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https://math.answers.com/other-math/What_is_5x5x5x5_using_an_exponent
|
math
|
54 there are four 5's in the problem so you write the exepont four
5^4 = 625
Here are a couple.An exponent is a number.Math taught us about using an exponent.
20000 using the exponent of 4 = 11.89214fourth root of 20000 = 11.8921
The exponent indicates the number of times the base is used as a factor.
2 with an exponent of 3 and 3. 2x2x2=8x8=24.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446708010.98/warc/CC-MAIN-20221126144448-20221126174448-00524.warc.gz
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CC-MAIN-2022-49
| 347 | 6 |
https://climateaudit.org/2007/11/22/ross-mckitrick-on-mann-et-al-2007/?like=1&source=post_flair&_wpnonce=d91e0e6d9a
|
math
|
First, on page 15, they say:
This result (as well as the separate study by Wahl and Ammann ) thus refutes the previously made claim by MM05 that the features of the MBH98 reconstruction are somehow an artifact arising from the use of PC summaries to represent proxy data.
Well, if we had argued that the only problem in MBH98/99 is the PC error, and that the PC error alone produces the MBH hockey stick, then this paper and its triumphant conclusion might count for something. But we argued something a tiny bit more complex (though not much). The PCs were done wrong, and this had 2 effects. (1) it overstated the importance of the bristlecones in the NOAMER network, justifying keeping them in even though theyre controversial for the purpose and their exclusion overturned the results. (2) It overstated the explanatory power of the model when checked against a null RE score (RE=0), since red noise fed into Manns PC algorithm yielded a much higher null value (RE>0.5) due to the fact that the erroneous PC algorithm bends the PC1 to fit the temperature data. Manns new paper doesnt mention the reliance on bristlecones. Nobody questions that you can get hockey sticks even if you fix the PC algorithm, as long as you keep the bristlecones. But if you leave them out, you dont get a hockey stick, no matter what method you use. Nor, as far as I can tell, does this paper argue that MBH98 actually is significant, and certainly the Wahl&Ammann recalculations (http://www.climateaudit.org/?p=564) should put that hope to rest.
Second, maybe Im missing a nuance, but the diatribe against r2 (and by extension, Steve et moi, para 64) is misguided on 2 counts. They say that it doesnt reward predicting out of sample changes in mean and variance (paragraph 39). Where I work, we dont talk about test statistics rewarding estimations, instead we talk about them penalizing specific failures. The RE and CE scores penalize failures that r2 ignores. We argued that r2 should be viewed as a minimum test, not the only test. You can get a good r2 score but fail the RE and CE tests, and as the NRC concluded, this means your model is unreliable. But if you fail the r2 test, and you pass the RE test, that suggests youve got a specification that artificially imposes some structure on the out of sample portion that conveniently follows the target data, even though the model has no real explanatory power. Another thing thats misguided is their use of the term nonstationary. They say that RE is much better because it takes account of the nonstationarity of the data. Again, where I work, if you told a group of econometricians you have nonstationary data, then proceeded to regress the series on each other in levels (rather than first differences) and brag about your narrow confidence intervals, nobody would stick around for the remainder of your talk. And youd hear very loud laughter as soon as the elevator doors closed up the hall.
Third, whats missing here is any serious thought about the statistical modeling. I get the idea that they came across an algorithm used to infill missing data, and somebody thoughtwhoah, that could be used for the proxy reconstruction problemand thus was born regEM. Chances are (just a guess on my part) people developing computer algorithms to fill in random holes in data matrices werent thinking about tree rings and climate when they developed the recursive data algorithm. You need to be careful when applying an algorithm developed for problem A to a totally different problem B, that the special features of B dont affect how you interpret the output of the algorithm.
For example, in the case of proxies and temperature, it is obvious that there is a direction of causality: tree growth doesnt drive the climate. In statistical modeling, the distinction between exogenous and endogenous variables matters acutely. If the model fails to keep the two apart, such that endogenous variables appear directly or indirectly the right-hand side of the equals sign, you violate the assumptions on which the identification of structural parameters and the distribution of the test statistics are derived. Among the specification errors in statistical modeling, failure to handle endogeneity bias is among the worst because it leads to both bias and inconsistency. A comment like (para 13) An important feature of RegEM in the context of proxy-based CFR is that variance estimates are derived in addition to expected values. would raise alarm bells in econometrics. It sounds like that guy in Spinal Tap who thinks his amp is better than the others because the dial on his goes up to 11. So, the stats package spits out a column of numbers called variances. My new program is better than the old one because the dial goes up to variances.
Its just a formula computer program, and it wont tell you if those numbers are gibberish in the context of your model and data (as they would be if you have nonstationary data). You need a statistical model to interpret the numbers and show to what extent the moments and test statistics approximate the scores you are interested in.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120206.98/warc/CC-MAIN-20170423031200-00399-ip-10-145-167-34.ec2.internal.warc.gz
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CC-MAIN-2017-17
| 5,107 | 7 |
https://www.elitetrack.com/forums/reply/reply-tomaxs-rep-scheme-8/
|
math
|
Well I got confused a bit with your week 1 and 2 setup…..but it's clear now.
How is 6×2 not unloading if week 1 you did 25 reps, week 2 you did 18, week 3 15 and then this? It's a smaller amount of reps so why would it not be unloading? I guess that's my biggest thing, if total volume is dropped, does it really matter how the volume is spread out?
And, also, you are increasing intensity each week with that setup you suggested, correct?
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334992.20/warc/CC-MAIN-20220927064738-20220927094738-00340.warc.gz
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CC-MAIN-2022-40
| 442 | 3 |
https://physics.stackexchange.com/questions/335182/is-the-electron-ever-a-wave/335204
|
math
|
I'm taking on an interesting task: explaining to an extremely advanced 11 year old that much of what he hears and reads from the "popular physics" videos and TV programs isn't accurate, and some of it is just wrong. I know I have a problem here however - once he moves on to high school physics he's going to come across many of these same incorrect explanations from his teachers. I need to decide if I go so far as to explain that there is ZERO evidence showing or theorizing that an electron (for instance) is ever a wave.
He's bright as heck, and I can certainly explain a probability function, and can explain that it's a solution to a wave equation. And I can explain that this probability function is NOT the electron itself. And that this probability function can be plotted in a way that it can look like a wave or a superposition of waves. But do I stop there or continue and say simply "this does not in any way imply that the electron itself is a wave!"
If I was that kid I would have LOVED it if someone explained this to me at an early age. But I know I would have upset some teachers. Maybe only once or twice though. I'd like to know what everyone else thinks.
From the comments I realize there's a need to modify my question with the following statement:
The probability function (or superposition thereof) has meaning because it is the solution to a very special wave equation that has proven to glean incredible meaning. The electron is and never has been a solution to such a special wave equation. We can learn things about electron behavior from the solutions to this wave equation, but it makes no sense to say that the electron is a solution to this same wave equation.
Furthermore, to posit that ANY localized entity is a wave is disingenuous and misleading: Propose any thing which can be plotted and/or described as a function, and I will show you a superposition of orthogonal solutions of SOME equation that will describe it to whatever precision that seems reasonable (let's say "measurable" for these purposes). If I choose that equation to be a wave equation then I allow myself the right to call the thing a wave and give it a wavelength which quantifies some average spatial distribution. Is that meaningful? Does that help us to understand an electron? I say no.
Yes - I do strongly believe that there is NO reason to call the electron a wave ever, because there is no evidence and no theory showing it to be such a thing. There IS however much evidence showing that an electron is a particle, and that's because we measure it to be such a thing. To state that it is also a wave is nothing but disingenuous babble. If it's an electron then it's measured always as a particle.
So do you think the electron IS ever a wave? If so please state your evidence and/or theories which prove it to be such a thing.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670921.20/warc/CC-MAIN-20191121153204-20191121181204-00103.warc.gz
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CC-MAIN-2019-47
| 2,839 | 8 |
http://www.paleotechnologist.net/?p=831
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math
|
Ohm’s Law is fundamental to understanding the three-way interaction between three of the most important concepts in electronics: Voltage, Current, and Resistance. Here is a quick overview of these concepts, from a DC-circuits viewpoint:
Voltage, measured in volts, is the difference in electrical potential between two points in a circuit. Voltage is always relative; giving the voltage at a certain point in the circuit is meaningless unless you specify what the reference point is. (Similarly, it would be meaningless to give your location as ten kilometers west, unless you also specified west of where.) The relativity of voltage is the reason that birds can safely land on power lines. The power line may be at hundreds or thousands of volts with respect to the ground, but if the birds don’t contact both points, the difference is meaningless.
Current, measured in amps, is the amount of electricity flowing through a certain point of the circuit (a connection, usually a wire) per second. Since current flow is specified at a certain point, it is absolute.
Resistance is a property of materials that relates current flow to the difference in voltage. The higher the resistance, the “harder” it is for current to flow through that material — and the higher the voltage will have to be in order to force a given amount of current through the material.
Intuitively, the three quantities can be thought of in terms of plumbing. This analogy works reasonably well, but don’t expect it to hold water (pardon the pun) for more complex circuits.
- Voltage can be thought of in terms of pressure — which is always a difference between two given points.
- Current can be thought of in terms of flow rate — how much electricity (or how much water) is going past a certain point in a given amount of time.
- Resistance can be thought of in terms of restricting the flow (of electricity, or water): if you want to move water at a given flow rate through a narrow, high-resistance pipe, it will take more pressure than if the pipe were larger. Similarly, it takes a greater voltage to move a given current through a high-resistance circuit than a low-resistance one.
The basic equation for Ohm’s Law, E=IR, can be expressed in three basic ways:
- E = IR : Voltage (E, for Electromotive force) is equal to Current (I) times Resistance (R). This means that if a current of I amps is flowing through a resistance of R ohms, there will be a difference in voltage of E volts across that resistor.
- I = E/R : Current is equal to Voltage divided by Resistance. This means that if you apply a voltage of E volts across a resistance of R ohms, the current flow will be equal to I amps.
- R = E/I : Resistance is equal to Voltage divided by Current. This means that if you pass a current of I amps through a resistor, and the voltage drop is E volts across that resistor, its resistance is equal to R ohms.
By solving the right form of this equation for the variable you want to find, you can determine what resistor to use, what the voltage will be for a given current and resistance, or what the current will be for a given voltage and resistance.
Perhaps the easiest way to remember Ohm’s Law is graphically:
Take the image above and cover the variable to solve for with your hand:
- Cover E, and you get I*R: Voltage is Current times Resistance.
- Cover I, and you get E/R: Current is Voltage over Resistance.
- Cover R, and you get E/I: Resistance is Voltage over Current.
Resistors actually turn out to be extremely useful when making electronic circuits. Some of their many uses will be covered in another article.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526386.37/warc/CC-MAIN-20190719223744-20190720005744-00160.warc.gz
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CC-MAIN-2019-30
| 3,625 | 19 |
https://sheetaki.com/imcsc-function-google-sheets/
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math
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The IMCSC function in Google Sheets is useful when you need to return the cosecant of a particular complex number.
Complex numbers require specialized functions like IMCSC since regular mathematical functions cannot deal with input in the form a + bi or a + bj.
The rules for using the
IMCSC function in Google Sheets are as follows:
- The function accepts a single argument which is the complex number we would like to obtain the cosecant of.
- The function then returns the cosecant of the provided complex number.
- The cosecant of a complex number is simply the reciprocal of its sine function.
Let’s look into how we can use this function with a brief use case!
Complex numbers are made up of two coefficients: the real coefficient and the imaginary coefficient. The horizontal axis is the real axis, and the vertical axis is the imaginary axis, forming a 2D plane. The coefficients allow us to use trigonometric functions within this plane.
In this example, we have a sample dataset of complex numbers. We would like to return various trigonometric values for each complex number, including its cosecant. Using the
IMCSC function, we can easily return the cosecant of each complex number in the dataset.
Now that we know when to use the
IMCSC function let’s dive into how to use it and work on an actual sample spreadsheet.
The Anatomy of the IMCSC Function
So the syntax (the way we write) of the
IMCSC function is as follows:
Let’s dissect this thing and understand what each of these terms means:
- = the equal sign is how we start any function in Google Sheets.
- IMCSC() is our
IMCSCfunction. It computes the cosecant of a given complex number
- number is the complex number which we would like to return the cosecant of. It usually comes in the a+bi or a+bj format.
A Real Example of Using IMCSC Function
Let’s look at an actual example of the
IMCSC function being used in a Google Sheet spreadsheet.
In the screenshot below, we have a data set of complex numbers along with their corresponding cosecant values.
To get the values in Column B, we just need to use the following formula:
You can make a copy of the spreadsheet above using the link I have attached below.
We have another example seen below. In this table, we have the same dataset but we also use additional functions to retrieve the real and imaginary coefficients of the cosecant. The
IMREAL functions are useful when you need to get these coefficients.
Now that you’ve seen the
IMCSC function in Google Sheets in action, let’s start learning how to write the function ourselves!
How to Use IMCSC Function in Google Sheets
- First, we need to prepare our input. In the table below, we have our complex numbers in the first column of our worksheet.
- To start using the
IMCSCfunction, first select the cell, we will first put our function’s output. In this example, we’ll be starting with cell B2.
- Next, we just have to type the equal sign ‘=‘ to begin the function, followed by ‘IMCSC(‘.
- As seen below, a tooltip box appears with info on the
IMCSCfunction. We can click on the arrow on the top-right-hand corner of the box to minimize it if necessary.
- Next, we’ll input the argument for our function. In this case, cell A2 is the location of our first complex number.
Afterward, simply hit Enter on your keyboard to let the function return the cosecant.
- Finally, drag down the formula in cell B2 to fill out the rest of the column.
Frequently Asked Questions (FAQ)
1. Why does my formula output a #NUM! error?
A #NUM! error indicates that the given arguments are invalid and prevents the calculation from evaluating. For the
IMCSC function, the formula returns a #NUM! error if the complex number doesn’t have lower case i or j as their variable for the imaginary coefficient.
That’s all you need to know to start using the
IMCSC function in Google Sheets. After going through this step-by-step guide, you should now know how to use the
IMCSC function in Google Sheets. It will come in handy whenever you need to find the cosecant of complex numbers.
You can now use the
IMCSC functions in Google Sheets together with the various other Google Sheets formulas available to create amazing worksheets.
Stay notified of new Google Sheets guides like this by subscribing to our newsletter!
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816879.72/warc/CC-MAIN-20240414130604-20240414160604-00150.warc.gz
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CC-MAIN-2024-18
| 4,300 | 51 |
https://www.coursehero.com/sitemap/schools/1431-New-England-College/departments/300535-MBS/
|
math
|
I NEED HELP WITH THESE 2 QUESTIONS.
1. A loan of 100,000 is payable over five years with monthly payments of 60,000
commencing one month after the inception date. The loan repayment is 2,000 per month
and the nominal rate 10 per cent. How much capital rem
M ultiple Choice (2 points each) 1. The revenue recognition principle provides that revenue is recognized when a. b. c. d. it is realized. it is realizable. it is realized or realizable and it is earned. none of these.
2. The percentage-of-completion meth
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s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824325.29/warc/CC-MAIN-20171020192317-20171020212317-00465.warc.gz
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CC-MAIN-2017-43
| 511 | 6 |
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=510324
|
math
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Practical Guide to Real Options in Discrete Time
28 Pages Posted: 6 Mar 2004
Date Written: February 24, 2004
Continuous time models in the theory of real options give explicit formulas for optimal exercise strategies when options are simple and the price of the underlying asset follows a geometric Brownian motion. This paper suggests a general, computationally simple approach to real options in discrete time. Explicit formulas are derived even for embedded options. Discrete time processes reflect the scarcity of observations in the data, and may account for fat tails and skewness of probability distributions of commodity prices. The method of the paper is based on the use of the expected present value operators.
