareSuitableModelTest] command was introduced in Maple 18.
Student/Statistics/ChiSquareSuitableModelTest/overview
|
Tail Recursion Explanation | Lesley Lai
Tail-recursion is an important concept to understand before we can analyse the behavior of a functional program. I will try to illustrate what tail recursion is with an Elm-like pseudocode. Though you don't need to know any Elm to understand this post.
From recursion to tail-recursion
factorial: Int -> Int
We can expand factorial(4) as
= if (4 == 0) 1 else 4 * factorial(4 - 1)
= 4 * factorial(4 - 1)
= 4 * (if (3 == 0) 1 else 3 * factorial(3 - 1))
Because we multiply numbers to the inner-function call result, we need a place to store those numbers 4, 3, 2, 1. Those numbers are stored in the stack frames. Since every function has its own frame, we need to create n + 1 stack frames for factorial(n).
Tail recursion is a space optimization for the recursive calls. Unlike most optimizations, it changes the asymptotic behavior of the memory usage from
\mathcal{O}(n)
\mathcal{O}(1)
. The idea is that if a recursive call itself is the last action in another function call, the function's stack frame can be reused. Function calls in the tail position of another function call are called tail call.
Accumulators - Technique for implement tail recursive functions
A nice technique to transform naive recursive functions to tail recursive counterparts is using accumulators. For example, here is a tail recursive version of factorial:
let helper acc n =
if n == 0 then acc else helper (acc * n) (n - 1)
helper 1 n
Using accumulators implies an iterative process that we use all the times with loops. Indeed, tail recursions will always transform into the same kind of low-level code as the loops by a compiler.
Accumulators are not always working. There is another technique called continuation-passing style (abbreviate as CPS) to transform more complex recursive functions. Here is our factorial() function in continuation-passing style:
factorial_k: Int -> (Int -> a) -> a
factorial_k n k =
factorial_k (n - 1) (\v -> k(v * n))
factorial_k n (\x -> x)
As you see, there is a lot of boilerplate with no apparent benefit. Writing code in CPS manually is tedious and error-prone, so it is probably not worthwhile to code every recursive function in CPS style. On the other hand, there are tools to translate normal functions into CPS.
Note that the Elm compiler cannot compile code like this at all and would generate infinite recursion at the time of writing, but you can try this function in some other languages.
Since tail recursion is an optimization, not all implementations of all programming language will implement them. For example, there is no mandatory tail-call elimination in the C++ Standard at the time of writing, though all the mainstream compilers (MSVC, Clang, and GCC) will do it anyway. The story is different in functional programming languages. Those languages usually will mandate tail-call elimination if you write a tail-recursive function. The reason is that those languages usually discourage loop or have no loop at all, so tail-call elimination is necessary to achieve a decent performance in a lot of cases. To be a good citizen in those languages, you should try to write recursive functions tail-recursive (at least on the easy cases where you can transform them with accumulators.)
Book Review: "Professional CMake: A Practical Guide"
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Discussion: “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes” (Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2004, ASME J. Appl. Mech., 71, pp. 622–631) | J. Appl. Mech. | ASME Digital Collection
Discussion: “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes” (Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2004, ASME J. Appl. Mech., 71, pp. 622–631)
J. Appl. Mech. Nov 2005, 72(6): 981 (1 pages)
This is a companion to: Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes
A correction has been published: Closure to “Discussion of ‘The Resistance of Clamped Sandwich Beams to Shock Loading’ ” (2005, ASME J. Appl. Mech., 72, pp.)
Simmonds, J. G. (November 1, 2005). "Discussion: “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes” (Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2004, ASME J. Appl. Mech., 71, pp. 622–631)." ASME. J. Appl. Mech. November 2005; 72(6): 981. https://doi.org/10.1115/1.2040451
carbon nanotubes, shells (structures), elasticity, buckling, torsion, Poisson ratio, variational techniques
Buckling, Carbon nanotubes, Shells, Poisson ratio, Elasticity, Torsion, Variational techniques
I wish to point out that there are equations for the vibration ((1), pp. 259–261) and buckling (2) of elastically isotropic circular cylindrical shells that are as accurate as, but much simpler than, the so-called Exact Flügge Equations (Model III) that the authors use as their standard of comparison for the two sets of approximate equations they analyze, namely, the (simplified) Donnell Equations (Model I) and the Simplified Flügge Equations (Model II). (I use the adjective “so-called” because there is no set of two-dimensional shell equations that is “exact.”) On pp. 225–230 of (1) Niordson presents one possible derivation of the Morley-Koiter equations in terms of midsurface displacements in which the two equations of tangential equilibrium (or motion) are identical to the simplified Donnell equations—that is, the first two of the authors’ Flügge equations (3) with the coefficients of the small parameter
(1−ν2)(D∕EhR2)
set to zero—whereas the equation of normal equilibrium (or motion) may be obtained from the third Flügge equation by replacing the coefficient of
(1−ν2)(D∕EhR2)
in brackets by
2R2∇2w+w
∇2=∂2∕∂x2+R−2∂2∕∂θ2
A simplified set of buckling equations for an elastically isotropic circular cylindrical shell under uniform axial, torsional, and internal pressure loads may be found in (2) where, as may be seen there from Eqs. (3.25)–(3.29), the equations for buckling of a simply supported cylinder under a uniform axial load or a uniform internal pressure are considerably simpler than the analogous Flügge equations yet free of the defects of the simplified Donnell equations. (A notable feature of these equations is that Poisson’s ratio
ν
appears only in the combined parameter
D∕EhR2
.)
It is also important to point out that these simple, accurate equations have been shown rigorously (3,4) to be as accurate as the Flügge equations for any problem that can be formulated as a variational principle using the Rayleigh quotient. The key is the demonstration that the modified strain-energy density that leads to the Morley-Koiter equations (and their analog for buckling) differs from the strain-energy density of the Flügge equations by terms of relative order
h∕R
—terms that are of the same order as the intrinsic errors in the Flügge equations.
Accurate Buckling Equations for Arbitrary and Cylindrical Elastic Shells
A Proof of the Accuracy of a Set of Simplified Buckling Equations for Circular Cylindrical Shells
Comments on the Paper ‘On an Accurate Theory for Circular Cylindrical Shells’ by S. Cheng
An Equivalent Orthotropic Representation of the Nonlinear Elastic Behavior of Multiwalled Carbon Nanotubes
Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes
Buckling Analyses of Double-Wall Carbon Nanotubes: A Shell Theory Based on the Interatomic Potential
Effects of DNA Encapsulation on Buckling Instability of Carbon Nanotube Based on Nonlocal Elasticity Theory
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Regression Performance - Evidently Documentation
TL;DR: The report analyzes the performance of a regression model
Works for a single model or helps compare the two
Displays a variety of plots related to the performance and errors
Helps explore areas of under- and overestimation
The Regression Performance report evaluates the quality of a regression model.
It can also compare it to the past performance of the same model, or the performance of an alternative model.
To run this report, you need to have input features, and both target and prediction columns available.
To generate a comparative report, you will need two datasets. The reference dataset serves as a benchmark. We analyze the change by comparing the current production data to the reference data.
You can also run this report for a single DataFrame , with no comparison performed. In this case, pass it as reference_data.
The report includes 12 components. All plots are interactive.
1. Model Quality Summary Metrics
We calculate a few standard model quality metrics: Mean Error (ME), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE).
For each quality metric, we also show one standard deviation of its value (in brackets) to estimate the stability of the performance.
Next, we generate a set of plots. They help analyze where the model makes mistakes and come up with improvement ideas.
2. Predicted vs Actual
Predicted versus actual values in a scatter plot.
3. Predicted vs Actual in Time
Predicted and Actual values over time or by index, if no datetime is provided.
4. Error (Predicted - Actual)
Model error values over time or by index, if no datetime is provided.
5. Absolute Percentage Error
Absolute percentage error values over time or by index, if no datetime is provided.
6. Error Distribution
Distribution of the model error values.
7. Error Normality
Quantile-quantile plot (Q-Q plot) to estimate value normality.
Next, we explore in detail the two segments in the dataset: 5% of predictions with the highest negative and positive errors. We refer to them as "underesimation" and "overestimation" groups. We refer to the rest of the predictions as "majority".
8. Mean Error per Group
We show a summary of the model quality metrics for each of the two groups: mean Error (ME), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE).
9. Predicted vs Actual per Group
We plot the predictions, coloring them by the group they belong to. It visualizes the regions where the model underestimates and overestimates the target function.
10. Error Bias: Mean/Most Common Feature Value per Group
This table helps quickly see the differences in feature values between the 3 groups:
OVER (top-5% of predictions with overestimation)
UNDER (top-5% of the predictions with underestimation)
MAJORITY (the rest 90%)
For the numerical features, it shows the mean value per group. For the categorical features, it shows the most common value.
If you have two datasets, the table displays the values for both REF (reference) and CURR (current).
If you observe a large difference between the groups, it means that the model error is sensitive to the values of a given feature.
To search for cases like this, you can sort the table using the column "Range(%)". It increases when either or both of the "extreme" groups are different from the majority.
Here is the formula used to calculate the Range %:
Range = 100*|(Vover-Vunder)/(Vmax-Vmin)|
Where: Vover = average feature value in the OVER group; Vunder = average feature value in the UNDER group; Vmax = maximum feature value; Vmin = minimum feature value
11. Error Bias per Feature
For each feature, we show a histogram to visualize the distribution of its values in the segments with extreme errors and in the rest of the data. You can visually explore if there is a relationship between the high error and the values of a given feature.
Here is an example where extreme errors are dependent on the "temperature" feature.
12. Predicted vs Actual per Feature
For each feature, we also show the Predicted vs Actual scatterplot. We use colors to show the distribution of the values of a given feature. It helps visually detect and explore underperforming segments which might be sensitive to the values of the given feature.
You can select which components of the reports to display or choose to show the short version of the report: Select Widgets.
Here are our suggestions on when to use it—you can also combine it with the Data Drift and Numerical Target Drift reports to get a comprehensive picture.
1. To analyze the results of the model test. You can explore the results of an online or offline test and contrast it to the performance in training. Though this is not the primary use case, you can use this report to compare the model performance in an A/B test, or during a shadow model deployment.
2. To generate regular reports on the performance of a production model. You can run this report as a regular job (e.g. weekly or at every batch model run) to analyze its performance and share it with other stakeholders.
3. To analyze the model performance on the slices of data. By manipulating the input data frame, you can explore how the model performs on different data segments (e.g. users from a specific region).
4. To trigger or decide on the model retraining. You can use this report to check if your performance is below the threshold to initiate a model update and evaluate if retraining is likely to improve performance.
5. To debug or improve model performance by identifying areas of high error. You can use the Error Bias table to identify the groups that contribute way more to the total error, or where the model under- or over-estimates the target function.
"regression_performance": {
"name": "regression_performance",
"prediction": "prediction"
"mean_error": mean_error,
"mean_abs_error": mean_abs_error,
"mean_abs_perc_error": mean_abs_perc_error,
"error_std": error_std,
"abs_error_std": abs_error_std,
"abs_perc_error_std": abs_perc_error_std,
"error_normality": {
"order_statistic_medians": [],
"slope": slope,
"intercept": intercept,
"r": r
"underperformance": {
"std_error": std_error
"underestimation": {
"overestimation": {
"error_bias": {
"feature_type": "num",
"ref_majority": ref_majority,
"ref_under": ref_under,
"ref_over": ref_over,
"ref_range": ref_range,
"prod_majority": prod_majority,
"prod_under": prod_under,
"prod_over": prod_over,
"prod_range": prod_range
"feature_type": "cat",
"ref_majority": 0,
"ref_under": 0,
"ref_over": 0,
"ref_range": 0,
"prod_majority": 0,
"prod_under": 0,
"prod_over": 1,
"prod_range": 1
See a tutorial "How to break a model in 20 days" where we create a demand prediction model and analyze its gradual decay.
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Disable Parse Rule - Maple Help
Home : Support : Online Help : Configure Maple : Customize the Maple System : 2-D Mathematics Display : Typesetting Package : Disable Parse Rule
EnableParseRule
enable use of rule for parsing 2-D input
DisableParseRule
disable use of rule for parsing 2-D input
QueryParseRule
query status of rule for parsing 2-D input
EnableParseRule(rule)
DisableParseRule(rule)
QueryParseRule(rule)
The EnableParseRule command turns on the specified parse rule(s) in the Typesetting package for use in 2-D input, while the DisableParseRule command turns off the specified parse rule(s) in the Typesetting package for use in 2-D input. The QueryParseRule command shows whether a rule is enabled or not. It returns a set of elements a=b where a is a rule name and b is true if a is enabled and false otherwise.
"BesselJ"
corresponds to the capability of parsing the function
J
v
) as a function a single variable (say
x
) as the BesselJ function (
\mathrm{BesselJ}\left(v,x\right)
The Typesetting[QueryParseRule] command was introduced in Maple 2017.
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Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : Line Segments : Equation
equation of a geometric object
Equation(obj, t )
Equation(obj, [x,y,z] )
(optional) the name of the parameter in the parametric equations of a line
(optional) the names of the axes
Users will be asked to provide the names of the axes or the name of the parameter in the parametric equations (when obj is a line) if they are not assigned to names and if the second optional argument is not given. See xname for an example of this.
The command with(geom3d,Equation) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{point}\left(A,0,0,0\right),\mathrm{point}\left(B,1,2,3\right):
Define the line that passes through two points A and B
\mathrm{line}\left(l,[A,B]\right)
\textcolor[rgb]{0,0,1}{l}
The equation of l is:
\mathrm{Equation}\left(l,m\right)
[\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}]
\mathrm{point}\left(C,3,7,11\right):
Define the plane that passes through three given points A, B, C
\mathrm{plane}\left(p,[A,B,C],[a,b,c]\right):
\mathrm{Equation}\left(p\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
geom3d[xname]
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Apollonian_gasket Knowpia
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga.[1]
An example of an Apollonian gasket
Mutually tangent circles. Given three mutually tangent circles (black), there are in general two other circles mutually tangent to them (red).
An Apollonian gasket can be constructed as follows. Start with three circles C1, C2 and C3, each one of which is tangent to the other two (in the general construction, these three circles may be different sizes, and they must have a common tangent). Apollonius discovered that there are two other non-intersecting circles, C4 and C5, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles. Adding the two Apollonian circles to the original three, we now have five circles.
Take one of the two Apollonian circles – say C4. It is tangent to C1 and C2, so the triplet of circles C4, C1 and C2 has its own two Apollonian circles. We already know one of these – it is C3 – but the other is a new circle C6.
In a similar way we can construct another new circle C7 that is tangent to C4, C2 and C3, and another circle C8 from C4, C3 and C1. This gives us 3 new circles. We can construct another three new circles from C5, giving six new circles altogether. Together with the circles C1 to C5, this gives a total of 11 circles.
Continuing the construction stage by stage in this way, we can add 2·3n new circles at stage n, giving a total of 3n+1 + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket.
The sizes of the new circles are determined by Descartes' theorem. Let ki (for i = 1, ..., 4) denote the curvatures of four mutually tangent circles. Then Descartes' Theorem states
{\displaystyle (k_{1}+k_{2}+k_{3}+k_{4})^{2}=2\,(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2}).}
The Apollonian gasket has a Hausdorff dimension of about 1.3057.[2]
The curvature of a circle (bend) is defined to be the reciprocal of its radius.
Negative curvature indicates that all other circles are internally tangent to that circle. This is bounding circle.
Zero curvature gives a line (circle with infinite radius).
Positive curvature indicates that all other circles are externally tangent to that circle. This circle lies in the interior of the circle with negative curvature.
In the limiting case (0,0,1,1), the two largest circles are replaced by parallel straight lines. This produces a family of Ford circles.
An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity.
Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. In this construction, the additional circles form a family of Ford circles.
The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing.
If two of the original generating circles have the same radius and the third circle has a radius that is two-thirds of this, then the Apollonian gasket has two lines of reflective symmetry; one line is the line joining the centres of the equal circles; the other is their mutual tangent, which passes through the centre of the third circle. These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is D2.
If all three of the original generating circles have the same radius then the Apollonian gasket has three lines of reflective symmetry; these lines are the mutual tangents of each pair of circles. Each mutual tangent also passes through the centre of the third circle and the common centre of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another, so the Apollonian gasket also has rotational symmetry of degree 3; the symmetry group of this gasket is D3.
Links with hyperbolic geometryEdit
The three generating circles, and hence the entire construction, are determined by the location of the three points where they are tangent to one another. Since there is a Möbius transformation which maps any three given points in the plane to any other three points, and since Möbius transformations preserve circles, then there is a Möbius transformation which maps any two Apollonian gaskets to one another.
Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry.
The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group.[3]
Integral Apollonian circle packingsEdit
Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)
Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)
Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)
If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.[4] Since the equation relating curvatures in an Apollonian gasket, integral or not, is
{\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=2ab+2ac+2ad+2bc+2bd+2cd,\,}
it follows that one may move from one quadruple of curvatures to another by Vieta jumping, just as when finding a new Markov number. The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three.
Integral Apollonian gaskets
Beginning curvatures
−1, 2, 2, 3, 3 D2
−3, 4, 12, 13, 13 D1
−3, 5, 8, 8, 12 D1
−6, 10, 15, 19, 19 D1
−6, 11, 14, 15, 23 C1
−10, 11, 110, 111, 111 D1
−10, 14, 35, 39, 39 D1
−10, 18, 23, 27, 35 C1
−13, 15, 98, 98, 102 D1
Symmetry of integral Apollonian circle packingsEdit
There are multiple types of dihedral symmetry that can occur with a gasket depending on the curvature of the circles.
No symmetryEdit
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C1; the gasket described by curvatures (−10, 18, 23, 27) is an example.
D1 symmetryEdit
Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D1 symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.
If two different curvatures are repeated within the first five, the gasket will have D2 symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D2 symmetry.
There are no integer gaskets with D3 symmetry.
If the three circles with smallest positive curvature have the same curvature, the gasket will have D3 symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2√3 − 3. As this ratio is not rational, no integral Apollonian circle packings possess this D3 symmetry, although many packings come close.
Almost-D3 symmetryEdit
(−15, 32, 32, 33)
The figure at left is an integral Apollonian gasket that appears to have D3 symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the D1 symmetry common to many other integral Apollonian gaskets.
The following table lists more of these almost-D3 integral Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the "a" disks obey the recurrence relation a(n) = 4a(n − 1) − a(n − 2) (sequence A001353 in the OEIS), from which it follows that the multiplier converges to √3 + 2 ≈ 3.732050807.
Integral Apollonian gaskets with near-D3 symmetry
−1 2 2 3 1×1 1×2 1×3 N/A N/A N/A N/A
−4 8 9 9 2×2 2×4 3×3 4.000000000 4.000000000 4.500000000 3.000000000
−15 32 32 33 3×5 4×8 3×11 3.750000000 4.000000000 3.555555556 3.666666667
−56 120 121 121 8×7 8×15 11×11 3.733333333 3.750000000 3.781250000 3.666666667
−209 450 450 451 11×19 15×30 11×41 3.732142857 3.750000000 3.719008264 3.727272727
−780 1680 1681 1681 30×26 30×56 41×41 3.732057416 3.733333333 3.735555556 3.727272727
−2911 6272 6272 6273 41×71 56×112 41×153 3.732051282 3.733333333 3.731112433 3.731707317
−10864 23408 23409 23409 112×97 112×209 153×153 3.732050842 3.732142857 3.732302296 3.731707317
−40545 87362 87362 87363 153×265 209×418 153×571 3.732050810 3.732142857 3.731983425 3.732026144
Sequential curvaturesEdit
Nested Apollonian gaskets
For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures:
(−n, n + 1, n(n + 1), n(n + 1) + 1).
For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20.
Descartes' theorem, for curvatures of mutually tangent circles
Ford circle, the special case of integral Apollonian gasket (0,0,1,1)
Apollonian network, a graph derived from finite subsets of the Apollonian gasket
^ Satija, I. I., The Butterfly in the Iglesias Waseas World: The story of the most fascinating quantum fractal (Bristol: IOP Publishing, 2016), p. 5.
^ McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018.
^ Counting circles and Ergodic theory of Kleinian groups by Hee Oh Brown. University Dec 2009
^ Ronald L. Graham, Jeffrey C. Lagarias, Colin M. Mallows, Alan R. Wilks, and Catherine H. Yan; "Apollonian Circle Packings: Number Theory" J. Number Theory, 100 (2003), 1-45
Benoit B. Mandelbrot: The Fractal Geometry of Nature, W H Freeman, 1982, ISBN 0-7167-1186-9
Paul D. Bourke: "An Introduction to the Apollony Fractal". Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136.
David Mumford, Caroline Series, David Wright: Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002, ISBN 0-521-35253-3
Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond the Descartes Circle Theorem, The American Mathematical Monthly, Vol. 109, No. 4 (Apr., 2002), pp. 338–361, (arXiv:math.MG/0101066 v1 9 Jan 2001)
The Wikibook Fractals has a page on the topic of: Apollonian fractals
Weisstein, Eric W. "Apollonian Gasket". MathWorld.
Alexander Bogomolny, Apollonian Gasket, cut-the-knot
An interactive Apollonian gasket running on pure HTML5 (the link is dead)
(in English) A Matlab script to plot 2D Apollonian gasket with n identical circles using circle inversion
Online experiments with JSXGraph
Apollonian Gasket by Michael Screiber, The Wolfram Demonstrations Project.
Interactive Apollonian Gasket Demonstration of an Apollonian gasket running on Java
Dana Mackenzie. A Tisket, a Tasket, an Apollonian Gasket. American Scientist, January/February 2010.
"Sand drawing the world's largest single artwork", The Telegraph, 16 Dec 2009 . Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles.
(in Italian)Dynamic apollonian gaskets ,Tartapelago by Giorgio Pietrocola, 2014.
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Discrete-time Luenberger observer - Simulink - MathWorks Deutschland
State-space parameterization
Discrete A matrix
Discrete B matrix
Discrete C matrix
Discrete D matrix
Continuous A matrix
Continuous B matrix
Continuous C matrix
Continuous D matrix
Observer gain
Desired eigenvalues
Discrete-time Luenberger observer
The Luenberger Observer block implements a discrete time Luenberger observer. Use this block to estimate the states of an observable system using:
The discrete inputs and outputs of the system.
A discrete state-space representation of the system.
The Luenberger Observer is also sometimes referred to as a state observer or simply an observer.
You can control multi-input, multi-output systems by passing the output state vector of this block to a State Feedback Controller block.
The block implements a discrete time Luenberger Observer using the backward Euler method due to its simplicity and stability.
The estimator is given by this difference equation:
\stackrel{^}{x}\left(k+1\right) ={A}_{d}\stackrel{^}{x}\left(k\right) +{B}_{d}u\left(k\right) + {L}_{d}\left(y\left(k\right) -\stackrel{^}{y}\left(k\right)\right),
\stackrel{^}{x}\left(k\right)
is the kth estimated state vector.
\stackrel{^}{y}\left(k\right)
is the kth estimated output vector.
u(k) is the kth input vector.
y(k) is the kth measured output vector.
Ad is the discretized state matrix.
Bd is the discretized input matrix.
Ld is the discretized observer gain matrix.
The dynamics of the estimation error are described by:
e\left(k+1\right)=\left({A}_{d}-{L}_{d}{C}_{d}\right)e\left(k\right),
e(k) is the kth error vector.
Cd is the output matrix.
The estimation error converges to zero when Ad-LdCd has its eigenvalues inside the unit circle. Therefore, the value of Ld should be such that this goal is achieved. The block computes the observer gain by solving
{L}_{d}^{T}=G{X}^{-1},
where G is an arbitrary matrix and X is obtained by solving the Sylvester equation:
{A}_{d}^{T}X-X\Lambda ={C}_{d}^{T}G.
Here, Λ is a matrix with the desired eigenvalues, which are not the same as the eigenvalues of Ad. This diagram shows the basic structure of a discrete time Luenberger Observer.
The system is observable, which is true if the state of the system can be determined from the input and output in a finite time. Mathematically, this means that the system observability matrix has full rank.
The desired eigenvalues are not the same as the eigenvalues of the open-loop model.
u — Control input
Input signal to the system whose state we want to estimate, specified as a vector.
Measured output of the system whose state we want to estimate, specified as a vector.
xhat — State estimate
Estimate of the state of the system, specified as a vector.
State-space parameterization — State-space parameterization
Select the strategy for parameterizing the state-space matrices and desired poles for the observer. The block implementation is discrete regardless of this parameterization.
Discrete A matrix — A matrix in discrete time
1 (default) | real scalar or matrix
State matrix of the discrete-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.
To enable this parameter, set State-space parameterization to Discrete-time.
Discrete B matrix — B matrix in discrete time
Input matrix of the discrete-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.
Discrete C matrix — C matrix in discrete time
Output matrix of the discrete-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.
Discrete D matrix — D matrix in discrete time
Feedthrough matrix of the discrete-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.
Continuous A matrix — A matrix in continuous time
State matrix of the continuous-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.
To enable this parameter, set State-space parameterization to Continuous-time.
Continuous B matrix — B matrix in continuous time
Input matrix of the continuous-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.
Continuous C matrix — C matrix in continuous time
Output matrix of the continuous-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.
Continuous D matrix — D matrix in continuous time
Feedthrough matrix of the continuous-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.
Observer design — State-space parameterization
Desired eigenvalues (default) | Observer gain
Select the strategy for parameterizing observer gain.
Observer gain — Observer gain
Specify the observer gain that puts all eigenvalues of the matrix Ad-LdCd inside the unit circle. The gain matrix must have the number of rows equal to number of system inputs and the number of columns equal to the order of the system.
State-space parameterization to Discrete-time.
Observer design to Observer gain.
Desired eigenvalues — Observer eigenvalues
0 (default) | real vector
Specify the location of the eigenvalues:
To have negative real part if State-space parameterization is set to Continuous-time. In this case, the eigenvalues of the continuous-time system are approximated to the discrete ones based on the Discretization sample time.
To lie within the unit circle if State-space parameterization is set to Discrete-time.
The Observer gain is then calculated based on these eigenvalues. The size of the vector should be the same as the system order.
0 (default) | real vector with length equal to system order
Select the initial condition of each state.
Discretization sample time — Discretization sample time
Value used to discretize the state space matrices and also approximate the discrete-time eigenvalues.
0.1 (default) | -1 or positive real number
Value used to simulate the dynamics of the model. Choose the same value as Discretization sample time, unless the block is placed within a triggered subsystem, in which case you must set it to -1.
[1] Luenberger, D. G. "An Introduction to Observers." IEEE Transactions on Automatic Control. Vol. 16, Number 6, 1971, pp. 596-602.
[2] Alessandri, A., and P. Coletta. "Design of Luenberger observers for a class of hybrid linear systems." In International Workshop on Hybrid Systems: Computation and Control, Berlin, March 2001.
[3] Varga, A. "Robust pole assignment via Sylvester equation based state feedback parametrization." In Computer-Aided Control System Design, pp. 13-18., Anchorage, Alaska, 2000.
State Feedback Controller | Induction Machine Flux Observer
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[[File:009A_SF_A_3.png|right|230px|frame|Both functions with constants chosen to provide continuity.]]
Both functions with constants chosen to provide continuity.
{\displaystyle f(x)={\begin{cases}{\sqrt {x}},&{\mbox{if }}x\geq 1,\\4x^{2}+C,&{\mbox{if }}x<1.\end{cases}}}
{\displaystyle C}
{\displaystyle f}
{\displaystyle x=1.}
{\displaystyle C}
{\displaystyle f}
{\displaystyle x=1}
{\displaystyle g(x)={\begin{cases}{\sqrt {x^{2}+3}},&\quad {\mbox{if }}x\geq 1\\{\frac {1}{4}}x^{2}+C,&\quad {\mbox{if }}x<1.\end{cases}}}
{\displaystyle C}
{\displaystyle f}
{\displaystyle x=1.}
{\displaystyle C}
{\displaystyle f}
{\displaystyle x=1}
{\displaystyle f}
{\displaystyle x_{0}}
{\displaystyle \lim _{x\rightarrow x_{0}}f(x)=f\left(x_{0}\right).}
This can be viewed as saying the left and right hand limits exist, and are equal. For problems like these, where we are trying to find a particular value for
{\displaystyle C}
, we can just set the two descriptions of the function to be equal at the value where the definition o{\displaystyle f}
When we speak of differentiability at such a transition point, being "motivated by the definition of the derivative" really means acknowledge that the derivative is a limit, and for a limit to exist it must agree from the left and the right. This means we must show the derivatives agree for both the descriptions o{\displaystyle f}
at the transition point.
(a) For continuity, we evaluate both rules for the function at the transition point
{\displaystyle x=1}
, set the results equal, and then solve for
{\displaystyle C}
. Since we want
{\displaystyle f(1)\,=\,1\,=\,4\cdot 1^{2}+C,}
we can set
{\displaystyle C=-3}
, and the function will be continuous (the left and right hand limits agree, and equal the function's value at the point
{\displaystyle x=1}
(b) To test differentiability, we note that for
{\displaystyle x>1}
{\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}},}
while for
{\displaystyle x<1}
{\displaystyle f'(x)=8x.}
{\displaystyle \lim _{x\rightarrow 1^{+}}f'(x)\,=\,{\frac {1}{2{\sqrt {1}}}}\,=\,{\frac {1}{2}},}
{\displaystyle \lim _{x\rightarrow 1^{-}}f'(x)\,=\,8\cdot 1\,=\,8.}
Since the left and right hand limit do not agree, the derivative does not exist at the point
{\displaystyle x=1}
(a) Like Version I, we begin by setting the two functions equal. We want
{\displaystyle f(1)\,=\,{\sqrt {1^{2}+3}}\,=\,2\,=\,{\frac {\,\,1^{2}}{4}}+C,}
{\displaystyle C=-7/4}
makes the function continuous.
(b) We again consider the derivative from each side of 1. For
{\displaystyle x>1}
{\displaystyle f'(x)={\frac {1}{2{\sqrt {x^{2}+3}}}}\cdot 2x\,=\,{\frac {x}{\sqrt {x^{2}+3}}},}
{\displaystyle x<1}
{\displaystyle f'(x)\,=\,{\frac {x}{2}}.}
{\displaystyle \lim _{x\rightarrow 1^{+}}f'(x)\,=\,{\frac {1}{\sqrt {1^{2}+3}}}\,=\,{\frac {1}{2}},}
{\displaystyle \lim _{x\rightarrow 1^{-}}f'(x)\,=\,{\frac {1}{2}}.}
Since the left and right hand limit do agree, the limit (which is the derivative) does exist at the point
{\displaystyle x=1}
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Section 37.21 (045Q): Cohen-Macaulay morphisms—The Stacks project
Section 37.21: Cohen-Macaulay morphisms (cite)
37.21 Cohen-Macaulay morphisms
Compare with Section 37.19. Note that, as pointed out in Algebra, Section 10.167 and Varieties, Section 33.13 “geometrically Cohen-Macaulay” is the same as plain Cohen-Macaulay.
Definition 37.21.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.
Let $x \in X$, and $y = f(x)$. We say that $f$ is Cohen-Macaulay at $x$ if $f$ is flat at $x$, and the local ring of the scheme $X_ y$ at $x$ is Cohen-Macaulay.
We say $f$ is a Cohen-Macaulay morphism if $f$ is Cohen-Macaulay at every point of $X$.
Lemma 37.21.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent
$f$ is Cohen-Macaulay, and
$f$ is flat and its fibres are Cohen-Macaulay schemes.
Proof. This follows directly from the definitions. $\square$
Lemma 37.21.3. Let $f : X \to Y$ be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point $x$ in $X$ with image $y$ in $Y$,
\[ \dim _ x(X) = \dim _ y(Y) + \dim _ x(X_ y), \]
where $X_ y$ denotes the fiber over $y$.
Proof. After replacing $X$ by an open neighborhood of $x$, there is a natural number $d$ such that all fibers of $X \to Y$ have dimension $d$ at every point, see Morphisms, Lemma 29.29.4. Then $f$ is flat, locally of finite type and of relative dimension $d$. Hence the result follows from Morphisms, Lemma 29.29.6. $\square$
Lemma 37.21.4. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in X$ with images $y \in Y$ and $z \in Z$.
If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$.
If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay.
If $g \circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.
If $g \circ f$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$.
Proof. Consider the map of Noetherian local rings
\[ \mathcal{O}_{Y_ z, y} \to \mathcal{O}_{X_ z, x} \]
and observe that its fibre is
\[ \mathcal{O}_{X_ z, x}/\mathfrak m_{Y_ z, y}\mathcal{O}_{X_ z, x} = \mathcal{O}_{X_ y, x} \]
Thus the lemma this follows from Algebra, Lemma 10.163.3. $\square$
Lemma 37.21.5. Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \in X$.
Proof. After translating into algebra this follows from Algebra, Lemma 10.163.3. $\square$
Lemma 37.21.6. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in X'$ be a point with image $x \in X$.
If $f$ is Cohen-Macaulay at $x$, then $f' : X' \to Y'$ is Cohen-Macaulay at $x'$.
If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.
If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.
Proof. Note that the assumption on $Y' \to Y$ implies that for $y' \in Y'$ mapping to $y \in Y$ the field extension $\kappa (y')/\kappa (y)$ is finitely generated. Hence also all the fibres $X'_{y'} = (X_ y)_{\kappa (y')}$ are locally Noetherian, see Varieties, Lemma 33.11.1. Thus the lemma makes sense. Set $y' = f'(x')$ and $y = f(x)$. Hence we get the following commutative diagram of local rings
\[ \xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{Y', y'} \ar[u] & \mathcal{O}_{Y, y} \ar[l] \ar[u] } \]
where the upper left corner is a localization of the tensor product of the upper right and lower left corners over the lower right corner.
Assume $f$ is Cohen-Macaulay at $x$. The flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ implies the flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay implies that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f'$ is Cohen-Macaulay at $x'$.
Assume $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f$ is Cohen-Macaulay at $x$.
Assume $Y' \to Y$ is flat at $y'$ and $f'$ is Cohen-Macaulay at $x'$. The flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$ and $\mathcal{O}_{Y, y} \to \mathcal{O}_{Y', y'}$ implies the flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f$ is Cohen-Macaulay at $x$. $\square$
Lemma 37.21.7. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let
\[ W = \{ x \in X \mid f\text{ is Cohen-Macaulay at }x\} \]
$W = \{ x \in X \mid \mathcal{O}_{X_{f(x)}, x}\text{ is Cohen-Macaulay}\} $,
$W$ is open in $X$,
$W$ dense in every fibre of $X \to S$,
the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : S' \to S$, consider the base change $f' : X' \to S'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.
Proof. As $f$ is flat with locally Noetherian fibres the equality in (1) holds by definition. Parts (2) and (3) follow from Algebra, Lemma 10.130.6. Part (4) follows either from Algebra, Lemma 10.130.8 or Varieties, Lemma 33.13.1. $\square$
Lemma 37.21.8. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \in X$ with image $s \in S$. Set $d = \dim _ x(X_ s)$. The following are equivalent
$f$ is Cohen-Macaulay at $x$,
there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite morphism $U \to \mathbf{A}^ d_ S$ over $S$ which is flat at $x$,
there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite flat morphism $U \to \mathbf{A}^ d_ S$ over $S$,
for any $S$-morphism $g : U \to \mathbf{A}^ d_ S$ of an open neighbourhood $U \subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\Rightarrow $ $g$ is flat at $x$.
Proof. Openness of flatness shows (2) and (3) are equivalent, see Theorem 37.15.1.
Choose affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $x \in U$ and $V = \mathop{\mathrm{Spec}}(R) \subset S$ with $f(U) \subset V$. Then $R \to A$ is a flat ring map of finite presentation. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$. After replacing $A$ by a principal localization we may assume there exists a quasi-finite map $R[x_1, \ldots , x_ d] \to A$, see Algebra, Lemma 10.125.2. Thus there exists at least one pair $(U, g)$ consisting of an open neighbourhood $U \subset X$ of $x$ and a locally1 quasi-finite morphism $g : U \to \mathbf{A}^ d_ S$.
Claim: Given $R \to A$ flat and of finite presentation, a prime $\mathfrak p \subset A$ and $\varphi : R[x_1, \ldots , x_ d] \to A$ quasi-finite at $\mathfrak p$ we have: $\mathop{\mathrm{Spec}}(\varphi )$ is flat at $\mathfrak p$ if and only if $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ is Cohen-Macaulay at $\mathfrak p$. Namely, by Theorem 37.16.2 flatness may be checked on fibres. The same is true for being Cohen-Macaulay (as $A$ is already assumed flat over $R$). Thus the claim follows from Algebra, Lemma 10.130.1.
The claim shows that (1) is equivalent to (4) and combined with the fact that we have constructed a suitable $(U, g)$ in the second paragraph, the claim also shows that (1) is equivalent to (2). $\square$
Lemma 37.21.9. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. For $d \geq 0$ there exist opens $U_ d \subset X$ with the following properties
$W = \bigcup _{d \geq 0} U_ d$ is dense in every fibre of $f$, and
$U_ d \to S$ is of relative dimension $d$ (see Morphisms, Definition 29.29.1).
Proof. This follows by combining Lemma 37.21.7 with Morphisms, Lemma 29.29.4. $\square$
Lemma 37.21.10. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Suppose $x' \leadsto x$ is a specialization of points of $X$ with image $s' \leadsto s$ in $S$. If $x$ is a generic point of an irreducible component of $X_ s$ then $\dim _{x'}(X_{s'}) = \dim _ x(X_ s)$.
Proof. The point $x$ is contained in $U_ d$ for some $d$, where $U_ d$ as in Lemma 37.21.9. $\square$
Lemma 37.21.11. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is Cohen-Macaulay” is local in the fppf topology on the target and local in the syntomic topology on the source.
Proof. We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f)$ where $\mathcal{P}_1(f)=$“$f$ is flat”, and $\mathcal{P}_2(f)=$“the fibres of $f$ are locally Noetherian and Cohen-Macaulay”. We know that $\mathcal{P}_1$ is local in the fppf topology on the source and the target, see Descent, Lemmas 35.22.15 and 35.26.1. Thus we have to deal with $\mathcal{P}_2$.
Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fppf covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that
\[ X_{i, y_ i} = \mathop{\mathrm{Spec}}(\kappa (y_ i)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} X_ y. \]
and that $\kappa (y_ i)/\kappa (y)$ is a finitely generated field extension. Hence if $X_ y$ is locally Noetherian, then $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. And if in addition $X_ y$ is Cohen-Macaulay, then $X_{i, y_ i}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Thus $\mathcal{P}_2$ is fppf local on the target.
Let $\{ X_ i \to X\} $ be a syntomic covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is a syntomic covering of the fibre. Hence the locality of $\mathcal{P}_2$ for the syntomic topology on the source follows from Descent, Lemma 35.16.2. Combining the above the lemma follows. $\square$
[1] If $S$ is quasi-separated, then $g$ will be quasi-finite.
Comment #5015 by Yujie Luo on April 05, 2020 at 23:35
Lemma \cite{0C0W} (4), a small typo
f\circ g
g\circ f
Comment #5876 by Sasha on January 10, 2021 at 14:30
In the statement of Lemma 054T it is not clear what is
W
@#5877: The lemma defines
W
in the statement! You can say more but the lemma does not claim more.
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Tarski's_undefinability_theorem Knowpia
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic.
The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
In 1931, Kurt Gödel published the incompleteness theorems, which he proved in part by showing how to represent the syntax of formal logic within first-order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, coding and, more generally, as arithmetization. In particular, various sets of expressions are coded as sets of numbers. For various syntactic properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences.
The undefinability theorem is conventionally attributed to Alfred Tarski. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1933 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann. Tarski had obtained almost all results of his 1933 monograph "The Concept of Truth in the Languages of the Deductive Sciences" between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result he did not obtain earlier. According to the footnote to the undefinability theorem (Twierdzenie I) of the 1933 monograph, the theorem and the sketch of the proof were added to the monograph only after the manuscript had been sent to the printer in 1931. Tarski reports there that, when he presented the content of his monograph to the Warsaw Academy of Science on March 21, 1931, he expressed at this place only some conjectures, based partly on his own investigations and partly on Gödel's short report on the incompleteness theorems "Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit", Akademie der Wissenschaften in Wien, 1930.
Let L be the language of first-order arithmetic. This is the theory of the natural numbers, including their addition and multiplication, axiomatized by the first-order Peano axioms. This is a "first-order" theory: the quantifiers extend over natural numbers, but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence.
Each formula φ in L has a Gödel number g(φ). This is a natural number that "encodes" φ. In that way, the language L can talk about formulas in L, not just about numbers. Let T denote the set of L-sentences true in N, and T* the set of Gödel numbers of the sentences in T. The following theorem answers the question: Can T* be defined by a formula of first-order arithmetic?
Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation." It is possible to define a formula True(n) whose extension is T*, but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language L. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.
To prove the theorem, we proceed by contradiction and assume that an L-formula True(n) exists which is true for the natural number n in N if and only if n is the Gödel number of a sentence in L that's true in N. We could then use True(n) to define a new L-formula S(m) which is true for the natural number m if and only if m is the Gödel number of a formula φ(x) (with a free variable x) such that φ(m) is false when interpreted in N (i.e. the formula φ(x), when applied to its own Gödel number, yields a false statement). If we now consider the Gödel number g of the formula S(m), and ask whether the sentence S(g) is true in N, we obtain a contradiction. (This is known as a diagonal argument.)
The theorem is a corollary of Post's theorem about the arithmetical hierarchy, proved some years after Tarski (1933). A semantic proof of Tarski's theorem from Post's theorem is obtained by reductio ad absurdum as follows. Assuming T* is arithmetically definable, there is a natural number n such that T* is definable by a formula at level
{\displaystyle \Sigma _{n}^{0}}
of the arithmetical hierarchy. However, T* is
{\displaystyle \Sigma _{k}^{0}}
Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems, such as ZFC.
The proof of Tarski's undefinability theorem in this form is again by reductio ad absurdum. Suppose that an L-formula True(n) as above existed, i.e., if A is a sentence of arithmetic, then True(g(A)) holds in N if and only if A holds in N. Hence for all A, the formula True(g(A)) ↔ A holds in N. But the diagonal lemma yields a counterexample to this equivalence, by giving a "liar" formula S such that S ↔ ¬True(g(S)) holds in N. This is a contradiction. QED.
The formal machinery of the proof given above is wholly elementary except for the diagonalization which the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way. The proof does assume that every L-formula has a Gödel number, but the specifics of a coding method are not required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated theorems of Gödel about the metamathematical properties of first-order arithmetic.
Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.
An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula A to its truth value ||A||, and the "semantic denotation function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows: No sufficiently powerful language is strongly-semantically-self-representational.
The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in N is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC.
G. Boolos, J. Burgess, and R. Jeffrey, 2002. Computability and Logic, 4th ed. Cambridge University Press.
J. R. Lucas, 1961. "Mind, Machines, and Gödel". Philosophy 36: 112–27.
R. Murawski, 1998. Undefinability of truth. The problem of the priority: Tarski vs. Gödel. History and Philosophy of Logic 19, 153–160
R. Smullyan, 1991. Godel's Incompleteness Theorems. Oxford Univ. Press.
R. Smullyan, 2001. "Gödel’s Incompleteness Theorems". In L. Goble, ed., The Blackwell Guide to Philosophical Logic, Blackwell, 72–89.
A. Tarski, 1933. Pojęcie Prawdy w Językach Nauk Dedukcyjnych. Nakładem Towarzystwa Naukowego Warszawskiego.
A. Tarski (1936). "Der Wahrheitsbegriff in den formalisierten Sprachen" (PDF). Studia Philosophica. 1: 261–405. Archived from the original (PDF) on 9 January 2014. Retrieved 26 June 2013.
A. Tarski, tr. J. H. Woodger, 1983. "The Concept of Truth in Formalized Languages". English translation of Tarski's 1936 article. In A. Tarski, ed. J. Corcoran, 1983, Logic, Semantics, Metamathematics, Hackett.
