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https://buttondown.email/jesper/archive/summer-arrived-all-suns-blazing/
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math
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Hey folks 🎉
this is such a lovely weather this week. I hope you had a good one and enjoyed some time outside!
NASA flew a helicopter on another planet!
Here you can see an image of the little guy jumping over its own shadow, and here’s a video of it taking off! The science behind building and testing an aircraft for another planet involves such a feat of engineering. Think about it, have you flown (and crashed because, of course, you did) a drone here on Earth? Now we’re doing this with minutes of latency under an atmosphere that is less than one per cent of the Earth. Basically, you’d have to move to a height of 35km in our atmosphere to get similar conditions, three times the height planes usually cruise at.
Still obsessed with my new bike. Been riding around Edinburgh almost every day and finally getting away a little bit from my flat ever since a year. It’s magic for my mind. 10/10 do recommend.
Would you believe it? I made another Youtube video! This time I talk about my most productive VS Code extensions. I have also written an accompanying blog post that gives some context for those that prefer reading: https://dramsch.net/posts/my-10-favourite-vs-code-extensions/.
Regression problems often depend on minimising some metric like the mean squared error. While this is a viable loss function, it can be very difficult to gauge the absolute performance of a regression model. Here’s where the Coefficient of Determinism or R^2 score comes in. It is a rather simple concept to all of you that have a more statistical background, but worth investigating regardless. A mean squared error of 30,000 or 3,000 doesn’t matter unless we know the scale and noise level of the data. The R^2 score, however, always returns a value up to 1, where 1 would describe a perfectly deterministic model. Generally, it’s a “higher is better” situation and it is possible for the R^2 score to be negative, this is the case for non-linear models that do not describe the unseen data sufficiently.
The calculation takes two main concepts. The first concept is the total sum of squares (TSS), proportional to the variance of the data. The second concept is the good ol’ residual sum of squares, equivalent to the unbiased mean squared error (RSS). Then the R^2 score is R^2 = 1 - TSS / RSS, so we explain the variance (ish) of the data and the residual error of the model prediction. Measuring whether a model describes the mean and the variance of a model is a simple and very intuitive way to explain the capability of a model to explain the data.
So how do we get negative values? The simplest way would be to “forget” the y-intercept in a linear model. For your (actually well implemented) models, it’s usually a sign that the mean of the data actually better explains the outcomes than your model, so a clear indicator that the model does not capture the complexity of the data accurately.
Send me your answers or post them on Twitter and Tag me. I’d love to see what you come up with. Then I can include them in the next issue!
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s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303884.44/warc/CC-MAIN-20220122194730-20220122224730-00710.warc.gz
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CC-MAIN-2022-05
| 3,063 | 10 |
https://forum.duolingo.com/comment/18750714/%CE%A4%CE%BF-%CF%86%CE%B1%CE%B3%CE%B7%CF%84%CF%8C-%CE%B5%CE%AF%CE%BD%CE%B1%CE%B9-%CE%B1%CF%81%CE%BA%CE%B5%CF%84%CF%8C
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math
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would "υπάρχει αρκετό φαγητό" properly express "There is enough food"? If not, would someone please explain why this construction is more natural?
Hello mods, or other native Greek speakers! Does anyone know the answer to Aling14’s question? Thanks in advance! :)
"The food is enough" should not be accepted?
Yes. Accepted per DL on Sep. 6, 2018.
How is this 'there is', when it is simply 'the food'?
I'm wondering the same thing...
I'm also waiting for an answer to Aling14's question
Why is this pronounced as stress-initial?
I believe that "Υπάρχει αρκετό φαγητό" should be accepted.
I'm waiting for the alignment too. The only option is "There is"
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571959.66/warc/CC-MAIN-20220813142020-20220813172020-00245.warc.gz
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CC-MAIN-2022-33
| 691 | 10 |
https://forum.solidworks.com/thread/45994
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math
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Does SW thermal simulation take into consideration the latent heat of fusion of water as it changes state from liquid to solid when calculating the temperature distribution?
I'm sure there's a convention to approximate latent heat, but I am unfamiliar with it. As for physically modeling latent heat, I'm quite sure the system doesn't do that.
Not sure I follow you on this one Kevin. Can you elaborate?
Like how in a convective thermal boundary condition using FEA, you supply it sort of an emperical value that's derived from physical tests that you find most suitable for your situation. Or how you would enter a thermal contact resistance BC.
Maybe you change some values around a little if you expect to have latent heat influence your results of interest.
I think the answer is no it doesn't, having played around with this on the weekend.
There is no provision in the material data base for the latent heat of fusion of a material. Since heat transfer is proportional to temperature trying to extrapolate results based on the sensible heat coefficient would be innacurate.
Retrieving data ...
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141717601.66/warc/CC-MAIN-20201203000447-20201203030447-00566.warc.gz
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CC-MAIN-2020-50
| 1,099 | 8 |
http://www.spankwire.com/categories/Straight/Brunette/Submitted/76?Page=7
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math
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- 18,606 views30:15 bikini, blonde, brunette, compilation, cumshot, facial, hardcore, teens, bigcock, bigtits, collection, cum-on-tits, doggystyle, smalltits, teamskeet, best-of
- 15,084 views26:28 big-dick, brunette, feet, foot, kink, oral, outside, public, cock-sucking, feet-licking, shaved-pussy, sole, toe, toe-sucking, sole-licking, 21natural
- 111,948 views18:32 3some, big-ass, cougar, doggy-style, hardcore, hd, huge-tits, milf, mom, mother, threesome, colombian, stepmom, hot-mom, moms-teach-sex, mom-and-son, mom-fucks-son
Blonde, brunette, redhead; everyone has their preference. At SpankWire.com we absolutely love sexy brunette babes. We have hundreds of free XXX videos featuring the hottest dark-haired babes from all around the world. Whether you like them tanned or pale, petit or voluptuous, these beauties never fail to please. Their appetite for sex is legendary, and they have no problem fucking all night long. Watch how their flowing hair and piercing eyes look as they perform incredible deep throat blowjobs. Enjoy how the long hair flows down one babes tanned back as she rides her man's thick cock while she shows off her luscious big boobs on camera for us. If you want to see amateur, MILF, or lesbian brunettes we have all you could ever desire, and far more to suit every fetish. They may say that blondes are more fun, but at SpankWire we know that brunettes know how to get it on in the sack!
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122933.39/warc/CC-MAIN-20170423031202-00201-ip-10-145-167-34.ec2.internal.warc.gz
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CC-MAIN-2017-17
| 1,472 | 4 |
https://brainmass.com/economics/demand-supply/market-equilibrium-price-output-combination-269015
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math
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1. The following relations describe monthly demand and supply relations for dry cleaning services in the metropolitan area:
QD = 500,000 - 50,000P (Demand)
QS = -100,000 + 100,000P (Supply)
where Q is quantity measured by the number of items dry cleaned per month and P is average price in dollars.
A. At what average price level would demand equal zero?
B. At what average price level would supply equal zero?
C. Calculate the equilibrium price/output combination.
2. The Creative Publishing Company (CPC) is a coupon book publisher with markets in several southeastern states. CPC coupon books are sold directly to the public, sold through religious and other charitable organizations, or given away as promotional items. Operating experience during the past year suggests the following demand function for CPC's coupon books:
Q = 5,000 - 4,000P + 0.02Pop + 0.25I + 1.5A,
where Q is quantity, P is price ($), Pop is population, I is disposable income per household ($), and A is advertising expenditures ($).
A.Determine the demand faced by CPC in a typical market in which P = $10, Pop = 1,000,000 persons, I = $60,000, and A = $10,000.
B.Calculate the level of demand if CPC increases annual advertising expenditures from $10,000 to $15,000.
C.Calculate the demand curves faced by CPC in parts A and B.
3. The Eastern Shuttle, Inc., is a regional airline providing shuttle service between New York and Washington, D.C. An analysis of the monthly demand for service has revealed the following demand relation:
Q = 26,000 - 500P - 250POG + 200IB - 5,000S,
where Q is quantity measured by the number of passengers per month, P is price ($), POG is a regional price index for other consumer goods (1967 = 1.00), IB is an index of business activity, and S, a binary or dummy variable, equals 1 in summer months and 0 otherwise.
A.Determine the demand curve facing the airline during the winter month of January if POG = 4 and IB = 250.
B.Determine the demand curve facing the airline, quantity demanded, and total revenues during the summer month of July if P = $100 and all other price-related and business activity variables are as specified previously.
4. Eye-de-ho Potatoes is a product of the Coeur d'Alene Growers' Association. Producers in the area are able to switch back and forth between potato and wheat production depending on market conditions. Similarly, consumers tend to regard potatoes and wheat (bread and bakery products) as substitutes. As a result, the demand and supply of Eye-de-ho Potatoes are highly sensitive to changes in both potato and wheat prices.
Demand and supply functions for Eye-de-ho Potatoes are as follows:
QD = -1,450 - 25P + 12.5PW + 0.1Y, (Demand)
QS = -100 + 75P - 25PW - 12.5PL + 10R, (Supply)
where P is the average wholesale price of Eye-de-ho Potatoes ($ per bushel), PW is the average wholesale price of wheat ($ per bushel), Y is income (GDP in $ billions), PL is the average price of unskilled labor ($ per hour), and R is the average annual rainfall (in inches). Both QD and QS are in millions of bushels of potatoes.
A. When quantity is expressed as a function of price, what are the Eye de ho Potatoes demand and supply curves if PW = $4, Y = $15,000 billion, PL = $8, and R = 20 inches?
B. Calculate the surplus or shortage of Eye-de-ho Potatoes when P = $1.50, $2, and $2.50.
C. Calculate the market equilibrium price/output combination.© BrainMass Inc. brainmass.com October 9, 2019, 11:49 pm ad1c9bdddf
a. Demand is zero when quantity demanded falls to zero. Given the demand function we need to solve:
0 = 500000 - 50000P
or P = 10.
So quantity demanded falls to zero when price goes to 10.
b. Similarly, quantity supplied falls to zero when
0 = -100000 + 100000P
or P = 1.
Hence quantity supplied falls to zero when price goes to 1.
c. To find the equilibrium we need to solve:
QD = QS
or 500000 - 50000P = -100000 + 100000P
or 600000 = 150000P
or P = 4.
Plug in either the demand or supply function and we get:
QD = 500000 - 50000*4 = 300000.
Hence, equilibrium price is 4, and equilibrium quantity is 300000.
2. Demand function for the company is given as
Calculate the market equilibrium price/output combination.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00348.warc.gz
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CC-MAIN-2020-10
| 4,175 | 44 |
https://www.lovebyname.com/power-of-a-name.php
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math
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Each name has two personalities and we are the result of both personalities. In turn, you are the combination of the personalities from all of your names. Each letter in a name is allocated a number then added together to calculate which of the 64 personality types you have for each one of your names. How we allocate a number to each letter is demonstrated in the Prime numbers chart and the Undertones chart below. The Prime number is the main personality of a name and the Undertone is the undercurrent or underlying personality.
Will it work for you?
The system will work for everyone as each and every one of us has a name. There are 26 letters of the alphabet from A-Z. Each letter is allocated a number ranging from 1-9. The letter A is given the number 1, the letter B is given the number 2, C is 3, D is 4, and so on. There are no double digits, so for example, the letter K would be 11 (1+1=2), which is reduced to the number 2. As each name is reduced to separate numbers, this allows names to be matched in any combination. Forenames can be matched with forenames, surname with surname, or surname with forenames names, Undertones can be matched with Prime numbers and Prime numbers with Undertones.
The charts below illustrate how each letter is allocated a number.
Prime Numbers Chart
(the undertone is the set of numbers required to bring the addition of each digit to 9)
(The undertone of 1 is 8; 2 is 7; 3 is 6; 4 is 5 etc)
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587877.85/warc/CC-MAIN-20211026103840-20211026133840-00670.warc.gz
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CC-MAIN-2021-43
| 1,441 | 7 |
https://ell.stackexchange.com/questions/217591/an-exactly-the-same-two-articles-ok
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math
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Does it make sense when two articles are used, for example “An exactly the same”? If the former conveys inspecific/abstract item, then what does the latter do?
You're asking if we use the phrase: An exactly the same?
Not normally. It's not generally grammatical to use an article to introduce an adverb (exactly), nor to use two articles to introduce a noun (same).
But it could be used for some kind of special effect, as could almost any string of words. A context might be that two people see a woman. One says she's the same woman they saw on the TV news because she robbed a bank:
A: I'm telling you, she's a completely different woman!
B: And I'm telling you: she's an exactly-the-same woman. Let's catch her!
That's something I can cook up in my imagination as possible. But whether or not it's grammatical depends on a careful definition of grammar.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301592.29/warc/CC-MAIN-20220119215632-20220120005632-00435.warc.gz
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CC-MAIN-2022-05
| 861 | 7 |
http://www.mywordsolution.com/question/a-screening-test-for-a-newly-discovered-disease/92044
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math
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problem 1: A screening test for a newly discovered disease is being computed. In order to find out the effectiveness of the new test, it was administered to 900 workers; 150 of the individuals diagnosed with the disease tested positive. A negative test finding out occurred in 60 people who had the disease. A total of 50 persons not diseased tested positive for it. Suppose that the prior probability is not known. By using TP as a true positive, FP as a false positive, FN as a false negative and TN as a true negative compute the given problem:
A) What was the sensitivity of the test?
B) What was the specificity of the test?
C) What is the total accuracy of the test?
D) What was the predictive value of a positive test?
E) What was the predictive value of a negative test?
Please illustrate your math.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122619.71/warc/CC-MAIN-20170423031202-00140-ip-10-145-167-34.ec2.internal.warc.gz
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CC-MAIN-2017-17
| 807 | 7 |
https://qa.answers.com/Q/How_many_on_fourths_equal_one_cup
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math
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That is 2 ounces.
Not really, its one quarter cup short.
one half cup, or two fourths cup.
Three-fourths of a cup of lentils would equal 100 grams of lentils. One hundred thirty grams is equal to one cup.
there are four (4) fourths in one (1) cup
There are 4 fourths in 1 cup, so there are 6 fourths in 1 and 1/2 cup.
one and one-fourths is equal to five-fourths
six fourths of a cup is one and a half cups.
One cup is 8 oz. So 3/4 cup is 6 oz.
No, two one-fourths (two-fourths) equal one half.
Two one-fourths is equal to one-half.
It takes 1 1/3 fourths to make one third. That equals 1.333... fourths.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506423.70/warc/CC-MAIN-20230922202444-20230922232444-00674.warc.gz
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CC-MAIN-2023-40
| 604 | 12 |
http://www.accounting4management.com/linear_programming_simplex_method.htm
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math
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Linear Programming and Maximization of Contribution Margin – Simplex Method:
Learning Objective of the Article:
Definition and Explanation of Simplex Method:
Simplex method is considered one of the basic techniques from which many linear programming techniques are directly or indirectly derived. The simplex method is an iterative, stepwise process which approaches an optimum solution in order to reach an objective function of maximization or minimization. Matrix algebra provides the deterministic working tools from which the simplex method was developed, requiring mathematical formulation in describing the problem. An example can help us explain the simplex method in detail.
Example of Linear Programming Simplex Method:
Assume that a small machine shop manufactures two models, standard and deluxe. Each standard model requires two hours of grinding and four hours of polishing; each deluxe module requires five hours of grinding and two hours of polishing. The manufacturer has three grinders and two polishers. Therefore in 40 hours week there are 120 hours of grinding capacity and 80 hours of polishing capacity. There is a contribution margin of $3 on each standard model and $4 on each deluxe model.
Before the simplex method can be applied, the following steps must be taken:
The relationship which establish the constraints or inequalities must be set up first. Letting x and y be respectively the quantity of items of the standard model and deluxe model that are to be manufactured, the system of inequalities or the set of constraints:
2x + 5y ≤ 120
Both x and y must be positive values or zero (x ≥ 0; y ≥ 0). Although this illustration involves only less-than-or-equal-to type constraints, equal-to and greater-than-or-equal-to type constraints can be encountered in maximization problems.
The objective function is the total contribution margin the manager can obtain from the two models. A contribution margin of $3 is expected for each standard model and $4 for each deluxe model. The objective function is CM = 3x + 4y. The problem is now completely described by mathematical notation.
The first tow steps are the same for the graphic method, the simplex method requires the use of equations, in contrast to the inequalities used by the graphic method. Therefore the set of inequalities (to-less-than-or-equal-to type constraints) must be transformed into a set of equations by introducing slack variables, s1 and s2. The use of slack variables involves the addition of an arbitrary variable to one side of the inequality, transforming it into an equality. This arbitrary variable is called a slack variable, since it takes up the slack in the inequality. The inequalities rewritten as equalities are:
2x + 5y + s1 = 120
The unit contribution margins of the fictitious products s1 and s2 are zero, and the objective equation becomes:
Maximize: CM = 3x + 4y + 0s1 + 0s2
At this point, the simplex method can be applied and the first matrix or tableau can be set up as shown below:
Explanation and Calculation for the First Tableau:
The simplex method records the pertinent data in a matrix form known as the simplex tableau. The components of a tableau are described in the following paragraphs.
The Objective row is made up of the notation of the variable of the problem including slack variables.
The problem rows in the first tableau contain the coefficients of the variables in the constraints. Each constraint adds an additional problem row. Variables not included in a constraints are assigned zero coefficients in the problem rows. In subsequent tableau, new problem row values will be computed.
At each iteration, the objective column receives different entries, representing the contribution margin per unit of the variable in the solution.
At each iteration, the variable column receives different notation by replacement. These notations are the variables used to find the total contribution margin of the particular iteration. In the first matrix, a situation of no production is is considered as a starting point in the iterative solution process. For this reason, only slack variables and artificial variables are entered in the objective column, and their coefficient in the objective function are recorded in the variable column. As the iterations proceed, by replacement, appropriate values and notations will be entered in the objective and variable column.
The quantity column shows the constant values of the constraints in the first tableau; in subsequent tableaus, it shows the solution mix.
The index row carries values computed by the following steps:
The index row for the illustration is determined as follows:
In this first tableau, the slack variables were introduced into the product mix variable column to find and initial feasible solution to the problem. It can be proven mathematically that beginning with positive slack and artificial variables assures a feasible solution. Hence, one feasible solution might have s1 take a value of 120 and s2 a value of 80. This approach satisfies the constraints but is undesirable since the resulting contribution margin is zero.
It is a rule of the simplex method that the optimum solution has not been reached if the index row carries any negative values at the completion of an iteration. Consequently, this first tableau does not carry the optimum solution since negative values appear in its index row. A second tableau or matrix must now be prepared, step by step, according to the rules of simplex method.
Explanation and Calculation for the Second Tableau:
The construction of the second tableau is accomplished through these six steps:
When these steps are completed for the contribution margin maximization illustration, the second tableau appears as follows:
This second matrix does not contain the optimum solution since a negative figure, -1.4, still remains in the index row. The contribution margin arising from this model mix is $96 [4(24) + 0(32)], which is an improvement. However, the second solution indicates that some standard models and $1.40 (or 7 / 5 dollars of contribution margin) can be added for each unit of the standard model substituted in this solution.
It is interesting to reflect on the significance of – 7 / 5 or – 1.4. The original statement of the problem had promised a unit contribution margin of $3 for the standard model. Now the contribution will increase by only $1.40 per unit. The significance of the -1.4 is that it measures the net increase of the per unit contribution margin after allowing for the reduction of the deluxe model represented by y units. That is, all the grinding hours have been committed to produce 24 deluxe models (24 units × 5 hours grinding time per unit = 120 hours capacity); the standard model cannot be made without sacrificing the deluxe model. The standard model requires 2 hours of grinding time; the deluxe model requires 5 hours of grinding time. To introduce one standard model unit into the product mix, the manufacture of 2 / 5 (0.4) of one deluxe model unit must be foregone. This figure, 0.4, appears in the column headed “x” on the row representing foregone variable (deluxe) models. If more non slack variables (i.e., more than two products) were involved , the figures for these variables, appearing in column x, would have the same meaning as 0.4 has for the deluxe models.
Thus, the manufacturer loses 2 / 5 of $4, or $1.60, by making 2 / 5 less deluxe models but gains $3 from the additional standard models. A loss of $1.60 and a gain of $3 results in a net improvement of $1.40. The final answer, calculated on graphical method page, adds $14 (10 standard models × 1.4) to the $96 contribution margin that results from producing 10 standard models (10 × $3 = $30) and (20 × $4 = $80). In summary, the -1.4 in the second tableau indicates the amount of increase possible in the contribution margin if one unit of the variable heading that column (x in this case) were added to the solution; and the 0.4 value in column x represents the production of the deluxe model that must be relinquished.
The quantity column was described for the first tableau as showing the constant values of the constraints, i.e., the maximum resources available (grinding and polishing hours in the illustration) for the manufacture of standard and deluxe models. In subsequent tableaus the quantity column shows the solution mix. Additionally, for a particular iteration in subsequent tableau, the quantity column shows the constraints that are utilized in an amount different from the constraints constant value. For example, in the second tableau’s quantity column, the number corresponding to the y variable denotes the number of y units in the solution mix (24), and its objective function coefficient of $4 when multiplied by 24 yields $96, the value of the solution at this iteration. The number corresponding to the s2 variable denotes the difference in total polishing hours and those used in the second tableau solution, i.e., 80 hours of available polishing hours less polishing hours used to produce 24 units of y (24 × 2), or 80 – 48 = 32. Thus the number of unused polishing hours is 32. No unused grinding hours, the s1 variable, are indicated because 24 units of y utilized the entire quantity of available grinding time (24 × 5 = 120 hours).
While this illustration is of less-than-or-equal-to type constraints, a similar interpretation can be made for equal-to type constraints; i.e., the quantity column denotes the difference in the constant value of the constraint and the value used in the tableau’s solution mix. For the greater-than-or-equal-to constraint, the quantity column denotes the amount beyond the constraint’s minimum requirement that is satisfied by the particular solution mix.
These constraint utilization of satisfaction differences provide useful information, especially in the optimal solution tableau, because management may wish to make decisions to reduce these differences, e.g., by plans to utilize presently unused capacity associated with less-than-or-equal-to constraints.
Explanation and Calculation for the Third Tableau:
The third tableau is computed by these steps:
Third tableau appears as follows:
There are no negative figures in the index row, which indicates that any further substitutions will not result in an increase in the contribution margin; the optimum solution has been obtained. The optimum strategy is to produce and sell 20 deluxe and 10 standard models for a contribution margin of $110.
You may also be interested in other articles from “linear programming technique” chapter
Other Related Accounting Articles:
- Linear Programming and Maximization of Contribution Margin – Graphical Method
- Linear Programming – Minimization of Cost – Simplex Method
- Shadow Prices
- Linear Programming Questions and Answers
- Linear Programming and Minimization of Cost-Graphical Method
- Linear Programming Techniques-General Observations
- Linear Programming Technique
- Dynamic Programming
- Linear Programming Solved Problems
Download E accounting book in MS-word format for just 20 $ - Click here to Download
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CC-MAIN-2014-42
| 11,176 | 52 |
https://all-free-download.com/free-vector/download/startup-banner-staff-book-stack-lightbulb-sketch_6844392.html
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math
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Startup banner staff book stack lightbulb sketch Free vector 2.02MB
File size: 2.02MB File type: Adobe Illustrator ai ( .ai ) format, Encapsulated PostScript eps ( .eps ) format Author:
Licence: Free for commercial with attribution. Please give a backlink to all-free-download.com. With out attribution just buy an commercial licence. Redistribute is forbidden. Please check author page for more information.
You can use this graphic design for commercial with attribution to all-free-download.com. Please buy a commercial licence for commercial use without attribution.
Tags: startup vector banner design business illustration poster background symbol advertising graphic backdrop staff book employment book stack sketch lightbulb bulb concept motivation knowledge aspirations successful light solution inspiration
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s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00476.warc.gz
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CC-MAIN-2021-21
| 1,819 | 8 |
https://www.proofwiki.org/wiki/Sun_Tzu_Suan_Ching/Examples
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math
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Sun Tzu Suan Ching/Examples
Jump to navigation Jump to search
Examples of Problems from Sun Tzu Suan Ching
- A woman was washing dishes in a river, when an official whose business was overseeing the waters demanded of her:
- "Why are there so many dishes here?"
- "Because a feasting was entertained in the house," the woman replied.
- Thereuon the official enquired the number of guests.
- "I don't know," the woman said, "how many guests there had been;
- but every $2$ used a dish for rice between them;
- every $3$ a dish for broth;
- every $4$ a dish for meat;
- and there were $65$ dishes in all."
- There are certain things whose number is unknown.
- Repeatedly divided by $3$, the remainder is $2$;
- by $5$ the remainder is $3$,
- and by $7$ the remainder is $2$.
- What will be the number?
- There are $3$ sisters, of whom the eldest comes home once every $5$ days,
- the middle every $4$ days,
- and the youngest in every $3$ days.
- In how many days will all the $3$ meet together?
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https://opentuition.com/topic/loan-notes-6/
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math
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- November 28, 2015 at 4:05 pm #285973
Hi Mr Moffat could you assist with this question from the mcq? I seem to be missing something.
A company has in issue loan notes with a nominal value of $100 each. Interest on the notes is 6% per year, payable annually. The loan notes will be redeemable in eight years time at a 5% premium to nominal value. The before-tax cost off debt to company is 7% per year. The rate of company tax is 30%.
What is te ex-interest market value of each loan notes?
Answer is $96.94
I am getting $92.53November 28, 2015 at 5:51 pm #285994
The market value is determined by the investors and is the present value of the expected receipts discounted at their required return of 7%.
On $100 nominal, the expected receipts are:
Interest of $6 per year from years 1 to 8
Redemption of $105 in 8 years time.
If you discount these flows at 7% you will come to $96.94
(Tax is irrelevant – it is only relevant when considering the cost of debt to the company)November 29, 2015 at 1:01 am #286036
Mr Moffat thanks I picked the wrong pv dcf in error for the 105 thanks again
I need help with one more thanks. Can I see the working for this one?
An investor plans to exchange $1,000.00 into euros now, invest the resulting euros for 12 months, and then exchange back into dollars at the end of the 12 months period. The spot exchange rate is EUR1.415 per $1 and the euro interest rate is 2 % per year. The dollar interest rate is 1.8 per year.
Compared to making a dollar investment for 12 months, at what 12 month forward exchange rate will the investor make neither a loss nor gain?
ANSWER EUR1.418November 29, 2015 at 8:07 am #286065
In future you must start a new thread when it is a question on a different topic – this question obviously has nothing to do with loan notes 🙂
You can get the same answer in two ways. Either you can work through the full money market hedging on this question and then calculate what rate effectively makes the amount in 12 months equal.
Alternatively (and quicker) you can use the interest rate parity formula on the formula sheet (because forward rates are determined from the relative interest rates).
So the forward rate = (1.02 / 1.018) x 1.415 = 1.418
(Our free lectures on forward rates and money market hedging will help you.)December 2, 2015 at 10:31 pm #287086
Thanks again for your help…December 3, 2015 at 7:37 am #287139
You are welcome 🙂
- You must be logged in to reply to this topic.
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http://www.pokeyplay.com/comunidad/index.php?showtopic=54033&p=717823&page=10
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math
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I'm afraid only Clan Leaders can get 1 mil, your a Clan member, not a clan leader. So you can only request 100,000.
my user name is ADITYAG i am a clan member can i get 1 million plz. i have all badges including sinnoh. ok i hope i can get the money i expect.
P.S. Page 10
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https://jzg.p1616jesucristo.site/tukey-post-hoc-test.html
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math
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A hospital wants to know how a homeopathic medicine for depression performs in comparison to alternatives.
Learn How to Calculate Tukey's Post HOC Test - Tutorial
They adminstered 4 treatments to patients for 2 weeks and then measured their depression levels. The data, part of which are shown above, are in depression. Before running any statistical test, always make sure your data make sense in the first place.
In this case, a split histogram basically tells the whole story in a single chart. We don't see many SPSS users run such charts but you'll see in a minute how incredibly useful it is. The screenshots below show how to create it. In step below, you can add a nice title to your chart. Clicking P aste results in the syntax below.
Running it creates our chart. We'll now take a more precise look at our data by running a means table. We could do so from A nalyze C ompare Means M eans but the syntax is so simple that just typing it is probably faster. Unsurprisingly, our table mostly confirms what we already saw in our histogram.
Well, for our sample we can. For our population all people suffering from depression we can't. The basic problem here is that samples differ from the populations from which they are drawn. If our four medicines perform equally well in our population, then we may still see some differences between our sample means.
However, large sample differences are unlikely if all medicines perform equally in our population. The question we'll now answer is: are the sample means different enough to reject the null hypothesis that the mean BDI scores in our populations are all equal?
However, it could be argued that you should always run post hoc tests. In some fields like market research, this is pretty common. Reversely, you could argue that you should never use post hoc tests because the omnibus test suffices: some analysts claim that running post hoc tests is overanalyzing the data.
Many social scientists are completely obsessed with statistical significance -because they don't understand what it really means- and neglect what's more interesting: effect sizes and confidence intervals. In any case, the idea of post hoc tests is clarified best by just running them.Since these are independent and not paired or correlated, the number of observations of each treatment may be different.
This calculator is hard-coded for a maximum of 10 treatments, which is more than adequate for most researchers. But it stops there in its tracks. This self-contained calculator, with flexibility to vary the number of treatments columns to be compared, starts with one-way ANOVA. However, it lacks the key built-in statistical function needed for conducting Excel-contained Tukey HSD. Continuing education in Statistics The hard-core statistical packages demand a certain expertise to format the input data, write code to implement the procedures and then decipher their s Old School Mainframe Era output.
This is the right tool for you!Scs extractor 2020
It was inspired by the frustration of several biomedical scientists with learning the software setup and coding of these serious statistical packages, almost like operating heavy bulldozer machinery to swat an irritating mosquito.
For code grandmasters, fully working code and setup instructions are provided for replication of the results in the serious academic-research-grade open-source and hence free R statistical package. Tukey originated his HSD test, constructed for pairs with equal number of samples in each treatment, way back in When the sample sizes are unequal, we the calculator automatically applies the Tukey-Kramer method Kramer originated in A decent writeup on these relevant formulae appear in the Tukey range test Wiki entry.
The NIST Handbook page mentions this modification but dooes not provide the formula, while the Wiki entry makes adequately specifies it.
However, this calculator is hard-coded for contrasts that are pairsand hence does not pester the user for additional input that defines generalized contrast structures. The Bonferroni and Holm methods of multiple comparison depends on the number of relevant pairs being compared simultaneously. This calculator is hard-coded for Bonferroni and Holm simultaneous multiple comparison of 1 all pairs and 2 only a subset of pairs relative to one treatment, the first column, deemed to be the control.
The post-hoc Bonferroni simultaneous multiple comparison of treatment pairs by this calculator is based on the formulae and procedures at the NIST Engineering Statistics Handbook page on Bonferroni's method. The original Bonferroni published paper in Italian dating back to is hard to find on the web. A significant improvement over the Bonferroni method was proposed by Holm Among the many reviews of the merits of the Holm method and its uniform superiority over the Bonferroni method, that of Aickin and Gensler is notable.
This paper is the also source of our algorithm to make comparisons according to the Holm method. All statistical packages today incorporate the Holm method. There is wide agreement that each of these three methods have their merits. The recommendation on the relative merits and advantages of each of these methods in the NIST Engineering Statistics Handbook page on comparison of these methods are reproduced below:.
The following excerpts from Aickin and Gensler makes it clear that the Holm method is uniformly superior to the Bonferroni method:. If only a subset of pairwise comparisons are required, Bonferroni may sometimes be better. Many computer packages include all three methods. So, study the output and select the method with the smallest confidence band.
No single method of multiple comparisons is uniformly best among all the methods.The idea behind the Tukey HSD Honestly Significant Difference test is to focus on the largest value of the difference between two group means.
The relevant statistic is. The statistic q has a distribution called the studentized range q see Studentized Range Distribution. Thus we can use the following t statistic. From these observations we can calculate confidence intervals in the usual way:. Since the difference between the means for women taking the drug and women in the control group is 5. The following table shows the same comparisons for all pairs of variables:. From Figure 1 we see that the only significant difference in means is between women taking the drug and men in the control group i.
In Figure 2 we compute the confidence interval for the comparison requested in the example as well as for the variables with maximum difference. These function are based on the table of critical values provided in Studentized Range q Table. The Real Statistics Resource Pack also provides the following functions which provide estimates for the Studentized range distribution and its inverse based on a somewhat complicated algorithm.
QDIST 4. To get the usual cdf value for the Studentized range distribution, you need to divide the result from QDIST by 2, which for this example is.
C n ,2 rows if the data in R1 contains n columns. The first two columns contain the column numbers in R1 from 1 to n that are being compared and the third column contains the p-values for each of the pairwise comparisons.
RSS - Posts. RSS - Comments. Real Statistics Using Excel. Everything you need to perform real statistical analysis using Excel. Skip to content. The critical value for differences in means is Since the difference between the means for women taking the drug and women in the control group is 5. Real Statistics Resources. Follow Real1Statistics. Search for:.
Proudly powered by WordPress.The Tukey HSD "honestly significant difference" or "honest significant difference" test is a statistical tool used to determine if the relationship between two sets of data is statistically significant — that is, whether there's a strong chance that an observed numerical change in one value is causally related to an observed change in another value. In other words, the Tukey test is a way to test an experimental hypothesis.
The Tukey test is invoked when you need to determine if the interaction among three or more variables is mutually statistically significant, which unfortunately is not simply a sum or product of the individual levels of significance. Simple statistics problems involve looking at the effects of one independent variable, like the number of hours studied by each student in a class for a particular test, on a second dependent variable, like the student's scores on the test.
Then you refer to a t-table that takes into account the number of data pairs in your experiment to see if your hypothesis was correct. Sometimes, however, the experiment may look at multiple independent or dependent variables simultaneously. For example, in the above example, the hours of sleep each student got the night before the test and his or her class grade going in might be included.
Such multivariate problems require something other than a t-test owing to the sheer number if independently varying relationships. ANOVA stands for "analysis of variance" and addresses precisely the problem just described. It accounts for the rapidly expanding degrees of freedom in a sample as variables are added.
For example, looking at hours vs. In an ANOVA test, the variable of interest after calculations have been run is F, which is the found variation of the averages of all of the pairs, or groups, divided by the expected variation of these averages.
The higher this number, the stronger the relationship, and "significance" is usually set at 0. John Tukey came up with the test that bears his name when he realized the mathematical pitfalls of trying to use independent P-values to determine the utility of a multiple-variables hypothesis as a whole. At the time, t-tests were being applied to three or more groups, and he considered this dishonest — hence "honestly significant difference. What his test does is compare the differences between means of values rather than comparing pairs of values.
The value of the Tukey test is given by taking the absolute value of the difference between pairs of means and dividing it by the standard error of the mean SE as determined by a one-way ANOVA test. The SE is in turn the square root of variance divided by sample size.
An example of an online calculator is listed in the Resources section. The Tukey test is a post hoc test in that the comparisons between variables are made after the data has already been collected.
This differs from an a priori test, in which these comparisons are made in advance. In the former case, you might look at the mile run times of students in three different phys-ed classes one year. In the latter case, you might assign students to one of three teachers and then have them run a timed mile.
Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs. More about Kevin and links to his professional work can be found at www. About the Author. Copyright Leaf Group Ltd.An ANOVA is a statistical test that is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups. The alternative hypothesis: Ha : at least one of the means is different from the others.
It simply tells us that not all of the group means are equal. If the p-value is not statistically significant, this indicates that the means for all of the groups are not different from each other, so there is no need to conduct a post hoc test to find out which groups are different from each other.