Note: An updated version of this abstract can be found at: http://ssrn.com/abstract=642262
Keywords: Real options, embedded options, expected present value operators
JEL Classification: D81, C61, G31
Suggested Citation: Suggested Citation
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s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371700247.99/warc/CC-MAIN-20200407085717-20200407120217-00399.warc.gz
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CC-MAIN-2020-16
| 960 | 8 |
https://thelawdictionary.org/graph/
|
math
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Depiction of data point relationships in a two-dimensional drawing. The relationship, most commonly between two set of numbers, displays in several different ways. It can be a line, a curve, a series of bars, or some other group of symbols. A common drawing starts with an independent variable represented on the horizontal line, called the X-axis. A dependent variable goes on the vertical line, or Y-axis. The X and Y axis intersect perpendicular at a point called the origin. Each axis is calibrated in some unit value for the quantities to be represented. Taken to mean the same as ‘chart’.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499966.43/warc/CC-MAIN-20230209112510-20230209142510-00515.warc.gz
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CC-MAIN-2023-06
| 598 | 1 |
https://www.reference.com/world-view/types-problem-solving-techniques-e92f748917f041b2
|
math
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Some problem-solving techniques include algorithms, heuristics, trial-and-error and insight. An algorithm is a step-by-step approach that always yields the correct solution but can be very time-consuming. Heuristics rely on assumptions and rules of thumb that may not be correct for all situations.
Trial-and-error may work if there are only a few possible solutions to the problem. If there are many solutions, using heuristics to reduce the possible solutions to a more manageable number can allow trial-and-error techniques to work quite well. Insight is often used to solve problems, but the underlying mental processes that occur are unknown. Brainstorming sessions are intended to facilitate the use of insight as a problem-solving technique.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672548.33/warc/CC-MAIN-20191017022259-20191017045759-00019.warc.gz
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CC-MAIN-2019-43
| 748 | 2 |
https://ideasna.com/a2-b2/
|
math
|
The Pythagorean theorem is one of the most fundamental and elegant concepts in mathematics. It provides a simple relationship between the sides of a right triangle, allowing us to calculate unknown lengths and understand the geometric properties of these triangles. At the heart of this theorem lies the expression a^2+b^2, which holds immense power and significance in various fields of study. In this article, we will delve into the depths of a^2+b^2, exploring its origins, applications, and the profound impact it has had on our understanding of the world.
The Origins of the Pythagorean Theorem
The Pythagorean theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. Born in the 6th century BCE, Pythagoras founded a school of thought that emphasized the importance of mathematics and its role in understanding the universe. The Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, was one of the key principles taught by Pythagoras and his followers.
The theorem itself can be expressed as:
a^2 + b^2 = c^2
Where ‘a’ and ‘b’ represent the lengths of the two shorter sides (the legs) of the right triangle, and ‘c’ represents the length of the hypotenuse.
Applications in Geometry
The Pythagorean theorem has numerous applications in geometry, allowing us to solve for unknown lengths and angles in right triangles. By rearranging the equation, we can isolate any of the variables to find the missing value. For example, if we know the lengths of the two legs, we can calculate the length of the hypotenuse using the equation c = √(a^2 + b^2).
Additionally, the Pythagorean theorem enables us to determine whether a triangle is a right triangle or not. If the equation a^2 + b^2 = c^2 holds true for the given side lengths, then the triangle is a right triangle. This property is particularly useful in geometry proofs and constructions.
While the Pythagorean theorem is a fundamental concept in mathematics, its applications extend far beyond the realm of geometry. This powerful equation finds practical use in various fields, including architecture, engineering, physics, and even computer graphics.
Architecture and Engineering
In architecture and engineering, the Pythagorean theorem is essential for ensuring structural stability and accuracy in building design. It allows architects and engineers to calculate the lengths of diagonal supports, determine the angles of intersecting walls, and ensure that structures are level and square.
For example, when constructing a staircase, the Pythagorean theorem is used to calculate the length of the diagonal stringer, which supports the steps. By applying the theorem, architects and engineers can ensure that the staircase is structurally sound and meets safety standards.
Physics and Mechanics
In the field of physics, the Pythagorean theorem plays a crucial role in understanding the motion of objects and the forces acting upon them. It is particularly relevant in mechanics, where it is used to analyze the components of forces and determine their resultant magnitude.
For instance, when studying projectile motion, the Pythagorean theorem is used to break down the initial velocity of an object into its horizontal and vertical components. By understanding these components, physicists can accurately predict the trajectory and landing point of a projectile.
In the realm of computer graphics, the Pythagorean theorem is employed to calculate distances and angles in three-dimensional space. This is essential for rendering realistic images, creating 3D models, and simulating virtual environments.
For example, in a video game, the Pythagorean theorem can be used to determine the distance between a player and an object in the virtual world. By calculating this distance, the game engine can apply appropriate visual and audio effects, enhancing the player’s immersion and overall gaming experience.
Case Studies and Examples
Let’s explore a few real-world examples that highlight the practical applications of the Pythagorean theorem:
Example 1: The Distance Formula
The distance formula, derived from the Pythagorean theorem, allows us to calculate the distance between two points in a coordinate plane. Given two points (x1, y1) and (x2, y2), the distance between them can be found using the equation:
d = √((x2 – x1)^2 + (y2 – y1)^2)
This formula is widely used in navigation systems, GPS technology, and map-making, enabling us to determine the shortest distance between two locations on Earth.
Example 2: Right Triangle Roof Design
When designing a roof with a triangular shape, the Pythagorean theorem is employed to ensure that the roof is structurally sound and aesthetically pleasing. By using the theorem, architects and engineers can calculate the length of the roof’s diagonal supports, ensuring that they are of sufficient strength to withstand external forces such as wind and snow.
Q: Can the Pythagorean theorem be applied to non-right triangles?
A: No, the Pythagorean theorem is only applicable to right triangles, where one angle measures 90 degrees.
Q: Are there any alternative forms of the Pythagorean theorem?
A: Yes, there are several alternative forms of the Pythagorean theorem, such as the Law of Cosines and the Law of Sines, which can be used to solve triangles that are not right triangles.
Q: Can the Pythagorean theorem be extended to higher dimensions?
A: While the Pythagorean theorem is specifically formulated for two-dimensional right triangles, it can be extended to higher dimensions using the concept of vector spaces and the dot product.
Q: Are there any historical applications of the Pythagorean theorem?
A: Yes, the ancient Egyptians and Babylonians were aware of the Pythagorean theorem long before Pythagoras. They used it to construct right angles in their architectural and surveying practices.
Q: How has the Pythagorean theorem influenced other areas of mathematics?
A: The Pythagorean theorem has had a profound impact on various branches of mathematics, including trigonometry, calculus, and linear algebra. It serves as a foundation for many mathematical concepts and proofs.
The Pythagorean theorem, encapsulated
|
s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817455.17/warc/CC-MAIN-20240419203449-20240419233449-00824.warc.gz
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CC-MAIN-2024-18
| 6,336 | 37 |
https://www.physicsforums.com/threads/loop-de-loop-normal-force-roller-coaster.300769/
|
math
|
hello, i'm trying to study for the mcat, and I have a conceptual question about normal force, mg, and centripetal force during a loop-de-loop on a roller coaster. Could you validate these force equations? 1. At the very bottom of the loop: N - mg = ma = mv^2 / r N = mg + ma 2. At the side of the loop: N = ma = mv^2 / r the normal force is providing all of the centripetal acceleration 3. At the very top of the loop: N + mg = ma = mv^2 / r N = ma - mg Thus the normal force would be the greatest at the bottom of the loop, and least at the top of the loop. Is all of this correct? also, for the very top of the loop, since normal force and weight are directed downward, what force prevents the cart from just dropping off the tracts? in relation to the previous question, what is happening on the side of the loop? thank you very much !
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s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201922.85/warc/CC-MAIN-20190319073140-20190319095140-00492.warc.gz
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CC-MAIN-2019-13
| 838 | 1 |
https://millerequip.net/pre-owned-equipment/
|
math
|
Please check out our current pre-owned agriculture equipment below.
Tag#G17735 SN:AT5A11924 A91 Package Joystick Controls 4** hours (subject to change with rental) Power Bobtach 80″ bucket
Tag#G17872 SN: A7PU11394 Open Station Standard Controls 2,205 hours (subject to change with rental) 68″ bucket
Tag#G17856 SN: 527717177 2,9** hours (subject to change with rental) Cab, Heat+A/C Standard Controls 72″ Tooth Bucket Power Bobtach
Tag#G17727 SN: YCZ50694 MY: 2012 35′ working width spare sickle
Tag#17758 MY: 2009 Air cooled gas engine 61″ deck 1,4** hours new seat, new tires
Tag#17592, 35′ working width, lift assist wheels on wings.
Tag#R2426, 2012, Kinze 3600 16/32, liquid fertilizer, electric pump, flow monitor, row clutches, corn meters, bean meters, no till coulters, rubber closing wheels, spring down pressure, ALL NEW BLADES AND SCRAPERS.
Tag#G17629, ALL NEW BLADES AND BEARINGS, 24.5′ working width, rear hitch, 3 bar spike Remlinger harrow.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402101163.62/warc/CC-MAIN-20200930013009-20200930043009-00028.warc.gz
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CC-MAIN-2020-40
| 969 | 9 |
https://www.coursehero.com/file/6760420/hw1/
|
math
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Unformatted text preview: p and tails with probability q = 1-p . We toss the coin n times. What is the probability that the number of heads and the number of tails are equal? 4. Let A and B be events with probabilities P ( A ) = 3 / 4 and P ( B ) = 1 / 3. ±ind the smallest and the largest possible values of P ( A ∩ B ) and of P ( A ∪ B ). 5. Let A 1 and A 2 be events. Prove that P ( A 1 ∩ A 2 ) ≥ P ( A 1 ) + P ( A 2 )-1. Prove that in general P ( A 1 ∩ . . . ∩ A n ) ≥ n s i =1 P ( A i )-( n-1) for events A 1 , . . . , A n in a probability space. 1...
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- Winter '12
- Probability, Probability theory, Prime number, Probability space
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CC-MAIN-2017-51
| 670 | 4 |
http://wow.joystiq.com/profile/3347802/
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math
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Dec 30th 2010 5:21PM :v
Dec 30th 2010 5:15PM :v
Sep 9th 2010 8:23PM pffft borean tundra was merely a setback
Sep 4th 2010 3:14PM i was thinking the EXACT same thing
Sep 1st 2010 5:56PM RAWRomg?
Aug 29th 2010 6:47PM a wizard did it?
Aug 25th 2010 12:29PM My favorite quote is probably from Mr Scourgelord Tyrannus in POS
"Alas, brave, brave adventurers..." and so on
and I always lol a bit when he gets to the part about the rabble :P
Aug 12th 2010 10:52PM I lol'd
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s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122276676.89/warc/CC-MAIN-20150124175756-00156-ip-10-180-212-252.ec2.internal.warc.gz
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CC-MAIN-2015-06
| 463 | 10 |
https://scholar.archive.org/work/dln3l5wdnvgfrl6d4x56orpzhq
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math
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A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2018; you can also visit the original URL.
The file type is
Why College or University Students Hate Proofs in Mathematics?
Journal of Mathematics and Statistics
Problem Statement: A proof is a notoriously difficult mathematical concept for students. Empirical studies have shown that students emerge from proof-oriented courses such as high-school geometry, introduction to proof, complex and abstract algebra unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof and cannot determine whether a purported proof is valid. A proof is a convincing method that demonstrates withdoi:10.3844/jms2.2009.32.41 fatcat:paglef2i3vevbphxknb3kfhp4y
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948858.7/warc/CC-MAIN-20230328104523-20230328134523-00124.warc.gz
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CC-MAIN-2023-14
| 835 | 5 |
https://simple.m.wikipedia.org/wiki/Exponent
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math
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For example, in the number , 5 is the base and 4 is the exponent. This can be read as "5 to the power of 4". Therefore, in this example, four copies of 5 are multiplied together, which means that .
Bases raised to the power of two, , can be read the base "squared", because the area of a square with a side length of x is .
Similarly, bases raised to the power of three, , can be called the base "cubed", because the area of a cube with a side length of x is .
Exponents are used in algebra. Exponentiation is a way of shortening the process/equation of repeatedly multiplying many copies of a number.
An exponent is a number or symbol, placed above and to the right of the expression to which it applies, that indicates the number of times the expression is used as a factor. For example. the exponent x in indicates x copies of are multiplied together.
There are some basic rules used in exponentiation
1. Product Rule:
Proof - When multiplying a base term by two different exponents , there are m number of a's and n number of a's.
Since exponentiation is simply repeated multiplication of a base term, we get that:
For example, . This is true because at first, we had 2 threes times 4 threes which, when multiplied together, gives us 3 multiplied by itself 6 times, or 4+2 times.
2. Quotient Rule:
Proof (when m > n) -
By grouping the corresponding a’s, we get:
Since corresponding a’s become 1, we are then left with
For example, . Notice that , which confirms the proof.
In fact, this property holds when m < n as well, but in that case, we get negative exponents instead. When m < n, the format of the problem becomes .
For more information on negative exponents, see § Negative exponents below.
3. Zero Rule:
Proof: we showed in #1 that .
This equation holds true for as well, so
If this statement is true where , then must equal one. That is:
Negative exponents change
A negative exponent is the reciprocal of a number with a positive exponent which can be mathematically represented as In the same way that regular exponents are considered repeated multiplication, negative exponents can be considered repeated division.
Negative exponents can also be different from -1. In this case the negative exponent can be separated from the positive exponent, so
The formula for a negative exponent in terms of a whole number exponent is There are more than one ways to prove this. The first proof involves a pattern form. It is not always a good idea to use patterns because in cases like exponents they may not be complete patterns for every case, but the proof still holds.
Proof #1: Why is ?
- ***(see zero rule for proof of this!)
divide both sides by
The second proof proves exactly the same thing but dives further into matching equalities
everything in the next row is proved by the rules in the exponent section
If we also have a multiplication, like in , it would be written as
Complex exponentiation change
This formula basically states that the multiplication of complex numbers is related to addition of angles.
There are some relationships between functions and “infinite degree polynomials”, that are called series representations of the functions. The important series representations that we will focus on are:
From these representations, you should notice that the terms of the sine and cosine series combine, along with some sign changes, to make the exponential series over factorial. By introducing the number i into the equation, we get
This result is often used as the beginning of the definition for complex number exponents.
There are some cases where we are raising other numbers to the power i. For example, we can use the following relationship . Here using base 2 and exponent i, we get
When using base i and exponent i, we get
Now we can look at a general formula.
We can rewrite the in the trigonometric form .
By the relationship we found between trigonometry and complex exponent, we can rewrite the equation in complex exponential form, .
Substituting this equation in the exponent in our original equation, we get
This can be simplified to be
The real and imaginary parts of the exponent can be simplified separately to obtain to result
Because there are many ways to describe the angle P, this formula can give many answers. If we allow the variable P to vary by multiples of , we get the final result
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474671.63/warc/CC-MAIN-20240227053544-20240227083544-00415.warc.gz
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CC-MAIN-2024-10
| 4,347 | 45 |
https://www.teacherspayteachers.com/Product/Perfect-Traders-Break-Even-Version-Algebra-Grades-8-12-615912
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math
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This is a quality math/economics trading game that teaches students how to
1) Evaluate a linear equation y = mx + b
2) Read Graphs
3) Read a table
4) Deductively figure out what the break even point is between their trading function and the other players.
30+ trader cards are included.
Rules are very easy to learn.