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Section 43.1 (0AZ7): Introduction—The Stacks project
In this chapter we construct the intersection product on the Chow groups modulo rational equivalence on a nonsingular projective variety over an algebraically closed field. Our tools are Serre's Tor formula (see [Chapter V, Serre_algebre_locale]), reduction to the diagonal, and the moving lemma.
We first recall cycles and how to construct proper pushforward and flat pullback of cycles. Next, we introduce rational equivalence of cycles which gives us the Chow groups $\mathop{\mathrm{CH}}\nolimits _*(X)$. Proper pushforward and flat pullback factor through rational equivalence to give operations on Chow groups. This takes up Sections 43.3, 43.4, 43.5, 43.6, 43.7, 43.8, 43.9, 43.10, and 43.11. For proofs we mostly refer to the chapter on Chow homology where these results have been proven in the setting of schemes locally of finite type over a universally catenary Noetherian base, see Chow Homology, Section 42.7 ff.
Since we work on a nonsingular projective $X$ any irreducible component of the intersection $V \cap W$ of two irreducible closed subvarieties has dimension at least $\dim (V) + \dim (W) - \dim (X)$. We say $V$ and $W$ intersect properly if equality holds for every irreducible component $Z$. In this case we define the intersection multiplicity $e_ Z = e(X, V \cdot W, Z)$ by the formula
\[ e_ Z = \sum \nolimits _ i (-1)^ i \text{length}_{\mathcal{O}_{X, Z}} \text{Tor}_ i^{\mathcal{O}_{X, Z}}(\mathcal{O}_{W, Z}, \mathcal{O}_{V, Z}) \]
We need to do a little bit of commutative algebra to show that these intersection multiplicities agree with intuition in simple cases, namely, that sometimes
\[ e_ Z = \text{length}_{\mathcal{O}_{X, Z}} \mathcal{O}_{V \cap W, Z}, \]
in other words, only $\text{Tor}_0$ contributes. This happens when $V$ and $W$ are Cohen-Macaulay in the generic point of $Z$ or when $W$ is cut out by a regular sequence in $\mathcal{O}_{X, Z}$ which also defines a regular sequence on $\mathcal{O}_{V, Z}$. However, Example 43.14.4 shows that higher tors are necessary in general. Moreover, there is a relationship with the Samuel multiplicity. These matters are discussed in Sections 43.13, 43.14, 43.15, 43.16, and 43.17.
Reduction to the diagonal is the statement that we can intersect $V$ and $W$ by intersecting $V \times W$ with the diagonal in $X \times X$. This innocuous statement, which is clear on the level of scheme theoretic intersections, reduces an intersection of a general pair of closed subschemes, to the case where one of the two is locally cut out by a regular sequence. We use this, following Serre, to obtain positivity of intersection multiplicities. Moreover, reduction to the diagonal leads to additivity of intersection multiplicities, associativity, and a projection formula. This can be found in Sections 43.18, 43.19, 43.20, 43.21, and 43.22.
Finally, we come to the moving lemmas and applications. There are two parts to the moving lemma. The first is that given closed subvarieties
\[ Z \subset X \subset \mathbf{P}^ N \]
with $X$ nonsingular, we can find a subvariety $C \subset \mathbf{P}^ N$ intersecting $X$ properly such that
\[ C \cdot X = [Z] + \sum m_ j [Z_ j] \]
and such that the other components $Z_ j$ are “more general” than $Z$. The second part is that one can move $C \subset \mathbf{P}^ N$ over a rational curve to a subvariety in general position with respect to any given list of subvarieties. Combined these results imply that it suffices to define the intersection product of cycles on $X$ which intersect properly which was done above. Of course this only leads to an intersection product on $\mathop{\mathrm{CH}}\nolimits _*(X)$ if one can show, as we do in the text, that these products pass through rational equivalence. This and some applications are discussed in Sections 43.23, 43.24, 43.25, 43.26, 43.27, and 43.28.
At the end of the second paragraph there seems to be an extra "ff".
See http://en.wiktionary.org/wiki/ff.
Ah ! The stacks project really is a amazing source of information :) Thanks a lot.
Comment #2339 by oliver on January 02, 2017 at 14:16
Why is the ground field in this chapter assumed to be algebraically closed?
@#2339: Mainly because in the proof of the moving lemma it is nice to have many rational points available. Burt Totaro pointed out to me that it is not necessary to do so. In the moving lemma one chooses always a rational point in a projective space or a grassmanian. Hence only finite fields cause trouble. In this case you can solve the problem after an extension of degree
2^n
and after an extension of degree
3^m
. Then you can combine the cycles you get to win. But you do have to be very careful with the order in which you do things so you don't get stuck so it isn't completely "free" to make the corresponding changes. Thanks for the comment!
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Implement quaternion representation of six-degrees-of-freedom equations of motion of simple variable mass with respect to wind axes - Simulink - MathWorks Deutschland
The translational motion of the wind-fixed coordinate frame is given below, where the applied forces [Fx Fy Fz]T are in the wind-fixed frame. Vrew is the relative velocity in the wind axes at which the mass flow (
\stackrel{˙}{m}
\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{w}+{\overline{\omega }}_{w}×{\overline{V}}_{w}\right)+\stackrel{˙}{m}\overline{V}r{e}_{w}\\ {\overline{V}}_{w}=\left[\begin{array}{l}V\\ 0\\ 0\end{array}\right],{\overline{\omega }}_{w}=\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\stackrel{˙}{\beta }\mathrm{sin}\alpha \\ {q}_{b}-\stackrel{˙}{\alpha }\\ {r}_{b}+\stackrel{˙}{\beta }\mathrm{cos}\alpha \end{array}\right],{\overline{w}}_{b}\left[\begin{array}{l}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\end{array}
\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{l}L\\ M\\ N\end{array}\right]=I{\stackrel{˙}{\overline{\omega }}}_{b}+{\overline{\omega }}_{b}×\left(I{\overline{\omega }}_{b}\right)+\stackrel{˙}{I}{\overline{\omega }}_{b}\\ I=\left[\begin{array}{lll}{I}_{xx}\hfill & -{I}_{xy}\hfill & -{I}_{xz}\hfill \\ -{I}_{yx}\hfill & {I}_{yy}\hfill & -{I}_{yz}\hfill \\ -{I}_{zx}\hfill & -{I}_{zy}\hfill & {I}_{zz}\hfill \end{array}\right]\end{array}
\stackrel{˙}{I}=\frac{{I}_{full}-{I}_{empty}}{{m}_{full}-{m}_{empty}}\stackrel{˙}{m}
\left[\begin{array}{l}{\stackrel{˙}{q}}_{0}\\ {\stackrel{˙}{q}}_{1}\\ {\stackrel{˙}{q}}_{2}\\ {\stackrel{˙}{q}}_{3}\end{array}\right]=-1/2\left[\begin{array}{llll}0\hfill & p\hfill & q\hfill & r\hfill \\ -p\hfill & 0\hfill & -r\hfill & q\hfill \\ -q\hfill & r\hfill & 0\hfill & -p\hfill \\ -r\hfill & -q\hfill & p\hfill & 0\hfill \end{array}\right]\left[\begin{array}{l}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]
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Modify a model predictive controller’s state estimator - MATLAB setEstimator - MathWorks América Latina
Design State Estimator by Pole Placement
Modify a model predictive controller’s state estimator
setEstimator(MPCobj,L,M)
setEstimator(MPCobj,'default')
setEstimator(MPCobj,'custom')
setEstimator(MPCobj,L,M) sets the gain matrices used for estimation of the states of an MPC controller. For more information, see State Estimator Equations.
setEstimator(MPCobj,'default') restores the gain matrices L and M to their default values. The default values are the optimal static gains calculated using kalmd for the plant, disturbance, and measurement noise models specified in MPCobj.
setEstimator(MPCobj,'custom') specifies that controller state estimation will be performed by a user-supplied procedure. This option suppresses calculation of L and M. When the controller is operating in this way, the procedure must supply the state estimate x[n|n] to the controller at the beginning of each control interval.
Design an estimator using pole placement, assuming the linear system
AM=L
Create a plant model.
G = tf({1,1,1},{[1 .5 1],[1 1],[.7 .5 1]});
To improve the clarity of this example, call mpcverbosity to suppress messages related to working with an MPC controller.
Create a model predictive controller for the plant. Specify the controller sample time as 0.2 seconds.
MPCobj = mpc(G, 0.2);
Obtain the default state estimator gain.
[~,M,A1,Cm1] = getEstimator(MPCobj);
Calculate the default observer poles.
e = eig(A1-A1*M*Cm1);
Specify faster observer poles.
new_poles = [.8 .75 .7 .85 .6 .81];
Compute a state-gain matrix that places the observer poles at new_poles.
L = place(A1',Cm1',new_poles)';
place returns the controller-gain matrix, whereas you want to compute the observer-gain matrix. Using the principle of duality, which relates controllability to observability, you specify the transpose of A1 and Cm1 as the inputs to place. This function call yields the observer gain transpose.
Obtain the estimator gain from the state-gain matrix.
M = A1\L;
Specify M as the estimator for MPCobj.
The pair, (
{A}_{1},{C}_{m1}
), describing the overall state-space realization of the combination of plant and disturbance models must be observable for the state estimation design to succeed. Observability is checked in Model Predictive Control Toolbox software at two levels: (1) observability of the plant model is checked at construction of the MPC object, provided that the model of the plant is given in state-space form; (2) observability of the overall extended model is checked at initialization of the MPC object, after all models have been converted to discrete-time, delay-free, state-space form and combined together.
Restore mpcverbosity.
A*M (default) | matrix
Kalman gain matrix for the time update, specified as a matrix. The dimensions of L are nx-by-nym, where nx is the total number of controller states, and nym is the number of measured outputs.
If L is empty, it defaults to L = A*M, where A is the state-transition matrix.
Kalman gain matrix for the measurement update, specified as a matrix. The dimensions of L are nx-by-nym, where nx is the total number of controller states, and nym is the number of measured outputs.
If M is omitted or empty, it defaults to a zero matrix, and the state estimator becomes a Luenberger observer.
{x}_{c}^{rev}\left(k\text{â}\text{|}kâ1\right)={x}_{c}\left(k\text{â}\text{|}kâ1\right)+{B}_{u}\left[{u}^{act}\left(kâ1\right)â{u}^{opt}\left(kâ1\right)\right]
e\left(k\right)={y}_{m}\left(k\right)â\left[{C}_{m}{x}_{c}^{rev}\left(k\text{â}\text{|}kâ1\right)+{D}_{mv}v\left(k\right)\right]
{x}_{c}\left(k\text{â}\text{|}k\right)={x}_{c}^{rev}\left(k\text{â}\text{|}kâ1\right)+Me\left(k\right)
{x}_{c}\left(k+1|k\right)=A{x}_{c}^{rev}\left(k|kâ1\right)+{B}_{u}{u}^{opt}\left(k\right)+{B}_{v}v\left(k\right)+Le\left(k\right)
getEstimator | mpc | mpcstate | kalman
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Big O notation is one of the most fundamental tools for computer scientists to analyze the time and space complexity of an algorithm.
Your choice of algorithm and data structure starts to matter when you’re tasked with writing software with strict SLAs (service level agreements) or for millions of users.
Big O Notation allows you to compare the efficiency of different algorithms, so when it comes to writing software, you can build a well designed and thought out application that will scale and perform better than competitors.
What are time and space complexity?
Common varieties of Big O Notation
Asymptotic analysis: How to find the complexity of an algorithm
Get hands-on practice with Big O
Be confident in any coding interview with dozens of practice interview questions.
Big O Notation is a mathematical function used in computer science to describe how complex an algorithm is — or more specifically, the execution time required by an algorithm. In software engineering, it’s used to compare the efficiency of different approaches to a problem.
With Big O Notation, you express the runtime in terms of how quickly it grows relative to the input, as the input gets arbitrarily large. Essentially, it’s a way to draw insights into how scalable an algorithm is.
It doesn’t tell you how fast or how slow an algorithm will go, but instead about how it changes with respect to its input size.
When talking about Big O Notation it’s important that you understand the concepts of time and space complexity, mainly because Big O Notation is a way to classify complexities.
Complexity is an approximate measure of the efficiency of an algorithm and is associated with every algorithm you write. This is something that all programmers must be aware of.
Typically, there are three tiers to solve for (best case, average case, and worst case) which are known as asymptotic notations. These notations allow us to answer questions such as:
Does it mostly maintain its quick run time as the input size increases?
Best case — represented as Big Omega or Ω(n)
Big-Omega, commonly written as Ω, is an Asymptotic Notation for the best case, or a floor growth rate for a given function. It provides us with an asymptotic lower bound for the growth rate of the runtime of an algorithm.
Average case — represented as Big Theta or Θ(n)
Theta, commonly written as Θ, is an Asymptotic Notation to denote the asymptotically tight bound on the growth rate of runtime of an algorithm.
Worst case — represented as Big O Notation or O(n)
Developers typically solve for the worst case scenario, Big O, because you’re not expecting your algorithm to run in the best or even average cases. It allows you to make analytical statements such as, “well, in the worst case scenario, my algorithm will scale this quickly”.
These varieties of Big-O Notation aren’t the only ones, but they’re the ones you’re most likely to encounter.
This translates to a constant runtime, meaning, regardless of the size of the input, the algorithm will have the same runtime.
Array: inserting or retrieving an element
O(log n) means that time goes up linearly, while the n goes up exponentially. So if it takes 1 second to compute 10 elements, it will take 2 seconds to compute 100 elements and so on. It is O(log n) when we use divide and conquer algorithms e.g binary search.
Binary tree: inserting or retrieving an element
Tree: Depth first search (DFS) of a tree
O(n²) - Quadratic time complexity
As the elements in your list increase, the time it will take for your algorithm to run will increase exponentially.
Sorting algorithm: Bubble sort and insertion sort
Learn Big O Notation without scrubbing through videos or documentation. Educative’s text-based courses are easy to skim and feature live coding environments — making learning quick and efficient.
Big O Notation for Coding Interviews and Beyond
How to find Big O of an algorithm
If you’re in an interview and are asked to find the Big O complexity of an algorithm here is a general rule:
Example: Find the Big O complexity of an algorithm with the time complexity 3n³ + 4n + 2.
This simplifies to O(n³).
How to find the time complexity of an algorithm
Here’s a simple example that measures the time complexity of a for loop of size “n”:
int n = 10; 1
int sum = 0; 1
int i = 0; 1
i < n; n + 1
i++; n
sum += 2; n
cout << sum; 1
After counting the number of operations and how many times each operation is executing, you just add all of these counts to get the time complexity of this program.
1 + 1 + 1 + (n + 1) + n + n + 1 = 3 + (n + 1) + 2n + 1 => 3n + 5
Every time a list or array gets iterated over c x length times, it is most likely in O(n) time.
When you see a problem where the number of elements in the problem space gets halved each time, it will most probably be in O(logn) runtime.
void printFirstItem(const vector<int>& items)
cout << items[0] << endl;
This function runs in O(1) time (or “constant time”) relative to its input. This means that the input array could be 1 item or 1,000 items, but this function would still just require one “step.”
void printAllItems(const vector<int>& items)
This function runs in O(n) time (or “linear time”), where n is the number of items in the vector. If the vector has 10 items, we have to print 10 times. If it has 1,000 items, we have to print 1,000 times.
void printAllPossibleOrderedPairs(const vector<int>& items)
for (int firstItem : items) {
for (int secondItem : items) {
cout << firstItem << ", " << secondItem << endl;
Here we’re nesting two loops. If our vector has n items, our outer loop runs n times and our inner loop runs n times for each iteration of the outer loop, giving us n^2 total prints. Thus this function runs in O(n^2) time (or “quadratic time”). If the vector has 10 items, we have to print 100 times. If it has 1,000 items, we have to print 1,000,000 times.
Array Insert: O(1) Retrieve: O(1) N/A
Linked List Insert at tail: O(n)
Insert at head: O(1)
Retrieve: O(n) Note that if new elements are added at the head of the linked list then insert becomes a O(1) operation
Binary Tree Insert: O(n)
Retrieve: O(n) In worst case, the binary tree becomes a linked-list.
Dynamic Array Insert: O(1)
Retrieve: O(1) Note by retrieving it is implied we are retrieving from a specific index of the array.
Stack Push: O(1)
Pop: O(1) There are no complexity trick questions asked for stacks or queues. We only mention them here for completeness. The two data-structures are more important from a last-in last-out (stack) and first in first out (queue) perspective.
Queue Enqueue: O(1)
Dequeue: O(1) N/A
Priority Queue (Binary Heap) Insert: O(logn)
Delete: O(logn)
Get Min/Max: O(1) N/A
Hash Table Insert: O(n)
Retrieve: O(n) Be mindful that a hash table’s average case for insertion and retrieval is O(1)
B-Trees Insert: O(logn)
Retrieve: O(logn) N/A
Red-Black Trees Insert: O(logn)
Sorting / Searching Algorithm Bubble sort: O(n²)
Insertion sort: O(n²)
Selection sort: O(n²)
Quick sort: O(n²)
Merge sort: O(logn) Note, even though worst case quicksort performance is O(n2) but in practice quicksort is often used for sorting since its average case is O(nlgn).
Tree Algorithm Depth first search: O(n)
Breadth first search: O(n)
Pre-order, in-order, post-order traversals: O(n) n is the total number of nodes in the tree. Most tree-traversal algorithms will end up seeing every node in the tree and their complexity in the worst case is thus O(n).
While there was quite a bit covered in this post, we’ve barely even scratched the surface.
Big O For Coding Interviews & Beyond is a simple and practical guide to algorithmic complexity. Learn what you need to know specifically to analyze any algorithm, and ace your next coding interview.
This course is designed for folks who aren’t math whizzes, or even super-experienced in programming. Written in a conversational style, chock full of real-world examples, this course is an antidote to the mountains of dry, technical Big-O references.
By the time you wrap up this course, you’ll have a working knowledge of Big-O, and the confidence to ace your programming interview at any tech company!
Continue reading about Big O and data structures
The insider’s guide to algorithm interview questions
Big-O Notation Cheat Sheet
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Extract vector of decision variables from matrix variable values - MATLAB mat2dec - MathWorks América Latina
mat2dec
Extract vector of decision variables from matrix variable values
decvec = mat2dec(lmisys,X1,X2,X3,...)
Given an LMI system lmisys with matrix variables X1, . . ., XK and given values X1,...,Xk of X1, . . ., XK, mat2dec returns the corresponding value decvec of the vector of decision variables. Recall that the decision variables are the independent entries of the matrices X1, . . ., XK and constitute the free scalar variables in the LMI problem.
This function is useful, for example, to initialize the LMI solvers mincx or gevp. Given an initial guess for X1, . . ., XK, mat2dec forms the corresponding vector of decision variables xinit.
An error occurs if the dimensions and structure of X1,...,Xk are inconsistent with the description of X1, . . ., XK in lmisys.
Consider an LMI system with two matrix variables X and Y such that
X is a symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block.
Y is a 2-by-3 rectangular matrix.
Particular instances of X and Y are
{X}_{0}=\left(\begin{array}{cccc}1& 3& 0& 0\\ 3& -1& 0& 0\\ 0& 0& 5& 0\\ 0& 0& 0& 5\end{array}\right),\text{ }{Y}_{0}=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)
and the corresponding vector of decision variables is given by
decv = mat2dec(lmisys,X0,Y0)
decv'
1 3 -1 5 1 2 3 4 5 6
Note that decv is of length 10 since Y has 6 free entries while X has 4 independent entries due to its structure. Use decinfo to obtain more information about the decision variable distribution in X and Y.
dec2mat | decinfo | decnbr
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Section 10.56 (00JL): Graded rings—The Stacks project
Section 10.56: Graded rings (cite)
10.56 Graded rings
A graded ring will be for us a ring $S$ endowed with a direct sum decomposition $S = \bigoplus _{d \geq 0} S_ d$ of the underlying abelian group such that $S_ d \cdot S_ e \subset S_{d + e}$. Note that we do not allow nonzero elements in negative degrees. The irrelevant ideal is the ideal $S_{+} = \bigoplus _{d > 0} S_ d$. A graded module will be an $S$-module $M$ endowed with a direct sum decomposition $M = \bigoplus _{n\in \mathbf{Z}} M_ n$ of the underlying abelian group such that $S_ d \cdot M_ e \subset M_{d + e}$. Note that for modules we do allow nonzero elements in negative degrees. We think of $S$ as a graded $S$-module by setting $S_{-k} = (0)$ for $k > 0$. An element $x$ (resp. $f$) of $M$ (resp. $S$) is called homogeneous if $x \in M_ d$ (resp. $f \in S_ d$) for some $d$. A map of graded $S$-modules is a map of $S$-modules $\varphi : M \to M'$ such that $\varphi (M_ d) \subset M'_ d$. We do not allow maps to shift degrees. Let us denote $\text{GrHom}_0(M, N)$ the $S_0$-module of homomorphisms of graded modules from $M$ to $N$.
At this point there are the notions of graded ideal, graded quotient ring, graded submodule, graded quotient module, graded tensor product, etc. We leave it to the reader to find the relevant definitions, and lemmas. For example: A short exact sequence of graded modules is short exact in every degree.
Given a graded ring $S$, a graded $S$-module $M$ and $n \in \mathbf{Z}$ we denote $M(n)$ the graded $S$-module with $M(n)_ d = M_{n + d}$. This is called the twist of $M$ by $n$. In particular we get modules $S(n)$, $n \in \mathbf{Z}$ which will play an important role in the study of projective schemes. There are some obvious functorial isomorphisms such as $(M \oplus N)(n) = M(n) \oplus N(n)$, $(M \otimes _ S N)(n) = M \otimes _ S N(n) = M(n) \otimes _ S N$. In addition we can define a graded $S$-module structure on the $S_0$-module
\[ \text{GrHom}(M, N) = \bigoplus \nolimits _{n \in \mathbf{Z}} \text{GrHom}_ n(M, N), \quad \text{GrHom}_ n(M, N) = \text{GrHom}_0(M, N(n)). \]
We omit the definition of the multiplication.
Lemma 10.56.1. Let $S$ be a graded ring. Let $M$ be a graded $S$-module.
If $S_+M = M$ and $M$ is finite, then $M = 0$.
If $N, N' \subset M$ are graded submodules, $M = N + S_+N'$, and $N'$ is finite, then $M = N$.
If $N \to M$ is a map of graded modules, $N/S_+N \to M/S_+M$ is surjective, and $M$ is finite, then $N \to M$ is surjective.
If $x_1, \ldots , x_ n \in M$ are homogeneous and generate $M/S_+M$ and $M$ is finite, then $x_1, \ldots , x_ n$ generate $M$.
Proof. Proof of (1). Choose generators $y_1, \ldots , y_ r$ of $M$ over $S$. We may assume that $y_ i$ is homogeneous of degree $d_ i$. After renumbering we may assume $d_ r = \min (d_ i)$. Then the condition that $S_+M = M$ implies $y_ r = 0$. Hence $M = 0$ by induction on $r$. Part (2) follows by applying (1) to $M/N$. Part (3) follows by applying (2) to the submodules $\mathop{\mathrm{Im}}(N \to M)$ and $M$. Part (4) follows by applying (3) to the module map $\bigoplus S(-d_ i) \to M$, $(s_1, \ldots , s_ n) \mapsto \sum s_ i x_ i$. $\square$
Let $S$ be a graded ring. Let $d \geq 1$ be an integer. We set $S^{(d)} = \bigoplus _{n \geq 0} S_{nd}$. We think of $S^{(d)}$ as a graded ring with degree $n$ summand $(S^{(d)})_ n = S_{nd}$. Given a graded $S$-module $M$ we can similarly consider $M^{(d)} = \bigoplus _{n \in \mathbf{Z}} M_{nd}$ which is a graded $S^{(d)}$-module.
Lemma 10.56.2. Let $S$ be a graded ring, which is finitely generated over $S_0$. Then for all sufficiently divisible $d$ the algebra $S^{(d)}$ is generated in degree $1$ over $S_0$.
Proof. Say $S$ is generated by $f_1, \ldots , f_ r \in S$ over $S_0$. After replacing $f_ i$ by their homogeneous parts, we may assume $f_ i$ is homogeneous of degree $d_ i > 0$. Then any element of $S_ n$ is a linear combination with coefficients in $S_0$ of monomials $f_1^{e_1} \ldots f_ r^{e_ r}$ with $\sum e_ i d_ i = n$. Let $m$ be a multiple of $\text{lcm}(d_ i)$. For any $N \geq r$ if
\[ \sum e_ i d_ i = N m \]
then for some $i$ we have $e_ i \geq m/d_ i$ by an elementary argument. Hence every monomial of degree $N m$ is a product of a monomial of degree $m$, namely $f_ i^{m/d_ i}$, and a monomial of degree $(N - 1)m$. It follows that any monomial of degree $nrm$ with $n \geq 2$ is a product of monomials of degree $rm$. Thus $S^{(rm)}$ is generated in degree $1$ over $S_0$. $\square$
Lemma 10.56.3. Let $R \to S$ be a homomorphism of graded rings. Let $S' \subset S$ be the integral closure of $R$ in $S$. Then
\[ S' = \bigoplus \nolimits _{d \geq 0} S' \cap S_ d, \]
i.e., $S'$ is a graded $R$-subalgebra of $S$.
Proof. We have to show the following: If $s = s_ n + s_{n + 1} + \ldots + s_ m \in S'$, then each homogeneous part $s_ j \in S'$. We will prove this by induction on $m - n$ over all homomorphisms $R \to S$ of graded rings. First note that it is immediate that $s_0$ is integral over $R_0$ (hence over $R$) as there is a ring map $S \to S_0$ compatible with the ring map $R \to R_0$. Thus, after replacing $s$ by $s - s_0$, we may assume $n > 0$. Consider the extension of graded rings $R[t, t^{-1}] \to S[t, t^{-1}]$ where $t$ has degree $0$. There is a commutative diagram
\[ \xymatrix{ S[t, t^{-1}] \ar[rr]_{s \mapsto t^{\deg (s)}s} & & S[t, t^{-1}] \\ R[t, t^{-1}] \ar[u] \ar[rr]^{r \mapsto t^{\deg (r)}r} & & R[t, t^{-1}] \ar[u] } \]
where the horizontal maps are ring automorphisms. Hence the integral closure $C$ of $S[t, t^{-1}]$ over $R[t, t^{-1}]$ maps into itself. Thus we see that
\[ t^ m(s_ n + s_{n + 1} + \ldots + s_ m) - (t^ ns_ n + t^{n + 1}s_{n + 1} + \ldots + t^ ms_ m) \in C \]
which implies by induction hypothesis that each $(t^ m - t^ i)s_ i \in C$ for $i = n, \ldots , m - 1$. Note that for any ring $A$ and $m > i \geq n > 0$ we have $A[t, t^{-1}]/(t^ m - t^ i - 1) \cong A[t]/(t^ m - t^ i - 1) \supset A$ because $t(t^{m - 1} - t^{i - 1}) = 1$ in $A[t]/(t^ m - t^ i - 1)$. Since $t^ m - t^ i$ maps to $1$ we see the image of $s_ i$ in the ring $S[t]/(t^ m - t^ i - 1)$ is integral over $R[t]/(t^ m - t^ i - 1)$ for $i = n, \ldots , m - 1$. Since $R \to R[t]/(t^ m - t^ i - 1)$ is finite we see that $s_ i$ is integral over $R$ by transitivity, see Lemma 10.36.6. Finally, we also conclude that $s_ m = s - \sum _{i = n, \ldots , m - 1} s_ i$ is integral over $R$. $\square$
In the proof of 077G, the bottom horizontal map in the diagram should be
r\mapsto t^{\deg(r)}r
Comment #6518 by Zhenhua Wu on August 26, 2021 at 23:29
In the first line, "ring
S
endowed with a direct sum decomposition
S=\bigoplus_{d\geq 0}S_d
", I was always wondering about which kind of structure are those
S_d
and which kind of direct sum is this? So I suggest it changes to "ring
S
endowed with a direct sum decomposition of abelian groups
S=\bigoplus_{d\geq 0}S_d
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Home : Support : Online Help : Mathematics : Group Theory : Degree
Degree( G )
The degree of a permutation group is the cardinality of the set upon which it acts. Since permutation groups act on sets of the form
\left\{1,2,\dots ,n\right\}
, the degree is the positive integer
n
The Degree( G ) command returns the degree of a permutation group G.
Some group constructors, such as those for the symmetric and alternating groups take a degree parameter as input, and the Degree command returns this value.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{Group}\left(\mathrm{Perm}\left([[1,2]]\right),\mathrm{Perm}\left([[2,3,4]]\right)\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}〈\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\right)〉
\mathrm{Degree}\left(G\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{Degree}\left(\mathrm{DihedralGroup}\left(14\right)\right)
\textcolor[rgb]{0,0,1}{14}
\mathrm{Degree}\left(\mathrm{AlternatingGroup}\left(10\right)\right)
\textcolor[rgb]{0,0,1}{10}
The GroupTheory[Degree] command was introduced in Maple 17.
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Jet Impingement Boiling From a Circular Free-Surface Jet During Quenching: Part 2—Two-Phase Jet | J. Heat Transfer | ASME Digital Collection
David E. Hall,
Michelin Americas Research Corporation, 515 Michelin Road, Greenville, SC 29602
Frank P. Incropera,
Notre Dame University, South Bend, IN 46556
Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division May 27, 1997; revision received March 22, 2001. Associate Editor: M. S. Sohal.
Hall, D. E., Incropera, F. P., and Viskanta, R. (March 22, 2001). "Jet Impingement Boiling From a Circular Free-Surface Jet During Quenching: Part 2—Two-Phase Jet ." ASME. J. Heat Transfer. October 2001; 123(5): 911–917. https://doi.org/10.1115/1.1389062
A proposed technique for controlling jet impingement boiling heat transfer involves injection of gas into the liquid jet. This paper reports results from an experimental study of boiling heat transfer during quenching of a cylindrical copper specimen, initially at a uniform temperature exceeding the temperature corresponding to maximum heat flux, by a two-phase (water-air), circular, free-surface jet. The second phase is introduced as small bubbles into the jet upstream of the nozzle exit. Data are presented for single-phase convective heat transfer at the stagnation point, as well as in the form of boiling curves, maximum heat fluxes, and minimum film boiling temperatures at locations extending from the stagnation point to a radius of ten nozzle diameters. For void fractions ranging from 0.0 to 0.4 and liquid-only velocities between 2.0 and 4.0 m/s
11,300⩽Red,fo⩽22,600,
several significant effects are associated with introduction of the gas bubbles into the jet. As well as enhancing single-phase convective heat transfer by up to a factor of 2.1 in the stagnation region, addition of the bubbles increases the wall superheat in nucleate boiling and eliminates the temperature excursion associated with cessation of boiling. The maximum heat flux is unaffected by changes in the void fraction, while minimum film boiling temperatures increase and film boiling heat transfer decreases with increasing void fraction. A companion paper (Hall et al., 2001) details corresponding results from the single-phase jet.
convection, two-phase flow, jets, boiling, bubbles, flow control, nozzles
Boiling, Bubbles, Convection, Heat flux, Heat transfer, Nozzles, Porosity, Temperature, Quenching (Metalworking), Nucleate boiling, Two-phase flow, Jets, Water, Film boiling
Viskanta, R. and Incropera, F. P., 1992, “Quenching with Liquid Jet Impingement,” I. Tanasawa and N. Lior, eds., Heat and Mass Transfer in Materials Processing, Hemisphere, New York, pp. 455–476.
Wagstaff, R. B. and Bowles, K. D., 1995, “Practical Low Head Casting (LHC) Mold for Aluminum Ingot Casting,” J. Evans, ed., Proceedings, TMS Light Metals Committee, The Minerals, Metals & Materials Society, Warrendale, PA, pp. 1071–1075.
Fischer, H., Wagstaff, F. E., and Ekenes, J. M., 1989, “Airslip and Turbo Development for Aluminum Sheet Ingot,” Proceedings, Ingot and Continuous Casting Process Technology Seminar for Flat Rolled Products, The Aluminum Association, pp. 417–426.
Serizawa, A., Takahashi, O., Kawara, Z., Komeyama, T., and Michiyoshi, I., 1990, “Heat Transfer Augmentation by Two-Phase Bubbly Flow Impinging Jet with a Confining Wall,” G. Hetsroni, et al., eds., Proceedings, 9th International Heat Transfer Conference, Hemisphere, New York, Vol., 4, pp. 93–98.
Chang, C. T., Kojasoy, G., Landis, F., and Downing, S., 1995, “Confined Single- and Multiple-Jet Impingement Heat Transfer—II. Turbulent Two-Phase Flow,” International journal of Heat and Mass transfer, Vol. 38, pp. 843–851.
Lockhart, R. W., and Martinelli, R. C., 1949, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes,” Chemical Engineering Progress, Vol. 45, pp. 39–48.
Zumbrunnen, D. A. and Balasubramanian, M., 1995, “Convective Heat Transfer Enhancement Due to Gas Injection Into an Impinging Liquid Jet,” ASME Journal of Heat Transfer. Vol. 117, pp. 1011–1017.
Jet Impingement Boiling From a Circular Free-Surface Jet During Quenching: 1—Single-Phase Jet
Bar-Cohen, A. and Simon, T. W., 1988, “Wall Superheat Excursions in the Boiling Incipience of Dielectric Fluids,” Heat Transfer Engineering, Vol. 9, pp. 19–31.
Webb, B. W. and Ma, C.-F., 1995, “Single-Phase Liquid Jet Impingement Heat Transfer,” J. P. Hartnett and T. F. Irvine, eds., Advances in Heat Transfer, Academic Press, New York, Vol. 26, pp. 105–217.
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The procedure TopologicalSort attempts to produce a linear ordering of a collection of elements that is consistent with a specified partial ordering of those elements. This means that an element
a
precedes an element
b
in the partial order only if
a
b
in the linear order.
The partial order is specified as a "relation" rel, which is a list or set of pairs
[a,b]
, each representing an ordering of the elements of the domain of the relation. A pair
[a,b]
belongs to the relation rel if
a
b
in the partial order it represents. The domain of the relation is the set of all expressions that occur as either a first or second entry (or both) in some pair in the relation rel.
Alternatively, you may think of the members of rel as directed edges in a graph whose vertices are the elements of the domain of the relation. In these terms, a topological sort of the vertices is a linear ordering of them such that a vertex
a
b
in the linear order only if there is a directed path from
a
b
in the graph. The two characterizations are equivalent.
In general, there may be many linear orderings of the vertices of a graph that are consistent with it. For example, the partial order
[[a,b],[a,c]]
(indicating that
a
is "less than"
b
a
c
has two consistent linear orderings:
[a,b,c]
[a,c,b]
. The TopologicalSort procedure produces one of them.
It is also possible that no linear ordering consistent with the given partial order exists. This is the case when the directed graph contains a cycle. If TopologicalSort detects a cycle in the graph, then an exception is raised. The simplest example of this is the relation
[[a,b],[b,a]]
, which clearly has no consistent linear order.
\mathrm{TopologicalSort}\left([[a,b],[a,c]]\right)
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]
\mathrm{TopologicalSort}\left([[a,b],[a,c],[b,d],[c,d]]\right)
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]
\mathrm{TopologicalSort}\left([[a,b],[b,a]]\right)
e≔F\left(G\left(x,y\right),H\left(G\left(u,v\right),a,b\right)\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)\right)
\mathrm{dom}≔\mathrm{indets}\left(e,'\mathrm{anything}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{dom}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)}
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dom}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{rel}[d]≔\mathrm{seq}\left([d,t],t=\mathrm{select}\left(\mathrm{has},\mathrm{dom},d\right)\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
\mathrm{rel}≔[\mathrm{seq}\left(\mathrm{rel}[d],d=\mathrm{dom}\right)]:
\mathrm{TopologicalSort}\left(\mathrm{rel}\right)
[\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)\right)]
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Have you ever wondered how much a full-stack serverless application costs; Especially at scale? Or when it starts to cost money?
When designing the initial prototype of Serv, I knew we had to use serverless architecture because of the success we found with Ajtima:
fast developer velocity and ramping
Easily automatable CICD pipelines
no infrastructure to manage (duh)
but one of the other things I took for granted was the free tier of AWS; ie "How much can we get for free"? And that's what I'm here to tell you about.
What will you get out of this reading ?
The awesome-ness of free tier providers such as AWS, Google Firebase, Netlify, etc
The stack we used when building a multi-tenant restaurant ordering app
Let's continue using Serv as the subject of the case study. Serv is a platform that lets restaurant patrons scan a QR code, browse a menu, order items, and pay for them.
We'll run the numbers at 3 different scales:
Having been a solutions architect for a few years, I hate to continue to preface everything with the horrible phrase "it depends", but it is true; YMMV depending on what tools your app is using, and how your users interact with your app.
With that said, Here are the generic numbers specific to the Serv use case you would see at the end of this guide. Continue if you want to see this number broken down by each service in the stack:
Running Monthly Costs
For 100 daily users, we don't pay a penny. We are completely leeching off of free SaaS platforms like AWS, GitHub, Netlify, and others.
For 10,000 daily users, or 300,000 unique monthly users & sessions, we pay $2.77 USD per MONTH. That is an absolutely, ridiculously low number, which only serves to highlight the absurd volumes that AWS is accustomed to dealing with. I mean, their pricing is laid out by the millions.
For 1 million unique daily users, or 30 million monthly users, it would only cost us $435.25 USD a month to support all of our infrastructure. How much does your app net you per user?
However, it's not all roses. It's like when Black Friday deals get you in the door to spend 10 bucks on a 20 dollar shirt, only to have you pay $1,000 for a full price TV while you're there. Vendor lock is diffiuclt to climb out of.
Each Service in the Stack
Serv has two interfaces. An end-user client for restaurant partons to place and pay for orders.
And a Dashboard view for restaurant staff to manage orders, menu, etc.
This is just React that gets compiled into HTML/JS/CSS. We can use any host for that. I've been using Netlify & GitHub for years, and never paid a penny (Thank-you Netlify & GitHub). Netlify has a direct integration with GitHub that allows you to build your dev/staging/production website on pushes and pull requests.
So far, our server bills are $0/month. Because Netlify doesn't charge for traffic, only for build hours in the CD pipeline. Yes, for 1,000,000 users of our front-end app, we don't pay a penny.
Back end API functions
The back-end is all in AWS, so slightly more difficult to calculate.
Serv uses the following:
AWS Lambda is invoked whenever somebody makes a payment, which is once a day per user. We use Lambda to avoid storing sensitive tokens in the front end.
1 daily user = 30 monthly requests per user
So our new running cost including Lambda
Using 50ms per request and 128MB memory consumed
Wow. That's right. To serve 1 MM users per day, or 30 MM requests a month, Lambda costs us....
To serve 10,000 users a day, it costs us a whopping NOTHING.
AWS Appsync is a managed GraphQL API that sits on top of Lambda/DynamoDB/other data sources.
Their pricing page lays out what is included in the free tier:
let's run the numbers. During a session ie single unique daily user:
Average group of customers is 5 people at one table
A user creates one order and receives 4 per session
Users spends 5 minutes ordering and paying.
Connection Minutes p/m
Query and Data Modification Ops p/m
Real-time Updates p/m
100 600 15,000 / $0.00 3,000 / $0.00 15,000 / $0.00 $0.00
10,000 60,000 1,500,000 / $0.072 300,000 / $0.20 1,500,000 / $2.50 $2.772 USD
1,000,000 6,000,000 150,000,000 / $11.952 30,000,000 / $119.00 150,000,000 / $299.50 $430.452
Note, we didn't add our data transfer charges, but it's a small minority of the overall Appsync charges
Math things: (can skip) - Daily Groups = Daily Users / 5 - Connections Per group = 5 mins * 5 users = 25 mins per group (
0.08 per million updates) - Query And Data Per Group = 1 order per user = 5 per group. (
4.00 per million operations) - Real Time Updates per group = 5 x 5 = 25 real time Updates per group ($2.00 per million updates)
Our running totals so far:
Still really affordable. I mean, 10,000 users is a whole community. And to support that across all our front-ends & APIs, we barely have to pay more than a shot of espresso PER MONTH.
With 1,000,000 unique daily users, it costs us $0.000435 per user. It's easy to come out ahead in this age of PAYG.
Let's continue with an example that really hurts the bank this time.
Text-Messaging on AWS SNS
We (initially) decided to use Phone number One-Time-Code sign up & sign in to make the ordering process as simple as possible. We used an awesome AWS Cloud Formation that spun up all the necessary lambdas and SES topics.
Each unique user will enter their phone number and receive an SMS verification code once per session.
The price of each text message sent by AWS Pinpoint/SNS is $0.00581 USD.
Oh and hey, we've left the free tier! Pinpoint is not included.
t = Daily Users
t X 30 X 0.00581 = AWS SNS cost per month
Whoa, what happened? Our costs just increased from leech level to beyond Enterprise.
Are we screwed? Even if we look at the industry-leading Twilio's pricing, we see that SMS is expensive. Maybe we can use email confirmation, or we can roll our own tools; We have other options.
But that's beside the point; The point is to highlight that (unlike Serv) you need to be careful, and can't expect all services to be affordable or you will be surprised month end.
What if you need to use a non-managed service or have your own APIs to run? We can run these workloads on EC2, FarGate, EKS, or one of the other compute services.
For Serv, we ran Tyk APIM and tried a few different compute services, but ended up settling on EC2 for the affordability.
This one is much more complex to calculate, as you have to figure out how many nodes you need at various RPS benchmarks. Here's a guide I wrote for Tyk to help us do that.
Long story short, it is extremely affordable or FREE to bootstrap your next application on AWS. As a matter of fact, we built Ajtima on Google's FireBase, which is another platform to run full-stack applications on, and found it equally affordable.
Does that mean it's free across the board? No. Some tools are more expensive than others. But all-in-all, a good developer picks the right tool for the job anyway.
Finally, is this blog all-encompassing? No, of course not. For example, I left out data storage costs, and DynamoDB costs which were behind the Appsync APIs. But they were besides the point; My point was to illustrate the generosity of PAYG Cloud platforms, not get into the weeds of each service.
Would you want to see more in-depth technical architecture diagrams to showcase each component in our tech stack or how it all fits together? Let me know, drop your comments below or reach me via contact options in my footer to start a conversation!
awsstartupscloudservajtima
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Option price and sensitivities by Merton76 model using FFT and FRFT - MATLAB optSensByMertonFFT - MathWorks Deutschland
\mathrm{max}\left(St-K,0\right)
\mathrm{max}\left(K-St,0\right)
\begin{array}{l}d{S}_{t}=\left(r-q-{\lambda }_{p}{\mu }_{j}\right){S}_{t}dt+\sigma {S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ \text{prob(}d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}
\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}
\frac{1}{\left(1+J\right)\delta \sqrt{2\pi }}\mathrm{exp}\left\{{\frac{-\left[\mathrm{ln}\left(1+J\right)-\left(\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}\right]}{2{\delta }^{2}}}^{2}\right\}
{f}_{Merton{76}_{j}}\left(\varphi \right)
\begin{array}{l}{f}_{Merton{76}_{j}}={f}_{B{S}_{j}}\mathrm{exp}\left({\lambda }_{p}\tau {\left(1+{\mu }_{j}\right)}^{{m}_{j}+\frac{1}{2}}\left[{\left(1+{\mu }_{j}\right)}^{i\varphi }{e}^{{\delta }^{2}\left({m}_{j}i\varphi +\frac{{\left(i\varphi \right)}^{2}}{2}\right)}-1\right]-{\lambda }_{p}\tau {\mu }_{j}i\varphi \right)\\ \text{where for }j=1,2:\\ {f}_{B{S}_{1}}\left(\varphi \right)=\frac{{f}_{B{S}_{2}}\left(\varphi -i\right)}{{f}_{B{S}_{2}}\left(-i\right)}\\ {f}_{BS2}\left(\varphi \right)=\mathrm{exp}\left(i\varphi \left[\mathrm{ln}{S}_{t}+\left(r-q-\frac{{\sigma }^{2}}{2}\right)\tau \right]-\frac{{\varphi }^{2}{\sigma }^{2}}{2}\tau \right)\\ {m}_{1}=\frac{1}{2},{m}_{2}=-\frac{1}{2}\end{array}
\begin{array}{l}Call\left(k\right)=\frac{{e}^{-\alpha k}}{\pi }{\int }_{0}^{\infty }\mathrm{Re}\left[{e}^{-iuk}\psi \left(u\right)\right]du\\ \psi \left(u\right)=\frac{{e}^{-r\tau }{f}_{2}\left(\varphi =\left(u-\left(\alpha +1\right)i\right)\right)}{{\alpha }^{2}+\alpha -{u}^{2}+iu\left(2\alpha +1\right)}\\ Put\left(K\right)=Call\left(K\right)+K{e}^{-r\tau }-{S}_{t}{e}^{-q\tau }\end{array}
\mathrm{ln}\left({S}_{t}\right)-\frac{N}{2}\Delta k
\mathrm{ln}\left({S}_{t}\right)+\left(\frac{N}{2}-1\right)\Delta k
{S}_{t}\mathrm{exp}\left(-\frac{N}{2}\Delta k\right)
{S}_{t}\mathrm{exp}\left[\left(\frac{N}{2}-1\right)\Delta k\right]
Call\left({k}_{n}\right)=\Delta u\frac{{e}^{-\alpha {k}_{n}}}{\pi }\sum _{j=1}^{N}\mathrm{Re}\left[{e}^{-i\Delta k\Delta u\left(j-1\right)\left(n-1\right){e}^{i{u}_{j}}\left[\frac{N\Delta k}{2}-\mathrm{ln}\left({S}_{t}\right)\right]}\psi \left({u}_{j}\right)\right]{w}_{j}
\Delta k\Delta u=\left(\frac{2\pi }{N}\right)
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Lemma 10.96.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $\varphi : M \to N$ be a map of $R$-modules.