As mentioned before, post hoc tests allow us to test for difference between multiple group means while also controlling for the family-wise error rate. In a hypothesis testthere is always a type I error rate, which is defined by our significance level alpha and tells us the probability of rejecting a null hypothesis that is actually true. When we perform one hypothesis test, the type I error rate is equal to the significance level, which is commonly chosen to be 0.
However, when we conduct multiple hypothesis tests at once, the probability of getting a false positive increases. For example, imagine that we roll a sided dice.
Tukey's Post HOC Test Calculator
If we roll five dice at once, the probability increases to For example, suppose we have four groups: A, B, C, and D. This means there are a total of six pairwise comparisons we want to look at with a post hoc test:.
If we have more than four groups, the number of pairwise comparisons we will want to look at will only increase even more. The following table illustrates how many pairwise comparisons are associated with each number of groups along with the family-wise error rate:. Notice that the family-wise error rate increases rapidly as the number of groups and consequently the number of pairwise comparisons increases.
This means we would have serious doubts about our results if we were to make this many pairwise comparisons, knowing that our family-wise error rate was so high.
Fortunately, post hoc tests provide us with a way to make multiple comparisons between groups while controlling the family-wise error rate. This means we have sufficient evidence to reject the null hypothesis that all of the group means are equal.
Next, we can use a post hoc test to find which group means are different from each other. We will walk through examples of the following post hoc tests:.Public library
R gives us two metrics to compare each pairwise difference:. Both the confidence interval and the p-value will lead to the same conclusion. In particular, we know that the difference is positive, since the lower bound of the confidence interval is greater than zero.Although ANOVA is a powerful and useful parametric approach to analyzing approximately normally distributed data with more than two groups referred to as 'treatments'it does not provide any deeper insights into patterns or comparisons between specific groups.
After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc analysis is Tukey's Test. The test is known by several different names.
Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal Wikipedia.
The outputs from two different but similar implementations of Tukey's Test will be examined along with how to manually calculate the test. Other methods of post-hoc analysis will be explored in future posts. ANOVA in this example is done using the aov function. The summary of the aov output is the same as the output of the anova function that was used in the previous example. To investigate more into the differences between all groups, Tukey's Test is performed.
The output gives the difference in means, confidence levels and the adjusted p-values for all possible pairs. The confidence levels and p-values show the only significant between-group difference is for treatments 1 and 2.
Note the other two pairs contain 0 in the confidence intervals and thus, have no significant difference. The results can also be plotted. Another way of performing Tukey's Test is provided by the agricolae package.
The HSD. The results from both tests can be verified manually.Mgsv nuclear disarmament reddit
We'll start with the latter test HSD. The MSE calculation is the same as the previous example. With the q-value found, the Honestly Significant Difference can be determined. The Honestly Significant Difference is defined as the q-value multiplied by the square root of the MSE divided by the sample size. As mentioned earlier, the Honestly Significant Difference is a statistic that can be used to determine significant differences between groups. If the absolute value of the difference of the two groups' means is greater than or equal to the HSD, the difference is significant.
The means of each group can be found using the tapply function. Since there's only three groups, I went ahead and just calculated the differences manually.
With the differences obtained, compare the absolute value of the difference to the HSD. I used a quick and dirty for loop to do this.
The output of the for loop shows the only significant difference higher than the HSD is between treatment 1 and 2.Google my maps hide legend
Since the test uses the studentized range, estimation is similar to the t-test setting. The Tukey-Kramer method allows for unequal sample sizes between the treatments and is, therefore, more often applicable though it doesn't matter in this case since the sample sizes are equal.
The Tukey-Kramer method is defined as:. Entering the values that were found earlier into the equation yields the same intervals as was found from the TukeyHSD output. The table from the TukeyHSD output is reconstructed below. Adjusted p-values are left out intentionally.Post hoc multiple comparison tests. Once you have determined that differences exist among the means, post hoc range tests and pairwise multiple comparisons can determine which means differ.
Comparisons are made on unadjusted values. These tests are used for fixed between-subjects factors only.Performing a One-way ANOVA in Excel with post-hoc t-tests
In GLM Repeated Measures, these tests are not available if there are no between-subjects factors, and the post hoc multiple comparison tests are performed for the average across the levels of the within-subjects factors. For GLM Multivariate, the post hoc tests are performed for each dependent variable separately. The Bonferroni and Tukey's honestly significant difference tests are commonly used multiple comparison tests.
The Bonferroni testbased on Student's t statistic, adjusts the observed significance level for the fact that multiple comparisons are made. Sidak's t test also adjusts the significance level and provides tighter bounds than the Bonferroni test.
Tukey's honestly significant difference test uses the Studentized range statistic to make all pairwise comparisons between groups and sets the experimentwise error rate to the error rate for the collection for all pairwise comparisons. When testing a large number of pairs of means, Tukey's honestly significant difference test is more powerful than the Bonferroni test.
For a small number of pairs, Bonferroni is more powerful. Hochberg's GT2 is similar to Tukey's honestly significant difference test, but the Studentized maximum modulus is used. Usually, Tukey's test is more powerful. Gabriel's pairwise comparisons test also uses the Studentized maximum modulus and is generally more powerful than Hochberg's GT2 when the cell sizes are unequal. Gabriel's test may become liberal when the cell sizes vary greatly.
Dunnett's pairwise multiple comparison t test compares a set of treatments against a single control mean. The last category is the default control category. Alternatively, you can choose the first category. You can also choose a two-sided or one-sided test. To test that the mean at any level except the control category of the factor is not equal to that of the control category, use a two-sided test.
Multiple step-down procedures first test whether all means are equal. If all means are not equal, subsets of means are tested for equality.
These tests are more powerful than Duncan's multiple range test and Student-Newman-Keuls which are also multiple step-down proceduresbut they are not recommended for unequal cell sizes.
When the variances are unequal, use Tamhane's T2 conservative pairwise comparisons test based on a t testDunnett's T3 pairwise comparison test based on the Studentized maximum modulusGames-Howell pairwise comparison test sometimes liberalor Dunnett's C pairwise comparison test based on the Studentized range.
Note that these tests are not valid and will not be produced if there are multiple factors in the model. Duncan's multiple range testStudent-Newman-Keuls S-N-Kand Tukey's b are range tests that rank group means and compute a range value. These tests are not used as frequently as the tests previously discussed. The Waller-Duncan t test uses a Bayesian approach.
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http://www.chegg.com/homework-help/questions-and-answers/direction-electric-field-origin-air-particular-region-attitude-500-m-ground-electric-field-q2030485
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math
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Show transcribed image textWhat is the direction of the electric field at the origin? In the air over a particular region at an attitude of 500 m above the ground, the electric field is 150 N/C directed downward. At 600 m above the ground, the electric field is 110 N/C downward. What is the average volume charge density in the layer of air between these two elevations? A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of lambda, and the cylinder has a net charge per unit of length of 2 lambda. From this information, use Gauss's law to find the charge per unit of length on the inner surface of the cylinder, the charge per unit of length on the outer surface of the cylinder, the electric field outside the cylinder a distance r from the axis.
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https://binaryoptionmaster.ml/2018/05/05/calendar-days-versus-trading-days/
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math
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The “calendar days” component in the formula in Table 7-5 is often debated. The question is, “Should the formula use calendar days or trading days?” Those in favor of trading days argue that volatility, that is, stock-price change, can only happen on trading days. Others counter that calendar days better reflect the actual amount of time until expiration. The answer is: In most cases, either can be used without much impact on the result.
The conversion formula uses square root of time in years, so the important question is whether calendar days or trading days best approximates time in years. It can be argued, generally, that it does not matter.Any percent of a full year is the same regardless of the number of days in a year. Choosing 252 (trading days) versus 365 (calendar days) for days per year for price-range estimates using volatility becomes an issue only when the time period is short. What is a “short” time period? Two examples follow that shed light on this issue.
First, consider a two-month time period in which there are 61 calendar days and 43 trading days. Also assume a stock price of 78.50 and 35 percent volatility. Calendar days are used to calculate a standard deviation for the period as follows: 61 calendar days divided by 365 calendar days in a year is 0.1671, the square root of which is 0.4087. The volatility for the period, therefore, is 14.3 percent (0.35 0.4087). And for a stock trading at 78.50, one standard deviation is 11.23 (78.50 0.143).
Trading days are used to calculate a standard deviation for the period as follows: 43 trading days divided by 252 trading days in a year is 0.1706, the square root of which is 0.4130. The volatility for the period, therefore, is 14.5 percent (0.35 .4130). And for a stock trading at 78.50, one standard deviation is 11.38 (78.50 0.145).The difference between using calendar days and trading days is 15 cents.
For a stock price of 78.50 and a period of two months, this difference is not significant.Second, consider a three-day time period, again assuming a stock price of 78.50 and volatility of 35 percent. Using calendar days, 3 divided by 365 calendar days in a year is 0.0082, the square root of which is 0.0906. The volatility for the period, therefore, is 3.2 percent (0.35 0.0906). And for a stock trading at 78.50, one standard deviation is 2.51 (78.50 0.032).
Using trading days, 3 divided by 252 trading days in a year is 0.0119,the square root of which is 0.1091. The volatility for the period, therefore,is 3.8 percent (0.35 0.1091). And for a stock trading at 78.50,one standard deviation is 2.98 (78.50 0.038).The difference between using calendar days and trading days is 47 cents (2.51 versus 2.98). This is approximately a 17 percent difference and arguably significant.
How much of a concern should the difference between using calendar days and trading days be to traders? For a two-month time period, given a stock price of 78.50, most traders would not consider 15 cents to be significant. For the three-day period, the difference of 47 cents might be significant depending on how often a trader uses strategies targeted at three days.In general, the answer also partly depends on how accessible the necessary information is.
Most traders have easy access to the number of calendar days to expiration because brokers supply it. In contrast, the number of trading days is more difficult to find and time-consuming to calculate. Many traders therefore use calendar days when converting annual volatility to shorter time periods because it is easier, and it usually does not make much difference.
The focus now will shift from volatility as it relates to stock-price movements to volatility as it relates to option prices. Remember, from Chapter 2, that volatility is one of the six components that influence option prices.Implied Volatility:Implied volatility is the volatility percentage that justifies the market price of an option. In other words, it is the volatility percentage that returns the option’s market price as the theoretical value.
This concept is best explained with an example.Consider Gary, who uses Op-Eval Pro to estimate the theoretical value of an XYZ March 70 Call. Figure 7-6 shows a Single Option Calculator screen from Op-Eval Pro with Gary’s inputs: current stock price of 68.00, strike price of 70, no dividend, interest rate of 4 percent,and 75 days to expiration. Gary chose a volatility of 26 percent because that percentage was the historic volatility based on the 30 most recent daily closing stock prices. Given Gary’s inputs, Op-Eval Pro calculates a value of 2.57 for the XYZ March 70 Call.
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http://www.scribd.com/doc/15644443/Understanding-Electronics-Components
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math
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of different resistors are shown in the photos. (The resistors are on millimeter paper, with 1cm spacing togive some idea of the dimensions).Photo 1.1ashows some low-power resistors, while photo1.1b shows some higher-power resistors. Resistors with power dissipation below 5 watt (most commonly usedtypes) are cylindrical in shape, with a wire protruding from each end for connecting to a circuit (photo 1.1-a). Resistors with power dissipation above 5 wattare shown below (photo 1.1-b).
Fig. 1.1a: Some low-power resistorsFig. 1.1b: High-power resistors and rheostats
The symbol for a resistor is shown in the following diagram (upper: American symbol, lower: European symbol.)
Fig. 1.2a: Resistor symbols
The unit for measuring resistance is the
Greek letter Ω
-called Omega). Higher resistance values are represented by "k" (kilo-ohms) and M (meg
ohms). For example, 120 000 Ω is represented as 120k, while 1 200 000 Ω is represented as 1M2. The dot is generally omitted a
s it can easily be lost in the printing process.In some circuit diagrams, a value such as 8 or 120 represents a resistance in ohms. Another common practice is to use the letter E for
resistance in ohms. The letter R can also be used. For example, 120E (120R) stands for 120 Ω, 1E2 s
tands for 1R2 etc.
1.1 Resistor Markings
Resistance value is marked on the resistor body. Most resistors have 4 bands. The first two bands provide the numbers for theresistance and the third band provides the number of zeros. The fourth band indicates the tolerance. Tolerance values of 5%, 2%, and 1% are most commonly available.The following table shows the colors used to identify resistor values:
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https://www.tutorpace.com/algebra/exponent-properties-online-tutoring
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math
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Exponents is the power or degree to a given variable or number. The exponent can be any real number. There are many different properties of the exponents in algebra which help in solving many types of question having exponents. Mentioned below are some properties of exponents.
Multiplication rule: am * an = a(m+n) (Here the base is the same value a)
Division rule: am / an = a(m-n) (Here the base is the same value a)
Power of a power: (am)n = amn
Example 1: Find the value of x in the equation 3(x+2) = 27.
Solution: Here the given equation is 3(x+2) = 27.
We need to simplify the 27 further.
The number 27 can be written as 27 = 3* 3 * 3
So, 27 = 33
Now we get 3(x+2) = 33.
Since the base number is 3 we can equate the exponents.
X + 2 = 3 (subtracting 2 on both sides.)
X = 3 – 2.
Hence the value of x = 1.
Example 2: Find the x in the equation 102 = 1/100.
Solution: Here the given equation is 102 = 1/100.
The fraction, 1/100 = 100-1.
We need to simplify 100 here further.
The number 100 can be written as 100 = 10* 10
So, 100 = 102
Now we get 10(x) = (102)-1.
Using the power of power rule.
10(x) = (10-2)
Since the base number is 10 we can equate the exponents.
Hence the value of x = -2.
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https://docsbay.net/theories-of-light
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math
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Theories of light
In the seventeenth century two rival theories of the nature of light were proposed, the wave theory and the corpuscular theory.
The Dutch astronomer Huygens (1629-1695) proposed a wave theory of light. He believed that light was a longitudinal wave, and that this wave was propagated through a material called the ‘aether’. Since light can pass through a vacuum and travels very fast Huygens had to propose some rater strange properties for the aether: for example; it must fill all space and be weightless and invisible. For this reason scientists were sceptical of his theory.
In 1690 Newton proposed the corpuscular theory of light. He believed that light was shot out from a source in small particles, and this view was accepted for over a hundred years.
The quantum theory put forward by Max Planck in 1900 combined the wave theory and the particle theory, and showed that light can sometimes behave like a particle and sometimes like a wave. You can find a much fuller consideration of this in the section on the quantum theory.
Wave theory of Huygens
As we have seen, Huygens considered that light was propagated in longitudinal waves through a material called the aether. We will now look at his ideas more closely.
Huygens published his theory in 1690, having compared the behaviour of light not with that of water waves but with that of sound. Sound cannot travel through a vacuum but light does, and so Huygens proposed that the aether must fill all space, be transparent and of zero inertia. Clearly a very strange material!
Even at the beginning of the twentieth century, however, scientists were convinced of the existence of the aether. One book states ‘whatever we consider the aether to be there can be no doubt of its existence’.
We now consider how Huygens thought the waves moved from place to place.
Consider a wavefront initially at position W, and assume that every point on that wavefront acts as a source of secondary wavelets. (Figure 1 shows some of these secondary sources). The new wavefront W1 is formed by the envelope of these secondary wavelets since they will all have moved forward the same distance in a time t (Figure 1).
There are however at least two problems with this idea and these led Newton and others to reject it:
(a) the secondary waves are propagated in the forward direction only, and
(b) they are assumed to destroy each other except where they form the new wavefront.
Newton wrote: ‘If light consists of undulations in an elastic medium it should diverge in every direction from each new centre of disturbance, and so, like sound, bend round all obstacles and obliterate all shadow.’ Newton did not know that in fact light does do this, but the effects are exceedingly small due to the very short wavelength of light.
Huygens’ theory also failed to explain the rectilinear propagation of light.
The reflection of a plane wavefront by a plane mirror is shown in Figure 2. Notice the initial position of the wavefront (AB), the secondary wavelets and the final position of the wavefront (CD).
Notice that he shape of the wavefront is not affected by reflection at a plane surface.
The lines below the mirror show the position that the wavefront would have reached if the mirror had not been there.
We will now show how Huygens’ wave theory can be used to explain reflection and refraction and the laws governing them.
Consider a parallel beam of monochromatic light incident on a plane surface, as shown in Figure 3. The wave fronts will be plane both before and after reflection, since a plane surface does not alter the shape of waves falling on it.
Consider a point where the wavefront AC has just touched the mirror at edge A. While the light travels from A to D, that from C travels to B. The new envelope for the wavefront AC will be BD after reflection. Therefore
AD = CB
Angle ACB = angle ADB = 90o
AB is common
Therefore ACB and BDA are similar and so angle CAB = angle BAD. Therefore i = r and the law of reflection is proved.
Consider a plane monochromatic wave hitting the surface of a transparent material of refractive index n. The velocity of light in the material is cm and that in air ca. Now in Figure 4,
CB = AB sin i
AD = AB sin r
The same argument applies about the new envelope as in the case of reflection:
time to travel CB = CB/ca = AB sin i/ca, time to travel AD AD/cm = AB sin i/cm
But these are equal and therefore:
ca/cm = sin i/sin r = anm.
This is Snell’s law, and it was verified later by Foucault and others.
Notice that since the refractive index of a transparent material is greater than 1, Huygens’ theory requires that the velocity of light in air should be greater than that in the material.
Corpuscular theory of Newton
Newton proposed that light is shot out from a source as a stream of particles. He argued that light could not be a wave because although we can hear sound from behind an obstacle we cannot see light - that is, light shows no diffraction. He stated that particles of different colours should be of different sizes, the red particles being larger than the blue.
Since these particles are shot out all the time, according to Newton’s theory, the mass of the source of light must get less!
We can use Newton’s theory to deduce the laws of reflection and refraction.
Consider a particle of light in collision with a mirror. The collision is supposed to be perfectly elastic, and so tile component of velocity perpendicular to the mirror is reversed while that parallel to the mirror remains unaltered. From Figure 5,
Component of velocity before collision parallel to the mirror = ca sin i
Component of velocity after collision parallel to the mirror = ca sin r
ca sin i = ca sin r
and so the law of reflection is proved.
Newton assumed that there is an attraction between the molecules of a solid and the particles of light, and that this attraction acts only perpendicularly to the surface and only at very short distances from the surface. (He explained total internal reflection by saying that the perpendicular component of velocity was too small to overcome the molecular attraction.) This has the effect of increasing the velocity of the light in the material.
Let the velocity of light in air be ca and the velocity of light in the material in Figure 6 be cm.
The velocity parallel to the material is unaltered and therefore:
ca sin i = cm sin r
cm/ca = sini/ sinr = anm
This ratio is the refractive index, but because n > 1 the velocity of light in the material must be greater than that in air. Newton accepted this result and other scientists preferred it to that of Huygens, mainly because of Newton’s eminence.
A problem of the corpuscular theory was that temperature has no effect on the velocity of light, although on the basis of this theory we would expect the particles to be shot out at greater velocities as the temperature rises.
Classical and modern theories of light
It is interesting to compare the two classical theories of light and see which phenomena can be explained by each theory. The following table does this.
Wave theory Corpuscular theory
Notice that neither theory can account for polarisation, since for polarisation to occur the waves must be transverse in nature.
Twentieth-century ideas have led us to believe that light is
(a) a transverse electromagnetic wave with a small wavelength, and
(b) emitted in quanta or packets of radiation of about 10-8 s duration with abrupt phase changes between successive pulses.
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CC-MAIN-2022-49
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https://www.coursehero.com/file/6045535/lab-sol-Chap012/
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math
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This preview shows pages 1–3. Sign up to view the full content.
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Unformatted text preview: Chapter 12 - Unemployment CHAPTER 12 12-1. Suppose 25,000 persons become unemployed. You are given the following data about the length of unemployment spells in the economy: Duration of Spell (in months) Exit Rate 1 0.60 2 0.20 3 0.20 4 0.20 5 0.20 6 1.00 where the exit rate for month t gives the fraction of unemployed persons who have been unemployed t months and who escape unemployment at the end of the month. The data can be used in the problem to calculate the number of workers who have 1 month of unemployment, the number who have 2 months of unemployment, and so on, and how many months of unemployment are associated with workers who get a job after a given duration. Duration (Months) Exit Rate # Unemp: Start of Month # of Exiters # of Stayers # Months For Duration 1 0.60 25,000 15,000 10,000 15,000 2 0.20 10,000 2,000 8,000 4,000 3 0.20 8,000 1,600 6,400 4,800 4 0.20 6,400 1,280 5,120 5,120 5 0.20 5,120 1,024 4,096 5,120 6 1.00 4,096 4,096 24,576 (a) How many unemployment-months will the 25,000 unemployed workers experience? The 25,000 workers will experience 58,616 months of unemployment, 2.34 months per worker. (b) What fraction of persons who are unemployed are long-term unemployed in that their unemployment spells will last 5 or more months? Only 5,120 of the 25,000 workers (20.5 percent) are in spells lasting 5 or more months. (c) What fraction of unemployment months can be attributed to persons who are long-term unemployed? Although only 20.5 percent of workers are unemployed for 5 or more months, they account for 29,696 of the 58,616 (50.7 percent) months of unemployment. (d) What is the nature of the unemployment problem in this example: too many workers losing their jobs or too many long spells? Most spells are short-lived, but workers in long spells account for most of the unemployment observed in this economy. Thus, the main problem is too many long spells. 12-1 Chapter 12 - Unemployment 12-2. Consider Table 610 of the 2008 U.S. Statistical Abstract . (a) How many workers aged 20 or older were unemployed in the United States during 2006? How many of these were unemployed less than 5 weeks, 5 to 14 weeks, 15 to 26 weeks, and 27 or more weeks? In total, 5,882,000 workers aged 20 years or older were unemployed in the U.S. in 2006. Of these, 35.5 percent, or 2,088,110 workers, were unemployed for less than 5 weeks; 29.7 percent, or 1,746,954 workers, were unemployed between 5 and 14 weeks; 15.5 percent, or 911,710 workers, were unemployed 15 to 26 weeks, and 19.2 percent, or 1,129,344 workers were unemployed for 27 or more weeks. (b) Assume that the average spell of unemployment is 2.5 weeks for anyone unemployed for less than 5 weeks. Similarly, assume the average spell is 10 weeks, 20 weeks, and 35 weeks for the remaining categories. How many weeks did the average unemployed worker remain unemployed? What percent of total months of unemployment are attributable to the workers that remained unemployed for at least 15 weeks?workers that remained unemployed for at least 15 weeks?...
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http://mathforum.org/kb/message.jspa?messageID=7889954
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math
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PT III says: >So what? You talk as if you have not heard of "without loss of generality" - look it up. It's standard in mathematics. (I used it a number of times when proving theorems during obtaining my math degree and when teaching geometry.)
Did you get through school by writing on tests about imaginary "processes" that produced answers in your imagination only?
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https://www.mediapop.ca/news/fibonacci-golden-ratio-video-film/
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math
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Influences in the way we shoot our productions range from directors we like, movies we cherish, and math. Math? Yes, math.
If you’re not familiar with Leonardo Fibonacci, he was an Italian mathematician born in the 12th century. He is known to have discovered the “Fibonacci numbers,” which are a sequence of numbers where each successive number is the sum of the two previous numbers.
e.g. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…1+1=2, 1+2=3, 2+3=5, 21+34=55 etc.
There is a special relationship between the Golden Ratio and the Fibonacci Sequence, here is a surprise: if you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio:
1.61803398875…and so on.
So how on earth does this relate to what we at MEDIAPOP do for our clients? First let’s begin with how that Golden Ratio and Fibonacci numbers can be visualized.
The Fibonacci Spiral:
The Fibonacci Spiral is evident in art, nature, and architecture to name a few…
For Video, Film, and Photography there is The Rule of Thirds:
With those ratios visualized we can now start to see how that can be applied to video and film, and subconsciously it makes for a more visually appealing frame when you’re watching. Ali Shirazi put together a wonderful video about the mathematical cinematography in one of my favourite movies, “There Will Be Blood” Directed by Paul Thomas Anderson.
Just as important as the robust content we aim for when shooting for our clients, is the visual beauty of any given frame. Using the Golden Ratio and the Fibonacci sequence we can mathematically calculate the beauty of a frame.
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https://www.hackmath.net/en/examples/area-of-shape?page_num=23
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math
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Examples of area of plane shapes - page 23
- Hamster cage
Ryan keeps his hamster cage on his dresser. The area of the top of Ryan's dresser is 1 2\3 as large as the area of the bottom of his hamster cage. The area of the dresser top is 960 square inches. How many square inches of his dresser top are not covered b
- Area of rectangle
Calculate the area of rectangle in square meters whose sides have dimensions a = 80dm and b = 160dm.
- Triangular prism
Calculate the volume of a triangular prism 10 cm high, the base of which is an equilateral triangle with dimensions a = 5 cm and height va = 4,3 cm
Calculate the area and circumference of the regular decagon when its radius of a circle circumscribing is R = 1m
Isosceles trapezium ABCD ABC = 12 angle ABC = 40 ° b=6. Calculate the circumference and area.
- Glass panel
A rectangular glass panel with dimensions of 72 cm and 96 cm will cut the glazier on the largest square possible. What is the length of the side of each square? How many squares does the glazier cut?
- The perimeter
The perimeter of equilateral △PQR is 12. The perimeter of regular hexagon STUVWX is also 12. What is the ratio of the area of △PQR to the area of STUVWX?
- Rectangular trapezoid
The ABCD rectangular trapezoid with the AB and CD bases is divided by the diagonal AC into two equilateral rectangular triangles. The length of the diagonal AC is 62cm. Calculate trapezium area in cm square and calculate how many differs perimeters of the.
- Equilateral triangle
Calculate the area of an equilateral triangle with circumference 72cm.
- Diameter to area
Find the area of a circle whose diameter is 26cm.
- Sand path
How many m3 of sand is needed to fill the 1.5m wide path around a rectangular flowerbed of 8m and 14m if the sand layer is 6cm high?
- The room
The room has a cuboid shape with dimensions: length 50m and width 60dm and height 300cm. Calculate how much this room will cost paint (floor is not painted) if the window and door area is 15% of the total area and 1m2 cost 15 euro.
- Coordinate axes
Determine the area of the triangle given by line -7x+7y+63=0 and coordinate axes x and y.
- Diagonal to area
Calculate the area of a rectangle in which the length of the diagonal is 10 cm.
- Base of house
Calculate the volume of the bases of a square house, if the base depth is 1.2 m, the width is 40 cm and their outer circumference is 40.7 m.
- Enlarged rectangle
The rectangle with dimensions of 8 cm and 12 cm is enlarged in a ratio of 5:4. What are the circumference and the area of the enlarged rectangle?
- Square prism
Calculate the volume of a foursided prism 2 dm high, the base is a trapezoid with bases 12 cm, 6 cm, height of 4 cm and 5 cm long arms.
The rectangular garden has dimensions of 27 m and 30 m. Peter and Katka split it in a ratio of 4:5. How many square meters did Katkin measure part of the garden?
- Two rectangles
I cut out two rectangles with 54 cm², 90 cm². Their sides are expressed in whole centimeters. If I put these rectangles together I get a rectangle with an area of 144 cm2. What dimensions can this large rectangle have? Write all options. Explain your calcu
- Triangular prism
Calculate the surface of a regular triangular prism with a bottom edge 8 of a length of 5 meters and an appropriate height of 60 meters and prism height is 1 whole 4 meters.
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https://www.physicsforums.com/threads/please-ive-been-stuck-on-this-inverse-laplace-for-awhile.94743/
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math
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I have to find the laplace inverse of a function y(s) which has repeated complex roots. Y(s)=s^2 / (s^2+4)^2 so s=2i, s=2i, s=-2i, s=-2i. My partial fraction is as follows: A/(s-2i) + B/(s-2i)^2 + C/(s+2j) + D/(s+2j)^2 I use the standard method for finding regular repeated roots but I get stuck trying to calculating C and D. My values are undefined. My work is below... A= d/ds[(s-2i)^2*Y(s)]=8s(3s^2-4)/(s^2+4)^3 + 4s^2(s^2-12)*i/(s^2+4)^3-->then you set s=2i which then results in A=-6i. And B=1 But now for C, when I use the same process as A but instead of 2i, I use -2i, my answer is a number over 0 which results in undefined. Am I even doing this problem correctly? Any help would be appreciated... Thanks!
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https://books.google.co.ve/books?id=DPo2AAAAMAAJ&pg=PA195&vq=%22line+drawn+through+the+centre,+and+terminated+both+ways+by+the+circumference.+A%22&dq=related:ISBN8474916712&lr=&output=html_text
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math
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« AnteriorContinuar »
To find the folidity of a sphere.
Multiply the cube of the diameter by --5236, and the product will be the folidity.
Rule 2. A globe may be considered as composed of an infinite number of cones, whose bases are in the surface of the sphere, and common vertex in the centre ; therefore the folidity of the globe may be found thus :-Multiply its surface by
the diameter, and the product will give the solidity.
Rule 3. Find the folidity of a cylinder, of equal diameter and altitude with the globe, and the result will give the folis dity of the globe.
Required the folidity of a globe, whose diameter is 50 incha
By RULE II.
By RULE III.
98175.0000 solid cylin,
65450 Anf. as above.
Ex. 2. Required the solidity of a sphere of 10 inches diame
Anf. 523.6 Ex. 3. Required the content of a sphere, whose diameter is
Anf. 8181 cubic feet. Сс
Ex. 4. What is the solidity of a sphere, whose diameter is 3 feet 1 inch ?
Ans. 15.3483 cubic feet. Ex. 5. Required the folidity of a globe, its diameter being 8 feet 4 inches.
Anf. 303.0092 Ex. 6. How many solid miles are in the terraqueous globe, its diameter being 7958 miles ? Anf. 263883017937.1232.
To find the furface of any zone, or fegment of a sphere.
Multiply the circumference of a great circle of the sphere by the segment's height, and the product will be the superficies.
Required the superficies of a zone, whose height is'3 inches, the diameter of the sphere being 12 inches.
3 the zone's height. 113.0976 Anf. in square inches:
Ex. 2. Required the surface of a figment of a sphere, whose height is 1 foot 9 inches, the diameter being 5 feet.
Anf. 27.489 19. feet. Ex. 3. How
many square inches will cover a segment, whose height is i inch, the diameter of the sphere being 3 inches ?
Anf. 9.4248 fq. inches.
To find the folidity of a spherical segment.
From the treple product of the diameter of the sphere, multiplied by the square of the segment's height, subtract twice the cube of the height, and the remainder, multiplied by .5236, will give the solidity.
Rule 2. To thrice the square of the radius of the fegment's base, add the square of its height; then multiply the sum by its height, and the product again by ·5236, the last product, is the solidity.
Required the solidity of a spherical segment, whose height is & inches, and the radius of its base 16 inches.
By Rule I,
120 treple prod, of diameter.
3485.08.6 folid inches as before. Ex. 2. Required the solidity of a segment, whose base dia meter is 100, and its height 13.5 inches.
Ans: 54302.75235 cubic inches. Ex. 3. How many folid miles are in either frigid zone, the height being 329 miles, and diameter of its base 3168 miles ?
Anf. 1315766512 folid miles.
To find the folidity of the middle zone of a sphere.
When the ends are unequal, add into one sum the squares of the radii of both ends, and the square of the zone's height;
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http://www.marcom-china.com/english/mva_tree.html
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math
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Classification tree model is a data mining technique that can select and prioritizes from among a large number of independent variables that are most important in determining the outcome of the dependent variable to be explained. The resulting model also show how the importance of an independent variable are affected by other independent variables. Marcom uses CART and CHAID algorithm to implement classification tree analysis.
The model is particular useful in segmentation study to group respondents into homogenous subgroups in terms of how they respond to different marketing mix.
What you need to give us?
You send us the data file. Tell us which value of the dependent variable you want to predict and optionally which variables most likely will affect the outcome of the dependent variable. In the Titanic example, "Y" is the value in the "survived" variable we want to predict.
What we give you back?
(A) A Tree Model indicating segments of people who are more likely to answer the predicted value. The tree diagram representing a classification system or a predictive model.
(B) Gain chart
the gain table of the Titanic case will indicates that:
a female passenger in 1st class will have 301% higher chance of survival than the average.
the first 4 nodes in total only account for 24.3 of the total sample, but has 52.5% of the total survived rather than just 24.3% of the total survived.
69.9% of the pasangers in the first 4 nodes have survived while only 32.3% of the total passengers have survived.
(C) Prioritize or rank independent variables by importance
A list of independent variables which are found to have significant effect on the outcome of the dependent variable are ranked in order of importance in their predictive power. For example, Sex is found to be the most factor in determining a passenager would survive or not.
(D) IF - THEN rules to predict the survival segments
In the Titanic case,
Segment/node 5 is ‘survived’ segment, and the rule is if female and 1st, then 97% survived.
Segment/node 6 is ‘survived’ segment, and the rule is If female and (2nd or crew), then 88% survived
From these rules and the tree model, we will observe that "first class, women and children first" policy and the fact that policy was not entirely successful in saving the women and children in the third class - are reflected
(E) A misclassification table indicating accuracy of the model
In the Titanic case, our generated tree model is correct in prediction 78% of the time.
How long does it take?
After clarification of the dependent and independent variables of interest, we take about 2 working days to finish your work and send you back the result.
Asian B2B Expert
China | Hong Kong | Japan | Korea | Taiwan | Singapore | India | Australia
Suppose you have a data file containing cases of passengers on the Titanic. In this example, "survived" is the variable of interest so it is called the dependent or target variable. The variables that may affect the dependent variable (sex, age, class) are known as independent variables.
Our analysis goal is:
How can we use our existing information to learn what variables would provide us with an indication of which persons are most likely to survive?
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https://www.risk.net/journal-of-risk/volume-4-number-3-spring-2002
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math
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University of California at Irvine
This issue of the Journal of Risk further advances the state of knowledge in risk management, with two papers on risk measurement and two others on options.
The first paper, “Value-at-Risk Risk Measures versus Traditional Risk Measures: an Analysis and Survey” by G. Kaplanski and Y. Kroll, compares available risk measures and discusses whether value-at-risk (VAR), or its variants, are consistent with expected utility maximization. The decision framework is evaluated in terms of stochastic dominance criteria and is compared to a series of other risk measures that have been analyzed in the literature. The paper confirms that VAR-based decision-making can be justified based on expected utility for normal distributions of returns. For general distributions, however, VAR implies a very unusual utility function, which has a discontinuity around VAR. In contrast, the utility function for expected tail-loss risk measures is much better behaved. This article gives a convincing argument for augmenting VAR measures with tail-loss measures (such as conditional VAR), or stress tests.
Next, in “Is Implied Volatility an Informationally Efficient and Effective Predictor of Future Volatility?”, L. Ederington and W. Guan extend previous work on regressions of realized volatility on implied volatility. Efficiency implies that the intercept should be zero and the slope unity. The setup attempts to minimize the effect of measurement errors, which has plagued most other studies of this topic. The authors use daily data from Standard & Poor’s 500 futures options, which is better synchronized with the underlying asset price. They also implement instrument variables, which should correct for measurement errors in option data. The authors find that implied volatilities subsume information in historical time-series models. Even so, the paper identifies a downward bias in the slope coefficient. Their conclusion is that this puzzling effect cannot be explained by measurement error.
The third paper, by A. Doffou and J. E. Hilliard, “Testing a Three-State Model in Currency Derivative Markets”, presents tests of a jump–diffusion/stochastic interest rates pricing model on Philadelphia Stock Exchange currency options. This class of models does substantially better than the simple Black model premised on a geometric Brownian motion, and even the jump–diffusion model developed by Bates. The authors show that the addition of parameters improves the in-sample fit but also out-of-sample performance. The paper also reports that stochastic interest rates do make a difference for short-dated options, contrary to what has been reported in the literature.