Record keeping sheet for easy student play, included.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159359.58/warc/CC-MAIN-20180923114712-20180923135112-00402.warc.gz
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CC-MAIN-2018-39
| 370 | 8 |
https://highschoolnotes.com.au/notes/773
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math
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COMPUTATION OF SQUARE ROOT The square root of a given number is another number which when multiplied with itself results in the given number. Similarly, the square root of a given polynomial P(x) is another polynomial Q(x) which when multiplied by itself gives P(x). In the earlier classes you learnt to find square root of polynomials by factorisation method. Here, we find square root by the method of division. For polynomials of higher degree the method of division is very much useful. This method is similar to the division method of finding the square root of numbers.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573029.81/warc/CC-MAIN-20220817153027-20220817183027-00202.warc.gz
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CC-MAIN-2022-33
| 575 | 1 |
http://www.jiskha.com/members/profile/posts.cgi?name=Albert&page=3
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math
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what is the force applied in the direction of the wire at a point attached to the patient in a traction system if the mass applied ius equal to 5kg.?
A rectangle field with area of 1,500,000 needs to be fenced. 2W+2L=5,000. solve for L and W
when 25.0 mL of a solution containing both Fe2+ and Fe3+ ions is titrated with 23.0 mL of 0.0200 M KMnO4 (in dilute sulfuric acid). As a result, all of the Fe2+ ions are oxidized to Fe3+ ions. Next, the solution is treated with Zn metal to convert all of the Fe3+ ions to Fe2+ ...
calculate the energy associated with a photon whose wavelength is 14.5*10^-8cm.
What generally accepted Accounting Principle(s) dominate the issue of asset valuation?
Estimate products of fractions. 2/9x26
were do the labels go for a 6th grade science project
f(x)=ln(e^x-e^-x), x>0 By applying in turn the Composite and Quotient Rules, find derivative and second derivative. Thanks.
The mean weight (1.0042kg)of the contents of samples of 30 bags of sugar has standard error 0.008kg. Choose the option that is closest to the probability, to three decimal places, that the mean weight of the contents of samples of 30 bags of sugar will be 1kg or more. Options ...
For Further Reading
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s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703293367/warc/CC-MAIN-20130516112133-00090-ip-10-60-113-184.ec2.internal.warc.gz
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CC-MAIN-2013-20
| 1,207 | 10 |
https://www.faadooengineers.com/threads/18932-Finite-element-analysis-ebook-download-pdf
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math
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Gender: : Male
Branch: : Mechanical Engineering
City : CoimbatoreSend Friend Request
The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to partial differential equations (PDE) and their systems, as well as integral equations. In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other. Its practical application is often known as finite element analysis (FEA).
Branch: : Mechanical EngineeringSend Friend Request
Very Useful for students like me... Thanks for the uploader. :p
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s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948579564.61/warc/CC-MAIN-20171215192327-20171215214327-00333.warc.gz
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CC-MAIN-2017-51
| 598 | 6 |
http://www.solutioninn.com/the-survey-of-drugstore-customers-mentioned-in-exercise-18-also
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math
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Question: The survey of drugstore customers mentioned in Exercise 18 also
The survey of drugstore customers mentioned in Exercise 18 also includes data about staff friendliness. Based on the survey, can Lynda Parks conclude that fewer than 5% of customers rate staff friendliness as "poor"? Use a 4% significance level.
Relevant QuestionsA gasoline refinery has developed a new gasoline that, theoretically, reduces gas consumption per 100 kilometres driven (or, to use a non-metric phrase, provides improved gas mileage).The company wants to gather evidence to ...A college surveys incoming students every year. An excerpt of one such survey, for Business students only, is available in MyStatLab. For this question, consider these particular incoming students as a random sample of all ...A survey of the morning beverage market shows that the primary breakfast beverage of 17% of Americans is milk. A Canadian dairy company believes the figure is higher in Canada. The company contacts a random sample of 500 ...Return to the confidence interval you constructed for Develop Your Skills 8.2, Exercise 6. The owner of the supermarket claims that the average household grocery bill is $95. Does the sample evidence support the owner's ...You want to know the average grade on a statistics test, but the teacher will not provide the information. You take a random sample of the marks of 20 students who wrote the test. The results are available in MyStatLab. ...
Post your question
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| 1,480 | 4 |
http://www.acperugiafootballacademy.it/library/a-handbook-of-real-variables-with-applications-to-differential-equations-and
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math
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Steven G. Krantz's A Handbook of Real Variables: With Applications to PDF
By Steven G. Krantz
This concise, well-written guide offers a distillation of genuine variable theory with a selected concentrate on the subject's major functions to differential equations and Fourier research. abundant examples and short explanations---with only a few proofs and little axiomatic machinery---are used to spotlight all of the significant result of actual research, from the fundamentals of sequences and sequence to the extra complex innovations of Taylor and Fourier sequence, Baire type, and the Weierstrass Approximation Theorem. Replete with real looking, significant purposes to differential equations, boundary price difficulties, and Fourier research, this specific paintings is a pragmatic, hands-on handbook of genuine research that's perfect for physicists, engineers, economists, and others who desire to use the end result of actual research yet who don't unavoidably have the time to understand the entire theory. invaluable as a entire reference, a learn consultant for college kids, or a brief assessment, "A guide of genuine Variables" will profit a large audience.
Read or Download A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis PDF
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This ebook serves as an advent to calculus on normed vector areas at a better undergraduate or starting graduate point. the must haves contain simple calculus and linear algebra, in addition to a definite mathematical adulthood. the entire vital topology and useful research subject matters are brought the place useful.
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A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis by Steven G. Krantz
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CC-MAIN-2018-51
| 3,069 | 11 |
https://soghdilar.home.blog/2019/05/07/the-cohen-haplotype/
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math
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Last week i had a look at the Cohen haplotypes and looked if we can deduce something from the origin of the Cohen haplotypes. I had a look to see if we can deduce something from the numbers. I chose to take the people of Cohen page of ftdna. This probably gives a representative impression of the people that call themselves Cohen. I determined from each person that has more than 12 markers to see what distance (using McGee) they have to the modals as i found for the different Ashkenazi groups. For each person i determined to which Ashkenazi group they are closest; in case their distance is more than 1500 years, i disregard the association to a Ashkenazi group.Â
So i got numbers of Cohen people for each group. I subsequently divided the numbers by the number of Ashkenazim that i found in my collection on my website (for each group). In case Cohen was a noise detection (not related to any group) these ratios should be constant (with some statistical noise). In case all Cohen descended from one person, those group should have equal numbers, and they should have a tmrca less than 3000 ybp.Â
I omitted the tiny groups (at least 10 in my collection), since the statistical noise is too large. I required that at least persons were found in the ftdna Cohen project. The distribution is shown in the diagram.
The result is that eight groups had Cohen presence:
All other large groups had low Cohen numbers:
Notice: on the ftdna project page were 92 persons that were not close (as defined above) to any of the Ashkenazim groups. A first impression suggests that they belong to other Jewish groups.
My conclusions: it is clear that the Cohen tradition was strong and was (to large degree) maintained for the Ashkenazi period. Given the numbers and diagram it seems reasonable that the Cohen tradition is strong in 8 Ashkenazi branches. These 8 groups have 5 ancestors at the start of Judaism: the three groups above of J1 have one ancestor at the start of Judaism (and are the majority of Cohen Ashkenazim). The two J2a-L25 groups have one ancestor at the start of Judaism. J2a-L70-2nd, J2b-L283 and R1b-U152 have their own ancestor at start of Judaism. The R1b-U152 ancestor was probably not in Judea at the start of Judaism.
It is well possible that four out of the five ancestors were in Judea at the start of Judaism (J1-ZS227, J2a-L70, J2a-L25 and J2b-L283). There is no reason why one branch is more likely to be the original Cohen branch than the others. A shared ancestors of these four branches was long before the start of Judaism.
A similar analysis is not possible for the Levi haplotype. The majority of Levi are member of R1a-M582. No ftdna project of the Levi group exists that has members of other haplotypes. Ashkenazi Levite is only for R1a-M582 members. A possible follow-up could be to use Levin at worldfamilies.net.
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CC-MAIN-2021-17
| 2,848 | 9 |
https://lists.defectivebydesign.org/archive/html/help-octave/2020-01/msg00030.html
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math
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[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: It is very difficult for Octave to calculate the inverse Laplace tra
Re: It is very difficult for Octave to calculate the inverse Laplace transform of some functions
Mon, 6 Jan 2020 01:45:58 -0800
Mozilla/5.0 (X11; Linux x86_64; rv:68.0) Gecko/20100101 Thunderbird/68.3.1
On 2020-01-06 12:28 a.m., billyandriam wrote:
I had no intention to torture test Gnu Octave. I had this Exercise from my
Circuit Analysis Textbook
which I was trying to solve. It is about circuit Analysis and one has to
find Mesh currents from a planar circuit.
In the last step, one has to find the inverse Laplace transform of the two
I only found out that Gnu Octave takes a huge amount of time to calculate
the inverse Laplace transform of some functions. Yesterday night, I was
going to bed and started the calculation only to find out in the morning
that it was not yet done calculating.
Here is a screenshot:
I got the job done using Wolfram Alpha but, that leaves me a big concern. I
don't know if I can rely on Octave's symbolic calculation abilities if I
happen to encounter a comparable question in my exam and if Octave stalls.
Octave's Symbolic package relies on Sympy. While it can do transforms,
its perhaps not its strongest feature.
Some things you could try:
1. Make sure you're running a recent version of Sympy (1.4 or 1.5)
2. Try to isolate the simplest problem for which it fails, then you or
I can report it upstream at https://github.com/sympy/sympy/issues
(For example, there are very large integers in the screenshot you
provided: does it also fail for rational expressions of quadrics with
Re: It is very difficult for Octave to calculate the inverse Laplace transform of some functions, Doug Stewart, 2020/01/06
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| 1,790 | 28 |
https://practice.freshcara.net/chapter-8-go-math-3rd-grade/
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math
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What fraction names point A on the number line. Each shape is divided into equal parts.
Which shape shows thirds.
Chapter 8 go math 3rd grade. They share some food. Common Core Grade 4 HMH Go Math Answer Keys. The main aim of providing the Go Math Answer Key for Grade 3 Chapter 8 is to make the students understand the concepts in an easy manner.
What fraction names the shaded part of the shape. The format is similar to that of the actual tests with different questions presented for practice. Chapter 4 Divide by 1-Digit Numbers.
This 60 slide powerpoint is designed to coordinate with each lesson in chapter 8 of the Go Math 3rd grade curriculum. This is a 4 page overview of the concepts presented in Chapter 8 of the Go Math series for third grade. They are clear and to the point and use the same format and directions from the program.
Preview 8 questions Show answers. Chapter 5 Factors Multiples and Patterns. They are based more on the on your own sections from each lesson.
Students of Grade 3 can get a strong foundation on mathematics concepts by referring to the Go Math Course Book. Which shape shows thirds. Mary and her 3 friends go on a picnic.
It was developed by highly professional mathematics educators and the solutions prepared by them are in a concise manner for easy grasping. 3 Student Edition grade 3 workbook answers help online. Chapter 2 Multiply by 1-Digit Numbers.
Go math grade 3 chapter 8 pdf. Show solution Question 2. Please share this page with your friends on FaceBook.
Go Math Answer Key for Grade 3. The format is similar to that of the actual tests with different questions presented for practice. 3_ 6 Reteach Chapter Resources 8-11 Reteach.
One sixth of the photos are of giraffes. The understanding fractions chapter includes various topics like Equal parts of a whole unit fractions relate fractions and whole numbers. This is a math test designed from the Go Math curriculum for third grade.
The Go Math Grade 3 Answer Key Chapter 8 Understand Fractions Extra Practice helps to test skills in this chapter. Mark all that apply. 183 6.
Enter the email address you signed up with and well email you a reset link. Show how they can make the fewest cuts possible. Houghton Mifflin Harcourt ISBN.
Go Math Grade 4 Answer Key. We have provided the images for the questions for understanding. We even considered images for a few questions for a better understanding of.
You can see the topics like Equal parts of a whole unit fractions relate fractions and whole numbers. Go Math Grade 3 Answer Key Chapter 8 Understand Fractions contains all the topics which help the students to score better marks in. Contains all the topics which help the students to score better marks in the exams.
Each shape is divided into equal parts. 31 Count Equal Groups 32 Relate Addition and Multiplication 33 Multiply with 2 34 Multiply with 4 35 Draw a Diagram – Multiplication 36 Model with Arrays 37 Commutative Property 38 Multiply with 1 and 0 39 Multiply with 5. Caleb took 18 photos at the zoo.
Chapter 1 Place Value Addition and Subtraction to One Million. We will learn to identify. The questions cover all the topics and lessons from the chapter but are much clearer than the current tests.
3 Student Edition Publisher. Chapter 3 Subpages 9. Go Math Grade 3 Answer Key Chapter 8 Understand Fractions Extra Practice helps you to test your preparation standards.
This is a 4 page overview of the concepts presented in Chapter 8 of the Go Math series for third grade. Chapter 3 Multiply 2-Digit Numbers. Patterns on a Hundred Chart Use the hundred chart.
Go Math Chapter 8 Study Guide Grade 3. Grade 3 Go Math Practice – Answer Keys. Grade 3 Chapter 8 Understand Fractions.
Topics included are81 Equal Parts of a Whole82 Equal Shares83 Unit Fractions of a Whole84 Fractions of a Whole85 Fractions on a Number Line86 Relate Fractions and W. Go Math Chapter 8 Study Guide Grade 3. Go Math Grade 3 Chapter 8 Answer Key Pdf.
Welcome to Go Math Grade 5. Question 1 request help Each shape is divided into equal parts. Select the shapes that show thirds.
This is a 4 page overview of the concepts presented in Chapter 8 of the Go Math series for third grade. Thus he walked 6 dogs today. During the next few weeks our math class will be learning about fractions.
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| 4,290 | 23 |
http://www.chegg.com/homework-help/definitions/incremental-cost-approach-37
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math
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The incremental-cost approach is a management approach focused on examining how costs change based on potential alternatives. For example, a company may want to know how much engineering costs could be reduced if a certain percentage of engineering work were to be outsourced. When analyzing outsourcing options, the company will try to look at only the relevant costs (those that actually differ between the alternatives). It will also ignore sunk costs, which are costs that have already been incurred. It should also include any opportunity costs, which are a benefit given up when choosing an alternative option.
Clarks Inc., a shoe retailer, sells boots in different styles. In early November the company starts selling “SunBoots” to customers for $75 per pair. When a customer purchases a pair of SunBoots, Clarks also gives the customer a 30% discount coupon for any additional future purchases made in the next 30 days. Customers can’t obtain the discount coupon otherwise. Clarks anticipates that approximately 20% of customers will utilize the coupon, and that on average those customers will purchase additional goods that normally sell for $100.
1. How many performance obligations are in a contract to buy a pair of SunBoots?
Prepare a journal entry to record revenue for the sale of 1,200 pairs of SunBoots, assuming that Clarks uses the residual method to estimate the stand-alone selling price of SunBoots sold without the discount coupon.
Chipper Corporation had sales of $540,000 which generated an income of $50,000. Chipper’s invested capital was $500,000. What was Chipper’s return on investment?
Alvin Company has sales of $280,000 and an income of $58,800. The company’s invested capital is $336,000. Calculate Alvin’s profit margin, turnover, and return on investment.
Teddy Company has NOPAT of $80,000. The company’s cost of capital is 12% and its invested capital totals $1,200,000. What is Teddy Company’s residual income?