If $M/IM \to N/IN$ is surjective, then $M^\wedge \to N^\wedge $ is surjective.
If $M \to N$ is surjective, then $M^\wedge \to N^\wedge $ is surjective.
If $0 \to K \to M \to N \to 0$ is a short exact sequence of $R$-modules and $N$ is flat, then $0 \to K^\wedge \to M^\wedge \to N^\wedge \to 0$ is a short exact sequence.
The map $M \otimes _ R R^\wedge \to M^\wedge $ is surjective for any finite $R$-module $M$.
Proof. Assume $M/IM \to N/IN$ is surjective. Then the map $M/I^ nM \to N/I^ nN$ is surjective for each $n \geq 1$ by Nakayama's lemma. More precisely, apply Lemma 10.20.1 part (11) to the map $M/I^ nM \to N/I^ nN$ over the ring $R/I^ n$ and the nilpotent ideal $I/I^ n$ to see this. Set $K_ n = \{ x \in M \mid \varphi (x) \in I^ nN\} $. Thus we get short exact sequences
\[ 0 \to K_ n/I^ nM \to M/I^ nM \to N/I^ nN \to 0 \]
We claim that the canonical map $K_{n + 1}/I^{n + 1}M \to K_ n/I^ nM$ is surjective. Namely, if $x \in K_ n$ write $\varphi (x) = \sum z_ j n_ j$ with $z_ j \in I^ n$, $n_ j \in N$. By assumption we can write $n_ j = \varphi (m_ j) + \sum z_{jk}n_{jk}$ with $m_ j \in M$, $z_{jk} \in I$ and $n_{jk} \in N$. Hence
\[ \varphi (x - \sum z_ j m_ j) = \sum z_ jz_{jk} n_{jk}. \]
This means that $x' = x - \sum z_ j m_ j \in K_{n + 1}$ maps to $x \bmod I^ nM$ which proves the claim. Now we may apply Lemma 10.87.1 to the inverse system of short exact sequences above to see (1). Part (2) is a special case of (1). If the assumptions of (3) hold, then for each $n$ the sequence
\[ 0 \to K/I^ nK \to M/I^ nM \to N/I^ nN \to 0 \]
is short exact by Lemma 10.39.12. Hence we can directly apply Lemma 10.87.1 to conclude (3) is true. To see (4) choose generators $x_ i \in M$, $i = 1, \ldots , n$. Then the map $R^{\oplus n} \to M$, $(a_1, \ldots , a_ n) \mapsto \sum a_ ix_ i$ is surjective. Hence by (2) we see $(R^\wedge )^{\oplus n} \to M^\wedge $, $(a_1, \ldots , a_ n) \mapsto \sum a_ ix_ i$ is surjective. Assertion (4) follows from this. $\square$
Comment #4201 by Qijun Yan on May 04, 2019 at 15:22
For Lemma 0315 (3), it seems to me that the operation of taking completion always preserves short exact sequences by Lemma 0AS0. Maybe I got something wrong.
In Lemma 86.4.5 in Section 86.4 the topology on the submodule
K
is the topology inherited from
M
(it is given by the submodules
K \cap I^nM
). But in the current Section 10.96 there is no mention whatsoever of topologies or completion with respect to any topology. We are just considering
I
-adic completion straight up. Maybe we should be a little bi more careful in the statement of Lemma 86.4.5. Thanks for the comment.
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Model concept, solver and details - GCAM - IAMC-Documentation
Revision as of 15:39, 2 September 2020 by Matthew Binsted (talk | contribs)
3 Economic Choice
3.1 Choice Functions
Supplied with input information from the GCAM Data System, the GCAM Core is the heart of the dynamic character of GCAM. GCAM takes in a set of assumptions and then processes those assumptions to create a full scenario of prices, energy and other transformations, and commodity and other flows across regions and into the future. GCAM represents five different interacting and interconnected systems. The interactions between these different systems all take place within the GCAM core; that is, they are not modeled as independent modules, but as one integrated whole.
The core operating principle for GCAM is that of market equilibrium. Representative agents in GCAM use information on prices, as well as other information that might be relevant, and make decisions about the allocation of resources. These representative agents exist throughout the model, representing, for example, regional electricity sectors, regional refining sectors, regional energy demand sectors, and land users who have to allocate land among competing crops within any given land region. Markets are the means by which these representative agents interact with one another. Agents indicate their intended supply and/or demand for goods and services in the markets. GCAM solves for a set of market prices so that supplies and demands are balanced in all these markets across the model. See the overview for more details.
At each time step, GCAM searches for a vector of prices that cause all markets to be cleared and all consistency conditions to be satisfied. The mapping from input prices to output market disequilibria is a vector function
{\textstyle {\vec {y}}=F({\vec {p}})}
. The GCAM solver is responsible for finding the root of this equation; that is, the point at which
{\displaystyle F({\vec {p}})=0}
. For more information on the solver, see the GCAM solver page.
Choice in GCAM is based on a single numerical value that orders the alternatives by preference. Generically, we call this the choice indicator, p. In practice the choice indicator is either cost or profit rate, though other indicators are possible in principle. In cases where multiple factors influence a choice, such as passenger transportation (where faster modes are more desirable), the additional factors are converted into a cost penalty and added to the basic cost to produce a single indicator that incorporates all of the relevant factors. Economic choice is described in more detail here.
GCAM provides a flexible system for specifying choice functions at runtime on a sector-by-sector basis. Choice functions are represented in the code by classes that implement the IDiscreteChoice interface. Two such classes, the Logit and the Modified Logit are currently provided. Descriptions of these classes and a comparison of the two can be found in the choice functions section of the documentation.
Retrieved from "https://www.iamcdocumentation.eu/index.php?title=Model_concept,_solver_and_details_-_GCAM&oldid=14443"
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Row exchange - MATLAB rowexch - MathWorks América Latina
rowexch
dRE = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns,model)
[dRE,X] = rowexch(...,param1,val1,param2,val2,...)
dRE = rowexch(nfactors,nruns) uses a row-exchange algorithm to generate a D-optimal design dRE with nruns runs (the rows of dRE) for a linear additive model with nfactors factors (the columns of dRE). The model includes a constant term.
[dRE,X] = rowexch(nfactors,nruns) also returns the associated design matrix X, whose columns are the model terms evaluated at each treatment (row) of dRE.
[dRE,X] = rowexch(nfactors,nruns,model) uses the linear regression model specified in model. model is one of the following:
The interaction terms in order (1, 2), (1, 3), ..., (1, n), (2, 3), ..., (n–1, n)
[dRE,X] = rowexch(...,param1,val1,param2,val2,...) specifies additional parameter/value pairs for the design. Valid parameters and their values are listed in the following table.
Initial design as an nruns-by-nfactors matrix. The default is a randomly selected set of points.
A structure that specifies whether to run in parallel, and specifies the random stream or streams. Create the options structure with statset. This option requires Parallel Computing Toolbox™. Option fields are:
Streams — A RandStream object or cell array of such objects. If you do not specify Streams, rowexch uses the default stream or streams. If you choose to specify Streams, use a single object except in the case
y={\beta }_{0}+{\beta }_{1}x{}_{1}+{\beta }_{2}x{}_{2}+{\beta }_{3}x{}_{3}+{\beta }_{12}x{}_{1}x{}_{2}+{\beta }_{13}x{}_{1}x{}_{3}+{\beta }_{23}x{}_{2}x{}_{3}+\epsilon
Use rowexch to generate a D-optimal design with seven runs:
[dRE,X] = rowexch(nfactors,nruns,'interaction','tries',10)
Columns of the design matrix X are the model terms evaluated at each row of the design dRE. The terms appear in order from left to right: constant term, linear terms (1, 2, 3), interaction terms (12, 13, 23). Use X to fit the model, as described in Linear Regression, to response data measured at the design points in dRE.
At each step, the row-exchange algorithm exchanges an entire row of X with a row from a design matrix C evaluated at a candidate set of feasible treatments. The rowexch function automatically generates a C appropriate for a specified model, operating in two steps by calling the candgen and candexch functions in sequence. Provide your own C by calling candexch directly. In either case, if C is large, its static presence in memory can affect computation.
candgen | candexch | cordexch
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Equations - Lobis Finance
Deposit = Withdrawal
Swaps between LOBI and LOBIS during staking and unstaking are always honored 1:1. The amount of LOBI deposited into the staking contract will always result in the same amount of LOBIs. And the amount of LOBIs withdrawn from the staking contract will always result in the same amount of LOBI.
rebase = 1 - (LOBI Deposits/sLOBI Outstanding)
The treasury deposits LOBI into the distributor. The distributor then deposits LOBI into the staking contract, creating an imbalance between LOBI and sLOBI. sLOBI is rebased to correct this imbalance between LOBI deposited and sLOBI outstanding. The rebase brings sLOBI outstanding back up to parity so that 1 sLOBI equals 1 staked LOBI.
bondPrice= Premium
In order to make a profit from minting, Lobis charges a premium for each minting action.
Premium = debtRatio * BCV
The premium determines profit due to the protocol and in turn, stakers. This is because new LOBI is minted from the profit and subsequently distributed among all stakers.
debtRatio = bondsOutstanding/LOBISupply
The debt ratio is the total of all LOBI promised to bonders divided by the total supply of LOBI. This allows us to measure the debt of the system.
bondPayoutreserveBond = marketValueasset / bondPrice
Bond payout determines the number of LOBI sold to a minter. For reserve mints, the market value of the assets supplied by the minter is used to determine the bond payout. For example, if a user supplies 1000 CRV and the mint price is 250 CRV, the user will be entitled 4 LOBI.
bondPayoutlpBond = marketValuelpToken/bondPrice
For liquidity mints, the market value of the LP tokens supplied by the minter is used to determine the bond payout. For example, if a user supplies 0.001 LOBI-OHM LP token which is valued at 1000 DAI (or any other stablecoin) at the time of bonding, and the bond price is 250 DAI (or any other stablecoin), the user will be entitled 4 LOBI.
LOBI Supply
LOBIsupplyGrowth = LOBIstakers + LOBIbonders + LOBIDAO +LOBIOlympus
LOBI supply does not have a hard cap. Its supply increases when:
LOBI is minted and distributed to the stakers.
LOBI is minted for the bonder. This happens whenever someone purchases a bond.
LOBI is minted for the DAO and for the Olympus DAO. This happens whenever someone purchases a bond. Additionally, 50% of the minted amount is redirected to the LOBIS DAO and an additional 1.1% of the minted amount goes to the Olympus DAO Treasury
LOBIstakers = LOBItotalSupply*rewardRate
At the end of each rebase, the treasury mints LOBI at a set reward rate. These LOBI will be distributed to all the stakers in the protocol.
LOBIbonders=bondPayout
Whenever someone purchases a bond, a set number of LOBI is minted. These LOBI will not be released to the minter all at once - they are vested to the bonder linearly over time. The bond payout uses a different formula for different types of bonds. Check the Minting section above to see how it is calculated.
Backing per LOBI
LOBIbacking = treasuryBalanceGovernanceTokens
Every LOBI in circulation is backed by the Lobis treasury. The assets in the treasury are Governance Tokens, so each LOBI is not just backed by the dollar value of the backing governance tokens but also by the governance rights derived by them.
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Dirichlet or periodic sinc function - MATLAB diric
Periodic and Aperiodic Sinc Functions
y = diric(x,n)
y = diric(x,n) returns the Dirichlet Function of degree n evaluated at the elements of the input array x.
Compute and plot the Dirichlet function between
-2\pi
2\pi
for N = 7 and N = 8. The function has a period of
2\pi
for odd N and
4\pi
for even N.
d7 = diric(x,7);
plot(x/pi,d7)
ylabel('N = 7')
title('Dirichlet Function')
xlabel('x / \pi')
The Dirichlet and sinc functions are related by
{D}_{N}\left(\pi x\right)=sinc\left(Nx/2\right)/sinc\left(x/2\right)
. Show this relationship for
N=6
. Avoid indeterminate expressions by specifying that the ratio of sinc functions is
{\left(-1\right)}^{\mathit{k}\left(\mathit{N}-1\right)}
\mathit{x}=2\mathit{k}
\mathit{k}
x = linspace(-xmax,xmax,1001)';
yd = diric(x*pi,N);
ys = sinc(N*x/2)./sinc(x/2);
ys(~mod(x,2)) = (-1).^(x(~mod(x,2))/2*(N-1));
plot(x,yd)
title('D_6(x*pi)')
plot(x,ys)
title('sinc(6*x/2) / sinc(x/2)')
Repeat the calculation for
\mathit{N}=13
title('D_{13}(x*pi)')
title('sinc(13*x/2) / sinc(x/2)')
real scalar | real vector | real matrix | real N-D array
Input array, specified as a real scalar, vector, matrix, or N-D array. When x is nonscalar, diric is an element-wise operation.
n — Function degree
Function degree, specified as a positive integer scalar.
Output array, returned as a real-valued scalar, vector, matrix, or N-D array of the same size as x.
The Dirichlet function, or periodic sinc function, is
{D}_{N}\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}\left(Nx/2\right)}{N\mathrm{sin}\left(x/2\right)}\hfill & x\ne 2\pi k,\text{ }k=0,±1,±2,±3,...\hfill \\ {\left(-1\right)}^{k\left(N-1\right)}\hfill & x=2\pi k,\text{ }k=0,±1,±2,±3,...\hfill \end{array}
for any nonzero integer N.
This function has period 2π for odd N and period 4π for even N. Its maximum value is 1 for all N, and its minimum value is –1 for even N. The magnitude of the function is 1/N times the magnitude of the discrete-time Fourier transform of the N-point rectangular window.
cos | gauspuls | pulstran | rectpuls | sawtooth | sin | sinc | square | tripuls
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Computational Design of Corrosion-Resistant Fe–Cr–Ni–Al Nanocoatings for Power Generation | J. Eng. Gas Turbines Power | ASME Digital Collection
, 6220 Culebra Road, San Antonio, TX 78238
Chan, K. S., Liang, W., Cheruvu, N. S., and Gandy, D. W. (March 4, 2010). "Computational Design of Corrosion-Resistant Fe–Cr–Ni–Al Nanocoatings for Power Generation." ASME. J. Eng. Gas Turbines Power. May 2010; 132(5): 052101. https://doi.org/10.1115/1.3204651
A computational approach has been undertaken to design and assess potential Fe–Cr–Ni–Al systems to produce stable nanostructured corrosion-resistant coatings that form a protective, continuous scale of alumina or chromia at elevated temperatures. The phase diagram computation was modeled using the THERMO-CALC® software and database (Thermo-Calc® Software, 2007, THERMO-CALC for Windows Version 4, Thermo-Calc Software AB, Stockholm, Sweden; Thermo-Calc® Software, 2007, TCFE5, Version 5, Thermo-Calc Software AB, Stockholm, Sweden) to generate pseudoternary Fe–Cr–Ni–Al phase diagrams to help identify compositional ranges without the undesirable brittle phases. The computational modeling of the grain growth process, sintering of voids and interface toughness determination by indentation, assessed microstructural stability, and durability of the nanocoatings fabricated by a magnetron-sputtering process. Interdiffusion of Al, Cr, and Ni was performed using the DICTRA® diffusion code (Thermo-Calc Software®, DICTRA, Version 24, 2007, Version 25, 2008, Thermo-Calc Software AB, Stockholm, Sweden) to maximize the long-term stability of the nanocoatings. The computational results identified a new series of Fe–Cr–Ni–Al coatings that maintain long-term stability and a fine-grained microstructure at elevated temperatures. The formation of brittle
σ
-phase in Fe–Cr–Ni–Al alloys is suppressed for Al contents in excess of
4 wt %
. The grain growth modeling indicated that the columnar-grained structure with a high percentage of low-angle grain boundaries is resistant to grain growth. Sintering modeling indicated that the initial relative density of as-processed magnetron-sputtered coatings could achieve full density after a short thermal exposure or heat-treatment. The interface toughness computation indicated that the Fe–Cr–Ni–Al nanocoatings exhibit high interface toughness in the range of
52–366J/m2
. The interdiffusion modeling using the DICTRA software package indicated that inward diffusion could result in substantial to moderate Al and Cr losses from the nanocoating to the substrate during long-term thermal exposures.
aluminium compounds, chromium compounds, corrosion protective coatings, corrosion resistance, gas turbines, iron compounds, mechanical engineering computing, nanostructured materials, nickel compounds, phase diagrams, steam turbines
Coatings, Computation, Corrosion, Design, Diffusion (Physics), Modeling, Phase diagrams, Sintering, Temperature, Fracture toughness, Coating processes, Grain boundaries, Aluminum coatings, Aluminum alloys
The Effect of Nanocrystallization on the Selective Oxidation and Adhesion of Al2O3 Scales
Oxidation Behaviour of Sputter-Deposited Ni–Cr–Al Micro-Crystalline Coatings
Oxidation Behaviours of Microcrystalline Ni–Cr–Al Alloy Coatings at 900°C
G. -F.
The Effect of Nanocrystallization on the Oxidation Resistance of Ni–5Cr–5Al Alloy
Oxidation Behavior of Sputtered Ni–3Cr–20 Al Nanocrystalline Coating
Improved Oxide Spallation Resistance of Microcrystalline Ni–Cr–Al Coatings
Oxidation Behavior of Sputtered Ni–Cr–Al–Ti Nanocrystalline Coating
Effect of Nanocrystallization on the Oxidation Behavior of a Ni–8Cr–3.5 Al Alloy
Ajdelsztajn
Shoenung
Synthesis and Oxidation Behavior of Nanocrystalline MCrAlY Bond Coats
Nanostructured Coatings by Pulsed Plasma Processing for Alloys used in Coal-Fired Environments
,” Department of Energy, Small Business Technology Transfer (STTR) Program, SwRI Fourth Quarterly Report to Karta Technologies, Report No. DE-FG02-5ER 86249.
M. -S.
Cyclic Oxidation of Sputter-Deposited Nanocrystalline Fe–Cr–Ni–Al Alloy Coatings
Oxidation Behaviour of Nanocrystaliine Fe–Ni–Cr–Al Alloy Coatings
Micro-Crystalline Fe–Cr–Ni–Al–Y2O3 ODS Alloy Coatings Produced by High Frequency Electric-Spaak Deposition
Thermo-Calc® Software
, 2007, THERMO-CALC for Windows Version 4, Thermo-Calc Software AB, Stockholm, Sweden.
, 2007, TCFE5, Version 5, Thermo-Calc Software AB, Stockholm, Sweden.
, DICTRA, Version 24, Thermo-Calc Software AB, Stockholm, 2007; Version 25, 2008.
Lifetimes Modeling of High-Temperature Corrosion Processes
, EFC Publications No.
High-Temperature Corrosion Resistance of Candidate FeAlCr Coatings in Low NOx Environments
Proceedings of the 19th Annual Conference on Fossil Energy Materials
, Knoxville, TN, May 9–11.
Oxidation Mechanism of Fe–Ni–20–25Cr–5Al Alloys-Influence of Small Amounts of Yttrium on Oxidation Kinetics and Oxide Adherence
Formation of Aluminum Oxide Scales in Sulfur-Containing High Temperature Environments
Karaminezhaad
Kordzadeh
The Effect of Nickel and Aluminum Addition on Oxidation Behavior of Austenitic Heat Resistance Steels
, Paper 4. 0002-7820
Alumina Surface Layer Formed by High Temperature Heat-Treatment of Fe-Cr-Ni-Al Alloy Intended for Blade Material
J. High-Temp. Soc.
Nano- and Microcrystal Coatings and Their High-Temperature Applications
The Use of Model Alloys to Develop Corrosion-Resistant Stainless Steels
Creep-Resistant Al2O3-Forming Austenitic Stainless Steels
Long-Term Performance of Aluminide Coatings on Fe-Based Alloys
On the Theory of Normal and Abnormal Grain Growth
Diffusion Sintering: I, Initial Stage Sintering Models and Their Application to Shrinkage of Powder Compacts
Measurement of the Adhesion of a Brittle Film on a Ductile Substrate by Indentation
Interfacial Toughness Measurements for Thin Films on Substrates
Krutenat
Diffusion Coatings of Steels: Formation Mechanism and Microstructure of Aluminized Heat-Resistant Stainless Steels
Interdiffusional Degradatino of Oxidation-Resistant Aluminide Coatings on Fe-Based Alloys
Mechanisms of Breakaway Oxidation and Application to a Chromia-Forming Steel
Effect of Cr and Ni Contents on the Oxidation Behavior of Ferritic and Austenitic Model Alloys in Air With Water Vapor
Laboratory-Simulated Fuel-Ash Corrosion of Superheater Tubes in Coal-Fired Ultra-Supercritical-Boilers
Update on Fireside Corrosion Resistance of Advanced Materials for Ultra-Supercritical Coal-Fired Power Plants
,” Presented at the 31st International Technical Conference on Coal Utilization & Fuel Systems, Sand Key Island, FL, May 1–26.
Properties and Corrosion Behavior of Chromium and Vanadium Carbide Composite Coatings Produced on Ductile Cast Iron by Thermoreactive Diffusion Technique
Formation of Highly Aligned Grooves on Inner Surface of Semipermeable Hollow Fiber Membrane for Directional Axonal Outgrowth
Emphasizing Conceptualization and Innovation in the Product Realization Process
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You see a curve describing a periodical function
f
2\pi
. Your task is to find the first terms of the Fourier series development of this function.
You can make several attempts to find these terms. For each attempt you will see the curve of your "development"
g
f
. You can modify your answer to make the curves as similar as possible. There is also the curve of remainder
f-g
, which allows you to see the errors.
Choose your type of development :
{A}_{n}\mathrm{cos}\left(nx+{p}_{n}\right)
{a}_{n}\mathrm{cos}\left(nx\right)+{b}_{n}\mathrm{sin}\left(nx\right)
Attention. for the same order, the development in the form
{a}_{n}\mathrm{cos}\left(nx\right)+{b}_{n}\mathrm{sin}\left(nx\right)
is more difficult to find than in the form amplitude and phase.
Description: graphical search of Fourier development of a function. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
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Laminar Natural Convection of Power-Law Fluids in a Square Enclosure With Differentially Heated Sidewalls Subjected to Constant Wall Heat Flux | J. Heat Transfer | ASME Digital Collection
Liverpool, L69 3GH
e-mail: osmanturan@ktu.edu.tr
e-mail: a.sachdeva@liv.ac.uk
Robert J. Poole,
e-mail: robpoole@liv.ac.uk
Nilanjan Chakraborty 1
e-mail: nilanjan.chakraborty@ncl.ac.uk
Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received July 24, 2011; final manuscript received April 3, 2012; published online October 8, 2012. Assoc. Editor: Sujoy Kumar Saha.
Turan, O., Sachdeva, A., Poole, R. J., and Chakraborty, N. (December 8, 2012). "Laminar Natural Convection of Power-Law Fluids in a Square Enclosure With Differentially Heated Sidewalls Subjected to Constant Wall Heat Flux." ASME. J. Heat Transfer. December 2012; 134(12): 122504. https://doi.org/10.1115/1.4007123
Two-dimensional steady-state laminar natural convection of inelastic power-law non-Newtonian fluids in square enclosures with differentially heated sidewalls subjected to constant wall heat flux (CHWF) are studied numerically. To complement the simulations, a scaling analysis is also performed to elucidate the anticipated effects of Rayleigh number (Ra), Prandtl number (Pr) and power-law index (n) on the Nusselt number. The effects of
n
in the range 0.6 ≤ n ≤ 1.8 on heat and momentum transport are investigated for nominal values Ra in the range 103–106 and a Pr range of 10–105. In addition the results are compared with the constant wall temperature (CWT) configuration. It is found that the mean Nusselt number
Nu¯
increases with increasing values of Ra for both Newtonian and power-law fluids in both configurations. However, the
Nu¯
values for the vertical walls subjected to CWHF are smaller than the corresponding values in the same configuration with CWT (for identical values of nominal Ra, Pr and n). The
Nu¯
values obtained for power-law fluids with
n<1
n>1
) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal value of Ra due to strengthening (weakening) of convective transport. With increasing shear-thickening (i.e., n > 1) the mean Nusselt number
Nu¯
settles to unity (
Nu¯=1.0
) as heat transfer takes place principally due to thermal conduction. The effects of Pr are shown to be essentially negligible in the range 10–105. New correlations are proposed for the mean Nusselt number
Nu¯
for both Newtonian and power-law fluids.
natural convection, power-law fluid, Nusselt number, Rayleigh number, Prandtl number
Fluids, Heat flux, Natural convection, Rayleigh number, Boundary-value problems, Temperature, Heat conduction, Convection
Natural Convection in Rectangular Enclosures Heated From One Side and Cooled From Above
John Wiley Sons Inc.
Hydrodynamic Stability and Natural Convection in Ostwald–De Waele and Ellis Fluids: The Development of a Numerical Solution
Lamsaadi
Natural Convection of Non-Newtonian Power Law Fluids in a Shallow Horizontal Rectangular Cavity Uniformly Heated From Below
A Numerical Study on Natural Convective Heat Transfer of Pseudoplastic Fluids in a Square Cavity
Numer. Heat Transf. A
Natural Convection Heat Transfer of Microemulsion Phase-Change-Material Slurry in Rectangular Cavities Heated From Below and Cooled From Above
Yield Stress Effects on Rayleigh–Bénard Convection
Weakly Nonlinear Viscoplastic Convection
J. Non-Newt. Fluid Mech.
Thermal Convection of a Viscoplastic Liquid with High Rayleigh and Bingham Numbers
Rayleigh–Bénard Convection of Viscoelastic Fluids in Finite Domains
Transient Buoyant Convection of a Power Law Non-Newtonian Fluid in an Enclosure
Natural Convection in a Vertical Rectangular Cavity Filled With a Non-Newtonian Power Law Fluid and Subjected to a Horizontal Temperature Gradient
Num. Heat Transf.
On a Natural-Convection Benchmark Problem in Non-Newtonian Fluids
On a Physically Realizable Benchmark Problem in Internal Natural Convection
Laminar Unsteady Flows of Bingham Fluids: A Numerical Strategy and Some Benchmark Results
Laminar Natural Convection of Bingham Fluids in a Square Enclosure With Differentially Heated Side Walls
Aspect Ratio Effects in Laminar Natural Convection of Bingham Fluids in Rectangular Enclosures With Differentially Heated Side Walls
Laminar Natural Convection of Bingham Fluids in a Square Enclosure With Vertical Walls Subjected to Constant Heat Flux
Laminar Natural Convection of Power-Law Fluids in a Square Enclosure With Differentially Heated Side Walls Subjected to Constant Temperatures
Natural Convection in Power Law Fluids
Int. Comm. Heat Mass Transf.
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If you are a programmer, you should be aware of which programming languages will be in demand in the future. Every year, programming languages evolve, and as a result, the market for programming abilities evolves as well. Knowing what companies want today and in 2022 might assist you in making an informed decision about your future schooling or career path. Here are the highest-paying programming languages that you should be aware of!
1. GO Lang:
Go is a simple, trustworthy, and efficient open-source programming language that allows you to create simple, dependable, and efficient programs. Google invented Go in 2009, and it has since been used by Uber, SoundCloud, Netflix, and Dropbox. This language was designed in the style of the well-known C programming language.
Despite the fact that this programming language is still relatively young, it has quickly gained popularity among many coders due to its ease of use. It's akin to Python and JavaScript, and it's just as powerful as C++.
Go programmers earn between
110,000 and
115,000 per year on average.
This language was the first to allow the development of programs for macOS and iOS-based devices. Objective-C was derived from C and, like Go, incorporated Smalltalk-style messaging components. As a result, Objective-C is an object-oriented programming language. Swift, a more sophisticated version of the same language, took its place in 2014.
Despite this, Objective-C is fairly popular since there are several IOS programs on the market that were created in it and require maintenance. As a result, if you hire an iOS developer, they will almost probably be familiar with Objective-C.
Objective-C programmers earn between
100,000 and
JavaScript is the most popular programming language at the moment. It's commonly used in web development, front-end design, and back-end programming. JavaScript programmers are in great demand and will remain so in 2022. With JS, you can create nearly anything, from a web application to a neural network.
Previously, JavaScript was seen as a "minor language" that may be used to enhance websites. However, with the advent of NodeJS and its enhanced capability, JS has evolved into more than simply a supplement. It is now feasible to construct full-fledged applications or even a web browser using contemporary frameworks such as React or VueJS using JS.
JavaScript programmers earn an annual salary of $100,000 on average.
Learn JavaScript: https://codingcafe.co.in/
Python is a powerful programming language that may be used for scripting, data analysis, machine learning, and other tasks. As more firms use Python for their projects, the need for Python developers is rising. This language allows you to construct server-side web applications, desktop programs with graphical user interfaces, and machine learning algorithms. In addition to being easier to use for developing scripts than other programming languages such as C++ or Java, it is also a very effective tool that allows you to write fewer lines of code.
Python programmers get paid an average of $110,000 per year.
Although C++ has been around since 1985, it remains one of the most popular programming languages today. Because of its ability to generate quick and efficient code, C++ is used to program many video games. C++ developers are in great demand now and will be in 2022.
C++ can be found everywhere, including your computer, mobile phone, and toaster. As a result, you may create your microcontrollers and fully customized desktop apps, among other things. This is a lower level than the same Python or Javascript, which means it virtually directly talks with the computer and has tens of times the speed of its competitors.
C++ programmers earn an annual salary of $105,000 on average.
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Carbon fiber reinforced polymer - Material DB - RoHS - Reach
Carbon fiber reinforced polymer (9291 views - Material Database)
Carbon fiber reinforced polymer, carbon fiber reinforced plastic, or carbon fiber reinforced thermoplastic (CFRP, CRP, CFRTP, or often simply carbon fiber, carbon composite, or even carbon), is an extremely strong and light fiber-reinforced plastic which contains carbon fibers. The alternative spelling 'fibre' is common in British Commonwealth countries. CFRPs can be expensive to produce but are commonly used wherever high strength-to-weight ratio and stiffness (rigidity) are required, such as aerospace, superstructure of ships, automotive, civil engineering, sports equipment, and an increasing number of consumer and technical applications. The binding polymer is often a thermoset resin such as epoxy, but other thermoset or thermoplastic polymers, such as polyester, vinyl ester, or nylon, are sometimes used. The composite material may contain aramid (e.g. Kevlar, Twaron), ultra-high-molecular-weight polyethylene (UHMWPE), aluminium, or glass fibers in addition to carbon fibers. The properties of the final CFRP product can also be affected by the type of additives introduced to the binding matrix (resin). The most common additive is silica, but other additives such as rubber and carbon nanotubes can be used. The material is also referred to as graphite-reinforced polymer or graphite fiber-reinforced polymer (GFRP is less common, as it clashes with glass-(fiber)-reinforced polymer).
Light, strong, and rigid composite material
Carbon fiber reinforced polymer, carbon fiber reinforced plastic, or carbon fiber reinforced thermoplastic (CFRP, CRP, CFRTP, or often simply carbon fiber, carbon composite, or even carbon), is an extremely strong and light fiber-reinforced plastic which contains carbon fibers. The alternative spelling 'fibre' is common in British Commonwealth countries. CFRPs can be expensive to produce but are commonly used wherever high strength-to-weight ratio and stiffness (rigidity) are required, such as aerospace, superstructure of ships, automotive, civil engineering, sports equipment, and an increasing number of consumer and technical applications.
The binding polymer is often a thermoset resin such as epoxy, but other thermoset or thermoplastic polymers, such as polyester, vinyl ester, or nylon, are sometimes used. The composite material may contain aramid (e.g. Kevlar, Twaron), ultra-high-molecular-weight polyethylene (UHMWPE), aluminium, or glass fibers in addition to carbon fibers. The properties of the final CFRP product can also be affected by the type of additives introduced to the binding matrix (resin). The most common additive is silica, but other additives such as rubber and carbon nanotubes can be used. The material is also referred to as graphite-reinforced polymer or graphite fiber-reinforced polymer (GFRP is less common, as it clashes with glass-(fiber)-reinforced polymer).
Reinforcement gives CFRP its strength and rigidity; measured by stress and elastic modulus respectively. Unlike isotropic materials like steel and aluminum, CFRP has directional strength properties. The properties of CFRP depend on the layouts of the carbon fiber and the proportion of the carbon fibers relative to the polymer.[2] The two different equations governing the net elastic modulus of composite materials using the properties of the carbon fibers and the polymer matrix can also be applied to carbon fiber reinforced plastics.[3] The following equation,
{\displaystyle E_{c}=V_{m}E_{m}+V_{f}E_{f}}
{\displaystyle E_{c}}
{\displaystyle V_{m}}
{\displaystyle V_{f}}
{\displaystyle E_{m}}
{\displaystyle E_{f}}
{\displaystyle E_{c}=\left({\frac {V_{m}}{E_{m}}}+{\frac {V_{f}}{E_{f}}}\right)^{-1}}
Despite its high initial strength-to-weight ratio, a design limitation of CFRP is its lack of a definable fatigue limit. This means, theoretically, that stress cycle failure cannot be ruled out. While steel and many other structural metals and alloys do have estimable fatigue or endurance limits, the complex failure modes of composites mean that the fatigue failure properties of CFRP are difficult to predict and design for. As a result, when using CFRP for critical cyclic-loading applications, engineers may need to design in considerable strength safety margins to provide suitable component reliability over its service life.
Environmental effects such as temperature and humidity can have profound effects on the polymer-based composites, including most CFRPs. While CFRPs demonstrate excellent corrosion resistance, the effect of moisture at wide ranges of temperatures can lead to degradation of the mechanical properties of CFRPs, particularly at the matrix-fiber interface.[5] While the carbon fibers themselves are not affected by the moisture diffusing into the material, the moisture plasticizes the polymer matrix.[4] The epoxy matrix used for engine fan blades is designed to be impervious against jet fuel, lubrication, and rain water, and external paint on the composites parts is applied to minimize damage from ultraviolet light.[4][6]
The primary element of CFRP is a carbon filament; this is produced from a precursor polymer such as polyacrylonitrile (PAN), rayon, or petroleum pitch. For synthetic polymers such as PAN or rayon, the precursor is first spun into filament yarns, using chemical and mechanical processes to initially align the polymer chains in a way to enhance the final physical properties of the completed carbon fiber. Precursor compositions and mechanical processes used during spinning filament yarns may vary among manufacturers. After drawing or spinning, the polymer filament yarns are then heated to drive off non-carbon atoms (carbonization), producing the final carbon fiber. The carbon fibers filament yarns may be further treated to improve handling qualities, then wound on to bobbins.[8] From these fibers, a unidirectional sheet is created. These sheets are layered onto each other in a quasi-isotropic layup, e.g. 0°, +60°, or −60° relative to each other.
The Airbus A350 XWB is built of 52% CFRP[9] including wing spars and fuselage components, overtaking the Boeing 787 Dreamliner, for the aircraft with the highest weight ratio for CFRP, which is 50%.[10] This was one of the first commercial aircraft to have wing spars made from composites. The Airbus A380 was one of the first commercial airliners to have a central wing-box made of CFRP; it is the first to have a smoothly contoured wing cross-section instead of the wings being partitioned span-wise into sections. This flowing, continuous cross section optimises aerodynamic efficiency.[citation needed] Moreover, the trailing edge, along with the rear bulkhead, empennage, and un-pressurised fuselage are made of CFRP.[11] However, many delays have pushed order delivery dates back because of problems with the manufacture of these parts. Many aircraft that use CFRP have experienced delays with delivery dates due to the relatively new processes used to make CFRP components, whereas metallic structures have been studied and used on airframes for years, and the processes are relatively well understood. A recurrent problem is the monitoring of structural ageing, for which new methods are constantly investigated, due to the unusual multi-material and anisotropic nature of CFRP.[12]
CFRP is now widely used in sports equipment such as in squash, tennis, and badminton racquets, sport kite spars, high quality arrow shafts, hockey sticks, fishing rods, surfboards, high end swim fins, and rowing shells. Amputee athletes such as Jonnie Peacock use carbon fiber blades for running. It is used as a shank plate in some basketball sneakers to keep the foot stable, usually running the length of the shoe just above the sole and left exposed in some areas, usually in the arch.
Controversially, in 2006, cricket bats with a thin carbon-fiber layer on the back were introduced and used in competitive matches by high-profile players including Ricky Ponting and Michael Hussey. The carbon fiber was claimed merely to increase the durability of the bats but was banned from all first-class matches by the ICC in 2007.[21]
Musical instruments, including violin bows, guitar picks and pick-guards, drum shells, bagpipe chanters, and entire musical instruments such as Luis and Clark's carbon fiber cellos, violas, and violins; and Blackbird Guitars' acoustic guitars and ukuleles; also audio components such as turntables and loudspeakers.
Archery, carbon fiber arrows and bolts, stock and rail.
In 2009, Zyvex Technologies introduced carbon nanotube-reinforced epoxy and carbon pre-pregs.[29] Carbon nanotube reinforced polymer (CNRP) is several times stronger and tougher than CFRP and is used in the Lockheed Martin F-35 Lightning II as a structural material for aircraft.[30] CNRP still uses carbon fiber as the primary reinforcement, but the binding matrix is a carbon nanotube filled epoxy.[31]
Mechanics of Oscar Pistorius's running blades
AeronauticaProgettazione di automobiliCarbonioFibra di carbonioNanotubo di carbonioCarbonato di cobalto(II)Discus Launch GliderSport estremoGlider (aircraft)AlianteVolo a velaRadio-controlled gliderNave
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Lemma 10.71.1 (00LP)—The Stacks project
Lemma 10.71.1. Let $R$ be a ring. Let $M$ be an $R$-module.
There exists an exact complex
\[ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0. \]
with $F_ i$ free $R$-modules.
If $R$ is Noetherian and $M$ finite over $R$, then we can choose the complex such that $F_ i$ is finite free. In other words, we can find an exact complex
\[ \ldots \to R^{\oplus n_2} \to R^{\oplus n_1} \to R^{\oplus n_0} \to M \to 0. \]
Proof. Let us explain only the Noetherian case. As a first step choose a surjection $R^{n_0} \to M$. Then having constructed an exact complex of length $e$ we simply choose a surjection $R^{n_{e + 1}} \to \mathop{\mathrm{Ker}}(R^{n_ e} \to R^{n_{e-1}})$ which is possible because $R$ is Noetherian. $\square$
In 2, replace "finite
R
" by "finite over
R
". Moreover, 2 seems to have too much hypotheses: If
R
is coherent, then
M
is finitely presented if and only if there is such an exact complex such that each
F_i
is finite free. The proof is the same as above, with "because
R
is Noetherian" replaced by "by 05CW and 05CX". (Unfortunalty, this needs the notion of and results about coherence appearing only in 05CU. It should probably be put after 05CX. The noetherian statement follows then from 05CY.)
Added "over" and made some cosmetic changes, see here.
Maybe the material about coherent modules over coherent rings should be put in the chapter More on Algebra? Somewhere in the section on pseudo-coherent modules? In that language it says that a coherent module over a coherent ring is pseudo-coherent. In any case I think we leave this lemma like it is now and we add an additional one to cover the coherent case. OK?
Oh, I have not seen that chapter. So, the coherent case might be incorporated in 066E.
Yes, 15.64.17 is a good place for it.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00LP. Beware of the difference between the letter 'O' and the digit '0'.
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Synthetic seismogram with multiples - SEG Wiki
Previously, we discussed the synthetic trace without multiples, and we have just looked at multiple water reverberations in the topmost water layer. We now turn to the more general case of a synthetic seismogram for a model which can produce multiples in any of its layers. The details of generating a synthetic trace with multiples were given in Robinson and Treitel (1978)[1], who also described an essential mathematical simplification that holds for the case of small reflection coefficients. The term small is defined in a relative manner that depends on the circumstances at hand. In some cases, small might mean a magnitude less than 0.2, whereas in other cases, it might mean a magnitude less than 0.05. Here, we present a method of generating a synthetic trace with multiples in such a way as to bring out what the mathematics is doing, rather than in terms of a strict mathematical derivation. The key concepts are feedforward and feedback.
We start with the case of only two interfaces (namely, the surface and the first subsurface interface). They are separated by a vertical distance with two-way traveltime T = 1. The reflectivity is (
{\displaystyle \varepsilon _{0}}
{\displaystyle \varepsilon _{1}}
). Its one-sided autocorrelation (with the zero-lag term set equal to unity) is (1,
{\displaystyle \varepsilon _{0}}
{\displaystyle \varepsilon _{1}}
), which we denote by (1,
{\displaystyle g_{1}}
{\displaystyle g_{l}={\varepsilon }_{0}{\varepsilon }_{l}}
We recognize that the current problem is the same as the reverberation problem treated above, but now with two-way time T = 1. Thus, the impulse response h (with multiples) in the case of two interfaces has the Z-transform
{\displaystyle H\left(Z\right)={\varepsilon }_{1}Z+{\varepsilon }_{1}\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)Z^{2}+{\varepsilon }_{1}{\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)}^{2}Z^{2}+}
{\displaystyle {\begin{aligned}={\varepsilon }_{1}Z\left[{1}+\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)Z+{\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)}^{2}Z^{2}+\dots \right]={\frac {{\varepsilon }_{1}Z}{{1}+{\varepsilon }_{0}{\varepsilon }_{1}Z}}.\end{aligned}}}
We recall that the impulse response (without multiples) is generated by a purely feedforward system (with the reflection coefficients on the feedforward loops). Here, the expression for the Z-transform shows that the impulse response (with multiples) is generated by a feedforward-feedback system (with the reflection coefficients on the feedforward loops and the autocorrelation coefficients on the feedback loops). Equation 22 , which is for the case of one layer, is exact. In other words, equation 22 gives the Z-transform of the impulse response for the dynamic model (equation 1).
Next let us consider the case of three interfaces. What is the impulse response (with multiples) for three interfaces? The reflectivity is then
{\displaystyle \left({\varepsilon }_{0}{,\ }{\varepsilon }_{1}{,\ }{\varepsilon }_{2}\right)}
. Its one-sided autocorrelation (with the zero-lag term set equal to unity) is
{\displaystyle {\begin{aligned}\left({1,\ }g_{1},g_{2}\right)=\left({1,\ }{\varepsilon }_{0}{\varepsilon }_{1}+{\varepsilon }_{1}{\varepsilon }_{2}{,\ }{\varepsilon }_{0}{\varepsilon }_{2}\right).\end{aligned}}}
By analogy with the result for two interfaces given above, the synthetic impulse response h (with multiples) in the case of three interfaces has the Z-transform
{\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}}{{1}+g_{1}Z+g_{2}Z^{2}}}.\end{aligned}}}
However, expression 24, obtained by analogy, is not exact but instead is an approximation to the true dynamic model. Equation 24 leads to the synthetic trace with multiples (equation 2).