Finally, the paper by C. Friedman, “Conditional Value-at-Risk in the Presence of Multiple Probability Measures”, extends the concept of conditional VAR (CVAR) to a situation where the probability space is unknown. This is an important question, as traditional risk measures assume a fixed probability space, or density function. In practice, the probability measures governing stochastic processes in financial markets may not be stable. Correlation patterns are prone to break down, for example. The paper derives risk management rules that are robust under multiple probability measures. This approach allows practitioners to discover dangerous risk measures, or risk holes.
The mission of the Journal of Risk is to further our understanding of risk management. Contributions to the journal are welcome from academics, practitioners, and regulators in the field. With this in mind, authors are encouraged to submit full-length papers. (See Guidelines for Authors for further details.)
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https://study.com/academy/exam/course/iseb-common-entrance-exam-at-13-geography-study-guide-test-prep.html
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math
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Take this practice test to check your existing knowledge of the course material. We'll review your answers and create a Test Prep Plan for you based on your results.
How Test Prep Plans work
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Free Practice Test Instructions:
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Question 1 1. Does history show that the Sahara Desert's landscape and climate has changed?
Question 2 2. Which of these is NOT a common export from Colombia?
Question 3 3. The Sahara Desert is the 3rd largest desert in the world. Which two deserts are larger than the Sahara?
Question 4 4. If you wanted to climb some really tall mountains, which of these Asian nations would you want to visit?
Question 5 5. Your family has decided to vacation in Europe during the early fall. If you wanted to know the climate during that time of year, which of the following data would you NOT want to use?
Question 6 6. When warm air gets pushed up a mountain or hill, what kind of rainfall do you get?
Question 7 7. Which of the following statements is true about climate?
Question 8 8. Cloud formation is an example of condensation because _____.
Question 9 9. Which of the following is NOT an environmental factor that affects where human settlements are located?
Question 10 10. What defines a sparsely populated area?
Question 11 11. What is the appropriate definition of suburban sprawl?
Question 12 12. Deduce the formula that represents the population growth rate.
Question 13 13. What denotes how far east or west a point is on the globe relative to the prime meridian?
Question 14 14. What do we call the study of how people interact with the environment?
Question 15 15. If I'm trying to identify a specific point in physical space using a map, what am I trying to find?
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CC-MAIN-2020-29
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https://www.terryjohnsonsflamingos.com/tag/method
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math
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My Mathematical Formulation is a horse racing system that truly has been around for a along time. The research of algebra meant mathematicians were fixing linear equations and methods, in addition to quadratics, and delving into optimistic and adverse solutions. Throughout this time, mathematicians started working with trigonometry. Taking part in games in the math classroom is an amazing technique to get multiple repetitions of details; a baby will happily learn the identical fact ten times when enjoying a sport, but groan when a flash drill card seems even a second time.
Three members of the division spoke in the Lie Principle Section of the 2014 International Congress of Mathematics. I do this to present you a deep understanding of fundamental mathematics and algebra...Read More
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CC-MAIN-2023-40
| 792 | 2 |
http://slideplayer.com/slide/3375102/
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math
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We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
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1.5 G RAPHING Q UADRATIC F UNCTIONS BY U SING T RANSFORMATIONS
Graph using the graph of
You move the key points of To shift to the right 3 spaces you add 3 to all of the x values! xy And then graph the new set of points!
(1, 4) (2, 1) (3, 0) (4, 1) (5, 4)
Graph This is a shift to the left 5 spaces. To shift to the left 5 spaces you subtract 5 from all of the x values! xy And then graph the new set of points!
Shift LEFT 5 Units (-7, 4) (-6, 1) (-5, 0) (-4, 1) (-3, 4)
Graph This is a shift down 4 spaces. To shift down 4 spaces you subtract 4 from all of the y values! xy And then graph the new set of points!
Shift DOWN 4 Units (-2, 0) (-1, -3) (0, -4) (1, -3) (2, 0)
Graph This is a shift up 6 spaces. To shift up 6 spaces you add 6 to all of the y values! xy And then graph the new set of points!
Shift UP 6 Units (-2,10) (-1, 7) (0, 6) (1, 7) (2, 10)
Graph This is a vertical stretch by a factor of 2. To stretch the parabola you multipy all of the y values by 2 ! xy And then graph the new set of points!
Strectch by a factor of 2 (-2,8) (-1, 2) (0, 0) (1, 2) (2, 8)
Graph This is a vertical compression by a factor of one half. To compress the parabola you multipy all of the y values by 0.5 ! (or divide them all by 2!) xy And then graph the new set of points!
Compress by a factor of (-2, 2) (-1, 0.5) (0, 0) (1, 0.5) (2, 2)
Graph This is a reflection in the x-axis. To reflect the parabola you multipy all of the y values by -1 ! xy And then graph the new set of points!
Reflect in the x-axis (-1, -1) (-2, -4) (0, 0) (2, -4) (1, -1)
Graph xy reflect in the x-axis and stretch by a factor of 2 shift the parabola up And then graph the new set of points!
(-1, 3) (-2, -3) (0, 5) (2, -3) (1, 3)
Graph xy shift the parabola left 7 shift the parabola down And then graph the new set of points!
(-9, 1) (-8, -2) (-7, -3) (-6, -2) (-5, 1)
Graph xy Compress by a factor of 0.25 shift the parabola right And then graph the new set of points!
(6, 1) (7, 0.25) (8, 0) (9, 0.25) (10, 1)
H OMEWORK : P AGE 47 #5 – 12
1.6 U SING M ULTIPLE T RANSFORMATIONS TO GRAPH QUADRATIC EQUATIONS.
Write equation or Describe Transformation. Write the effect on the graph of the parent function down 1 unit1 2 3 Stretch by a factor of 2 right 1 unit.
5-3 T RANSFORMING PARABOLAS ( PART 1) Big Idea: -Demonstrate and explain what changing a coefficient has on the graph of quadratic functions.
Your Transformation Equation y = - a f(-( x ± h)) ± k - a = x-axis reflection a > 1 = vertical stretch 0 < a < 1 = vertical compression -x = y-axis reflection.
Chapter 4 Quadratics 4.3 Using Technology to Investigate Transformations.
Order of function transformations Horizontal shifts Horizontal stretch/compression Reflection over y-axis Vertical stretch/compression Reflection over.
Transformations Review Vertex form: y = a(x – h) 2 + k The vertex form of a quadratic equation allows you to immediately identify the vertex of a parabola.
Section 3.5 Graphing Techniques: Transformations.
How would you sketch the following graph? ◦ y = 2(x – 3) 2 – 8 You need to perform transformations to the graph of y = x 2 Take it one step at a.
Graphing Quadratics. Parabolas x y=x 2 We can see the shape looks like: Starting at the vertex Out 1 up 1 2 Out 2 up 2 2 Out 3 up 3 2 Out 4 up.
Section 1.4 Transformations and Operations on Functions.
Transforming reciprocal functions. DO NOW Assignment #59 Pg. 503, #11-17 odd.
Transformations of Functions. The vertex of the parabola is at (h, k).
C HAPTER Using transformations to graph quadratic equations.
EXAMPLE 1 Compare graph of y = with graph of y = a x 1 x 1 3x3x b. The graph of y = is a vertical shrink of the graph of. x y = 1 = y 1 x a. The graph.
TRANSFORMATIONS OF FUNCTIONS Shifts and stretches.
Types of Functions. Type 1: Constant Function f(x) = c Example: f(x) = 1.
G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a 0.
Summary of 2.1 y = -x 2 graph of y = x 2 is reflected in the x- axis Note: Negative in front of x 2 makes parabola “frown”. y = (-x) 2 graph of y = x 2.
Algebra-2 Graphical Transformations. Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.)
Warm Up Find five points and use them to graph Hint, use an x-y table to help you.
And the Quadratic Equation……. Parabola - The shape of the graph of y = a(x - h) 2 + k Vertex - The minimum point in a parabola that opens upward or.
S TUDY P AGES MAT170 SPRING 2009 Material for 1 st Quiz.
1. Write the parabola in vertex form:. October 7 th.
G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a 0. The graph is “U-shaped” and is called a parabola. The.
Section 2.5 Transformations of Functions. Overview In this section we study how certain transformations of a function affect its graph. We will specifically.
QUADRATIC EQUATIONS in VERTEX FORM y = a(b(x – h)) 2 + k.
Section 3.3 Graphing Techniques: Transformations.
Standard 9.0 Determine how the graph of a parabola changes as a, b, and c vary in the equation Students demonstrate and explain the effect that changing.
Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted up k units (k > 0) or down k units.
If you take any shape, you can transform it: SQUARE STRETCH IT COMPRESS IT TRIANGLE STRETCH IT COMPRESS IT.
Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations Objectives Graph Functions Using Horizontal and Vertical Shifts Graph Functions Using.
Remember this example… Example If g(x) = x 2 + 2x, evaluate g(x – 3) g( ) = x x (x -3) g(x-3) = (x 2 – 6x + 9) + 2x - 6 g(x-3) = x 2 – 6x + 9 +
© The Visual Classroom Transformation of Functions Given y = f(x), we will investigate the function y = af [k(x – p)] + q for different values of a, k,
TRANSFORMATIONS Shifts Stretches And Reflections.
1 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x.
Graphical Transformations Vertical and Horizontal Translations Vertical and Horizontal Stretches and Shrinks.
Section 2.5 Transformations Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Graphing Absolute Value Functions using Transformations.
Essential Question: In the equation f(x) = a(x-h) + k what do each of the letters do to the graph?
Graph Absolute Value Functions using Transformations.
Math-3 Lesson 1-3 Quadratic, Absolute Value and Square Root Functions.
10.1 Quadratic GRAPHS! – Quadratic Graphs Goals / “I can…” Graph quadratic functions of the form y = ax Graph quadratic functions of the form.
4-1 Quadratic Functions Unit Objectives: Solve a quadratic equation. Graph/Transform quadratic functions with/without a calculator Identify function.
Precalculus Functions & Graphs Notes 2.5A Graphs of Functions TerminologyDefinitionIllustration Type of Symmetry of Graph f is an even function f(-x) =
Square Root Function Graphs Do You remember the parent function? D: [0, ∞) R: [0, ∞) What causes the square root graph to transform? a > 1 stretches vertically,
Section 3-2: Analyzing Families of Graphs A family of graphs is a group of graphs that displays one or more similar characteristics. A parent graph is.
2.5 Shifting, Reflecting, and Stretching Graphs. Shifting Graphs Digital Lesson.
Transformations xf(x) Domain: Range:. Transformations Vertical Shifts (or Slides) moves the graph of f(x) up k units. (add k to all of the y-values) moves.
Lesson 1-6 Graphical Transformations Graphical Transformations Transformation: an adjustment made to the parent function that results in a change to.
© 2017 SlidePlayer.com Inc. All rights reserved.
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CC-MAIN-2017-09
| 8,019 | 78 |
http://forums.wolfram.com/mathgroup/archive/2005/Jul/msg00529.html
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math
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Setting gridlines thickness in Plot
- To: mathgroup at smc.vnet.net
- Subject: [mg58939] Setting gridlines thickness in Plot
- From: carlos at colorado.edu
- Date: Sun, 24 Jul 2005 01:21:58 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
The problem: The default thickness of framelines, gridlines and tickmarks in Plot is 0.25pt. This is too thin for typical figure reductions in journals & books, and may disappear from the printed version. The question: I want to set the thickness of frameline, gridlines and ticks to 2, 1.25 and 1 pt, respectively. These are good values for typical paper figures. How do I do that? (At the moment I do it with Illustrator postprocessing) Note: the online help is not much help. There a passing mention on setting Gridlines thickness with PlotStyle, but no examples. What would be nice is to reset the defaults once and for all. Thanks for any help.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376006.87/warc/CC-MAIN-20210307013626-20210307043626-00324.warc.gz
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CC-MAIN-2021-10
| 897 | 7 |
https://vegandivasnyc.com/isosceles-and-equilateral-triangles-worksheet-answer-key-with-work/
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math
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Isosceles And Equilateral Triangles Worksheet Answer Key With Work
What are Isosceles and Equilateral Triangles?
A triangle is a three-sided polygon. Isosceles and equilateral triangles are two special types of triangles. Isosceles triangles have two equal sides and two equal angles. An equilateral triangle has three equal sides and three equal angles. Both of these types of triangles are special because all their angles and sides are the same.
Why Is It Important to Know About Isosceles and Equilateral Triangles?
Knowing the properties of isosceles and equilateral triangles is important for anyone working in mathematics or engineering. These types of triangles are used in many different fields, including geometry, architecture, and engineering. They can be used to measure distances, calculate angles, and create shapes.
What is a Worksheet Answer Key?
A worksheet answer key is a document that contains the answers to a set of questions or problem sets. It is often used in school and college classes to help students understand the material better and check their work. The answer key can also help teachers grade student work quickly and accurately.
How Can a Worksheet Answer Key Help With Isosceles and Equilateral Triangles?
A worksheet answer key with work can help students understand the material better and check their work. It can also help teachers grade student work quickly and accurately. With an answer key, students can practice solving problems and checking their work to make sure they are on the right track.
Where Can You Find an Isosceles and Equilateral Triangles Worksheet Answer Key With Work?
An isosceles and equilateral triangles worksheet answer key with work can be found online, in textbooks, or in teacher resource books. Many websites offer free worksheets with answer keys, and some textbooks may include them as well. Teacher resource books are also a great source for worksheets with answer keys.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224657735.85/warc/CC-MAIN-20230610164417-20230610194417-00050.warc.gz
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CC-MAIN-2023-23
| 1,943 | 11 |
https://financial-dictionary.thefreedictionary.com/Squared+multiple+correlation
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math
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Table 5 Squared Multiple Correlations
: (Group number 1 - Default model)
Researchers can now compute a confidence interval for a squared multiple correlation using a simple R command (see Kelley, 2007).
To obtain [w.sub.1] use the ci * R2 function in the "MBESS" R package with the sample size set to n and the sample squared multiple correlation set to its expected value.
Squared multiple correlations of subscales of the Urinary Incontinence Attitude Scale were 0.96 (ATTITUDEA) and 0.85 (ATTITUDEB).
Subscale squared multiple correlations also served as a reliability estimate.
First, it is well known that the distribution of sample squared multiple correlation is generally skewed.
The sample squared multiple correlation coefficient [R.sup.2] is a prevailing strength of association effect size measure for the population squared multiple correlation coefficient [[rho].sup.2] between the criterion variable and the set of predictor variables.
The adequacy of the proposed structural model was evaluated by testing the significance of the parameters and by estimating the reliabilities of the factors and the average variances extracted from the factors--the squared multiple correlations
([R.sup.2]) for each item (Joreskog and Sorbom, 1993)--which provide a direct index of item performance for each factor.
Regression coefficients, correlation coefficients, squared multiple correlations
, and model fit indices for the path model.
% Bus 410 1.10 Lorry 2993 8.00 Motorcar 13,630 36.44 Motorcycle 17,827 47.66 Others 2,548 6.81 Total 37,408 100.00 TABLE 3: Regression weights, standardized regression weights, variance, and squared multiple correlations
. (a) Regression weights and standardized regression weights Unstandardized Standardized coefficient coefficient est.
Squared multiple correlations
for the two endogenous variables, individualized consideration and class grade, were 0.04 and 0.13 respectively.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735958.84/warc/CC-MAIN-20200805124104-20200805154104-00016.warc.gz
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CC-MAIN-2020-34
| 1,923 | 16 |
https://brainsanswer.com/question/404803
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math
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f(x) = (x + 3)^2
Graph: see image
Translation just means shift or slide. To show a LEFT shift in the equation, go in close to the x, inside the parenthesis with the x, for LEFT shift, insert +3.
Left and right shifts might be the opposite of what you may think. Minus a number shifts right and plus a number shifts left.
For the graph, see image. The answer is the green curve.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711376.47/warc/CC-MAIN-20221209011720-20221209041720-00859.warc.gz
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CC-MAIN-2022-49
| 377 | 5 |
http://matric.shrishti.org/?p=10168
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math
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Back in trigonometry, we learn that what is tangent to the ideal angle is equal to the square of the hypotenuse.
That is of use when doing geometry, since in the event that you understand just how to do it and know the notion of the square root of two, you can make utilize of the exact purpose to figure along a triangle out.
So, as a way to come across the duration of a triangle, whatever essay writer service you want to do is subtract it from the hypotenuse and additionally choose the negative that is contrary the hypotenuse of this triangle. The equation for this can be
As soon as you have is straightforward. In order to get an answer, you need touse the line rule to fix the distance between the points and divide this from the distance between the things.
What’s tangent in math is not the most widely used, but also almost certainly the easiest type of space immersion. This usually means that it is used to calculate many https://internationaloffice.berkeley.edu/students/new different things in your life. If you are searching for, it is possible to figure out the number of calories you will burn up , together with how much a footwear will cost you much, and lots other activities.
What’s tangent in mathematics comes in handy whenever you’ve got to make a lot of dimensions. For instance, in the event that you are shopping for trousers, then you can calculate just how much you’ll pay to get these by simply measuring their width and length .
One particular essential thing to keep in mind is that the exact distance between the points will not continually be corresponding to the length. It follows you will need to be able to come across the length of the line to be sure you do not end up by having a wrong measurement. Do you understand expert-writers what’s tangent in mathematics. It is going to certainly come in handy in all types of situations within your life.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145966.48/warc/CC-MAIN-20200224163216-20200224193216-00442.warc.gz
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CC-MAIN-2020-10
| 1,899 | 7 |
https://uncommondescent.com/intelligent-design/does-bayesian-fuzziness-add-to-the-analysis/
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math
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In comment 30 to this post Elizabeth Liddle writes:
I can think of lots of ways of testing specific design hypotheses, but they all involve a hypothesis involving a postulated designer. And IDists insist that this is irrelevant – that “Design detection” should only involve the observed pattern, not any hypothesis about the designer. This is ludicrous, frankly.
Let’s explore one of Lizzie’s prior forays into design detection, and we’ll leave it up to the onlookers to decide which side is “ludicrous.”
In a prior post I posed the following question to Dr. Liddle:
If you were to receive a radio signal from outer space that specified the prime numbers between 1 and 100 would you conclude (provisionally pending the discovery a better theory, of course) that the best theory to account for the data is ‘the signal was designed and sent by an intelligent agent?’
Dr. Liddle responded:
Yes. And I’ve explained why.
She expanded on her explanation:
Barry, I did NOT make the inference ‘based upon nothing but the existence of CSI’!
My inference had nothing to do with CSI.
It was a Bayesian inference based on two priors:
My priors concerning the probability that other parts of the universe host intelligent life forms capable of sending radio signals (high)
My priors concerning the probability that a non-intelligent process might generate such a signal (low).
Dr. Liddle’s problem can be summarized as follows:
1. Denying the design inference based on the prime number sequence is not an option. The inference is so glaringly obvious that to deny it would be absurd. Even arch-atheist Carl Sagan admitted this signal was obviously designed (when he used it as the basis of his book “Contact”). Therefore, were Dr. Liddle to deny the obvious design inference she would instantly lose all credibility.
2. So she asks herself: “How can I admit the design inference while continuing to deny the methods of ID proponents?”
3. Her solution: “I know. I’ll admit the design inference but cover up my admission with Bayesian fuzziness, and that will obscure the fact that I used the methods of the ID proponents while I continue to denounce those very methods.”
Notice how Dr. Liddle’s Bayesian “priors” add absolutely nothing to the design detection methods advocated by ID proponents. Here is a graph of the explanatory filter:
Let’s run the prime number sequence through the explanatory filter to see how.
1. We observe an event (i.e. a radio signal specifying the prime numbers between 1 and 100).
2. Is it highly contingent? Yes. We can exclude mechanical necessity.
3. Is it highly complex and specified? Yes. We can exclude chance.
4. The best explanation for the data: Design.
Now let’s see if Dr. Liddle’s Bayesian analysis adds anything to what we already have.
Prior 1: Estimate of the probability that other parts of the universe host intelligent life forms capable of sending radio signals: High
It is obvious that one’s prior estimate of the probability of the existence of intelligent life forms in other parts of the universe is utterly irrelevant to the design inference. How do I know? By supposing the exact opposite of course. Let’s assume that a person believes there is practically zero chance that other parts of the universe have intelligent life (as we have seen on this site, there is very good reason to believe this). If that person were to receive this signal he would have to revise his conclusion, because the signal is obviously designed.
We see, therefore, that whether one’s Bayesian prior regarding the probability of the existence of intelligence life forms is 0% or 100% makes absolutely no difference to the design inference. From this we conclude that Dr. Liddle’s first prior adds nothing to the analysis.
Prior 2: Estimate of the probability that a non-intelligent process might generate such a signal: Low
This, of course, is the explanatory filter by another name. How do we know that the probability that a non-intelligent process might generate such a signal is low? Because it is highly continent, complex and specified.
It is important to see two things:
1. When Dr. Liddle correctly inferred design from the prime number sequence she had one and only one data point: A radio signal specifying the primes between 1 and 100.
2. Dr. Liddle knew nothing about the provenance of the radio signal. In other words she made a design inference based on nothing but the pattern itself while knowing absolutely nothing about the designer. When she made her design inference she did not first make a hypothesis based on the “postulated designer,” for the simple reason that there was not a scintilla of data upon which to base that hypothesis other than the pattern itself.
Conclusion: Though she tried to cover it up with Bayesian fuzziness, Dr. Liddle did the very thing she now says is “ludicrous.”
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100540.62/warc/CC-MAIN-20231205010358-20231205040358-00141.warc.gz
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CC-MAIN-2023-50
| 4,902 | 33 |
https://difattamagic.com/fontana-di-carte-con-telecomando.html
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math
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Card Fountain - With remote control
By using his manipulative skills the magician produces many cards from different places, and throws them in a hat on the table. He then makes a magical gesture towards the hat, and all the cards inside, fly out of it like a fountain of cards.
• Only the mechanism that allows you to produce a fountain of cards is supplied (you can use your own cards and hat).
• The fountain is remote controlled, and works with eight 1,5 Volt batteries (not supplied). It measures cm 14,2 (5,59") x 14,5 (5,7") x 9,9 (3,89").
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506045.12/warc/CC-MAIN-20230921210007-20230922000007-00009.warc.gz
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CC-MAIN-2023-40
| 550 | 4 |
http://www.thevillager.com.na/articles/13597/cabinet-endorses-rewarding-athletes-who-win-internationally/
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math
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Cabinet endorses rewarding athletes who win internationally
The Cabinet has taken a decision to endorse the categorisation of sports codes for the period of 2018-2020 as well as to monetarily reward excelling athletes, a motion submitted by the minister of sports, Erastus Uutoni last month.
In the national sports codes category, Football; Netball and rugby have been endorsed under the category, while in the priority sports codes category Athletics; Paralympics; Wrestling and Boxing were endorsed.
Cricket; Gymnastics; Swimming; Archery and Hockey have been endorsed under the Development Sports code category.
Furthermore, Uutoni has submitted sport recognition medals reward and preparation grant system for all sport codes.
Wining athletes of gold medals in Olympic and Paralympic games will be awarded N$200 000 while the coach or personal trainer will be awarded an amount of N$80 000.
Silver medal winners will be awarded N$150 000 and the coach N$ 60 000. Bronze medal winners will be awarded N$100 000.
Gold medal winners of World Championship will be awarded an amount of N$N$100 000; Silver medal winners (N$80 000) and Bronze medal winners (N$50 000).
African Championships with more than 30 countries participating gold medal winners will be awarded N$ 50 000; Silver medal winners (N$30 000) and Bronze medal winners (N$20 000).
Special Olympics World Summer and Winter Games Gold medal winners will be a awarded N$40 000; Silver medal winners (N$30 000) and Bronze medal winners and amount of N$20 000.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527907.70/warc/CC-MAIN-20190722092824-20190722114824-00334.warc.gz
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CC-MAIN-2019-30
| 1,520 | 10 |
https://yolpusulasi.com/ask-560
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math
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Pre calc homework help
There are a lot of Pre calc homework help that are available online. Math can be a challenging subject for many students.
The Best Pre calc homework help
We'll provide some tips to help you choose the best Pre calc homework help for your needs. In order to solve any problem, you have to start by identifying the problem itself. This is a key first step because it allows you to identify what exactly is wrong with your situation and how best to go about solving it. Once you've done this, then you can start looking for a solution that will work well in your situation. The solution must be a step by step one so you can keep track of the progress. It's best to start off slow and increase the pressure gradually so that you don't get discouraged or give up too soon. Once you find a solution that works well for you, you should implement it as quickly as possible so that you can see results sooner rather than later.
How to solve factorials? There are a couple different ways to do this. The most common way is to use the factorial symbol. This symbol looks like an exclamation point. To use it, you write the number that you want to find the factorial of and then put the symbol after it. For example, if you wanted to find the factorial of five, you would write 5!. The other way to solve for factorials is to use multiplication. To do this, you would take the number that you want to find the factorial of and multiply it by every number below it until you reach one. Using the same example from before, if you wanted to find the factorial of five using multiplication, you would take 5 and multiply it by 4, 3, 2, 1. This would give you the answer of 120. So, these are two different ways that you can solve for factorials!
There are many ways to solve a quadratic equation, but one of the most popular methods is using a quadratic solver. This is a tool that helps you to find the roots of a quadratic equation, which are the values of x that make the equation equal to zero. There are many different quadratic solvers available online and in math textbooks.
Math Homework help is something every math student needs at some point during their academic career. Math can be a difficult subject for some students, and doing homework can be a tedious and time-consuming process. Luckily, there are a number of resources available to help math students with their homework. Online resources such as Mathway and Khan Academy offer step-by-step solutions to problems, as well as practice exercises and video lessons. In addition, many teachers offer after-school homework help sessions, and there are often tutors available through school districts or local organizations. With a little effort, any math student can get the help they need to succeed.
There are two methods that can be used to solve quadratic functions: factoring and using the quadratic equation. Factoring is often the simplest method, and it can be used when the equation can be factored into two linear factors. For example, the equation x2+5x+6 can be rewritten as (x+3)(x+2). To solve the equation, set each factor equal to zero and solve for x. In this case, you would get x=-3 and x=-2. The quadratic equation can be used when factoring is not possible or when you need a more precise answer. The quadratic equation is written as ax²+bx+c=0, and it can be solved by using the formula x=−b±√(b²−4ac)/2a. In this equation, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, if you were given the equation 2x²-5x+3=0, you would plug in the values for a, b, and c to get x=(5±√(25-24))/4. This would give you two answers: x=1-½√7 and x=1+½√7. You can use either method to solve quadratic functions; however, factoring is often simpler when it is possible.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499845.10/warc/CC-MAIN-20230131055533-20230131085533-00471.warc.gz
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CC-MAIN-2023-06
| 3,814 | 8 |
https://link.springer.com/chapter/10.1007/978-3-540-74448-1_4
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math
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Local Asymptotic Mixed Normality for Nonhomogeneous Diffusions
- 2.2k Downloads
We study the asymptotic properties of various estimators of the parameter appearing nonlinearly in the nonhomogeneous drift coefficient of a functional stochastic differential equation when the corresponding solution process, called the diffusion type process, is observed over a continuous time interval [0, T]. We show that the maximum likelihood estimator, maximum probability estimator and regular Bayes estimators are strongly consistent and when suitably normalised, converge to a mixture of normal distribution and are locally asymptotically minimax in the Hajek-Le Cam sense as T → ∞ under some regularity conditions. Also we show that posterior distributions, suitably normalised and centered at the maximum likelihood estimator, converge to a mixture of normal distribution. Further, the maximum likelihood estimator and the regular Bayes estimators are asymptotically equivalent as T → ∞. We illustrate the results through the exponential memory Ornstein-Uhlenbeck process, the nonhomogeneous Ornstein-Uhlenbeck process and the Kalman- Bucy filter model where the limit distribution of the above estimators and the posteriors is shown to be Cauchy.
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US 5165008 A
A method for synthesizing human speech using a linear mapping of a small set of coefficients that are speaker-independent. Preferably, the speaker-independent set of coefficients are cepstral coefficients developed during a training session using a perceptual linear predictive analysis. A linear predictive all-pole model is used to develop corresponding formants and bandwidths to which the cepstral coefficients are mapped by using a separate multiple regression model for each of the five formant frequencies and five formant bandwidths. The dual analysis produces both the cepstral coefficients of the PLP model for the different vowel-like sounds and their true formant frequencies and bandwidths. The separate multiple regression models developed by mapping the cepstral coefficients into the formant frequencies and formant bandwidths can then be applied to cepstral coefficients determined for subsequent speech to produce corresponding formants and bandwidths used to synthesize that speech. Since less data are required for synthesizing each speech segment than in conventional techniques, a reduction in the required storage space and/or transmission rate for the data required in the speech synthesis is achieved. In addition, the cepstral coefficients for each speech segment can be used with the regressive model for a different speaker, to produce synthesized speech corresponding to the different speaker.
1. A method for synthesizing human speech, comprising the steps of:
a. for a given human vocalization, determining a set of Perceptual Line Predictive (PLP) coefficients defining an auditory-like, speaker-independent spectrum of the vocalization;
b. mapping the set of PLP coefficients to a vector in a vocal tract resonant vector space, where the vector is defined by a plurality of vector elements; and
c. using the vector in the vocal tract resonant space to produce a synthesized speech signal simulating the given human vocalization.
2. The method of claim 1, wherein fewer PLP coefficients are required in the set of coefficients than the plurality of vector elements that define the vector in the vocal tract resonant vector space.
3. The method of claim 2, wherein the set of coefficients is stored for later use in synthesizing speech.
4. The method of claim 2, wherein the set of coefficients comprises data that are transmitted to a remote location for use in synthesizing speech at the remote location.
5. The method of claim 1, further comprising the steps of determining speaker-dependent variables that define qualities of the given human vocalization specific to a particular speaker; and using the speaker-dependent variables in mapping the set of coefficients to produce the vector in the vocal tract resonant space, which is used in producing a simulation of that speaker uttering the given vocalizations.
6. The method of claim 5, wherein the speaker-dependent variables remain constant and are used with successive different human vocalizations to produce a simulation of the speaker uttering the successive different vocalizations.
7. The method of claim 1, wherein the set of coefficients represents a second formant, F2', corresponding to a speaker's mouth cavity shape during production of the given vocalization.
8. The method of claim 1, wherein the step of mapping comprises the step of determining a weighting factor for each coefficient of the set so as to minimize a mean squared error of each element of the vector in the vocal tract resonant space.
9. The method of claim 8, wherein each element of the vector in the vocal tract resonant space is defined by: ##EQU9## where ei is the i-th element, ai0 is a constant portion of that element, aij is the weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient for the i-th element; and N is the number of coefficients.
10. A method for synthesizing human speech, comprising the steps of:
a. repetitively sampling successive short segments of a human utterance so as to produce a unique frequency domain representation for each segment;
b. transforming the unique frequency domain representations into auditory-like, speaker-independent spectra, by representing a human psychophysical auditory response to the short segments of speech with the transformation;
c. defining each of the speaker-independent spectra using a limited set of Perceptual Line Predictive (PLP) coefficients for each segment;
d. mapping each limited set of PLP coefficients that define the speaker-independent spectra into one of a plurality of vectors in a vocal tract resonant vector space of a dimension greater than a cardinality of the limited set of PLP coefficients; and
e. producing a synthesized speech signal from the plurality of vectors in the vocal tract resonant space, taken in succession, thereby simulating the human utterance.
11. The method of claim 10, wherein the transforming step comprises the steps of:
a. warping the frequency domain representations into their Bark frequencies;
b. convolving the Bark frequencies with a power spectrum of a simulated critical-band masking curve, producing critical band spectra;
c. pre-emphasizing the critical band spectra with a simulated equal-loudness function, producing pre-emphasized, equal loudness spectra; and
d. compressing the pre-emphasized, equal loudness spectra with a cubic-root amplitude function, producing the auditory-like, speaker-independent spectra.
12. The method of claim 10, wherein the step of defining each of the auditory-like, speaker-independent spectra comprises the step of applying an inverse frequency transformation, using an all-pole model, wherein the limited set of coefficients comprise autoregression coefficients of the inverse frequency transformation.
13. The method of claim 10, wherein the limited set of coefficients that define each speaker-independent spectrum comprise cepstral coefficients of a perceptual linear prediction model.
14. The method of claim 10, wherein the vocal tract resonant vector space represents a linear predictive model.
15. The method of claim 10, further comprising the step of determining speaker-dependent variables that define qualities of a vocal tract in a speaker that produced the human utterance; and using the speaker-dependent variables in mapping each of the limited set of coefficients that define the speaker-independent spectra to produce the vectors in the vocal tract resonant space, thereby enabling simulation of the speaker producing the utterance.
16. The method of claim 15, wherein the speaker-dependent variables remain constant and are used to simulate additional different human utterances by that speaker.
17. The method of claim 16, the limited set of coefficients for each segment of the utterance and the speaker-dependent variables comprise data that are transmitted to a remote location for use in synthesizing the utterance at the remote location.
18. The method of claim 15, wherein the step of mapping comprises the step of determining a weighting factor for each coefficient so as to minimize a means squared error of each element of the vectors in the vocal tract resonant space.
19. The method of claim 10, wherein the coefficients represent a second formant, F2', corresponding to a speaker's mouth cavity shape during the utterance of each segment.
20. The method of claim 10, wherein each element comprising the vectors in the vocal tract resonant space is defined by: ##EQU10## where ei is the i-th element, ai0 is a constant portion of that element, aij is the weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient of the i-th element; and N is the number of coefficients.
This invention generally pertains to speech synthesis, and particularly, speech synthesis from parameters that represent short segments of speech with multiple coefficients and weighting factors.
Speech can be synthesized using a number of very different approaches. For example, digitized recordings of words can be reassembled into sentences to produce a synthetic utterance of a telephone number. Alternatively, a phonetic representation of the telephone number can be produced using phonemes for each sound comprising the utterance. Perhaps the dominant technique used in speech synthesis is linear predictive coding (LPC), which describes short segments of speech using parameters that can be transformed into positions (frequencies) and shapes (bandwidths) of peaks in the spectral envelope of the speech segments. In a typical 10th order LPC model, ten such parameters are determined, the frequency peaks defined thereby corresponding to resonant frequencies of the speaker's vocal tract. The parameters defining each segment of speech (typically, 10-20 milliseconds per segment) represent data that can be applied to conventional synthesizer hardware to replicate the sound of the speaker producing the utterance.
It can be shown that for a given speaker, the shape of the front cavity of the vocal tract is the primary source of linguistic information. The LPC model includes substantial information that remains approximately constant from segment to segment of an utterance by a given speaker (e.g., information reflecting the length of the speaker's vocal chords). As a consequence, the data representing each segment of speech in the LPC model include considerable redundancy, which creates an undesirable overhead for both storage and transmission of that data.
It is desirable to use the smallest number of parameters required to represent a speech segment for synthesis, so that the requirements for storing such data and the bit rate for transmitting the data can be reduced. Accordingly, it is desirable to separate the speaker-independent linguistic information from the superfluous speaker-dependent information. Since the speaker-independent information that varies with each segment of speech conveys the data necessary to synthesize the words embodied in an utterance, considerable storage space can potentially be saved by separately storing and transmitting the speaker-dependent information for a given speaker, separate from the speaker-independent information. Many such utterances could be stored or transmitted in terms of their speaker-independent information and then synthesized into speech by combination with the speaker-dependent information, thereby greatly reducing storage media requirements and making more channels in an assigned bandwidth available for transmittal of voice communications using this technique. Furthermore, different speaker-dependent information could be combined with the speaker-independent information to synthesize words spoken in the voice of another speaker, for example, by substituting the voice of a female for that of a male or the voice of a specific person for that of the speaker. By reducing the amount of data required to synthesize speech, data storage space and the quantity of data that must be transmitted to a remote site in order to synthesize a given vocalization are greatly reduced. These and other advantages of the present invention will be apparent from the drawings and from the Detailed Description of the Preferred Embodiment that follows.