Top management of the Pearl Corp. is trying to construct a performance evaluation system to use to evaluate each of its three divisions. Financial data are as follows:
Division A Division B Division C
Total assets $830,000 $10,700,000 $6,375,000
Noninterest-bearing current liabilities 30,000 1,250,000 500,000
NOPAT 102,000 950,000 780,000
Tax rate 40% 40% 40%
Required rate of return 9% 12% 15%
How would the divisions be ranked (from best to worst performance) if the evaluation were based on net income compared to return on investment? Is management more likely to over or underinvest under each of these?•
Wiengot Antennas, Inc., produces and sells a unique type of TV antenna. The company has just opened a new plant to manufacture the antenna, and the following cost and revenue data have been provided for the first month of the plant�s operation. Beginning inventory 0 Units produced 45,000 Units sold 40,000 Selling price per unit $79 Selling and administrative expenses: Variable per unit $3 Fixed (total) $ 449,000 Manufacturing costs Direct materials cost per unit $15 Direct labor cost per unit $8 Variable manufacturing overhead cost per unit $3 Fixed manufacturing overhead cost (total) $ 765,000 Because the new antenna is unique in design, management is anxious to see how profitable it will be and has asked that an income statement be prepared for the month. Required: 1. Assume that the company uses absorption costing. a. Determine the unit product cost. (Omit the "$" sign in your response.) Unit product cost $ b. Prepare an income statement for the month. (Input all amounts as positive values except losses which should be indicated by a minus sign. Omit the "$" sign in your response.) Absorption Costing Income Statement Sales $ ? Cost of goods sold $ ? Gross margin $ ? Selling and administrative expenses $ ? Net operating income (loss) $ ? 2. Assume that the company uses variable costing. a. Determine the unit product cost. (Omit the "$" sign in your response.) Unit product cost $ ? b. Prepare a co•
Payback Period and NPV: Taxes and Straight-Line Depreciation
Assume that United Technologies is evaluating a proposal to change the company's manual design system to a computer-aided design (CAD) system. The proposed system is expected to save 9,000 design hours per year; an operating cost savings of $45 per hour. The annual cash expenditures of operating the CAD system are estimated to be $200,000. The CAD system requires an initial investment of $550,000. The estimated life of this system is five years with no salvage value. The tax rate is 35 percent, and United Technologies uses straight-line depreciation for tax purposes. United Technologies has a cost of capital of 14 percent.
(a) Compute the annual after-tax cash flows related to the CAD project.
(b) Compute each of the following for the project:
1. Payback period (in years). Round your answer to 2 decimal places.
2. Net present value. (Round answer to the nearest whole number.)
Please explain how you got the answers.•
NPV and IRR:
Equal Annual Net Cash Inflows
Apache Junction Company is evaluating a capital expenditure proposal that requires an initial investment of $15,666, has predicted cash inflows of $3,000 per year for 17 years, and has no salvage value.
(a) Using a discount rate of 16 percent, determine the net present value of the investment proposal. (Round to the nearest whole number.)
(b) Determine the proposal's internal rate of return.
(c) What discount rate would produce a net present value of zero?
Marlene Grady and Pauline Monroe are partners engaged in operating The G&M Doll Shop, which has employed the following persons since the beginning of the year:Grady and Monroe are each paid a weekly salary allowance of $950.
The doll shop is located in a state that requires unemployment compensation contributions of employers of one or more individuals. The company is subject to state contributions at a rate of 3.1% for wages not in excess of $8,100. Compute each of the following amounts based upon the 41st weekly payroll period for the week ending October 10, 2014
V. Hoffman (general office worker) $1,700 per month
A. Drugan (saleswoman) $15,000
G Beiter (stock clerk) $180 per week
S. Egan (deliveryman) $220 per week
B. Lin (cleaning and maintenance Part-time) $160 per week•
On January 1, 2014, Park Corporation sold a $600,000, 7.5 percent bond issue (8.5 percent market rate). The bonds were dated January 1, 2014, pay interest each June 30 and December 31, and mature in four years.
Show how the bond interest expense and the bonds payable should be reported on the June 30, 2014, income statement and balance sheet. (Amounts to be deducted should be indicated by a minus sign.)
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https://www.physicsforums.com/threads/blue-skies.16631/
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math
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Our perception of a blue sky has more to do with Rayleigh scattering than refraction. Due to dipole moments in the molecules that make up the atmosphere, incident light (from the sun) is scattered. The intensity of this scattering is proportional to (1/λ)4, so that shorter wavelengths are scattered more strongly. (Since the eye has cones that detect red, green, and blue light, the net effect is that we perceive the sky to be blue.)
Thanks for the reminder. Can you further detail the dipole mechanism of Rayleigh scattering? (Isn't scattering a type of refraction?)
I guess from the standpoint of photon-electron interaction, reflection, refraction, and scattering are all similar/the same. But yeah, I think that refraction is the result of the combined scattering by the individual molecules of some material. Since these molecules are arranged in some structure, that combination results in the refracted beam and the reflected beam (radiation in other directions destructively interferes). Beyond that, I'm not really qualified to comment; what I said may even be off base.
As far as the scattering in the skies goes, here's what I remember from my undergrad EM course:
we're dealing with a non-relativistic situation where the scattering radius is on the order of the Bohr radius, which << than the wavelength of visible light. The dipoles in the molecules of the atmosphere are driven by the electric field in the incident wave, and the scattered light comes out polarized. So you end up with the Larmor radiation formula with that ω4 dependence. I was going to try to transcirbe my old notes, but this has the same stuff only with better organization.
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https://www.aiche.org/conferences/aiche-annual-meeting/2017/proceeding/paper/711b-model-parameterization-through-data-mining
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math
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(711b) Model Parameterization through Data-Mining
Methodology: Principal Component Analysis (PCA) is the well-established approach to find parameters which are spanning the low-dimensional (linear) subspace of the system, or better say the good global reduced coordinates (parameters). However, it is known that in physical and chemical kinetics such a linear low-dimensional subspace is a poor description of the system and nonlinear embedding approaches should be considered. Diffusion maps (Dmap), as one of nonlinear model reduction techniques have been repeatedly applied to find low-dimensional, nonlinear manifolds underlying high-dimensional datasets and offers the potential of finding âgoodâ global reduced coordinates .
By applying diffusion distance, Dmaps offers more meaning-full distance (metric) than Euclidean metric specifically in nonlinear manifolds. However the metric and kernel in Dmap can be modified further to organize the intrinsic parameters even better than its original formulation. For example, if the observed data on parameter space are locating on neighborhood of level-sets of a function then one can deduce that similarity between two data on the level set is more than similarity of one of them with other point somewhere else in the parameter space but with the same metric. This issue is addressed and solution is proposed in this work, lead us to find identifiable, effective global parameterization of low order model, while the problem set-up can be treated as a black box which we have only access on the observed output. The solution can be considered as combination of parameterization and sensitivity analysis while the numerical cost is considerably lower. The method is applied on datasets organized in the form of an nsamples´nvariables matrix which can be an output of numerical simulation or experiments.
Results: In the first example a dataset was found by Michaelis-Menten enzyme kinetics which the parameterization and low-dimensional model are well-established analytically. Then the methods is applied on the filtered highly resolved Euler-Lagrange gas-particle simulation to identify optimal set of filtered variables and sub-grid correlations, seeking constitutive models for two-phase flow setup.
Summary: We use a modification of Diffusion Maps, a manifold learning technique, that employs an output-inspired metric in the input (i.e. parameter) space of a dynamic model. This approach allows us to identify the number and type of relevant parameter combinations, and to identify singular as well as regular perturbation regimes without formulas, in a purely data-driven manner.
Coifman, Ronald R., and Stéphane Lafon. "Diffusion maps." Applied and computational harmonic analysis 21.1 (2006): 5-30.
Singer, Amit, Radek Erban, Ioannis G. Kevrekidis, and Ronald R. Coifman. "Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps. " Proceedings of the National Academy of Sciences 106, no. 38 (2009): 16090-16095.
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http://slideplayer.com/slide/225093/
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math
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Presentation on theme: "Warm-up: 1. What is the kinetic energy of a charging rhino that has a mass of 1500 kg and a velocity of 5 m/s? 2. What is the potential energy of a 2 kg."— Presentation transcript:
Warm-up: 1. What is the kinetic energy of a charging rhino that has a mass of 1500 kg and a velocity of 5 m/s? 2. What is the potential energy of a 2 kg water balloon held off of an 80 m tall building? 3. Is it possible for one object to have both KE and PE at the same time? If so, give an example.
A quick recap… Energy is possessed by an object. Work is done when energy is transferred from one object to another. In other words, work is a measurement of how much energy was gained or lost by an object. Energy transferred into work: When would W = KE? When would W = PE?
Example #1: A 1 kg soccer ball reaches a maximum height of 30 m after a goalies drop kick. How much work could the ball do on a midfielders head? W = PE (the work it can do is equal to its energy) W = mgh = (1kg)(10m/s 2 )(30m) = (1kg)(10m/s 2 )(30m) = 300J = 300J
When a certain car locks up the brakes and slides, it has a frictional stopping force of 2500 N. If the car has a mass of 1000 kg, how far will the car slide if it slams on the brakes traveling at 10 m/s? W = KE (the brakes have to do work to stop the cars KE) F·d = ½ mv 2 F·d = ½ mv 2 (2500N) d = (1/2)(1000kg)(10m/s) 2 2500 kgm/s 2 d = 50,000 kgm 2 /s 2 2500 kgm/s 2 d = 50,000 kgm 2 /s 2 d = 20m d = 20m Example #2:
Practice problems… A car smashes into a stationary cow while traveling at 10 m/s and does 62,500 J of work on the cow. How much mass does the car have? W = KE (the work the car can do is equal to its KE) W = KE (the work the car can do is equal to its KE) 62,500J = ½mv 2 62,500kgm 2 /s 2 = ½ m(10 m/s) 2 m = 1250kg m = 1250kg A baseball catcher applies 640 N of force to stop the incoming fastballs thrown by his pitcher. When the pitcher throws a fastball, it knocks the catchers glove back.25 m before completely stopping the ball. If a baseball has a mass of.2 kg, how fast is the pitchers fastball? W = KE (the work required to stop the ball is equal to its KE) W = KE (the work required to stop the ball is equal to its KE) F·d = ½mv 2 F·d = ½mv 2 v = 40 m/s v = 40 m/s
Internal Energy Definition: energy due to the random motion of molecules. Also called thermal energy, because: High temperature = large internal energy Low temperature = small internal energy Even ice at 0 C has a fairly large amount of internal energy. Zero internal energy = -273 C (this is absolute zero - the lowest possible temperature)
Internal energy is a combination of KE and PE on a microscopic scale, which makes it impossible for us to measure individually…we need to use temperature as an indicator of internal energy. Cold is simply the absence of internal energy.Cold is simply the absence of internal energy. Cold cannot flow into something, you can only remove energy from it to lower its temperature. Example: A refrigerator doesnt put coldness into the food, it removes energy from the food.
Heat Definition: a transfer of energy that results in a change in temperature. Heat is another measurement of energy, so it has units of Joules. Just like we said with Work, an object cannot contain heat. Instead, we say that heat is transferred whenever there is a temperature change. Heat always flows from an object with higher internal energy to something with less internal energy. An object has energy, an object can do work. An object has energy, an object can transfer heat. (Biological Energy)
Application Problems 1) How many times would you have to lift a 10 pound weight (equivalent to about 50 N of force) through a distance of 1 m in order to burn 1 Calorie? (1Cal. = 4,186 J) (about 84 times) 2) How many Calories did you burn in the Harvard Step Test? (your work/4186 J) 3) How many minutes would you have to keep stepping in order to burn off a 280 Calorie Snickers bar? (280 Cal./above answer) This would be pretty bad news if we had to burn all our Calories by doing mechanical work. What are some other ways that our bodies use energy? (Think about the fact that our body temperature is maintained at 98.6 degrees.)
Human Metabolism Even when inactive, an average adult male has a basal metabolism rate of 90 W!!! That means he burns over 1850 Cal per day just by being alive!!! Calculating Basal Metabolic Rate (BMR) (Enter pounds, inches, and years into your equation): Males: 66 + (6.23*W) + (12.7*H) - (6.8*A) = Cal/day Females: 655 + (4.35*W) + (4.7*H) – (4.7*A) Multiplication Factors: 1.2 – little or no exercise 1.375 – (light exercise, 1-3 days per week) 1.55 – (moderate exercise, 3-5 days per week) 1.725 – (heavy exercise, 6-7 days per week)
The Biological Process Whenever fats, carbs, or proteins enter our body, they are broken down into the chemical glucose. Everything else gets dumped to the large intestine for elimination. Glucose is the greatest common denominator of biological energy. Whenever we do not use all of the energy that we have consumed, our body builds the extra glucose into fat, because fat is the most efficient means of storing energy. If the bodys extra energy was stored as carbs instead of fat, then cellulite would take up 3 to 4 times as much space as it does now!! (…but it would also burn off at a faster rate).
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https://axse.it/1991/12559875bf6eaaca5d4ca54
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We get the ratio from John. Before students log in to Imagine Math Facts Though we haven't officially done the work to solve the problem, this is a very good guess, just based on approximation and process of elimination. When identifying the parts of the word problem, distance is typically given in units of miles, meters, kilometers, or inches. 2nd grade math worksheets, including multiplication charts, printable math worksheets, graph paper and other problems for 2nd grade math. This is great for students because they not only receive the solutions, by they know the "how" and the "why" behind their exercises. * Lay the foundation for higher-order thinking. We use the same ratio for Jane. An example of one-step equations with word problems. Here's a simple-sounding problem: Imagine a circular fence that encloses one acre of grass. Consider this problem: "In history class, the boy to girl ratio is 5 to 8. * Build confidence and pre-algebra readiness. A set of fun and engaging math worksheets for 5th grade students. 4. Imagine Math. Counting each able to average 285 days a year, at the end of 10 years from the time they leave . p6q7(p2q)3 (p1q4)2p10 3 ( p 6 q 7 ( p 2 q) 3 ( p 1 q 4) 2 . 5th Grade Math Worksheets PDF. The "order of operations" convention (see Mathworld) is important for this problem. We will do your Math homework even if it's offline! First, use the function y= 2xto calculate the values to complete the table, graph the function, and label the graph. 3Add/Edit Students Option 1 1.
2. During this maintenance, students and educators may experience interruptions when using the Imagine Robotify platform. Interviewer Rebecca Cotton-Barratt even mentions this in Oxford's article: "A nice extension is what happens when we look at a point 1/3 or 2/3 up the ladder." If you studied music in school, this . . more What's New Version History Version 9.190 Various bug fixes and improvements App Privacy See Details Turn on Spanish Language Support 1. Starting temperature; Electrics - conductor Let us imagine that you are working as a computer programmer in a company that makes computer games. . This innovative motivation system inspires students to learn by helping develop problem-solving skills, perseverance, and confidence. Option 2 1. Select Espaol. Imagine Math's special Summer Pathways become available in-product on May 1 for teachers to assign. The problems in Guided Learning are designed to address student misconceptions head-on in order to provide opportunities for learning. 5. Imagine Math Facts combines high-end gameplay with cutting-edge curriculum and insightful reports to ensure students develop complete fluency with their core math facts. Imagine a cable fastened snug around the earth along the equator. Related math problems and questions: Air thermal Imagine that a unit of air rises at 3000 meters high, if temperature decreases 6 degrees celcius for every 1000 meter, what will be its temperature at 1400 meters, 2000 meters, 2500 meters and when it reaches the 3000 meter elevation. 3. Imagine Math PreK-2 Imagine Math Grade 3+ Imagine Math Facts. Kids (and adults!) Some are easier, some are harder. . Most worksheets have an answer key attached on . The n -th light bulb is toggled once for every factor of n. Squares are the only numbers with an odd number of factors, which can be seen because every factor J of a number, has a co-factor K for which JK=n. Analyze the problem situation a. . 1000 + 20 = 1020 (Right . Simple projects like no-bake cookies or snack mixes that require measuring and mixing but no potentially dangerous activities like ovens or knives can be fun while reinforcing math concepts.