Now let us consider the case of four interfaces. What is the impulse response (with multiples) for four interfaces? The reflectivity is then
{\displaystyle \left({\varepsilon }_{0}{,\ }{\varepsilon }_{1}{,\ }{\varepsilon }_{2}{,\ }{\varepsilon }_{2}\right)}
{\displaystyle {\begin{aligned}\left({1,\ }g_{1},g_{2},g_{3}\right)=\left({1,\ }{\varepsilon }_{0}{\varepsilon }_{1}+{\varepsilon }_{1}{\varepsilon }_{2}+{\varepsilon }_{2}{\varepsilon }_{3}{,\ }{\varepsilon }_{0}{\varepsilon }_{2}+{\varepsilon }_{1}{\varepsilon }_{3}{,\ }{\varepsilon }_{0}{\varepsilon }_{3}\right).\end{aligned}}}
By analogy, the synthetic impulse response (with multiples) in the case of four interfaces has the Z-transform
{\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+{\varepsilon }_{3}Z^{3}}{{1}+g_{1}Z+g_{2}Z^{2}+g_{3}Z^{3}}}.\end{aligned}}}
However, expression 26, obtained by analogy, is not exact but is an approximation to the true dynamic model. Expression 26 can be diagrammed as the feedforward-feedback system shown in Figure 10.
Figure 10. The feedforward-feedback filter for a four-interface case.
By analogy, the synthetic impulse response (with multiples) for N interfaces has the Z-transform
{\displaystyle {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots {.+.}{\varepsilon }_{N}Z^{N}}{{1+}g_{1}Z+g_{2}Z^{2}+.+g_{N}Z^{N}}}.\end{aligned}}}
As before, this expression is an approximation to the dynamic model. Let us define E(Z) and G(Z) as
{\displaystyle {\begin{aligned}E\left(Z\right)={\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots +{\varepsilon }_{N}Z^{N}\mathrm {\ and\ } G\left(Z\right)={1}+g_{1}Z+g_{2}Z^{2}+\dots +g_{N}Z^{N}.\end{aligned}}}
{\displaystyle {\begin{aligned}H\left(Z\right)={\frac {E\left(Z\right)}{G\left(Z\right)}}=E\left(Z\right)M\left(Z\right)\mathrm {\ where\ } M\left(Z\right)={\frac {1}{G\left(Z\right)}}=1+m_{1}Z+m_{2}Z^{2}+\ldots .\end{aligned}}}
For equation 29 to be true, the denominator polynomial G(Z) must be a minimum-delay polynomial. In such a case, we have the approximation
{\displaystyle h=m*\varepsilon }
{\displaystyle x=s*m*\varepsilon }
. Thus, equation 2 holds for the synthetic seismogram with multiples.
Let us now discuss the approximation that is required. It is called the small-reflection-coefficient approximation, and it was introduced by Robinson and Treitel (1978[1]) and further developed by Robinson (1982[2], 1999). The approximation requires that the reflection coefficients be small enough to make the denominator polynomial G(Z) a minimum-delay polynomial. Thus, before this approximation can be used, the denominator polynomial must be tested for minimum delay. If the denominator polynomial is minimum delay, polynomial division can be used to find the coefficients
{\displaystyle m=(1,m_{1},m_{2},m_{2},....)}
{\displaystyle M\left(Z\right)}
. If the denominator polynomial is not minimum delay, the approximation fails, and the exact dynamic expression (i.e., the expression obtained without the small-reflection-coefficient approximation) must be used. In fact, it is always prudent to use the exact dynamic model (equation 1) in computations (Robinson, 1999[3]).
If the small-reflection-coefficient approximation holds, then the approximation
{\displaystyle H\left(Z\right)=E\left(Z\right)M\left(Z\right)}
holds, and the multiples
{\displaystyle M\left(Z\right)}
can be removed by ordinary (i.e., linear time-invariant) deconvolution. In other words, the small-reflection-coefficient approximation justifies the use of ordinary deconvolution. For this reason, a sequence of time gates is chosen on the actual seismic trace. The choice is made on the supposition that within each gate, the small-reflection-coefficient approximation holds. A deconvolution operator is computed for each gate. The deconvolved trace is made up of all the deconvolved gates along with appropriate interpolation between any two adjacent gates.
Water reverberations Examples
Wavelets Wavelet Processing
Synthetic seismogram with multiples/en
↑ 1.0 1.1 Robinson, E. A., and S. Treitel, 1978, The fine structure of the normal incidence synthetic seismogram: Geophysical Journal International, 53, no. 2, 289–309.
↑ Robinson, E. A., 1982, Spectral approach to geophysical inversion by Lorentz, Fourier, and Radon transforms: Proceedings of the IEEE, 70, 1039–1054.
↑ Robinson, E. A., 1999, Seismic inversion and deconvolution: Handbook of geophysical exploration, 4B: Elsevier.
Retrieved from "https://wiki.seg.org/index.php?title=Synthetic_seismogram_with_multiples/en&oldid=168772"
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Modeling Election Problem by a Stochastic Differential Equation
1Faculty of Fundamental Science, Military Academy of Logistics, Hanoi, Vietnam
2Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Hanoi, Vietnam
\frac{\text{d}n\left(t\right)}{\text{d}t}=E\left(N-n\left(t\right)\right)+Mn\left(t\right)\left(N-n\left(t\right)\right)
x\left(t\right)=\frac{n\left(t\right)}{N}\in \left[0;1\right]
b=M\cdot N
\sigma a\text{d}W\left(t\right)
\text{d}x\left(t\right)=\left[a\left(1-x\left(t\right)\right)+bx\left(t\right)\left(1-x\left(t\right)\right)\right]\text{d}t+\sigma a\left(1-x\left(t\right)\right)\text{d}W\left(t\right)
\text{d}x\left(t\right)=\left[a\left(1-x\left(t\right)\right)+bx\left(t\right)\left(1-x\left(t\right)\right)\right]\text{d}t+\sigma a\left(1-x\left(t\right)\right)\text{d}W\left(t\right)
1-x\left(t\right)=X\left(t\right)
-\text{d}x\left(t\right)=\text{d}X\left(t\right)
\text{d}X\left(t\right)=-X\left(t\right)\left[a+b\left(1-X\left(t\right)\right)\right]\text{d}t+\sigma aX\left(t\right)\text{d}W\left(t\right)
\text{d}X\left(t\right)=\left[b{X}^{2}\left(t\right)-\left(a+b\right)X\left(t\right)\right]\text{d}t-\sigma aX\left(t\right)\text{d}W\left(t\right)
\text{d}X\left(t\right)=\left[\alpha {X}^{2}\left(t\right)+\beta X\left(t\right)\right]\text{d}t+\lambda X\left(t\right)\text{d}W\left(t\right)
\alpha ,\beta
\lambda
X\left(t\right)
Y\left(t\right)={X}^{-1}\left(t\right)=\frac{1}{X\left(t\right)}
\begin{array}{c}\text{d}Y\left(t\right)=-\frac{1}{{X}^{2}\left(t\right)}\text{d}X\left(t\right)+\frac{1}{2}\frac{2}{{X}^{3}\left(t\right)}{\lambda }^{2}{X}^{2}\left(t\right)\text{d}t\\ =-\frac{1}{{X}^{2}\left(t\right)}\left[\left(\alpha {X}^{2}\left(t\right)+\beta X\left(t\right)\right)\text{d}t+\lambda X\left(t\right)\text{d}W\left(t\right)\right]+\frac{{\lambda }^{2}}{X\left(t\right)}\text{d}t\\ =-\left(\alpha +\frac{\beta }{X\left(t\right)}\right)\text{d}t-\frac{\lambda }{X\left(t\right)}\text{d}W\left(t\right)+\frac{{\lambda }^{2}}{X\left(t\right)}\text{d}t\\ =\left(\frac{{\lambda }^{2}-\beta }{X\left(t\right)}-\alpha \right)\text{d}t-\frac{\lambda }{X\left(t\right)}\text{d}W\left(t\right)\end{array}
Y\left(t\right)
\text{d}Y\left(t\right)=\left(\left({\lambda }^{2}-\beta \right)Y\left(t\right)-\alpha \right)\text{d}t-\lambda Y\left(t\right)\text{d}W\left(t\right)
\text{d}Y\left(t\right)=\left({a}_{1}Y\left(t\right)+{c}_{1}\right)\text{d}t+\left({b}_{1}Y\left(t\right)+{d}_{1}\right)\text{d}W\left(t\right)
Y\left(t\right)=\varphi \left(t\right)\left[Y\left(0\right)+\left({c}_{1}-{b}_{1}{d}_{1}\right){\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}+{d}_{1}{\int }_{0}^{t}\frac{\text{d}W\left(s\right)}{\varphi \left(s\right)}\right]
\varphi \left(t\right)=\mathrm{exp}\left\{\left({a}_{1}-\frac{{b}_{1}^{2}}{2}\right)t+{b}_{1}W\left(t\right)\right\}
\varphi \left(t\right)
\text{d}\varphi \left(t\right)={a}_{1}\varphi \left(t\right)\text{d}t+{b}_{1}\varphi \left(t\right)\text{d}W\left(t\right)
\varphi \left(t\right)
\varphi \left(t\right)=\mathrm{exp}\left\{\left({a}_{1}-\frac{{b}_{1}^{2}}{2}\right)t+{b}_{1}W\left(t\right)\right\}
Z\left(t\right)
Z\left(t\right)=\frac{Y\left(t\right)}{\varphi \left(t\right)}
\begin{array}{c}\text{d}Z\left(t\right)=\frac{1}{\varphi \left(t\right)}\text{d}Y\left(t\right)-\frac{1}{{\varphi }^{2}\left(t\right)}Y\left(t\right)\text{d}\varphi \left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{2}\frac{2}{{\varphi }^{2}\left(t\right)}\text{d}Y\left(t\right)\text{d}\varphi \left(t\right)+\frac{1}{2}\frac{2Y\left(t\right)}{{\varphi }^{3}\left(t\right)}\text{d}\varphi \left(t\right)\text{d}\varphi \left(t\right)\\ =\frac{1}{\varphi \left(t\right)}\text{d}Y\left(t\right)-\frac{1}{{\varphi }^{2}\left(t\right)}Y\left(t\right)\left[{a}_{1}\varphi \left(t\right)\text{d}t+{b}_{1}\varphi \left(t\right)\text{d}W\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{{\varphi }^{2}\left(t\right)}\left[\left({a}_{1}Y\left(t\right)+{c}_{1}\right)\text{d}t+\left({b}_{1}Y\left(t\right)+{d}_{1}\right)\text{d}W\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times \left[{a}_{1}\varphi \left(t\right)\text{d}t+{b}_{1}\varphi \left(t\right)\text{d}W\left(t\right)\right]+\frac{Y\left(t\right)}{{\varphi }^{3}\left(t\right)}\text{d}\varphi \left(t\right)\text{d}\varphi (t)\end{array}
\begin{array}{c}\text{d}Z\left(t\right)=\frac{1}{\varphi \left(t\right)}\text{d}Y\left(t\right)-\frac{{a}_{1}Y\left(t\right)}{\varphi \left(t\right)}\text{d}t-\frac{{b}_{1}Y\left(t\right)}{\varphi \left(t\right)}\text{d}W\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{\varphi \left(t\right)}\left[0+0+0+\left({b}_{1}^{2}Y\left(t\right)+{b}_{1}{d}_{1}\right)\text{d}t\right]+\frac{Y\left(t\right)}{\varphi \left(t\right)}{b}_{1}^{2}\text{d}t\end{array}
\begin{array}{c}\text{d}Z\left(t\right)=\frac{\left(\left({a}_{1}Y\left(t\right)+{c}_{1}\right)\text{d}t+\left({b}_{1}Y\left(t\right)+{d}_{1}\right)\text{d}W\left(t\right)\right)}{\varphi \left(t\right)}-\frac{{a}_{1}Y\left(t\right)}{\varphi \left(t\right)}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{b}_{1}Y\left(t\right)}{\varphi \left(t\right)}\text{d}W\left(t\right)-\frac{\left({b}_{1}^{2}Y\left(t\right)+{b}_{1}{d}_{1}\right)}{\varphi \left(t\right)}\text{d}t+\frac{{b}_{1}^{2}Y\left(t\right)}{\varphi \left(t\right)}\text{d}t\\ =\frac{{d}_{1}}{\varphi \left(t\right)}\text{d}W\left(t\right)+\frac{\left({c}_{1}-{b}_{1}{d}_{1}\right)}{\varphi \left(t\right)}\text{d}t\end{array}
Z\left(t\right)
Z\left(t\right)=Z\left(0\right)+\left({c}_{1}-{b}_{1}{d}_{1}\right){\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}+{d}_{1}{\int }_{0}^{t}\frac{\text{d}W\left(s\right)}{\varphi \left(s\right)}
Z\left(0\right)=\frac{Y\left(0\right)}{\varphi \left(0\right)}=Y\left(0\right)
Y\left(t\right)=\varphi \left(t\right)\left[Y\left(0\right)+\left({c}_{1}-{b}_{1}{d}_{1}\right){\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}+{d}_{1}{\int }_{0}^{t}\frac{\text{d}W\left(s\right)}{\varphi \left(s\right)}\right]
Y\left(t\right)=\varphi \left(t\right)\left[Y\left(0\right)-\alpha {\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}\right]
\varphi \left(t\right)=\mathrm{exp}\left\{\left(\frac{{\lambda }^{2}}{2}-\beta \right)t-\lambda W\left(t\right)\right\}
{a}_{1}
{\lambda }^{2}-\beta
{c}_{1}
-\alpha
{b}_{1}
-\lambda
{d}_{1}=0
X\left(t\right)={\varphi }^{-1}\left(t\right){\left[\frac{1}{X\left(0\right)}-\alpha {\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}\right]}^{-1}
\varphi \left(t\right)=\mathrm{exp}\left\{\left(\frac{{\lambda }^{2}}{2}-\beta \right)t-\lambda W\left(t\right)\right\}
Y\left(t\right)
\frac{1}{X\left(t\right)}
X\left(t\right)={\varphi }^{-1}\left(t\right){\left[\frac{1}{X\left(0\right)}-b{\int }_{0}^{t}\frac{\text{d}s}{\varphi \left(s\right)}\right]}^{-1}
\varphi \left(t\right)=\mathrm{exp}\left\{\left(\frac{{\sigma }^{2}{a}^{2}}{2}+a+b\right)t+\sigma aW\left(t\right)\right\}
x\left(t\right)
a=0.5;b=0.2
\sigma =0.3
x\left(t\right)
a=0.3;b=-0.6
\sigma =0.3
x\left(t\right)
a=-0.3;b=0.6
\sigma =0.3
x\left(t\right)
a=-0.3;b=-0.6
\sigma =0.1
Trung, N.T. (2018) Modeling Election Problem by a Stochastic Differential Equation. American Journal of Operations Research, 8, 441-447. https://doi.org/10.4236/ajor.2018.86024
1. Chen, E., Simonovits, G., Krosnick, J.A. and Pasek, J. (2014) The Impact of Candidate Name Order on Election Outcomes in North Dakota. Electoral Studies, 35, 115-122. https://doi.org/10.1016/j.electstud.2014.04.018
2. Ksendal, B. (2003) Stochastic Differential Equations: An Introduction with Applications. Vol. XXVII, Springer, Berlin, 379 p. https://doi.org/10.1007/978-3-642-14394-6
3. Arnold, L. (1974) Stochastic Differential Equations: Theory and Applications. Wiley, Hoboken, New Jersey.
4. Malliaris, A.G. (1983) Ito’s Calculus in Financial Decision making. SIAM Review, 25, 481-496. https://doi.org/10.1137/1025121
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Inverse_Pythagorean_theorem Knowpia
In geometry, the inverse Pythagorean theorem is as follows:[1]
Comparison of the inverse Pythagorean theorem with the Pythagorean theorem using the smallest positive integer inverse-Pythagorean triple in the table below
(3, 4, 5) 20 = 4× 5 15 = 3× 5 12 = 3× 4 25 = 52
(5, 12, 13) 156 = 12×13 65 = 5×13 60 = 5×12 169 = 132
(8, 15, 17) 255 = 15×17 136 = 8×17 120 = 8×15 289 = 172
(20, 21, 29) 609 = 21×29 580 = 20×29 420 = 20×21 841 = 292
All positive integer primitive inverse-Pythagorean triples having up to
three digits, with the hypotenuse for comparison
Let A, B be the endpoints of the hypotenuse of a right triangle ABC. Let D be the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse. Then
{\displaystyle {\frac {1}{CD^{2}}}={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}.}
This theorem should not be confused with proposition 48 in book 1 of Euclid's Elements, the converse of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle.
The area of triangle ABC can be expressed in terms of either AC and BC, or AB and CD:
{\displaystyle {\begin{aligned}{\tfrac {1}{2}}AC\cdot BC&={\tfrac {1}{2}}AB\cdot CD\\(AC\cdot BC)^{2}&=(AB\cdot CD)^{2}\\{\frac {1}{CD^{2}}}&={\frac {AB^{2}}{AC^{2}\cdot BC^{2}}}\end{aligned}}}
given CD > 0, AC > 0 and BC > 0.
{\displaystyle {\begin{aligned}{\frac {1}{CD^{2}}}&={\frac {BC^{2}+AC^{2}}{AC^{2}\cdot BC^{2}}}\\&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\\quad \therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}}
Special case of the cruciform curveEdit
The cruciform curve or cross curve is a quartic plane curve given by the equation
{\displaystyle x^{2}y^{2}-b^{2}x^{2}-a^{2}y^{2}=0}
where the two parameters determining the shape of the curve, a and b are each CD.
Substituting x with AC and y with BC gives
{\displaystyle {\begin{aligned}AC^{2}BC^{2}-CD^{2}AC^{2}-CD^{2}BC^{2}&=0\\AC^{2}BC^{2}&=CD^{2}BC^{2}+CD^{2}AC^{2}\\{\frac {1}{CD^{2}}}&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\\therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}}
Inverse-Pythagorean triples can be generated using integer parameters t and u as follows.[2]
{\displaystyle {\begin{aligned}AC&=(t^{2}+u^{2})(t^{2}-u^{2})\\BC&=2tu(t^{2}+u^{2})\\CD&=2tu(t^{2}-u^{2})\end{aligned}}}
If two identical lamps are placed at A and B, the theorem and the inverse-square law imply that the amount of light received at C is the same as when a single lamp is placed at D.
Pythagorean theorem – Relation between sides of a right triangle
^ Johan Wästlund, "Summing inverse squares by euclidean geometry", http://www.math.chalmers.se/~wastlund/Cosmic.pdf, pp. 4–5.
^ "Diophantine equation of three variables".
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Volume 105 Issue 2B | Bulletin of the Seismological Society of America | GeoScienceWorld
February - Volume 105, Number 1 April - Volume 105, Number 2A May - Volume 105, Number 2B June - Volume 105, Number 3 August - Volume 105, Number 4 October - Volume 105, Number 5 December - Volume 105, Number 6
Introduction to the Special Issue on the 2012 Haida Gwaii and 2013 Craig Earthquakes at the Pacific–North America Plate Boundary (British Columbia and Alaska)
Thomas S. James; John F. Cassidy; Garry C. Rogers; Peter J. Haeussler
Kristin M. M. Rohr
Maureen A. L. Walton; Sean P. S. Gulick; Peter J. Haeussler; Emily C. Roland; Anne M. Tréhu
Shear‐Wave Velocity Structure in the Vicinity of the 2012 Mw 7.8 Haida Gwaii Earthquake from Receiver Function Inversion
Jeremy M. Gosselin; John F. Cassidy; Stan E. Dosso
Anne M. Tréhu; Maren Scheidhauer; Kristin M. M. Rohr; Basil Tikoff; Maureen A. L. Walton; Sean P. S. Gulick; Emily C. Roland
Bulletin of the Seismological Society of America March 03, 2015, Vol.105, 1114-1128. doi:https://doi.org/10.1785/0120140159
K. Aderhold; R. E. Abercrombie
Stephen Holtkamp; Natalia Ruppert
Coseismic and Early Postseismic Deformation of the 5 January 2013 Mw 7.5 Craig Earthquake from Static and Kinematic GPS Solutions
Kaihua Ding; Jeffrey T. Freymueller; Qi Wang; Rong Zou
Triggered Seismic Events along the Eastern Denali Fault in Northwest Canada Following the 2012 Mw 7.8 Haida Gwaii, 2013 Mw 7.5 Craig, and Two Mw>8.5 Teleseismic Earthquakes
Chastity Aiken; Jessica P. Zimmerman; Zhigang Peng; Jacob I. Walter
Impacts of the October 2012 Magnitude 7.8 Earthquake near Haida Gwaii, Canada
Alison L. Bird; Maurice Lamontagne
Assessment of Ground‐Motion Models for Use in the British Columbia North Coast Region, Canada
Trevor I. Allen; Camille Brillon
Source Characteristics of the 2012 Haida Gwaii Earthquake Sequence
Honn Kao; Shao‐Ju Shan; Amir Mansour Farahbod
Rupture Process of the 2012 Mw 7.8 Haida Gwaii Earthquake from an Empirical Green’s Function Method
Tiegan E. Hobbs; John F. Cassidy; Stan E. Dosso
Spatiotemporal Distribution of Events during the First Week of the 2012 Haida Gwaii Aftershock Sequence
Amir Mansour Farahbod; Honn Kao
GPS Observations of Crustal Deformation Associated with the 2012 Mw 7.8 Haida Gwaii Earthquake
Lisa Nykolaishen; Herb Dragert; Kelin Wang; Thomas S. James; Michael Schmidt
Coulomb Stress Changes Following the 2012 Mw 7.8 Haida Gwaii, Canada, Earthquake: Implications for Seismic Hazard
T. E. Hobbs; J. F. Cassidy; S. E. Dosso; C. Brillon
Peter J. Haeussler; Robert C. Witter; Kelin Wang
The Preservation Potential of Coastal Coseismic and Tsunami Evidence Observed Following the 2012 Mw 7.8 Haida Gwaii Thrust Earthquake
Lucinda J. Leonard; Jan M. Bednarski
Thermal Condition of the 27 October 2012 Mw 7.8 Haida Gwaii Subduction Earthquake at the Obliquely Convergent Queen Charlotte Margin
Kelin Wang; Jiangheng He; Franziska Schulzeck; Roy D. Hyndman; Michael Riedel
Mw
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Section 5.9 (0050): Noetherian topological spaces—The Stacks project
Section 5.9: Noetherian topological spaces (cite)
5.9 Noetherian topological spaces
Definition 5.9.1. A topological space is called Noetherian if the descending chain condition holds for closed subsets of $X$. A topological space is called locally Noetherian if every point has a neighbourhood which is Noetherian.
Lemma 5.9.2. Let $X$ be a Noetherian topological space.
Any subset of $X$ with the induced topology is Noetherian.
The space $X$ has finitely many irreducible components.
Each irreducible component of $X$ contains a nonempty open of $X$.
Proof. Let $T \subset X$ be a subset of $X$. Let $T_1 \supset T_2 \supset \ldots $ be a descending chain of closed subsets of $T$. Write $T_ i = T \cap Z_ i$ with $Z_ i \subset X$ closed. Consider the descending chain of closed subsets $Z_1 \supset Z_1\cap Z_2 \supset Z_1 \cap Z_2 \cap Z_3 \ldots $ This stabilizes by assumption and hence the original sequence of $T_ i$ stabilizes. Thus $T$ is Noetherian.
Let $A$ be the set of closed subsets of $X$ which do not have finitely many irreducible components. Assume that $A$ is not empty to arrive at a contradiction. The set $A$ is partially ordered by inclusion: $\alpha \leq \alpha ' \Leftrightarrow Z_{\alpha } \subset Z_{\alpha '}$. By the descending chain condition we may find a smallest element of $A$, say $Z$. As $Z$ is not a finite union of irreducible components, it is not irreducible. Hence we can write $Z = Z' \cup Z''$ and both are strictly smaller closed subsets. By construction $Z' = \bigcup Z'_ i$ and $Z'' = \bigcup Z''_ j$ are finite unions of their irreducible components. Hence $Z = \bigcup Z'_ i \cup \bigcup Z''_ j$ is a finite union of irreducible closed subsets. After removing redundant members of this expression, this will be the decomposition of $Z$ into its irreducible components (Lemma 5.8.4), a contradiction.
Let $Z \subset X$ be an irreducible component of $X$. Let $Z_1, \ldots , Z_ n$ be the other irreducible components of $X$. Consider $U = Z \setminus (Z_1\cup \ldots \cup Z_ n)$. This is not empty since otherwise the irreducible space $Z$ would be contained in one of the other $Z_ i$. Because $X = Z \cup Z_1 \cup \ldots Z_ n$ (see Lemma 5.8.3), also $U = X \setminus (Z_1\cup \ldots \cup Z_ n)$ and hence open in $X$. Thus $Z$ contains a nonempty open of $X$. $\square$
Lemma 5.9.3. Let $f : X \to Y$ be a continuous map of topological spaces.
If $X$ is Noetherian, then $f(X)$ is Noetherian.
If $X$ is locally Noetherian and $f$ open, then $f(X)$ is locally Noetherian.
Proof. In case (1), suppose that $Z_1 \supset Z_2 \supset Z_3 \supset \ldots $ is a descending chain of closed subsets of $f(X)$ (as usual with the induced topology as a subset of $Y$). Then $f^{-1}(Z_1) \supset f^{-1}(Z_2) \supset f^{-1}(Z_3) \supset \ldots $ is a descending chain of closed subsets of $X$. Hence this chain stabilizes. Since $f(f^{-1}(Z_ i)) = Z_ i$ we conclude that $Z_1 \supset Z_2 \supset Z_3 \supset \ldots $ stabilizes also. In case (2), let $y \in f(X)$. Choose $x \in X$ with $f(x) = y$. By assumption there exists a neighbourhood $E \subset X$ of $x$ which is Noetherian. Then $f(E) \subset f(X)$ is a neighbourhood which is Noetherian by part (1). $\square$
Lemma 5.9.4. Let $X$ be a topological space. Let $X_ i \subset X$, $i = 1, \ldots , n$ be a finite collection of subsets. If each $X_ i$ is Noetherian (with the induced topology), then $\bigcup _{i = 1, \ldots , n} X_ i$ is Noetherian (with the induced topology).
Example 5.9.5. Any nonempty, Kolmogorov Noetherian topological space has a closed point (combine Lemmas 5.12.8 and 5.12.13). Let $X = \{ 1, 2, 3, \ldots \} $. Define a topology on $X$ with opens $\emptyset $, $\{ 1, 2, \ldots , n\} $, $n \geq 1$ and $X$. Thus $X$ is a locally Noetherian topological space, without any closed points. This space cannot be the underlying topological space of a locally Noetherian scheme, see Properties, Lemma 28.5.9.
Lemma 5.9.6. Let $X$ be a locally Noetherian topological space. Then $X$ is locally connected.
Proof. Let $x \in X$. Let $E$ be a neighbourhood of $x$. We have to find a connected neighbourhood of $x$ contained in $E$. By assumption there exists a neighbourhood $E'$ of $x$ which is Noetherian. Then $E \cap E'$ is Noetherian, see Lemma 5.9.2. Let $E \cap E' = Y_1 \cup \ldots \cup Y_ n$ be the decomposition into irreducible components, see Lemma 5.9.2. Let $E'' = \bigcup _{x \in Y_ i} Y_ i$. This is a connected subset of $E \cap E'$ containing $x$. It contains the open $E \cap E' \setminus (\bigcup _{x \not\in Y_ i} Y_ i)$ of $E \cap E'$ and hence it is a neighbourhood of $x$ in $X$. This proves the lemma. $\square$
Comment #955 by Antoine Chambert-Loir on August 28, 2014 at 09:15
Lemma 5.8.2 should state that every irreducible subset of a noetherian topological space
X
is contained in an irreducible component, equivalently that
X
is the union of its irreducible components. This is more or less proved: Define
A
as the set of closed subspaces of
X
which are not the union of finitely many irreducible subsets.
By our definition of irreducible components (as maximal irreducible subsets) this is true for every topological space. See Lemma 5.8.3.
Comment #996 by Antoine Chambert-Loir on September 06, 2014 at 09:57
Sorry to have overlooked that Lemma 004W. But you will agree that it should state: ``every irreducible subset of
X
is contained in an irreducible component''. (With the same proof, starting from the given irreducible subset instead of a singleton.)
Comment #997 by Johan on September 06, 2014 at 13:15
Yes, indeed, sorry for misunderstanding your point and thanks for persisting. The corresponding change is here. Thanks!
Comment #5560 by Patrick Rabau on October 29, 2020 at 02:54
Lemma 0052, part (2): At the end of the proof of part (2),
Z
has been written as a finite union of closed irreducible sets, and by removing any redundant member it is claimed that this is the decomposition of
Z
into its irreducible components. That step is not entirely clear and I think it would be worthwhile to expand it, or even better, make it a lemma in section 5.8 about irreducible components. Something like this maybe:
X
be a topological space and suppose
X=\bigcup_{i=1,\ldots,n}X_i
X_i
is an irreducible closed subset of
X
X_i
is contained in the union of the other members. Then each
X_i
is an irreducible component of
X
and each irreducible component of
X
X_i
Y
X
Y=\bigcup_{i=1,\ldots,n}(Y\cap X_i)
Y\cap X_i
Y
X_i
X
. By irreducibility of
Y
Y
is equal to one of the
Y\cap X_i
Y\subseteq X_i
. By maximality of irreducible components,
Y=X_i
Conversely, take one of the
X_i
and expand it to an irreducible component
Y
, which we have already shown is one of the
X_j
X_i\subseteq X_j
and since the original union does not have redundant members,
X_i=X_j
, which is an irreducible component.
Thanks very much. I added your lemma with your proof. See this commit.
View Section 5.9 as pdf
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Dictionary:Pseudogeometric factor - SEG Wiki
A coefficient used for estimating the response of a resistivity measurement Ra at different invasion depths:
{\displaystyle R_{a}=R_{xo}J+R_{t}(1-J)}
where Rxo=flushed-zone resistivity, Rt=uncontaminated-zone resistivity, and J=pseudogeometric factor, a function of invasion depth.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Pseudogeometric_factor/en&oldid=106026"
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(Redirected from Program Evaluation and Review Technique)
Statistical tool used in project management
"PERT" redirects here. For other uses, see PERT (disambiguation).
The program evaluation and review technique (PERT) is a statistical tool used in project management, which was designed to analyze and represent the tasks involved in completing a given project.
First developed by the United States Navy in 1958, it is commonly used in conjunction with the critical path method (CPM) that was introduced in 1957.
4.2 Next step, creating network diagram by hand or by using diagram software
4.3 Next step, determination of critical path and possible slack
4.4 Avoiding loops
5 As project scheduling tool
5.3 Uncertainty in project scheduling
PERT is a method of analyzing the tasks involved in completing a given project, especially the time needed to complete each task, and to identify the minimum time needed to complete the total project. It incorporates uncertainty by making it possible to schedule a project while not knowing precisely the details and durations of all the activities. It is more of an event-oriented technique rather than start- and completion-oriented, and is used more in those projects where time is the major factor rather than cost. It is applied on very large-scale, one-time, complex, non-routine infrastructure and on Research and Development projects.
PERT offers a management tool, which relies "on arrow and node diagrams of activities and events: arrows represent the activities or work necessary to reach the events or nodes that indicate each completed phase of the total project."[1]
PERT and CPM are complementary tools, because "CPM employs one time estimation and one cost estimation for each activity; PERT may utilize three time estimates (optimistic, expected, and pessimistic) and no costs for each activity. Although these are distinct differences, the term PERT is applied increasingly to all critical path scheduling."[1]
PERT was developed primarily to simplify the planning and scheduling of large and complex projects. It was developed for the U.S. Navy Special Projects Office in 1957 to support the U.S. Navy's Polaris nuclear submarine project.[2] It found applications all over industry. An early example is when it was used for the 1968 Winter Olympics in Grenoble which applied PERT from 1965 until the opening of the 1968 Games.[3] This project model was the first of its kind, a revival for scientific management, founded by Frederick Taylor (Taylorism) and later refined by Henry Ford (Fordism). DuPont's critical path method was invented at roughly the same time as PERT.
PERT Summary Report Phase 2, 1958
Initially PERT stood for Program Evaluation Research Task, but by 1959 was renamed.[2] It had been made public in 1958 in two publications of the U.S. Department of the Navy, entitled Program Evaluation Research Task, Summary Report, Phase 1.[4] and Phase 2.[5] In a 1959 article in The American Statistician the main Willard Fazar, Head of the Program Evaluation Branch, Special Projects Office, U.S. Navy, gave a detailed description of the main concepts of the PERT. He explained:
The concept of PERT was developed by an operations research team staffed with representatives from the Operations Research Department of Booz Allen Hamilton; the Evaluation Office of the Lockheed Missile Systems Division; and the Program Evaluation Branch, Special Projects Office, of the Department of the Navy.[6]
PERT Guide for management use, June 1963
Ten years after the introduction of PERT in 1958 the American librarian Maribeth Brennan published a selected bibliography with about 150 publications on PERT and CPM, which had been published between 1958 and 1968. The origin and development was summarized as follows:
PERT originated in 1958 with the ... Polaris missile design and construction scheduling. Since that time, it has been used extensively not only by the aerospace industry but also in many situations where management desires to achieve an objective or complete a task within a scheduled time and cost expenditure; it came into popularity when the algorithm for calculating a maximum value path was conceived. PERT and CPM may be calculated manually or with a computer, but usually they require major computer support for detailed projects. A number of colleges and universities now offer instructional courses in both.[1]
For the subdivision of work units in PERT[7] another tool was developed: the Work Breakdown Structure. The Work Breakdown Structure provides "a framework for complete networking, the Work Breakdown Structure was formally introduced as the first item of analysis in carrying out basic PERT/COST."[8]
In a PERT diagram, the main building block is the event, with connections to its known predecessor events and successor events.
Besides events, PERT also knows activities and sub-activities:
PERT has defined four types of time required to accomplish an activity:
{\displaystyle te={\frac {o+4m+p}{6}}}
{\displaystyle TE=\sum _{i=1}^{n}te_{i}}
{\displaystyle {\begin{aligned}&\sigma _{te}={\frac {p-o}{6}}\\[8pt]&\sigma _{TE}={\sqrt {\sum _{i=1}^{n}{\sigma _{te_{i}}}^{2}}}\end{aligned}}}
PERT supplies a number of tools for management with determination of concepts, such as:
critical activity: An activity that has total float equal to zero. An activity with zero free float is not necessarily on the critical path since its path may not be the longest.
The first step for scheduling the project is to determine the tasks that the project requires and the order in which they must be completed. The order may be easy to record for some tasks (e.g., when building a house, the land must be graded before the foundation can be laid) while difficult for others (there are two areas that need to be graded, but there are only enough bulldozers to do one). Additionally, the time estimates usually reflect the normal, non-rushed time. Many times, the time required to execute the task can be reduced for an additional cost or a reduction in the quality.
A Gantt chart created using OmniPlan. Note (1) the critical path is highlighted, (2) the slack is not specifically indicated on task 5 (d), though it can be observed on tasks 3 and 7 (b and f), (3) since weekends are indicated by a thin vertical line, and take up no additional space on the work calendar, bars on the Gantt chart are not longer or shorter when they do or don't carry over a weekend.
Next step, creating network diagram by hand or by using diagram software[edit]
A node like this one (from Microsoft Visio) can be used to display the activity name, duration, ES, EF, LS, LF, and slack.
The expected duration time
Next step, determination of critical path and possible slack[edit]
Avoiding loops[edit]
Depending upon the capabilities of the data input phase of the critical path algorithm, it may be possible to create a loop, such as A -> B -> C -> A. This can cause simple algorithms to loop indefinitely. Although it is possible to "mark" nodes that have been visited, then clear the "marks" upon completion of the process, a far simpler mechanism involves computing the total of all activity durations. If an EF of more than the total is found, the computation should be terminated. It is worth saving the identities of the most recently visited dozen or so nodes to help identify the problem link.
As project scheduling tool[edit]
The network charts tend to be large and unwieldy, requiring several pages to print and requiring specially-sized paper.
The lack of a timeframe on most PERT/CPM charts makes it harder to show status, although colours can help, e.g., specific colour for completed nodes.
Uncertainty in project scheduling[edit]
During project execution a real-life project will never execute exactly as it was planned due to uncertainty. This can be due to ambiguity resulting from subjective estimates that are prone to human errors or can be the result of variability arising from unexpected events or risks. The main reason that PERT may provide inaccurate information about the project completion time is due to this schedule uncertainty. This inaccuracy may be large enough to render such estimates as not helpful.
^ a b c Brennan, Maribeth, PERT and CPM: a selected bibliography, Monticello, Ill., Council of Planning Librarians, 1968. p. 1.
^ a b Malcolm, D. G., J. H. Roseboom, C. E. Clark, W. Fazar. "Application of a Technique for Research and Development Program Evaluation," Operations Research, Vol. 7, No. 5, September–October 1959, pp. 646–669
^ 1968 Winter Olympics official report. p. 49. Accessed 1 November 2010. (in English and French)
^ U.S. Dept. of the Navy. Program Evaluation Research Task, Summary Report, Phase 1. Washington, D.C., Government Printing Office, 1958.
^ Willard Fazar cited in: B. Ralph Stauber, H. M. Douty, Willard Fazar, Richard H. Jordan, William Weinfeld and Allen D. Manvel. "Federal Statistical Activities." The American Statistician 13(2): 9-12 (Apr., 1959) , pp. 9-12
^ Desmond L. Cook (1966), Program Evaluation and Review Technique. p. 12
^ Harold Bright Maynard (1967), Handbook of Business Administration. p. 17
Sapolsky, Harvey M. (1971). The Polaris System Development: Bureaucratic and Programmatic Success in Government. Harvard University Press. ISBN 0674682254.
Media related to PERT charts at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Program_evaluation_and_review_technique&oldid=1087182715"
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Astronomical Journal (New York, N.Y. Online) (458)
MOTION (17537)
ROTATION (12486)
Consideration on the relation between dynamic seismic motion and static seismic coefficient for the earthquake proof design of slope around nuclear power plant
Ito, Hiroshi; Kitahara, Yoshihiro; Hirata, Kazuta
[en] When the large cutting slopes are constructed closed to around nuclear power plants, it is important to evaluate the stability of the slopes during the strong earthquake. In the evaluation, it may be useful to clarify relationship between the static seismic coefficient and dynamic seismic force corresponded to the basic seismic motion which is specified for designing the nuclear power facilities. To investigate this relation some numerical analyses are conducted in this paper. As the results, it is found that dynamic forces considering the amplified responses of the slopes subjected to the basic seismic motion with a peak acceleration of 500 gals at the toe of the slopes, are approximately equal to static seismic force which generates in the slopes when the seismic coefficients of k = 0.3 is applied. (author)
Denryoku Chuo Kenkyusho Hokoku; CODEN DCKHD; (no.385022); p. 1-3, 1-36
ACCELERATION, GROUND MOTION, LANDSLIDES, MATHEMATICAL MODELS, REACTOR SITES, SEISMIC EFFECTS, STABILITY
[en] A method to amplify the rotation angle of a mirror, based on multiple reflections between two quasi-parallel mirrors, is presented. The method allows rotations of fractions of nanoradians to be measured with a simple setup. The working principle, the experimental setup, and the results are presented
Applied Optics; ISSN 0003-6935; ; CODEN APOPAI; v. 45(8); p. 1725-1729
AMPLIFICATION, MIRRORS, NANOSTRUCTURES, REFLECTION, ROTATION
Comment on ‘Wigner function for a particle in an infinite lattice’
Bizarro, João P S, E-mail: bizarro@ipfn.ist.utl.pt
[en] It is pointed out that in a recent paper (2012 New J. Phys. 14 103009) in which a Wigner function for a particle in an infinite lattice (a system described by an unbounded discrete coordinate and its conjugate angle-like momentum) has been introduced, no reference is made to previous, pioneering work on discrete Wigner distributions (more precisely, on the rotational Wigner function for a system described by a rotation angle and its unbounded discrete-conjugate angular momentum). Not only has the problem addressed in essence been solved for a long time (the discrete coordinate and angle-like conjugate momentum are the perfect dual of the rotation angle and discrete-conjugate angular momentum), but the solution advanced only in some distorted manner obeys two of the fundamental properties of a Wigner distribution (that, when integrated over one period of the momentum variable, it should yield the correct marginal distribution on the discrete position variable, and that it should be invariant with respect to translation). (comment)
ANGULAR MOMENTUM, COORDINATES, DISTRIBUTION, FUNCTIONS, PARTICLES, ROTATION, WIGNER DISTRIBUTION
Erratum: “How accurate are stochastic rotation dynamics simulations of polymer dynamics?” [J. Rheol. 57, 1177–1194 (2013)]
Jiang, Lei; Watari, Nobuhiko; Larson, Ronald G., E-mail: rlarson@umich.edu
(c) 2014 The Society of Rheology; Country of input: International Atomic Energy Agency (IAEA)
Journal of Rheology; ISSN 0148-6055; ; CODEN JORHD2; v. 58(2); p. 563-563.1
POLYMERS, ROTATION, SIMULATION, STOCHASTIC PROCESSES
Stauffer, John; Rebull, Luisa; Bouvier, Jerome; Hillenbrand, Lynne A.; David, Trevor; Collier-Cameron, Andrew; Pinsonneault, Marc; Somers, Garrett; Aigrain, Suzanne; Barrado, David; Bouy, Herve; Ciardi, David; Cody, Ann Marie; Micela, Giusi; Soderblom, David; Valenti, Jeff; Stassun, Keivan G.; Vrba, Frederick J.
[en] We use high-quality K2 light curves for hundreds of stars in the Pleiades to better understand the angular momentum evolution and magnetic dynamos of young low-mass stars. The K2 light curves provide not only rotational periods but also detailed information from the shape of the phased light curve that was not available in previous studies. A slowly rotating sequence begins at
{\left(V-{K}_{\mathrm{s}}\right)}_{0}
∼ 1.1 (spectral type F5) and ends at
{\left(V-{K}_{\mathrm{s}}\right)}_{0}
∼ 3.7 (spectral type K8), with periods rising from ∼2 to ∼11 days in that interval. A total of 52% of the Pleiades members in that color interval have periods within 30% of a curve defining the slow sequence; the slowly rotating fraction decreases significantly redward of
{\left(V-{K}_{\mathrm{s}}\right)}_{0}
= 2.6. Nearly all of the slow-sequence stars show light curves that evolve significantly on timescales less than the K2 campaign duration. The majority of the FGK Pleiades members identified as photometric binaries are relatively rapidly rotating, perhaps because binarity inhibits star–disk angular momentum loss mechanisms during pre-main-sequence evolution. The fully convective late M dwarf Pleiades members (5.0 <
{\left(V-{K}_{\mathrm{s}}\right)}_{0}
< 6.0) nearly always show stable light curves, with little spot evolution or evidence of differential rotation. During pre-main-sequence evolution from ∼3 Myr (NGC 2264 age) to ∼125 Myr (Pleiades age), stars of 0.3
{M}_{\odot }
shed about half of their angular momentum, with the fractional change in period between 3 and 125 Myr being nearly independent of mass for fully convective stars. Our data also suggest that very low mass binaries form with rotation periods more similar to each other and faster than would be true if drawn at random from the parent population of single stars.
Available from http://dx.doi.org/10.3847/0004-6256/152/5/115; Country of input: International Atomic Energy Agency (IAEA)
ANGULAR MOMENTUM, MASS, ROTATION, STAR CLUSTERS, STARS
Gueroult, R.; Evans, E. S.; Zweben, S. J.; Fisch, N. J.; Levinton, F.
Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States). Funding organisation: USDOE (United States)
[en] High throughput plasma mass separation requires rotation control in a high density multi-species plasmas. A preliminary mass separation device based on a helicon plasma operating in gas mixtures and featuring concentric biasable ring electrodes is introduced. Plasma profile shows strong response to electrode biasing. In light of floating potential measurements, the density response is interpreted as the consequence of a reshaping of the radial electric field in the plasma. This field can be made confining or de-confining depending on the imposed potential at the electrodes, in a way which is consistent with single particle orbit radial stability. In conclusion, concurrent spatially resolved spectroscopic measurements suggest ion separation, with heavy to light ion emission line ratio increasing with radius when a specific potential gradient is applied to the electrodes
PPPL--5197; OSTIID--1257876; AC02-09CH11466; Available from http://www.osti.gov/pages/biblio/1257876; Country of input: United States
Plasma Sources Science and Technology; ISSN 0963-0252; ; v. 25(3); vp
ELECTRIC FIELDS, PLASMA, PLASMA RADIAL PROFILES, ROTATION
http://dx.doi.org/10.1088/0963-0252/25/3/035024, http://www.osti.gov/pages/biblio/1257876
ROTATION–ACTIVITY CORRELATIONS IN K AND M DWARFS. I. STELLAR PARAMETERS AND COMPILATIONS OF v sin i AND P /sin i FOR A LARGE SAMPLE OF LATE-K AND M DWARFS
Houdebine, E. R.; Mullan, D. J.; Paletou, F.; Gebran, M., E-mail: eric_houdebine@yahoo.fr
[en] The reliable determination of rotation–activity correlations (RACs) depends on precise measurements of the following stellar parameters: T eff, parallax, radius, metallicity, and rotational speed v sin i . In this paper, our goal is to focus on the determination of these parameters for a sample of K and M dwarfs. In a future paper (Paper II), we will combine our rotational data with activity data in order to construct RACs. Here, we report on a determination of effective temperatures based on the ( R – I )C color from the calibrations of Mann et al. and Kenyon and Hartmann for four samples of late-K, dM2, dM3, and dM4 stars. We also determine stellar parameters ( T eff, log( g ), and [M/H]) using the principal component analysis–based inversion technique for a sample of 105 late-K dwarfs. We compile all effective temperatures from the literature for this sample. We determine empirical radius–[M/H] correlations in our stellar samples. This allows us to propose new effective temperatures, stellar radii, and metallicities for a large sample of 612 late-K and M dwarfs. Our mean radii agree well with those of Boyajian et al. We analyze HARPS and SOPHIE spectra of 105 late-K dwarfs, and we have detected v sin i in 92 stars. In combination with our previous v sin i measurements in M and K dwarfs, we now derive P /sin i measures for a sample of 418 K and M dwarfs. We investigate the distributions of P /sin i , and we show that they are different from one spectral subtype to another at a 99.9% confidence level.
CALIBRATION, CORRELATIONS, DISTRIBUTION, METALLICITY, ROTATION, SPECTRA, STARS, VELOCITY
To the theory of origin of globular cluster's subsystem of galaxies
Tadzhibaev, I.U.
[en] We study the origin problems of a subsystem of globular clusters of galaxies in the background of nonlinear time-dependent model of collapsing galaxies. The stability of the model for two modes of high order perturbation is studied. The dependencies of initial virial ratio on the parameter of rotation are drawn for all modes. The dependences of instability increments on physical parameters of the model are found. (authors)
K teorii proiskhozhdeniya podsistemy sharovykh skoplenij galaktik
11 refs., 2 figs.
Uzbekiston Fizika Zhurnali; ISSN 1025-8817; ; v. 18(1); p. 3-10
DISTURBANCES, GALAXIES, GALAXY CLUSTERS, INSTABILITY, NONLINEAR PROBLEMS, ORIGIN, PERTURBATION THEORY, ROTATION, STABILITY, TIME DEPENDENCE
Yang, Jie; Li, Yang; Liu, Yu-Xiao; Li, Yun-Liang; Wei, Shao-Wen
Funding organisation: SCOAP3, CERN, Geneva (Switzerland)
[en] We study the effect of ultrahigh energy collisions of two particles with different energies near the horizon of a 2+1 dimensional BTZ black hole (BSW effect). We find that the particle with the critical angular momentum could exist inside the outer horizon of the BTZ black hole regardless of the particle energy. Therefore, for the nonextremal BTZ black hole, the BSW process is possible on the inner horizon with the fine tuning of parameters which are characterized by the motion of particle, while, for the extremal BTZ black hole, the particle with the critical angular momentum could only exist on the degenerated horizon, and the BSW process could also happen there
Available from http://dx.doi.org/10.1155/2014/204016; Available from http://repo.scoap3.org/record/1663; PUBLISHER-ID: 204016; OAI: oai:repo.scoap3.org:1663; This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.; Country of input: International Atomic Energy Agency (IAEA)
Advances in High Energy Physics (Online); ISSN 1687-7365; ; v. 2014; [7 p.]
ANGULAR MOMENTUM, BLACK HOLES, COSMOLOGICAL CONSTANT, INTERACTIONS, ROTATION, SPACE-TIME
http://dx.doi.org/10.1155/2014/204016, https://repo.scoap3.org/record/1663
Problems of non-linear dynamics of plates at short-term force influence
Abdulloev, A.
Academy of Scinces of the Republic of Tajikistan (Tajikistan)
Conference 'Conference materials of young scientists of Academy ofSciences of Republic of Tajikistan' Proceedings
Voprosi nelineynoy dinamiki plastin pri kratkovremennikh silovikhvozdeystviyakh
Academy of Scinces of the Republic of Tajikistan, Russian Academy ofSciences(Tajikistan); 86 p; Feb 1987; p. 93-94; Conference on materials of young scientists of Academy of Sciences of Republic of Tajikistan; Konferentsiya 'Materiali konferentsii molodikh uchenikh Akademii NaukRespubliki Tajikistan'; Dushanbe (Tajikistan); Nov 1986; Available from the library of Academy of Sciences of the Republic ofTajikistan
CALCULATION METHODS, DEFORMATION, MOMENT OF INERTIA, PLATES, ROTATION
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InverseGaussian - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : InverseGaussian
InverseGaussian(mu, lambda)
InverseGaussianDistribution(mu, lambda)
The inverse Gaussian distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)={\begin{array}{cc}0& t<0\\ \frac{\sqrt{2}\sqrt{\frac{\mathrm{\lambda }}{\mathrm{\pi }{t}^{3}}}{ⅇ}^{-\frac{\mathrm{\lambda }{\left(t-\mathrm{\mu }\right)}^{2}}{2{\mathrm{\mu }}^{2}t}}}{2}& \mathrm{otherwise}\end{array}
0<\mathrm{\mu },0<\mathrm{\lambda }
Note that the InverseGaussian command is inert and should be used in combination with the RandomVariable command.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{InverseGaussian}\left(\mathrm{\mu },\mathrm{\lambda }\right)\right):
\mathrm{PDF}\left(X,u\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\frac{\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}}{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{3}}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{\mu }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}}}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{PDF}\left(X,0.5\right)
\textcolor[rgb]{0,0,1}{1.128379166}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1.000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{0.5}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{{\textcolor[rgb]{0,0,1}{\mathrm{\mu }}}^{\textcolor[rgb]{0,0,1}{2}}}}
\mathrm{Mean}\left(X\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{Variance}\left(X\right)
\frac{{\textcolor[rgb]{0,0,1}{\mathrm{\mu }}}^{\textcolor[rgb]{0,0,1}{3}}}{\textcolor[rgb]{0,0,1}{\mathrm{\lambda }}}
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Lemma 15.11.3 (0CT7)—The Stacks project
Lemma 15.11.3. Let $A = \mathop{\mathrm{lim}}\nolimits A_ n$ where $(A_ n)$ is an inverse system of rings whose transition maps are surjective and have locally nilpotent kernels. Then $(A, I_ n)$ is a henselian pair, where $I_ n = \mathop{\mathrm{Ker}}(A \to A_ n)$.
Proof. Fix $n$. Let $a \in A$ be an element which maps to $1$ in $A_ n$. By Algebra, Lemma 10.32.4 we see that $a$ maps to a unit in $A_ m$ for all $m \geq n$. Hence $a$ is a unit in $A$. Thus by Algebra, Lemma 10.19.1 the ideal $I_ n$ is contained in the Jacobson radical of $A$. Let $f \in A[T]$ be a monic polynomial and let $\overline{f} = g_ nh_ n$ be a factorization of $\overline{f} = f \bmod I_ n$ with $g_ n, h_ n \in A_ n[T]$ monic generating the unit ideal in $A_ n[T]$. By Lemma 15.11.2 we can successively lift this factorization to $f \bmod I_ m = g_ m h_ m$ with $g_ m, h_ m$ monic in $A_ m[T]$ for all $m \geq n$. At each step we have to verify that our lifts $g_ m, h_ m$ generate the unit ideal in $A_ n[T]$; this follows from the corresponding fact for $g_ n, h_ n$ and the fact that $\mathop{\mathrm{Spec}}(A_ n[T]) = \mathop{\mathrm{Spec}}(A_ m[T])$ because the kernel of $A_ m \to A_ n$ is locally nilpotent. As $A = \mathop{\mathrm{lim}}\nolimits A_ m$ this finishes the proof. $\square$
On line 3 of the proof, replace "radical" with "Jacobson radical". Near the end of the proof, it should be explained via the "locally nilpotent" hypothesis on the kernels of the surjective maps that the condition on
g_m
h_m
generating the unit ideal for
m=0
is inherited for
m=1
and so on for invoking Lemma 0ALI to "successively" lift as indicated.
OK, I replaced "radical" by "Jacobson radical" in all places where appropriate and whenever we used the notation
\text{rad}(R)
I have added text saying this denotes the Jacobson radical. See the corresponding changes here.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CT7. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0CT7, in case you are confused.
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(Redirected from Centrifugal force (rotating reference frame))
{\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}
{\displaystyle F=m\omega ^{2}r}
{\displaystyle {\frac {\operatorname {d} {\boldsymbol {P}}}{\operatorname {d} t}}=\left[{\frac {\operatorname {d} {\boldsymbol {P}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {P}}\ ,}
{\displaystyle \times }
{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {P}}}
{\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}
{\displaystyle {\boldsymbol {a}}={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\ ,}
{\displaystyle {\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}}
{\displaystyle {\frac {\operatorname {d} }{\operatorname {d} t}}\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]}
{\displaystyle {\begin{aligned}{\boldsymbol {a}}&={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}={\frac {\operatorname {d} }{\operatorname {d} t}}{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times {\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times \left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\ .\end{aligned}}}
{\displaystyle \left[{\frac {d^{2}{\boldsymbol {r}}}{dt^{2}}}\right]}
{\displaystyle {\frac {d^{2}{\boldsymbol {r}}}{dt^{2}}}}
{\displaystyle {\boldsymbol {F}}-m{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}-2m{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}
{\displaystyle =m\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]\ .}
{\displaystyle -m\operatorname {d} {\boldsymbol {\omega }}/\operatorname {d} t\times {\boldsymbol {r}}}
{\displaystyle -2m{\boldsymbol {\omega }}\times \left[\operatorname {d} {\boldsymbol {r}}/\operatorname {d} t\right]}
{\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}
{\displaystyle ({\boldsymbol {\omega }}=0)}
{\displaystyle (r,\ \theta )}
{\displaystyle {\boldsymbol {\dot {q}}}}
{\displaystyle {\boldsymbol {q}}}
{\displaystyle {{\dot {q}}_{i}}^{2}}
{\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}
{\displaystyle {\boldsymbol {q}}}
{\displaystyle {\dot {\phi }}}
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Francis_Birch_(geophysicist) Knowpia
Francis Birch (August 22, 1903 – January 30, 1992) was an American geophysicist. He is considered one of the founders of solid Earth geophysics. He is also known for his part in the atomic bombing of Hiroshima and Nagasaki.
Barbara Channing
During World War II, Birch participated in the Manhattan Project, working on the design and development of the gun-type nuclear weapon known as Little Boy. He oversaw its manufacture, and went to Tinian to supervise its assembly and loading into Enola Gay, the Boeing B-29 Superfortress tasked with dropping the bomb.
A graduate of Harvard University, Birch began working on geophysics as a research assistant. He subsequently spent his entire career at Harvard working in the field, becoming an Associate Professor of Geology in 1943, a professor in 1946, and Sturgis Hooper Professor of Geology in 1949, and professor emeritus in 1974.
Birch published over 100 papers. He developed what is now known as the Birch-Murnaghan equation of state in 1947. In 1952 he demonstrated that Earth's mantle is chiefly composed of silicate minerals, with an inner and outer core of molten iron. In two 1961 papers on compressional wave velocities, he established what is now called Birch's law.
Albert Francis Birch was born in Washington, D.C., on August 22, 1903, the son of George Albert Birch, who was involved in banking and real estate, and Mary Hemmick Birch, a church choir singer and soloist at St. Matthew's Cathedral in Washington, D.C. He had three younger brothers: David, who became a banker; John, who became a diplomat; and Robert, who became a songwriter. He was educated at Washington, D.C., schools, and Western High School, where he joined the High School Cadets in 1916.[1][2]
In 1920 Birch entered Harvard University on a scholarship. While there he served in Harvard's Reserve Officers' Training Corps Field Artillery Battalion. He graduated magna cum laude in 1924, and received his Bachelor of Science (S.B.) degree in electrical engineering.[1]
Birch went to work in the Engineering Department of the New York Telephone Company. He applied for and received an American Field Service Fellowship in 1926, which he used to travel to Strasbourg, and study at the University of Strasbourg's Institut de Physique under the tutelage of Pierre Weiss.[3] There, he wrote or co-wrote four papers, in French, on topics such as the paramagnetic properties of potassium cyanide, and the magnetic moment of Cu++ ions.[4]
On returning to the United States in 1928, Birch went back to Harvard to pursue physics. He was awarded his Master of Arts (A.M.) degree in 1929, and then commenced work on his 1932 Doctor of Philosophy (Ph.D.) degree under the supervision of Percy Bridgman,[3] who would receive the Nobel Prize for Physics in 1946. For his thesis, Birch measured the vapor-liquid critical point of mercury. He determined this as 1460±20 °C and 1640±50 kg/cm2, results he published in 1932 in the Physical Review.[4][5]
Around this time, there was an increased interest in geophysics at Harvard University, and Reginald Aldworth Daly established a Committee for Experimental Geology and Geophysics that included Bridgman, astronomer Harlow Shapley, geologists Louis Caryl Graton and D. H. McLaughlin and chemist G. P. Baxter. William Zisman, another one of Bridgman's Ph.D. students, was hired as the committee's research associate, but, having little interest in the study of rocks, he resigned in 1932. The position was then offered to Birch, who had little interest or experience in geology either, but with the advent of the Great Depression, jobs were hard to find, and he accepted. [6][7]
On July 15, 1933, Birch married Barbara Channing, a Bryn Mawr College alumna, and a collateral descendant of the theologian William Ellery Channing. They had three children: Anne Campaspe, Francis (Frank) Sylvanus and Mary Narcissa. Frank later became a professor of geophysics at the University of New Hampshire.[8][9]
Birch (left) works on the Little Boy bomb while Norman F. Ramsey (right) looks on
In 1942, during World War II, Birch took a leave of absence from Harvard, in order to work at the Massachusetts Institute of Technology Radiation Laboratory, which was developing radar. He worked on the proximity fuze, a radar-triggered fuze that would explode a shell in the proximity of a target. The following year he accepted a commission in the United States Navy as a lieutenant commander, and was posted to the Bureau of Ships in Washington, D.C.[10]
Later that year he was assigned to the Manhattan Project, and moved with his family to Los Alamos, New Mexico. There he joined the Los Alamos Laboratory's Ordnance (O) Division, which was under the command of another Naval officer, Captain William S. Parsons. Initially the goal of the O Division was to design a gun-type nuclear weapon known as Thin Man. This proved to be impractical due to contamination of the reactor-bred plutonium with plutonium-240, and in February 1944, the Division switched its attention to the development of the Little Boy, a smaller device using uranium-235. Birch used unenriched uranium to create scale models and later full-scale mock-ups of the device.[11]
Birch supervised the manufacture of the Little Boy, and went to Tinian to supervise its assembly and loading it onto Enola Gay, the Boeing B-29 Superfortress tasked with dropping the bomb. He devised the 'double plug' system that allowed for actually arming the bomb after Enola Gay took off so that if it crashed, there would not be a nuclear explosion.[11] He was awarded the Legion of Merit. His citation read:
for exceptionally meritorious conduct in the performance of outstanding services to the Government of the United States in connection with the development of the greatest military weapon of all time, the atomic bomb. His initial assignment was the instrumentation of laboratory and field tests. He carried out this assignment in such outstanding fashion that he was placed in charge of the engineering and development of the first atomic bomb. He carried out this assignment with outstanding judgment and skill, and finally, went with the bomb to the advanced base where he insured, by his care and leadership, that the bomb was adequately prepared in every respect. Commander Birch's engineering ability, understanding of all principles involved, professional skill and devotion to duty throughout the development and delivery of the atomic bomb were outstanding and were in keeping with the highest traditions of the United States Naval Service.[12]
Birch returned to Harvard after the war ended, having been promoted to Associate Professor of Geology in 1943 while he was away. He would remain at Harvard for the rest of his career, becoming a professor in 1946, and Sturgis Hooper Professor of Geology in 1949, and professor emeritus in 1974. Professor Birch published over 100 papers.[2][3] He served as president of The Geological Society of America in 1964 and was awarded their Penrose Medal in 1969.[13]
In 1947, he adapted the isothermal Murnaghan equation of state, which had been developed for infinitesimal strain, for Eulerian finite strain, developing what is now known as the Birch-Murnaghan equation of state.[14]
Albert Francis Birch is known for his experimental work on the properties of Earth-forming minerals at high pressure and temperature, in 1952 he published a well-known paper in the Journal of Geophysical Research, where he demonstrated that the mantle is chiefly composed of silicate minerals, the upper and lower mantle are separated by a thin transition zone associated with silicate phase transitions, and the inner and outer core are alloys of crystalline and molten iron. His conclusions are still accepted as correct today. The most famous portion of the paper, however, is a humorous footnote he included in the introduction:[15][failed verification]
Unwary readers should take warning that ordinary language undergoes modification to a high-pressure form when applied to the interior of the Earth. A few examples of equivalents follow:
Certain Dubious
Undoubtedly Perhaps
Positive proof Vague suggestion
Unanswerable argument Trivial objection
Pure iron Uncertain mixture of all the elements
In 1961, Birch published two papers on compressional wave velocities establishing a linear relation of the compressional wave velocity Vp of rocks and minerals of a constant average atomic weight
{\displaystyle {\bar {M}}}
{\displaystyle \rho }
as:[16][17]
{\displaystyle V_{p}=a({\bar {M}})+b\rho }
This relationship became known as Birch's law. Birch was elected to the National Academy of Sciences in 1950,[18] and served as the president of the Geological Society of America in 1963 and 1964.[3] He received numerous honors in his career, including the Geological Society of America's Arthur L. Day Medal on 1950 and Penrose Medal in 1969, the American Geophysical Union's William Bowie Medal in 1960, the National Medal of Science from President Lyndon Johnson in 1967, the Vetlesen Prize (shared with Sir Edward Bullard) in 1968, the Gold Medal of the Royal Astronomical Society in 1973, and the International Association for the Advancement of High Pressure Research's Bridgman Award in 1983.[19] Since 1992, the American Geophysical Union's Tectonophysics section has sponsored a Francis Birch Lecture, given at its annual meeting by a noted researcher in this field.[20]
Birch died of prostate cancer at his home in Cambridge, Massachusetts, on January 30, 1992. He was survived by wife Barbara, his three children and his three brothers.[2][18] His papers are in the Harvard University Archives.[3]
^ a b Ahrens 1998, p. 4.
^ a b c Sullivan, Walter (February 5, 1992). "Francis Birch, Is Dead at 88; Was a Professor". The New York Times.
^ a b c d e "Birch, Francis, 1903-1992. Papers of Francis Birch : an inventory ( HUGFP 132 )". Harvard University. Retrieved February 4, 2014.
^ Birch, Francis (September 1932). "The Electrical Resistance and the Critical Point of Mercury". Physical Review. 41 (5): 641–648. Bibcode:1932PhRv...41..641B. doi:10.1103/PhysRev.41.641. ISSN 1050-2947.
^ Ahrens 1998, p. 7.
^ Birch, Francis (1979). "Reminiscences and digressions". Annual Review of Earth and Planetary Sciences. 7: 1–10. Bibcode:1979AREPS...7....1B. doi:10.1146/annurev.ea.07.050179.000245. ISSN 0084-6597.
^ Sedgwick 1961, p. 245.
^ Ahrens 1998, pp. 10–11.
^ a b Ahrens 1998, pp. 11–13.
^ "Valor awards for Albert Francis Birch". Military Times. Archived from the original on February 21, 2014. Retrieved February 6, 2014.
^ Birch, Francis (June 1947). "Finite Elastic Strain of Cubic Crystals". Physical Review. 71 (11): 809–824. Bibcode:1947PhRv...71..809B. doi:10.1103/PhysRev.71.809. ISSN 1050-2947.
^ Birch, Francis (June 1952). "Elasticity and Constitution of the Earth's Interior". Journal of Geophysical Research. 57 (2): 227–286. Bibcode:1952JGR....57..227B. doi:10.1029/JZ057i002p00227. ISSN 0148-0227.
^ Birch, Francis (July 1961). "The velocity of Compressional Waves in Rocks to 10 kilobars. Part 2". Journal of Geophysical Research. 66 (7): 2199–2224. Bibcode:1961JGR....66.2199B. doi:10.1029/JZ066i007p02199. ISSN 0148-0227.
^ Birch, Francis (December 1961). "Composition of the Earth's Mantle". Geophysical Journal of the Royal Astronomical Society. 4: 295–311. Bibcode:1961GeoJ....4..295B. doi:10.1111/j.1365-246X.1961.tb06821.x. ISSN 1365-246X.
^ a b Shankland, Thomas; O'Connell, Richard (November 1992). "Obituary: Francis Birch". Physics Today. 45 (11): 105–106. Bibcode:1992PhT....45k.105S. doi:10.1063/1.2809890. ISSN 0031-9228. Archived from the original on October 4, 2013.
^ Ahrens 1998, p. 20.
^ Olsen, Kenneth. "Francis Birch (1903–1992)". American Geophysical Union. Retrieved February 6, 2014.
Ahrens, Thomas J. (1998). Albert Francis Birch (PDF). Biographical Memoirs. Vol. 74. Washington, D.C.: National Academy of Sciences. pp. 1–24. Retrieved February 4, 2014.
Sedgwick, Hubert M. (1961). A Sedgwick Genealogy: Descendants of Deacon Benjamin Sedgwick. New Haven, Connecticut: New Haven Colony Historical Society. Retrieved February 4, 2014.
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Medical Equipment Research and Development Division (MERADD), ICCC, Lahore, Pakistan.
Abstract: A new method is proposed here aiming at designing a shielding wall with the efficiency significantly higher than that of traditional designs. This new design arises from the idea of using channeling in multilayered shielding wall structure, each layer composed of bent crystallites distributed in a way that each layer covers a small section of 2π angular range of which wall is exposed. Part of the incident charged particles will get channeled in bent crystallites in each layer. Bending of channeled particles in bent crystallites will change their directions in the wall increasing their path lengths in the wall which would enhance its shielding efficiency for charged particle radiations. Proposed design is useful for radiation shielding in fission power plants, future fusion reactors and air travel.
Keywords: Radiation Shielding, Bent Crystals, Channeling, Nuclear and Thermonuclear Reactors, Cosmic Rays
Radiation shielding is a great concern in environments of nuclear technology and air travel [1] [2] . These environments include various radiation fields with a considerable component of charged particle rays. Charged particle rays include fission fragments, heavy and light ions and nuclei, and electrons. A major fraction of these particles from nuclear/thermonuclear reactors and cosmic rays has high energy loss rate in tissue and other materials. Shielding of human beings and material devices against their radiation is highly desirable. There are several radiation shielding designs using numerous materials of special characteristics, but none of them have made use of the concept presented here with the theme as “bending charged particle radiations in the shields to increase their path lengths in them”. Bending of charged particle rays would be achieved by employing a multilayered shield with each layer composed of pieces of bent crystals. Strong fields of bent crystalline planes and axes have an ability of bending charged particles entering planar and axial channels under certain conditions [3] [4] [5] [6] . Details of the method and its implementation are given below. Section 2 describes channeling condition in bent crystals whereas proposed method for enhancement of shielding efficiency is given in Section 3. Section 4 includes conclusion of the study.
2. Channeling in Bent Crystals
Ion channeling is the passage of a particle beam (particles travelling in parallel) through open spaces in materials, e.g., crystal planar and axial channels or possibly nanotubes. Traditionally, this process is studied in single crystals where a variety of channels, both with similar and very different geometric and electronic structures, are available. The channeled beam in a crystal travels on a very specific path with a statistical variation. Ion channeling was discovered in early 1960s and it continued developing due to a number of its present [7] [8] and potential uses including channeling in bent crystals [3] [9] [10] and defect measurement in crystals [11] [12] [13] [14] [15] .
Steering of channelled particles in a crystal continues even if the crystal is slightly bent deviating from their original direction as in strong magnetic fields of strength several thousand tesla. This fact provides us with the possibility of designing a “crystalline kicker” (Uggerhøj, 2005) [3] . This crystalline kicker for charged particles is shown in Figure 1.
Lindhard [16] has shown that incident angle (
{\theta }_{i}
) of the particle trapped in transverse field of the crystalline channel must not exceed the limiting angle, called Lindhard angle (
{\theta }_{L}
), as given below,
{\theta }_{i}\le {\theta }_{L}=\sqrt{\frac{2{U}_{T}}{{\epsilon }_{||}}}
{U}_{T}
is the transverse potential of the channel and
{\epsilon }_{||}
is longitudinal kinetic energy of the particle in the channel and is given by,
{\epsilon }_{||}={m}_{o}{c}^{2}\gamma
\gamma
is the well known Lorentz factor. Under the condition given in Equation (1), transverse kinetic energy of the channeled particle will remain below the transverse potential
{U}_{T}
{\theta }_{SO}
is the angle of the curvature in the bent crystal subtended on length of one transverse oscillation of the planar channeled particle (
{L}_{SO}
{L}_{SO}=r{\theta }_{SO}
, where r is radius of curvature of the bent crystal. Channeling in the bent crystal will persist if,
{\theta }_{SO}\le {\theta }_{L}
Figure 1. A schematic drawing showing bending of the channeled particles in the bent crystal. Typical dimensions are shown.
Maximum bending angle of the channeled particle (
{\theta }_{\mathrm{max}}
) in the bent crystal is given by the following condition [17] ,
{\theta }_{\mathrm{max}}\le {\theta }_{L}{N}_{Osc}
{N}_{Osc}
is the number of transverse oscillations made by the channeled particle in the bent channel of the crystal. Above Equations (1)-(4) describe completely the condition of channeling in the bent crystal for the purpose being served in this paper.
3. Proposed Method for Enhancement of Shielding Efficiency
It is shown in Section 2 that charged particles entering channels of a bent crystal under certain conditions would bend by an angle up to
{\theta }_{\mathrm{max}}
(Equation (4)) which depends on Lindhard angle
{\theta }_{L}
and number of transverse oscillations made by the channeled particle
{N}_{Osc}
. Value of
{\theta }_{L}
is determined by properties of the crystalline material and used channel along with characteristics of the channeled particle like atomic and mass numbers, charge state and energy. The number of oscillations
{N}_{Osc}
can be controlled by length of the bent crystal. Let us now consider a test piece of a bent crystal (like Ge) with length
{L}_{T}
and Lindhard critical angle
{\theta }_{T}^{L}
for a certain radiation (say proton, alpha particle or a fission fragment). The new scientific idea being presented here is that a multilayered radiation shield with each layer composed of bent crystalline pieces arranged in a certain manner will serve as a kicker for charged particle rays changing direction of the radiations and hence increasing their path lengths in the shield giving rise to the enhanced shielding efficiency. The maximum bending angle (
{\theta }_{T}^{\mathrm{max}}
) of a charged particle in the above mentioned bent crystal piece is given by,
{\theta }_{T}^{\mathrm{max}}\le {\theta }_{T}^{L}\cdot \frac{{L}_{T}}{{L}_{T}^{SO}}
where labels “T” and “SO” refer to test crystal piece and single transverse oscillation of the channeled particle, respectively.
Figure 2 shows the detail of the design pictorially. Left section shows a straight crystal whereas right section two layers of the proposed shield. This scheme can be employed for the cases when radiation field falling on the shield wall has at least a local directionality which means charged particles falling on a small surface area of the shield wall will incident in a an angular range as shown in Figure 3. In Figure 3, a 2D shielding box of a radiation source is shown. A wall of infinite dimensions will have 2π directions in which radiation can reach it. For a wall of limited dimensions, angular range will be much narrower than 2π. Angular range of radiations reaching each wall from the source may be divided into several sections, as divided into four sections in Figure 3. Particles falling in each such a section will have a limited angular range. In such cases above mentioned design will widen incident angular range of radiations falling on the wall increasing path lengths of radiations in the wall, hence enhancing shielding efficiency of the wall. In this scheme, bent crystalline pieces will be arranged in each layer in each section (I to IV) covering a section of the entire particle incidence range. Bent channels in bent crystals will offer entry into them to a part of radiations bending them in the shield. If a number of such layers, covering all the incidence range in a section, are employed in a structure, major fraction of incident charged rays will bend in the shield increasing path lengths of radiations in the shield. Two layers covering two small angular segments in the incidence range are shown in right column of Figure 2. In real case with 2D
Figure 2. Design of the shielding wall of the charged particle radiations. Left part shows straight crystal structure whereas right part two layers of a multilayered shielding design with each layer composed of bent crystal pieces to bend a part of falling radiations.
Figure 3. A charged particle radiation source enclosed by a shielding box (2D case for simplicity). Radiations falling on right wall are classified into four angular sections, each of them with a limited range of incidence angle of radiations falling on it.
shielding wall, wall surface will be divided in 2D grid with each grid element having a multilayered bent crystal structures. Alignment of crystal pieces will be arranged according to the range of incidence angles of charged particle rays. These bent crystal pieces will have length from fraction of a millimeter to tens of a millimeter depending upon charge and energy of radiations. Cross-sectional dimension of these elements will have inverse relationship with bending efficiency. This shield design will show efficiency better than the traditional shield wall. High efficiency amorphous material layers can also be joined behind this new multilayered structure for better shielding.
A new charged particle radiation shield design is proposed here. This design will have a higher shielding efficiency than the tradition designs. In the new design, multilayered wall structure with each wall composed of bent crystal pieces is proposed. Each layer will bent a part of charged particle rays entering in bent crystals under channeling condition increasing path lengths of radiations in the shield. If a sufficient number of layers are used, major fraction of charged particles will be bent in the shield travelling longer distance in the shield compared with the traditional design, hence enhancing shielding efficiency. This shielding design can be employed in a number of radiation environments with situations comparable to that described in Figure 3. It may be noticed that the present design is different but has some similarity with a proposal by Breese [18] . This design will consume shielding materials in a smaller quantity compared to the traditional design.
I express my gratitude to my Ph.D. supervisors/guides Prof. Breese, M.B.H., Assoc. Prof. Osipowicz, T., Prof. Watt, F. and Dr. Ren, M.Q. at the National University of Singapore, for guidance and discussions. This idea was initially discussed with Dr. Valery Biryukov (Russia) in 2008. Email discussions with several colleagues worldwide are gratefully acknowledged for understanding basics of ion channeling. Leading colleagues in this regard include Dr. Scandale, W., Dr. Biryukov, V.M., Dr. Taratin, A.M., Moller, S.P., Dr. Mazzolari, A., Greiner, W. and Fromm, M.
*A new method for radiation shielding.
Cite this paper: Rana, M. (2018) Channeling in Bent Crystallites: A New Method to Enhance the Radiation Shielding Efficiency*. Modern Instrumentation, 7, 35-41. doi: 10.4236/mi.2018.73004.
[1] Ewing, R.C. (2007) Materials Science: Displaced by Radiation. Nature, 445, 161-162.
[2] Farnan, I., Cho, H. and Weber, W.J. (2007) Quantification of Actinide α-Radiation Damage in Minerals and Ceramics. Nature, 445, 190-193.
[3] Uggerhøj, U.I. (2005) The Interaction of Relativistic Particles with Strong Crystalline Fields. Review of Modern Physics, 77, 1131-1172.
[4] Møller, S.P. (1995) High-Energy Channeling—Applications in Beam Bending and Extraction. Nuclear Instruments and Methods in Physics Research Section A, 361, 403-420.
[5] Baurichter, A., et al. (2000) Channeling of High-Energy Particles in Bent Crystals—Experiments at the CERN SPS. Nuclear Instruments and Methods in Physics Research Section B, 164-165, 27-43.
[6] Biryukov, V.M., et al. (1994) On Measuring 70 GeV Proton Dechanneling Lengths in Silicon Crystals (110) and (111). Nuclear Instruments and Methods in Physics Research Section B, 86, 245-250.
[7] Feldman, L.C., Mayer, J.W. and Picraux, S.T. (1982) Materials Analysis by Ion Channeling: Submicron Crystallography. Academic Press, New York.
[8] Breese, M.B.H., Jemieson, D.N. and King, P.J.C. (1996) Materials Analysis Using a Nuclear Microprobe. John Wiley & Sons, New York.
[9] Scandale, W., et al. (2008) Apparatus to Study Crystal Channeling and Volume Reflection Phenomena at the SPS H8 Beamline. Review of Scientific Instruments, 79, Article ID: 023303.
[10] Rana, M.A. (2008) A New Method for Monitoring the Radiation Damage in Nuclear Waste Containers Using Ion Channeling. Annals of Nuclear Energy, 35, 1580-1583.
[11] Breese, M.B.H., Teo, E.J., Rana, M.A., Huang, L., van Kan, J.A., Watt, F. and King, P.J.C. (2004) Observation of Many Coherent Oscillations for MeV Protons Transmitted through Stacking Faults. Physical Review Letters, 92, Article ID: 045503.
[12] Rana, M.A., Breese, M.B.H. and Osipowicz, T. (2004) A Monte Carlo Simulation Study of Channelling and Dechannelling Enhancement Due to Lattice Translations. Nuclear Instruments and Methods in Physics Research Section B, 222, 53-60.
[13] Lu, Y., Cong, G.W., Liu, X.L., Lu, D.C. and Wang, Z.G. (2004) Depth Distribution of the Strain in the GaN Layer with Low-Temperature AlN Interlayer on Si(111) Substrate Studied by Rutherford Backscattering/Channeling. Applied Physics Letters, 85, 5562.
[14] Rana, M.A. (2008) Planar Channelling Criticalities of MeV Protons in Si Crystal: Simulations, Evaluation and Applications. Chinese Physics Letters, 25, 3724-3727.
[15] Wang, K., Ding, Z. and Yao, S. (2007) Elastic Strain in Mg0.28Zn0.72O Layer: Combined Rutherford Backscattering/Channeling and X-Ray Diffraction. Nuclear Instruments and Methods in Physics Research Section B, 259, 966-968.
[16] Lindhard, J. (1965) Influence of Crystal Lattice on Motion of Energetic Charged Particles. Kongel. Dan. Vidensk. Selsk., Mat.-Fys. Medd, 34, 1.
[17] Solov’yov, A.V., Schäfer, A. and Greiner, W. (1996) Channeling Process in a Bent Crystal. Physical Review E, 53, 1129-1137.
[18] Breese, M.B.H. (2007) A Large-Area Bent Crystal Shield for Deflection of High-Energy Ions. Applied Physics Letters, 91, Article ID: 261901.
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Mass ratio of air to a fuel
Air–fuel ratio (AFR) is the mass ratio of air to a solid, liquid, or gaseous fuel present in a combustion process. The combustion may take place in a controlled manner such as in an internal combustion engine or industrial furnace, or may result in an explosion (e.g., a dust explosion, gas or vapor explosion or in a thermobaric weapon).
The air–fuel ratio determines whether a mixture is combustible at all, how much energy is being released, and how much-unwanted pollutants are produced in the reaction. Typically a range of fuel to air ratios exists, outside of which ignition will not occur. These are known as the lower and upper explosive limits.
In an internal combustion engine or industrial furnace, the air–fuel ratio is an important measure for anti-pollution and performance-tuning reasons. If exactly enough air is provided to completely burn all of the fuel, the ratio is known as the stoichiometric mixture, often abbreviated to stoich. Ratios lower than stoichiometric are considered "rich". Rich mixtures are less efficient, but may produce more power and burn cooler. Ratios higher than stoichiometric are considered "lean". Lean mixtures are more efficient but may cause higher temperatures, which can lead to the formation of nitrogen oxides. Some engines are designed with features to allow lean-burn. For precise air–fuel ratio calculations, the oxygen content of combustion air should be specified because of different air density due to different altitude or intake air temperature, possible dilution by ambient water vapor, or enrichment by oxygen additions.
2 Engine management systems
3 Other types of engines
4 Other terms used
4.2 Fuel–air ratio (FAR)
4.3 Air–fuel equivalence ratio (λ)
4.4 Fuel–air equivalence ratio (ϕ)
4.5 Mixture fraction
4.6 Percent excess combustion air
In theory, a stoichiometric mixture has just enough air to completely burn the available fuel. In practice, this is never quite achieved, due primarily to the very short time available in an internal combustion engine for each combustion cycle. Most of the combustion process is completed in approximately 2 milliseconds at an engine speed of 6,000 revolutions per minute. (100 revolutions per second; 10 milliseconds per revolution of the crankshaft - which for a four-stroke engine would mean typically 5 milliseconds for each piston stroke). This is the time that elapses from the spark plug firing until 90% of the fuel–air mix is combusted, typically some 80 degrees of crankshaft rotation later. Catalytic converters are designed to work best when the exhaust gases passing through them are the result of nearly perfect combustion.
A perfectly stoichiometric mixture burns very hot and can damage engine components if the engine is placed under high load at this fuel–air mixture. Due to the high temperatures at this mixture, the detonation of the fuel-air mix while approaching or shortly after maximum cylinder pressure is possible under high load (referred to as knocking or pinging), specifically a "pre-detonation" event in the context of a spark-ignition engine model. Such detonation can cause serious engine damage as the uncontrolled burning of the fuel-air mix can create very high pressures in the cylinder. As a consequence, stoichiometric mixtures are only used under light to low-moderate load conditions. For acceleration and high-load conditions, a richer mixture (lower air–fuel ratio) is used to produce cooler combustion products (thereby utilizing evaporative cooling), and so avoid overheating of the cylinder head, and thus prevent detonation.
Engine management systems[edit]
The stoichiometric mixture for a gasoline engine is the ideal ratio of air to fuel that burns all fuel with no excess air. For gasoline fuel, the stoichiometric air–fuel mixture is about 14.7:1[1] i.e. for every one gram of fuel, 14.7 grams of air are required. For pure octane fuel, the oxidation reaction is:
25 O2 + 2 C8H18 → 16 CO2 + 18 H2O + energy
Any mixture greater than 14.7:1 is considered a lean mixture; any less than 14.7:1 is a rich mixture – given perfect (ideal) "test" fuel (gasoline consisting of solely n-heptane and iso-octane). In reality, most fuels consist of a combination of heptane, octane, a handful of other alkanes, plus additives including detergents, and possibly oxygenators such as MTBE (methyl tert-butyl ether) or ethanol/methanol. These compounds all alter the stoichiometric ratio, with most of the additives pushing the ratio downward (oxygenators bring extra oxygen to the combustion event in liquid form that is released at the time of combustions; for MTBE-laden fuel, a stoichiometric ratio can be as low as 14.1:1). Vehicles that use an oxygen sensor or other feedback loops to control fuel to air ratio (lambda control), compensate automatically for this change in the fuel's stoichiometric rate by measuring the exhaust gas composition and controlling fuel volume. Vehicles without such controls (such as most motorcycles until recently, and cars predating the mid-1980s) may have difficulties running certain fuel blends (especially winter fuels used in some areas) and may require different carburetor jets (or otherwise have the fueling ratios altered) to compensate. Vehicles that use oxygen sensors can monitor the air–fuel ratio with an air–fuel ratio meter.
Other types of engines[edit]
In the typical air to natural gas combustion burner, a double-cross limit strategy is employed to ensure ratio control. (This method was used in World War II).[citation needed] The strategy involves adding the opposite flow feedback into the limiting control of the respective gas (air or fuel). This assures ratio control within an acceptable margin.
Other terms used[edit]
There are other terms commonly used when discussing the mixture of air and fuel in internal combustion engines.
Mixture is the predominant word that appears in training texts, operation manuals, and maintenance manuals in the aviation world.
Air–fuel ratio is the ratio between the mass of air and the mass of fuel in the fuel–air mix at any given moment. The mass is the mass of all constituents that compose the fuel and air, whether combustible or not. For example, a calculation of the mass of natural gas—which often contains carbon dioxide (CO
2), nitrogen (N
2), and various alkanes—includes the mass of the carbon dioxide, nitrogen and all alkanes in determining the value of mfuel.[2]
For pure octane the stoichiometric mixture is approximately 15.1:1, or λ of 1.00 exactly.
In naturally aspirated engines powered by octane, maximum power is frequently reached at AFRs ranging from 12.5 to 13.3:1 or λ of 0.850 to 0.901.[citation needed]
The air-fuel ratio of 12:1 is considered as the maximum output ratio, whereas the air-fuel ratio of 16:1 is considered as the maximum fuel economy ratio.[citation needed]
Fuel–air ratio (FAR) [edit]
Fuel–air ratio is commonly used in the gas turbine industry as well as in government studies of internal combustion engine, and refers to the ratio of fuel to the air.[citation needed]
{\displaystyle \mathrm {FAR} ={\frac {1}{\mathrm {AFR} }}}
Air–fuel equivalence ratio (λ)[edit]
Air–fuel equivalence ratio, λ (lambda), is the ratio of actual AFR to stoichiometry for a given mixture. λ = 1.0 is at stoichiometry, rich mixtures λ < 1.0, and lean mixtures λ > 1.0.
{\displaystyle \lambda ={\frac {\mathrm {AFR} }{\mathrm {AFR} _{\text{stoich}}}}}
Because the composition of common fuels varies seasonally, and because many modern vehicles can handle different fuels when tuning, it makes more sense to talk about λ values rather than AFR.
Most practical AFR devices actually measure the amount of residual oxygen (for lean mixes) or unburnt hydrocarbons (for rich mixtures) in the exhaust gas.
Fuel–air equivalence ratio (ϕ)[edit]
The fuel–air equivalence ratio, ϕ (phi), of a system is defined as the ratio of the fuel-to-oxidizer ratio to the stoichiometric fuel-to-oxidizer ratio. Mathematically,
{\displaystyle \phi ={\frac {\mbox{fuel-to-oxidizer ratio}}{({\mbox{fuel-to-oxidizer ratio}})_{\text{st}}}}={\frac {m_{\text{fuel}}/m_{\text{ox}}}{\left(m_{\text{fuel}}/m_{\text{ox}}\right)_{\text{st}}}}={\frac {n_{\text{fuel}}/n_{\text{ox}}}{\left(n_{\text{fuel}}/n_{\text{ox}}\right)_{\text{st}}}}}
where m represents the mass, n represents a number of moles, subscript st stands for stoichiometric conditions.