In accordance with the present invention, a method for synthesizing human speech comprises the steps of determining a set of coefficients defining an auditory-like, speaker-independent spectrum of a given human vocalization, and mapping the set of coefficients to a vector in a vocal tract resonant vector space. Using this vector, a synthesized speech signal is produced that simulates the linguistic content (the string of words) in the given human vocalization. Substantially fewer coefficients are required than the number of vector elements produced (the dimension of the vector). These coefficients comprise data that can be stored for later use in synthesizing speech or can be transmitted to a remote location for use in synthesizing speech at the remote location.
The method further comprises the steps of determining speaker-dependent variables that define qualities of the given human vocalization specific to a particular speaker. The speaker-dependent variables are then used in mapping the coefficients to produce the vector of the vocal resonant tract space, to effect a simulation of that speaker uttering the given vocalization. Furthermore, the speaker-dependent variables remain substantially constant and are used with successive different human vocalizations to produce a simulation of the speaker uttering the successive different vocalizations.
Preferably, the coefficients represent a second formant, F2', corresponding to a speaker's mouth cavity shape during production of the given vocalization. The step of mapping comprises the step of determining a weighting factor for each coefficient so as to minimize a mean squared error of each element of the vector in the vocal tract resonant space (preferably determined by multivariate least squares regression). Each element is preferably defined by: ##EQU1## where ei is the i-th element, ai0 is a constant portion of that element, aij is a weighting factor associated with a j-th coefficient for the i-th element, cij is the j-th coefficient for the i-th element; and N is the number of coefficients.
FIG. 1 is a schematic block diagram illustrating the principles employed in the present invention for synthesizing speech;
FIG. 2 is a block diagram of apparatus for analyzing and synthesizing speech in accordance with the present invention;
FIG. 3 is a flow chart illustrating the steps implemented in analyzing speech to determine its characteristic formants, associated bandwidths, and cepstral coefficients;
FIG. 4 is a flow chart illustrating the steps of synthesizing speech using the speaker-independent cepstral coefficients, in accordance with the present invention;
FIG. 5 is flow chart showing the steps of a subroutine for analyzing formants;
FIG. 6 is a flow chart illustrating the subroutine steps required to perform a perceptive linear predictive (PLP) analysis of speech, to determine the cepstral coefficients;
FIG. 7 graphically illustrates the mapping of speaker-independent cepstral coefficients and a bias value to formant and bandwidth that is implemented during synthesis of the speech;
FIGS. 8A through 8C illustrate vocal tract area and length for a male speaker uttering three Russian vowels, compared to a simulated female speaker uttering the same vowels;
FIGS. 9A and 9B are graphs of the F1 and F2 formant vowel spaces for actual and modelled female and male speakers;
FIGS. 10A and 10B graphically illustrate the trajectories of complex pole predicted by LPC analysis of a sentence, and the predicted trajectories of formants derived from a male speaker-dependent model and the first five cepstral coefficients from the 5th order PLP analysis of that sentence, respectively; and
FIGS. 11A and 11B graphically illustrate the trajectories of formants predicted using a regressive model for a male and the first five cepstral coefficients from a sentence uttered by a male speaker, and the trajectories of formants predicted using a regressive model for a female and the first five cepstral coefficients from that same sentence uttered by a male speaker.
The principles employed in synthesizing speech according to the present invention are generally illustrated in FIG. 1. The process starts in a block 10 with the PLP analysis of selected speech segments that are used to "train" the system, producing a speaker-dependent model. (See the article, "Perceptual Linear Predictive (PLP) Analysis of Speech", by Hynek Hermansky, Journal of the Acoustical Society of America, Vol 87, pp 1738-1752 April 1990.) This speaker-dependent model is represented by data that are then transmitted in real time (or pre-transmitted and stored) over a link 12 to another location, indicated by a block 14. The transmission of this speaker-dependent model may have occurred sometime in the past or may immediately precede the next phase of the process, which involves the PLP analysis of current speech, separating its substantially constant speaker-dependent content from its varying speaker-independent content. The speaker-independent content of the speech that is processed after the training phase is transmitted over a link 16 to block 14, where the speech is reconstructed or synthesized from the speaker-dependent information, at a block 18. If a different speaker-dependent model, for example, speaker-dependent model for a female, is applied to speaker-independent information produced from the speech (of a male) during the process of synthesizing speech, the reconstructed speech will sound like the female from whom the speaker-dependent model was derived. Since the speaker-independent information for a given vocalization requires only about one-half the number of data points of the conventional LPC model typically used to synthesize speech, storage and transmission of the speaker-independent data are substantially more efficient. The speaker-dependent data can potentially be updated as rarely as once each session, i.e., once each time that a different speaker-dependent model is required to synthesize speech (although less frequent updates may produce a deterioration in the nonlinguistic parts of the synthesized speech).
Apparatus for synthesizing speech in accordance with the present invention are shown generally in FIG. 2 at reference numeral 20. A block 22 represents either speech uttered in real time or a recorded vocalization. Thus, a person speaking into a microphone may produce the speech indicated in block 22, or alternatively, the words spoken by the speaker may be stored on semi-permanent media, such as on magnetic tape. Whether produced by a microphone or by playback from a storage device (neither shown), the analog signal produced is applied to an analog-to-digital (A-D) converter 24, which changes the analog signal representing human speech to a digital format. Analog-to-digital converter 24 may comprise any suitable commercial integrated circuit A-D converter capable of providing eight or more bits of digital resolution through rapid conversion of an analog signal.
A digital signal produced by A-D converter 24 is fed to an input port of a central processor unit (CPU) 26. CPU 26 is programmed to carry out the steps of the present method, which include the both the initial training session and analysis of subsequent speech from block 22, as described in greater detail below. The program that controls CPU 26 is stored in a memory 28, comprising, for example, a magnetic media hard drive or read only memory (ROM), neither of which is separately shown. Also included in memory 28 is random access memory (RAM) for temporarily storing variables and other data used in the training and analysis. A user interface 30, comprising a keyboard and display, is connected to CPU 26, allowing user interaction and monitoring of the steps implemented in processing the speech from block 22.
Data produced during the initial training session through analysis of speech are converted to a digital format and stored in a storage device 32, comprising a hard drive, floppy disk, or other nonvolatile storage media. For subsequently processing speech that is to be synthesized, CPU 26 carries out a perceptual linear predictive (PLP) analysis of the speech to determine several cepstral coefficients, C1 . . . Cn that comprise the speaker-independent data. In the preferred embodiment, only five cepstral coefficients are required for each segment of the speaker-independent data used to synthesize speech (and in "training" the speaker-dependent model).
In addition, CPU 26 is programmed to perform a formant analysis, which is used to determine a plurality of formants F1 through Fn and corresponding bandwidths B1 through Bn. The formant analysis produces data used in formulating a speaker-dependent model. The formant and bandwidth data for a given segment of speech differ from one speaker to another, depending upon the shape of the vocal tract and various other speaker-dependent physiological parameters. During the training phase of the process, CPU 26 derives multiple regressive speaker-dependent mappings of the cepstral coefficients of the speech segments spoken during the training exercise, to the corresponding formants and bandwidths Fi and Bi for each segment of speech. The speaker-dependent model resulting from mapping the cepstral coefficients to the formants and bandwidths for each segment of speech is stored in storage device 32 for later use.
Alternatively, instead of storing this speaker-dependent model, the data comprising the model can be transmitted to a remote CPU 36, either prior to the need to synthesize speech, or in real time. Once remote CPU 36 has stored the speaker-dependent model required to map between the speaker-independent cepstral coefficients and the formants and bandwidths representing the speech of a particular speaker, it can apply the model data to subsequently transmitted cepstral coefficients to reproduce any speech of that same speaker.
The speaker-dependent model data are applied to the speaker-independent cepstral coefficients for each segment of speech that is transmitted from CPU 26 to CPU 36 to reproduce the synthesized speech, by mapping the cepstral coefficients to corresponding formants and bandwidths that are used to drive a synthesizer 42. A user interface 40 is connected to remote CPU 36 and preferably includes a keyboard and display for entering instructions that control the synthesis process and a display for monitoring its progression. Synthesizer 42 preferably comprises a Klsyn88™ cascade/parallel formant synthesizer, which is a combination software and hardware package available from Sensimetrics Corporation, Cambridge, Mass. However, virtually any synthesizer suitable for synthesizing human speech from LPC formant and bandwidth data can be used for this purpose. Synthesizer 42 drives a conventional loudspeaker 44 to produce the synthesized speech. Loudspeaker 44 may alternatively comprise a telephone receiver or may be replaced by a recording device to record the synthesized speech.
Remote CPU 36 can also be controlled to apply a speaker-dependent model mapping for a different speaker to the speaker-independent cepstral coefficients transmitted from CPU 26, so that the speech of one speaker is synthesized to sound like that of a different speaker. For example, speaker-dependent model data for a female speaker can be applied to the transmitted cepstral coefficients for each segment of speech from a male speaker, causing synthesizer 42 to produce synthesized speech, which on loudspeaker 44, sounds like a female speaker speaking the words originally uttered by the male speaker. CPU 36 can also modify the speaker-dependent model in other ways to enhance, or otherwise change the sound of the synthesized speech produced by loudspeaker 44.
One of the primary advantages of the technique implemented by the apparatus in FIG. 1 is the reduced quantity of data that must be stored and/or transmitted to synthesize speech. Only the speaker-dependent model data and the cepstral coefficients for each successive segment of speech must be stored or transmitted to synthesize speech, thereby reducing the number of bytes of data that need be stored by storage device 32, or transmitted to remote CPU 36.
As noted above, the training steps implemented by CPU 26 initially determine the mapping of cepstral coefficients for each segment of speech to their corresponding formants and bandwidths to define how subsequent speaker-independent cepstral coefficients should be mapped to produce synthesized speech. In FIG. 3, a flow chart 50 shows the steps implemented by CPU 26 in this training procedure and the steps later used to derive the speaker-independent cepstral coefficients for synthesizing speech. Flow chart 50 starts at a block 52. In a block 54, the analog values of the speech are digitized for input to a block 56. In block 56, a predefined time interval of approximately 20 milliseconds in the preferred embodiment defines a single segment of speech that is analyzed according to the following steps. Two procedures are performed on each digitized segment of speech, as indicated in flow chart 50 by the parallel branches to which block 56 connects.
In a block 58, a subroutine is called that performs formant analysis to determine the F1 through Fn formants and their corresponding bandwidths, B1 through Bn for each segment of speech processed. The details of the subroutine used to perform the formant analysis are shown in FIG. 5 in a flow chart 60. Flow chart 60 begins at a block 62 and proceeds to a block 64, wherein CPU 26 determines the linear prediction coefficients for the current segment of speech being processed. Linear predictive analysis of digital speech signals is well known in the art. For example, J. Makhoul described the technique in a paper entitled "Spectral Linear Prediction: Properties and Applications," IEEE Transaction ASSP-23, 1975, pp. 283-296. Similarly, in U.S. Pat. No. 4,882,758 (Uekawa et al.), an improved method for extracting formant frequencies is disclosed and compared to the more conventional linear predictive analysis method.
In block 64, CPU 26 processes the digital speech segment by applying a pre-emphasis and then using a window with an autocorrelation calculation to obtain linear prediction coefficients by the Durbin method. The Durbin method is also well known in the art, and is described by L. R. Rabiner and R. W. Schafer in Digital Processing of Speech Signals, a Prentice-Hall publication, pp. 411-413.
In a block 66, a constant Z0 is selected for an initial value as a root Zi. In a block 68, CPU 26 determines a value of A(z) from the following equation: ##EQU2## where ak are linear prediction coefficients. In addition, the CPU determines the derivative A'(Zi) of this function. A decision block 70 then determines if the absolute value of A(Zi)/A'(Zi) is less than a specified tolerance threshold value K. If not, a block 72 assigns a new value to Zi, as shown therein. The flow chart then returns to block 68 for redetermination of a new value for the function A(Zi) and its derivative. As this iterative loop continues, it eventually reaches a point where an affirmative result from decision block 70 leads to a block 74, which assigns Zi and its complex conjugate Zi * as roots of the function A(z). A block 76 then divides the function A(z) by the quadratic expression of Zi and its complex conjugate, as shown therein.
A decision block 78 determines whether Zi is a zero-order root of the function A(Z) and if not, loops back to block 64 to repeat the process until a zero order value for the function A(Z) is obtained. Once an affirmative result from decision block 78 occurs, a block 80 determines the corresponding formants Fk for all roots of the equation as defined by:
Fk =(f8 /2π)tan-1 [Im(Zi)/Re(Zi)](2)
Similarly, a block 82 defines the bandwidth corresponding to the formants for all the roots of the function as follows:
Bk =(fs /π)1n|Z1 | (3).
A block 84 then sets all roots with Bk less than a constant threshold T equal to formants Fi having corresponding bandwidths Bi. A block 86 then returns from the subroutine to the main program implemented in flow chart 50.
Following a return from the subroutine called in block 58 of FIG. 3, a block 90 stores the formants F1 through FN and corresponding bandwidths B1 through BN in memory 28 (FIG. 2).
The other branch of flow chart 50 following block 56 in FIG. 3 leads to a block 92 that calls a subroutine to perform PLP analysis of the digitized speech segment to determine its corresponding cepstral coefficients. The subroutine called by block 92 is illustrated in FIG. 6 by a flow chart 94.
Flow chart 94 begins at a block 96 and proceeds to a block 98, which performs a fast Fourier transform of the digitized speech segment. In carrying out the fast Fourier transform, each speech segment is weighted by a Hamming window, which is a finite duration window represented by the following equation:
W(n)=0.54+0.46cos [2πn/(T-1)] (4)
where T, the duration of the window, is typically about 20 milliseconds. The Fourier transform performed in block 98 transforms the speech segment weighted by the Hamming window into the frequency domain. In this step, the real and imaginary components of the resulting speech spectrum are squared and added together, producing a short-term power spectrum P(ω), which can be represented as follows:
P(ω)=Re[S(ω)]2 +Im[S(ω)]2 (5).
Typically, for a 10 KHz sampling frequency, a 256-point fast Fourier transform is applied to transform 200 speech samples (from the 20-millisecond window that was applied to obtain the segment), with the remaining 56 points padded by zero-valued samples.
In a block 100, critical band integration and resampling is performed, during which the short-term power spectrum P(ω) is warped along its frequency access ω into the Bark frequency Ω as follows: ##EQU3## wherein ω is the angular frequency in radians per second, resulting in a Bark-Hz transformation. The resulting warped power spectrum is then convolved with the power spectrum of the simulated critical band masking curve ψ(ω). Except for the particular shape of the critical-band curve, this step is similar to spectral processing in mel cepstral analysis. The critical band curve is defined as follows: ##EQU4## The piece-wise shape of the simulated critical-band masking curve is an approximation to an asymmetric masking curve. The intent of this step is to provide an approximation (although somewhat crude) of an auditory filter based on the proposition that the shape of auditory filters is approximately constant on the Bark scale and that the filter skirts are generally truncated at -40dB.
Convolution of ψ(ω) with (the even symmetric and periodic function) P(ω) yields samples of the critical-band power spectrum: ##EQU5## This convolution significantly reduces the spectral resolution of θ(Ω) in comparison with the original P(ω), allowing for the down-sampling of θ(Ω). In the preferred embodiment, θ(Ω) is sampled at approximately one-Bark intervals. The exact value of the sampling interval is chosen so that an integral number of spectral samples covers the entire analysis band. Typically, for a bandwidth of 5 KHz, corresponding to 16.9-Bark, 18 spectral samples of θ(Ω) are used, providing 0.994-Bark steps.
In a block 102, a logarithm of the computed critical-band spectrum is performed, and any convolutive constants appear as additive constants in the logarithm.
A block 104 applies an equal-loudness response curve to pre-emphasize each of the segments, where the equal-loudness curve is represented as follows:
In this equation, the function E(ω) is an approximation to the human sensitivity to sounds at different frequencies and simulates the unequal sensitivity of hearing at about the 40dB level. Under these conditions, this function is defined as follows: ##EQU6## The curve approximates a transfer function for a filter having asymptotes of 12dB per octave between 0 and 400 Hz, 0 dB per octave between 400 Hz and 1,200 Hz, 6 dB per octave between 1,200 Hz and 3,100 Hz, and zero dB per octave between 3,100 Hz and the Nyquist frequency (10 KHz in the preferred embodiment). In applications requiring a higher Nyquist frequency, an additional term can be added to the preceding expression. The values of the first (zero-Bark) and the last samples are made equal to the values of their nearest neighbors to ensure that the function resulting from the application of the equal loudness response curve begins and ends with two equal-valued samples.
In a block 106, a power-law of hearing function approximation is performed, which involves a cubic-root amplitude compression of the spectrum, defined as follows:
This compression is an approximation that simulates the nonlinear relation between the intensity of sound and its perceived loudness. In combination, the equal-loudness pre-emphasis of block 104 and the power law of hearing function applied in block 106 reduce the spectral-amplitude variation of the critical-band spectrum to produce a relatively low model order.
A block 108 provides for determining an inverse logarithm (i.e., determines an exponential function) of the compressed log critical-band spectrum. The resulting function approximates a relatively auditory spectrum.
A block 110 determines an inverse discrete Fourier transform of the auditory spectrum Φ(Ω). Preferably, a 34-point inverse discrete Fourier transform is used. The inverse discrete Fourier transform is a better choice than the fast Fourier transform in this case, because only a few autocorrelation values are required in the subsequent analysis.
In linear predictive analysis, a set of coefficients that will minimize a mean-squared prediction error over a short segment of speech waveform is determined. One way to determine such a set of coefficients is referred to as the autocorrelation method of linear prediction. This approach provides a set of linear equations that relate autocorrelation coefficients of the signal representing the processed speech segment with the prediction coefficients of the autoregressive model. The resulting set of equations can be efficiently solved to yield the predictor parameters. The inverse Fourier transform of a non-negative spectrum-like function resulting from the preceding steps can be interpreted as the autocorrelation function, and an appropriate autoregressive model of such a spectrum can be found. In the preferred embodiment of the present method, the equations for carrying out this solution apply Durbin's recursive procedure, as indicated in a block 112. This procedure is relatively efficient for solving specific linear equations of the autoregressive process.
Finally, in a block 114, a recursive computation is applied to determine the cepstral coefficients from the autoregressive coefficients of the resulting all-pole model.
If the overall LPC system has a transfer function H(z) with an impulse response h(n) and a complex cepstrum h(n), then h(n) can be obtained from the recursion: ##EQU7## (as shown by L. R. Rabiner and R. W. Schafer in Digital Processing of Speech Signals, a Prentice-Hall publication, page 442.) The complex cepstrum cited in this reference is equivalent to the cepstral coefficients C1 through C5.
After block 114 produces the cepstral coefficients, a block 116 returns to flow chart 50 in FIG. 3. Thereafter, a block 120 provides for storing the cepstral coefficients C1 through C5 in nonvolatile memory. Following blocks 90 or 120, a decision block 122 determines if the last segment of speech has been processed, and if not, returns to block 56 in FIG. 3.
After all segments of speech have been processed, a block 124 provides for deriving multiple regressive speaker-dependent mappings from the cepstral coefficients Ci using the corresponding formants Fi and bandwidths Bi. The mapping process is graphically illustrated in FIG. 7 generally at reference numeral 170, where five cepstral coefficients 176 and a bias value 178 are linearly combined to produce five formants and corresponding bandwidths 180 according to the following relationship: ##EQU8## where ei are elements representing the respective formants and their bandwidths (i=1 through 10, corresponding to F1 through F5 and B1 through B5, in succession), ai0 is the bias value, and aij are weighting factors for the j-th cepstral coefficient and the i-th element (formant or bandwidth) that are applied to the cepstral coefficients Cij. Mapping of the cepstral coefficients and bias value corresponds to a linear function that estimates the relationship between the formants (and their corresponding bandwidths) and the cepstral coefficients.
The linear regression analysis performed in this step is discussed in detail in An Introduction to Linear Regression and Correlation, by Allen L. Edwards (W. H. Freeman & Co., 1976), ch. 3. Thus, for each segment of speech, linear regression analysis is applied to map the cepstral coefficients 176 and bias value 178 into the formants and bandwidths 180. The mapping data resulting from this procedure are stored for subsequent use, or immediately used with speaker-independent cepstral coefficients to synthesize speech, as explained in greater detail below. A block 128 ends this first training portion of the procedure required for developing the speaker-dependent model for mapping of speaker-independent cepstral coefficients into corresponding formants and bandwidths.
Turning now to FIG. 4, the speaker-dependent model defined by mapping data developed from the training procedure implemented by the steps of flow chart 50 can later be applied to speaker-independent data to synthesize vocalizations by that same speaker, as briefly noted above. Alternatively, the speaker-independent data (represented by cepstral coefficients) of one speaker can be modified by the model data of a different speaker to produce synthesized speech corresponding to the vocalization of the different speaker. Steps required for carrying out either of these scenarios are illustrated in a flow chart 140 in FIG. 4, starting at a block 142.
In a block 143, signals representing the analog speech of an individual (from block 22 in FIG. 2) are applied to an A-D converter, producing corresponding digital signals that are processed one segment at a time. Digital signals are input to CPU 36 in a block 144. A block 146 calls a subroutine to perform PLP analysis of the signal to determine the cepstral coefficients for the speech segment, as explained above with reference to flow chart 94 in FIG. 6. This subroutine returns the cepstral coefficients for each segment of speech, which are alternatively either stored for later use in a block 148, or transmitted, for example, by telephone line, to a remote location for use in synthesizing the speech represented by the speaker-independent cepstral coefficients. Transmission of the cepstral coefficients is provided in a block 150.
In a block 152, the speaker-dependent model represented by the mapping data previously developed during the training procedure is applied to the cepstral coefficients, which have been stored in block 148 or transmitted in block 150, to develop the formants F1 through Fn and corresponding bandwidths B1 through Bn needed to synthesize that segment of speech. As noted above, the linear combination of the cepstral coefficients to produce the formants and bandwidth data in block 152 is graphically illustrated in FIG. 7.
A block 154 uses the formants and bandwidths developed in block 152 to produce a corresponding synthesized segment of speech, and a block 156 stores the digitized segment of speech. A decision block 158 determines if the last segment of speech has been processed, and if not, returns to block 144 to input the next speech segment for PLP analysis. However, if the last segment of speech has been processed, a block 160 provides for digital-to-analog (D-A) conversion of the digital signals. Referring back to FIG. 2, block 160 produces the analog signal used to drive loudspeaker 44, producing an auditory response synthetically reproducing the speech of either the original speaker or speech sounding like another person, depending upon whether the original speaker's model (mapping data) or the other person's model is used in block 152 to map the cepstral coefficients into corresponding formants and bandwidths. A block 162 terminates flow chart 140 in FIG. 4.
Experiments have shown that there is a relatively high correlation between the estimated formants and bandwidths used to synthesize speech in the present invention and the formants and bandwidths determined by conventional LPC analysis of the original speech segment. Table 1, below, shows correlations between the true and model-predicted form of these parameters, the root mean square (RMS) error of the prediction, and the maximum prediction error. For comparison, values from the 10th order LPC formant estimation are shown in parentheses. The RMS error of the PLP-based formant frequency prediction is larger than the LPC estimation RMS error. LPC exhibits occasional gross errors in the estimation of lower formants, which show in larger values of the maximum LPC error. In fact, formant bandwidths are far better predicted by the PLP-based technique.
TABLE 1__________________________________________________________________________FORMANT AND BANDWIDTH COMPARISONSPARAM.__________________________________________________________________________F1 F2 F3 F4 F5__________________________________________________________________________CORR. 0.94 (0.98) 0.98 (0.99) 0.91 (0.98) 0.64 (0.98) 0.86 (0.99)RMS[Hz] 23.6 (15.5) 48.1 (37.0) 48.2 (21.2) 46.1 (12.6) 52.4 (13.1)MAX[Hz] 131 (434) 344 (2170) 190 (1179) 190 (610) 220 (130)__________________________________________________________________________B1 B2 B3 B4 B5__________________________________________________________________________CORR. 0.86 (0.05) 0.92 (0.17) 0.96 (0.43) 0.64 (0.24) 0.86 (0.33)RMS[Hz] 2.2 (45) 1.6 (35) 4.1 (37) 4.1 (50) 5.5 (52)MAX[Hz] 29.3 (3707) 6.23 (205) 32.0 (189) 18.0 (119) 22.0 (354)__________________________________________________________________________
A significant advantage of the present technique for synthesizing speech is the ability to synthesize a different speaker's speech using the cepstral coefficients developed from low-order PLP analysis, which are generally speaker-independent. To evaluate the potential for voice modification, the vocal tract area functions for a male voicing three vowels /i/, /a/, and /u/ were modified by scaling down the length of the pharyngeal cavity by 2 cm and by linearly scaling each pharyngeal area by a constant. This constant was chosen for each vowel by a simple search so that the differences between the log of a male and a female-like PLP spectra are minimized. It has been observed that to achieve similar PLP spectra for both the longer and the shorter vocal tracts, the pharyngeal cavity for the female-like tracts need to be slightly expanded.
FIGS. 8A through 8C show the vocal tract functions for the three Russian vowels /i/, /a/, and /u/, using solid lines to represent the male vocal tract and dashed lines to represent the simulated female-like vocal tract. Thus, for example, solid lines 192, 196, and 200 represent the vocal tract configuration for a male, whereas dashed lines 190, 194, and 198 represent the simulated vocal tract voicing for a female.
Both the original and modified vocal tract functions were used to generate vowel spaces. The training procedure described above was used to obtain speaker-dependent models, one for the male and one for the simulated female-like vowels. PLP vectors (cepstral coefficients) derived from male speech were used with a female-regressive model, yielding predicted formants, as shown in FIG. 9A. Similarly, PLP vectors derived from female speech were used with the male-regressive models to yield predicted formants depicted in FIG. 9B. In FIG. 9A, boundaries of the original male vowel space are indicated by a solid line 202, while boundaries of the original female space are indicated by a dashed line 204. Similarly, in FIG. 9B, boundaries of the original female vowel space are indicated by a solid line 206, and boundaries of the original male vowel space are indicated by a dashed line 208. Based on a comparison of the F1 and F2 formants for the original and the predicted models, both male and female, it is evident that the range of predicted formant frequencies is determined by the given regression model, rather than by the speech signals from which the PLP vectors are derived.
Further verification of the technique for synthesizing the speech of a particular speaker in accordance with the present invention was provided by the following experiment. The regression speaker-dependent model for a particular speaker was derived from four all-voiced sentences: "We all learn a yellow line roar;" "You are a yellow yo-yo;" "We are nine very young women;" and "Hello, how are you?" each uttered by a male speaker. The first five cepstral coefficients (log energy excluded) from the fifth order PLP analysis of the first utterance, "I owe you a yellow yo-yo," together with the regressive model derived from training with the four sentences were used in predicting formants of the test utterance, as shown in FIG. 10B.
An estimated formant trajectory represented by poles of a 10th order LPC analysis for the same sentence, "I owe you a yellow yo-yo," uttered by a male speaker are shown in FIG. 10A. Comparing the predicted formant trajectories of FIG. 10B with the estimated formant trajectories represented by poles of the 10th order LPC analysis shown in FIG. 10A, it is clear that the first formant is predicted reasonably well. On the second formant trajectory, the largest difference is in /oh/ of "owe . . .," where the predicted second formant frequency is about 50% higher than the LPC estimated one. Furthermore, the predicted frequencies of the /j/s in "you" and "yo-yo," and of /e/ and /u/ in "yellow" are 15-20% lower than the LPC estimated ones. The predicated third order trajectory is again reasonably close to the LPC estimated trajectory. The LPC estimated fourth and fifth formants are generally unreliable, and comparing them to the predicted trajectories is of little value.
A similar experiment was done to determine whether synthetic speech can yield useful speaker-dependent models. In this case, speaker-dependent models derived from synthetic speech vowels were used, to produce a male regressive model for the same sentence. The trajectories of the formants predicted using the male regressive model in the first five cepstral coefficients from the fifth order PLP analysis of the sentence "I owe you a yellow yo-yo" uttered by a male speaker were then compared to the trajectories of formants predicted using the female regressive model (also derived from the synthetic vowel-like samples) in the first five cepstral coefficients from the fifth order PLP analysis of the same sentence, uttered by the male speaker.
Within the 0 through 5 KHz frequency band of interest, the male regressive model yields five formants, while the female-like model yields only four. By comparison of FIGS. 11A and 11B, it is apparent that the formant trajectories for both genders are approximately the same. The frequency span of the female second formant trajectory is visibly larger than the frequency span of the male second formant trajectory, almost coinciding with the third male formants in extreme front semi-vowels, such as the /j/s in "yo-yo" and being rather close to the male second formants in the rounded /u/ of "you." The male third formant trajectory is very similar to the female third formant trajectory, except for approximately a 400 Hz constant downward frequency shift. However, the male fourth formant trajectory bears almost no similarity to any of the female formant trajectories. Finally, the fifth formant trajectory for the male is quite similar to the female fourth formant trajectory.
Although the preferred embodiment uses PLP analysis to determine a speaker-dependent model for a particular speaker during the training process and for producing the speaker-independent cepstral coefficients that are used with that or another speaker's model for speech synthesis, it should be apparent that other speech processing techniques might be used for this purpose. These and other modifications and changes that will be apparent to those of ordinary skill in this art fall within the scope of the claims that follow. While the preferred embodiment of the invention has been illustrated and described, it will be appreciated that such changes can be made therein without departing from the spirit and scope of the invention defined by these claims.
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s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928562.33/warc/CC-MAIN-20150521113208-00113-ip-10-180-206-219.ec2.internal.warc.gz
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CC-MAIN-2015-22
| 48,172 | 106 |
https://www.crosswordgenius.com/clue/religious-words-just-audible
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math
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Religious words just audible (4)
I believe the answer is:
'religious words' is the definition.
Although both the answer and definition are singular nouns, I can't see how one could define the other.
'just audible' is the wordplay.
'just' becomes 'right' (associated in meaning).
'audible' indicates a 'sounds like' (homophone) clue.
'right' is a homophone of 'RITE'.
Can you help me to learn more?
(Other definitions for rite that I've seen before include "Ceremonial procedure", "something like mass", "Custom", "Tire of ceremony", "Established religious ceremony".)
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s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487607143.30/warc/CC-MAIN-20210613071347-20210613101347-00075.warc.gz
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CC-MAIN-2021-25
| 567 | 10 |
https://www.thecoursehero.com/explain-the-process-and-the-results/
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math
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Given the following data, use Excel’s syntax (and showthe formulas) to calculate the MIRR for a project with an expectedeconomic life six years and a salvage value of $1,000 (the lastdata entry). The end-of-period cash flows are expected to increaseat the rate of 10% per annum. Project data: outlay cost: -$10,000 Estimated end-of-period cash flows: -$2,000, $5,000,$4,000, $3,000, $5,000, and $1,000 To estimate the MIRR, compute the Future Value of thecash flows at the rate of ten per cent and solve for the discountrate that discounts this future value back to the present value of$10,000. Please explain the process and the results
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s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988753.97/warc/CC-MAIN-20210506114045-20210506144045-00152.warc.gz
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CC-MAIN-2021-21
| 944 | 10 |
https://hghreviews.to/reply/2672
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math
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Written by WhiteBenjamin on . Posted in
hey there guys, I am also having a question and I would really appreciate if you could help me with it, please. I really wanted to know after how much time you are starting to retain the water after you have started out with the GH? With this being said, I would really appreciate if you could answer the following: is it possible at all that in the bulking phase the water retention doesn’t actually increase very much the bodyweight due to the fact that we are already maintaing a reasonable higher level of water than we do in our cutting phase? I would appreciate a lot if you could help me with this. Thanks!
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00204.warc.gz
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CC-MAIN-2021-49
| 655 | 2 |
http://newton.kias.re.kr/~mueller/
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math
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My primary interests and areas of research are
- Topological Hamiltonian and contact dynamics
- Symplectic and contact geometry and topology
- Topological dynamics
and the deeper topological structure underlying symplectic and contact topology. Topological Hamiltonian dynamics and topological contact dynamics are natural and genuine extensions of smooth Hamiltonian and contact dynamics to topological dynamics, and of symplectic and contact diffeomorphisms to homeomorphisms as transformations preserving the additional geometric structure. These new theories have numerous applications to their smooth counterparts, as well as to other areas of mathematics, such as topological dynamics, in particular in low dimensions, and to geodesic flows on Riemannian manifolds. Applications are also expected to billiard dynamics. I also have an interest in action selectors in Hamiltonian Floer theory and related subjects.
A topological Hamiltonian or contact dynamical system consists of a topological Hamiltonian or contact isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying symplectic or contact manifold, and a topological conformal factor (a continuous function) in the contact case. By definition, this Hamiltonian function is the limit of a sequence of smooth Hamiltonian functions with respect to the usual Hofer metric. Moreover, the corresponding sequence of smooth Hamiltonian or contact isotopies converges uniformly to the above continuous isotopy, and their smooth conformal factors converge uniformly to the above continuous function associated to the limit isotopy. In other words, we consider limits of smooth Hamiltonian and contact dynamical systems with respect to a metric that combines topological and dynamical information. Similarly, topological automorphisms of a symplectic or contact structure or form are C^0-limits of symplectic or contact diffeomorphisms (together with their conformal factors in the contact case).
A topological Hamiltonian or contact isotopy is not generated by a vector field, and may not even be Lipschitz continuous; nonetheless, it is uniquely determined by its associated topological Hamiltonian function, and in turn, in the contact case it determines uniquely its topological conformal factor. Composition and inversion can therefore be defined as in the smooth case, and the usual transformation law continues to hold. Conversely, every topological Hamiltonian or contact isotopy possesses a unique topological Hamiltonian function. The topological automorphism groups of a contact structure and a contact form exhibit surprising rigidity properties analogous to the well-known Eliashberg-Gromov rigidity in the case of a symplectic structure.