The North Carolina Department of Public Instruction (NCDPI) is excited to announce a new partnership with Imagine Learning to assist students, educators, and parents with math in grades 3-8, NC Math 1, and NC Math 2. LoginAsk is here to help you access Imagine Math Login quickly and handle each specific case you encounter. Dear math folks, I'm a math teacher, and I've been playing around with trying to make some educational games in Desmos. The Lily Pad Problem. Select Management> Classrooms > Edit Classroom. Spanish Language Arts and Literacy. These math sheets can be printed as extra teaching material for teachers, extra math practice for kids or as homework material parents can use. Click on the Management tab. If it takes 48 days for the patch to cover the entire lake, how long would it take for the . c. Select a diagram that models the problem situation. Page 7 The Anatomy of an Imagine Math Lesson Guided Learning This is the activity in which students begin to engage in meaningful instruction that facilities their learning of the lesson's skills, concepts, and goals. Here you'll find a selection of problems and challenges to keep you sharp. Elementary Brain Teaser, Middle School Madness, & Problem of the Week: New problems will be posted each Monday at 8am Central . Date of the Problem. If you can imagine a furry humanoid seven feet tall, with the face of an intelligent gorilla and the braincase of a man, you'll have a rough idea of what they looked like . The Answer: 4100. Ratios are one area where math meets the real world, and sometimes kids struggle with this particular aspect of math class. Pre-built and customizable benchmark and formative assessments. This pairs up all the factors of n, unless J=K, which only occurs when n is a square. Galileo K-12 Galileo Birth-PreK. These 6th grade Multiplying Fractions Worksheets with Answers are made of the following Math skills for kids: multiply fraction with whole numbers, fraction of a number word problems, estimate products of fractions and whole numbers, multiplication of two fractions, multiplying fractions word problems, scaling whole numbers by fractions, scale whole number by mixed numbers, multiply three or . I can calculate the unit rate for real life situations by breaking down the ratio (fractions) by dividing to solve the problem to find the relationship between two units. Section 1-1 : Integer Exponents. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. Here's a couple problems from the book: 1. Your prefrontal cortex is actually working. Search the student name 4. We guarantee an overall grade of an "A" or "B" or your money back. The total profit in the first month was $438. The 5 steps in Imagine Math's problem-solving model are grounded in research in cognitive thinking processes and intervention methods for struggling students. You can also enable Friendly Mode from the Imagine Math Facts Teacher Portal. Imagine that you are filling treat bags with jellybeans. Daily Math Challenge: A new problem will appear at 10am Central every Tuesday-Friday . In a nutshell, we asked student/teacher teams to send us an Imagine Math journal page that illustrated how students broke down a math problem and solved for the correct answer. In a lake, there is a patch of lily pads. b. Every hundreds chart you could imagine! For those who love math, I promise fun and challenges with this 10th grade math test. Identify what the problem is asking. Imagine Math's special Summer Pathways become available in-product on May 1 for teachers to assign. (7.RP.A.1) For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per . Imagine that you are filling treat bags with jellybeans. Plus, Imagine Math Facts - a 2021 CODiE award winner for Best Gamification in Learning . If you cannot install a supported browser, contact your district IT support or IM customer support. . Two classmates leave the country school, one for work for 75 a day with board; the other borrows $250 and goes away for 3 years to a trade school and learns a trade which pays him $1.75 a day with board. The outbound process is outlined in the process flow below. b. Furthermore, you can find the "Troubleshooting Login Issues" section which can answer your unresolved problems and equip you with a lot . How many scoops will she need to fill the bird . Dear math folks, I'm a math teacher, and I've been playing around with trying to make some educational games in Desmos. 3. 1. Math is a muscle. The quiz items are pulled from a database of thousands of questions. Unfortunately, it turns out that proving them is a little harder.
Su, Francis E., et al. One of the greatest unsolved mysteries in math is also very easy to write. Time. 2. Algebra I: Example Problems Part I Lesson 1: Foundations for Algebra Lesson 1.1 Lesson 1.1 Problem 1 and Got It #1 Lesson 1.1 Problem 2 and Got It #2 Lesson 1.1 Problem 3 and Got It #3 Lesson 1.1 P Here's one that I think people might potentially find useful, especially in Grades 6-8, and I'd be very interested to hear what people think. Some values are entered for you. Science, Technology, and Engineering. Click Save. b. Related math problems and questions: The bird The bird feeder holds 1 and 1/2 cups of birdseed. So what is Math Journaling all about? Assign variables: Let x = number of blue marbles for Jane. A comprehensive database of more than 2022 math quizzes online, test your knowledge with math quiz questions. At least that's what she thought when she picked the spot. But as Avery Thompson points out at Popular Mechanics, from the outset at least, some of these problems seem surprisingly simple - so simple, in fact, that anyone with some basic maths knowledge can understand them. Rate is distance per time, so its units could be mph, meters per second, or inches per year. 5. Jul 25, 2017 - Using a math journal helps students think about problem solving in new ways. I've seen . The Imagine Learning supplemental math programs provide adaptive, age-appropriate learning environments for students in PreK-8. 2. Scheduled - Thursday, June 30th 10:00pm Pacific time / July 1st,1:00 am Eastern time, Imagine Learning will be performing extended planned system maintenance to improve our customer experience. Outbound Process Flow: When a customer's order is placed, a set of associates called "Pickers" pull the customer's order from the shelves and place the items on a cart. The solution for each Problem of the Week (POTW) is posted here the following week (included in the downloadable POTW file). Now loosen the cable so that it is 3 feet longer and thus not so perfectly snug anymore. Select Optionson pause screen. You'll find quite a variety among these problems too. Imagine Math Login will sometimes glitch and take you a long time to try different solutions. If the multiplication of the insides of the radicals was done first [sqrt(36); sqrt(36) = 6)], an erroneous result would occur. Reagan is using a scoop that holds 1/8 of a cup. Free Math Worksheets is a math related website that contains pre-algebra, algebra and geometry worksheets and . Imagine Math Facts teaches students math facts through a series of powerful, engaging activities on multiplication, division, addition, and subtractionhelping each child quickly gain automaticity and math fluency along with a new interest in math. 1. 2. imagine Math PreK-2 PREK-2: Offers engaging, effective math instruction designed to help early-learners learn and love math imagine Math 3+ How high off the ground will the cable be? Options 7 6Student Pre-Test and Post-Test Imagine Math does not support the browser you are using. Write the sentence as a conditional and identify the hypothesis and the conclusion. Imagine Learning Twig Science. Starting temperature; Electrics - conductor Getting Started.
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https://essayschief.com/service/more-services/math-and-finance-problem
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math
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Math And Finance Problem
If you are a math or finance student, you will be required to deal with essay on math or finance every now and then. It can be so tough for the students to deal with assignment in math and finance. Generally, math and finance turn out to be a toughest assignment for almost all the students and they used to struggle with problem solving at all times. However, by practice the students can improve their problem solving skills and in order to make your math essay successful you should have better problem solving skills.
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MATHEMATICAL FINANCE, ITS APPLICATION, ITS VARIOUS PROBLEMS AND A BRIEF CONCLUSION
Students used to get plenty of assignment writing tasks during their school, college and university years. Mathematics finance problems essay is one of the assignments that students are asked to write when they are in college or university. It is important for the students to develop skills in writing and especially, to write the mathematics finance problems essay. Most of the students find writing mathematics finance problems challenging and tough because they don’t know how to approach the essay and how to write it in a way that catches the attention of readers. But, students can find custom essay writing service online and they have expert writers who can effectively manage mathematics finance problems paper.
Introduction to Mathematics Finance Problems
Mathematics has been over the century a great menace of fear and disaster of our results slips during the school time era. Who loves mathematics after all? It involved and will progressively involve a lot of complex, ambiguous calculations that are very hard to understand or explain. Mathematics and finance based tutors in learning institutions were highly rejected by most of the students in major institutions. History, geography or other related humanities subjects were highly favored, and students used to score highly remarkable marks. I bet very sure this is also evident in our current institutions. Mathematics has been for a few chosen individuals who are regarded as mathematics genius or gurus. This type of unevenly distributed students exposed their ability all over the institution. They were highly regarded despite their heights, weights or even race. They used to be loved by each and every teacher and were rarely punished because of their high level of intelligence.
Notably, mathematics finance problems circulate its major concepts on stochastic optimization problems. This is the most severe problem most of the student come across while studying finance. It involves major calculations and extreme steep process and procedures of deriving a solution to the mathematical problem. Furthermore, tools from mathematical variables such as probability, statistics, and economic theory have been a menace in financial mathematics. A critical analysis and clear understanding of the concepts is a key necessary requirement in evading and succeeding from the pangs of that menace. Mathematics finance problems paper can be managed with ease if you ensure to get help from cheap essay writing service online.
Nevertheless, every problem has a solution despite its magnitude. Mathematical finance should not and has never been an exception. Through some strategically aided techniques, scholars in the field of mathematical finance have sprouted out some of the curbing mechanisms to this calculation menace. Therefore, the article below will discuss the definition of mathematical problems, types of mathematical problems evident in current settings, and the proven strategies to cater for the menace.
Definition of Mathematical Finance
Mathematical finance has a greater scope of scrutiny thus it has various wide definitions in reference to different understanding from different scholars. However, this has not been such big issue in definition as it major concepts circulates around the various definition. As earlier stated above, financial mathematics involves calculations all over its scope and little theory hence it is quantitative finance. In simple terms, it is the employment of mathematical concepts and methods to various financial related menaces. You can come across several writing companies on the web who can tackle your mathematics or finance problems paper, and therefore, it is sensible to get assistance from them.
Application of Mathematical Finance
Mathematical finance has been of importance all over the globe due to its effect upon serious implementations of its skills and technical know-how. The mathematical methodologies such as probability, statistics, stochastic processes and economic theories are integrated into finance to aid reasoning. Most of the successful trending businesses apply these concepts and methods found in financial mathematics in coming up with the solutions to problems such as derivative securities valuation, portfolio structuring, risk management and scenario simulations. Notably, manufacturing and processing industries also integrate mathematical finance in their day in and day out activities. Due to its strictness and efficiency in its workouts, it has aided in streamlining and brought about equilibrium in financial markets thus it's increasingly beneficial in terms of regulatory themes.
Notably, we had highlighted issues concerning the application of economics to mathematical problems. As usual, economics majorly circulates its concepts on valuations of various assets and equal distribution of resources. Some of the variables and concepts found in economics such as asset price regulations concerns, various market movements, and varying interest rates allow scholars in mathematical finance in making viable and rational decisions which might be a great problem in ignorance of those variables. In a more clearly elaborate example to aid reasoning, stress- testing is among the major application of knowledge and skills in banking. Some incorporated concepts in mathematical finance such as numerical analysis, Monte Carlo simulation analysis; optimizations and so on are of major guidance in financial mathematics.
In summary, it is evident that financial mathematics incorporates other concepts from other fields that assist in policy formulation and rational decision making. It is not an independent discipline in terms of its roles in financial problems thus making it a rigid area of concerns on its capability, severity, and efficiency upon application. Mathematical participation in every related field is and has been a fundamental principle on the evident roles in most of the millions of industries. You don’t have to be worried about writing a mathematics finance problems paper since you have the choice of hiring experts from a best essay writing service, who can do it upon your request.
Mathematical Financial Problems
In each and every successful object, it must incorporate some of both inevitable and evitable difficulties. Similarly, mathematical finance is not an exception. It incorporates some of the hindrances that usually dim its progress upon implementations of its various methods and concepts. These problems might be either internal or external, and nevertheless, they both accumulate to a large blockage in financial growth and development. In mathematics finance, there exists basic model which contains various unsolved financial problems.
Some of the mathematical finance problems include;
- Stefan’s problem for American options.
- Lack of an analogue of Clark- Haussmann formula in determinist calculus.
- Explicit formulas for replicating process.
- Optimal investment problems for observable but unhedgeable parameters.
- Discrete-time market
1. Stefan’s Problem For American Options
Stefan’s problem is a specific type of boundary value problem for a partial differential equation in which a phase boundary can allocate with time. The free boundary provides an adequate decomposition of binomial tree constituents which makes decisions for American put options with a very long lifespan. This is expressed by the curve that divides the two decisions of either to input the action or not goes through the asymptotic expansion in which the cooperating of the significant term is expressed as an inherent term of the free boundary. It is obvious that custom essay writing service will let you make your mathematics finance problems paper effective and high in quality.
However, if these concepts are infused for the American options, then option holders will possess the capability of exercising their options at any time in advance to and inclusive of the maturity date. The upheaved options for the American policy option holder, in this case, can only be in a simulacrum to the European options, and this stage of the heightened option is only realized at the maturity date.
For an American option holder to ascend to this stage, he or she needs to embrace the perspective that he or she could put into use the numerical simulation to find the probability of the integral that consequently will automatically derive an effective way of making correct and rational decisions for long life options. This simply entails the holder to purchase or to procure sales of an elaborate underlying asset on or before a fore determined expiration due date.
2. Lack Of An Analogue Of Clark- Haussmann Formula In Determinist Calculus
Haussmann formula is simply a theory of stochastic analysis. It implies that the estimation value of some function which in this case is a price on the classical Weiner space of uninterrupted paths commencing at the zero vectors as the sum of its average value and an Ito integral that will be in respect to that path. The elucidation of stochastic integrals as a divergence accrues to the concepts such as the Skorokhod integral and the tools of the Malliavin calculus.
Haussmann formula has a decisive claim that the mapping of the function is an isometric bisection that lacks the analogue in the deterministic calculus. This property of omitting analogue of Clarks permits Ito’s process to possess the properties that ensure it has a full representation of the ultimate model for stock prices. The process involves creating a non-anticipative calculus for operations of a progressive semi martingale that is an elongation of its formula to a self-dependent functional which attain a number of directional derivatives. The results culminated from finding the derivatives that will extrapolate the stock prices to their specified details that are highly required and recommended.
3. Explicit Formulas for Replicating Process
An explicit formula creates a sequence by putting into use the number of location of each item. It designates a connoted ‘n’ as the abbreviation of its location to provide quick and easy access of any term in the sequence. The process of writing an explicit formula entails finding out if the sequence is arithmetic followed by determining the common difference then creating the formula. The formula is created using the pattern that is depicted in the sequence like the addition of the multiple numbers or separating the terms in their ratio and conjugates that will result in per mutative analysis.
On the other hand, replicating strategy is exhibited in the process of financial accounting where it is used as a financial instrument. As a set of liquid, commonly it is exchange-traded assets that accumulate to the same net profit. It imposes dynamic trading strategy that can inevitably change a portfolio to be exposed between two dynamics of either total security or an open-ended that incorporates a lot of various risks. This can result in the production of similar payoff function as an added advantage strategy. It is majorly used by dealers to impede risks exposure that culminates from ascribed options. Therefore, the explicit formula for replicating strategy enumerates the optimal claims in an admissible strategy that inputs historical prices.
4. Optimal Investment Problems For Observable But Unhedgeable Parameters
Optimal investment problem gives an elaborate utility function which brings out a vivid description of risk preferences. It inputs into use the most common utility such as logarithms and powers that are raised for the numerical values. In the current era, there are a number of modifications to the generic optimal investment problem such as optimal investment and consumption problem that comes at the forefront of the list. This involves portfolio problems that an investor faces when they try to increase their utility from their terminal wealth. This problem can easily be solved through a thorough implication of logarithms utility of special constant. It then proceeds to optimal hedging of non-replicable claims that usually entails multiple assets that are assumed to follow a vector Ito differential equation since it is a continuous time process that results in hedging errors.
This is preceded by a problem with the constraint that involves designing a specific function to assets so that it can bring out about maximum profits. This results in designing and assigning faulty measures to the assets hence outlaying negative feedback. It can also be resolved by introducing some of the different dummy variables to incorporate a constant measure. When this is done, the right retrospective objective is properly ascertained. Cheap writing service will take away your tensions and stress when it comes to writing your paper.