The advantage of using equivalence ratio over fuel–oxidizer ratio is that it takes into account (and is therefore independent of) both mass and molar values for the fuel and the oxidizer. Consider, for example, a mixture of one mole of ethane (C
6) and one mole of oxygen (O
2). The fuel–oxidizer ratio of this mixture based on the mass of fuel and air is
{\displaystyle {\frac {m_{{\ce {C2H6}}}}{m_{{\ce {O2}}}}}={\frac {1\times (2\times 12+6\times 1)}{1\times (2\times 16)}}={\frac {30}{32}}=0.9375}
and the fuel-oxidizer ratio of this mixture based on the number of moles of fuel and air is
{\displaystyle {\frac {n_{{\ce {C2H6}}}}{n_{{\ce {O2}}}}}={\frac {1}{1}}=1}
Clearly the two values are not equal. To compare it with the equivalence ratio, we need to determine the fuel–oxidizer ratio of ethane and oxygen mixture. For this we need to consider the stoichiometric reaction of ethane and oxygen,
C2H6 + 7⁄2 O2 → 2 CO2 + 3 H2O
{\displaystyle ({\text{fuel-to-oxidizer ratio based on mass}})_{\text{st}}=\left({\frac {m_{{\ce {C2H6}}}}{m_{{\ce {O2}}}}}\right)_{\text{st}}={\frac {1\times (2\times 12+6\times 1)}{3.5\times (2\times 16)}}={\frac {30}{112}}=0.268}
{\displaystyle ({\text{fuel-to-oxidizer ratio based on number of moles}})_{\text{st}}=\left({\frac {n_{{\ce {C2H6}}}}{n_{{\ce {O2}}}}}\right)_{\text{st}}={\frac {1}{3.5}}=0.286}
Thus we can determine the equivalence ratio of the given mixture as
{\displaystyle \phi ={\frac {m_{{\ce {C2H6}}}/m_{{\ce {O2}}}}{\left(m_{{\ce {C2H6}}}/m_{{\ce {O2}}}\right)_{\text{st}}}}={\frac {0.938}{0.268}}=3.5}
{\displaystyle \phi ={\frac {n_{{\ce {C2H6}}}/n_{{\ce {O2}}}}{\left(n_{{\ce {C2H6}}}/n_{{\ce {O2}}}\right)_{\text{st}}}}={\frac {1}{0.286}}=3.5}
Another advantage of using the equivalence ratio is that ratios greater than one always mean there is more fuel in the fuel–oxidizer mixture than required for complete combustion (stoichiometric reaction), irrespective of the fuel and oxidizer being used—while ratios less than one represent a deficiency of fuel or equivalently excess oxidizer in the mixture. This is not the case if one uses fuel–oxidizer ratio, which takes different values for different mixtures.
The fuel–air equivalence ratio is related to the air–fuel equivalence ratio (defined previously) as follows:
{\displaystyle \phi ={\frac {1}{\lambda }}}
Mixture fraction[edit]
Further information: Mixture fraction
The relative amounts of oxygen enrichment and fuel dilution can be quantified by the mixture fraction, Z, defined as
{\displaystyle Z=\left[{\frac {sY_{\mathrm {F} }-Y_{\mathrm {O} }+Y_{\mathrm {O,0} }}{sY_{\mathrm {F,0} }+Y_{\mathrm {O,0} }}}\right]}
{\displaystyle s=\mathrm {AFR} _{\mathrm {stoich} }={\frac {W_{\mathrm {O} }\times v_{\mathrm {O} }}{W_{\mathrm {F} }\times v_{\mathrm {F} }}}}
YF,0 and YO,0 represent the fuel and oxidizer mass fractions at the inlet, WF and WO are the species molecular weights, and vF and vO are the fuel and oxygen stoichiometric coefficients, respectively. The stoichiometric mixture fraction is
{\displaystyle Z_{\mathrm {st} }=\left[{\frac {1}{1+{\frac {Y_{\mathrm {F,0} }\times W_{\mathrm {O} }\times v_{\mathrm {O} }}{Y_{\mathrm {O,0} }\times W_{\mathrm {F} }\times v_{\mathrm {F} }}}}}\right]}
The stoichiometric mixture fraction is related to λ (lambda) and ϕ (phi) by the equations
{\displaystyle Z_{\text{st}}={\frac {\lambda }{1+\lambda }}={\frac {1}{1+\phi }}}
{\displaystyle \mathrm {AFR} ={\frac {Y_{\mathrm {O,0} }}{Y_{\mathrm {F,0} }}}}
Percent excess combustion air[edit]
Ideal stoichiometry
In industrial fired heaters, power plant steam generators, and large gas-fired turbines, the more common terms are percent excess combustion air and percent stoichiometric air.[5][6] For example, excess combustion air of 15 percent means that 15 percent more than the required stoichiometric air (or 115 percent of stoichiometric air) is being used.
A combustion control point can be defined by specifying the percent excess air (or oxygen) in the oxidant, or by specifying the percent oxygen in the combustion product.[7] An air–fuel ratio meter may be used to measure the percent oxygen in the combustion gas, from which the percent excess oxygen can be calculated from stoichiometry and a mass balance for fuel combustion. For example, for propane (C
8) combustion between stoichiometric and 30 percent excess air (AFRmass between 15.58 and 20.3), the relationship between percent excess air and percent oxygen is:
{\displaystyle {\begin{aligned}\mathrm {Mass\%\ O_{2}\ in\ propane\ combustion\ gas} &\approx -0.1433(\mathrm {\%\ excess\ O_{2}} )^{2}+0.214(\mathrm {\%\ excess\ O_{2}} )\\\mathrm {Volume\%\ O_{2}\ in\ propane\ combustion\ gas} &\approx -0.1208(\mathrm {\%\ excess\ O_{2}} )^{2}+0.186(\mathrm {\%\ excess\ O_{2}} )\end{aligned}}}
Stoichiometric air-to-fuel ratio of common fuels
^ Hillier, V.A.W.; Pittuck, F.W. (1966). "Sub-section 3.2". Fundamentals of Motor Vehicle Technology. London: Hutchinson Educational. ISBN 0 09 110711 3.
^ See Example 15.3 in Çengel, Yunus A.; Boles, Michael A. (2006). Thermodynamics: An Engineering Approach (5th ed.). Boston: McGraw-Hill. ISBN 9780072884951.
^ Kumfer, B.; Skeen, S.; Axelbaum, R. (2008). "Soot inception limits in laminar diffusion flames with application to oxy-fuel combustion" (PDF). Combustion and Flame. 154: 546–556. doi:10.1016/j.combustflame.2008.03.008.
^ Introduction to Fuel and Energy: 1) MOLES, MASS, CONCENTRATION AND DEFINITIONS, accessed 2011-05-25
^ "Energy Tips – Process Heating – Check Burner Air to Fuel Ratios" (PDF). U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy. November 2007. Retrieved 29 July 2013.
^ "Stoichiometric combustion and excess of air". The Engineering ToolBox. Retrieved 29 July 2013.
^ Eckerlin, Herbert M. "The Importance of Excess Air in the Combustion Process" (PDF). Mechanical and Aerospace Engineering 406 - Energy Conservation in Industry. North Carolina State University. Archived from the original (PDF) on 27 March 2014. Retrieved 29 July 2013.
HowStuffWorks: fuel injection, catalytic converter
University of Plymouth: Engine Combustion primer
Kamm, Richard W. "Mixed Up About Fuel Mixtures?". Aircraft Maintenance Technology (February 2002). Archived from the original on 2010-11-20. Retrieved 2009-03-18.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Air–fuel_ratio&oldid=1078714159"
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Pons_asinorum Knowpia
In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (Latin: [ˈpõːs asɪˈnoːrũː], English: /ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-i-NOR-əm), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The term is also applied to the Pythagorean theorem.[1]
Pons asinorum is also used metaphorically for a problem or challenge which acts as a test of critical thinking in a field, separating capable and incapable reasoners; it represents a test of ability or understanding. Its first known usage in this was in 1645.[2]
A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.[3][4] In fact, Marvin Minsky recounts that he had rediscovered the Pappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.[5][6]
Euclid and ProclusEdit
Euclid's statement of the pons asinorum includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's proof involves drawing auxiliary lines to these extensions. But, as Euclid's commentator Proclus points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way.
There has been much speculation and debate as to why Euclid added the second conclusion to the theorem, given that it makes the proof more complicated. One plausible explanation, given by Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions where Euclid does not cover every case.[7] The proof relies heavily on what is today called side-angle-side, the previous proposition in the Elements.
Proclus' variation of Euclid's proof proceeds as follows:[8]
Let ABC be an isosceles triangle with AB and AC being the equal sides. Pick an arbitrary point D on side AB and construct E on AC so that AD = AE. Draw the lines BE, DC and DE.
Consider the triangles BAE and CAD; BA = CA, AE = AD, and
{\displaystyle \angle A}
is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal.
{\displaystyle \angle ABE=\angle ACD}
{\displaystyle \angle ADC=\angle AEB}
, and BE = CD.
Since AB = AC and AD = AE, BD = CE by subtraction of equal parts.
Now consider the triangles DBE and ECD; BD = CE, BE = CD, and
{\displaystyle \angle DBE=\angle ECD}
have just been shown, so applying side-angle-side again, the triangles are congruent.
{\displaystyle \angle BDE=\angle CED}
{\displaystyle \angle BED=\angle CDE}
{\displaystyle \angle BDE=\angle CED}
{\displaystyle \angle CDE=\angle BED}
{\displaystyle \angle BDC=\angle CEB}
by subtraction of equal parts.
Consider a third pair of triangles, BDC and CEB; DB = EC, DC = EB, and
{\displaystyle \angle BDC=\angle CEB}
, so applying side-angle-side a third time, the triangles are congruent.
In particular, angle CBD = BCE, which was to be proved.
PappusEdit
Proclus gives a much shorter proof attributed to Pappus of Alexandria. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.[9][6] This method is lampooned by Charles Lutwidge Dodgson in Euclid and his Modern Rivals, calling it an "Irish bull" because it apparently requires the triangle to be in two places at once.[10]
The proof is as follows:[11]
Let ABC be an isosceles triangle with AB and AC being the equal sides.
Consider the triangles ABC and ACB, where ACB is considered a second triangle with vertices A, C and B corresponding respectively to A, B and C in the original triangle.
{\displaystyle \angle A}
is equal to itself, AB = AC and AC = AB, so by side-angle-side, triangles ABC and ACB are congruent.
{\displaystyle \angle B=\angle C}
A standard textbook method is to construct the bisector of the angle at A.[13] This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of the Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.
The proof proceeds as follows:[14]
As before, let the triangle be ABC with AB = AC.
Construct the angle bisector of
{\displaystyle \angle BAC}
and extend it to meet BC at X.
AB = AC and AX is equal to itself.
{\displaystyle \angle BAX=\angle CAX}
, so, applying side-angle-side, triangle BAX and triangle CAX are congruent.
It follows that the angles at B and C are equal.
In inner product spacesEdit
{\displaystyle x+y+z=0{\text{ and }}\|x\|=\|y\|,}
{\displaystyle \|x-z\|=\|y-z\|.}
{\displaystyle \|x-z\|^{2}=\|x\|^{2}-2x\cdot z+\|z\|^{2},}
{\displaystyle x\cdot z=\|x\|\|z\|\cos \theta }
Etymology and related termsEdit
Another medieval term for the pons asinorum was Elefuga which, according to Roger Bacon, comes from Greek elegia "misery", and Latin fuga "flight", that is "flight of the wretches". Though this etymology is dubious, it is echoed in Chaucer's use of the term "flemyng of wreches" for the theorem.[17]
Similarly, the name Dulcarnon was given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theorem, after the Arabic Dhū 'l qarnain ذُو ٱلْقَرْنَيْن, meaning "the owner of the two horns", because diagrams of the theorem showed two smaller squares like horns at the top of the figure. The term is also used as a metaphor for a dilemma.[17] The theorem was also sometimes called "the Windmill" for similar reasons.[20]
Metaphorical usageEdit
Uses of the pons asinorum as a metaphor for a test of critical thinking include:
Pons Asinorum is the name given to a particular configuration[23] of a Rubik's Cube.
Eric Raymond referred to the issue of syntactically-significant whitespace in the Python programming language as its pons asinorum.[24]
^ Smith, David Eugene (1925). History Of Mathematics. Vol. II. Ginn And Company. pp. 284. It formed at bridge across which fools could not hope to pass, and was therefore known as the pons asinorum, or bridge of fools.¹
1. The term is something applied to the Pythagorean Theorem.
^ Pons asinorum — Definition and More from the Free Merriam
^ Jaakko Hintikka, "On Creativity in Reasoning", in Ake E. Andersson, N.E. Sahlin, eds., The Complexity of Creativity, 2013, ISBN 9401587884, p. 72
^ A. Battersby, Mathematics in Management, 1966, quoted in Deakin
^ Jeremy Bernstein, "Profiles: A.I." (interview with Marvin Minsky), The New Yorker December 14, 1981, p. 50-126
^ a b Michael A.B. Deakin, "From Pappus to Today: The History of a Proof", The Mathematical Gazette 74:467:6-11 (March 1990) JSTOR 3618841
^ Heath pp. 251–255
^ Following Proclus p. 53
^ For example F. Cuthbertson Primer of geometry (1876 Oxford) p. 7
^ Charles Lutwidge Dodgson, Euclid and his Modern Rivals Act I Scene II §6
^ Heath p. 254 for section
^ For example J.M. Wilson Elementary geometry (1878 Oxford) p. 20
^ Following Wilson
^ A. M. Legendre Éléments de géométrie (1876 Libr. de Firmin-Didot et Cie) p. 14
^ J. R. Retherford, Hilbert Space, Cambridge University Press, 1993, page 27.
^ a b c d A. F. West & H. D. Thompson "On Dulcarnon, Elefuga And Pons Asinorum as Fanciful Names For Geometrical Propositions" The Princeton University bulletin Vol. 3 No. 4 (1891) p. 84
^ D.E. Smith History of Mathematics (1958 Dover) p. 284
^ Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p. 202. ISBN 0-309-08549-7. first-class mathematician. {{cite book}}: CS1 maint: location (link)
^ W.E. Aytoun (Ed.) The poetical works of Thomas Campbell (1864, Little, Brown) p. 385 Google Books
^ John Stuart Mill Principles of Political Economy (1866: Longmans, Green, Reader, and Dyer) Book 2, Chapter 16, p. 261
^ Reid, Michael (28 October 2006). "Rubik's Cube patterns". www.cflmath.com. Archived from the original on 12 December 2012. Retrieved 22 September 2019.
^ Eric S. Raymond, "Why Python?", Linux Journal, April 30, 2000
^ Aasinsilta on laiskurin apuneuvo | Yle Uutiset | yle.fi
Pons asinorum at PlanetMath.
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Lemma 26.7.8 (01IE)—The Stacks project
Section 26.7: Quasi-coherent sheaves on affines
Lemma 26.7.8. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be an affine scheme. Suppose that
is a short exact sequence of sheaves of $\mathcal{O}_ X$-modules. If two out of three are quasi-coherent then so is the third.
Proof. This is clear in case both $\mathcal{F}_1$ and $\mathcal{F}_2$ are quasi-coherent because the functor $M \mapsto \widetilde M$ is exact, see Lemma 26.5.4. Similarly in case both $\mathcal{F}_2$ and $\mathcal{F}_3$ are quasi-coherent. Now, suppose that $\mathcal{F}_1 = \widetilde M_1$ and $\mathcal{F}_3 = \widetilde M_3$ are quasi-coherent. Set $M_2 = \Gamma (X, \mathcal{F}_2)$. We claim it suffices to show that the sequence
\[ 0 \to M_1 \to M_2 \to M_3 \to 0 \]
is exact. Namely, if this is the case, then (by using the mapping property of Lemma 26.7.1) we get a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \widetilde M_1 \ar[r] \ar[d] & \widetilde M_2 \ar[r] \ar[d] & \widetilde M_3 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1 \ar[r] & \mathcal{F}_2 \ar[r] & \mathcal{F}_3 \ar[r] & 0 } \]
and we win by the snake lemma.
The “correct” argument here would be to show first that $H^1(X, \mathcal{F}) = 0$ for any quasi-coherent sheaf $\mathcal{F}$. This is actually not all that hard, but it is perhaps better to postpone this till later. Instead we use a small trick.
Pick $m \in M_3 = \Gamma (X, \mathcal{F}_3)$. Consider the following set
\[ I = \{ f \in R \mid \text{the element }fm\text{ comes from }M_2\} . \]
Clearly this is an ideal. It suffices to show $1 \in I$. Hence it suffices to show that for any prime $\mathfrak p$ there exists an $f \in I$, $f \not\in \mathfrak p$. Let $x \in X$ be the point corresponding to $\mathfrak p$. Because surjectivity can be checked on stalks there exists an open neighbourhood $U$ of $x$ such that $m|_ U$ comes from a local section $s \in \mathcal{F}_2(U)$. In fact we may assume that $U = D(f)$ is a standard open, i.e., $f \in R$, $f \not\in \mathfrak p$. We will show that for some $N \gg 0$ we have $f^ N \in I$, which will finish the proof.
Take any point $z \in V(f)$, say corresponding to the prime $\mathfrak q \subset R$. We can also find a $g \in R$, $g \not\in \mathfrak q$ such that $m|_{D(g)}$ lifts to some $s' \in \mathcal{F}_2(D(g))$. Consider the difference $s|_{D(fg)} - s'|_{D(fg)}$. This is an element $m'$ of $\mathcal{F}_1(D(fg)) = (M_1)_{fg}$. For some integer $n = n(z)$ the element $f^ n m'$ comes from some $m'_1 \in (M_1)_ g$. We see that $f^ n s$ extends to a section $\sigma $ of $\mathcal{F}_2$ on $D(f) \cup D(g)$ because it agrees with the restriction of $f^ n s' + m'_1$ on $D(f) \cap D(g) = D(fg)$. Moreover, $\sigma $ maps to the restriction of $f^ n m$ to $D(f) \cup D(g)$.
Since $V(f)$ is quasi-compact, there exists a finite list of elements $g_1, \ldots , g_ m \in R$ such that $V(f) \subset \bigcup D(g_ j)$, an integer $n > 0$ and sections $\sigma _ j \in \mathcal{F}_2(D(f) \cup D(g_ j))$ such that $\sigma _ j|_{D(f)} = f^ n s$ and $\sigma _ j$ maps to the section $f^ nm|_{D(f) \cup D(g_ j)}$ of $\mathcal{F}_3$. Consider the differences
\[ \sigma _ j|_{D(f) \cup D(g_ jg_ k)} - \sigma _ k|_{D(f) \cup D(g_ jg_ k)}. \]
These correspond to sections of $\mathcal{F}_1$ over $D(f) \cup D(g_ jg_ k)$ which are zero on $D(f)$. In particular their images in $\mathcal{F}_1(D(g_ jg_ k)) = (M_1)_{g_ jg_ k}$ are zero in $(M_1)_{g_ jg_ kf}$. Thus some high power of $f$ kills each and every one of these. In other words, the elements $f^ N \sigma _ j$, for some $N \gg 0$ satisfy the glueing condition of the sheaf property and give rise to a section $\sigma $ of $\mathcal{F}_2$ over $\bigcup (D(f) \cup D(g_ j)) = X$ as desired. $\square$
Typo in statement: sheaves of
\mathcal{O}_X
5 comment(s) on Section 26.7: Quasi-coherent sheaves on affines
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Theoretical Calculation of Lift Force for Ideal Electric Asymmetric Capacitor Loaded by High Voltage
Engineering > Vol.12 No.1, January 2020
Theoretical Calculation of Lift Force for Ideal Electric Asymmetric Capacitor Loaded by High Voltage ()
Xiangyu Cheng1,2, Guangli Kuang1*, Yan Zhang3, Pengcheng Huang1, Donghui Jiang1, Shili Jiang1, Xinxing Qian1, Hangwei Ding1, Wenge Chen1
1Hefei Institute of Physical Science, Chinese Academy of Science, Hefei, China.
2No. 38 Research Institute of CETC, Hefei, China.
3Anhui Technical College of Water Resources and Hydroelectric Power, Hefei, China.
Asymmetric capacitor like the so-called lifter can fly up from the ground. Some common characteristics exist in the asymmetric capacitor: high-voltage, capacitor, lift force. What are the accurate quantitative relations among them and how can we figure the lift force out? It’s a thorny problem so far. In this article, we attempt to establish a model that can match the actual experimental data and theory derivation result. After checking, the calculation result is verified to be correct.
High-Voltage, Asymmetric Capacitor, Lift Force, Calculation
Cheng, X. , Kuang, G. , Zhang, Y. , Huang, P. , Jiang, D. , Jiang, S. , Qian, X. , Ding, H. and Chen, W. (2020) Theoretical Calculation of Lift Force for Ideal Electric Asymmetric Capacitor Loaded by High Voltage. Engineering, 12, 33-40. doi: 10.4236/eng.2020.121003.
Hairs loaded high-voltage [1] can float up in air. Consisted of balsa wood, fine cooper, and aluminum foil, so-called lifter (refer Figure 1) can get off the ground after loading more than a high-voltage of 20 kV [2] [3]. We attempt to find out the rules hid behind the phenomena about what are the dominant factors that affect the lift force [4]. So we can further adjust the factors to enhance the force.
It is clear that the lift force tends to be positively correlated with the high-voltage in condition of under the breakdown voltage loaded between the 2 electrodes. There are two respects included in the words: 1) higher voltage input means greater force output; 2) the high-voltage loaded between the electrodes
Figure 1. A lifter flying up loaded with high-voltage power.
can’t be too high leading to breakdown. The voltage is limited by the breakdown voltage in air, and so does the lift force. The important work is how to get the maximum force at the highest voltage below breakdown. Are there any other methods to break through the limit, so that we can enhance the output of force?
Different from the methods that exist nowadays [5], we provided a simpler and more accurate way. The method is established on hypotheses. By a series of theoretical derivation, we turn out a brief result for calculating the lift force. After comparing with experimental results, the validity of the calculation result is verified.
2. Description of Problem and Overview of Result
The original problem we described is as follows:
When we load 20 kV high-voltage on an asymmetric capacitor like as lifter (refer Figure 2), a thrust [6] is produced on the lifter. The force direction normally is from the big electrode plate to the small one [7]. It can make the asymmetric capacitor to move ahead. Just like the thruster of MIT ionocraft (Ion-propelled aircraft). But what is the detail value of the thrust force and how to figure it out, which is the task we want to solve in next content.
When the two electrode plates of asymmetric capacitor are loaded within a high-voltage circuit, the electric charge carried on the two plates is equal. The different surface areas, lead to the different the surface charged densities. Because of the surface charge distributions can be affected by their shapes and the space between the 2 plates, the calculation is very complex and difficult to establish a quantitative model.
For convenience sake, we plan to simplify the charged asymmetric capacitor model (refer Figure 3) into an ideal state firstly. After figuring out the quantitative
Figure 2. A lifter composed by asymmetric capacitors.
Figure 3. A simple asymmetric capacitor model.
relation by the ideal model, we further expand it into the general cases of complex distributed charged densities. The calculation and test result indicated that this is a feasible method to solve the complicated problem.
After simplification, we successfully figured out the force of uniform distribution of surface charge of asymmetric capacitor. Subsequently, we will give the deduced result firstly, and then show the derived process.
2.2. Result Overview
The result of the problem is as follows:
If the surface electric density is uniform under the ideal condition, the thrust force of an asymmetric capacitor charged with high-voltage electricity can be expressed as Equation (1):
f=\frac{{q}^{2}}{\epsilon }\left(\frac{1}{{S}_{1}}-\frac{1}{{S}_{2}}\right)
where f is the thrust force of asymmetric capacitor in a static environment [8], q is the quantity of electric charge on the capacitor plate,
{S}_{1}
is superficial area of the small plate,
{S}_{2}
is superficial area of the large plate and
\epsilon
is the dielectric constant.
In Equation (1), the dielectric constant
\epsilon
can also be written into
\epsilon ={\epsilon }_{r}{\epsilon }_{0}
{\epsilon }_{r}
is relative dielectric constant and
{\epsilon }_{0}
is vacuum dielectric constant.
In above content, the method of calculating thrust force is performed in ideal static environment [9] [10]. To further illustrate Equation (1), the details will be derived subsequently.
We use two ways to prove the thrust force calculation formula separately.
For an ideal equal charge density distribution model, from surface charge density
\sigma =\frac{q}{S}
, we can get the surface charge density of the plate 1 (we normally call the small one plate 1 and the large one plate 2) is
{\sigma }_{1}=\frac{q}{{S}_{1}}
, and the plate 2 is
{\sigma }_{2}=\frac{q}{{S}_{2}}
3.1. The First Way
The first way to calculate the thrust force of an asymmetric capacitor is shown as follows:
Assuming the charge density of the plate 1 is equal to plate 2, the positive charge carried by plate 1 is
{q}_{1h}={S}_{1}{\sigma }_{2}={S}_{1}\frac{q}{{S}_{2}}
At this moment, the electric field strength nearby plate 1 is equal to plate 2
{E}_{1h}={E}_{2}=\frac{{\sigma }_{2}}{\epsilon }=\frac{q}{\epsilon {S}_{2}}
Because of actual electric charge q is
n=\frac{q}{{q}_{1h}}=\frac{{S}_{2}}{{S}_{1}}
times of the hypothetical electric charge
{q}_{1h}
, it needs n electric fields
{E}_{1h}
to superpose each other to compose the actual field strength on plate 1.
The residual electric charge
{q}_{1r}
removed the assuming charge
{q}_{1h}
from actual charge q is
{q}_{1r}=q-{q}_{1h}=q-{S}_{1}{\sigma }_{2}=q\left(1-\frac{{S}_{1}}{{S}_{2}}\right)
We need n residual electric charges
{q}_{1r}
to superpose each other to compose the actual field strength on plate 1
{E}_{1r}=q\left(1-\frac{{S}_{1}}{{S}_{2}}\right)\cdot n
. So we can get the electric field force f of the n residual electric charge
{q}_{1r}
in assuming electric field
{E}_{1h}={E}_{2}
\begin{array}{c}f=q\left(1-\frac{{S}_{1}}{{S}_{2}}\right)\cdot {E}_{2}\cdot n\\ =q\left(1-\frac{{S}_{1}}{{S}_{2}}\right)\cdot \frac{q}{\epsilon {S}_{2}}\cdot \frac{{S}_{2}}{{S}_{1}}\\ =\frac{{q}^{2}}{\epsilon }\left(\frac{1}{{S}_{1}}-\frac{1}{{S}_{2}}\right)\end{array}
Therefore we get the right result of the thrust force of practical asymmetric capacitor.
3.2. The Second Way
We assume the air medium between the two plates of the asymmetric capacitor as a whole block in physical shape, and electrical non-conduction at the inner (refer Figure 4). The gaps between the air block surfaces near to the plates are infinitesimal. So the molecular collision between them causes to charge exchange, and then leads to the upper surface positively charged and the lower surface negative charged. As the reason of like charges repelling, there is a pair of reaction forces produced on the surfaces of air block and the plates. You can further find the force loaded on the upper plate is larger than the lower plate for reason of different plate areas causing to different electric densities. That is to say their sum is positive. It is exactly the reason that an asymmetric capacitor like as lifter loaded by high voltage DC power can produce an upward lift force. Then, we can attempt to figure out the strength of the levitation force.
The forces of the top and bottom surfaces of the air block exerted by the asymmetric capacitor are different. Because of different areas S lead to different
charge densities
\sigma
, combining
E=\frac{q}{\sigma }
f=qE
, we can find that the sum
force of the air block is not equal to zero. It gets a total trend to downward by the electric field force exerted, and makes the asymmetric capacitor to get the trend to upward which is from the large plate pointing to the small plate. So, the lift force produced by the asymmetric capacitor is
Figure 4. Air block analytical approach of asymmetric capacitor.
\begin{array}{c}f={f}_{1u}-{f}_{2d}={f}_{1d}-{f}_{2u}=q{E}_{1}-q{E}_{2}\\ =q\frac{q}{\epsilon {S}_{1}}-q\frac{q}{\epsilon {S}_{2}}=\frac{{q}^{2}}{\epsilon }\left(\frac{1}{{S}_{1}}-\frac{1}{{S}_{2}}\right)\end{array}
4. Brief Analysis of Result
The results by above methods are the same. Except q, the other three variables
\epsilon
{S}_{1}
{S}_{\text{2}}
are known. So, before we can calculate the final lift force, we must to figure out the carried charge firstly. If when
\epsilon =\text{8}\text{.854187817}\times {\text{10}}^{-12}\text{F}/\text{m}
{S}_{1}\in \left[1.8\text{9}\times {10}^{-4},240\times {10}^{-4}\right]{\text{m}}^{2}
{S}_{\text{2}}\in \left[1.8\text{9}\times {10}^{-4},240\times {10}^{-4}\right]{\text{m}}^{2}
q=1.8\times {10}^{-8}\text{C}
and using above formula
f=\frac{{q}^{2}}{\epsilon }\left(\frac{1}{{S}_{1}}-\frac{1}{{S}_{2}}\right)
, we can get the lift force produced by the asymmetric capacitor that is between the maximum 0.2 N and the minimum −0.2 N. The trend of
f\left({S}_{1},{S}_{2}\right)
is drawn in Figure 5.
We can see that smaller small-plate than the large-plate means stronger force output. But too smaller dimension may lead to structural strength problem. In other works, by reference to kinds of lifters, we found that thinner copper wire can produce more lift force. The trend reflected by the formula is verified to be correct.
From Equation (1) and Figure 5, we can find the lift force f is sensitive to amount of the charges carried on asymmetric capacitor. As
q=UC
, the amount of the charges q is proportional to the load voltage U. Through an asymmetric capacitor in lifter form (refer Figure 6), we test the relation between load voltage U and force f, and achieve a data table (refer Table 1). This trend of force change reflects (refer Figure 7) that f is assuredly sensitive to U. As the voltage is increased, the lift force increases quickly with charge growing on the plates.
Figure 6. Test lifter of asymmetric capacitor.
Table 1. The relation table between voltage and lift force of a lifter.
Figure 7. Relative changing trend between voltage and lift force of a lifter.
Through two methods, we achieved lift force formula of asymmetric capacitor successfully. But how to obtain the value q of the asymmetric capacitor, which needs much more content to present the derivation details? So including the experiment verification of the calculating method for the lift force of asymmetric capacitor, we plan to write them in the follow-up papers. In this one, we mainly completed the lift force calculation established on some theoretical assumptions by two ways, the results by both ways point to the same form formula. It hints us that the calculation may be right, as for the exact verification of the formula, we will continue to achieve it at next papers to thoroughly resolve the lift force problem of asymmetric capacitor loaded by high voltage.
The authors gratefully acknowledge the support of the Thirteenth Five-Year Plan of Hefei Institute of Physical Science of Chinese Academy of Science (Grant No. Y86CT21051 “Electric and Magnetic Propulsion System”), the Research Activity Funding of Postdoctoral Fellow of Anhui Province (Grant No. 2018B250 “High-energy Ions Accelerated Thruster”), the Natural Science Research Project of Anhui Education Department (Grant No. KJ2018A0725 “the Uniformity Optimization and Software Development for MRI Magnets”), and a portion of this work was supported by the High Magnetic Field Laboratory of Anhui Province.
[1] Koike, K., et al. (2004) Structural Analysis of Human Hair Fibers under the Ultra-High Voltage Electron Microscope. Journal of Cosmetic Science, 55, S25-S27.
[2] Einat, M. and Kalderon, R. (2014) High Efficiency Lifter Based on the Biefeld-Brown Effect. AIP Advances, 4, 337-350.
[3] Ren, J.X., Liu, Y., Jiang, X.L., Le, X.Y. and Wang, L.Y. (2010) Investigation of the Lifter’s Lift Mechanism and Efficiency. Journal of Aerospace Power, 25, 1395-1400.
[4] Bahder, T. and Fazi, Ch. (2003) Force on an Asymmetrical Capacitor. Army Report ARL-TR-3005. Army Res. Lab., Adelphi.
[5] Canning, F.X., Melcher, C. and Winet, E. (2004) Asymmetrical Capacitors for Propulsion. NASA/CR, 2004-213312.
[6] Brown, T.T. (1928) A Method of and an Apparatus or Machine for Producing Force or Motion. UK Patent No. 300311.
[7] Long, K. (2012) The Role of Speculative Science in Driving Technology. In: Deep Space Propulsion, Springer, New York, NY, 287-304.
[8] Primas, J., Malík, M., Jašíková, D. and Kopecky, V. (2010) Force on High Voltage Capacitor with Asymmetrical Electrodes. Processing of the WASET 2010 Conference, Amsterdam, 335-339.
[9] Malík, M., Primas, J., Kotek, M., Jašíková, D. and Kopecky, V. (2019) Influence of an External Air Flow on a System of High-Voltage Asymmetrical Electrodes. Journal of Applied Mechanics and Technical Physics, 60, 1-6.
[10] Tajmar, M. (2004) Biefeld-Brown Effect: Misinterpretation of Corona Wind Phenomena. AIAA Journal, 42, 315-318.
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3 Ways to Account for Stock Based Compensation - wikiHow
1 Calculating Compensation Value
3 Recording Compensation As an Employee
Stock compensation is a way for companies to pay employees in shares of stock or stock options. Stock options are the most common type of stock compensation and allow an employee to purchase the company's stock at a set price during a set vesting period. Accounting for stock compensation is significantly more complex than doing so for traditional compensation. The company is required to properly value the stock or stock options and then make accounting entries to record stock compensation expense.
Calculating Compensation Value
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Distinguish between important dates. There are several important dates associated with stock compensation plans. Each one is essential to properly recording and reporting options plans. In order, they are:
The grant date. This represents when the date at which employee is compensated.
The vesting date. The date at which, in a stock option plan, an employee can exercise their options (to buy stock shares).
The exercise date. The date at which the employee chooses to exercise his or her options. If they choose to not exercise their options, there will not be an exercise date recorded.
Expiration date. The date at which any remaining, unexercised options expire.[1] X Research source
Choose a method for determining the value of the stock-based compensation. In order to be recorded in journal entries, the stock compensation must be appropriately valued. The two most common methods recognized by the Financial Accounting Standards Board (FASB) are intrinsic value and fair value methods.
Intrinsic value refers to the difference between the stock price when the stock is granted and the price of the stock at the earliest date the stock vests and can be sold.
Fair value bases the value of stock on a complex model of factors that estimates the value of the stock or option at the time of the grant.[2] X Research source
Publicly-traded companies are required to use the fair value method. Non-public companies may use either method.[3] X Research source
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Find the value of restricted stock. Restricted stock, also known as non-vested stock, includes stock compensation that has not yet become vested. This means that employees given this stock are currently unable to exercise their options or sell the stock that they have been compensated with. The fair value of this stock is recorded as the market price of a share of the stock on the grant date. The total value of each plan's restricted stock is the number of shares multiplied by the fair value.[4] X Research source
Calculate stock option value. Stock option fair values are somewhat more complicated to calculate than the fair values of stock shares. Option values are calculated using a model that takes into consideration the market price at the grant date, the price at which the option is exercised, volatility, expected dividends, and the risk-free interest rate.[5] X Research source The Black-Scholes model is one of the more common methods for fair-value estimation of options. This calculation is typically handled by accounting or financial modeling software.[6] X Research source
Make an entry to record compensation. Original stock compensation is recorded according to when the stocks or options become vested (available to the employee). The specifics of when this occurs are specific to individual employee stock compensation plans and are created at the discretion of the company. The entries made on the vesting date(s) are a debit to Compensation Expense and a credit to Additional Paid-In Capital, Stock Options, both for the fair value of the vested options or stocks.
For example, imagine that an employee is granted a stock option plan on the first day of 2014 that gives them the option to purchase 1,000 shares of stock after a 2-year vesting period.
The options included in the plan are valued at $35,000 through the use of a fair value model.
The entries made on the vesting date, which would be the last day of 2015 (12/31/2015) are a debit of $35,000 to Compensation Expense and a Credit of $35,000 to Additional Paid-In Capital, Stock Options.[7] X Research source
Record exercised options. All other entries for stock compensation plans will likely be made on the expiration date. Any exercised options will be recorded to reflect the increase in cash and change in common stock and options accounts. Continuing with the previous example, imagine that the employee decides to exercise 400 of his options. This would mean that he buys 400 shares of the stock at the option price.
The option price is $50, this would represent $20,000 (
{\displaystyle 400\times \$50}
) in cash coming in to the company. In addition, it would represent 40 percent (400 of 1000 total) of the stock options originally granted leaving the company.
However, this also means that the common stock shares created in the purchase must be recorded. This will be done at the par value. So, if the par value of the shares is $5, this would mean that the company has gained $2,000 (
{\displaystyle \$5\times 400}
) in common stock.
This transaction would be recorded at the expiration date of the options as a debit to Cash for $20,000, a debit to Additional Paid-In Capital, Stock Options, for $14,000, a credit to Common Stock for $2,000, and, finally, a balancing credit to Additional Paid-In Capital, Common Stock, for $32,000.[8] X Research source
The balancing entry at the end represents the difference between the debits to Cash and Stock Options ($34,000 total) and the common stock credit ($2,000).
Write off expired options. At the expiration date, any unexercised options are also recorded. In this case, having exercised 40 percent of their options over the vesting period, the employee has elected not to exercise the remaining 60 percent. This means that 60 percent of the original $35,000 value, or $21,000, will be written off as expired stock options. Specifically, a debit to Additional Paid-In Capital, Stock Options, will be made along with a credit to Additional Paid-In Capital, Expired Stock Options, both for the $21,000 fair value of the expired options.[9] X Research source
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Account for the employee stock-based compensation when completing your financial statements. How financial statements are presented is your prerogative, but you must include all stock-based compensation when distributing statements to your stockholders. Stock compensation should be recorded as an expense on the income statement. However, stock compensation expenses must also be included on the company's balance sheet and statement of cash flows.[10] X Research source
Recording Compensation As an Employee
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Find your grant price. Determine the price at which you could purchase a share under the terms of your employee stock-based compensation plan. This is known as the par value or the grant price for stock options. Repeat this process for each "batch" of stock if you received stock-based compensation at different price points. This information should be listed in your employment contract or can be found by contacting your HR department.
For example, an employee might have a grant price of $10. This represents how much he or she would pay for a share, regardless of the current market price.[11] X Research source
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Calculate the difference between the grant price and market price at the exercise date. If you choose to exercise your options at any point, you need to record the full market value of the stock shares at the time at which you exercised the options. This is because the difference between the market price and the grant date at the exercise date is taxable as income if you do not hold the stock for a long enough period of time.
This period of time, generally one or two years, is determined by federal and state law and varies between states and options plans.[12] X Research source
For example, if your grant price is $10 and the current market price at the date of exercise is $50, you would need to calculate the difference, which here is $40 per share.
If you sell before the waiting period is over, you will be responsible for paying income tax on that difference. This would be calculated as your marginal tax rate times the total amount of the compensation.
So, if you exercised 100 options, you would need to pay income tax on the per share difference ($40) times 100 shares, which would be $4,000.[13] X Research source
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Determine capital gains on the sale of the stock. When you sell the stock provided by your stock compensation plan, you must pay capital gains on your returns from the sale. These taxes are all that you will owe on your stock compensation gains if you have already reached the end of the required waiting period when you sell. Capital gains are determined as the difference between the market value at exercise and the market value at the selling date.[14] X Research source
For example, if you sold the 100 shares from the previous example when the price hit $70, you would experience a taxable capital gain or $20 per share, or $2,000.[15] X Research source
File your taxes properly. Make sure to disclose all capital gains and losses on your income taxes. You should also include any stock sold before the required waiting period. Speak to a financial professional if you are unsure of when this waiting period ends. Failure to properly to report these gains can result in fines or criminal penalties.
The accounting procedures and rules described in this article are specific to U.S. generally accepted accounting procedures (U.S. GAAP). Those entities using International Financial Reporting Standards (IFRS) for their accounting practices will need to follow the rules specific to IFRS.[16] X Research source
↑ http://accounting.utep.edu/sglandon/c19/c19a.pdf
↑ http://rsmus.com/pdf/stock-based-compensation-at-a-glance.pdf
↑ http://www.fasb.org/summary/stsum123.shtml
↑ http://www.quickmba.com/finance/black-scholes/
↑ http://www.investopedia.com/articles/06/fas123r.asp
↑ http://personal.fidelity.com/products/stockoptions/exercise.shtml
↑ http://www.foundersworkbench.com/hiring/stock-based-compensation/
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Systems of Inequalities and Linear Programming | Boundless Algebra | Course Hero
The non-graphical method is much more complicated, and is perhaps much harder to visualize all the possible solutions for a system of inequalities. However, when you have several equations or several variables, graphing may be the only feasible method.
Application of Systems of Inequalities: Linear Programming
Linear programming involves finding an optimal solution for a linear equation, given a number of constraints.
Explain the steps of the Simplex Method to solve applications of systems of linear inequalities
The standard form for a linear program is: minimize
c\cdot x
Ax=b, x_{i}\geq0
. c is the coefficients of the objective function, x is the variables, A is the left-side of the constraints and b is the right side.
The Simplex Method involves choosing an entering variable from the nonbasic variables in the objective function, finding the corresponding leaving variable that maintains feasibility, and pivoting to get a new feasible solution, repeating until you find a solution.
In the Simplex Method, if there are no positive coefficients corresponding to the nonbasic variables in the objective function, then you are at an optimal solution.
In the Simplex Method, if there are no choices for the leaving variable, then the solution is unbounded.
objective function: A function to be maximized or minimized in optimization theory.
canonical form: The format in which a linear program in standard form can be represented, if the columns of A are rearranged so that it contains the the number of rows in A.
pivot: Moving from one basic feasible solution to an adjacent basic feasible solution.
constraint: A condition that a solution to a problem must satisfy.
the simplex method: An algorithm that optimizes a system of linear inequalities.
A common application of systems of inequalities is linear programming. Linear programming is a mathematical method for determining a way to achieve the best outcome for a list of requirements represented as linear relationships.
An example where linear programming would be helpful to optimize a system of inequalities is as follows:
A factory makes three types of chairs, A, B, and C. The factory makes a profit of $2 on chair A, $3 on chair B, and $4 on chair C. Chair A requires 30 man-hours, chair B requires 20, and chair C requires 10. Chair A needs 2m2 of wood, chair B needs 5m2, and chair C needs 3m2. Given 100 man-hours and 15m2 of wood per week, how many chairs of each type should be made each week to maximize profit?
The most common method in linear programming is the Simplex Method, or the Simplex Algorithm. To use the Simplex Method, we need to represent the problem using linear equations. Let a be the number of A chairs, b the B chairs, and c the C chairs. Then, we can write two linear inequalities where three variables must be non-negative, and all constraints must be satisfied. One linear inequality will show a relationship between the man-hours required for the project, and the other will show the amount of wood needed in the project:
First, an inequality for for man-hours, simplified:
30a+20b+10c\leq100 \\3a+2b+c\leq10
Then, an inequality for materials:
2a+5b+3c\leq15
The function to be maximized (the objective function, and in this case, the profit on the chairs) is:
P=2a+3b+4c
The standard form for the Simplex Method is:
c\cdot x
Ax=b, x_{i}\geq0
x=[x_{1}, x_{2},..., x_{n}]^{T}
c=[c_{1}, c_{2},..., c_{n}]
are the coefficients of the objective function, A is the left-side of the constraints, and
b=[b_{1}, b_{2},..., b_{p}]^{T}
The solution of a linear program is accomplished in two steps. In the first step, Phase I, a starting extreme point is found. Phase I either gives a basic feasible solution or no solution. If there is no solution, the linear program is considered infeasible.
In the second step, Phase II, the Simplex Algorithm is applied using the solution found in Phase I as a starting point. The possible results from Phase II are either an optimal solution or an unbounded solution.
Achieving Standard Form
You may have noticed that we had been given inequalities, such as
3a+2b+c \leq 10
, but standard form calls for equalities, or equations. We therefore introduce a slack variable that represents the difference between the two sides of the inequality and is non-negative.
This gives us the new equality:
3a+2b+c+s=10
The other inequality,
2a+5b+3c \leq 15
2a+5b+3c+t=15
Standard form also requires the objective function to be a minimization. If the problem calls for maximization, multiply the objective function by
-1
Here are the pieces for standard form:
x=[a, b, c, s, t]^{T}
c=[-2, -3, -4, 0, 0]
A=\begin{bmatrix} 3 & 2&1&1& 0\\ 2& 5&3&0&1 \end{bmatrix}
b=\begin{bmatrix} 10 \\ 15 \end{bmatrix}
Canonical Tableaux
A linear program in standard form can be represented as a tableau of the form
\begin{bmatrix} 1 & -c&0 \\ 0&A&b \end{bmatrix}
where the first row defines the objective function and the remaining rows specify the constraints. If the columns of A can be rearranged so that it contains the p-by-p identity matrix (the number of rows in A), then the tableau is said to be in canonical form. The variables corresponding to the columns of the identity matrix are called basic variables, while the remaining variables are called nonbasic or free variables. If the nonbasic variables are assumed to be
0
, then the values of the basic variables are easily obtained as entries in b, and this solution is a basic feasible solution.