The uniqueness theorems for topological Hamiltonian and contact dynamical systems, rigidity, the group structures, and the transformation law, provide ample evidence that Hamiltonian and contact dynamics are in fact topological theories, and a priori smooth invariants can be extended to topological Hamiltonian and contact dynamical systems. The study of these topological theories is to a large extend motivated by physical considerations and by various continuity phenomena already present in their smooth counterparts, and in turn, they have applications to smooth Hamiltonian and contact dynamics as well as to topological dynamics and low-dimensional topology. For example, one can answer important questions of V. I. Arnold regarding the continuity of the helicity of a volume preserving isotopy, and its behavior under conjugation by volume preserving homeomorphisms, provided the isotopy and the transformation can be described as lifts from a manifold one dimension lower. The relation between topological Hamiltonian dynamics and topological contact dynamics is … [More]
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125841.92/warc/CC-MAIN-20170423031205-00172-ip-10-145-167-34.ec2.internal.warc.gz
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CC-MAIN-2017-17
| 3,866 | 8 |
http://earthguide.ucsd.edu/virtualmuseum/Glossary_Climate/newtonsi.html
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math
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Newton, Sir Isaac -
(1642-1727): English physicist and mathematician. He was also a bible scholar and was given the post of “Master of the British Mint” in 1695. Considered by many to be one of the greatest scientists who ever lived, Newton discovered the laws of gravity and explored the nature of light. He invented calculus independently of Gottfried Leibniz (1646-1716). He formulated the basic laws of physics, fundamental to celestial mechanics, which describe the
gravitational attraction between two bodies (i.e. product of masses divided by square of distance). One of his most important books was Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy first published in 1687. This text contains the three laws of motion: a body at rest or in uniform motion will retain that state unless a force is applied; force equals the mass of a body multiplied by the acceleration produced by application of the force; if a body exerts a force on another, that body exerts an equal and opposite force on the first body. Newton used these laws to explain a wide variety of motions, from the Moon and planets to tides. Newton's laws are an integral part of describing all motions on Earth, including winds and ocean currents.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583515029.82/warc/CC-MAIN-20181022092330-20181022113830-00292.warc.gz
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CC-MAIN-2018-43
| 1,264 | 3 |
https://sonichours.com/how-many-millimeters-are-in-9-4-centimeters/
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math
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To know how many millimeters are in 9.4, you need to multiply the length in centimeters by 0.1. The formula for this is 9.4 * 0.1, and you’ll end up with 94 millimeters. To make this conversion even simpler, we’ve put together a visual chart that you can use on any screen. It shows the relative value of inches and centimeters in different colors and lengths.
In the metric system, a millimeter is equal to 1/1000 metre (one-thousandth of a kilometer). In engineering, a millimeter is the standard measurement for length and is equivalent to 25.4 millimeters. In this way, you can easily convert a centimeter into any other length. The following table shows the millimeter to centimeter conversion.
Nine-four centimeters is the same as nine-four inches. In other words, you get 94 millimeters for every 9.4 centimeters you have. You can use a centimeter to convert a metric measurement. If you need to convert millimeters to inches, you can use a conversion calculator to help you. Once you know the metric conversion formula, you can use it to calculate any other lengths.
The metric system uses millimeters as the unit of length. One millimeter is equal to 1/1000 of an inch, and one inch is equal to 255.4 mm. This rule of thumb works for both metric and imperial measurements. For example, a ten-centimeter-long length would be a ten-thousand-four-thousand-four-centimeter-inch length.
The metric system also uses the millimeter as the unit of length. A centimeter is a 1/1000 of a kilometer. A millimeter is one 1/1000 of an inch. An ounce is twenty-five mm. This means that nine-four centimeters is equal to 0.9 inches. It is very important to know this ratio to calculate a conversion.
To convert 9.4 centimeters to inches, you need to multiply the length in inches by 0.1. This is done by using the metric scale. The metric system also uses a millimeter. Its weight is equivalent to one-eighty-three pounds. This is the shortest unit of length. Once you’ve calculated the mass of a hammer, divide it by the weight in centimeters.
Using the metric scale, the centimeter is the smallest unit of length in the metric system. The metric system uses the gram as its base unit. In other words, a centimeter is 1E-three meter. In the metric system, a millimeter is equal to twenty-four millimeters. In the metric system, an inch is equal to a quarter of a foot.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100568.68/warc/CC-MAIN-20231205204654-20231205234654-00610.warc.gz
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CC-MAIN-2023-50
| 2,369 | 7 |
http://www.wyzant.com/Montgomery_MA_calculus_tutors.aspx
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math
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Springfield, MA 01109
Great asian math tutor is here
...I am from China and I have been studying math at the University of Hartford for past 3 years. I had been a successful tutor in our university's math lab, covering topics including differential equation, probability theory, calculus
II&III, and I've always been the...
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s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997883858.16/warc/CC-MAIN-20140722025803-00166-ip-10-33-131-23.ec2.internal.warc.gz
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CC-MAIN-2014-23
| 323 | 4 |
https://www.txstate.edu/mathworks/about/news/2022-Kodosky-Grant.html
|
math
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Kodosky Foundation Grants Mathworks $4,000
Mathworks is pleased to announce that we have received a donation of $4,000 from the Kodosky Foundation to support our 2022 summer math camps. This donation helps to ensure that all students can attend camp regardless of their financial status.
Half-Day Junior Summer Math Camp is a two-week commuter program for grades 3-8. During this camp students learn math in a fun way, using the Mathworks Math Quest curriculum written by Texas State University faculty. The Residential Junior Summer Math Camp is a two-week program for grades 6-8, and the Honors Summer Math Camp is a six-week residential camp for outstanding high school students. The Mathworks residential camps are advance level programs that expose students to higher level math concepts. These camps are taught by university faculty and attract students from all over the world.
Mathworks is a center for innovation in mathematics education at Texas State University with core programs of Summer Math Camps, Teacher Training, and Curriculum Development. The Mathworks Junior Summer Math Camp and Honors Summer Math Camp are nationally recognized as two of the most outstanding programs in the country. For more information about the camps and other Mathworks programs, see www.txstate.edu/mathworks.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103344783.24/warc/CC-MAIN-20220627225823-20220628015823-00515.warc.gz
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CC-MAIN-2022-27
| 1,305 | 4 |
https://biology.stackexchange.com/questions/28618/relationship-between-nerves-and-axons
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math
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I just wanted to get a realistic viewpoint of our nervous system. I understand arteries and veins, but I wanted to know how similar our nervous system is to that?
I understand we have neurons (please correct me if I am wrong) all over the surface of body. Whenever we feel a touch a neuron fires a response, and that response travels through axons (myelin sheath).
My main question is what a nerve exactly is. Is it a long axon? How many axons (same thing as neuron body?) are in a nerve? I am sure it depends on different nerves.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816875.61/warc/CC-MAIN-20240414064633-20240414094633-00849.warc.gz
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CC-MAIN-2024-18
| 530 | 3 |
https://nrich.maths.org/public/leg.php?code=31&cl=1&cldcmpid=7301&setlocale=en_US
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math
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A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole?
Use these head, body and leg pieces to make Robot Monsters which are different heights.
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
Find all the numbers that can be made by adding the dots on two dice.
As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?
Can you find 2 butterflies to go on each flower so that the numbers on each pair of butterflies adds to the same number as the one on the flower?
Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?
Can you each work out the number on your card? What do you notice? How could you sort the cards?
Investigate the different distances of these car journeys and find out how long they take.
In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?
I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to?
This is an adding game for two players.
Investigate what happens when you add house numbers along a street in different ways.
Leah and Tom each have a number line. Can you work out where their counters will land? What are the secret jumps they make with their counters?
There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket?
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
These two group activities use mathematical reasoning - one is numerical, one geometric.
Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?
These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?
In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?
If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?
Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest?
In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?
Twizzle, a female giraffe, needs transporting to another zoo. Which route will give the fastest journey?
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.
I throw three dice and get 5, 3 and 2. Add the scores on the three dice. What do you get? Now multiply the scores. What do you notice?
Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?
Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.
Vera is shopping at a market with these coins in her purse. Which things could she give exactly the right amount for?
Sam got into an elevator. He went down five floors, up six floors, down seven floors, then got out on the second floor. On what floor did he get on?
Woof is a big dog. Yap is a little dog. Emma has 16 dog biscuits to give to the two dogs. She gave Woof 4 more biscuits than Yap. How many biscuits did each dog get?
There were 22 legs creeping across the web. How many flies? How many spiders?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Can you work out how many flowers there will be on the Amazing Splitting Plant after it has been growing for six weeks?
Can you substitute numbers for the letters in these sums?
In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?
Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?
The value of the circle changes in each of the following problems. Can you discover its value in each problem?
Can you score 100 by throwing rings on this board? Is there more than way to do it?
Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7?
Where can you draw a line on a clock face so that the numbers on both sides have the same total?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?
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s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824618.72/warc/CC-MAIN-20171021062002-20171021082002-00863.warc.gz
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CC-MAIN-2017-43
| 6,285 | 50 |
https://www.engineeringexcelspreadsheets.com/2011/03/
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math
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Where to Find Partially Full Pipe Flow Calculations Spreadsheets
To obtain Excel spreadsheets for partially full pipe flow calculations, click here to visit our spreadsheet store for partially full pipe flow calculations spreadsheets. Read on for information about Excel spreadsheets that can be used as partially full pipe flow calculators.
The Manning equation can be used for flow in a pipe that is partially full, because the flow will be due to gravity rather than pressure. the Manning equation [Q = (1.49/n)A(R2/3)(S1/2) for (U.S. units) or Q = (1.0/n)A(R2/3)(S1/2) for (S.I. units)] applies if the flow is uniform flow For background on the Manning equation and open channel flow and the conditions for uniform flow, see the article, “Manning Equation/Open Channel Flow Calculations with Excel Spreadsheets.”
Direct use of the Manning equation as a partially full pipe flow calculator, isn’t easy, however, because of the rather complicated set of equations for the area of flow and wetted perimeter for partially full pipe flow. There is no simple equation for hydraulic radius as a function of flow depth and pipe diameter. As a result graphs of Q/Qfull and V/Vfull vs y/D, like the one shown at the left are commonly used for partially full pipe flow calculations. The parameters, Q and V in this graph are flow rate an velocity at a flow depth of y in a pipe of diameter D. Qfull and Vfull can be conveniently calculated using the Manning equation, because the hydraulic radius for a circular pipe flowing full is simply D/4.
With the use of Excel formulas in an Excel spreadsheet, however, the rather inconvenient equations for area and wetted perimeter in partially full pipe flow become much easier to work with. The calculations are complicated a bit by the need to consider the Manning roughness coefficient to be variable with depth of flow as discussed in the next section.
Is the Manning Roughness Coefficient Variable for Partially Full Pipe Flow Calculations?
Using the geometric/trigonometric equations discussed in the next couple of sections, it is relatively easy to calculate the cross-sectional area, wetted perimeter, and hydraulic radius for partially full pipe flow with any specified pipe diameter and depth of flow. If the pipe slope and Manning roughness coefficient are known, then it should be easy to calculate flow rate and velocity for the given depth of flow using the Manning Equation [Q = (1.49/n)A(R2/3)(S1/2)], right? No, wrong! As long ago as the middle of the twentieth century, it had been observed that measured flow rates in partially full pipe flow aren’t the same as those calculated as just described. In a 1946 journal article (ref #1 below), T. R. Camp presented a method for improving the agreement between measured and calculated values for partially full pipe flow. The method developed by Camp consisted of using a variation in Manning roughness coefficient with depth of flow as shown in the graph above.
Although this variation in Manning roughness due to depth of flow doesn’t make sense intuitively, it does work. It is well to keep in mind that the Manning equation is an empirical equation, derived by correlating experimental results, rather than being theoretically derived. The Manning equation was developed for flow in open channels with rectangular, trapezoidal, and similar cross-sections. It works very well for those applications using a constant value for the Manning roughness coefficient, n. Better agreement with experimental measurements is obtained for partially full pipe flow, however, by using the variation in Manning roughness coefficient developed by Camp and shown in the diagram above.
The graph developed by Camp and shown above appears in several publications of the American Society of Civil Engineers, the Water Pollution Control Federation, and the Water Environment Federation from 1969 through 1992, as well as in many environmental engineering textbooks (see reference list at the end of this article). You should beware, however that there are several online calculators and websites with equations for making partially full pipe flow calculations using the Manning equation with constant Manning roughness coefficient, n. The equations and Excel spreadsheets presented and discussed in this article use the variation in n that was developed by T.R. Camp.
Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe Less than Half Full
The parameters used in partially full pipe flow calculations with the pipe less than half full are shown in the diagram at the right. K is the circular segment area; S is the circular segment arc length; h is the circular segment height; r is the radius of the pipe; and θ is the central angle.
The equations below are those used, together with the Manning equation and Q = VA, in the partially full pipe flow calculator (Excel spreadsheet) for flow depth less than pipe radius, as shown below.
- h = y
- θ = 2 arccos[ (r – h)/r ]
- A = K = r2(θ – sinθ)/2
- P = S = rθ
The equations to calculate n/nfull, in terms of y/D for y < D/2 are as follows
- n/nfull = 1 + (y/D)(1/3) for 0 < y/D < 0.03
- n/nfull = 1.1 + (y/D – 0.03)(12/7) for 0.03 < y/D < 0.1
- n/nfull = 1.22 + (y/D – 0.1)(0.6) for 0.1 < y/D < 0.2
- n/nfull = 1.29 for 0.2 < y/D < 0.3
- n/nfull = 1.29 – (y/D – 0.3)(0.2) for 0.3 < y/D < 0.5
The Excel template shown below can be used as a partially full pipe flow calculator to calculate the pipe flow rate, Q, and velocity, V, for specified values of pipe diameter, D, flow depth, y, Manning roughness for full pipe flow, nfull; and bottom slope, S, for cases where the depth of flow is less than the pipe radius. This Excel spreadsheet and others for partially full pipe flow calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.
Excel Spreadsheet/Partially Full Pipe Flow Calculator for Pipe More than Half Full
The parameters used in partially full pipe flow calculations with the pipe more than half full are shown in the diagram at the right. K is the circular segment area; S is the circular segment arc length; h is the circular segment height; r is the radius of the pipe; and θ is the central angle.
The equations below are those used, together with the Manning equation and Q = VA, in the partially full pipe flow calculator (Excel spreadsheet) for flow depth more than pipe radius, as shown below.
- h = 2r – y
- θ = 2 arccos[ (r – h)/r ]
- A = πr2 – K = πr2 – r2(θ – sinθ)/2
- P = 2πr – S = 2πr – rθ
The equation used for n/nfull for 0.5 < y//D < 1 is: n/nfull = 1.25 – [(y/D – 0.5)/2]
An Excel spreadsheet like the one shown above for less than half full flow, and others for partially full pipe flow calculations, are available in either U.S. or S.I. units at a very low cost at www.engineeringexceltemplates.com.
1. Bengtson, Harlan H., Uniform Open Channel Flow and The Manning Equation, an online, continuing education course for PDH credit.
2. Camp, T.R., “Design of Sewers to Facilitate Flow,” Sewage Works Journal, 18 (3), 1946
3. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959.
4. Steel, E.W. & McGhee, T.J., Water Supply and Sewerage, 5th Ed., New York, McGraw-Hill Book Company, 1979
5. ASCE, 1969. Design and Construction of Sanitary and Storm Sewers, NY
6. Bengtson, H.H., “Manning Equation Partially Filled Circular Pipes,” An online blog article
7. Bengtson, H.H., “Partially Full Pipe Flow Calculations with Spreadsheets“, available as an Amazon Kindle e-book and as a paperback.
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http://peacockcandi.blogspot.com/2006/10/wow-wow-sauce_8105.html
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math
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(civilian, non-fulminating version)
# 2oz butter
# 1oz plain flour
# half pint stock*
# 1 tablespoon vinegar
# 1 teaspoon ready mixed English mustard
# 1 tablespoon mushroom ketchup or port
# 1 tablespoon finely chopped parsley
# 6 pickled walnuts, diced
Melt the butter in the pan over a low heat then stir in the flour and cook gently for 2 to 3 minutes.
Add the stock gently, stirring all the while to prevent lumps forming.When the sauce is smooth and creamy add the vinegar, mustard and ketchup (or port).
Simmer the whole until it reaches a consistency that you like and then stir in the chopped parsley and walnuts.
Heat through for another minute or so and serve hot.
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CC-MAIN-2018-30
| 675 | 13 |
http://www.ascd.org/publications/educational-leadership/apr14/vol71/num07/What-Does-Multiplying-Two-Candy-Bars-Really-Mean%C2%A2.aspx
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math
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April 2014 | Volume 71 | Number 7
Writing: A Core Skill
Jihwa Noh and Karen Sabey
When students write their own math word problems, teachers get immediate feedback about which concepts they do and don't understand.
How do you go about calculating 2/3 × 1/4? You may have multiplied the two numbers on top (namely, numerators) to get 2 and multiplied the two numbers on the bottom (namely, denominators) to get 12, so you would get the answer of 2/12 (or 1/6 if simplified). Or you may have begun by simplifying the 2 in 2/3 and the 4 in 1/4 and then followed the previously described method. If you got the correct answer, congratulate yourself for remembering the procedure.
Now we have a different question for you: Can you write a word problem in which you would calculate 2/3 × 1/4 to find the answer? Hmm, do you feel like you need to brush up on your math skills? Perhaps your teachers never asked you to write a word problem when you were in school.
You can use at least four different contextual situations for your word problem, each of which embodies a different interpretation for multiplication of fractions. First, there's looking at fraction multiplication as taking a part of a part. For example, using that same problem, "There was 1/4 of a pan of brownies left from yesterday's party. If you ate 2/3 of the leftover brownies, what fractional part of the pan of brownies would you have eaten?"
Second, you might make a scale drawing in which one foot is represented by 1/4 of an inch. To determine how long a 2/3-foot-long table would be on the drawing, you'd need to multiply 2/3 × 1/4.
Third, perhaps you want to determine the probability of rain on both days when there's a 66 percent chance of rain on Saturday (approximately 2/3) and a 25 percent chance on Sunday. You'd multiply the two numbers in fraction or decimal form (2/3 × 1/4).
Finally, you might wish to find the area of a rectangular region whose dimensions are 2/3 of a yard by 1/4 of a yard. You'd do that by multiplying 2/3 × 1/4.
In our study involving 140 college freshmen, only two students were able to write a word problem that correctly represented a given fraction multiplication problem. Fifty-seven (41 percent) wrote a problem that would be answered by operations other than multiplication, such as addition. Fifty-three (38 percent) didn't even attempt to write a word problem, or their responses contained no substantial information, making them unanalyzable. Not only was the students' mathematical understanding disappointing, but there also were language issues, such as spelling, semantic, and syntactical errors, and issues using inappropriate or unrealistic contexts.
The Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and the National Council of Teachers of Mathematics (2000) advocate using multiple methods that give students opportunities to learn mathematical ideas and demonstrate their understandings. By having students write word problems that encompass a variety of contextual situations, teachers gain insight into how students have interpreted a mathematical idea as well as their preferences for problem-solving strategies (National Mathematics Advisory Panel, 2008; Newton, 2008).
We want our students to make sense of the mathematics they're learning and solve problems in sense-making ways, rather than merely applying rules and formulas. In our study, one of the common errors that students made concerning the multiplication of fractions was the result of a misconception—that multiplication always makes numbers bigger.
One student wrote, "Tom has 2/3 of an apple and needs to make a big batch for a recipe by 5/7. How many apples will he have now?" In addition to the readability issue this problem presents, adjusting a recipe by 5/7 would not yield a bigger batch, but rather a smaller one. Having students write word problems gives teachers immediate feedback about student misconceptions as well as the opportunity to develop lesson plans to both address student weaknesses and bolster student strengths.
When students write their own word problems, they typically make use of situations and contexts with which they're familiar. This can make the problems far more meaningful and comprehensible (Barwell, 2003; Chapman, 2006). This can be particularly helpful for English language learners (ELLs), who may find word problems difficult because they lack appropriate background knowledge, such as knowledge about U.S. currency or American football rules. By having ELLs write their own word problems using situations familiar to them, as well as language they can manage, teachers can more easily assess their mathematical abilities.
There's a prevalent myth that ELLs cannot be successful in solving word problems until they're more fluent in English (Martiniello, 2008). Because of this misunderstanding, many teachers might limit the teaching of mathematics to computation exercises instead of engaging the students in problem-solving efforts using word problems. This approach yields missed opportunities for ELLs to work toward overcoming the language demands of mathematics.
As students share their word problems with the class and invite their peers to solve those problems, they're led into discussions, both in small groups and as part of a whole-class discussion, about the meaning of their problems and how best to solve them. These discussions give students practice using mathematics language, which both the Common Core State Standards in Mathematics and the National Council of Teachers of Mathematics emphasize as an essential component in learning mathematics. A problem-posing activity can bring in many forms of communication, such as writing, speaking, reading, and listening, which benefit not just ELLs but all students.
Mathematics language is semantically and syntactically specialized. Students may be familiar with the common uses of words like even, odd, and improper, but these words have a different meaning when used in mathematics. Sometimes the same mathematical word is used in more than one way within the field itself. The word square, for example, can refer to a shape and also to a number times itself (Rubenstein & Thompson, 2002). Also, mathematics language makes use of certain syntactic structures—such as greater than/less than, n times as much as, divided by as opposed to divided into, if/then, and so on (Chamot & O'Malley, 1994).
Student-written math problems help students connect the mathematics they're learning to other mathematical ideas and with ideas outside the mathematics classroom. These problems also help students understand how the theoretical language of mathematics and the everyday language of word problems are related.
When using fractions, it's important to clarify what the unit whole is. In the following example, the whole is unclear: "Bill has 2/3 of oranges left in his bag. Bill would like 5/7 more. How many oranges would Bill end up with?" It's unclear what whole is associated with the 5/7: the number of oranges that Bill currently has or the number of oranges that were in the bag originally. In either case, we can't answer the question because we don't know what the original number of oranges was. Asking "what fraction of a bag" instead of "how many oranges" would be more appropriate.
In the following example, the measurements the student used are unrealistic, although the problem is correct mathematically: "Jamie made a pan of brownies. The pan's length was 2/3cm and its width was 5/7cm. How big is the pan?" In addition, the word big is ambiguous because it could mean either the area or perimeter of the pan.
Also, a number of students merely translated the multiplication sign into words. For example, "Ben has 2/3 of his candy bar left, and Sally has 5/7 of her candy bar left. If you multiply their candy bars together, how much would they have?" But what does multiplying two candy bars really mean?
Teachers need to find ways of helping students overcome the difficulties they encounter in writing word problems. Let's say you ask students to write a word problem in which they need to do the calculation from the opening of this article—2/3 × 1/4—to find the answer. A student may respond with the following: Julie ate 1/4 of a pizza. Janet ate 2/3 more. How much pizza did Janet eat?
The first problem we encounter is with the words "2/3 more." Is this (a) 2/3 of a whole pizza, or is this (b) 2/3 of what Julie ate? If the meaning is (a), then the answer to the problem—How much pizza did Janet eat?—is 1/4 + 2/3, or 11/12 of a pizza. If the meaning is (b), then the answer is 1/4 + 2/3(1/4), or 5/12 of a pizza. Neither of these interpretations uses 2/3 × 1/4 to calculate the answer.
To help students gain conceptual understanding of the word problem in question, teachers can provide a visual representation. To illustrate the problem in a way that requires a calculation of 2/3 × 1/4, you'll need to start by restating the problem correctly: Julie ate 1/4 of a pizza. Janet ate 2/3 as much pizza as Julie did. How much pizza did Janet eat?
Draw a picture of a circle (pizza) cut into 4 equal pieces; one shaded piece represents the 1/4 pizza that Julie ate. Then divide that 1/4 into 3 equal pieces (each piece now represents 1/3 of 1/4, which is 1/12). Shading two of those pieces gives the answer of 2/12 (or simplified, 1/6), which is the correct answer to 2/3 × 1/4. Illustrating the problem with an image like this one can increase students' ability to write meaningful word problems.
Here are some things that teachers can do to help students write good word problems:
Under the new Common Core standards, mathematics instruction emphasizes conceptual understanding, procedural fluency, multiple approaches to and models of mathematical problems, and problems requiring analysis and explanation. Having students write, solve, and talk about their own word problems is an enjoyable way to integrate communication skills, including writing, into instruction while deepening students' mathematical knowledge.
Barwell, R. (2003). Working on word problems. Mathematics Teaching, 185, 6–8.
Chamot, A., & O'Malley, J. M. (1994). CALLA handbook: Implementing the cognitive academic language learning approach. MA: Addison-Wesley.
Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211–230.
Martiniello, M. (2008). Language and the performance of ELLs in math word problems. Harvard Educational Review, 78(2), 333–368.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington DC: Author.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
Newton, K. J. (2008). An extensive analysis of pre-service elementary teachers' knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.
Rubenstein, R. N., & Thompson, D. R. (2002). Understanding and supporting children's mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107–112.
Jihwa Noh is associate professor and Karen Sabey is assistant professor in the Department of Mathematics, University of Northern Iowa, Cedar Falls.
Copyright © 2014 by ASCD
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https://www.onlinemathlearning.com/solve-problems-fractions-illustrative-math.html
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math
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Lesson 16: Solving Problems Involving Fractions
Let’s add, subtract, multiply, and divide fractions.
Illustrative Math Unit 6.4, Lesson 16 (printable worksheets)
Lesson 16 Summary
The following diagram shows how to add, subtract, multiply, and divide both whole numbers and fractions.
Lesson 16.1 Operations with Fractions
Without calculating, order the expressions according to their values from least to greatest. Be prepared to explain or show your reasoning.
¾ + ⅔
¾ - ⅔
¾ · ⅔
¾ ÷ ⅔
Lesson 16.2 Situations with ¾ and ½
Here are four situations that involve ¾ and ½.
- Before calculating, decide if each answer is greater than 1 or less than 1.
- Write a multiplication equation or division equation for the situation.
- Answer the question. Show your reasoning. Draw a tape diagram, if needed.
- There was ¾ liter of water in Andre’s water bottle. Andre drank ½ of the water. How many liters of water did he drink?
- The distance from Han’s house to his school is ¾ kilometer. Han walked ½ kilometer. What fraction of the distance from his house to the school did Han walk?
- Priya’s goal was to collect ½ kilogram of trash. She collected ¾ kilogram of trash. How many times her goal was the amount of trash she collected?
- Mai’s class volunteered to clean a park with an area of ½ square mile. Before they took a lunch break, the class had cleaned ¾ of the park. How many square miles had they cleaned before lunch?
Lesson 16.3 Pairs of Problems
- Work with a partner to write equations for the following questions. One person should work on the questions labeled A1, B1, . . . , E1 and the other should work on those labeled A2, B2, . . . , E2.
A1. Lin’s bottle holds 3¼ cups of water. She drank 1 cup of water. What fraction of the water in the bottle did she drink?
B1. Plant A is 16/3 feet tall. This is 4/5 as tall as Plant B. How tall is Plant B?
C1. 8/9 kilogram of berries is put into a container that already has 7/3 kilogram of berries. How many kilograms are in the container?
D1. The area of a rectangle is 14½ sq cm and one side is 4½ cm. How long is the other side?
E1. A stack of magazines is 4⅔ inches high. The stack needs to fit into a box that is 2⅛ inches high. How many inches too high is the stack?
A2. Lin’s bottle holds 3¼ cups of water. After she drank some, there were 1½ cups of water in the bottle. How many cups did she drink?
B2. Plant A is 16/3 feet tall. Plant C is 4/5 as tall as Plant A. How tall is Plant C?
C2. A container with 8/9 kilogram of berries is 2/3 full. How many kilograms can the container hold?
D2. The side lengths of a rectangle are 4½ cm and 2⅖ cm. What is the area of the rectangle?
E2. A stack of magazines is 4⅖ inches high. Each magazine is ⅖-inch thick. How many magazines are in the stack?
- Trade papers with your partner, and check your partner’s equations. If there is a disagreement about what an equation should be, discuss it until you reach an agreement.
- Your teacher will assign 2–3 questions for you to answer. For each question:
a. Estimate the answer before calculating it.
b. Find the answer, and show your reasoning.
Lesson 16.4 Baking Cookies
Mai, Kiran, and Clare are baking cookies together. They need ¾ cup of flour and ½ cup of butter to make a batch of cookies. They each brought the ingredients they had at home.
- Mai brought 2 cups of flour and ¼ cup of butter.
- Kiran brought 1 cup of flour and ½ cup of butter.
- Clare brought 1¼ cups of flour and ¾ cup of butter.
If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of cookies can they make? Explain your reasoning.
Lesson 16 Practice Problems
- An orange has about ¼ cup of juice. How many oranges are needed to make 2½ cups of juice? Select all equations that represent this question.
A. ? · ¼ = 2½
B. ¼ ÷ 2½ = ?
C. ? ÷ 2½ = ¼
D. 2½ ÷ ¼ = ?
- Mai, Clare, and Tyler are hiking from a parking lot to the summit of a mountain. They pass a sign that gives distances.
- Parking lot: ¾ mile
- Summit: 1½ miles
Mai says: “We are one third of the way there.” Clare says: “We have to go twice as far as we have already gone.” Tyler says: “The total hike is three times as long as what we have already gone.”
Can they all be correct? Explain how you know.
3. Priya’s cat weighs 5½ pounds and her dog weighs 8¼ pounds. Estimate the missing number in each statement before calculating the answer. Then, compare your answer to the estimate and explain any discrepancy.
The cat is _______ as heavy as the dog.
Their combined weight is _______ pounds.
The dog is _______ pounds heavier than the cat.
4. Before refrigerators existed, some people had blocks of ice delivered to their homes. A delivery wagon had a storage box in the shape of a rectangular prism that was feet by 6 feet by 6 feet. The cubic ice blocks stored in the box had side lengths feet. How many ice blocks fit in the storage box?
5. Fill in the blanks with 0.001, 0.1, 10, or 1000 so that the value of each quotient is in the correct column.
close to 1/100
- ____ ÷ 9
- 12 ÷ ____
close to 1
- ____ ÷ 0.12
- ⅛ ÷ ____
greater than 100
- ____ ÷ ⅓
- 700.7 ÷____
- A school club sold 300 shirts. 31% were sold to fifth graders, 52% were sold to sixth graders, and the rest were sold to teachers. How many shirts were sold to each group—fifth graders, sixth graders, and teachers? Explain or show your reasoning.
- Jada has some pennies and dimes. The ratio of Jada’s pennies to dimes is 2 to 3.
a. From the information given above, can you determine how many coins Jada has?
b. If Jada has 55 coins, how many of each kind of coin does she have?
c. How much are her coins worth?
The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
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https://physics.stackexchange.com/questions/593131/how-do-we-know-that-1s-is-the-ground-state-of-the-helium-atom
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math
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Let $\psi=a_1\phi(1s(2) \ ^1S)+a_2\phi(1s(1)2s(1) \ ^1S)+a_3\phi(2s(2) \ ^1S) +... $ be a state of the helium atom. Applying variationally calculus we can found the energy expectation value of this state is almost exact to the experimental value. Is by this comparation that we know that the total angular momentum $J$ of the helium atom is zero or can we proof it theoretically alone, or experimentally alone ?
The answer to this question can be given with a much simpler explanation. Hund's rules state that:
1)For a given electron configuration, the term with maximum multiplicity has the lowest energy. The multiplicity is equal to $2S+1$ , where $S$ is the total spin angular momentum for all electrons. The multiplicity is also equal to the number of unpaired electrons plus one. Therefore, the term with lowest energy is also the term with maximum $S$, and maximum number of unpaired electrons.
2)For a given multiplicity, the term with the largest value of the total orbital angular momentum quantum number $L$ , has the lowest energy.
3)For a given term, in an atom with outermost subshell half-filled or less, the level with the lowest value of the total angular momentum quantum number $J$ (for the operator $J=L+S$ lies lowest in energy. If the outermost shell is more than half-filled, the level with the highest value of $J$ is lowest in energy.
Since the total spin is 0 and the angular momentum is 0, you have the ground state that you have written.
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https://www.pgrmseducation.com/ncert-solutions-for-class-6-maths-chapter-12-exercise-12-3/
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math
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Ncert Solutions for Class 6 Maths Chapter 12 Ratio and Proportion Exercise 12.3:-
Exercise 12.3 Class 6 maths NCERT solutions Chapter 12 Ratio And Proportion pdf download:-
Ncert Solution for Class 6 Maths Chapter 11 Ratio And Proportion Exercise 12.2 Tips:-
Consider the following situations:
Two friends Reshma and Seema went to the market to purchase
notebooks. Reshma purchased 2 notebooks for ` 24. What is the
price of one notebook?
A scooter requires 2 litres of petrol to cover 80 km. How many
litres of petrol is required to cover 1 km?
These are examples of the kind of situations that we face
in our daily life. How would you solve these?
Reconsider the first example: Cost of 2 notebooks is 24.
Therefore, cost of 1 notebook = 24 2 = 12.
Now, if you were asked to find the cost of 5 such notebooks. It would be
= ` 12 × 5 = ` 60
Reconsider the second example: We want to know how many litres are
needed to travel 1 km.
For 80 km, petrol needed = 2 litres.
Therefore, to travel 1 km, petrol needed =
80 = 1
Now, if you are asked to find how many litres of petrol are required to cover
Then petrol needed = 1
×120 litres = 3 litres.
The method in which first we find the value of one unit and then the
value of the required number of units is known as the Unitary Method.
What have we discussed?
1. For comparing quantities of the same type, we commonly use the method of
taking the difference between the quantities.
2. In many situations, a more meaningful comparison between quantities is made
by using division, i.e. by seeing how many times one quantity is to the other
quantity. This method is known as comparison by ratio.
For example, Isha’s weight is 25 kg and her father’s weight is 75 kg. We
say that Isha’s father’s weight and Isha’s weight are in the ratio 3 : 1.
3. For comparison by ratio, the two quantities must be in the same unit. If they are not,
they should be expressed in the same unit before the ratio is taken.
4. The same ratio may occur in different situations.
5. Note that the ratio 3: 2 is different from 2: 3. Thus, the order in which quantities are
taken to express their ratio is important.
6. A ratio may be treated as a fraction, thus the ratio 10 : 3 may be treated as10/3
7. Two ratios are equivalent if the fractions corresponding to them are equivalent. Thus,
3:2 is equivalent to 6:4 or 12:8.
8. A ratio can be expressed in its lowest form. For example, ratio 50:15 is treated as 50/15 in its lowest form50/15 =10/3. Hence, the lowest form of ratio 50 : 15 is 10 : 3.
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https://researchrepository.wvu.edu/etd/1190/
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math
|
Date of Graduation
Statler College of Engineering and Mineral Resources
Lane Department of Computer Science and Electrical Engineering
The discovery of fractal phenomenon in computer-related areas such as network traffic flow leads to the hypothesis that many computer resources display fractal characteristics. The goal of this study is to apply fractal analysis to computer memory usage patterns. We devise methods for calculating the Holder exponent of a time series and calculating the fractal dimension of a plot of a time series. These methods are then applied to memory-related data collected from a Unix server. We find that our methods for calculating the Holder exponent of a time series yield results that are independently confirmed through calculation of the fractal dimension of the time series, and that computer memory use does indeed display multifractal behavior. In addition, it is hypothesized that this multifractal behavior may be useful in making certain predictions about the future behavior of an operating system.
Crowell, Jonathan Browning, "Multifractal analysis of memory usage patterns" (2001). Graduate Theses, Dissertations, and Problem Reports. 1190.
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| 1,183 | 5 |
https://mindrightdetroit.com/interesting/what-are-descriptive-statistics.html
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math
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What do you mean by descriptive statistics?
Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread).
What are the four types of descriptive statistics?
There are four major types of descriptive statistics:
- Measures of Frequency: * Count, Percent, Frequency.
- Measures of Central Tendency. * Mean, Median, and Mode.
- Measures of Dispersion or Variation. * Range, Variance, Standard Deviation.
- Measures of Position. * Percentile Ranks, Quartile Ranks.
What is descriptive statistics in research?
Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Descriptive statistics are typically distinguished from inferential statistics. With descriptive statistics you are simply describing what is or what the data shows.
What are the five descriptive statistics?
There are a variety of descriptive statistics. Numbers such as the mean, median, mode, skewness, kurtosis, standard deviation, first quartile and third quartile, to name a few, each tell us something about our data.
What are the two major types of descriptive statistics?
Measures of central tendency and measures of dispersion are the two types of descriptive statistics. The mean, median, and mode are three types of measures of central tendency. Inferential statistics allow us to draw conclusions from our data set to the general population.
What is the importance of descriptive statistics?
Descriptive statistics are very important because if we simply presented our raw data it would be hard to visualize what the data was showing, especially if there was a lot of it. Descriptive statistics therefore enables us to present the data in a more meaningful way, which allows simpler interpretation of the data.