5. Discrete-Time Market
Most of the major market models manifest from a set of the axiom, and thus it derives its definition simply as a mapping from the classes of adopted strategies to the set of unanimous variables illustrating the outcome of trading. Under sequential upper semi continuity, the model can be depicted as a normal integrand. Prices that are variables in a market model are given as a time series which is applicable only when it is being put into use in the time series to resolve it. The resolution attains optimal solution for the continuous model at the continuous limit. Optimal solution requires solutions of Bellman’s discrete-time equation that will consequently calculate conditional densities at any step.
In summary, mathematical finance has been a tough subject for most of us due to its large and extreme calculations. However, some of the major unsolved and solved mathematical finance problems have been highlighted above. Those problems have been a menace to the entire fraternity of mathematics. However, upon knowledge and understanding of those highlighted problems, one can be able to avoid them. We have noted mathematical finance has a very large scope of its application mostly in matters concerning finance issues. Industries that deal with calculations in their entire strategy focus on experts who have elaborate knowledge and skills in mathematical finance. Furthermore, it is evident that mathematical finance exposes its concepts to almost all over the industries. Therefore, this discipline aids in efficiency and severity when it comes to management and rational decision making.
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CC-MAIN-2023-06
| 16,613 | 41 |
http://iucee.org/iucee/iucee-mathworks-webinar-series/
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math
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Welcome to the IUCEE-MathWorks Webinar Series.
The webinar series will cover topics ranging from technical computing, modeling, and simulation with MATLAB and Simulink in engineering courses within various departments like Electronics and Communication, Electrical, Computer Science, Instrumentation Control, Mechanical and Automobile.
Webinars will be led by MathWorks subject matter experts who will cover range of topics starting from Computational Thinking, Mathematics and Statistics, Control System Modeling, Deep Learning, IoT and HDL code generation workflows using MATLAB and Simulink.
The webinars also highlight the various resources available for academia that can help create a more active classroom learning experience like Live Editor, App Designer, MATLAB Grader, Onramp courses and MATLAB Academic Online Training Suite. Most importantly, provide you with the opportunity to ask any relevant questions which will help you in learning.
Schedule: Mondays at 3:30 pm IST (10:00 am GMT)
1. June 17, Monday, 3:30 pm IST: Integrating computational thinking and interactive learning in curriculum using MATLAB: Link: https://attendee.gotowebinar.com/register/7815090455659889676
2. July 15, Monday, 3:30 pm IST: Teaching Basic Statistics and Visualization in MATLAB Link: https://attendee.gotowebinar.com/register/2852748188902240780
3. Aug 19, Monday, 3:30 pm IST: Learning Systems Modelling and Control with Simulink and Simscape Link: https://attendee.gotowebinar.com/register/4143335146376481036
4. Sept 16, Monday, 3:30 pm IST: Introduction to Deep Learning using MATLAB Link: https://attendee.gotowebinar.com/register/1366913255102573068
5. Oct 14, Monday, 3:30 pm IST: Enabling IoT applications using MATLAB and ThingSpeak Link: https://attendee.gotowebinar.com/register/6243484818800737548
6. Nov 11, Monday, 3:30 pm IST: Generating and Verifying HDL Code using Simulink and deployment on FPGA Link: https://attendee.gotowebinar.com/register/4339143873632247820
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CC-MAIN-2020-10
| 1,979 | 11 |
https://www.hackmath.net/en/math-problem/2061
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math
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Tram no. 3,7,10,11 rode together from the depot at 5am. Tram No. 3 returns after 2 hours, tram No. 7 an hour and half, no. 10 in 45 minutes and no. 11 in 30 minutes. For how many minutes and when these trams meet again?
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https://blog.wolfram.com/2019/11/26/new-wolfram-books-adventures-lessons-and-computations-to-spark-your-curiosity/
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math
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New Wolfram Books: Adventures, Lessons and Computations to Spark Your Curiosity
It’s been another big year of exploration with the Wolfram Language. CEO Stephen Wolfram’s new book takes us on a tour of his computational adventures throughout the years. We’re also excited to introduce a Spanish-language version of the popular An Elementary Introduction to the Wolfram Language, as well as books to enhance the mathematics and engineering curricula. There’s something new for everyone, from students to lifelong adventurers, to discover with the Wolfram Language. Just in time for the holidays, find the perfect read for those who love learning new things—including yourself!
Join Stephen Wolfram as he brings the reader along on some of his most surprising and engaging intellectual adventures, showcasing his own signature way of thinking about an impressive range of subjects. From science consulting for a Hollywood movie, solving problems of AI ethics, hunting for the source of an unusual polyhedron, communicating with extraterrestrials, to finding the fundamental theory of physics and exploring the digits of pi, this lively book of essays captures the infectious energy and curiosity of one of the great pioneers of the computational world.
¡Ahora en español por primera vez! Stephen Wolfram’s introduction to modern computational thinking is now available in Spanish, bringing the leading-edge computation of the Wolfram Language to an even larger audience. The Wolfram Language scales from a single line of easy-to-understand, interactive code to million-line production systems. The book does not assume any programming knowledge, making it suitable for students of all levels, as well as anyone interested in learning the latest technology and the world’s only large-scale computational language. You can also find the second edition of the English version here.
Look inside the machines that power our civilization and provide innumerable goods and services, and you will find mechanisms at the core. This book is designed to supplement the existing curriculum and lab work by students. Building on the foundations from the classroom, Máquinas y mecanismos, implementación con Wolfram Mathematica enhances this branch of engineering for both sides of the instructor’s desk. Students engage more deeply with the material; examples and small Wolfram Language programs embedded within the text encourage the dynamic exploration of the behavior of any linkage. The Manipulate function in the Wolfram Language allows students to easily examine effects produced by variations in the lengths of the links or their relationships, the linear and angular speeds and accelerations of the input link, and the inertial parameters in the behavior of a mechanism.
Mathematical Analysis and Instrumental Methods for Solving Problems (Russian)
Compiled into two books, materials from the Mathematical Analysis course by authors Vladimir Grigorievich Chirsky and Kirill Yuryevich Shilin bring theoretical mathematics closer to instrumental methods. Sit in on the first and second semesters of the course (originally taught at the Department of Economics of the Institute of Economics, Mathematics and Information Technology, RANEPA), with each book chapter thoughtfully divided into four sections: 1) basic definitions, statement of theorems and simple proofs; 2) evidence of theorems and more in-depth readings; 3) solutions to common problems; and 4) how to solve the problems using Mathematica.
Discrete mathematics is as essential for a programmer as an applied mathematician, but many students struggle with the subject. Discrete Mathematics: Learning to Program in Wolfram Mathematica combines an introduction to the main concepts and methods of discrete mathematics with the basics of programming in Mathematica. This book takes you through a range of theoretical and applied discrete mathematics, including combinatorics and enumerative combinatorics, data structures such as binary heaps and binary search trees, sorting algorithms, and operations in residue rings and modern encryption methods. Twelve Mathematica “programming lessons” show the actual code to implement all the algorithms introduced throughout the chapters, making this book a valuable resource for students, university professors and anyone interested in learning how to program in Mathematica.
This compact, new edition of The Student’s Introduction to Mathematica and the Wolfram Language is an excellent companion to the standard mathematics curriculum. A brief introduction to the aspects of the Mathematica software program most useful to students, this third edition includes significant updated material to account for the latest developments in the Wolfram Language. As a result, this book also serves as a great tutorial for former students looking to learn about new features such as natural language queries, 3D printing and computational geometry, as well as the vast stores of real-world data now integrated through the cloud.
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https://studybay.com/math/tags/recreational-mathematics/
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math
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A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)
Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..
There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.Group-theoretically, let be the modular group Gamma0, and let be the compactification (by adding cusps) of , where is the upper half-plane. Also define to be the Fricke involution defined by the block matrix . For a prime, define . Then is a supersingular prime if the genus of .The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field . They have their j-invariant in .Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group...
The least common multiple of two numbers and , variously denoted (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), (Gellert et al. 1989, p. 25; Graham et al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West 2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325; Wolfram Language), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that(1)The least common multiple of more than two numbers is similarly defined.The least common multiple of , , ... is implemented in the Wolfram Language as LCM[a, b, ...].The least common multiple of two numbers and can be obtained by finding the prime factorization of each(2)(3)where the s are all prime factors of and , and if does not occur in one factorization, then the corresponding exponent is taken as 0. The least..
A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.
Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to the large sphere on the extremities of one of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then , 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (OEIS A046160). The sizes and positions of the first few rings of spheres are given in the table below.100--220--3546059611071481591801021112312270, 1330143315380Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using the equation of five tangent spheres. Letting givesFor example, , , , , , and so on, giving the sequence , 2, 3, 11, 15, 27, 35, 47,..
The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..
The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid. Two cylinders intersecting at right angles are called a bicylinder or mouhefanggai (Chinese for "two square umbrellas"), and three intersecting cylinders a tricylinder. Half of a bicylinder is called a vault.For two cylinders of radius oriented long the - and -axes gives the equations(1)(2)which can be solved for and gives the parametric equations of the edges of the solid,(3)(4)The surface area can be found as , where(5)(6)Taking the range of integration as a quarter or one face and then multiplying by 16 gives(7)The volume common to two cylinders was known to Archimedes (Heath 1953, Gardner 1962) and the Chinese mathematician Tsu Ch'ung-Chih (Kiang 1972), and does not require calculus to derive. Using calculus provides a simple derivation, however. Noting that the solid has a square cross section..
The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).Two additional types of Penrose tiles known as the rhombs (of which there are two varieties: fat and skinny) and the pentacles (or which there are six type) are sometimes also defined that have slightly more complicated matching conditions (McClure 2002).In 1997, Penrose sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling (Mirsky 1997). The suit was apparently settled out of court.To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into..
The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...
A (general, asymmetric) lens is a lamina formed by the intersection of two offset disks of unequal radii such that the intersection is not empty, one disk does not completely enclose the other, and the centers of curvatures are on opposite sides of the lens. If the centers of curvature are on the same side, a lune results.The area of a general asymmetric lens obtained from circles of radii and and offset can be found from the formula for circle-circle intersection, namely(1)(2)Similarly, the height of such a lens is(3)(4)A symmetric lens is lens formed by the intersection of two equal disk. The area of a symmetric lens obtained from circles with radii and offset is given by(5)and the height by(6)A special type of symmetric lens is the vesica piscis (Latin for "fish bladder"), corresponding to a disk offset which is equal to the disk radii.A lens-shaped region also arises in the study of Bessel functions, is very important in the theory of..
A fact noticed by physicist G. Gamow when he had an office on the second floor and physicist M. Stern had an office on the sixth floor of a seven-story building (Gamow and Stern 1958, Gardner 1986). Gamow noticed that about 5/6 of the time, the first elevator to stop on his floor was going down, whereas about the same fraction of time, the first elevator to stop on the sixth floor was going up. This actually makes perfect sense, since 5 of the 6 floors 1, 3, 4, 5, 6, 7 are above the second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the sixth. However, the situation takes some unexpected turns if more than one elevator is involved, as discussed by Gardner (1986). Furthermore, even worse, the analysis by Gamow and Stern (1958) turns out to be incorrect (Knuth 1969)!Main character Charles Eppes discusses the elevator paradox in the Season 4 episode "Chinese Box" of the television crime drama NUMB3RS...
A schematic diagram used in logic theory to depict collectionsof sets and represent their relationships.The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing a total of four regions, , , , and (the empty set, represented by none of the regions occupied). Here, denotes the intersection of sets and .The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled , , and consist of members which are only in one set and no others, the three regions labelled , , and consist of members which are in two sets but not the third, the region consists of members which are simultaneously in all three, and no regions occupied represents .In general, an order- Venn diagram is a collection of simple closed curves in the plane such that 1. The curves partition the plane into connected..
The knight graph is a graph on vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other).The number of edges in the knight graph is (8 times the triangular numbers), so for , 2, ..., the first few values are 0, 0, 8, 24, 48, 80, 120, ... (OEIS A033996).Knight graphs are bipartite and therefore areperfect.The following table summarizes some named graph complements of knight graphs.-knight graph-queen graph-knight graph-queen graphThe knight graph is implemented in the Wolfram Language as KnightTourGraph[m, n], and precomputed properties are available in using GraphData["Knight", m, n].Closed formulas for the numbers of -graph cycles of the knight graph are given by for odd and(1)(E. Weisstein, Nov. 16, 2014).A knight's path is a sequence of moves by a..
The king graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a king.The number of edges in the king graph is , so for , 2, ..., the first few values are 0, 6, 20, 42, 72, 110, ... (OEIS A002943).The order graph has chromatic number for and for . For , 3, ..., the edge chromatic numbers are 3, 8, 8, 8, 8, ....King graphs are implemented in the Wolfram Language as GraphData["King", m, n].All king graphs are Hamiltonian and biconnected. The only regular king graph is the -king graph, which is isomorphic to the tetrahedral graph . The -king graphs are planar only for (with the case corresponding to path graphs) and , some embeddings of which are illustrated above.The -king graph is perfect iff (S. Wagon, pers. comm., Feb. 22, 2013).Closed formulas for the numbers of -cycles of with are given by(1)(2)(3)(4)where the formula for appears in Perepechko and Voropaev.The..
The queen graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a queen. The -queen graphs have nice embeddings, illustrated above. In general, the default embedding with vertices corresponding to squares of the chessboard has degenerate superposed edges, the only nontrivial exception being the -queen graph.Queen graphs are implemented in the Wolfram Language as GraphData["Queen", m, n].The following table summarized some special cases of queen graphs.namecomplete graph tetrahedral graph The following table summarizes some named graph complements of queen graphs.-queen graph-knight graph-queen graph-queen graph-knight graphAll queen graphs are Hamiltonian and biconnected. The only planar and only regular queen graph is the -queen graph, which is isomorphic to the tetrahedral graph .The only perfect queen graphs are , , and .A closed formula..
The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. Now compute (discarding any initial 0s) and iterate, where is sometimes called the Kaprekar function. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending on the number of digits in and the value of . The list of values is sometimes called a Kaprekar sequence, and the result is sometimes called a Kaprekar number (Deutsch and Goldman 2004), though this nomenclature should be deprecated because of confusing with the distinct sort of Kaprekar number.In base-10, the numbers for which are given by 495, 6174, 549945, 631764, ... (OEIS A099009). Similarly, the numbers for which iterating gives a cycle of length are given by 53955, 59994, 61974, 62964, 63954, 71973, ... (OEIS..
The quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."
1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):(K. MacMillan, pers. comm., Apr. 29, 2007).This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.1729 was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character..
The second Mersenne prime , which is itself the exponent of Mersenne prime . It gives rise to the perfect number It is a Gaussian prime, but not an Eisenstein prime, since it factors as , where is a primitive cube root of unity. It is the smallest non-Sophie Germain prime. It is also the smallest non-Fermat prime, and as such is the smallest number of faces of a regular polygon (the heptagon) that is not constructible by straightedge and compass.It occurs as a sacred number in the Bible and in various other traditions. In Babylonian numerology it was considered as the perfect number, the only number between 2 and 10 which is not generated (divisible) by any other number, nor does it generate (divide) any other number.Words referring to number seven may have the prefix hepta-, derived from the Greek -) (heptic), or sept- (septuple), derived from the Latin septem...
According to the novel The Hitchhiker's Guide to the Galaxy (Adams 1997), 42 is the ultimate answer to life, the universe, and everything. Unfortunately, it is left as an exercise to the reader to determine the actual question.On Feb. 18, 2005, the 42nd Mersenne prime was discovered (Weisstein 2005), leading to humorous speculation that the answer to life, the universe, and everything is somehow contained in the 7.8 million decimal digits of that number.It is also amusing that 042 occurs as the digit string ending at the 50 billionth decimal place in each of and , providing another excellent answer to the ultimate question (Berggren et al. 1997, p. 729).