Moving from one basic feasible solution to an adjacent basic feasible solution is called a pivot. First, a nonzero pivot element is selected in a nonbasic column. The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the others to change the other entries in the column to
0
. The result is that if the pivot is in row
r
, then the column becomes the
r
-th column of the identity matrix. The variable for this column is now basic, replacing the variable which corresponded to the
r
-th column of the identity matrix. The variable corresponding to the pivot column enters the set of basic variables, and the variable being replaced leaves the set of basic variables.
Now, the Simplex Method proceeds by performing successive pivot operations which each improve the basic feasible solution; the choice of pivot element at each step is largely determined by the requirement that this pivot improves the solution.
For the entering variable, choose any column in which the entry in the objective row is positive. If all the entries in the objective row are less than or equal to
0
, then no choice of entering variable can be made and the solution is optimal.
For the choice of pivot row, only positive entries in the pivot column are considered. This guarantees that the value of the entering variable will be non-negative. If there are none in the pivot column, then the entering variable can take any non-negative value with the solution remaining feasible. Therefore, the objective function is unbounded.
Next, the pivot row must be selected so that all the other basic variables remain positive. This occurs when the resulting value of the entering variable is at a minimum. If the pivot column is c, then the pivot row r is chosen so that
b_{r}/a_{cr}
Using our example, the canonical tableau is:
\begin{bmatrix} 1&2&3&4&0&0&0 \\ 0&3&2&1&1&0&10\\ 0&2&5&3&0&1&15 \end{bmatrix}
Columns 5 and 6 are the basic variables s and t, and the basic feasible solution is
a=b=c=0, s=10, t=15
Columns 2, 3, and 4 can be selected as pivot columns; for this example column 4 is selected. The values of x resulting from the choice of rows 2 and 3 as pivot rows are
\frac{10}{1}=10
\frac{15}{3}=5
respectively. Of these, the minimum is 5, so row 3 must be the pivot row. Performing the pivot produces:
\begin{bmatrix} 1&\frac{-2}{3}&\frac{-11}{3}&0&0&\frac{-4}{3}&-20 \\ 0&\frac{7}{3}&\frac{1}{3}&0&1&\frac{-1}{3}&5\\ 0&\frac{2}{3}&\frac{5}{3}&1&0&\frac{1}{3}&5 \end{bmatrix}
Now columns 4 and 5 represent the basic variables c and s and the corresponding basic feasible solution is:
a=b=t=0, s=5, c=5
For the next step, there are no positive entries in the objective row, and in fact:
-P=-20+\frac{2}{3}a+\frac{11}{3}b+\frac{4}{3}t
So, we should make 5 chairs of type C to maximize our profits with 20 dollars.
Algebra/Graphing Inequalities. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/Algebra/Graphing_Inequalities. License: CC BY-SA: Attribution-ShareAlike
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inequality. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/inequality. License: CC BY-SA: Attribution-ShareAlike
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Original figure by Julien Coyne. Licensed CC BY-SA 4.0. Provided by: Julien Coyne. License: CC BY-SA: Attribution-ShareAlike
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GetTables - Maple Help
Home : Support : Online Help : Connectivity : Database Package : Connection : GetTables
get a list of tables from the database
connection:-GetTables( opts )
(optional) equation(s) of the form option=value where option is one of catalog, schema, table, or output
GetTables returns an Array where each row of the Array contains matches from one table in the database connected to using connection. A column in the Array is data associated with the table. The output option controls the columns and their order in the returned Array.
output = list containing a selection of symbols from: name, catalog, schema, and type
The output option specifies the columns to return. The order of the columns in the output is the same as the order of the symbols in the list. By default, only the name is returned. Some databases do not support all of these fields.
- name returns the name of the table.
- catalog returns the name of the catalog that contains the table.
- schema returns the name of the schema that the table uses.
- type returns the type of the table. These are a database-specific values. Typical results are "TABLE", "VIEW", "SYSTEM TABLE","GLOBAL TEMPORARY", "LOCAL TEMPORARY", "ALIAS", and "SYNONYM".
catalog = string
Return only columns from the specified catalog. To return columns from tables not in a catalog, specify the empty string (""). By default, no restrictions are applied.
schema = string
Return only columns from databases in which the schema matches the specified pattern. The pattern can consist of any valid schema name characters and the special characters % and _. A % matches any string and an _ matches any character. To match a literal _ or %, you must prepend the escape character \. For example to match a _ use \_. By default, no restrictions are applied.
Return only columns for which the table matches the specified pattern. The pattern can consist of any valid table name characters and the special characters % and _. A % matches any string and an _ matches any character. To match a literal _ or %, you must prepend the escape character \. For example to match a _ use \_. By default, no restrictions are applied.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{conn}:-\mathrm{GetTables}\left('\mathrm{output}'=['\mathrm{schema}','\mathrm{name}']\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{"Schema1"}& \textcolor[rgb]{0,0,1}{"Table1"}\\ \textcolor[rgb]{0,0,1}{"Schema1"}& \textcolor[rgb]{0,0,1}{"Table2"}\\ \textcolor[rgb]{0,0,1}{"Schema2"}& \textcolor[rgb]{0,0,1}{"Table3"}\\ \textcolor[rgb]{0,0,1}{"Schema2"}& \textcolor[rgb]{0,0,1}{"Table4"}\\ \textcolor[rgb]{0,0,1}{"Schema3"}& \textcolor[rgb]{0,0,1}{"Table5"}\end{array}]
\mathrm{conn}:-\mathrm{GetTables}\left('\mathrm{schema}'="%1",'\mathrm{output}'=['\mathrm{name}']\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{"Table1"}\\ \textcolor[rgb]{0,0,1}{"Table2"}\end{array}]
\mathrm{conn}:-\mathrm{GetTables}\left('\mathrm{table}'="%1",'\mathrm{output}'=['\mathrm{name}','\mathrm{schema}','\mathrm{type}']\right)
[\textcolor[rgb]{0,0,1}{"Table1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Schema1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"TABLE"}]
Database[Connection][GetCatalogs]
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Balanced flow - Wikipedia
(Redirected from Cyclostrophic flow)
Model of atmospheric motion
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Balanced flow is often an accurate approximation of the actual flow, and is useful in improving the qualitative understanding and interpretation of atmospheric motion. In particular, the balanced-flow speeds can be used as estimates of the wind speed for particular arrangements of the atmospheric pressure on Earth's surface.
The Momentum Equations in Natural Coordinates[edit]
The momentum equations are written primarily for the generic trajectory of a packet of flow travelling on a horizontal plane and taken at a certain elapsed time called t. The position of the packet is defined by the distance on the trajectory s=s(t) which it has travelled by time t. In reality, however, the trajectory is the outcome of the balance of forces upon the particle. In this section we assume to know it from the start for convenience of representation. When we consider the motion determined by the forces selected next, we will have clues of which type of trajectory fits the particular balance of forces.
The trajectory at a position s has one tangent unit vector s that invariably points in the direction of growing s's, as well as one unit vector n, perpendicular to s, that points towards the local centre of curvature O. The centre of curvature is found on the 'inner side' of the bend, and can shift across either side of the trajectory according to the shape of it. The distance between the parcel position and the centre of curvature is the radius of curvature R at that position. The radius of curvature approaches an infinite length at the points where the trajectory becomes straight and the positive orientation of n is not determined in this particular case (discussed in geostrophic flows). The frame of reference (s,n) is shown by the red arrows in the figure. This frame is termed natural or intrinsic because the axes continuously adjust to the moving parcel, and so they are the most closely connected to its fate.
In the balanced-flow idealization we consider a three-way balance of forces that are:
Pressure force. This is the action on the parcel arising from the spatial differences of atmospheric pressure p around it. (Temporal changes are of no interest here.) The spatial change of pressure is visualised through isobars, that are contours joining the locations where the pressure has a same value. In the figure this is simplistically shown by equally spaced straight lines. The pressure force acting on the parcel is minus the gradient vector of p (in symbols: grad p) – drawn in the figure as a blue arrow. At all points, the pressure gradient points to the direction of maximum increase of p and is always normal to the isobar at that point. Since the flow packet feels a push from the higher to the lower pressures, the effective pressure vector force is contrary to the pressure gradient, whence the minus sign before the gradient vector.
Governing equations[edit]
{\displaystyle {\frac {DV}{Dt}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial s}}-KV}
{\displaystyle {\frac {V^{2}}{R}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial n}}\pm fV,}
in the forward and sideway directions respectively, where ρ is the density of air.
{\displaystyle {DV}/{Dt}}
is the temporal rate of speed change to the parcel (tangential acceleration);
{\displaystyle -{\partial p}/{\partial s}}
is the component of the pressure force per unit volume along the trajectory;
{\displaystyle -KV}
is the deceleration due to friction;
{\displaystyle {V^{2}}/{R}}
is the centripetal acceleration;
{\displaystyle -{\partial p}/{\partial n}}
is the component of the pressure force per unit volume perpendicular to the trajectory;
{\displaystyle \pm fV}
is the Coriolis force per unit mass (the sign ambiguity depends on the mutual orientation of the force vector and n).
Steady-state assumption[edit]
In the following discussions, we consider steady-state flow. The speed cannot thus change with time, and the component forces producing tangential acceleration need to sum up to zero. In other words, active and resistive forces must balance out in the forward direction in order that
{\displaystyle DV/Dt=0}
. Importantly, no assumption is made yet on whether the right-hand side forces are of either significant or negligible magnitude there. Moreover, trajectories and streamlines coincide in steady-state conditions, and the pairs of adjectives tangential/normal and streamwise/cross-stream become interchangeable. An atmospheric flow in which the tangential acceleration is not negligible is called allisobaric.
The schematisations[edit]
The limitations[edit]
Vertical differences of air properties[edit]
One difference of air speed in every air column invariably occurs, however, near the ground/sea, also if the air density is the same anywhere and no vertical motion occurs. There, the roughness of the contact surface slows down the air motion above, and this retarding effect peters out with height. See, for example, planetary boundary layer. Frictional antitriptic flow applies near the ground, while the other schematisations apply far enough from the ground not to feel its "braking" effect (free-air flow). This is a reason to keep the two groups conceptually separated. The transition from low-quote to high-quote schematisations is bridged by Ekman-like schematisations where air-to-air friction, Coriolis and pressure forces are in balance.
Horizontal differences of air properties[edit]
Even if air columns are homogeneous with height, the density of each column can change from location to location, firstly since air masses have different temperatures and moisture content depending on their origin; and then since air masses modify their properties as they flow over Earth's surface. For example, in extra-tropical cyclones the air circulating around a pressure low typically comes with a sector of warmer temperature wedged within colder air. The gradient-flow model of cyclonic circulation does not allow for these features.
Balanced-flow schematisations can be used to estimate the wind speed in air flows covering several degrees of latitude of Earth's surface. However, in this case assuming a constant Coriolis parameter is unrealistic, and the balanced-flow speed can be applied locally. See Rossby waves as an example of when changes of latitude are dynamically effective.
Unsteadiness[edit]
Antitriptic flow[edit]
The name comes from the Greek words 'anti' (against, counter-) and 'triptein' (to rub) – meaning that this kind of flow proceeds by countering friction.
In the streamwise momentum equation, friction balances the pressure gradient component without being negligible (so that K≠0). The pressure gradient vector is only made by the component along the trajectory tangent s. The balance in the streamwise direction determines the antitriptic speed as
{\displaystyle V=-{\frac {1}{K\rho }}{\frac {\partial p}{\partial s}}}
A positive speed is guaranteed by the fact that antitriptic flows move along the downward slope of the pressure field, so that mathematically
{\displaystyle {\partial p}/{\partial s}<0}
. Provided the product KV is constant and ρ stays the same, p turns out to vary linearly with s and the trajectory is such that the parcel feels equal pressure drops while it covers equal distances. (This changes, of course, when using a non-linear model of friction or a coefficient of friction that varies in space to allow for different surface roughness.)
In the cross-stream momentum equation, the Coriolis force and normal pressure gradient are both negligible, leading to no net bending action. As the centrifugal term
{\displaystyle {V^{2}}/{R}}
vanishes while the speed is non-zero, the radius of curvature goes to infinity, and the trajectory must be a straight line. In addition, the trajectory is perpendicular to the isobars since
{\displaystyle \partial p/\partial n=0}
. Since this condition occurs when the n direction is that of an isobar, s is perpendicular to the isobars. Thus, antitriptic isobars need to be equispaced circles or straight lines.
Because the Coriolis effects are neglected, antitriptic flow occurs either near the equator (irrespective of the motion's length scale) or elsewhere whenever the Ekman number of the flow is large (normally for small-scale processes), as opposed to geostrophic flows.
Geostrophic flow[edit]
See also: Geostrophic current
Nearly parallel isobars supporting quasi-geostrophic conditions
The speed cannot be determined by this balance. However,
{\displaystyle \partial p/\partial s=0}
entails that the trajectory must run along isobars, else the moving parcel would experience changes of pressure like in antitriptic flows. No bending is thus only possible if the isobars are straight lines in the first instance. So, geostrophic flows take the appearance of a stream channelled along such isobars.
{\displaystyle V={\frac {1}{\rho }}\left|{\frac {1}{f}}{\frac {\partial p}{\partial n}}\right|.}
Modelers, theoreticians, and operational forecasters frequently make use of geostrophic/quasi-geostrophic approximation. Because friction is unimportant, the geostrophic balance fits flows high enough above the Earth's surface. Because the Coriolis force is relevant, it normally fits processes with small Rossby number, typically having large length scales. Geostrophic conditions are also realised for flows having small Ekman number, as opposed to antitriptic conditions.
Cyclostrophic flow[edit]
Cyclostrophic flow describes a steady-state flow in a spatially varying pressure field when
{\displaystyle {\frac {V^{2}}{R}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial n}}.}
{\displaystyle V={\sqrt {-{\frac {R}{\rho }}{\frac {\partial p}{\partial n}}}}.}
Another implication of the cross-stream momentum equation is that a cyclostrophic flow can only develop next to a low-pressure area. This is implied in the requirement that the quantity under the square root is positive. Recall that the cyclostrophic trajectory was found to be an isobar. Only if the pressure increases from the centre of curvature outwards, the pressure derivative is negative and the square root is well defined – the pressure in the centre of curvature must thus be a low. The above mathematics gives no clue whether the cyclostrophic rotation ends up to be clockwise or anticlockwise, meaning that the eventual arrangement is a consequence of effects not allowed for in the relationship, namely the rotation of the parent cell.
Inertial flow[edit]
As before, frictionless flow in steady-state conditions implies that
{\displaystyle \partial p/\partial s=0}
. However, in this case isobars are not defined in the first place. We cannot draw any anticipation about the trajectory from the arrangement of the pressure field.
In the cross-stream momentum equation, after omitting the pressure force, the centripetal acceleration is the Coriolis force per unit mass. The sign ambiguity disappears, because the bending is solely determined by the Coriolis force that sets unchallenged the side of curvature – so this force has always a positive sign. The inertial rotation will be clockwise (anticlockwise) in the northern (southern) hemisphere. The momentum equation
{\displaystyle {\frac {V^{2}}{R}}=\left|f\right|V,}
{\displaystyle V=\left|f\right|R.}
A nearly uniform pressure field covers Central Europe and Russia with pressure differences smaller than 8 mbar over several tens of degrees of latitude and longitude. (For the conditions over the Atlantic Ocean see geostrophic and gradient flow)
Gradient flow[edit]
Like in all but the antitriptic balance, frictional and pressure forces are neglected in the streamwise momentum equation, so that it follows from
{\displaystyle \partial p/\partial s=0}
that the flow is parallel to the isobars.
{\displaystyle V=\pm {\frac {fR}{2}}\pm {\sqrt {{\frac {f^{2}R^{2}}{4}}-{\frac {R}{\rho }}{\frac {\partial p}{\partial n}}}}.}
Not all solutions of the gradient wind speed yield physically plausible results: the right-hand side as a whole needs be positive because of the definition of speed; and the quantity under square root needs to be non-negative. The first sign ambiguity follows from the mutual orientation of the Coriolis force and unit vector n, whereas the second follows from the square root.
Pressure lows and cyclones[edit]
{\displaystyle {\frac {V^{2}}{R}}={\frac {1}{\rho }}\left|{\frac {\partial p}{\partial n}}\right|-\left|f\right|V.}
{\displaystyle {\frac {V_{\text{geostrophic}}}{V_{\text{cyclone}}}}=1+{\frac {V_{\text{cyclone}}}{V_{\text{inertial}}}}>1,}
{\displaystyle V_{\text{cyclone}}=-{\frac {V_{\text{inertial}}}{2}}+{\sqrt {{\frac {V_{\text{inertial}}^{2}}{4}}+V_{\text{cyclostrophic}}^{2}}}.}
Pressure highs and anticyclones[edit]
{\displaystyle {\frac {V^{2}}{R}}=-{\frac {1}{\rho }}\left|{\frac {\partial p}{\partial n}}\right|+\left|f\right|V.}
{\displaystyle {\frac {V_{\text{geostrophic}}}{V_{\text{anticyclone}}}}=1-{\frac {V_{\text{anticyclone}}}{V_{\text{inertial}}}}<1,}
{\displaystyle V_{\text{anticyclone}}={\frac {V_{\text{inertial}}}{2}}-{\sqrt {{\frac {V_{\text{inertial}}^{2}}{4}}-V_{\text{cyclostrophic}}^{2}}}}
that requires that
{\displaystyle V_{\text{inertial}}\geq 2V_{\text{cyclostrophic}}}
to be meaningful. This condition can be translated in the requirement that, given a high-pressure zone with a constant pressure slope at a certain latitude, there must be a circular region around the high without wind. On its circumference the air blows at half the corresponding inertial speed (at the cyclostrophic speed), and the radius is
{\displaystyle R^{*}={\frac {4}{\rho f^{2}}}\left|{\frac {\partial p}{\partial n}}\right|,}
Balanced-flow speeds compared[edit]
The gradient speed comes with two curves valid for the speeds around a pressure low (blue) and a pressure high (red). The wind speed in cyclonic circulation grows from zero as the radius increases and is always less than the geostrophic estimate.
In the anticyclonic-circulation example, there is no wind within the distance of 260 km (point R*) – this is the area of no/low winds around a pressure high. At that distance the first anticyclonic wind has the same speed as the cyclostrophic winds (point Q), and half of that of the inertial wind (point P). Farther away from point R*, the anticyclonic wind slows down and approaches the geostrophic value with decreasingly larger speeds.
^ Schaefer Etling, J.; C. Doswell (1980). "The Theory and Practical Application of Antitriptic Balance". Monthly Weather Review. 108 (6): 746–756. Bibcode:1980MWRv..108..746S. doi:10.1175/1520-0493(1980)108<0746:TTAPAO>2.0.CO;2. ISSN 1520-0493.
^ Rennó, N.O.D.; H.B. Bluestein (2001). "A Simple Theory for Waterspouts". Journal of the Atmospheric Sciences. 58 (8): 927–932. Bibcode:2001JAtS...58..927R. doi:10.1175/1520-0469(2001)058<0927:ASTFW>2.0.CO;2. ISSN 1520-0469.
^ Winn, W.P.; S.J. Hunyady G.D. Aulich (1999). "Pressure at the ground in a large tornado". Journal of Geophysical Research. 104 (D18): 22, 067–22, 082. Bibcode:1999JGR...10422067W. doi:10.1029/1999JD900387.
Plymouth State Weather Center Balanced Flows Tutorial Archived 8 July 2007 at the Wayback Machine
Retrieved from "https://en.wikipedia.org/w/index.php?title=Balanced_flow&oldid=1082958446#Cyclostrophic_flow"
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INFRARED THERMOGRAPHY (1013)
MEASURING METHODS (796)
Determination of an Test Condition for IR Thermography to Inspect a Wall-Thinning Defect in Nuclear Piping Components
Kim, Jin Weon; Yun, Won Kyung; Jung, Hyun Chul; Kim, Kyeong Suk
[en] This study conducted infrared (IR) thermography tests using pipe and plate specimens with artificial wall-thinning defects to find an optimal condition for IR thermography test on the wall-thinned nuclear piping components. In the experiment halogen lamp was used to heat the specimens. The distance between the specimen and the lamp and the intensity of halogen lamp were regarded as experimental parameter. When the distance was set to 1∼2 m and the lamp intensity was above 60 % of full power, a single scanning of IR thermography detected all artificial wall-thinning defects, whose minimum dimension was 2θ = 90 .deg., d/t=0.5, and L/Do, within the pipe of 500 mm in length. Regardless of the distance between the specimen and the lamp, the image of wall-thinning defect in IR thermography became distinctive as the intensity of halogen lamp increased. The detectability of IR thermography was similar for both plate and pipe specimens, but the optimal test condition for IR thermography depended on the type of specimen
Journal of the Korean Society for Nondestructive Testing; ISSN 1225-7842; ; v. 32(1); p. 12-19
DEFECTS, IMAGES, INFRARED THERMOGRAPHY, PIPES
MEASURING METHODS, THERMOGRAPHY, TUBES
Wavenumber Selection by Bénard-Marangoni Convection at High Supercritical Number
Wu Di; Duan Li; Kang Qi, E-mail: duanli@imech.ac.cn, E-mail: kq@imech.ac.cn
[en] Marangoni-Bénard convection, which is mainly driven by the thermocapillary (Marangoni) effect, occurs in a thin liquid layer heated uniformly from the bottom. The wavenumber of supercritical convection is studied experimentally in a 160×160
m{m}^{2}
cavity that is heated from the bottom block. The convection pattern is visualized by an infrared thermography camera. It is shown that the onset of the Bénard cell is consistent with theoretical analysis. The wavenumber decreases obviously with increasing temperature, except for a slight increase near the onset. The wavenumber gradually approaches the minimum when the supercritical number ε is larger than 10. Finally, a formula is devised to describe the wavenumber selection in supercritical convection. (paper)
CAMERAS, CONVECTION, INFRARED THERMOGRAPHY, LIQUIDS
ENERGY TRANSFER, FLUIDS, HEAT TRANSFER, MASS TRANSFER, MEASURING METHODS, THERMOGRAPHY
Bianco, A., E-mail: alessia.bianco@unirc.it
[en] The paper explains some opportunities and limitations of thermographic investigations in terms of their capability to define the conservative conditions of architectural heritage and in terms of the historical recollection for a technical diagnosis. Different approaches are demonstrated in two case studies: the first integrates thermography with other investigative methods; the second combines thermographic monitoring with hygrothermal monitoring. (author)
CINDE Journal; ISSN 1700-2729; ; v. 34(2); p. 9-15
ARCHITECTURE, BUILDINGS, INFRARED THERMOGRAPHY, MONITORING
MEASURING METHODS, THERMOGRAPHY
Defect Sizing and Location by Lock-in Photo-Infrared Thermography
Choi, Man Yong; Kang, Ki Soo; Park, Jeong Hak; Kim, Won Tae; Kim, Koung Suk
[en] In lock-in thermography, a phase difference between the defect area and the healthy area indicates the qualitative location and size of the defect. To accurately estimate these parameters, the shearing-phase technique has been employed which gives the shearing-phase distribution. The shearing-phase distribution has maximum, minimum, and zero points that help determine quantitatively the size and location of the subsurface defect. In experiment, the proposed technique is verified with artificial specimen and these related factors are analyzed
Journal of the Korean Society for Nondestructive Testing; ISSN 1225-7842; ; v. 27(4); p. 321-327
DEFECTS, INFRARED THERMOGRAPHY, SHEAR, SIZE
Arts-of-the-states of Infrared Thermography in Domestic Field Application and Applied Study
Kim, Won Tae; Choi, Man Yong; Park, Jung Hak; Kang, Ki Soo
[en] This paper aims to understand current domestic arts-of-the-states of infrared thermography(IRT) through e-mail survey. This survey is conducted for relevant employers and researchers working in the area of IRT. By thoroughly analyzing some difficulties occurring in on-site fields and application study, it is assumed to be utilized into future research planning. In surveys, usefulness of IR camera, level of utilization, and correction status of IR detector including systems are also considered. for results, 1) analysis in domestic states, 2) satisfaction and problems of technology, and 3) improvement of IRT are described.
The Korean Society for Nondestructive Testing, Seoul (Korea, Republic of); 197 p; Nov 2006; p. 75-80; 2006 Fall Meeting of the Korean Society for Nondestructive Testing; Seoul (Korea, Republic of); 24 Nov 2006; Available from KSNT, Seoul (KR); 3 refs, 13 figs
DETECTION, INFRARED THERMOGRAPHY, PLANNING, USES
A study on the health evaluation in spot welded zone by using optical pulse and lock-in phase infrared thermography
Park, Hee Sang; Choi, Mang Yog; Kwon, Koo Ahn; Park, Jeong Hak; Kim, Ki Tae; Lee, Bo Young
[en] The non-destructive testing using infrared thermography is extended to a variety of industries and non-destructive testing of welds using infrared thermography is also in progress in various ways. Currently, a non-destructive testing of electrical resistance spot welds which is mainly used is Radiography Testing. This study detected area of spot welds nugget using optical-infrared thermography. In the results, it is possible for detecting defects of nugget in a short period of time using pulse-infrared thermography.
5 refs, 14 figs, 3 tabs
DETECTION, INFRARED THERMOGRAPHY, NONDESTRUCTIVE TESTING, PULSES
MATERIALS TESTING, MEASURING METHODS, TESTING, THERMOGRAPHY
Application defects detection in the small bore pipe using infrared thermography technique
Yun, Kyung Won; Kim, Dong Lyul; Jung, Hyun Chul; Hong, Dong Pyo; Kim, Kyeong Suk
[en] In the advanced research deducted infrared thermography (IRT) test using 4 inch pipe with artificial wall thinning defect to measure on the wall thinned nuclear pipe components. This study conducted for defect detection condition of nuclear small bore pipe research using deducted condition in the advanced research. Defect process is processed by change for defect length, circumferential direction angle, wall thinning depth. In the used equipment IR camera and two halogen lamps, whose full power capacity is 1 kW, halogen lamps and Target pipe experiment performed to the distance of the changed 1 m, 1.5 m, 2 m. To analysis of the experimental results ensure for the temperature distribution data, by this data measure for defect length. Artificial defect of 4 inch pipe is high reliability in the 2 m, but small bore pipe is in the 1.5 m from defect clearly was detected
CAMERAS, DEFECTS, DETECTION, INFRARED THERMOGRAPHY, PIPES
Thermal resolution analysis of lock-in infrared microscope
Kim, Ghi Seok; Lee, Kye Sung; Kim, Geon Hee; Hur, Hwan; Chang, Ki Soo; Kim, Dong Ik
[en] In this study, we analyzed and showed the enhanced thermal resolution of a lock-in infrared thermography system by employing a blackbody system and micro-register sample. The noise level or thermal resolution of an infrared camera system is usually expressed by a noise equivalent temperature difference (NETD), which is the mean square of the deviation of the different values measured for one pixel from its mean values obtained in successive measurements. However, for lock-in thermography, a more convenient quantity in the phase-independent temperature modulation amplitude can be acquired. On the basis of results, it was observed that the NETD or thermal resolution of the lock-in thermography system was significantly enhanced, which we consider to have been caused by the averaging and filtering effects of the lock-in technique.
17 refs, 6 figs
FILTERS, INFRARED THERMOGRAPHY, MICROSCOPES, NOISE, RESOLUTION
Nondestructive Testing Technique using Infrared Thermography
Park, Moon Ho; Lee, Ik Hwan
[en] Infrared(IR) thermography method was developed as a result of an investigation into a means of deriving a more visual method of temperature analysis. It recently provides an excellent nondestructive testing(NDT) technique in a variety of industries such as nuclear power plant, fossil plants, etc.. This paper offers a basic principles of infrared radiation, the nature of instruments used for measurement and the applications
5 refs, 5 figs
INFRARED THERMOGRAPHY, NONDESTRUCTIVE TESTING, RADIATIONS, USES
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Solve boundary value problem — fifth-order method - MATLAB bvp5c - MathWorks Deutschland
Solve boundary value problem — fifth-order method
{{\mathit{y}}^{\prime }}^{\prime }+\mathit{y}=0
\left[0,\pi /2\right]
\mathit{y}\left(0\right)=0
\mathit{y}\left(\pi /2\right)=2
{\mathit{y}}_{1}=\mathit{y}
{\mathit{y}}_{2}={\mathit{y}}^{\prime }
{{\mathit{y}}_{1}}^{\prime }={\mathit{y}}_{2}
{{\mathit{y}}_{2}}^{\prime }=-{\mathit{y}}_{1}
\mathit{g}\left(\mathit{y}\left(\mathit{a}\right),\mathit{y}\left(\mathit{b}\right)\right)=0
\mathit{y}\left(0\right)=0
\mathit{y}\left(\pi /2\right)-2=0
{{\mathit{y}}^{\prime }}^{\prime }
\mathit{y}
\mathit{x}
{\mathit{y}}_{1}
{\mathit{y}}_{2}
{{\mathit{y}}^{\prime }}^{\prime }+\frac{2}{\mathit{x}}{\mathit{y}}^{\prime }+\frac{1}{{\mathit{x}}^{4}}\mathit{y}=0
\left[\frac{1}{3\pi },1\right]
\mathit{y}\left(\frac{1}{3\pi }\right)=0
\mathit{y}\left(1\right)=\mathrm{sin}\left(1\right)
{\mathit{y}}_{1}=\mathit{y}
{\mathit{y}}_{2}={\mathit{y}}^{\prime }
{{\mathit{y}}_{1}}^{\prime }={\mathit{y}}_{2}
{{\mathit{y}}_{2}}^{\prime }=-\frac{2}{\mathit{x}}{\mathit{y}}_{2}-\frac{1}{{\mathit{x}}^{4}}{\mathit{y}}_{1}
\mathit{g}\left(\mathit{y}\left(\mathit{a}\right),\mathit{y}\left(\mathit{b}\right)\right)=0
\mathit{y}\left(\frac{1}{3\pi }\right)=0
\mathit{y}\left(1\right)-\mathrm{sin}\left(1\right)=0
\mathit{J}=\frac{\partial {\mathit{f}}_{\mathit{i}}}{\partial \mathit{y}}=\left[\begin{array}{cc}\frac{\partial {\mathit{f}}_{1}}{\partial {\mathit{y}}_{1}}& \frac{\partial {\mathit{f}}_{1}}{\partial {\mathit{y}}_{2}}\\ \frac{\partial {\mathit{f}}_{2}}{\partial {\mathit{y}}_{1}}& \frac{\partial {\mathit{f}}_{2}}{\partial {\mathit{y}}_{2}}\end{array}\right]=\left[\begin{array}{cc}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ -\frac{1}{{\mathit{x}}^{4}}& -\frac{2}{\mathit{x}}\end{array}\right]
\left[1/3\pi ,1\right]
{\mathit{y}}_{1}
{\mathit{y}}_{1}=\mathrm{sin}\left(\frac{1}{\mathit{x}}\right)
{\mathit{y}}_{2}=-\frac{1}{{\mathit{x}}^{2}}\mathrm{cos}\left(\frac{1}{\mathit{x}}\right)
y\text{'}=5y-3
\begin{array}{l}y{\text{'}}_{1}={y}_{1}+2{y}_{2}\\ y{\text{'}}_{2}=3{y}_{1}+2{y}_{2}\end{array}
\begin{array}{l}y\text{'}=S\frac{y}{x}+f\left(x,y,p\right),\\ 0=bc\left(y\left(0\right),y\left(b\right),p\right).\end{array}
bvp5c is a finite difference code that implements the four-stage Lobatto IIIa formula [1]. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fifth-order accurate uniformly in [a,b]. The formula is implemented as an implicit Runge-Kutta formula. Some of the differences between bvp5c and bvp4c are:
bvp5c solves the algebraic equations directly. bvp4c uses analytical condensation.
bvp4c handles unknown parameters directly. bvp5c augments the system with trivial differential equations for the unknown parameters.
[1] Shampine, L.F., and J. Kierzenka. "A BVP Solver that Controls Residual and Error." J. Numer. Anal. Ind. Appl. Math. Vol. 3(1-2), 2008, pp. 27–41.
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Creator Staking Pool - Only1
Social Activities indicator and Content Control
Each creator will have a staking pool created for them when they onboard Only1. Users and supporters can stake LIKE tokens to earn Annual Percentage Yield (APY) Rewards. The APY of each creator will adjust according to their social activities across various platforms.
Creator Staking Pool APY serves to exercise content control. APY of a creator is ‘slashed’ when bad content is flagged from DAO Community (we call this “Proof-of-Content”) and vice versa when the creator is actively creating content that abides by the DAO’s community guidelines.
Annual Percentage Yield (APY) Algorithm
Factors and Metrices
APY of a creator’s staking pool should correlate to his/her social activities across different platforms. The APY calculation algorithm takes into account the following factors, and will be continually reviewed and updated to optimize for maximum social engagement and fair rewards for all users;
A. Following (F)
Number of followers/friends/subscribers
B. Content (C)
Twitter - Number of Tweets
Facebook - Posts, Albums, Live Streams, Groups or Forums joined
Youtube - Number of of Videos, Live Streams
Instagram - Number of Posts, Stories, Reels
TikTok - Number of Videos
C. Engagement (E)
D. Tokenomics (T)
Tokens release should be in accordance with our vesting scheduled for Community Pool outlined in here [Insert Link]
The APY of the k-th creator can be calculated as:
APY_k = Logistic(m_k) \space\space\space\space\space\space\space 0 \le m_k \le1
The Logistic function is defined as:
Logistic(x)= \frac {APY \scriptstyle max}{1+e^{{-b(x-a)}}}
APY_{max} = Max \space yield \space a \space fan \space can \space earn \space from \space a \space creator
a = A \space value \space where \space the \space APY_{max} \space is \space halfed
b = A \space constant \space to \space control \space the \space steepness \space of \space the \space logistic \space curve
m_k = w_FF + w_CC +w_EE +w_TT
0\le w_F,w_C,w_E,w_T,F,C,E,T\le1 \space and \space w_F+w_C+w_F+w_T=1
w_F = weighting \space assigned \space to \space the \space overall \space following \space factor
w_C = weighting \space assigned \space to \space the \space overall \space content \space factor
w_E = weighting \space assigned \space to \space the \space overall \space engagement \space factor
w_T = weighting \space assigned \space to \space the \space overall \space tokenomics \space factor
Overall Social Factor
The overall social factors, F, C, E, and T, can be calculated as the weighted average of all the sub-social factors within the corresponding overall social factor categories.
Overall \space Social \space Factor = \sum_{j=1}^\N w_{Sub \space Social \space Factor,j} Sub \space Social \space Factor_j
0\le \forall \space w_{Sub \space Social \space Factor,j} \space \le1
0\le \forall \space Sub \space Social \space Factor_j\le1
\sum_{j=1}^M w_{Sub \space Social \space Factor,j} =1
w_{Sub \space Social \space Factor,j} = Weighting \space assigned \space to \space the \space jth \space sub \space social \space actor \space within \space an \space overall \space social \space factor
Sub \space Social \space Factor_j =The \space jth \space sub \space social \space factor \space within \space an \space overall \space social \space factor
M = The \space total \space number \space of \space the \space sub\space social \space factors \space within \space an \space overall \space social \space factor
Sub Social Factor
The sub social factor in each overall social factor can be defined as:
Sub \space Social \space Factor = \sum_{p=1}^Q w_{Platform,p} Platform \space Variable_p
0\le \forall \space w_{Platform,p}\le1
0\le \forall \space Platform \space Variable _p\le1
\sum_{p=1}^Q w_{Platform,p} =1
w_{Platform,p} = Weighting \space assigned \space to \space the \space pth \space platform
Platform \space Variable _p = pth \space platform \space variable
Q = Total \space number \space of \space the \space social \space platforms \space managed \space by \space a \space creator
The weightings assigned to each social platform of the creator is depending on the “active status” of the platforms, i.e. whether the creator is actively managing his or her social platforms.
Each platform variable is following a pre-defined distribution so that:
0\le \forall \space Platform \space Variable \space Values \le 1
The above data will be extracted from socials using open APIs, transformed, and analyzed for real-time APY calculation. More about the technology behind here.
Creators will get 0.4% of the total value locked in their creator staking pool by seconds as a reward for being active and producing quality content that abides with the community guidelines.
These rewards will be given out in LIKE from the Community Pool according to the pre-set vesting schedule.
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A role for external Ca2+ in maintaining muscle contractility in periodic paralysis | Journal of General Physiology | Rockefeller University Press
Correspondence to Stephen Cannon: sccannon@mednet.ucla.edu
Stephen C. Cannon; A role for external Ca2+ in maintaining muscle contractility in periodic paralysis. J Gen Physiol 6 July 2020; 152 (7): e202012615. doi: https://doi.org/10.1085/jgp.202012615
Periodic paralysis is an ion channel disorder of skeletal muscle wherein recurrent episodes of severe weakness are caused by anomalous depolarization of the resting potential,
Vrest,
that persists for minutes to hours, with associated inactivation of voltage-gated sodium channels and loss of fiber excitability (Cannon, 2015). Clinical management of periodic paralysis is symptomatic; that is to say, it minimizes provocative maneuvers that trigger attacks of weakness or use interventions that may reduce attack frequency and severity (Statland et al., 2018). Administration of calcium gluconate has been used empirically in an attempt to hasten recovery from an ongoing attack of weakness (Lehmann-Horn et al., 2004), and in this issue of the Journal of General Physiology,Uwera et al (2020) use a mouse model of hyperkalemic periodic paralysis (HyperKPP; Hayward et al., 2008) to systematically assess the efficacy of Ca2+ in reducing the susceptibility to high-K+ induced loss of force and explore the mechanistic basis for protection.
The empirical use of calcium gluconate as abortive therapy for an episode of HyperKPP dates back to the 1950s (Gamstorp, 1956), before it was known that this dominantly inherited disorder is caused by gain-of-function missense mutations in the skeletal muscle isoform of the α subunit of the voltage-gated sodium channel, NaV1.4 (Cannon, 2015; Lehmann-Horn et al., 2004). Controlled trials on the effectiveness of Ca2+ in HyperKPP have never been performed, and anecdotal reports describe mixed results (reviewed in Samaha, 1965), although there is one convincing example wherein low serum total Ca2+ (<2.1 mM; normal 2.1–2.6) and Mg2+ (<0.5 mM, normal 0.6–1.1 mM) secondary to chemotherapy dramatically worsened the symptoms of HyperKPP (Mankodi et al., 2015). To address the question of a role for extracellular Ca2+ in modulating susceptibility to weakness in HyperKPP, Uwera et al (2020) performed ex vivo contraction studies and microelectrode measurements of
Vm
in an established mouse model for HyperKPP (NaV1.4-M1592V knock-in; Hayward et al., 2008).
This study convincingly demonstrates that reducing Ca2+ aggravates the susceptibility to high-K+ induced loss of force in HyperKPP muscle. The force–
Ke+
relation has a midpoint (50% loss) of ∼11–12 mM for HyperKPP muscle in 2.4 mM Ca2+, and this shifted leftward to ∼8 mM in 1.3 mM Ca2+. Moreover, the tetanic force decreased to 20–30% of baseline in 0.3 mM Ca2+, even while K+ remained at a control level of 4.7 mM (e.g., Fig. 1 in Uwera et al., 2020). In contrast, WT muscle tolerates a 12 mM K+ challenge in 2.4 mM Ca2+ (∼75% of baseline force). At the lowest concentration of Ca2+ tested (0.3 mM), WT muscle also had a pronounced loss of force during a high-K challenge (e.g., 50% reduction in 10 mM K+). These observations led the authors to propose several mechanisms that may contribute to enhanced K+-sensitivity in low Ca2+: (1) the gating of voltage-dependent channels will have an apparent left (hyperpolarized) shift caused by the reduced screening of negative surface charge on the external face of the plasma membrane in low divalent cation solutions (Hille, 1968); (2) impaired Ca2+ release, which is an intrinsic dependence of excitation–contraction on extracellular Ca2+ that is not alleviated by substitution with Mg2+ (Brum et al., 1988); and (3) enhanced depolarization of
Vrest.
The latter is more complex than appears at first glance because it includes possible contributions from (i) a depolarized shift of the equilibrium potential for K+; (ii) a hyperpolarized shift of NaV1.4 activation in low divalent cation solutions; and (iii) for HyperKPP fibers gain-of-function defects manifest as impaired inactivation and a hyperpolarized shift of activation. Taken together, it is proposed these effects cause a depolarization-dependent loss of force in low Ca2+ that occurs in WT fibers only in when K+ is increased (e.g., 10 mM), but happens in HyperKPP fibers even in normal K+ because the NaV1.4 gain-of-function defect increases the propensity for depolarization.
An indirect method was used to assess whether the variations of extracellular Ca2+ used in the contractility studies caused a shift in the voltage-dependence of sodium channel availability. The peak amplitude of the Na+ current was estimated from the maximum
dVm/dt
during the upstroke of the action potential (AP; Hodgkin and Katz, 1949). The limitations of using this approach to determine the voltage-dependence of availability are well known: (i)
dVm/dt
is proportional to the total sum of ionic currents and therefore is representative of
INa
only when the relative contribution of other currents is much smaller, as normally occurs during the maximum rate of rise for normal APs, but less so for attenuated APs; and (ii) changes in
[Ke+]
are used to vary
Vm,
but precise control is not possible and so binning of data over a range of
Vm
is required for the analysis. Even with these caveats, the authors show for WT muscle that reducing extracellular Ca2+ from 2.4 to 0.3 mM (with constant Mg2+ of 3.1 mM) caused a 5 mV leftward shift of Na+ channel availability (visually estimated from Fig. 14 A in Uwera et al. (2020)). This shift was prevented by maintaining a constant divalent concentration (2.4 mM Ca2+ + 3.1 mM Mg2+
→
0.3 mM Ca2+ + 5.2 mM Mg2+), consistent with the expectation of a negative surface charge effect. For HyperKPP muscle (NaV1.4-M1592V), the
dVm/dt
technique revealed the previously reported impairment of slow inactivation (Hayward et al., 1997), such that in 2.4 mM Ca2+ availability was barely reduced even for the largest test depolarization of
−62
mV. Again, in support of a surface charge effect, a large decrease in availability was observed in 0.3 mM Ca2+, consistent with a left shift of gating, and which was also reversed by increased Mg2+.
The relation between low Ca2+ and depolarization of
Vrest
follows the same trend observed for low Ca2+ and the loss of contractility. Namely, for WT fibers in 4.7 mM K+, a reduction of Ca2+ from 2.4 to 0.3 mM did not cause depolarization or a loss of force. Only when external K+ was increased to 10 mM did WT fibers have an additional loss of force and depolarization in response to reducing Ca2+. Conversely, HyperKPP muscle always depolarized and had lower tetanic force in response to a reduction of Ca2+ (2.4 mM
→
0.3 mM), regardless of whether external K+ was 4.7 or 10 mM. This pattern is qualitatively consistent with their proposed mechanism for loss of contractility in low Ca2+, wherein depolarization (from high K+ or from the HyperKPP mutation) is necessary to exhibit the Ca2+ sensitivity. It would be interesting to test whether the depolarization induced by low Ca2+ would be prevented if the total divalent concentration were held constant. These data might provide additional insight on whether the left shift of gating contributes to depolarization, perhaps by enhancing a small subthreshold Na+ current.
The major finding of this study, that low Ca2+ clearly exacerbates the K+-induced loss of force in HyperKPP, has important translational value to the management of this muscle channelopathy. The robust demonstration of the deleterious effect of low Ca2+ would have been impractical to establish in clinical studies or with human biopsy material, which demonstrates the power of high-fidelity mouse models of human disease. Another important point is that both WT and HyperKPP muscle show this Ca2+ sensitivity in the proper context. As the authors point out, this implies the exacerbation of weakness for HyperKPP in low Ca2+ is not because of a specific mechanism imparted by the NaV1.4 mutation. Instead, the loss of force in low Ca2+ is a fundamental property of skeletal muscle under conditions where
Vrest
is depolarized.