What are the three types of descriptive statistics?
The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.
- Univariate statistics summarize only one variable at a time.
- Bivariate statistics compare two variables.
- Multivariate statistics compare more than two variables.
How do you write the results of descriptive statistics?
Interpret the key results for Descriptive Statistics
- Step 1: Describe the size of your sample.
- Step 2: Describe the center of your data.
- Step 3: Describe the spread of your data.
- Step 4: Assess the shape and spread of your data distribution.
- Compare data from different groups.
How do you do descriptive statistics?
To generate descriptive statistics for these scores, execute the following steps.
- On the Data tab, in the Analysis group, click Data Analysis.
- Select Descriptive Statistics and click OK.
- Select the range A2:A15 as the Input Range.
- Select cell C1 as the Output Range.
- Make sure Summary statistics is checked.
- Click OK.
How do you show descriptive statistics?
Choose Stat > Basic Statistics > Display Descriptive Statistics.
What are the limitations of descriptive statistics?
Descriptive statistics are limited in so much that they only allow you to make summations about the people or objects that you have actually measured. You cannot use the data you have collected to generalize to other people or objects (i.e., using data from a sample to infer the properties/parameters of a population).
Is Anova a descriptive statistics?
2. Descriptive statistics: Summarization of a collection of data in a clear and understandable way. One-way ANOVA stands for Analysis of Variance Purpose: Extends the test for mean difference between two independent samples to multiple samples.
What are the major types of statistics?
The two major areas of statistics are known as descriptive statistics, which describes the properties of sample and population data, and inferential statistics, which uses those properties to test hypotheses and draw conclusions.
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http://www.freepatentsonline.com/article/American-Journal-Applied-Sciences/182424954.html
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math
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Several research and industrial applications concentrated their
efforts on providing simple and easy control algorithms to cope with the
increasing complexity of the controlled processes/systems (1). The
design method for a controller should enable full flexibility in the
modification of the control surface (2). The systems involved in
practice are, in general, complex and time variant, with delays and
nonlinearities, and often with poorly defined dynamics. Consequently,
conventional control methodologies based on linear system theory have to
simplify/linearize the nonlinear systems before they can be used, but
without any guarantee of providing good performance. To control
nonlinear systems satisfactorily, nonlinear controllers are often
developed. The main difficulty in designing nonlinear controllers is the
lack of a general structure (3). In addition, most linear and nonlinear
control solutions developed during the last three decades have been
based on precise mathematical models of the systems. Most of those
systems are difficult/impossible to be described by conventional
mathematical relations, hence, these model-based design approaches may
not provide satisfactory solutions (4). This motivates the interest in
using FLC; FLCs are based on fuzzy logic theory (5) and employ a mode of
approximate reasoning that resembles the decision making process of
humans. The behavior of a FLC is easily understood by a human expert, as
knowledge is expressed by means of intuitive, linguistic rules.
In contrast with traditional linear and nonlinear control theory, a
FLC is not based on a mathematical model and is widely used to solve
problems under uncertain and vague environments, with high
nonlinearities (6), (7). Since their advent, FLCs have been implemented
successfully in a variety of applications such as insurance and robotics
(8), (9), (10), (11). Fuzzy logic provides a certain level of artificial
intelligence to the conventional PID controllers. Fuzzy PID controllers
have self-tuning ability and on-line adaptation to nonlinear, time
varying, and uncertain systems Fuzzy PID controllers provide a promising
option for industrial applications with many desirable features.
MATERIALS AND METHODS
Fuzzy Controllers includes in their structure the following main
A. Fuzzification: Enabling the input physical signal to use the
rule base, the approach is using membership functions. Four membership
functions are given for the signals e and e in Fig. 1.
B. Programmable Rule Base:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To implement the FLC on a digital computer according to the
u(t) = u(kT) and u(t+) = u((/k+1)T)
Where, T is the sampling time. The following rule base is applied
u(t) = u(kT) and u(t+) = u((k+ 1)T)
Where, T is the sampling time. The following rule base is applied
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where, e (kT) [approximately equal to] l/T [e(k-1)T)], with initial
y(0) = 0, e(-T) = e(0) = r - y(0),
e(0) = 1/T[e(0) - e(-T)] = 0
C. Defuzzification: Select membership functions for the different
control outputs from the rule base
In Figure 2 typical membership functions for u is given. The
overall control signal, u, is generated by a weighted average formula:
u(k + 1)T) = [[[N.summation over (i =
1)][[mu].sub.i][u.sub.i](kT)]/[[N.summation over (i =
1)][[mu].sub.i]]].([[mu].sub.i][greater than or equal
Where control outputs [u.sub.i] (kT), i = 1, N=8 are from the rule
D. Discretization of Conventional PID Controllers:
Digitization of the conventional analog PID controllers by:
S = [2/T][[z-1]/[z+1]]
Where, T > 0 is the sampling time for the PI controller, in Fig.
1 the block diagram for PI digital controller is given:
u(nT) = u(nT - T) + T[DELTA]u(nT) [DELTA]u(nT) = [[~.K].sub.p](nT)
Modeling of the controlling unit: As an example, consider the
voltage raising type-pulse controller. The detailed characteristics of
which are given in (12). The equivalent circuit in view of parasitic
parameters of filtering elements is shown in Fig 4.
Fig.4: Equivalent scheme of boost-converter
Similar structure may be considered as a dynamic system with
external disturbance, in particular, periodic. Using state variables,
the system may be described as
[[dY]/[dt]] = A([S.sub.f])Y + b (1)
[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[S.sub.f] is the pulse function which describes a state of the
switch on the specified period of regulation. This function may be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Where, T is the period, [t.sub.k]-the moment of transition of the
switch from one state to another on the specified period of regulation.
As an initial parameters of the model, the range of variation for
the input voltage [U.sub.in] are set with triple overlapping from 20 V
up to 60 V, the range of variation of target resistance [R.sub.0] with
tenfold overlapping from 100 Ohm up to 1000 Ohm and the parasitic
parameters of elements of the filter which define the losses and quality
factor, accordingly, for inductance L = 2 mH; capacitance C = 100
[micro]F; [R.SUB.L] = 0,7 Ohm and [R.sub.c] = 0,2 Ohm.
The block diagram (see Fig.5) of the generalized indistinct
controller consists of four elements (13:
(1.) 1 Fuzzification block, transforming input physical values
[y.sub.i] into corresponding linguistic variables
(2.) Knowledge base, containing rules table for logic output block;
(3.) Logic output block, transforming input linguistic variables
into output with some belonging functions Con;
(4.) Defuzzification block, transforming output linguistic
variables into physical control influence.
Figure 6 shows the structure of P-type a fuzzy controller. In this
case, the error of regulation [epsilon] may be taken as the input
information. The output information is the signal of the relative
duration of conducting state of the switch Con = [t.sub.k]/T-(k-1). The
structure of PI Fuzzy controller is shown in Fig. 7 (13). The input
variables of this controller are, accordingly, the error of regulation
[epsilon] and its derivative [epsilon]. The output is the gain of
relative duration of the switch conducting state [delta] Con. The
membership functions of the input linguistic variables are shown if Fig.
It is expedient to divide a range of values of the normalized input
variables into five linguistic terms: negative big (NB), negative
small (NS), zero equal (ZE), positive small (PS) and positive big (PB).
With the application of indistinct logic, the logic choice for a P-type
controller can be obtained on the basis of table -1 (the definition
rules of the normalized error of regulation). The specified table is
filled on the basis of the following logic expression:
[epsilon] is Ai, then [Con.sub.k] is [C.sub.j], (3)
Where, [A.sub.i], B-terms of indistinct variables, [C.sub.j] - the
centre of j- accessory function.
Calculation of output signal Con of P-type controller is carried
out according to the following equation:
[Con.sub.k] = [[[n.summation over (j =
1)][[mu].sub.j](k[U.sub.in][epsilon])[C.sub.j]]/[[n.summation over (j =
Where, k[U.sub.in] is the weighting factor which normalizes the
input error [epsilon] to the unit.
The logic choice for the PI controllers with the application of
indistinct logic can be lead on the basis of table-2 (the definition
rules for the normalized error of regulation). The specified table is
filled on the basis of following logic expression:
[epsilon] is [A.sub.i] and )[EPSILON] is [B.sub.i], then
[Con.sub.k] is [Cj. (5)
Calculation for the target signal Con is carried out according to
the following equation:
[Con.sub.k]= [Con.sub.K-1] + 0[delta] [Con.sub.K]. (6)
Where, * * is a weighting factor which normalizes the target value
Con to unity.
[delta]Co[n.sub.k] = [[[n.summation over (j =
1)][[mu].sub.j](y)[C.sub.j]]/[[n.summation over (j -
Where, y - The input linguistic variable. The next values (0.1,
0.2, 0.3) of 0-coefficient were used when indistinct PI-regulator was
Comparison for quality parameters of P and PI controllers: The
following values were taken for comparison: [U.Sub.ref] = 3; [BETA]
=0,04, k[U.sub.in]: 0,25; 0,5; 1,0; 2,0; 4,0; 0: 0,1; 0,2; 0,3. The
Simulation of the structure of fig. 4 allows defining the value of the
static regulation error > and the values of overcorrection 8. For
that, it was necessary to vary the parameters of an input voltage in the
above-mentioned range and the factor of error scaling k[U.sub.in] The
results given in tables 3, 4 are obtained at a value of loading
resistance [R.sub.0] = 300 Ohm. It is found that with the increasing of
error scaling factor k[U.sub.in], the static error is decreased and the
overregulation is increased. The value of static error was defined for
the input voltage [U.sub.in] = 60 V only, quasiperiodic oscillations
were observed for other values of the input voltage. The estimation of
the specified parameters of the controller structure of Fig. 7
isn't given, as it is practically static (>[approximately equal
to] 0,1 %) with a periodic transient.
Two-parametrical diagrams of synchronous mode existence areas are
given for the structures of controllers on Fig. 6 and Fig. 7 accordingly
in Fig. 9 and Fig. 10 for two values of k[U.sub.in] and 0. The area of
existence of a synchronous mode is shaded. Time-domain diagrams of a
current [i.sub.1] flowing in the coil and voltage across the capacitor
[u.sub.c], are presented on Fig. 11 and Fig. 12, respectively. For a
fuzzy P-type controller a value of k[U.sub.in]=1 is chosen, and for
PI-type a value of * * = 0.1 is chosen.
Fuzzy logic provides a certain level of artificial intelligence to
the conventional controllers, leading to the effective fuzzy
controllers. Process loops that can benefit from a non-linear control
response are excellent candidates for fuzzy control. Since fuzzy logic
provides fast response times with virtually no overshoot. Loops with
noisy process signals have better stability and tighter control when
fuzzy logic control is applied.
P Fuzzy controller has smaller sensitivity to the change in the
input voltage, however, more sensitivity is observed to load changes.
PI- Fuzzy controller has less sensitivity to load changes, where, higher
sensitivity to the change of the input voltage is observed.
Analysis of transient and static error of regulation has shown
advantage of an indistinct PI- controller for the output voltage over
the P-type fuzzy controller.
P Fuzzy controller has faster transient as compared to PI
controller, while, transient for PI Fuzzy controller is almost periodic.
(1.) Verbruggen, H. B. and Bruijn, P. M., 1997. Fuzzy control and
conventional control: What is (And Can Be) the Real Contribution of
Fuzzy Systems Fuzzy Sets Systems, Vol. 90, 151-160.
(2.) Kowalska, T. O., Szabat, K. and Jaszczak, K., 2002. The
Influence of Parameters and Structure of PI-Type Fuzzy-Logic Controller
on DC Drive System Dynamics, Fuzzy Sets and Systems, Vol. 131, 251-264.
(3.) Ahmed, M. S., Bhatti, U. L., Al-Sunni, F. M. and El-Shafei,
M., 2001. Design of a Fuzzy Servo-Controller, Fuzzy Sets and Systems,
vol. 124: 231-247.
(4.) Zilouchian, A., Juliano, M., Healy, T., 2000. Design of Fuzzy
Logic Controller for a Jet Engine Fuel System, Control and Engineering
Practices, Vol. 8: 873-883.
(5.) Zadeh, L. A., 1965. Fuzzy sets, Information Control, Vol. 8,
(6.) Liu, B. D., 1997. Design and Implementation of the Tree-Based
Fuzzy Logic Controller, IEEE Transactions on Systems, Man, and
Cybernetics, Part B: Cybernetics., Vol.27, No. 3, 475-487.
(7.) Zhiqiang, G., 2002. A Stable Self-Tuning Fuzzy Logic Control
System for Industrial Temperature Regulation, IEEE 1886 Transactions on
Industry Applications. Vol.38, No.2: 414-424.
(8.) Shapiro, A. F., 2004. Fuzzy Logic in Insurance, Insurance:
Mathematics and Economics, Vol.35, No.2, 399-424.
(9.) Hayward, G. and Davidson, V., 2003. Fuzzy Logic Applications,
Analyst, Vol.128, 1304-1306.
(10.) Peri, V. M. and Simon, D., 2005. Fuzzy Logic Control for an
Autonomous Robot, North American Fuzzy Information Processing Society,
NAFIPS 2005 Annual Meeting, 337- 342.
(11.) Sofiane Achiche, Wang Wei, Zhun Fan and others 2007:
Genetically generated double-level fuzzy controller with a fuzzy
adjustment strategy. GECCO'07, July 7-11.
(12.) Severns R., Bloom G., 1985. Modern DC to DC switchmode
converter circuits. Van Nostrand Rainhold Co. NY.
(13.) So W. C., Tse C. K., 1996. Development of a Fuzzy Logic
Controller for DC/DC Converters: Design, Computer Simulation and
Experimental Evaluation. IEEE Trans. on Power Electronics, vol. PE11,
(14.) Parker D., 1987. Second order back propagation Implementation
of an optimal 0(n) approximation, IEEE Trans. on PA & MI.
Electrical Engineering Department. Faculty of Engineering Mutah
Abdullah I. Al-Odienat, Department of Electrical Engineering,
Faculty of Engineering, Mutah University
Table 1: The definition rules of [epsilon] for P controller
NB NS ZE PS PB
[C.sub.j] 0 0.225 0.45 0.675 0.9
Table 2: The definition rules of [EPSILON] for controller
NB NS ZE PS PB
* [epsilon] PB -0.3 -0.35 -0.45 -0.65 -1.0
PS 0.0 -0.1 -0.2 -0.35 -0.5
ZE 0.2 0.1 0.0 -0.1 -0.2
NS 0.5 0.35 0.2 0.1 0.0
NB 1.0 0.65 0.45 0.35 0.3
Table 3. The Static error of regulation >, %
V 0.25 0.5 1.0 2.0 4.0
20 37.30 27.5 17.0 9.6 *
30 18.10 13.3 8.5 4.8 *
40 1.80 1.3 0.8 0.5 *
50 -12.70 -9.2 -6.0 -3.5 *
60 -25.90 -18.7 -12,3.0 -7.3 -4.0
* - a quasiperiodic mode.
Table 4. An overcorrection 8, %
Uin, V 0.25 0.5 1.0 2.0 4.0
20 0.30 29.0 41.0 47.0 50.0
30 38.00 73.0 88.0 95.0 98.0
40 71.00 111.0 127.0 139.0 143.0
50 102.00 148.0 171.0 180.0 185.0
60 129.00 183.0 208.0 220.0 225.0
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https://www.jvejournals.com/article/17197
|
math
|
Stability and slow-fast oscillation in fractional-order Belousov-Zhabotinsky reaction with two time scales
Jingyu Hou1 , Xianghong Li2 , Jufeng Chen3
1, 2, 3Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China
Journal of Vibroengineering, Vol. 18, Issue 7, 2016, p. 4812-4823.
Received 23 May 2016; received in revised form 13 August 2016; accepted 6 September 2016; published 15 November 2016
The fractional-order Belousov-Zhabotinsky (BZ) reaction with different time scales is investigated in this paper. Based on the stability theory of fractional-order differential equation, the critical condition of Hopf bifurcation with two parameters in fractional-order BZ reaction is discussed. By comparison of the fractional-order and integer-order systems, it is found that they will behave in different stabilities under some parameter intervals, and the parameter intervals may become larger with the variation of fractional order. Furthermore, slow-fast effect is firstly studied in fractional-order BZ reaction with two time scales coupled, and the Fold/Fold type slow-fast oscillation with jumping behavior is found, whose generation mechanism is explained by using the slow-fast dynamical analysis method. The influences of different fractional orders on the slow-fast oscillation behavior as well as the internal mechanism are both analyzed.
Keywords: fractional-order system, slow-fast oscillation, Belousov-Zhabotinsky reaction, stability.
One of the best-studied chemical oscillation systems is the Belousov-Zhabotinsky (BZ) reaction, which was elucidated by 20 chemical equations to explain the reaction mechanism and was simplified to three-variable differential equations [1, 2]. Subsequently, many works about physical and chemical mechanism, numerical simulation and experimental research on BZ reaction appeared abundantly [3-5]. After the 1990s, the slow-fast oscillation was found in many chemical reactions [6, 7], the reason was that the catalyst could make the reaction process involve in different time scales with large gap. One of the classical slow-fast oscillation, i.e. bursting oscillation was observed by Strizhak in the experiment of BZ reaction . However, most of the researches on the slow-fast oscillation in chemical reaction were limited to numerical simulation and experimental investigation.
The better method to qualitatively reveal the bifurcation mechanism of the slow-fast phenomenon should be the slow-fast dynamical analysis method proposed by Rinzel . By use of the method, multiple time-scale systems had been developed in overwhelming growth in last three decades [10-12]. For example, Izhikevich provided all co-dimensional one bifurcation about slow-fast oscillation. Shilnikov had summarized the qualitative methods on Hindmarsh-Rose model, and presented the different bursting oscillations under Hopf bifurcation. Chumakov established a kinetic model of catalytic hydrogen oxidation, and studied the generation mechanism of slow-fast oscillation. Simpson analyzed bursting behavior in a stochastic piecewise system, and discussed the influence of noise on slow-fast oscillation. The bifurcation patterns about neuron system with three time scales were studied by Lu . Li and Bi proposed enveloping slow-fast analysis method to reveal the bursting oscillation mechanism of the system with periodic excitation. Zhang used differential inclusion theory to analyze the bifurcation mechanism of non-smooth systems with multiple time scales.
Actually slow-fast oscillation could be involved not only in the integer-order system but also in the fractional-order system. The reason lies in that fractional calculus would make the dynamical system more complicated, and those slow-fast oscillations have been found in various fields such as chemical, physical, mechanical, electrical, biological, economical and control engineering .
In the last three decades, fractional-order dynamical systems had been developed rapidly. Ahmad analyzed the fractional-order Wien-bridge oscillator associated with the periodic oscillations. Ahmed et al. proved the existence and uniqueness of solutions in fractional-order predator-prey system. Wang gave the similarities and differences of the feedbacks between fractional-order and integer-order SDOF linear damped oscillator. Shen et al. [24-29] studied analytically and numerically the primary resonances of van der Pol (VDP) and Duffing oscillators with fractional-order derivative, and investigated the dynamical of linear single degree-of-freedom oscillator with fractional-order derivative. Liu and Duan studied a fractional-order oscillator by Laplace transform and its complex inversion integral formula. Elouahab extended the nonlinear feedback control in fractional-order financial system to eliminate the chaotic behaviors. Many works on stochastic dynamical system with fractional-order derivative had also been done [32-34]. Furthermore, the classical Brusselator with fractional-order derivative was investigated by Gafiychuk and Li , in which the stability conditions and limit cycle were discussed.
However, to our best knowledge, the fractional-order BZ reaction is rarely studied, especially when the different time scales is involved. Here we will focus on the fractional-order BZ reaction with two time scales. The paper is organized as follow. In Section 2, the mathematical model of fractional-order BZ reaction is given and the bifurcation condition is discussed. The stabilities of integer-order and fractional-order systems are analyzed in details in Section 3. Then the slow-fast oscillation phenomenon and the corresponding generation mechanism are discussed by use of slow-fast dynamical analysis method in Section 4. At last, the main conclusions of this paper are made.
2. Mathematical model and bifurcation analysis
The photosensitive version of BZ reaction, i.e. Oregonator, was proposed by Seliguchi et al , and it was completed by the well-known reaction steps from the FKN mechanism , described as:
where is the light-excited molecule of The corresponding mathematical model was given in the form:
where , and represent the concentrations of , , and respectively, and , , and are the dimensionless constants related to the reaction rates. Because these parameters are closely related to the reaction condition such as temperature, pressure, feed rate, etc., the dynamical behaviors of reaction process caused by the parameter variation are very important.
The fractional-order version of BZ reaction could be established as:
in which is the operator of fractional derivative with the order . Obviously, Eq. (1) is a special case of Eq. (2). Here, we adopt Caputo’s definition as:
The equilibria of fractional-order BZ reaction can be obtained as:
The corresponding Jacobian matrix at the equilibrium is:
The stabilities of , and can be determined by the associated characteristic equation:
For convenience, some expressions are defined as follow:
Based on Shengjin’s formulas , a method to solve the univariate cubic equation, three real roots can exist in Eq. (3) for . While for , there are a single real root and a pair of conjugate complex roots, shown as:
The absolute values of arguments of the complex roots are expressed as:
Based on the stability theory of fractional-order differential equation , the stability condition for the equilibria of Eq. (2) should be:
In the range of , the stability of Eq. (2) would keep unchanged for , whereas it may be related to the fractional order for . The critical condition for losing stability of Eq. (2) can be expressed as:
Because of , the critical condition Eq. (4) can be .
If we fix the parameters as , and in Eq. (2), three equilibria are calculated and denoted as follow respectively:
is not appropriate in practice and is always unstable, so that only the equilibrium point will be discussed for practical significance. The coefficients of Eq. (3) can be obtained:
The double-parameter bifurcation diagram of Eq. (2) with respect to the parameters and is plotted in Fig. 1(a), where the equilibria are stable in region (I). When the parameters pass across the critical curve into region (II), the equilibria will become unstable, and a stable limit cycle will appear in the system. It is obviously that the parameter range of stable equilibria is bigger than stable limit cycles. However, if , the parameter interval of stable limit cycle is larger than stable equilibria. The phase diagram of the stable limit cycle for and is presented in Fig. 1(b).
Fig. 1. The bifurcation diagram and the phase diagram of limit cycle: a) the bifurcation diagram with respect to parameters and ; b) the phase diagram for the parameters and
3. Stability analysis of the integer-order and fractional-order system
The parameter is closely related to the stability of the system (2) although it is irrelevant to the equilibrium point of this system. Therefore, the variation of the parameter will result into the change of the dynamical behaviors. The stability of the system for and when will be analyzed in details in the following parts.
Considering , the values of other parameters are the same as those in Section 2. The real and imaginary parts of the eigenvalues of Eq. (2) are plotted in Fig. 2, denoted by the solid line and stars respectively. For , there are two positive and one negative real eigenvalues, so that is unstable. A negative real eigenvalue and two complex conjugate roots are found for . Here we would like to point that the real part of complex conjugate roots undergoes variation with the increase of the parameter . For , the real part of conjugate complex roots is larger than zero, while it is less than zero for . This means that the equilibrium is unstable for and stable for . Therefore, Hopf bifurcation happens at the critical parameter value denoted by point A in Fig. 2(a). On the other hand, the necessary condition of Hopf bifurcation is that a pair of pure imaginary roots appear in the system, which can be obtained by , and . Because there is stable limit cycle when and stable equilibrium for , the Hopf bifurcation at critical point A is subcritical. This means that stable periodic reaction could exist when the reaction rate is small enough, while large will make the stable periodic reaction disappear, and the concentrations , and will approach constants.
For , there are a negative real eigenvalue and a pair of complex conjugate roots , and all the arguments are shown in Fig. 3. The argument of real eigenvalue is always , and the arguments of the complex conjugate roots may vary with the increase of parameter . The equilibria are unstable for because of . And they are locally asymptotical stable for because the stable condition is met. The critical point of losing stability takes place at , which can also be verified by Fig. 1.
Comparing the abovementioned two situations, it could be found that the stabilities of the fractional-order and integer-order systems are uniform in the most cases. However, there is a parameter interval, (0.1049, 0.1782), in which the stabilities of the two systems are totally different. In this interval, is unstable for , while it is stable for . The numerical simulations of time history for are plotted in Fig. 4, which coincide with the abovementioned theoretical analysis. Furthermore, with the decrease of fractional order , the interval length of different stability may become larger and larger, which can be found from Fig. 1.
Fig. 2. The stability of the integer-order system: a) The real and imaginary parts of the eigenvalues of Eq. (2) for ; b) The enlarge figure near 3.75×10-3
Fig. 3. The critical value and the absolute values of eigenvalue’s argument for 0.95
Fig. 4. The stability of Eq. (2): a) the time history of the fractional-order system; b) the time history of the integer-order system
4. Fold/Fold slow-fast oscillation and bifurcation mechanism
Considering , Eq. (2) may involve in two time scales, so that the whole system may behave in the typical slow-fast phenomenon. For , the periodic slow-fast oscillation appears in the whole system. The corresponding phase diagram and time history are plotted in Fig. 5. In the periodic process, there are twice instantaneous jumping behaviors, denoted by the arrows in Fig. 5(a). Furthermore, the fast and slow variables present different dynamical features. For the fast variable , the instantaneous jumping behaviors form the spiking state, and the quiet state takes up the most time in the periodic oscillation, which can be found in Fig. 5(b). While the instantaneous jumping phenomenon doesn’t happen in the slow variable , and it will change uniformly, which is presented in Fig. 5(c). Here we would like to point that the periodic oscillation is stable, which is produced by subcritical Hopf bifurcation. The time histories under different initial values can illustrate the stability of the slow-fast periodic oscillation, as shown in Fig. 6.
Fig. 5. The periodic slow-fast oscillation for
a) The phase diagram
b) The time history for the fast variable
c) The time history for the slow variable
Fig. 6. The limit cycle for with different initial values
In order to reveal the generation mechanism, we will analyze the system by use of the bifurcation theory. Obviously, the whole system Eq. (2) can be divided into a fast subsystem (FS) and a slow subsystem (SS). The FS is given by the fast variables and , while the SS is modeled by the slow variable . Slow variable is taken as the bifurcation parameter of the fast variables. The FS can be written as:
The equilibria of FS can be determined by:
one could obtain the extreme points. The extreme points of the equilibrium line is used to analyze the bifurcation of FS. For , and , we can obtain two extreme points, denoted by LP1(2.7425, 3.1569) and LP2(8.6777, 3.8461) respectively. Then the equilibrium line of FS is divided into three branches and plotted in Fig. 7, where the points on the branch (I) and (III) are stable nodes, and the ones on the branch (II) are unstable saddle points. The details can be verified by Fig. 8, where the eigenvalues of the equilibrium curve are presented. Therefore, LP1 and LP2 are the critical points of the Fold bifurcation of FS. Here we would like to point out that the equilibria are either nodes or saddles under the taken parameter condition. The imaginary parts of eigenvalues are always zero, so that the stabilities of FS may keep unchanged with the variation of the fractional order.
Fig. 7. Bifurcation diagram of FS
Fig. 8. The eigenvalues of FS in Fig. 7
a) The eigenvalues for branch (I)
b) The eigenvalues for branch (II)
c) The eigenvalues for branch (III)
Because the system possesses fast and slow subsystems, the slow-fast dynamical analysis method is used here. By overlapping the bifurcation diagram of FS with the phase diagram of the whole system, Fig. 9 is obtained, which can be used to explain the generation mechanism of the periodic slow-fast oscillation.
Fig. 9. Overlapping of phase diagram of the fractional-order system with bifurcation diagram of FS for
Now we describe the periodic oscillation in details. The trajectory beginning at point D may keep in quiescent state (QS), because the trajectory may move slowly along the stable equilibrium line branch (I) of FS. The QS will be interrupted at point A, where the trajectory moves to the minimal value, i.e. the critical point LP1 of Fold bifurcation of FS. At the same time, the system may be attracted by the stable attractors on branch (III), which results into the jumping phenomenon called as spiking state (SP). When the trajectory jumps to point B, the system response may enter QS again, shown as the slow movement along the stable equilibrium line branch (III). When the system response arrives at the critical point LP2 of FS, the QS terminates and is followed by the SP characterized by jumping to point D because of the attraction from the stable equilibrium line branch (I). The whole procedure forms one period of oscillation. In a word, the twice Fold bifurcations result into twice transitions between the QS and FS, so that the periodic oscillation should belong to Fold/Fold type slow-fast oscillation.
Fig. 10. The generation mechanisms of periodic slow-fast oscillation for different fractional orders
Furthermore, the effect of fractional order on the periodic slow-fast oscillation is compared here. The generation mechanisms for and are shown in Fig. 10. From the figure it could be observed that the generation mechanisms of the periodic oscillation are almost the same with the variation of the fractional order. The primary reason should be that the type of the equilibrium point of the subsystem keeps unchanged with the variation of the fractional order. However, the oscillation period may become much longer with the decrease of fractional order . The differences between the periods of integer-order and fractional-order systems lie in the fact, power law stability is used to define the asymptotical stability of fractional-order system instead of traditional exponential stability. The details associated with the time histories can be found in Fig. 11.
Fig. 11. The time histories of periodic slow-fast oscillation for different fractional orders
The fractional-order BZ reaction is investigated by use of theoretical analysis and numerical simulation. Based on the stability condition of fractional-order system, the double-parameter bifurcation diagram with respect to fractional order is firstly given, and subcritical Hopf bifurcation is found in fractional-order BZ reaction. By comparing factional-order and integer-order systems, we present the parameter interval about different stability of the two systems, and find that the different features may become more obvious with the increase of fractional order.
Furthermore, the slow-fast oscillation phenomenon is firstly discussed in fractional-order BZ reaction with two time scales coupled. Based on the slow-fast dynamical analysis method, it is found that the fast subsystem possesses twice Fold bifurcations, which leads the system to jump transiently. The switch between different stable equilibrium line branches results into the transition between QS and SP. Accordingly, the slow-fast oscillation belongs to Fold/Fold type. It is also found that the effect of fractional order on trajectory shape and generation mechanism is small, because the type of the equilibrium points of FS keeps unchanged with the variation of the fractional order. However, power law stability in fractional-order system will make the period become longer.
The authors are grateful to the support by National Natural Science Foundation of China (Nos. 11302136 and 11672191), Natural Science Foundation of Hebei Province (A2014210062), and the Training Program for Leading Talent in University Innovative Research Team in Hebei Province (LJRC006)
- Field R. J., Koros E., Noyes R. M. Oscillations in chemical systems II Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system. Journal of the American Chemical Society, Vol. 94, Issue 25, 1972, p. 8649-8664. [Publisher]
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- Yang S. P., Chen L. Q., Li S. H. Dynamics of Vehicle-Road Coupled System. Springer Berlin Heidelberg, 2015. [Publisher]
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- Meng P., Lu Q. S., Wang Q. Y. Dynamical analysis of bursting oscillations in the Chay-Keizer model with three time scales. Science China Technological Sciences, Vol. 54, Issue 8, 2011, p. 2024-2032. [Publisher]
- Li X. H., Bi Q. S. Bursting oscillation in CO oxidation with small excitation and the enveloping slow-fast analysis method. Chinese Physics B, Vol. 21, Issue 6, 2012, p. 100-106. [Publisher]
- Zhang Z. D., Bi Q. S. Bifurcation in a piecewise linear circuit with switching boundaries. International Journal of Bifurcation and Chaos, Vol. 22, Issue 2, 2012, p. 240-265. [Publisher]
- Zhou Y., Ionescu C., Machado J. A. T. Fractional dynamics and its applications. Nonlinear Dynamics, Vol. 80, 2015, p. 1661-1664. [Publisher]
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- Ahmed E., El-Sayed A. M. A., El-Saka H. A. A. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications, Vol. 325, Issue 1, 2007, p. 542-553. [Publisher]
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- Shen Y. J., Wei P., Yang S. P. Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dynamics, Vol. 77, Issue 4, 2014, p. 1629-1642. [Publisher]
- Shen Y. J., Yang S. P., Sui C. Y. Analysis on limit cycle of fractional-order van der Pol oscillator. Chaos, Solitons and Fractals, Vol. 67, 2014, p. 94-102. [Publisher]
- Shen Y. J., Yang S. P., Xing H. J. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative. Acta Physica Sinica, Vol. 61, Issue 11, 2012, p. 110505. [Search CrossRef]
- Shen Y. J., Wen S. F., Li X. H., Yang S. P., Xing H. J. Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method. Nonlinear Dynamics, Vol. 85, Issue 3, 2016, p. 1457-1467. [Publisher]
- Liu L. L., Duan J. S. A detailed analysis for the fundamental solution of fractional vibration equation. Open Mathematics, Vol. 13, 2015, p. 826-838. [Publisher]
- Abd-Elouahab M. S., Hamri N. E., Wang J. Chaos control of a fractional-order financial system. Mathematical Problems in Engineering, Vol. 17, Issue 4, 2010, p. 270646-1-18. [Publisher]
- Huang Z., Jin X. Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. Journal of Sound and Vibration, Vol. 319, Issue 3, 2009, p. 1121-1135. [Publisher]
- Chen L., Hu F., Zhu W. Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping. Fractional Calculus and Applied Analysis, Vol. 16, Issue 1, 2013, p. 189-225. [Publisher]
- Xu Y., Li Y. G., Liu D. A method to stochastic dynamical systems with strong nonlinearity and fractional damping. Nonlinear Dynamics, Vol. 83, Issue 4, 2016, p. 2311-2321. [Publisher]
- Gafiychuk V., Datsko B. Stability analysis and limit cycle in fractional system with Brusselator nonlinearities. Physics Letters A, Vol. 372, Issue 29, 2008, p. 4902-4904. [Publisher]
- Zhou T. S., Li C. P. Synchronization in fractional-order differential systems. Physica D: Nonlinear Phenomena, Vol. 212, Issues 1-2, 2005, p. 111-125. [Publisher]
- Sekiguchi T., Mori Y., Hanazaki I. Photoresponse of the (Ru(bpy)3)2+ / BrO3- / H+ system in a continuous-flow stirred tank reactor. Chemistry Letters, Vol. 8, Issue 1993, 1993, p. 1309-1312. [Search CrossRef]
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Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback
Indian Journal of PhysicsJufeng Chen, Yongjun Shen, Xianghong Li, Shaopu Yang, Shaofang Wen
Modern Physics Letters BJingyu Hou, Shaopu Yang, Qiang Li, Yongqiang Liu
Pitchfork-bifurcation-delay-induced bursting patterns with complex structures in a parametrically driven Jerk circuit system
Journal of Physics A: Mathematical and TheoreticalXindong Ma, Shuqian Cao
Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations
Chaos: An Interdisciplinary Journal of Nonlinear ScienceXiujing Han, Yi Zhang, Qinsheng Bi, Jürgen Kurths
Chinese Physics BQing-Shuang Han, Di-Yi Chen, Hao Zhang
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https://reference.wolfram.com/language/guide/SolidMechanicsPDEModels.html
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math
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Solid Mechanics PDEs and Boundary Conditions
Solid mechanics is the field of physics that models mechanical deformation, strain and stress of solids under load.
SolidMechanicsPDEComponent — model solid mechanics
SolidMechanicsStrain — computes strain from displacement
SolidMechanicsStress — computes stress from strain
SolidFixedCondition — model fixed constraints
SolidDisplacementCondition — model prescribed displacements
SolidBoundaryLoadValue — model boundary loads
Solid Mechanics — monograph about modeling solid mechanics
Solid Mechanics Model Verification — test suite with solid mechanics model verification
Disc Brake — model heat and thermal expansion in a disc brake
Hyperelastic Tissue — model a biaxial tension test of tissue with a neo-Hookean model
Thermal Load — model the effect of temperature on a structure
Vascular Vessel — model a vascular artery with a Yeoh hyperelastic model
Comparative overview of the solid mechanics models.