The base 8 notational system for representing real numbers. The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that (8 in base 10) is represented as () in base 8. The following table gives the octal equivalents of the first few decimal numbers.11111321252212142226331315232744141624305515172531661620263277172127338101822283491119232935101220243036The song "New Math" by Tom Lehrer (That Was the Year That Was, 1965) explains how to compute in octal. (The answer is .)
In simple algebra, multiplication is the process of calculating the result when a number is taken times. The result of a multiplication is called the product of and , and each of the numbers and is called a factor of the product . Multiplication is denoted , , , or simply . The symbol is known as the multiplication sign. Normal multiplication is associative, commutative, and distributive.More generally, multiplication can also be defined for other mathematical objects such as groups, matrices, sets, and tensors.Karatsuba and Ofman (1962) discovered that multiplication of two digit numbers can be done with a bit complexity of less than using an algorithm now known as Karatsuba multiplication.Eddy Grant's pop song "Electric Avenue" (Electric Avenue, 2001) includes the commentary: "Who is to blame in one country; Never can get to the one; Dealin' in multiplication; And they still can't feed everyone, oh no."..
Long division is an algorithm for dividing two numbers, obtaining the quotient one digit at a time. The example above shows how the division of 123456/17 is performed to obtain the result 7262.11....The term "long division" is also used to refer to the method of dividing one polynomial by another, as illustrated above. This example illustrates the resultThe symbol separating the dividend from the divisor seems to have no established name, so can be simply referred to as the long division symbol (or sometimes the division bracket).The chorus of the song "Singular Girl" by Rhett Miller (The Believer, 2006) contains the slightly cryptic line "Talking to you girl is like long division, yeah." Coincidentally, Long Division (1995) is also the name of the second album by the band Low...
An irreducible fraction is a fraction for which , i.e., and are relatively prime. For example, in the complex plane, is reducible, while is not.The figure above shows the irreducible fractions plotted in the complex plane (Pickover 1997; Trott 2004, p. 29).
The Farey sequence for any positive integer is the set of irreducible rational numbers with and arranged in increasing order. The first few are(1)(2)(3)(4)(5)(OEIS A006842 and A006843). Except for , each has an odd number of terms and the middle term is always 1/2.Let , , and be three successive terms in a Farey series. Then(6)(7)These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of terms, insert the mediant fraction between terms and when (Hardy and Wright 1979, pp. 25-26; Conway and Guy 1996; Apostol 1997). Given with , let be the mediant of and . Then , and these fractions satisfy the unimodular relations(8)(9)(Apostol 1997, p. 99).The number of terms in the Farey sequence for the integer is(10)(11)where is the totient function and is the summatory function of , giving 2, 3, 5, 7, 11, 13, 19, ... (OEIS A005728). The asymptotic limit..
The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth..
The hypercube is a generalization of a 3-cube to dimensions, also called an -cube or measure polytope. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted and has Schläfli symbol .The following table summarizes the names of -dimensional hypercubes.object1line segment2square3cube4tesseractThe number of -cubes contained in an -cube can be found from the coefficients of , namely , where is a binomial coefficient. The number of nodes in the -hypercube is therefore (OEIS A000079), the number of edges is (OEIS A001787), the number of squares is (OEIS A001788), the number of cubes is (OEIS A001789), etc.The numbers of distinct nets for the -hypercube for , 2, ... are 1, 11, 261, ... (OEIS A091159; Turney 1984-85).The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope vertices, 32 polytope edges, 24 squares, and eight cubes.The dual of the tesseract..
Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the mice in the mice problem, and form logarithmic spirals.The square whirl appears on the cover of Freund (1993).
A star polygon , with positive integers, is a figure formed by connecting with straight lines every th point out of regularly spaced points lying on a circumference. The number is called the polygon density of the star polygon. Without loss of generality, take . The star polygons were first systematically studied by Thomas Bradwardine.The circumradius of a star polygon with and unit edge lengths is given by(1)and its characteristic polynomial is a factor of the resultant with respect to of the polynomials(2)(3)where is a Chebyshev polynomial of the first kind (Gerbracht 2008).The usual definition (Coxeter 1969) requires and to be relatively prime. However, the star polygon can also be generalized to the star figure (or "improper" star polygon) when and share a common divisor (Savio and Suryanaroyan 1993). For such a figure, if all points are not connected after the first pass, i.e., if , then start with the first unconnected point..
The Star of Lakshmi is the star figure , that is used in Hinduism to symbolize Ashtalakshmi, the eight forms of wealth. This symbol appears prominently in the Lugash national museum portrayed in the fictional film The Return of the Pink Panther.The interior of a Star of Lakshmi with edges of length is a regular octagon with side lengths(1)The areas of the intersection and union of the two constituent squares are(2)(3)
The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon .It is a pagan religious symbol that is one of the oldest symbols on Earth and is known to have been used as early as 4000 years B.C. It represents the "sacred feminine" or "divine goddess" (Brown 2003, pp. 35-37). However, in modern American pop culture, it more commonly represents devil worship. In the novel The Da Vinci Code, dying Louvre museum curator Jacque Saunière draws a pentagram on his abdomen with his own blood as a clue to identify his murderer (Brown 2003, p. 35).In the above figure, let the length from one tip to another connected tip be unity, the length from a tip to an adjacent dimple be , the edge lengths of the inner pentagon be , the inradius of the inner pentagon be , the circumradius of the inner pentagon be , the circumradius of the pentagram be , and the additional horizontal and vertical..
A star polygon-like figure for which and are not relatively prime. Examples include the hexagram , star of Lakshmi , and nonagram .
Also called Chvátal's art gallery theorem. If the walls of an art gallery are made up of straight line segments, then the entire gallery can always be supervised by watchmen placed in corners, where is the floor function. This theorem was proved by Chvátal (1975). It was conjectured that an art gallery with walls and holes requires watchmen, which has now been proven by Bjorling-Sachs and Souvaine (1991, 1995) and Hoffman et al. (1991).In the Season 2 episode "Obsession" (2006) of the television crime drama NUMB3RS, Charlie mentions the art gallery theorem while building an architectural model.
The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since . A quantity taken to the power 2 is said to be squared. The number of times a given binary number is divisible by 2 is given by the position of the first , counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.The only known solutions to the congruenceare summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below on Jan. 27 2007, finding no other solutions in this range.reference4700063497Guy (1994)3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)8365386194032363Crump (pers. comm., 2000)10991007971508067Crump (2007)63130707451134435989380140059866138830623361447484274774099906755Montgomery (1999)In general,..
A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..
An antimagic square is an array of integers from 1 to such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966. Antimagic squares of orders 4-9 are illustrated above (Madachy 1979). For the square, the sums are 30, 31, 32, ..., 39; for the square they are 59, 60, 61, ..., 70; and so on.Let an antimagic square of order have entries 0, 1, ..., , , and letbe the magic constant. Then if an antimagic square of order exists, it is either positive with sums , or negative with sums (Madachy 1979).Antimagic squares of orders one, two, and three are impossible. In the case of the square, there is no known method of proof of this fact except by case analysis or enumeration by computer. There are 18 families of..
A heterosquare is an array of the integers from 1 to such that the rows, columns, and diagonals have different sums. (By contrast, in a magic square, they have the same sum.) There are no heterosquares of order two, but heterosquares of every odd order exist. They can be constructed by placing consecutive integers in a spiral pattern (Fults 1974, Madachy 1979).An antimagic square is a special case of a heterosquare for which the sums of rows, columns, and main diagonals form a sequence of consecutive integers.
The binomial coefficient is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number. The symbols and are used to denote a binomial coefficient, and are sometimes read as " choose ." therefore gives the number of k-subsets possible out of a set of distinct items. For example, The 2-subsets of are the six pairs , , , , , and , so . The number of lattice paths from the origin to a point ) is the binomial coefficient (Hilton and Pedersen 1991).The value of the binomial coefficient for nonnegative and is given explicitly by(1)where denotes a factorial. Writing the factorial as a gamma function allows the binomial coefficient to be generalized to noninteger arguments (including complex and ) as(2)For nonnegative integer arguments, the gamma function reduces to factorials, leading to(3)which is Pascal's triangle. Using the symmetryformula(4)for integer , and complex , this..
The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.Rather amazingly, the Costa surface belongs to the dihedral group of symmetries.The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon1999).On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was..
Scherk's two minimal surfaces were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful images of wood sculptures of Scherk surfaces are illustrated by Séquin.Scherk's first surface is doubly periodic and is defined by the implicit equation(1)(Osserman 1986, Wells 1991, von Seggern 1993). It has been observed to form in layers of block copolymers (Peterson 1988).Scherk's second surface is the surface generated by Enneper-Weierstrassparameterization with(2)(3)It can be written parametrically as(4)(5)(6)(7)(8)(9)for , and . With this parametrization, the coefficients of the first fundamental form are(10)(11)(12)and of the second fundamental form are(13)(14)(15)The Gaussian and mean curvatures are(16)(17)
A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations(1)(2)(3)where .The line element is(4)The first fundamental form has coefficients(5)(6)(7)and the second fundamental form has coefficients(8)(9)(10)The principal curvatures are(11)(12)The mean curvature of the catenoid is(13)and the Gaussian curvature is(14)The helicoid can be continuously deformed into a catenoid with by the transformation(15)(16)(17)where corresponds to a helicoid and to a catenoid.This deformation is illustrated on the cover of issue 2, volume 2 of The MathematicaJournal.
The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.The gyroid is the only known embedded triply periodic minimal surface with triple junctions. In addition, unlike the five triply periodic minimal surfaces studied by Anderson et al. (1990), the gyroid does not have any reflectional symmetries (Große-Brauckmann 1997).The image above shows a metal print of the gyroid created by digital sculptor BathshebaGrossman (https://www.bathsheba.com/).
Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as therefore satisfies Lagrange's equation,(1)(Gray 1997, p. 399).Finding a minimal surface of a boundary with specified constraints is a problem in the calculus of variations and is sometimes known as Plateau's problem. Minimal surfaces may also be characterized as surfaces of minimal surface area for given boundary conditions. A plane is a trivial minimal surface, and the first nontrivial examples (the catenoid and helicoid) were found by Meusnier in 1776 (Meusnier 1785). The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz in 1890 (Schwarz 1972).Note that while a sphere is a "minimal surface" in the sense that it minimizes the surface area-to-volume ratio, it does not qualify as a minimal surface in the sense used by mathematicians.Euler proved that a minimal surface is planar..
A Markov chain is collection of random variables (where the index runs through 0, 1, ...) having the property that, given the present, the future is conditionally independent of the past.In other words,If a Markov sequence of random variates take the discrete values , ..., , thenand the sequence is called a Markov chain (Papoulis 1984, p. 532).A simple random walk is an example of a Markovchain.The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.
The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).As if this were..
A regular polygon is an -sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.A regular -gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[x, y, rspec, n], etc.The sum of perpendiculars from any point to the sides of a regular polygon of sides is times the apothem.Let be the side length, be the inradius, and the circumradius..
RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..
An invariant defined using the angles of a three-dimensional polyhedron. It remains constant under solid dissection and reassembly. Solids with the same volume can have different Dehn invariants.Two polyhedra can be dissected into each other only if they have the same volume and the same Dehn invariant. In 1902, Dehn showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems, and Sydler (1965) showed that two polyhedra with the same Dehn invariants are interdissectable.
A Mrs. Perkins's quilt is a dissection of a square of side into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect square dissection, however, the smaller squares need not be all different sizes. In addition, only prime dissections are considered so that patterns which can be dissected into lower-order squares are not permitted.The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (OEIS A005670), the first few of which are illustrated above.On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly..
French curves are plastic (or wooden) templates having an edge composed of several different curves. French curves are used in drafting (or were before computer-aided design) to draw smooth curves of almost any desired curvature in mechanical drawings. Several typical French curves are illustrated above.While an undergraduate at MIT, Feynman (1997, p. 23) used a French curve to illustrate the fallacy of learning without understanding. When he pointed out to his colleagues in a mechanical drawing class the "amazing" fact that the tangent at the lowest (or highest) point on the curve was horizontal, none of his classmates realized that this was trivially true, since the derivative (tangent) at an extremum (lowest or highest point) of any curve is zero (horizontal), as they had already learned in calculus class...
The points of intersection of the adjacent angle trisectors of the angles of any triangle are the polygon vertices of an equilateral triangle known as the first Morley triangle. Taylor and Marr (1914) give two geometric proofs and one trigonometric proof.A line is parallel to a side of the first Morley triangle if and only ifin directed angles modulo (Ehrmann and Gibert 2001).An even more beautiful result is obtained by taking the intersections of the exterior, as well as interior, angle trisectors, as shown above. In addition to the interior equilateral triangle formed by the interior trisectors, four additional equilateral triangles are obtained, three of which have sides which are extensions of a central triangle (Wells 1991).A generalization of Morley's theorem was discovered by Morley in 1900 but first published by Taylor and Marr (1914). Each angle of a triangle has six trisectors, since each interior angle trisector has two associated..
A hyperboloid is a quadratic surface which may be one- or two-sheeted. The one-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci, while the two-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the line joining the foci (Hilbert and Cohn-Vossen 1991, p. 11).
The partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other. A Voronoi diagram is sometimes also known as a Dirichlet tessellation. The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons.Voronoi diagrams were considered as early at 1644 by René Descartes and were used by Dirichlet (1850) in the investigation of positive quadratic forms. They were also studied by Voronoi (1907), who extended the investigation of Voronoi diagrams to higher dimensions. They find widespread applications in areas such as computer graphics, epidemiology, geophysics, and meteorology. A particularly notable use of a Voronoi diagram was the analysis of the 1854 cholera epidemic in London, in which physician John Snow determined a strong correlation of deaths with proximity to a particular..
The hexagram is the star polygon , also known as the star of David or Solomon's seal, illustrated at left above.It appears as one of the clues in the novel TheDa Vinci Code (Brown 2003, p. 455).For a hexagram with circumradius (red circle), the inradius (green circle) is(1)and the circle passing through the intersections of the triangles has radius(2)The interior of a hexagram is a regular hexagon with side lengths equal to 1/3 that of the original hexagram. Given a hexagram with line segments of length , the areas of the intersection and union of the two constituent triangles are(3)(4)There is a "nonregular" hexagram that can be obtained by spacing the integers 1 to 6 evenly around a circle and connecting . The resulting figure is called a "unicursal hexagram" and was evidently discovered in the 19th century. It is not regular because there are some edges going from to (mod 6) and some edges going from to (mod 6). However,..
A Greek cross, also called a square cross, is a cross inthe shape of a plus sign. It is a non-regular dodecagon.A square cross appears on the flag of Switzerland, and also on the key to the Swiss Bank deposit box in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 146 and 171-172).Greek crosses can tile the plane, as noted by the protagonist Christopher in The Curious Incident of the Dog in the Night-Time (Haddon 2003, pp. 203-204).
A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical. Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).The number of demiregular tessellations is commonly given as 14 (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by "demiregular tessellation."A more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65)...
Regular tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations. In the plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp. 37-41; Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227). Williams (1979, pp. 37-41) also illustrates the dual tessellations of the semiregular tessellations. The dual tessellation of the tessellation of squares and equilateral triangles is called the Cairo tessellation (Williams 1979, p. 38; Wells 1991, p. 23).