This work was supported by a grant from the National Institute of Arthritis and Musculoskeletal and Skin Diseases of the National Institutes of Health (R01-AR063182).
Acta Paediatr. (Stockh.)
Charges and potentials at the nerve surface. Divalent ions and pH
. Nondystrophic Myotonias and Periodic Paralyses. In
Hyperkalemic Periodic Paralysis. A Genetic Study, Clinical Observations, and Report of a New Method of Therapy
Arch. Neurol
Review of the Diagnosis and Treatment of Periodic Paralysis
Lower Ca2+ enhances the K+-induced force depression in normal and HyperKPP mouse musclesDepressive effect of low Ca2+ in HyperKPP muscles
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Steinhaus–Moser notation - Wikipedia
Notation for extremely large numbers
In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation.[1]
4 Moser's number
a number n in a triangle means nn.
a number n in a square is equivalent to "the number n inside n triangles, which are all nested."
a number n in a pentagon is equivalent with "the number n inside n squares, which are all nested."
etc.: n written in an (m + 1)-sided polygon is equivalent with "the number n inside n nested m-sided polygons". In a series of nested polygons, they are associated inward. The number n inside two triangles is equivalent to nn inside one triangle, which is equivalent to nn raised to the power of nn.
Steinhaus defined only the triangle, the square, and the circle , which is equivalent to the pentagon defined above.
Steinhaus defined:
mega is the number equivalent to 2 in a circle: ②
megiston is the number equivalent to 10 in a circle: ⑩
Moser's number is the number represented by "2 in a megagon". Megagon is here the name of a polygon with "mega" sides (not to be confused with the polygon with one million sides).
Alternative notations:
use the functions square(x) and triangle(x)
let M(n, m, p) be the number represented by the number n in m nested p-sided polygons; then the rules are:
{\displaystyle M(n,1,3)=n^{n}}
{\displaystyle M(n,1,p+1)=M(n,n,p)}
{\displaystyle M(n,m+1,p)=M(M(n,1,p),m,p)}
mega =
{\displaystyle M(2,1,5)}
megiston =
{\displaystyle M(10,1,5)}
moser =
{\displaystyle M(2,1,M(2,1,5))}
Mega[edit]
A mega, ②, is already a very large number, since ② = square(square(2)) = square(triangle(triangle(2))) = square(triangle(22)) = square(triangle(4)) = square(44) = square(256) = triangle(triangle(triangle(...triangle(256)...))) [256 triangles] = triangle(triangle(triangle(...triangle(256256)...))) [255 triangles] ~ triangle(triangle(triangle(...triangle(3.2 × 10616)...))) [254 triangles] = ...
Using the other notation:
mega = M(2,1,5) = M(256,256,3)
With the function
{\displaystyle f(x)=x^{x}}
we have mega =
{\displaystyle f^{256}(256)=f^{258}(2)}
where the superscript denotes a functional power, not a numerical power.
We have (note the convention that powers are evaluated from right to left):
M(256,2,3) =
{\displaystyle (256^{\,\!256})^{256^{256}}=256^{256^{257}}}
{\displaystyle (256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}}
{\displaystyle 256^{\,\!256^{256^{257}}}}
M(256,4,3) ≈
{\displaystyle {\,\!256^{256^{256^{256^{257}}}}}}
{\displaystyle {\,\!256^{256^{256^{256^{256^{257}}}}}}}
{\displaystyle {\,\!256^{256^{256^{256^{256^{256^{257}}}}}}}}
{\displaystyle M(256,256,3)\approx (256\uparrow )^{256}257}
{\displaystyle (256\uparrow )^{256}}
{\displaystyle f(n)=256^{n}}
Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈
{\displaystyle 256\uparrow \uparrow 257}
, using Knuth's up-arrow notation.
After the first few steps the value of
{\displaystyle n^{n}}
is each time approximately equal to
{\displaystyle 256^{n}}
. In fact, it is even approximately equal to
{\displaystyle 10^{n}}
(see also approximate arithmetic for very large numbers). Using base 10 powers we get:
{\displaystyle M(256,1,3)\approx 3.23\times 10^{616}}
{\displaystyle M(256,2,3)\approx 10^{\,\!1.99\times 10^{619}}}
{\displaystyle \log _{10}616}
is added to the 616)
{\displaystyle M(256,3,3)\approx 10^{\,\!10^{1.99\times 10^{619}}}}
{\displaystyle 619}
is added to the
{\displaystyle 1.99\times 10^{619}}
, which is negligible; therefore just a 10 is added at the bottom)
{\displaystyle M(256,4,3)\approx 10^{\,\!10^{10^{1.99\times 10^{619}}}}}
{\displaystyle M(256,256,3)\approx (10\uparrow )^{255}1.99\times 10^{619}}
{\displaystyle (10\uparrow )^{255}}
{\displaystyle f(n)=10^{n}}
{\displaystyle 10\uparrow \uparrow 257<{\text{mega}}<10\uparrow \uparrow 258}
Moser's number[edit]
It has been proven that in Conway chained arrow notation,
{\displaystyle \mathrm {moser} <3\rightarrow 3\rightarrow 4\rightarrow 2,}
and, in Knuth's up-arrow notation,
{\displaystyle \mathrm {moser} <f^{3}(4)=f(f(f(4))),{\text{ where }}f(n)=3\uparrow ^{n}3.}
Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:[2]
{\displaystyle \mathrm {moser} \ll 3\rightarrow 3\rightarrow 64\rightarrow 2<f^{64}(4)={\text{Graham's number}}.}
^ Hugo Steinhaus, Mathematical Snapshots, Oxford University Press 19693, ISBN 0195032675, pp. 28-29
^ Proof that G >> M
Factoid on Big Numbers
Megistron at mathworld.wolfram.com (Steinhaus referred to this number as "megiston" with no "r".)
Circle notation at mathworld.wolfram.com
Steinhaus-Moser Notation - Pointless Large Number Stuff
Retrieved from "https://en.wikipedia.org/w/index.php?title=Steinhaus–Moser_notation&oldid=1037885930"
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Polynomial-Time Algorithms for Multiple-Arm Identification with Full-Bandit Feedback | Neural Computation | MIT Press
Yuko Kuroki,
University of Tokyo, Bunkyo-ku, Tokyo, 113-0333, Japan, and RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103-0027, Japan ykuroki@ms.k.u-tokyo.ac.jp
Liyuan Xu,
University of Tokyo, Bunkyo-ku, Tokyo, 113-0333, Japan, and RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103-0027, Japan liyuan@ms.k.u-tokyo.ac.jp
Atsushi Miyauchi,
RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103-0027, Japan atsushi.miyauchi.hv@riken.jp
Junya Honda,
University of Tokyo, Bunkyo-ku, Tokyo, 113-0333, Japan, and RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103-0027, Japan honda@edu.k.u-tokyo.ac.jp
RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103-0027, Japan, and University of Tokyo, Bunkyo-ku, Tokyo, 113-0333, Japan sugi@k.u-tokyo.ac.jp
L.X. is now at Gatsby Computational Neuroscience Unit, University College London.
Yuko Kuroki, Liyuan Xu, Atsushi Miyauchi, Junya Honda, Masashi Sugiyama; Polynomial-Time Algorithms for Multiple-Arm Identification with Full-Bandit Feedback. Neural Comput 2020; 32 (9): 1733–1773. doi: https://doi.org/10.1162/neco_a_01299
We study the problem of stochastic multiple-arm identification, where an agent sequentially explores a size-
k
subset of arms (also known as a super arm) from given
arms and tries to identify the best super arm. Most work so far has considered the semi-bandit setting, where the agent can observe the reward of each pulled arm or assumed each arm can be queried at each round. However, in real-world applications, it is costly or sometimes impossible to observe a reward of individual arms. In this study, we tackle the full-bandit setting, where only a noisy observation of the total sum of a super arm is given at each pull. Although our problem can be regarded as an instance of the best arm identification in linear bandits, a naive approach based on linear bandits is computationally infeasible since the number of super arms
K
is exponential. To cope with this problem, we first design a polynomial-time approximation algorithm for a 0-1 quadratic programming problem arising in confidence ellipsoid maximization. Based on our approximation algorithm, we propose a bandit algorithm whose computation time is
O
(log
K
), thereby achieving an exponential speedup over linear bandit algorithms. We provide a sample complexity upper bound that is still worst-case optimal. Finally, we conduct experiments on large-scale data sets with more than 10
super arms, demonstrating the superiority of our algorithms in terms of both the computation time and the sample complexity.
A Normative Account of Confirmation Bias During Reinforcement Learning
Behaviors Coordination Using Restless Bandits Allocation Indexes
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IsMapleAssignedName - Maple Help
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : C Application Programming Interface : IsMapleAssignedName
return the number of arguments in an EXPSEQ object
test if an object is an assigned name
test if an object is complex numeric
test if an object is an equation
test if an object is numeric
test if an object is a fraction
test if an object is an integer
test if an object is an 8-bit integer
test if an object is a 16-bit integer
test if an object is a list
test if an object is a name
test if an object is a logical operation
test if an object is NULL
test if an object is a generated pointer
test if an object is the string "NULL"
test if an object is a relation
test if an object is a Maple procedure
test if an object is an rtable
test if an object is a set
test for an "end of session" object
test if an object is a string
test if an object is a table
test if an object is an unassigned name
test if an object is 0
MapleNumArgs(kv, s)
IsMapleAssignedName(kv, s)
IsMapleComplexNumeric(kv, s)
IsMapleEquation(kv, s)
IsMapleNumeric(kv, s)
IsMapleFraction(kv, s)
IsMapleInteger(kv, s)
IsMapleInteger8(kv, s)
IsMapleInteger16(kv, s)
IsMapleList(kv, s)
IsMapleLogical(kv, s)
IsMapleName(kv, s)
IsMapleNULL(kv, s)
IsMaplePointer(kv, s)
IsMaplePointerNULL(kv, s)
IsMapleProcedure(kv, s)
IsMapleRelation(kv, s)
IsMapleRTable(kv, s)
IsMapleSet(kv, s)
IsMapleStop(kv, s)
IsMapleString(kv, s)
IsMapleTable(kv, s)
IsMapleUnassignedName(kv, s)
IsMapleUnnamedZero(kv, s)
The IsMaple* functions test the specified type of the given Maple object. These functions all return TRUE (1) when the Maple DAG, s, fits the description given by the function name. If s is not of the correct type, FALSE (0) is returned.
MapleNumArgs returns the length of any Maple object. It is most useful for computing the length of the argument expression sequence passed out to the call_external dll entry point. An expression sequence object can be treated as a simple array with indexing starting at 1 (not 0). For example, if args is an expression sequence and MapleNumArgs(kv,args) returns 3, then args[1], args[2], and args[3] are all Maple objects. In the case of an external entry point, these are the arguments given during the function call.
There are several functions for determining what kind of NULL an object is. These are primarily used by automatic wrapper generation, and are not commonly be seen in hand-written code. IsMapleUnnamedZero looks for a Maple zero object, but not a name assigned the value zero. IsMapleNULL tests for the empty expression sequence denoted by NULL in the Maple language. IsPointerNULL tests for the C version of NULL (hardware zero), or the Maple string, "NULL".
IsMapleStop can be used with OpenMaple to detect the evaluation of the quit command.
The IsMaple...Numeric routines use the Maple type numeric definition. All other tests use the object type definition as defined by the type command. The only significant exception is IsMapleName, which returns TRUE only for NAME objects, while type(t[1], name) returns true even if it is testing a TABLEREF object.
Integer query routines, with the bit size specified in the name, test to ensure that the given Maple object, s, is a Maple integer and also that it can fit into the specified number of bits if converted to a hardware integer.
ALGEB M_DECL MyRand( MKernelVector kv, ALGEB *args )
M_INT argc, val;
if( !IsMapleInteger32(kv,args[1]) ) {
MapleRaiseError2(kv,
"integer expected for %-1 argument, instead got %2",
ToMapleInteger(kv,1),args[1]);
val = MapleToInteger32(kv,args[1]);
return( ToMapleInteger(kv,(rand() % val)) );
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{myrand}≔\mathrm{DefineExternal}\left(\mathrm{MyRand},\mathrm{dll}\right):
\mathrm{myrand}\left(100\right)
\textcolor[rgb]{0,0,1}{27}
\mathrm{myrand}\left(\mathrm{xx}\right)
Error, (in myrand) integer expected for 1st argument, instead got xx
The IsMapleEquation, IsMapleFraction, IsMapleLogical and IsMapleRelation commands were introduced in Maple 2018.
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{\displaystyle f(x)}
{\displaystyle \displaystyle f'(x)=2\cos(x)-e^{x}}
{\displaystyle f(0)=0}
We wish to find a function with derivative as given. So what function gives a derivative of
{\displaystyle \cos(x)}
? What about for
{\displaystyle e^{x}}
? Lastly, don't forget that adding a constant to the function also gives an antiderivative. Use the information ƒ(0) = 0 to find this constant.
{\displaystyle {\tfrac {d}{dx}}\sin(x)=\cos(x)}
{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}
, we have that the function given by
{\displaystyle f(x)=2\sin(x)-e^{x}+C}
where C is any constant are functions that have the given derivative.
Since we are also given that
{\displaystyle f(0)=0}
, we plug this information in to see that
{\displaystyle 0=f(0)=2\sin(0)-e^{0}+C=0-1+C}
. Isolating for C gives
{\displaystyle C=1}
. Thus the required function is given by
{\displaystyle \displaystyle f(x)=2\sin(x)-e^{x}+1}
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Fundamental theorem of calculus, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag
MER Tag Fundamental theorem of calculus
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Generic motor and drive with closed-loop torque control - MATLAB - MathWorks Nordic
Modeling Electrical Losses
Motor and driver overall efficiency (percent)
Generic motor and drive with closed-loop torque control
The Motor & Drive block represents a generic brushless motor and drive with closed-loop torque control. It is a simplified version of the Motor & Drive (System Level) (Simscape Electrical) block. The Motor & Drive block is useful if you need a generic or low-fidelity motor implementation in your system. It is also suited for cases when you do not know all of your motor specifications or you want to use the block to find an appropriate motor for your system.
To enable faster simulation, the block abstracts the motor, drive electronics, and control. The block generates a torque-speed envelope that saturates the input torque, and it permits only the range of torques and speeds that the envelope defines.
The Motor & Drive block models first-order losses based on the overall efficiency for a given speed and torque, which you specify as Motor and driver overall efficiency (percent), Speed at which efficiency is measured, and Torque at which efficiency is measured, respectively. The block uses the speed and torque to generate a torque-speed envelope. The envelope saturates the input torque, which yields the torque that the motor responds to, τelec. This is also the torque that the block uses to compute the electrical losses.
The block only considers torque-dependent resistive losses such that
{P}_{losses}=k{\tau }_{elec}^{2},
k=\frac{{\omega }_{\eta }\left(1-\eta /100\right)}{{\tau }_{\eta }\cdot \eta /100}.
Resistive losses are also known as Ohmic losses and occur due to the tendency of the armature windings to resist the flow of electrons. The electrical power includes these losses such that
{P}_{elec}={P}_{losses}+\omega {\tau }_{elec}.
The rate of conversion from electrical energy to heat energy is defined by Joule's law:
I=\frac{{P}_{elec}}{V},
Pelec is the electrical power that the block calculates and uses in the governing equation.
Plosses is the electrical power lost during operation. When you model the effects of heat flow and temperature change, this value represents the rate of heat flow that gets distributed into the thermal mass or out port H.
ω is the angular velocity of the rotor. This is equivalent to the W output port value.
τelec is the saturated torque demand.
k is the proportionality constant for resistance losses, which has the units (energy*time)-1.
η is the efficiency of the motor and driver for a given speed and torque. This value is equivalent to the Motor and driver overall efficiency (percent) parameter.
ωη is the angular velocity that corresponds to the overall efficiency. This value is equivalent to the Speed at which efficiency is measured parameter.
τη is the torque that corresponds to the overall efficiency. This value is equivalent to the Torque at which efficiency is measured parameter.
V is the voltage across the terminals.
I is the current through the terminals.
When you enable thermal modeling, Plosses represents the contribution from the block to the heat flow.
To include series resistance, fixed losses, and iron losses, you can add blocks to your model or use the Motor & Drive (System Level) (Simscape Electrical) block.
You can add damping and inertia with the Rotational Damper block and Inertia block, respectively.
When you model the effects of heat flow and temperature change, the electrical losses from the motor contribute to these effects.
To enable this setting, set Thermal port to Model.
The motor driver tracks a torque demand with the time constant Tc.
Motor speed fluctuations due to mechanical load do not affect the motor torque tracking.
Tr — Reference torque demand, N*m
Physical signal input port associated with the reference torque demand.
W — Mechanical rotational speed, rad/s
Physical signal output port associated with the mechanical rotational speed.
+ — Positive electrical DC supply
Electrical conserving port associated with the positive electrical DC supply.
- — Negative electrical DC supply
Electrical conserving port associated with the negative electrical DC supply.
Thermal conserving port associated with heat flow. Electrical losses from the motor contribute to the heat flow through this port.
Maximum torque — Torque to define torque-speed envelope boundary
400 N*m (default) | positive scalar
Maximum permissible torque value. The block uses this value and the Maximum power parameter to define the torque-speed envelope.
Maximum power — Power to define torque-speed envelope boundary
Maximum permissible power value. The block uses this value and the Maximum torque parameter to define the torque-speed envelope.
Torque control time constant, Tc — Output torque time step
Torque controller output time interval. Use this parameter to tell the block how long to wait between issuing torque value output information.
Motor and driver overall efficiency (percent) — Energy efficiency for given speed and torque
100 (default) | scalar in the range (0,100]
Efficiency to convert from electrical to mechanical rotational energy.
Speed at which efficiency is measured — Given speed for efficiency measurement
Speed that the block uses to calculate torque-dependent electrical losses.
Torque at which efficiency is measured — Given torque for efficiency measurement
Torque that the block uses to calculate torque-dependent electrical losses.
Thermal port — Simulate thermal losses in the mass of the electrical wiring
Option to enable the thermal port and include thermal losses in your simulation.
Thermal mass — Tendency of the wiring to retain heat
Thermal mass of the electrical winding, defined as the energy required to raise the temperature by one temperature unit.
To enable this parameter, set Thermal port to Model.
Motor & Drive (System Level) (Simscape Electrical)
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networks(deprecated)/maxdegree - Maple Help
Home : Support : Online Help : networks(deprecated)/maxdegree
finds maximum vertex degree in a graph
maxdegree(G)
maxdegree(G, vname)
vertex of maximum degree (returned)
Important:The networks package has been deprecated. Use the superseding command GraphTheory[MaximumDegree]instead.
The total number of edges (undirected, including loops) incident at each vertex is computed, and the maximum returned.
If vname is specified, a vertex of max degree is returned as vname.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[maxdegree](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{complete}\left(4\right):
\mathrm{addedge}\left([{1},{1},{2},{3},{4}],G\right)
\textcolor[rgb]{0,0,1}{\mathrm{e7}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e9}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e10}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e11}}
\mathrm{maxdegree}\left(G,\mathrm{large}\right)
\textcolor[rgb]{0,0,1}{5}
\mathrm{large}
\textcolor[rgb]{0,0,1}{1}
\mathrm{mindegree}\left(G,\mathrm{small}\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{small}
\textcolor[rgb]{0,0,1}{2}
GraphTheory[MaximumDegree]
|
The optional filter parameter, passed as the index to the Map or Map2 command, restricts the application of
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔\mathrm{Matrix}\left([[1,2,3],[0,1,4]],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper},\mathrm{unit}]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
M≔\mathrm{Map}\left(x↦x+1,A\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{evalb}\left(\mathrm{addressof}\left(A\right)=\mathrm{addressof}\left(M\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
B≔〈〈1,2,3〉|〈4,5,6〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map2}[\left(i,j\right)↦\mathrm{evalb}\left(i=1\right)]\left(\left(x,a\right)↦a\cdot x,3,B\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map}\left(x↦x+1,g\left(3,A\right)\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\end{array}]\right)
C≔\mathrm{Matrix}\left([[1,2],[3]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{symmetric}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{Map}\left(x↦x+1,C\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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An Analysis of Modified Emden-Type Equation ẍ + αxẋ + βx3 = 0: Exact Explicit Analytical Solution, Lagrangian, Hamiltonian for Arbitrary Values of α and β ()
Department of Physics (Formerly), Kalyani Mahavidyalaya, Kalyani, India.
The modified Emden-type is being investigated by mathematicians as well as physicists for about a century. However, there exist no exact explicit solution of this equation, ẍ + αxẋ + βx3 = 0 for arbitrary values of α and β. In this work, the exact analytical explicit solution of modified Emden-type (MEE) equation is derived for arbitrary values of α and β. The Lagrangian and Hamiltonian of MEE are also worked out. The solution is also utilized to find exact explicit analytical solution of Force-free Duffing oscillator-type equation. And exact explicit analytical solution of two-dimensional Lotka-Volterra System is also worked out.
Exact Analytical Solution, Lagrangian, Hamiltonian
Biswas, D. (2019) An Analysis of Modified Emden-Type Equation ẍ + αxẋ + βx3 = 0: Exact Explicit Analytical Solution, Lagrangian, Hamiltonian for Arbitrary Values of α and β. Natural Science, 11, 8-16. doi: 10.4236/ns.2019.111002.
\stackrel{¨}{x}+\alpha x\stackrel{\dot{}}{x}+\beta {x}^{3}=0
x={x}_{1}^{m}
m to be determined later. (2.1)
\stackrel{˙}{x}=m{x}_{1}^{m-1}{\stackrel{˙}{x}}_{1}
dot represents time derivative (2.2)
\stackrel{¨}{x}=m\left(m-1\right){x}_{1}^{m-2}{\stackrel{\dot{}}{x}}_{1}^{2}+m{x}_{1}^{m-1}{\stackrel{¨}{x}}_{1}
m{x}_{1}^{m-1}{\stackrel{¨}{x}}_{1}+m\left(m-1\right){x}_{1}^{m-2}{\stackrel{\dot{}}{x}}_{1}^{2}+\alpha m{x}_{1}^{2m-1}{\stackrel{\dot{}}{x}}_{1}+\beta {x}_{1}^{3m}=0
{\stackrel{¨}{x}}_{1}+\alpha {x}_{1}^{m}{\stackrel{\dot{}}{x}}_{1}+\left(m-1\right)\frac{{\stackrel{\dot{}}{x}}_{1}^{2}}{{x}_{1}}+\frac{\beta }{m}{x}_{1}^{2m+1}=0
{\stackrel{˙}{x}}_{1}=u\left({x}_{1}\right)
{\stackrel{¨}{x}}_{1}={u}^{\prime }\left({x}_{1}\right)u\left({x}_{1}\right)
prime denotes derivative w.r. to x1 (2.7)
{u}^{\prime }\left({x}_{1}\right)+\alpha {x}_{1}^{m}+\left(m-1\right)\frac{u\left({x}_{1}\right)}{{x}_{1}}+\frac{\beta }{m}\frac{{x}_{1}^{2m+1}}{u\left({x}_{1}\right)}=0
u=A{x}_{1}^{K}
A and K are constants and K ≠ 1 (2.9)
AK{x}_{1}^{K-1}+\alpha {x}_{1}^{m}+A\left(m-1\right){x}_{1}^{K-1}+\frac{\beta }{mA}{x}_{1}^{2m+1-K}=0
2m+1-K=K-1
m=K-1
AK{x}_{1}^{K-1}+\alpha {x}_{1}^{K-1}+A\left(m-1\right){x}_{1}^{K-1}+\frac{\beta }{mA}{x}_{1}^{K-1}=0
Cancelling
{x}_{1}^{K-1}
in (2.13) assuming
{x}_{1}^{K-1}\ne 0
AK+\alpha +A\left(m-1\right)+\frac{\beta }{mA}=0
A{m}^{2}+\left(AK+\alpha -A\right)m+\frac{\beta }{A}=0
m=\frac{\left(A-\alpha -AK\right)\pm \sqrt{{\left(A-\alpha -AK\right)}^{2}-4\beta }}{2A}
K-1=\frac{\left(A-\alpha -AK\right)\pm \sqrt{{\left(A-\alpha -AK\right)}^{2}-4\beta }}{2A}
2A\left(K-1\right)-A+\alpha +AK=\pm \sqrt{{\left(A-\alpha -AK\right)}^{2}-4\beta }
3AK-3A+\alpha =\pm \sqrt{{\left(A-\alpha -AK\right)}^{2}-4\beta }
2{A}^{2}{K}^{2}+KA\left(\alpha -4A\right)+\left(2{A}^{2}-A\alpha +\beta \right)=0
K=\frac{\left(4A-\alpha \right)\pm \sqrt{{\alpha }^{2}-8\beta }}{4A}
m=K-1=\frac{-\alpha \pm \sqrt{{\alpha }^{2}-8\beta }}{4A}
{\stackrel{˙}{x}}_{1}=A{x}_{1}^{K}
\frac{{x}_{1}^{1-K}}{1-K}=At+BB
= Constant of integration
{x}_{1}={\left[A\left(1-K\right)t+B\left(1-K\right)\right]}^{\frac{1}{1-K}}
x={x}_{1}^{m}={\left[A\left(1-K\right)t+B\left(1-K\right)\right]}^{\frac{m}{1-K}}
={\left[A\left(1-K\right)t+B\left(1-K\right)\right]}^{-1}
using (2.12) (2.22)
x=\frac{1}{A\left(1-K\right)t+B\left(1-K\right)}
x=\frac{1}{\frac{A\left\{\alpha \mp \sqrt{{\alpha }^{2}-8\beta }\right\}t}{4A}+\frac{B\left\{\alpha \mp \sqrt{{\alpha }^{2}-8\beta }\right\}}{4A}}
using (2.20)
x=\frac{4A}{A\left\{\alpha \mp \sqrt{{\alpha }^{2}-8\beta }\right\}t+B\left\{\alpha \mp \sqrt{{\alpha }^{2}-8\beta }\right\}}
\stackrel{¨}{x}+\frac{\left(n-2\right)}{\left(n-1\right)}{R}^{\prime }\left(x\right)\stackrel{\dot{}}{x}+\frac{1}{\left(1-n\right)}R\left(x\right){R}^{\prime }\left(x\right)=0
L={\left\{\stackrel{˙}{x}+R\left(x\right)\right\}}^{n}
R\left(x\right)=\frac{{A}_{0}}{2}{x}^{2}
A0 is a constant (3.3)
{R}^{\prime }\left(x\right)={A}_{0}x
\stackrel{¨}{x}+\frac{\left(n-2\right)}{\left(n-1\right)}{A}_{0}x\stackrel{\dot{}}{x}+\frac{1}{\left(1-n\right)}\frac{{A}_{0}^{2}}{2}{x}^{3}=0
(n ≠ 1, 2) (3.5)
\frac{\left(n-2\right)}{\left(n-1\right)}{A}_{0}=\alpha
\frac{1}{\left(1-n\right)}\frac{{A}_{0}^{2}}{2}=\beta
\stackrel{¨}{x}+\alpha x\stackrel{\dot{}}{x}+\beta {x}^{3}=0
\frac{2{\left(n-2\right)}^{2}}{\left(1-n\right)}=\frac{{\alpha }^{2}}{\beta }
2\left({n}^{2}-4n+4\right)=\frac{{\alpha }^{2}}{\beta }-n\frac{{\alpha }^{2}}{\beta }
2{n}^{2}-n\left(8-\frac{{\alpha }^{2}}{\beta }\right)+8-\frac{{\alpha }^{2}}{\beta }=0
n=\frac{8-\frac{{\alpha }^{2}}{\beta }\pm \sqrt{{\left(8-\frac{{\alpha }^{2}}{\beta }\right)}^{2}-8\left(8-\frac{{\alpha }^{2}}{\beta }\right)}}{4}
\frac{\alpha \left(n-1\right)}{\left(n-2\right)}={A}_{0}
, where n is given by (3.10) (3.11)
L={\left[\stackrel{˙}{x}+\frac{{A}_{0}}{2}{x}^{2}\right]}^{n}
{L}_{1}=\frac{1}{\stackrel{˙}{x}+K{x}^{2}}
{L}_{2}={\left(2\stackrel{˙}{x}+K{x}^{2}\right)}^{\frac{1}{2}}
n=\frac{-1\pm \sqrt{1+8}}{4}=\frac{-1\pm 3}{4}=-1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}
\begin{array}{l}\text{when}\text{\hspace{0.17em}}n=-1,\text{\hspace{0.17em}}{A}_{0}=2K\\ \text{when}\text{\hspace{0.17em}}n=\frac{1}{2},\text{\hspace{0.17em}}{A}_{0}=K\end{array}\right\}
L={\left[\stackrel{˙}{x}+K{x}^{2}\right]}^{-1}=\frac{1}{\stackrel{˙}{x}+K{x}^{2}}
L={\left[\stackrel{˙}{x}+\frac{K}{2}{x}^{2}\right]}^{1/2}=\frac{{\left[2\stackrel{˙}{x}+K{x}^{2}\right]}^{1/2}}{\sqrt{2}}
Rejecting the factor
\sqrt{2}
in the denominator of R. H. S of (4.6) we get
L={\left[2\stackrel{˙}{x}+K{x}^{2}\right]}^{1/2}
\frac{\partial L}{\partial \stackrel{˙}{x}}=p=n{\left[\stackrel{˙}{x}+\frac{{A}_{0}}{2}{x}^{2}\right]}^{n-1}
\stackrel{˙}{x}={\left(\frac{p}{n}\right)}^{\frac{1}{n-1}}-\frac{{A}_{0}}{2}{x}^{2}
{\left(\frac{p}{n}\right)}^{\frac{1}{n-1}}=\stackrel{˙}{x}+\frac{{A}_{0}}{2}{x}^{2}
H=p\stackrel{˙}{x}-L
H=p\left[{\left(\frac{p}{n}\right)}^{\frac{1}{n-1}}-\frac{{A}_{0}}{2}{x}^{2}\right]-{\left[\stackrel{˙}{x}+\frac{{A}_{0}}{2}{x}^{2}\right]}^{n}
using (5.2) and (3.12)
H=p\left[{\left(\frac{p}{n}\right)}^{\frac{1}{n-1}}-\frac{{A}_{0}}{2}{x}^{2}\right]-{\left(\frac{p}{n}\right)}^{\frac{n}{n-1}}
using (5.3) (5.4)
{\omega }^{″}+\left(\alpha \omega +\lambda \right){\omega }^{\prime }+\beta {\omega }^{3}+\frac{\alpha \gamma }{3}{\omega }^{2}+\frac{2{\gamma }^{2}\omega }{9}=0
{\omega }^{\prime }=\frac{\text{d}\omega }{\text{d}\tau }
\omega =x{\text{e}}^{-\frac{\gamma }{3}\tau }
t=-\frac{3}{\gamma }{\text{e}}^{-\frac{\gamma }{3}\tau }
, γ = arbitrary parameter (6.2)
\omega =\frac{4A{\text{e}}^{-\frac{\gamma }{3}\tau }}{-\frac{3A}{\gamma }\left\{\alpha \pm \sqrt{{\alpha }^{2}-8\beta }\right\}{\text{e}}^{-\frac{\gamma }{3}\tau }+B\left\{\alpha \pm \sqrt{{\alpha }^{2}-8\beta }\right\}}
\stackrel{˙}{x}=x\left({a}_{1}+{a}_{2}x+{a}_{3}y\right),\stackrel{˙}{y}=y\left({b}_{1}+{b}_{2}x+{b}_{3}y\right)
{a}_{1},{a}_{2},\cdots
{b}_{1},{b}_{2},\cdots
etc are real parameters. This equation has been studied for a long time and its solution is important in mathematical biology [ 16 ].
\stackrel{˙}{x}=x\left({a}_{1}+{a}_{2}x+{a}_{3}y\right),\stackrel{˙}{y}=y\left({a}_{1}+{b}_{2}x-{a}_{3}y\right)
Equation (6.5) can be written by eliminating y and
\stackrel{˙}{y}
\stackrel{¨}{x}-\left\{\left(3{a}_{2}+{b}_{2}\right)x+3{a}_{1}\right\}\stackrel{\dot{}}{x}+{a}_{2}\left({a}_{2}+{b}_{2}\right){x}^{3}+{a}_{1}\left(3{a}_{2}+{b}_{2}\right){x}^{2}+2{a}_{1}^{2}x=0
\alpha =–\left(3{a}_{2}+{b}_{2}\right)
\beta ={a}_{\text{2}}\left({a}_{\text{2}}+{b}_{\text{2}}\right)
\gamma =-\text{3}{a}_{\text{1}}
x=\frac{4A{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}\left\{-\left(3{a}_{2}+{b}_{2}\right)\pm \sqrt{{\left(3{a}_{2}+{b}_{2}\right)}^{2}-8{a}_{2}\left({a}_{2}+{b}_{2}\right)}\right\}{\text{e}}^{{a}_{1}t}+B\left\{-\left(3{a}_{2}+{b}_{2}\right)\pm \sqrt{{\left(3{a}_{2}+{b}_{2}\right)}^{2}-8{a}_{2}\left({a}_{2}+{b}_{2}\right)}\right\}}
From (6.4),
{a}_{1}+{a}_{2}x+{a}_{3}y=\frac{\stackrel{˙}{x}}{x}
y=\frac{1}{{a}_{3}}\left[\frac{\stackrel{˙}{x}}{x}-{a}_{2}x-{a}_{1}\right]
x=\frac{4A{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ}
Z=-\left(3{a}_{2}+{b}_{2}\right)\pm \sqrt{{\left(3{a}_{2}+{b}_{2}\right)}^{2}-8{a}_{2}\left({a}_{2}+{b}_{2}\right)}
\stackrel{˙}{x}=\frac{4A{a}_{1}{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ}-\frac{4A{\text{e}}^{{a}_{1}t}\frac{A}{{a}_{1}}Z{a}_{1}{\text{e}}^{{a}_{1}t}}{{\left(\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ\right)}^{2}}
\frac{\stackrel{˙}{x}}{x}={a}_{1}-\frac{AZ{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ}
\begin{array}{c}y=\frac{1}{{a}_{3}}\left[-\frac{AZ{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ}-\frac{4{a}_{2}A{\text{e}}^{{a}_{1}t}}{\frac{A}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}+BZ}\right]\\ =\frac{1}{{a}_{3}}\left[\frac{{A}_{0}Z{\text{e}}^{{a}_{1}t}}{BZ-\frac{{A}_{0}}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}}+\frac{4{a}_{2}{A}_{0}{\text{e}}^{{a}_{1}t}}{BZ-\frac{{A}_{0}}{{a}_{1}}Z{\text{e}}^{{a}_{1}t}}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{0}=-A\end{array}
[1] Painleve, P. (1902) Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme. Acta Mathematica, 25, 1-85.
[2] Kamke, E. (1983) Differentialgleichungen Losungsmethoden und Losungen. Tenbner, Stuttgart.
[3] Ince, E.L. (1956) Ordinary Differential Equations. Dover Publications, Mineola, NY.
[4] Gobulevev, V.V. (1950) Lectures on Analytical Theory of Differential Equations. Gostek hizdat, Moscow.
[5] Erwin, V.J., Ames, W.F. and Adams, E. (1984) Wave Phenomena: Modern Theory and Applications. Rogers, C. and Moodie, J.B. (Eds.), North-Holland, Amsterdam.
[6] Chisholm, J.S.R. and Common, A.K. (1987) A Class of Second-Order Differential Equations and Related First-Order Systems. Journal of Physics A: Mathematical and General, 20, 5459-5472.
[7] Yang, C.N. and Mills, R.L. (1954) Conservation of Isotopic Spin and Isotopic Gauge Invariance. Physical Review, 96, 191.
[8] Leach, P.G.L. (1985) First Integrals for the Modified Emden Equation q..+α(t) q.+qn =0. Journal of Mathematical Physics, 26, 2510-2514.
[9] Chandrasekhar, S. (1957) An Introduction to Study of Stellar Structure. Dover Publications, Mineola, NY.
[10] Chandrasekar, V.K., Senthivelan, M. and Lakshmanan, M. (2004) On the General Solution for the Modified Emden Type Equation ẍ + αxẋ + βx3 = 0. Journal of Physics A: Mathematical and Theoretical, 40, 4717-4727.
[11] Vujanovic, B.D. and Jones, S.E. (1989) Variational Methods in Nonconservative Phenomena. Academic Press, Cambridge, MA.
[12] Cariena, J.F., Raada, M.F. and Santander, F. (2005) Lagrangian Formalism for Nonlinear Second-Order Riccati Systems: One-Dimensional Integrability and Two-Dimensional Superintegrability. Journal of Mathematical Physics, 46, 062703.
[13] Chandrasekhar, V.K., Pandey, S.N., Senthivelan, M. and Lakshmanan, M. (2006) A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems. Journal of Mathematical Physics, 47, 023508.
[14] Bluman, G.W. and Anco, S. (2002) Symmetries and Integration Methods for Differential Equations. Springer-Verlag, New York.
[15] Lakshmenan, M. and Rajsekhar, S. (2003) Nonlinear Dynamics: Integrability Chaos and Patterns. Springer-Verlag.
[16] Murray, J.D. (1989) Mathematical Biology. Springer-Verlag.
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Measurement_of_a_Circle Knowpia
Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis)[1] is a treatise that consists of three propositions by Archimedes, ca. 250 BCE.[2][3] The treatise is only a fraction of what was a longer work.[4][5]
Proposition oneEdit
The circle and the triangle are equal in area.
Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion.[6]
Proposition twoEdit
Proposition two states:
The area of a circle is to the square on its diameter as 11 to 14.
This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition.[6]
Proposition threeEdit
Proposition three states:
The ratio of the circumference of any circle to its diameter is greater than
{\displaystyle 3{\tfrac {10}{71}}}
{\displaystyle 3{\tfrac {1}{7}}}
This approximates what we now call the mathematical constant π. He found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons.[7]
Approximation to square rootsEdit
This proposition also contains accurate approximations to the square root of 3 (one larger and one smaller) and other larger non-perfect square roots; however, Archimedes gives no explanation as to how he found these numbers.[5] He gives the upper and lower bounds to √3 as 1351/780 > √3 > 265/153.[6] However, these bounds are familiar from the study of Pell's equation and the convergents of an associated continued fraction, leading to much speculation as to how much of this number theory might have been accessible to Archimedes. Discussion of this approach goes back at least to Thomas Fantet de Lagny, FRS (compare Chronology of computation of π) in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen. In the early 1880s, Friedrich Otto Hultsch (1833–1906) and Karl Heinrich Hunrath (b. 1847) noted how the bounds could be found quickly by means of simple binomial bounds on square roots close to a perfect square modelled on Elements II.4, 7; this method is favoured by Thomas Little Heath. Although only one route to the bounds is mentioned, in fact there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by an iterative geometrical construction suggested by Archimedes' Stomachion in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12.
^ Knorr, Wilbur R. (1986-12-01). "Archimedes' dimension of the circle: A view of the genesis of the extant text". Archive for History of Exact Sciences. 35 (4): 281–324. doi:10.1007/BF00357303. ISSN 0003-9519. S2CID 119807724.
^ Lit, L.W.C. (Eric) van. "Naṣīr al-Dīn al-Ṭūsī's Version of The Measurement of the Circle of Archimedes from his Revision of the Middle Books". Tarikh-e Elm. The measurement of the circle was written by Archimedes (ca. 250 B.C.E.)
^ Knorr, Wilbur R. (1986). The Ancient Tradition of Geometric Problems. Courier Corporation. p. 153. ISBN 9780486675329. Most accounts of Archimedes' works assign this writing to a time relatively late in his career. But this view is the consequence of a plain misunderstanding.
^ Heath, Thomas Little (1921), A History of Greek Mathematics, Boston: Adamant Media Corporation, ISBN 978-0-543-96877-7, retrieved 2008-06-30
^ a b "Archimedes". Encyclopædia Britannica. 2008. Retrieved 2008-06-30.
^ a b c Heath, Thomas Little (1897), The Works of Archimedes, Cambridge University: Cambridge University Press., pp. lxxvii , 50, retrieved 2008-06-30
^ Heath, Thomas Little (1931), A Manual of Greek Mathematics, Mineola, N.Y.: Dover Publications, p. 146, ISBN 978-0-486-43231-1
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Lines of Best Fit - Course Hero
College Algebra/Linear Functions and Modeling/Lines of Best Fit
Linear regression is a statistical method that calculates a line of best fit for a given set of data points. The line of best fit has the minimum value for the sum of the squares of the distances from the data points to the line.
The line of best fit is also called a regression line for the data in a scatterplot. The distance from each point to the line is calculated as the sum of the squares of each point's distance, which is minimized by the process of linear regression.
A correlation is a relationship between two variables. When the points in a scatterplot are very close to a line of best fit, there is a strong correlation. When they show a general linear pattern, but are not close to the line, there is a weak correlation. Both positive and negative trends may exhibit strong or weak correlations. When the points show no linear pattern at all, there is no linear correlation.
The correlation coefficient,
r
, of a line of best fit is a value between –1 and 1, inclusive, that indicates the strength and direction of the correlation of the line.
r
-value of 1 indicates that the line has a positive slope and all the points lie on the line.
r
close to 1 indicates a strong positive correlation.
A positive
r
-value closer to zero than to 1 indicates a weak positive correlation.
r
-value of zero indicates no correlation.
A negative
r
-value closer to zero than to –1 indicates a weak negative correlation.
r
close to –1 indicates a strong negative correlation.
r
-value of –1 indicates that the line has a negative slope and all the points lie on the line.
Scatterplots show how the correlation coefficient indicates the strength and direction of a linear correlation. The sign of
r
indicates whether the correlation is positive or negative, and the absolute value of
r
indicates the strength of the correlation. The greater the absolute value, the stronger the correlation.
Interpreting Lines of Best Fit
For very small data sets, a line of best fit can be calculated by hand. Most often, technology such as graphing calculators, spreadsheets, or online tools is used to determine the equation of the line.
Determining the Equation of a Line of Best Fit
Employees at a company start with an average salary of $40,500 at year zero. The table shows the average salaries of the company's employees for selected years of service. Graph and interpret the line of best fit.
Calculate the line of best fit by using the linear regression function of a graphing calculator.
y \approx 1\rm{,}061x + 41\rm{,}660
Graph the line of best fit on the scatterplot.
Next, interpret the line of best fit.
The correlation coefficient of
r\approx 0.98
means that there is a very strong positive correlation between the years and salaries. When employees work at the company for a number of years, their average salaries have increased.
The slope of the line is about 1,061, which means that salaries have increased by about $1,061 per year.
y
-intercept is about 41,660, which indicates that the average salary in year zero was about $41,660. Notice that the value is slightly more than the actual average salary in year zero of $40,500. The difference is the result of the line of best fit as an approximation of the data set rather than an actual intersection through each data point.
Making Predictions Using Lines of Best Fit
The line of best fit can be used to predict values that are not in the data set.
Interpolation is predicting a data value between given data points.
Extrapolation is predicting a data value outside the set of given data points.
Predictions using the line of best fit may not always be accurate because the trend may not continue into the future. Thus, extrapolation from the line of best fit is associated with a greater degree of uncertainty than interpolation and is more likely to produce inaccurate results. By contrast, interpolation is quite useful for making accurate predictions between measured values.
The line of best fit for the average salaries at a company, where
y
is the average salary in dollars and
x
is the number of years the company has been in operation, is:
y = 1\rm{,}061.340641x + 41\rm{,}660.20236
To use interpolation to predict the average salary at the company after 7 years, substitute 7 for
x
in the equation of the line of best fit.
\begin{aligned}y &= 1\rm{,}061.340641(7) + 41\rm{,}660.20236 \\&= 49\rm{,}089.58685\end{aligned}
Based on the line of best fit, the average salary at the company after it has been operating for 7 years is about $49,100.
<Linear Modeling
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