More examples for boundary conditions can be found on the respective reference pages.
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https://mathspace.co/textbooks/syllabuses/Syllabus-409/topics/Topic-7249/subtopics/Subtopic-96851/?activeTab=interactive
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math
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The price for playing arcade games is shown in the graph.
What does the point $\left(2,3\right)$(2,3) represent on the graph?
$2$2 games cost $\$$$$3$3.
$3$3 games cost $\$$$$2$2.
The number of eggs farmer Joe's chickens produce each day are shown in the graph.
The amount of time it takes Kate to make beaded bracelets is shown on the graph.
Apply direct and inverse relationships with linear proportions
Apply numeric reasoning in solving problems
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| 449 | 8 |
https://www.free-online-aptitude-test.com/quizzes/math-aptitude-test-48/
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math
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This math aptitude Test #48 will measure your aptitude in a variety of math subjects. The questions could be addition, subtraction, multiplication, division, algebra, geometry, square roots or measurements. The questions could be expressed in numbers or written as word problems.
There are 10 questions on the test. You have TWO MINUTES to answer all 10 questions. You may not skip any of the questions on this math aptitude test. You must answer all 10 questions in two minutes in order to receive your score. Your score will be shown immediately after you complete the test.
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Math Aptitude Test 48
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Question 1 of 10
If 2x – 6 = 0, then x = ?CorrectIncorrect
Question 2 of 10
If 5x + 4x = 54, then x = ?CorrectIncorrect
Question 3 of 10
If 4/5 = a/20, then a = ?CorrectIncorrect
Question 4 of 10
8 + (1 + 5)² ÷ 4 = ?CorrectIncorrect
Question 5 of 10
If a parallelogram has a base of 5 inches and height of 4 inches, what is the area in square inches?CorrectIncorrect
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Solve for x: 5/6 = x/12CorrectIncorrect
Question 7 of 10
If x + 3 – 3 = -6 then x = ?CorrectIncorrect
Question 8 of 10
When a coin is flipped it has equal chances of landing on either heads or tails on each flip. If a coin is flipped 3 times, what is the probability that it will be heads all three times?CorrectIncorrect
Question 9 of 10
David is half as old as his uncle Robert. In 4 years David will be 6 years younger than Robert. How old are David and Robert now?CorrectIncorrect
Question 10 of 10
A gardener is building a raised flower bed that measures 16′ long, 5′ wide and 1′ deep. If the gardener wants to fill the flower bed 3/4 full, how many cubic feet of gardening soil does he need to buy?CorrectIncorrect
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| 2,557 | 35 |
https://essays4sale.com/2021/03/09/how-to-solve-proportion-problems_5e/
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math
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Componendo dividendo applied on number system and ratio proportion problems. fortunately, information technology dissertation topics the proportion method will work for all three types of free research essays questions:. the pre-algebra concept of resume with military experience proportions builds upon knowledge of fractions, ratios, variables and basic facts. then the long piece, being the total piece less what was cut off how to cite a thesis in mla for the short piece, must have a length of how to solve proportion problems 21 — x. yes, you can solve an incorrectly set up how to solve proportion problems theatre essay topics equation and find an answer. the distance ali runs in 40 minutes is 3 miles. if it is reduced to a width of 3 in then how tall will dnp capstone projects it be? Ratios rates proportions add to my workbooks (2) add to google classroom add to microsoft teams share through need help writing a essay whatsapp:. proportion problems and answers, but end up in infectious how to solve proportion problems downloads. this tutorial let's you see the steps to take in order how to write an scholarship essay to turn a word dhcp not assigning ip address problem involving a blueprint into a proportion. law research paper child abuse of exponents 10. a proportion on the other hand is an equation that says that two ratios are equivalent.
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https://stat.ethz.ch/pipermail/r-help/2006-March/100427.html
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math
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[R] Maximally independent variables
ggrothendieck at gmail.com
Wed Mar 1 05:05:15 CET 2006
Are there any R packages that relate to the
following data reduction problem fo finding
maximally independent variables?
Currently what I am doing is solving the following
minimax problem: Suppose we want to find the
three maximally independent variables. From the
full n by n correlation matrix, C, of all n variables
chooose three variables and form their 3 by 3 correlation
submatrix, C1, finding the offdiagonal entry of C1
which is largest in absolute value. Call that z. Thus for
each set of 3 variables we can associate such a z.
Now for each possible set of three variables find the one for
which its value of z is least.
I only give the above formulation because that is
what I am doing now but I would be happy to
consider other different formulations.
More information about the R-help
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https://projecteuclid.org/proceedings/advanced-studies-in-pure-mathematics/Various-Aspects-of-Multiple-Zeta-Functions--in-honor-of/Chapter/An-overview-and-supplements-to-the-theory-of-functional-relations/10.2969/aspm/08410263
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math
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We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types $B_r$, $D_r$, $A_3$ and $C_2$. To show those new results, we use two different methods. The first method, for $B_r$, $D_r$, $A_3$, is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for $C_2$ is more elementary, using partial fraction decompositions.
Digital Object Identifier: 10.2969/aspm/08410263
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| 634 | 2 |
https://www.physicsforums.com/threads/wave-comparison.898310/
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math
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1. The problem statement, all variables and given/known data Flat harmonic electromagnetic wave propagates in the positive direction in vacuo axis y. Vector electromagnetic energy flux density is given by: S(y,t)=Sm *cos(wt-ky)2.Wave value: k=(2*π)/λ=0.41 m-1,Amplitude Sm=26 W/m2.Compare this wave with another wave. 2. Relevant equations ƒ=V/λ k=(2*π)/λ E=h*ƒ 3. The attempt at a solution I find wavelenght from wave value λ=15m ⇒ it is a radio short wave ⇒next i find frequency from ƒ=V/λ=2*107⇒and finally i find photon energy E=h*ƒ=1,3*10-26 J =81*10-9neV. I compared the result with the table, and he enters the period of short radio waves, ie, this wave is the same radio signal from your phone or radio?Maybe I made a mistake somewhere?
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| 761 | 1 |
http://www.chegg.com/homework-help/questions-and-answers/algebra-archive-2008-july-26
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math
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Algebra Archive: Questions from July 26, 2008
Anonymous askedThere are three kinds of apples all mixed up i...Please answer the following question including steps:There are three kinds of apples all mixed up in a basket. Howmany apples must you draw (without looking) from the basket to besure of getting two apples of one kind.Any assistance is appreciated. Thank you in advance.•1 answer
Anonymous askedAn example of a polynomial fu...
Many different kinds of data can be modeled using polynomialfunctions.
An example of a polynomial function would be gas mileage for anautomobile. If we compare gas mileage at two different speeds, V1and V2, the gas required varies as (V1/V2), raised to the thirdpower, (V1/V2)3.
Rational functions are also useful. For example, a cubic/cubicmodel can be used to explain the thermal expansion of metals withtemperature. Rational functions have been used to describe problemsas diverse as the movement of blood through the body to how toproduce items at the lowest possible cost.
I must create a set of data that can be modeled as a polynomialfunction. Can anyone provide a reference to thedata? Please help me plot the data using Microsoft Excelincluding the equation for the fit. I need to discuss how closelythe data seems to match to the best fit line. Do the same for datathat can be modeled using a rational function. I must include in my answer how this can be used in a real-lifeapplication.•1 answer
Anonymous asked1 answer
Anonymous askedA parabolic arch has a span of 120 feet and a maximum heightof 25 feet. Choose a suitable rectangula...A parabolic arch has a span of 120 feet and a maximum heightof 25 feet. Choose a suitable rectangular coordinateaxes and find the equation of the parabola. Afterwardcalculate the height of the arch at 10 feet from thecenter.•2 answers
Anonymous asked2 answers
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https://www.barnesandnoble.com/w/mathematicians-under-the-nazis-sanford-l-segal/1119938014
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math
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Contrary to popular belief--and despite the expulsion, emigration, or death of many German mathematicians--substantial mathematics was produced in Germany during 1933-1945. In this landmark social history of the mathematics community in Nazi Germany, Sanford Segal examines how the Nazi years affected the personal and academic lives of those German mathematicians who continued to work in Germany.
The effects of the Nazi regime on the lives of mathematicians ranged from limitations on foreign contact to power struggles that rattled entire institutions, from changed work patterns to military draft, deportation, and death. Based on extensive archival research, Mathematicians under the Nazis shows how these mathematicians, variously motivated, reacted to the period's intense political pressures. It details the consequences of their actions on their colleagues and on the practice and organs of German mathematics, including its curricula, institutions, and journals. Throughout, Segal's focus is on the biographies of individuals, including mathematicians who resisted the injection of ideology into their profession, some who worked in concentration camps, and others (such as Ludwig Bieberbach) who used the "Aryanization" of their profession to further their own agendas. Some of the figures are no longer well known; others still tower over the field. All lived lives complicated by Nazi power.
Presenting a wealth of previously unavailable information, this book is a large contribution to the history of mathematics--as well as a unique view of what it was like to live and work in Nazi Germany.
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About the Author
Sanford L. Segal is Professor of Mathematics at the University of Rochester and the author of Nine Introductions in Complex Analysis.
Read an Excerpt
Mathematicians under the Nazis
By Sanford L. Segal
PRINCETON UNIVERSITY PRESSCopyright © 2003 Princeton University Press
All rights reserved.
Mathematics under the Nazi regime in Germany? This seems at first glance a matter of no real interest. What could the abstract language of science have to say to the ideology that oppressed Germany and pillaged Europe for twelve long years? At most, perhaps, unseemly (or seemly) anecdotes about who behaved badly (or well) might be offered. While such biographical material, when properly evaluated to sift out gossip and rumor, is of interest—history is made by human beings, and their actions affect others and signify attitudes—there is much more to mathematics and how it was affected under Nazi rule. Indeed, there are several areas of interaction between promulgated Nazi attitudes and the life and work of mathematicians. Thus this book is an attempt at a particular investigation of the relationships between so-called pure (natural) science and the extra-scientific culture. That there should be strong cultural connections between the technological applications of pure science (including herein the social applications of biological theory) and various aspects of the Industrial Revolution is obvious. Social Darwinism, and similar influences of science on social thought and action, have been frequently studied. It is not at all clear at the outset, however, that theoretical science and the contemporary cultural ambience have much to do with one another. Belief in this nonconnection is strengthened by the image of science proceeding in vacuo, so to speak, according to its own stringent rules of logic: the scientific method. In the past thirty years, however, this naive assumption of the autonomy of scientific development has begun to be critically examined.
A general investigation of this topic is impossible, even if the conclusion were indeed the total divorce of theoretical science from other aspects of culture. Hence the proposal to study one particular microcosm: the relationship between mathematics and the intensity of the Nazi Weltanschauung (or "worldview") in Germany. Although 1939 is a convenient dividing line in the history of Hitler's Reich, nonetheless the prewar Nazi period must also be viewed as a culmination; the Germany of those years was prepared during the Weimar Republic, and both the cultural and scientific problems that will concern us have their origins at the turn of the century. World War I symbolized the conclusion of an era whose end had already come. Similarly, World War II was a continuation of what had gone before, and a terminal date of 1939 is even more artificial and will not be adhered to.
The concentration on mathematics may perhaps need some justification. At first glance, a straw man has been set up—after all, what could be more culture-free than mathematics, with its strict logic, its axiomatic procedures, and its guarantee that a true theorem is forever true. Disputes might arise about the validity of a theorem in certain situations: whether all the hypotheses had been explicitly stated; whether in fact the logical chain purporting to lead to a certain conclusion did in fact do so; and similar technical matters; but the notion of mathematical truth is often taken as synonymous with eternal truth. Nor is this only a contemporary notion, as the well-known apocryphal incident involving Euler and Diderot at the court of Catherine the Great, or the Platonic attitude toward mathematics, indicate. Furthermore, there is the "unreasonable effectiveness" of mathematics in its application to the physical and social scientific world. Even so-called applied mathematics, concerning which Carl Runge remarked that it was merely pure mathematics applied to astronomy, physics, chemistry, biology, and the like, proceeds by abstracting what is hypothesized as essential in a problem, solving a corresponding mathematical problem, and reinterpreting the mathematical results in an "applied" fashion.
Mathematics also has a notion of strict causality: if A, then B. It is true that the standards of rigor, the logical criteria used to determine whether or not a proof is valid, that is, to determine whether or not B truly follows from A, have changed over time; nevertheless, the notion that it is conceivable that B can be shown always to follow from A is central to mathematics. As the prominent American mathematician E. H. Moore remarked, "Sufficient unto the day is the rigor thereof." Both the necessary process of abstraction and the idea of mathematical causality separate mathematics from more mundane areas. Somewhat paradoxically, perhaps, they are also partly responsible for the great power of mathematics in application. Mathematical abstraction and mathematical causality seem to elevate mathematics above the sphere of the larger culture.
Twin popular illusions incorrectly elaborate upon this view and make mathematics seem even more remote from the general culture. The first of these is that the doing of mathematics is only a matter of calculation, or, more sophisticatedly, of logical step-by-step progress from one eternal truth to another via intermediate truths. This view is enhanced by the way mathematicians publicly present the results of their investigations: exactly as such logical progressions. In fact, however, the discovery of mathematics, as opposed to the presentation of it, is more like the reconnoitering of some unknown land. Various probes in various directions each contribute to the forming of a network of logical connections, often even unconsciously. The realization of this network, the a posteriori checking for logical flaws, and the orderly presentation of the results, do not reflect the process of mathematical creativity, whatever that may be, and however ill it is understood.
The second illusion is that all that counts for a mathematician is to distinguish the correct from the incorrect. Correctness is indeed the sine qua non of mathematics, but aesthetic considerations are of great importance.
Among the various aesthetic factors influencing mathematical activity are economy of presentation, and the logic (inevitability) of often unexpected conclusions. While correctness is indeed the mathematical essential, some correct proofs are preferable to others. Proofs should be as clear and transparent as possible (to those cognizant of the prerequisite knowledge). A good notation, a good arrangement of the steps in a proof, are essential, not only to aid the desired clarity, but also because, by indicating fundamentals in the problem area, they actually incline toward new results. Clarity, arrangement, and logical progression of thought leading to an unexpected conclusion are well illustrated in an incident concerning no less a personage than the philosopher Thomas Hobbes:
He was 40 yeares old before he looked on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the Proposition. By G——, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that truth. This made him in love with Geometry.
A simple example of an "unbeautiful truth" is a list of positive integers. Mathematics is not frozen in time like a Grecian urn; solutions of old problems lead to new considerations. Though truth may not necessarily be beauty, beauty is truth, and for the mathematician impels to its own communications. As Helmut Hasse (who will be met again) remarks:
Sometimes it happens in physics again and again, that after the discovery of a new phenomenon, a theory fitted out with all the criteria of beauty must be replaced by a quite ugly one. Luckily, in most cases, the course of further development indeed reveals that this ugly theory was only provisional....
In mathematics this idea leads in many instances to the truth. One has an unsolved problem, and, at first, has no insight at all how the solution should go, even less, how one might find it. Then the thought comes to describe for oneself what the sought-for truth must look like were it beautiful. And see, first examples show that it really seems to look that way, and then one is successful in confirming the correctness of what was envisaged by a general proof.... In general we find a [mathematical] formulation all the more beautiful, the clearer, more lucid, and more precise it is.
As Hasse puts it elsewhere, truth is necessary, but not sufficient for real (echt) mathematics—what is also needed is beautiful form and organic harmony.
One result of this aesthetic is that the mathematician thinks of himself as an artist, as G. H. Hardy did:
The case for my life, then, or that of anyone else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.
Or, as Hasse says even more forcefully, "The true mathematician who has found something beautiful, senses in it the irresistible pressure to communicate his discovery to others."
Mathematics is the "basic science" sine qua non. At the same time, it is quite different from basic experimental science by being divorced from laboratory procedures. Even so-called applied mathematics only takes place on paper with pencil. The hallmark of mathematics is logical rigor. However important or suggestive or helpful heuristic or analogical arguments may be, it is only the mathematical proof according to accepted standards of logical rigor that establishes a mathematical result. Those logical standards may be and are disputed (and were in Nazi Germany), but given an accepted set of such standards, mathematical proofs according to them establish mathematical results that are true without qualification. On the one hand, a mathematical result is "sure"; on the other, however, all but the final results with proofs are, at best, incomplete mathematics: the mathematician's "experiments" are usually eminently unpublishable as such. This removal of mathematics from the concrete world contributes to the mathematical aesthetic. While there are notions of a "beautiful" experiment in the experimental sciences, in mathematics the aesthetic is purer for its removal from the natural irregularities of concrete life. "As for music, it is audible mathematics," writes the biologist Bentley Glass, and perhaps the traditional musical aesthetic is the one most closely resembling the mathematical; here, too, given the underlying assumptions, there is a purity of form that is part of the notion of beautiful. Deviations like Mozart's Musikalischer Spass or some of the less slapstick efforts of P.D.Q. Bach (Peter Schickele) are jokes because of their introduction of irregularities into a presumed form. Similarly, Littlewood presents as humorous an unnecessarily cumbrous presentation of a proof that can be expressed quite clearly and elegantly. The papers of Hasse and Archibald cited earlier also stress the analogy between the musical and the mathematical aesthetic.
In some sense, then, mathematics is an ideal subject matter; it is, however, made real by the actions of mathematicians. In Russell's well-known words:
Mathematics possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Nevertheless, mathematicians make tremendous emotional investments in the doing of mathematics. Mathematicians, despite their pure aesthetic, the divorce of their actual work from concrete reality, and the surety of their results, are not like petty gods in ivory towers playing at abstruse and difficult, but meaningless, games. The final piece of mathematics is abstract, aesthetically beautiful, and certain; but it is not (nor could it be) an instantaneous or automatic creation. The doing of mathematics is as emotionally involved, often clumsy, and uncertain as any other work that has not been reduced to a purely automatic procedure.
Thus, the nature of mathematical abstraction and mathematical causality, coupled with the popular ignorance of the nature of mathematical research and the removal of mathematics from the everyday world, seem to make mathematics one of the least likely subjects for the sort of investigation proposed. Yet some Nazi mathematicians and psychologists stood this reasoning on its head. At the same time, they emphasized with a peculiarly Nazi bias the often neglected roles of aesthetics and inspiration in creating mathematics. They argued that exactly the apparent culture-free nature of mathematical abstraction and mathematical causality makes mathematics the ideal testing-ground for theories about racially determined differences in intellectual attitudes. As E. R. Jaensch wrote in 1939:
Mathematics can simply have no other origin than rational thinking and mental activity (Verstandestätigkeit). "Irrational" mathematics would be a wooden iron, a self-contradiction. If, therefore, one discovers something worth exposure about the ways of thought (Verstandeskräfte) that still command the field on this area—and that happens in many respects with complete justice—so one can hereby only obtain help by bringing other forms of rational thought in more strongly—in no case however, through the conjuring up of irrationality. This way is simply excluded in mathematical thought. Even if in other scientific and educational disciplines it is possible artificially still to maintain the appearance that Reason (Verstand), as treated through that radical cure, still lives—in Mathematics it is impossible.
Hereby, the question of mathematical thought attains the character of an especially instructive example—an "illuminating case" in Baco's [sic] sense—for the forms of logical thought and rationalism above all, but also in other areas of knowledge and in everyday life.
What is important to note here is the insistence that the supposed autonomy of mathematics from irrational influence makes it exactly appropriate for investigating various intellectual types. Just because of the rationality of its results, mathematics was deemed an excellent medium for perceiving the various important differences between different peoples' ways of thought. It did not prove difficult to discover, for example, a Nordic type, a Romance or Latin type, a Jewish type, and, in fact, several subvarieties of these. Jaensch's theory of types could be elaborated independent of or in conjunction with Rassenseelenkunde, or the theory of the "racial soul." This was done most prominently by the distinguished mathematician Ludwig Bieberbach, who will be discussed particularly in chapters 6 and 7. By delineating a "Nordic" mathematics distinct from French or Jewish mathematics, great emphasis could be placed (necessarily) on the mode of intellectual discovery as opposed to its fruits, and, therefore, on feeling and attitude toward the world.
However important this inversion of the usual attitude toward mathematics may be for investigation, there are at least two other reasons arguing for a study of mathematics in the Nazi period. The first is that among the substantial number of mathematicians who were sympathetic in varying degrees to the Nazi cause were several who attempted to associate the political argument with various philosophical differences within mathematics. This did not alter the truth of any mathematical fact, but it did declare that certain mathematical disciplines were "more equal" than other varieties. Nor was this simply a question of "pure" versus "applied," of theory versus immediately usable results. Both these beliefs and the ones about the salience of psycho-racial differences within mathematics also argued for the distinction in differences of pedagogical style. Put succinctly, a Nazi argument promoted by Bieberbach was that because Jews thought differently, and were "suited" to do mathematics in a different fashion, they could not be proper instructors of non-Jews. Indeed, their presence in the classroom caused a perversion of instruction. Thus an elaborate intellectual rationale for the dismissal of Jews was established, discussed, and defended.
Excerpted from Mathematicians under the Nazis by Sanford L. Segal. Copyright © 2003 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
- Frontmatter, pg. i
- Contents, pg. ix
- PREFACE, pg. xi
- ACKNOWLEDGMENTS, pg. xix
- ABBREVIATIONS, pg. xxi
- CHAPTER ONE. Why Mathematics?, pg. 1
- CHAPTER TWO. The Crisis in Mathematics, pg. 14
- CHAPTER THREE. The German Academic Crisis, pg. 42
- CHAPTER FOUR. Three Mathematical Case Studies, pg. 85
- CHAPTER FIVE. Academic Mathematical Life, pg. 168
- CHAPTER SIX. Mathematical Institutions, pg. 229
- CHAPTER SEVEN. Ludwig Bieberbach and “Deutsche Mathematik”, pg. 334
- CHAPTER EIGHT. Germans and Jews, pg. 419
- APPENDIX, pg. 493
- BIBLIOGRAPHY, pg. 509
- INDEX, pg. 523
What People are Saying About This
Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany.
Walter Noll, Carnegie Mellon University
"Segal must be commended for the enormous amount of research he has done in arriving at an accurate picture of the complexity of the situation faced by mathematicians during the Nazi regime. Avoiding stereotypes and oversimplification, he presents fascinating information and valuable insights to those interested in mathematicians and to people interested in the history of Nazi Germany."Walter Noll, Carnegie Mellon University
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http://wcplfriends.org/index.php/epub/analysis-of-variance-for-random-models-vol-ii-unbalanced-data
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math
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By H. Sahai, M. Ojeda
Read or Download Analysis of Variance for Random Models [Vol II - Unbalanced Data] PDF
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6) have to be solved for the elements of α, σe2 , and the ρi s contained in H with the constraints that the σe2 and ρi s be nonnegative. 6), which are difficult equations to handle. The difficulty arises because the ML equations may yield multiple roots or the ML estimates may be on the boundary points. 5) can be readily solved in terms of ρi s. 7a) and 1 ˆ H −1 (Y − Xα) ˆ (Y − Xα) N 1 = [Y H −1 Y − (X H −1 Y ) (X H −1 X)−1 (X H −1 Y )]. 8) where R = I − X(X H −1 X)−1 X H −1 . 8). For some alternative formulations of the likelihood functions and the ML equations, see Hocking (1985, pp.
21), the weights σi2 s are, of course, unknown. Rao (1972) suggested the following two amendments to this problem: (i) If we have a priori knowledge of the approximate ratios σ12 /σp2 , . . 21) and use the W thus comσp−1 p puted. 21) and obtain MINQUEs of σi2 s. 21) and the MINQUE procedure repeated. 1). 10. Minimum-Norm/-Variance Quadratic Unbiased Estimation 45 the property of unbiasedness is usually lost; but the estimates thus obtained may have some other interesting properties. Rao (1971a) also gives the conditions under which the MINQUE is independent of a priori weights σi2 s.
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Analysis of Variance for Random Models [Vol II - Unbalanced Data] by H. Sahai, M. Ojeda
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https://www.jiskha.com/display.cgi?id=1294018553
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posted by Brianna .
The weather channel has predicted that there is a 70% chance of rain today and 20% chance of rain tomorrow. A. What is the probability it wont rain tomorrow? B. What is the probability it wont rain tomorrow given that it rained today?
a. wont rain tomorrow? .8
B cannot be answered, there is no indication that rain on successive days are independent events.
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https://www.ask-rk.com/can-you-take-the-gradient-of-a-vector.html
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math
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- 1 What is the gradient of a vector?
- 2 How do you find the gradient of a vector field?
- 3 Is gradient always vector?
- 4 Is the gradient of a vector a scalar?
- 5 Is a vector a function?
- 6 What if the gradient is zero?
- 7 What is gradient tool?
- 8 What are the examples of vector field?
- 9 What is the formula for calculating gradient?
- 10 Is gradient always positive?
- 11 What are two synonyms for gradient?
- 12 What is the gradient in simple terms?
- 13 Is gradient only for scalar?
- 14 Can you take gradient of a scalar?
- 15 What is the value of curl of a gradient vector?
- 16 People also ask:
No, gradient of a vector does not exist. Gradient is only defined for scaler quantities. Gradient converts a scaler quantity into a vector.
What is the gradient of a vector?
The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that. Points in the direction of greatest increase of a function (intuition on why) Is zero at a local maximum or local minimum (because there is no single direction of increase)
How do you find the gradient of a vector field?
Is gradient always vector?
A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals).
Is the gradient of a vector a scalar?
Gradient is a scalar function. The magnitude of the gradient is equal to the maxium rate of change of the scalar field and its direction is along the direction of greatest change in the scalar function.
Is a vector a function?
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector.
What if the gradient is zero?
Straight Across. A line that goes straight across (Horizontal) has a Gradient of zero.
What is gradient tool?
The Gradient tool creates a gradual blend between multiple colors. You can choose from preset gradient fills or create your own. … You cannot use the Gradient tool with bitmap or indexed-color images. To fill part of the image, select the desired area.
What are the examples of vector field?
- A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point.
- Velocity field of a moving fluid.
What is the formula for calculating gradient?
To calculate the gradient of a straight line we choose two points on the line itself. The difference in height (y co-ordinates) ÷ The difference in width (x co-ordinates). If the answer is a positive value then the line is uphill in direction. If the answer is a negative value then the line is downhill in direction.
Is gradient always positive?
The gradient changes from negative to positive here, so the graph of y=g′(x) will pass through the point (−2,0). The gradient of y=g′(x) is always increasing, and the graph of y=g(x) is always bending to the left as x increases. Therefore g″(x) is always positive.
What are two synonyms for gradient?
What is the gradient in simple terms?
1a : the rate of regular or graded (see grade entry 2 sense transitive 2) ascent or descent : inclination. b : a part sloping upward or downward.
Is gradient only for scalar?
The gradient is most often defined for scalar fields, but the same idea exists for vector fields – it’s called the Jacobian. Taking the gradient of a vector valued function is a perfectly sensible thing to do.
Can you take gradient of a scalar?
The gradient of a scalar field is the derivative of f in each direction. Note that the gradient of a scalar field is a vector field. An alternative notation is to use the del or nabla operator, ∇f = grad f.
What is the value of curl of a gradient vector?
The curl of a gradient is zero.
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https://www.projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Clustering-and-percolation-of-point-processes/10.1214/EJP.v18-2468.full
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<p>We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices and more generally, negatively associated point processes. Examples of such coverage models are k-coverage in the Boolean model (coverage by at least k grains), and SINR-coverage (coverage if the signal to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of k-faces in the Cech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered” than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always “worsen” percolation, as well as that upper-bounding this cluster-ing by a Poisson process is a consequential assumption for the phase transition to hold.
"Clustering and percolation of point processes." Electron. J. Probab. 18 1 - 20, 2013. https://doi.org/10.1214/EJP.v18-2468
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https://www.ecologycenter.us/thermodynamic-equilibrium/info-wnp.html
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math
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We see that the amount of information decreases as it is transmitted from lower to higher levels. This implies that at each level there is redundant information. Certainly, only non-redundant information has a cost, but the repeating of information, its redundancy, provides the reliability of its transmission defending it from errors and destruction of the text by noise. The redundancy of information at rth level of perception (or description) can be defined as (Klix, 1974)
where max T(r) = r log2n. For r = 0 the redundancy is defined as R(0) = 0. Then the corresponding redundancies in English will be equal to R(0) = 0, R(1) = 0.15, R(2) = 0.30 and R(3) = 0.35. The latter, for instance, implies that only 35% of letters are redundant at the third level, i.e. 65% of randomly distributed letters are sufficient for the understanding of the text. The cost of information can be defined as the degree of non-redundancy (Volkenstein, 1988):
Then for each level we have C(0) = 1, C(1) = 1.18, C(2) = 1.43, C(3) = 1.54.
We used here one of the simplest definitions of the cost of information. In fact, this problem "What is the cost of information?", in spite of continuing discussion, is still far from its completion. This discussion falls outside the framework of our book, but nevertheless we shall cite one example.
So, there is some aim. Let probabilities of its attainment before and after receiving information be equal to P0 and P1, respectively. Then the cost of information is equal to C = log2(P1/P0) (Kharkevich, 1963). However, if the aim is unattained without information (P0 = 0), then the cost of any finite information is equal to infinity. This is not properly understandable.
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Hi, I'd like to model a box floating in open water, that means that wave generation and especially wave absorption should be modelised. I've modeled this on Hypermesh with the solver Radioss In this model, there are 2 sorts of interaction : - fluid/fluid : air with water - fluid/structure : both fluids with the box. First, I've used an ALE model for the fluids. I've tried to use both MAT/LAW51 (multimaterial) and MAT/LAW37 (biphasic material) in order to model this, but everytime there is a leakage and the computation can't run until the end (negative rho error). I've also tried to use MAT/LAW11 (bound) because it doesn't work properly with a biphasic material (negative rho error). Then, I've tried with an SPH model with IN and OUT condition, so that the computaiton doesn't take into account the air/water interaction, the computation also stops because of a leakage. For the fluid/structure interaction, I've used INTER/TYPE18 and INTER/TYPE7 respectively for ALE and SPH model. I have to set a STfval value for the stiffness. I use the formula of the Help page : stfval=rho*v²*A_el/GAP = 66.667avec - rho=rho_water = 0.001g/mm3; - v=10 mm/ms , initial velocity of the box when it is dropped in the water; - A_el = lagragien element surface =500 * 500 mm² ; - GAP= 1.5 * E_el = 1.5 * 250 = 375mm With this value, the box is floating in the air, the only way i've found a value that models reality was by dichotomy.. That's not convenient for further simulation with different initial data.... Should you have any suggestion to model an infinite water volume (so wave absorption) and for the stiffness value, please contact me. Sincerly yours, Emmanuelle
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https://history.stackexchange.com/questions/53412/where-was-this-destroyed-bridge-in-south-vietnam
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math
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Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
History Stack Exchange is a question and answer site for historians and history buffs. It only takes a minute to sign up.
Can anyone provide the location of this destroyed bridge during the Vietnam War? I believe the sign at the end of the bridge refers to the Big Red One, 1st Infantry Division
Required, but never shown
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http://www.urbandictionary.com/define.php?term=Donnarific
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math
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Often used to describe exceptional math talent, especially that having been inspired by the great 70's and 80's disco/dance/pop singer Donna Summer. May describe a noun causing lack of care towards math homework, due to Donna's mad beats.
#2: Yeah, that proof was so Donnarific!
#1: Too bad I still have no idea what's going on in that class.
#2: Who cares about Real Analysis when you've got Donna!?!
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https://www.physicsforums.com/threads/simple-differential-quesion.588094/
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math
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1. The problem statement, all variables and given/known data 1. ty'+2y = 4t^2 , y(1) = 2 2. Relevant equations 3. The attempt at a solution 1. I know how to get the answer but i have a trivial question. The answer is y=t^2 + t^(-2) and t>0. I do not get why t>0. Why can't t be R except 0?
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https://forum.duolingo.com/comment/21322662/Maembe-manne
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math
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Do numbers have to agree with the classes? I'm confused because some numbers do and some don't.
Good question. Only the numbers 1, 2, 3, 4, 5 and 8 take agreements with the nouns they describe. All other numbers take no agreements.
The reason for this is that the aforementioned numbers are of Bantu origin and so inflect according to standard patterns. The remaining numbers are of Arabic origin, and so like most loan words are not inflected. So for 1 through 10:
-moja, -wili, -tatu, -nne, -tano, sita, saba, -nane, tisa, kumi
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https://brainsanswers.co.uk/mathematics/the-length-of-the-fraser-river-is-a-14030138
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math
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hello : y =(2/3)x+5
hello : let a(3,7) b(6,9)
the slope is : (yb - ya)/(xb -xa)
(9-7)/(6-3) = 2/3
an equation is : y=ax+b a is a slope
y = (2/3)x +b
the line through point (3,7) : 7 = (2/3)(3)+b
b = 5
the equation is : y =(2/3)x+5
1/2, line rises
we are given the following pair of points and and we are to find the slope of line passing through these points:
(-1, 3) and (3, 5)
slope = = = =
the slope of this line is 1/2 which is positive so the line rises from bottom left towards top right.
it is on the y-axis because the eight is in the 2nd part of the ordered pair (0,8).
well there is no real question here but cody is 165 which means you have to multiply that by 10% or 0.10 which gives you 16.5 and subtract that from the total
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https://nilogen.com/2020/06/02/what-is-a-square-root/
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math
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What’s a Square Root?
The number’s square, Writer at its most straightforward form, has something to do with algebra and trigonometry. That is just one area of this mathematics.
There are when understanding mathematics, you must learn: surgeries, exponents, and examples. For instance, the square of a number usually means the total amount of these squared sides. It doesn’t have anything to do with the square of any variety that is additional.
Certainly one is the number’s square. You don’t require a calculator or a graphing calculator to do that, but you’ll want to know the particular formula. For instance, the square of the number could be prepared as one times 2. If you have just a trigonometry knowledge and would like to know how many areas that are per side of a triangle, then you can use the right-hand-side of the formula above.
By first working with a small number of numbers the very perfect method to discover this sort of knowledge is always to practice it. You could find in the event the quantity you’ve got readily available is that a repeating pattern of ones and zeros, that this is easy. You could practice your knowledge of addition and subtraction just before you move ahead to higher figures by using a quantity of examples.
It is usually more convenient to compose an equation than just accomplish just a tiny amount of math While it can be easier to compose an alternative expression compared simply just check it up at a cubicle, once you need to learn a theory. Thus, in the example above, the easiest way c and also the result is your formula for your own square.
If you prefer to be familiar with range of sides of the triangle, you will need to know the period of their angles the components, and also the exact distance between the two 3 points. The length will be called the hypotenuse help on writing a critical essay as well as also the angle is known as the diagonal. To understand these amounts, whatever you need to do is find the exact distance between the hypotenuse and also your center point.
A method for finding the area of a circle is exactly like the one for a square foot. For example, if you may determine the radius of the circle, you are able to find out the location of the square. This really is the notion of this triangle area formula.
Then you can utilize the formulation above to find the area of the triangle if you are working with a rectangle which has sides which do not need https://meetings.cshl.edu/courses.aspx?course=C-ION&year=20 to repeat. For example, in the event the triangle’s surfaces aren’t right, then you will need to get to the angle into the traces drawn from the ends of their triangle in the midpoint of this triangle.
An example of this would be a quadrilateral. That the triangles within this case are not right, as you may observe. They have just a little curve at the same finish.
Inside this case, the method can be used by us again, yet this moment, we’ll subtract the side’s duration by the length of the angle. This will give us the angle.
The angle from the centre point of the triangle’s corners is equivalent to the hypotenuse divided company website by the hypotenuse. Additionally, you will need to be aware of the region of the corner and also the base angle. These can seem like challenging formulas, nonetheless it is crucial to try to remember which you’re currently dealing with angles and also should not try and remedy them for your answers.
Most teachers can explain to you the root of a few is simply the square of the number. In mathematics, a root is definitely a side of the number.
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https://practice.geeksforgeeks.org/problems/find-second-largest-element/0
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math
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Given an array of elements. Your task is to find the second maximum element in the array.
The first line of input contains an integer T which denotes the number of test cases. Then T test cases follows. First line of each test case contains a single integer N which denotes the number of elements in the array. Second line of each test case contains N space separated integers which denotes the elements of the array.
For each test case in a new line print the second maximum element in the array. If there does not exist any second largest element, then print -1.
2 4 5 6 7
7 8 2 1 4 3
If you have purchased any course from GeeksforGeeks then please ask your doubt on course discussion forum. You will get quick replies from GFG Moderators there.
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https://e-booksdirectory.com/details.php?ebook=9558
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math
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by Vladlen Koltun
Publisher: Stanford University 2008
Number of pages: 89
Contents: Sets and Notation; Induction; More Proof Techniques; Divisibility; Prime Numbers; Modular Arithmetic; Relations and Functions; Mathematical Logic; Counting; Binomial Coefficients; The Inclusion-Exclusion Principle; The Pigeonhole Principle; Asymptotic Notation; Graphs; Trees; etc.
Home page url
Download or read it online for free here:
by Rupinder Sekhon - Connexions
Applied Finite Mathematics covers topics including linear equations, matrices, linear programming (geometrical approach and simplex method), the mathematics of finance, sets and counting, probability, Markov chains, and game theory.
by A.F. Pixley - Harvey Mudd College
This text is an introduction to a selection of topics in discrete mathematics: Combinatorics; The Integers; The Discrete Calculus; Order and Algebra; Finite State Machines. The prerequisites include linear algebra and computer programming.
by Ken Bogart, Cliff Stein - Dartmouth College
It gives thorough coverage to topics that have great importance to computer scientists and provides a motivating computer science example for each math topic. Contents: Counting; Cryptography and Number Theory; Reflections on Logic and Proof.
by W W L Chen - Macquarie University
Logic and sets, the natural numbers, division and factorization, languages, finite state machines, finite state automata, Turing machines, groups and modulo arithmetic, introduction to coding theory, group codes, public key cryptography, etc.
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https://demonstrations.wolfram.com/CombinationsOfSinesInTheComplexPlane/
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math
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Combinations of Sines in the Complex Plane
Initializing live version
Combinations of two sine functions must always have their zeros on the real line. Combinations of three need not. The height here is the absolute value of the sum of sine functions; the hue is the phase.
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https://alljobs.co.in/study-material/electricity-class-10-science-question/
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math
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Chapter 12 of Science Subject of Class 10 is “Electricity”. Practice Questions [ Extra Questions ] for Electricity Class 10 Science Chapter 12 are provided on this page.
Class 10 Science Chapter 12 : Electricity
Chapter 12 : Electricity ( Practice Questions )
|Chapter||Chapter 12 : Electricity|
|Study Material||Practice Questions ( Extra Question for Practice )|
|Total Questions||35 Questions|
|Text Book||NCERT Text Book|
Practice Question of Electricity Class 10 Science Chapter 12 (Extra Question Answer )
Ques. 1 : What is Electricity ?
Answer : It is a type / form of energy . Flow of electrons is called as Electricity.
Ques. 2 : Define Electric Current .
Answer : Rate of flow of charge through a conductor . Electric Current is denoted by Capital ” I “. The SI Unit of Electric Current is Ampere.
Ques. 3 : What is Coulomb ?
Answer : It is the SI Unit of Electric Charge. It is denoted by ‘C’.
I = Q/t = Coulomb / charge
Q = I.t
Ques. 4 : What is 1 Ampere ?
Answer : The current passing through a conductor is said to be 1 Ampere when a charge of 1 coulomb flows through it in 1 second. It is denoted by A . It is SI Unit of Electric Current.
Ques. 5 : Define Electric Current, Mathematically .
Answer : Mathematically electric current can be defined as
Ques. 6 : What does 1C/1s represent ?
Answer : It represent that 1 Coulomb of charge is flowing in 1 second. It represent the rate of flow of charge.
Ques. 7 : How smaller are the units of Current ?
Answer : Milliampere and Microampere are related to an ampere.
1mA = 10-3 A
1μA = 10-6 A
Ques. 8 : What is the amount of charge in 1 electron ?
Answer : 1.6 * 10-19 C of charge is present in 1 electron
Ques. 9 : How many electrons carry 1 coulomb of charge ?
Answer : 6 * 1018 electrons carry 1 coulomb of charge.
Ques. 10 : What is the direction of flow of electron ?
Answer : The direction of flow of current is opposite to flow of electrons.
Ques. 11 : Name the instrument that is used for measuring Electric Current .
Answer : Ammeter is the instrument which is used to measure the electric current in an electric circuit.
Ques. 12 : How ammeter is connected in the circuit ?
Answer : Ammeter is connected in series in the circuit to measure the electric current in the circuit.
Ques. 13 : What is an Electric Circuit ?
Answer : A continuous and closed path of an electric current is called an Electric Circuit.
Ques. 14 : What is Resistance ?
Answer : Resistance is an opposition to the flow of an electric current to a substance . The SI Unit of Resistance is Ohm ( )
Ques. 15 : Draw the symbols of Various electric components
Answer : Various Electric Components are –
- Electric Cell
- A battery
- Plug key ( Open )
- Plug Key ( Closed )
- A Wire Joint
- Wire Crossing without joining
- An Electric Bulb
- A resistor
- Rheostat ( Variable Resistance )
Ques. 16 : What is Potential Difference ?
Answer : The difference in electric potential between any two points in an electrical field is known as Potential Difference. It is the amount of work done in carrying a unipositive charge from a point at lower potential to a point at higher potential. It is denoted by ‘ V ‘ .
V = W/ Q
The SI unit of Potential Difference is Volts.
Ques. 17 : What is Ohm’s Law ?
Answer : According to Ohm’s Law , the amount of current passing through a conductor is directly proportional to the potential difference across its different points.
V proportional I
V = IR
Ques. 18 : What are the three factors on which resistance depend ?
Answer : Factors on which resistance depends are –
- Length of a Conductor
- Area of Cross section
- Nature of its material
Ques. 19 : What is Resistivity ?
Answer : Resistivity can be defined as the resistance of conductor having length equals to 1 cm and area of cross section equals to 1cm2.
Ques. 20 : What is the symbol of Resistivity ?
Answer : Symbol of Resistivity is Rho ( ρ ) .
Ques. 21 : Derive the relation between Resistance , length and area of cross section of a conductor ?
Answer : R proportional L
R proportional 1/A
R proportional L/A
R = ρ L /A.
Ques. 22 : What is the Heating effect of Electric Current ?
Answer : If the electric circuit is purely resistive, i.e. a configuration of resistors only connected to a battery, the source of energy continuously dissipated entirely , in the form of heat. It is known as Heating effect of Electric Current.
Ques. 23 : Name one source of electricity ?
Answer : Battery/ Cell is one of the source of electricity.
Ques. 24 : What are the two principle factor which determines the heat produced in a wire of a given material ?
Answer : The two principle factors which determines the heat produced in a wire of a given material are-
Ques. 25 : What property of electricity is used in heater ,etc.
Answer : Heating effect of electric current.
Ques. 26 : Define Electric Power .
Answer : Electric Power is the amount of work done by an electrical appliance in one second. It is also defined as the rate at which the electrical appliance consumes energy.
P = Work / Time = Energy Consumed / Time
Ques. 27 : What is the Unit of Power ?
Answer : The Unit of Power is ” Watt “
Ques. 28 : Name the bigger unit of Power .
Answer : Kilowatt ( kW).
1 kW = 1000 W
Ques. 29 : What is the commercial unit of electric energy ?
Answer : Commercial unit of electric energy is ” KWh “
Ques. 30 : How KWh is related to Joule ?
Answer : 1 KWh = 1000 watt* 60 *60 sec
= 1000 * 3600 watt sec
= 3.6 * 106 Joules
Q. What is the direction of flow of current ?
Ans. The direction of flow of current is opposite to the flow of electrons.
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http://mathhelpforum.com/calculus/75255-volume-spherical-cap-section.html
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math
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I'm not sure I understand the shape itself. Can you give a "real world" example of this shape? Picture would be really helpful too!
In calculating the volume of a partially-filled horizontal cylindrical tank which has ends similar to spherical caps, how do you calculate the volume of a partially-filled spherical cap (from the side of a sphere?)??
The cylinder I have solved, and the volume of a full spherical cap has been easy to find but the partial volume of the cap ends is tough.
This has been so hard for me to find information on, so I hope someone here has an idea!
The dimensions of this particular problem are: Cylinder Length = 8.16m, Diameter = 3.7m, Cap "height" (although horizontal) = 0.53m. I hope this is enough detail...
I guess the end caps put together could be called a squashed sphere or a "spheroid ovoid" .. There are many english sweets this shape but I don't know the names of any american candy for comparison. Anyway, it's circular in 1 axis, ovoid in 2 axes. Split in half down the circular face and stuck on the end of a horizontal cylinder.
That link looks like just the job, I've just got to pick it to pieces so I know that I can trust it now!
Wow... unfortunately I really have no idea how to help you...
Just an idea about how to think of it... think of it as two separate shapes. One is the cylinder. The other is the merged end caps (which would have a shape similar to an American football). I don't know if that makes it any easier or not. Sorry I couldn't be more help!
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https://labs.ebay.com/category/mathematics?type=All&page=1
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math
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We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.
In this paper we present recent existence, uniqueness, and stability results for the motion of immersed hypersurfaces driven by surface diffusion. We provide numerical simulations for curves and surfaces that exhibit the creation of singularities.
Moreover, our numerical simulations show that the flow causes a loss of embeddedness for some initially embedded configurations.
The (two-sided) Mullins-Sekerka model is a nonlocal evolution model for closed hypersurfaces, which was originally proposed as a model for phase transitions of materials of negligible specific heat. Under this evolution the propagating interfaces maintain the enclosed volume while the area of the interfaces decreases.
We will show by means of an example that the Mullins-Sekerka flow does not preserve convexity in two space dimensions, where we consider both the Mullins-Sekerka model on a bounded domain, and the Mullins-Sekerka model defined on the whole plane.
The Mullins-Sekerka model is a nonlocal evolution model for hypersurfaces, which arises as a singular limit for the Cahn-Hilliard equation. Assuming the existence of sufficiently smooth solutions we will show that the one-sided Mullins-Sekerka flow does not preserve convexity. The main tool is the strong maximum principle for elliptic second order differential equations.
Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions.
The proposed numerical scheme will allow to treat such free boundary problems in both R2 and R3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.
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| 2,419 | 8 |
https://www.clubtouareg.com/threads/new-here-2014-tdi-execline-r.293748/page-17#post-2144985
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math
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Plus $1875 USD for this...I paid 930$ USD all in for both parts in November 2020.
Now I just looked at the price, 878$ USD for the DPF and 768$ USD for the downpipe (1646$ USD!!) outch
I'm not gonna say anything... I'll just leave this here (with you being lowered and cops being clueless, no one would ever know visually)The only problem I see with the aftermarket mufflers is the extra attention from cops. I am trying as much as possible to keep an OEM+ look; although I would LOVE a full 3in line and replace the suitcase with other mufflers.
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https://www.mixbook.com/photo-books/education/question-17-6840698
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math
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FC: Hill Word Problem | By Corey Andrade
1: Question 17: Calvin Butterball is driving along the highway. He starts up a long, straight hill. 114 meters from the bottom of the hill Calvin's car runs out of gas. He doesn't put on the brakes. So the car keeps rolling for a while, coasts to a stop, then starts rolling backwards. He finds that his distances from the bottom of the hill 4 and 6 seconds after he runs out of gas are 198 and 234 meters respectively. - There are three ordered pairs of time and distance in the information given above. What are they? - Write an equation expressing Calvin's distance from the bottom of the hill in terms of the number of seconds which have elapsed since he ran out of gas. Assume that a quadratic function is a reasonable mathematical model. - Use the model to predict Calvin's distance from the bottom of the hill 10 and 30 seconds after he ran out of gas - Last Chance Texaco Station is located 400 meters from the bottom of the hill. Based on your model, will Calvin's car reach the station before it stops and starts rolling backwards? Justify your answer.
2: Three ordered pairs of time | The three ordered pairs of time are: 0, 114 4,198 and 6, 234. The problem is only concerned with when the car runs out of gas, which is at 114 meters from the bottom of the hill. Because 114 is where the car first runs out of gas, that will be the starting point. At the starting point, the car has not moved therefore it's time will be 0 seconds at 114 meters. The problem mentions two more critical points of distance and time, which will be the other two ordered pairs. After 4 seconds, the car has traveled 198 meters and after 6 seconds, it traveled 234 meters.
3: The equation | In y=, Y1 should have the equation and Y2 should equal zero | To get the equation that expresses Calvin's distance from the bottom of the hill, put your ordered pairs in a graphing calculator. On your calculator click Stat, Edit..., put the X values (0,4,6) into L1, put the Y values (114,198,234) into L2, Stat, Scroll to Calc, Quadreg, 2nd 1 (L1), comma, 2nd 2 (L2), comma, Vars, scroll to Y-Vars, function..., Y1, enter You should get a=-.5, b=23, and c=114 therefore your equation is: y=-.5x^2+23x+114
4: Making a prediction | By zooming out enough, you can notice the graph of this equation is a parabola. By looking at the table of this graph on your graphing calculator (2nd, graph) you can see the distance traveled under Y1 and the amount of seconds elapsed under X. To find Calvin's distance from the bottom of the hill after 10 seconds, just scroll down to 10 in the X column. To find his distance after 30 seconds, scroll down to 30 under the X column. You should have that after 10 seconds, Calvin traveled 294 meters and after 30 seconds, he traveled 354 meters. By using this table, you can predict how far Calvin traveled in any amount of time.
5: Last Chance Texaco station | Last Chance Texaco Station is located 400 meters from the bottom of the hill. Based on the quadratic model, Calvin will not travel the distance of 400 meters. By going back to the table (2nd, graph) and scrolling down the X column, at 23 seconds, Calvin's car has traveled 378.5 meters. At 24 seconds, the car has now traveled 378 meters and then as time goes on, the distance keeps decreasing, showing that at this point, his car is rolling backwards. Since the station is 400 meters away and the car's maximum distance is 378.5 meters, Calvin never reaches the station before his car stops and starts rolling backwards.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540481076.11/warc/CC-MAIN-20191205141605-20191205165605-00082.warc.gz
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https://www.stevenredhead.com/jokes/GeorgeCarlin.html
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math
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• How come wrong numbers are never busy?
• Do people in Australia call the rest of the world "up over"?
• Can a stupid person be a smart-ass?
• Does killing time damage eternity?
• Why is it that night falls but day breaks?
• Why is the third hand on the watch called a second hand?
• Why is lemon juice made with artificial flavor, and dishwashing liquid made with real lemons?
• Are part-time band leaders semi-conductors?
• Can you buy an entire chess set in a pawn-shop?
• Daylight savings time. Why are they saving it and where do they keep it?
• Did Noah keep his bees in ArcHives?
• Do pilots take crash-courses?
• Do stars clean themselves with meteor showers?
• Do you think that when they asked George Washington for ID that he just whipped out a quarter?
• Have you ever imagined a world with no hypothetical situations?
• Have you ever seen a toad on a toadstool?
• How can there be self-help "groups"?
• How do you get off a non-stop flight?
• How do you write zero in Roman numerals?
• How many weeks are there in a light year?
• If Barbie's so popular, why do you have to buy all her friends?
• If blind people wear dark glasses, why don't deaf people wear earmuffs?
• If space is a vacuum, who changes the bags?
• If tin whistles are made out of tin, what do they make fog horns out of?
• If you shouldn't drink and drive, why do bars have parking lots?
• If you jog backwards, will you gain weight?
• Why do the signs that say "Slow Children" have a picture of a running child?
• Why do they call it "chili" if it's hot?
• Why is the time of day with the slowest traffic called rush hour?
• One tequila, two tequila, three tequila, floor.
• The main reason Santa is so jolly is because he knows where all the bad girls live.
• I went to a bookstore and asked the saleswoman, "Where's the self-help section?" She said if she told me, it would defeat the purpose.
• Could it be that all those trick-or-treaters wearing sheets aren't going as ghosts but as mattresses?
• If a mute swears, does his mother wash his hands with soap?
• If a man is standing in the middle of the forest speaking and there is no woman around to hear him...is he still wrong?
• If someone with multiple personalities threatens to kill himself, is it considered a hostage situation?
• Isn't it a bit unnerving that doctors call what they do "practice?"
• Where do forest rangers go to "get away from it all?"
• Why do they lock gas station bathrooms? Are they afraid someone will clean them?
• If the police arrest a mime, do they tell him he has the right to remain silent?
• Why do they put Braille on the drive-through bank machines?
• How do blind people know when they are done wiping?
• How do they get the deer to cross at that yellow road sign?
• Is it true that cannibals don't eat clowns because they taste funny?
• One nice thing about egotists: they don't talk about other people.
• Does the Little Mermaid wear an algebra?
• Do infants enjoy infancy as much as adults enjoy adultery?
• If you try to fail, and succeed, which have you done?
• Why is it called tourist season if we can't shoot at them?
• If the "black box" flight recorder is never damaged during a plane crash, why isn't the whole damn airplane made out of that stuff?
• If you spin an oriental man in a circle three times, does he become disoriented?
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https://www.johndcook.com/blog/2023/08/06/swish-swiss/
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The previous post looked at the swish function and related activation functions for deep neural networks designed to address the “dying ReLU problem.”
Unlike many activation functions, the function f(x) is not monotone but has a minimum near x0 = -1.2784. The exact location of the minimum is
where W is the Lambert W function, named after the Swiss mathematician Johann Heinrich Lambert .
The minimum value of f is -0.2784. I thought maybe I made a mistake, confusing x0 and f(x0). If you look at more decimal place, the minimum value of f is
and occurs at
That can’t be a coincidence.
It turns out you can prove that f(x0) − x0 = 1 without explicitly finding x0. Take the derivative of f using the quotient rule and set the numerator equal to zero. This shows that at the minimum,
The fourth equation is where we use the equation satisfied at the minimum.
Lambert is sometimes considered Swiss and sometimes French. The plot of land he lived on belonged to Switzerland at the time, but now belongs to France. I wanted him to be Swiss so could use “swish” and “Swiss” together in the title.
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https://cloudflare-ipfs.com/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Stress_(mechanics).html
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In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting a weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface (such as a piston) push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules.
Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like gravity) or to its surface (like contact forces, external pressure, or friction). Any strain (deformation) of a solid material generates an internal elastic stress, analogous to the reaction force of a spring, that tends to restore the material to its original non-deformed state. In liquids and gases, only deformations that change the volume generate persistent elastic stress. However, if the deformation is gradually changing with time, even in fluids there will usually be some viscous stress, opposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress.
Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass. Stress may also be imposed on a material without the application of net forces, for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials).
The relation between mechanical stress, deformation, and the rate of change of deformation can be quite complicated, although a linear approximation may be adequate in practice if the quantities are small enough. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition.
In some branches of engineering, the term stress is occasionally used in a looser sense as a synonym of "internal force". For example, in the analysis of trusses, it may refer to the total traction or compression force acting on a beam, rather than the force divided by the area of its cross-section.
Since ancient times humans have been consciously aware of stress inside materials. Until the 17th century the understanding of stress was largely intuitive and empirical; and yet it resulted in some surprisingly sophisticated technology, like the composite bow and glass blowing.
Over several millennia, architects and builders, in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as the capitals, arches, cupolas, trusses and the flying buttresses of Gothic cathedrals.
Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: Galileo Galilei's rigorous experimental method, René Descartes's coordinates and analytic geometry, and Newton's laws of motion and equilibrium and calculus of infinitesimals. With those tools, Augustin-Louis Cauchy was able to give the first rigorous and general mathematical model for stress in a homogeneous medium. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum).
The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow.
Stress is defined as the force across a "small" boundary per unit area of that boundary, for all orientations of the boundary. Being derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.
Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them.:p.90–106 Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like the grains of a metal rod or the fibers of a piece of wood.
Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S, divided by the area of S.:p.41–50 In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure. In a solid, or in a flow of viscous liquid, the force F may not be perpendicular to S; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S. Thus the stress state of the material must be described by a tensor, called the (Cauchy) stress tensor; which is a linear function that relates the normal vector n of a surface S to the stress T across S. With respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field.
Normal and shear stress
In general, the stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S. The vector T may be regarded as the sum of two components: the normal stress (compression or tension) perpendicular to the surface, and the shear stress that is parallel to the surface.
If the normal unit vector n of the surface (pointing from Q towards P) is assumed fixed, the normal component can be expressed by a single number, the dot product T · n. This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress) The shear component is then the vector T − (T · n)n.
The dimension of stress is that of pressure, and therefore its coordinates are commonly measured in the same units as pressure: namely, pascals (Pa, that is, newtons per square metre) in the International System, or pounds per square inch (psi) in the Imperial system. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for mega pascal, is a common unit of stress.The dimensional formula for stress is ML^-1T^-2
Causes and effects
Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in temperature and phase, and electromagnetic fields) act on the bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines, or points; and possibly also on very short time intervals (as in the impulses due to collisions). In general, the stress distribution in the body is expressed as a piecewise continuous function of space and time.
Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence, polarization, and permeability. The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring, tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas) by definition can only oppose deformations that would change their volume. However, if the deformation is changing with time, even in fluids there will usually be some viscous stress, opposing that change.
The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition.
In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress, the simple shear stress, and the isotropic normal stress.
Uniaxial normal stress
A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to tension by opposite forces of magnitude along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force F. Therefore, the stress throughout the bar, across any horizontal surface, can be described by the number = F/A, where A is the area of the cross-section.
On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between the two halves across the cut.
This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress change sign, and the stress is called compressive stress.
This analysis assumes the stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value = F/A will be only the average stress, called engineering stress or nominal stress. However, if the bar's length L is many times its diameter D, and it has no gross defects or built-in stress, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle).
Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to the cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid.
Simple shear stress
Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool. Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F. Assuming that the direction of the forces is known, the stress across M can be expressed by the single number = F/A, where F is the magnitude of those forces and A is the area of the layer.
However, unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that is perpendicular to the layer, the net internal force across S, and hence the stress, will be zero.
As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F/A will only be an average ("nominal", "engineering") stress. However, that average is often sufficient for practical purposes.:p.292 Shear stress is observed also when a cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges").
Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected.
In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or just pressure. Gases by definition cannot withstand tensile stresses, but some liquids may withstand surprisingly large amounts of isotropic tensile stress under some circumstances. see Z-tube.
Parts with rotational symmetry, such as wheels, axles, pipes, and pillars, are very common in engineering. Often the stress patterns that occur in such parts have rotational or even cylindrical symmetry. The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor.
Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction , and zero across any surfaces that are parallel to . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial, and can be viewed as the sum of two normal or shear stresses. In the most general case, called triaxial stress, the stress is nonzero across every surface element.
The Cauchy stress tensor
Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way.
However, Cauchy observed that the stress vector across a surface will always be a linear function of the surface's normal vector , the unit-length vector that is perpendicular to it. That is, , where the function satisfies
for any vectors and any real numbers . The function , now called the (Cauchy) stress tensor, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus, is classified as second-order tensor of type (0,2).
Like any linear map between vectors, the stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered or named , the matrix may be written as
The stress vector across a surface with normal vector with coordinates is then a matrix product (where T in upper index is transposition) (look on Cauchy stress tensor), that is
The linear relation between and follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric, that is , , and . Therefore, the stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written
where the elements are called the orthogonal normal stresses (relative to the chosen coordinate system), and the orthogonal shear stresses.
Change of coordinates
The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution.
As a symmetric 3×3 real matrix, the stress tensor has three mutually orthogonal unit-length eigenvectors and three real eigenvalues , such that . Therefore, in a coordinate system with axes , the stress tensor is a diagonal matrix, and has only the three normal components the principal stresses. If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame.
Stress as a tensor field
In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.
Stress in thin plates
Man-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies.
In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. However, these simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of the plate).
Stress in thin beams
The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis).
Other descriptions of stress
The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations where the differences in stress distribution in most cases can be neglected. For large deformations, also called finite deformations, other measures of stress, such as the first and second Piola–Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor, are required.
Solids, liquids, and gases have stress fields. Static fluids support normal stress but will flow under shear stress. Moving viscous fluids can support shear stress (dynamic pressure). Solids can support both shear and normal stress, with ductile materials failing under shear and brittle materials failing under normal stress. All materials have temperature dependent variations in stress-related properties, and non-Newtonian materials have rate-dependent variations.
Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics, vulcanism and avalanches; and in biology, to understand the anatomy of living beings.
Goals and assumptions
Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium. By Newton's laws of motion, any external forces are being applied to such a system must be balanced by internal reaction forces,:p.97 which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle, creating a stress distribution throughout the body.
The typical problem in stress analysis is to determine these internal stresses, given the external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material;:p.42–81 or concentrated loads (such as friction between an axle and a bearing, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point.
In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known constitutive equations.
Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach is often used for safety certification and monitoring. However, most stress analysis is done by mathematical methods, especially during design.
The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum) and the Euler-Cauchy stress principle, together with the appropriate constitutive equations. Thus one obtains a system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem.
Stress analysis for elastic structures is based on the theory of elasticity and infinitesimal strain theory. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (plastic flow, fracture, phase change, etc.).
However, engineered structures are usually designed so that the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke’s law for continuous media); that is, the deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear.
Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc.
Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as the finite element method, the finite difference method, and the boundary element method.
Alternative measures of stress
Piola–Kirchhoff stress tensor
In the case of finite deformations, the Piola–Kirchhoff stress tensors express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration. For infinitesimal deformations and rotations, the Cauchy and Piola–Kirchhoff tensors are identical.
Whereas the Cauchy stress tensor relates stresses in the current configuration, the deformation gradient and strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1st Piola–Kirchhoff stress tensor, is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state.
The 1st Piola–Kirchhoff stress tensor, relates forces in the present configuration with areas in the reference ("material") configuration.
In terms of components with respect to an orthonormal basis, the first Piola–Kirchhoff stress is given by
Because it relates different coordinate systems, the 1st Piola–Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The 1st Piola–Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress.
If the material rotates without a change in stress state (rigid rotation), the components of the 1st Piola–Kirchhoff stress tensor will vary with material orientation.
The 1st Piola–Kirchhoff stress is energy conjugate to the deformation gradient.
2nd Piola–Kirchhoff stress tensor
Whereas the 1st Piola–Kirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2nd Piola–Kirchhoff stress tensor relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.
In index notation with respect to an orthonormal basis,
This tensor, a one-point tensor, is symmetric.
If the material rotates without a change in stress state (rigid rotation), the components of the 2nd Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation.
The 2nd Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange finite strain tensor.
- Compressive strength
- Kelvin probe force microscope
- Mohr's circle
- Residual stress
- Shear strength
- Shot peening
- Strain tensor
- Strain rate tensor
- Stress–energy tensor
- Stress–strain curve
- Stress concentration
- Transient friction loading
- Tensile strength
- Virial stress
- Yield (engineering)
- Yield stress
- Yield surface
- Virial theorem
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- Peter Chadwick (1999), "Continuum Mechanics: Concise Theory and Problems". Dover Publications, series "Books on Physics". ISBN 0-486-40180-4. pages
- I-Shih Liu (2002), "Continuum Mechanics". Springer ISBN 3-540-43019-9
- (2009) The art of making glass. Lamberts Glashütte (LambertsGlas) product brochure. Accessed on 2013-02-08.
- Ronald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages. ISBN 9781574447132
- Walter D. Pilkey, Orrin H. Pilkey (1974), "Mechanics of solids" (book)
- Donald Ray Smith and Clifford Truesdell (1993) "An Introduction to Continuum Mechanics after Truesdell and Noll". Springer. ISBN 0-7923-2454-4
- Fridtjov Irgens (2008), "Continuum Mechanics". Springer. ISBN 3-540-74297-2
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506045.12/warc/CC-MAIN-20230921210007-20230922000007-00459.warc.gz
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CC-MAIN-2023-40
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https://www.mql5.com/en/code/7471
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math
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If the red graph is greater than zero and grows - go bullish (if at the same time the blue graph is less than zero and falls - the uptrend is getting stronger)
If the red graph is less than zero and falls - go bearish (if at the same time the blue graph is greater than zero and grows - the downtrend is getting stronger)
Common parameters - parameters that affect both lines:
Translated from Russian by MetaQuotes Software Corp.
Original code: https://www.mql5.com/ru/code/7471
Trend filter. In short, the RSIFilter fails on strong fluctuations, but handles the trend good enough, but if it get improved a bit, it could be able to give clear answers.EVWMA
Elastic Volume Weighted Moving Average (EVWMA), a natural replacement for the standard moving average.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320869.68/warc/CC-MAIN-20170626221252-20170627001252-00427.warc.gz
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CC-MAIN-2017-26
| 759 | 7 |
http://media-central.jiodjremix.in/tag/1207755.html
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math
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1 Gajam=9 Square Feet (sq ft) Gajam to Square Feet. So Now you know what is Gajam and what we call 1 Gajam in English. Since 1 Gajam is equal to 1 yard and 1 Yard is equal to 9 Square feet. so we can say that 1 Gajam is equal to 9 square feet.
In mathematical expression, 1 Gajam = 9 Square Feet (sq ft). Gajam ( గజం ) is a telugu word, is a unit of land measurement widely used in Andhra Pradesh (AP). Gajam is called Yard in English. The Gajam is composed of 9 sq ft or 1296 sq inches. It is 0.836 sq meters per Gajam.
1 Answers. Popular. Latest. Add Answer. Gajam is the unit measurement of length. It is a telugu word, mostly used in Andhra Pradesh for measuring the land area. In English, 1 Gajam = 1 Yard. In Imperial and U.S. customary units, Feet is used for measuring the length. 1 Gajam = 9 sq feet.
Gajam Unit Conversion Calculator. Gajam is one of the units used for land measurements. Given below is the online conversion calculator which is used to convert from gajam to other land measurement units. The word 'Gajam' was derived from the language Telugu and in English it is termed as Square Yard. (i.e.,) 1 Gajam = 1 Sq Yard. 1 square yard is equivalent to 9 sq feet and hence 100 gajams = 100 x 9 sq feet.
Its value is equal to the value of Sq. Yard. Generally people always ask the question like. 1 Gajam = 9 Sq. Feet. 1 Gajam = 0.83612736 Sq. Meters. 1 Gajam = 0.00000083612736 Sq. Kilometers. 1 Gajam = 1 Sq.
GAJ in Hindi, is referred to as square yard. 1 GAJ = 9 Square feet. If you want to be precise then 1 GAJ = 8.91 sq. ft. Normal practice is that, when referred to plots or empty land where there is nothing constructed, then we measure it in yards or square yards.
Therefore, one Gajam is equal to decimal point six Cent in Survey System. In mathematical expression, 1 Gajam = 0.06198347107438016 Cent . Converter / Calculator.
Land Measurements : 1 Gajam = 1 Sq Yard = 9 sq feet. 1 cent = 435.600142084 sq.feet (ft2) 1 cent = 6.05 Ankana / 48 Sq.Yards/ 48 gajams. 100 cents = 1 acre. 1 foot = 12 inch = 30.48 cm. 1 Gajam = 1 Sq Yard = 9 sq feet. 100 gajams =100Sq.yard = 900 Sq.ft. at October 16, 2015.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363405.77/warc/CC-MAIN-20211207170825-20211207200825-00289.warc.gz
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CC-MAIN-2021-49
| 2,136 | 8 |
https://breldigital.com/a-mass-on-a-string-of-unknown-length-oscillates-as-a-pendulum-with-a-period-of-6-0-s/
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math
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A mass on a string of unknown length oscillates as a pendulum with a period of 1.8 s. What is the period in the following situations?
(a) The mass is doubled?
(b) The string length is doubled?
(c) The string length is halved?
(d) The amplitude is doubled?
Period ( P) is given as
a) since mass has no effect on the period of a pendulum. So, the period will remain 1.8seconds
b) using the formula above ,period varies with the square root of the length. Thus , when the length doubles, the period is multiplied by √2. So, the period is 1.8s*√2 = 2.54s
c) in this case, the period is multiplied by √(1/2).
d) amplitude of the pendulum doesn’t affect the period (unless itsvery high, so, the period is still 1.8s
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233511364.23/warc/CC-MAIN-20231004084230-20231004114230-00041.warc.gz
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CC-MAIN-2023-40
| 717 | 10 |
https://openwetware.org/wiki/Physics307L_F07:People/Knockel/Lab2
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math
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Electron's e/m ratio summarySJK 01:44, 10 October 2007 (CDT)
The e/m ratio (charge to mass ratio) for an electron can, in theory, be measured by seeing how an electron beam behaves in a magnetic field. Using a Helmholtz coil to generate the field and an electron gun in a vacuum tube filled with helium, a beam can be created, formed into a circle with the magnetic field, and measured (since the electrons makes the helium glow).
This experiment was a disaster since there was ridiculous drag from the helium and for other reasons. The only thing I could conclude was that
Taking e/m = 3.13x1011 C/kg, there is a relative error of 78% from the actual value of 1.76x1011 C/kg.
In my lab notebook, I go into great detail explaining the theory, why there is so much systematic error, and alternative methods for finding e/m without so much error. I, of course, also explain the setup, equipment, procedure, etc., and I give my data and calculations.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948517845.16/warc/CC-MAIN-20171212173259-20171212193259-00439.warc.gz
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CC-MAIN-2017-51
| 947 | 5 |
https://geeksquad.fixya.com/support/p2520871-gemini_sound_products_gemini_xga_2000_20/page-4
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math
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20 Most Recent
- Page 4 Questions & Answers
I need to know the value of the Resistor "R1" in
Let us use a bit of logic to resolve this... I have an Axiom but it would be a lot of screws to take out to get to the part... so let us think about this. The resistor value starts with a "1" and we know it is 5% tolerance (gold). we know the resistor burnt up with likely 5 volts on it... Value could be 1000 ohms, however the voltages present (12 volts max) would NOT have burned up a resistor of 1000 ohms. The USB area is mostly 5 volts and across 100 ohms is only a quarter watt... not enough to really burn up a resistor... First thing is to MEASURE that the resistor s open... in spite of being burned, it MAY still be OK... these resistors now usually either open or remain close to their value. The resistor ALSO may be a fusible resistor intended to act as a fuse... Is the resistor open or not? is it near a resistance with a "1" as the first digit? If it is not open I suspect it is not all of the problem. In that case, look for a burned circuit trace on the board. If it is open, then MEASURE the voltage across the resistor with the power applied. If you find 5 volts, then it MIGHT be a 10 ohm used to limit the USB current to 500ma. Get back to me with your findings... I have unraveled many of these things...
on Jun 25, 2019
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s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250593295.11/warc/CC-MAIN-20200118164132-20200118192132-00199.warc.gz
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CC-MAIN-2020-05
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https://plus.maths.org/content/comment/reply/node/2179/comment_node_article/5989
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math
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943625.81/warc/CC-MAIN-20230321033306-20230321063306-00061.warc.gz
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