Consider a two-dimensional tessellation with regular -gons at each polygon vertex. In the plane,(1)(2)so(3)(Ball and Coxeter 1987), and the only factorizations are(4)(5)(6)Therefore, there are only three regular tessellations (composed of the hexagon, square, and triangle), illustrated above (Ghyka 1977, p. 76; Williams 1979, p. 36; Wells 1991, p. 213).There do not exist any regular star polygon tessellations in the plane. Regular tessellations of the sphere by spherical triangles are called triangular symmetry groups.
An aperiodic tiling is a non-periodic tiling in which arbitrarily large periodic patches do not occur. A set of tiles is said to be aperiodic if they can form only non-periodic tilings. The most widely known examples of aperiodic tilings are those formed by Penrose tiles.The Federation Square buildings in Melbourne, Australia feature an aperiodic pinwheel tiling attributed to Charles Radin. The tiling is illustrated above in a pair of photographs by P. Bourke.
A plane tiling is said to be isohedral if the symmetry group of the tiling acts transitively on the tiles, and -isohedral if the tiles fall into n orbits under the action of the symmetry group of the tiling. A -anisohedral tiling is a tiling which permits no -isohedral tiling with .The numbers of anisohedral polyominoes with , 9, 10, ... are 1, 9, 44, 108, 222, ... (OEIS A075206), the first few of which are illustrated above (Myers).
The P versus NP problem is the determination of whether all NP-problems are actually P-problems. If P and NP are not equivalent, then the solution of NP-problems requires (in the worst case) an exhaustive search, while if they are, then asymptotically faster algorithms may exist.The answer is not currently known, but determination of the status of this question would have dramatic consequences for the potential speed with which many difficult and important problems could be solved.In the Season 1 episode "Uncertainty Principle" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes uses the game minesweeper as an analogy for the P vs. NP problem.
Percolation theory deals with fluid flow (or any other similar process) in random media.If the medium is a set of regular lattice points, then there are two main types of percolation: A site percolation considers the lattice vertices as the relevant entities; a bond percolation considers the lattice edges as the relevant entities. These two models are examples of discrete percolation theory, an umbrella term used to describe any percolation model which takes place on a regular point lattice or any other discrete set, and while they're most certainly the most-studied of the discrete models, others such as AB percolation and mixed percolation do exist and are reasonably well-studied in their own right.Contrarily, one may also talk about continuum percolation models, i.e.,models which attempt to define analogous tools and results with respect to continuous, uncountable domains. In particular, continuum percolation theory involves notions..
A permutation problem invented by Cayley. Let the numbers 1, 2, ..., be written on a set of cards, and shuffle this deck of cards. Now, start counting using the top card. If the card chosen does not equal the count, move it to the bottom of the deck and continue counting forward. If the card chosen does equal the count, discard the chosen card and begin counting again at 1. The game is won if all cards are discarded, and lost if the count reaches .The number of ways the cards can be arranged such that at least one card is in the proper place for , 2, ... are 1, 1, 4, 15, 76, 455, ... (OEIS A002467).
A generalization to a quartic three-dimensional surface is the quartic surface of revolution(1)illustrated above. With , this surface is termed the "zeck" surface by Hauser. It has volume(2)geometric centroid(3)(4)(5)and inertia tensor(6)for constant density and mass .
Consider the plane quartic curve defined bywhere homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve for a number of values of between and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve is isomorphic to but the natural map is purely inseparable.The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation(Thurston 1999, p. 3; right figure) is more properly known as the Klein quarticor Klein curve. It has constant zero Gaussian curvature.Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston..
A dragon curve is a recursive nonintersecting curve whose name derives from its resemblance to a certain mythical creature.The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to the end, then append the string of preceding digits with its middle digit complemented. For example, the second-order curve is generated as follows: , and the third as .Continuing gives 110110011100100... (OEIS A014577), which is sometimes known as the regular paperfolding sequence and written with s instead of 0s (Allouche and Shallit 2003, p. 155). A recurrence plot of the limiting value of this sequence is illustrated above.Representing the sequence of binary digits 1, 110, 1101100, 110110011100100, ... in octal gives 1, 6, 154, 66344, ...(OEIS A003460; Gardner 1978, p. 216).This procedure is equivalent to drawing a right angle and subsequently..
A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..
A cube can be divided into subcubes for only , 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, p. 297). This sequence provides the solution to the so-called Hadwiger problem, which asks for the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts, and has the answer 47 (Gardner 1992, pp. 297-298).If and are in the sequence, so is , since -dissecting one cube in an -dissection gives an -dissection. The numbers 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations(1)(2)(3)(4)(5)(6)(7)Combining these facts gives the remaining terms in the sequence, and all numbers , and it has been shown that no other numbers occur.It is not possible to cut a cube into subcubes that are all different sizes (Gardner 1961, p. 208; Gardner 1992, p. 298).The seven pieces used to construct..
Gardner showed how to dissect a square into eight and nine acute scalene triangles.W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, illustrated above (pers. comm. to Ed Pegg, Jr., Oct 25, 2002). The coordinates can be found from solving the four simultaneous equations (1)(2)(3)(4)for the four unknowns and picking the solutions for which . The solutions are roots of 12th order polynomials with numerical values given approximately by(5)(6)(7)(8)Pegg has constructed a dissection of a square into 22 acute isosceles triangles.Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).
The number of different triangles which have integer side lengths and perimeter is(1)(2)(3)where is the partition function giving the number of ways of writing as a sum of exactly terms, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly complicated closed form is given by(4)The values of for , 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.The generating function for is given by(5)(6)(7) also satisfies(8)It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational..
A Nash equilibrium of a strategic game is a profile of strategies , where ( is the strategy set of player ), such that for each player , , , where and .Another way to state the Nash equilibrium condition is that solves for each . In words, in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players.The Season 1 episode "Dirty Bomb" (2005) of the television crime drama NUMB3RS mentions Nash equilibrium.
The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve..
The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann in 1928.Formally, let and be mixed strategies for players A and B. Let be the payoff matrix. Thenwhere is called the value of the game and and are called the solutions. It also turns out that if there is more than one optimal mixed strategy, there are infinitely many.In the Season 4 opening episode "Trust Metric" (2007) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that he used the minimax theorem in an attempt to derive an equation describing friendship.
For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..
A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by(1)where is the identity matrix.A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graphA complete set of Walsh..
The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle. The word derives from the Greek hypo- ("under") and teinein ("to stretch").The length of the hypotenuse of a right trianglecan be found using the Pythagorean theorem.Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta The Pirates of Penzance impresses the pirates with his knowledge of the hypotenuse in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of..
An -gonal trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron (not to be confused with a deltahedron), is a dual polyhedra of an -antiprism. Unfortunately, the name for these solids is not particularly well chosen since their faces are not trapezoids but rather kites. The trapezohedra are isohedra.The 3-trapezohedron (trigonal trapezohedron) is a rhombohedron having all six faces congruent. A special case is the cube (oriented along a space diagonal), corresponding to the dual of the equilateral 3-antiprism (i.e., the octahedron).A 4-trapezohedron (tetragonal trapezohedron) appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The trapezohedra generated by taking the duals of the equilateral antiprisms have side length , half-heights (half the peak-to-peak distance) , surface areas , and volumes..
"Escher's solid" is the solid illustrated on the right pedestal in M. C. Escher's Waterfall woodcut (Bool et al. 1982, p. 323). It is obtained by augmenting a rhombic dodecahedron until incident edges become parallel, corresponding to augmentation height of for a rhombic dodecahedron with unit edge lengths.It is the first rhombic dodecahedron stellation and is a space-filling polyhedron. Its convex hull is a cuboctahedron.It is implemented in the Wolfram Languageas PolyhedronData["EschersSolid"].It has edge lengths(1)(2)surface area and volume(3)(4)and moment of inertia tensor(5)The skeleton of Escher's solid is the graph of the disdyakis dodecahedron.Escher's solid can also be viewed as a polyhedron compound of three dipyramids (nonregular octahedra) with edges of length 2 and ...
The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid . It is also the dual polyhedron of the pentagonal prism and is an isohedron.It is implemented in the Wolfram Language as PolyhedronData["Dipyramid", 5].A pentagonal dipyramid appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).For a pentagonal dipyramid having a base with unit edge lengths, the circumradiusof the base pentagon is(1)In order for the top and bottom edges to also be of unit length, the polyhedron must be of height(2)The ratio of is therefore given by(3)where is the golden ratio.The surface area and volume of a unit pentagonal dipyramid are(4)(5)
The elongated square dipyramid with unit edge lengths is Johnson Solid .An elongated square dipyramid (having a central ribbon composed of rectangles instead of squares) appears in the top center as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).
Dürer's solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which Dürer's magic square appears, which depicts a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Although Dürer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted cube which is first stretched to give rhombic faces with angles of , and then truncated on top and bottom to yield bounding triangular faces whose vertices lie on the circumsphere of the azimuthal cube vertices.It is implemented in the Wolfram Languageas PolyhedronData["DuererSolid"].The skeleton of Dürer's solid is the Dürer graph (i.e., generalized Petersen graph ).Starting..
The stella octangula is a polyhedron compound composed of a tetrahedron and its dual (a second tetrahedron rotated with respect to the first). The stella octangula is also (incorrectly) called the stellated tetrahedron, and is the only stellation of the octahedron. A wireframe version of the stella octangula is sometimes known as the merkaba and imbued with mystic properties.The name "stella octangula" is due to Kepler (1611), but the solid was known earlier to many others, including Pacioli (1509), who called it the "octaedron elevatum," and Jamnitzer (1568); see Cromwell (1997, pp. 124 and 152).It is implemented in the Wolfram Languageas PolyhedronData["StellaOctangula"].A stella octangula can be inscribed in a cube, deltoidal icositetrahedron, pentagonal icositetrahedron, rhombic dodecahedron, small triakis octahedron, and tetrakis hexahedron, (E. Weisstein, Dec. 24-25,..
The uniform polyhedron whose dual is the great dirhombicosidodecacron. This polyhedron is exceptional because it cannot be derived from Schwarz triangles and because it is the only uniform polyhedron with more than six polygons surrounding each polyhedron vertex (four squares alternating with two triangles and two pentagrams). This unique polyhedron has features in common with both snub forms and hemipolyhedra, and its octagrammic faces pass through the origin.It has pseudo-Wythoff symbol . Its faces are , and its circumradius for unit edge length isThe great dirhombicosidodecahedron appears on the cover of issue 4, volume 3 of TheMathematica Journal.
The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of conics when rolled upon a line are sections of minimal surfaces (i.e., they yield minimal surfaces when revolved about the line) known as unduloids.A particularly interesting case of a roulette is a regular -gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000). For the rolling square, the shape of the road is the catenary truncated at (Wagon 2000). For a regular -gon, the Cartesian equation of the corresponding catenary iswhereThe roulette consisting of a square..
The great rhombic triacontahedron, also called the great stellated triacontahedron, is a zonohedron which is the dual of the great icosidodecahedron and Wenninger model . It is one of the rhombic triacontahedron stellations.It appears together with an isometric projection of the 5-hypercube on the cover (and p. 103) of Coxeter's well-known book on polytopes (Coxeter 1973).The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of , where is the golden ratio (Kabai 2002, p. 183) and keeping the pieces illustrated in the above stellation diagram.
The uniform polyhedron whose dual polyhedron is the small dodecicosacron. It has Wythoff symbol . Its faces are . Its circumradius for unit edge lengths is
The Desargues graph is the cubic symmetric graph on 20 vertices and 30 edges illustrated above in several embeddings. It is isomorphic to the generalized Petersen graph and to the bipartite Kneser graph . It is the incidence graph of the Desargues configuration. It can be represented in LCF notation as (Frucht 1976). It can also be constructed as the graph expansion of with steps 1 and 3, where is a path graph. It is distance-transitive and distance-regular graph and has intersection array .The Desargues graph is one of three cubic graphs on 20 nodes with smallest possible graph crossing number of 6 (the others being two unnamed graphs denoted CNG 6B and CNG 6C by Pegg and Exoo 2009), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).The Desargues is an integral graph with graph spectrum . It is cospectral with another nonisomorphic graph (Haemers and Spence 1995, van Dam and Haemers 2003).It is also a unit-distance..
The curve produced by fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .To derive the equations of the hypocycloid, call the angle by which a point on the small circle rotates about its center , and the angle from the center of the large circle to that of the small circle . Then(1)so(2)Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are(3)(4)(5)(6)If instead so the first point is at maximum radius (on the circle), then the equations of the hypocycloid are(7)(8)The curvature, arc length, and tangential angle of a hypocycloid are given by(9)(10)(11)An -cusped hypocycloid has . For an integer and with , the equations of the hypocycloid therefore become(12)(13)and the arc length and area are therefore(14)(15)A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145;..
The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations(1)(2)A polar equation can be derived by computing(3)(4)so(5)But(6)so(7)(8)Note that is the parameter here, not the polar angle. The polar angle from the center is(9)To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.(10)(11)(12)(13)so(14)(15)An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps times as large. After traveling around the circle once,..
The Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedTetrahedron"].The dual of the truncated tetrahedron is the triakis tetrahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and hexagonal faces are given by(4)(5)The surface area and volumeare(6)(7)
The snub cube, also called the cubus simus (Kepler 1619, Weissbach and Martini 2002) or snub cuboctahedron, is an Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms known as laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron , and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubCube"].Surprisingly, the tribonacci constant is intimately related to the metric properties of the snub cube.It can be constructed by snubification of a unit cube with outward offset(1)(2)and twist angle(3)(4)(5)(6)Here, the notation indicates a polynomial root and is the tribonacci constant.An attractive dual of the two enantiomers superposed on one another is illustrated above.Its dual polyhedron is the pentagonalicositetrahedron.The..
The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube and Wenninger dual . The mineral cuprite () forms in pentagonal icositetrahedral crystals (Steinhaus 1999, pp. 207 and 209).Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.A cube, octahedron, and stella octangula can all be inscribed on the vertices of the pentagonal icositetrahedron (E. Weisstein, Dec. 25, 2009).Surprisingly, the tribonacci constant is intimately related to the metric properties of the pentagonal icositetrahedron cube.Its irregular pentagonal faces have vertex angles of(1)(2)(3)(four times) and(4)(5)(6)(once), where is a polynomial root and is the tribonacci constant.The dual formed from a snub cube with..
The truncated octahedron is the 14-faced Archimedean solid , with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol . It was called the "mecon" by Buckminster Fuller (Rawles 1997).The dual polyhedron of the truncated octahedron is the tetrakis hexahedron. The truncated octahedron has the octahedral group of symmetries. The form of the fluorite () resembles the truncated octahedron (Steinhaus 1999, pp. 207-208).It is implemented in the Wolfram Languageas PolyhedronData["TruncatedOctahedron"].The solid of edge length can be formed from an octahedron of edge length via truncation by removing six square pyramids, each with edge slant height , base on a side, and height . The height and base area of the square pyramid are then(1)(2)(3)and its volume is(4)(5)The volume of the truncated octahedron is then given bythe volume of the octahedron(6)(7)minus..
In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a trisoctahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. The small triakis octahedron, called simply the triakis octahedron by Holden (1971, p. 55), is the 24-faced dual polyhedron of the truncated cube and is Wenninger dual . The addition of the word "small" is necessary to distinguish it from the great triakis octahedron, which is the dual of the stellated truncated hexahedron. The small triakis octahedron It can be constructed by augmentation of a unit edge-length octahedron by a pyramid with height .A small triakis octahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's..
The pentagonal hexecontahedron is the 60-faced dual polyhedron of the snub dodecahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentagonal hexecontahedron (E. Weisstein, Dec. 25-27, 2009).Its irregular pentagonal faces have vertex angles of(1)(2)(four times) and(3)(4)(once), where is a polynomial root.Because it is the dual of the chiral snub dodecahedron, the pentagonal hexecontahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.Starting with a snub dodecahedron with unit edge lengths, the edges lengths of the pentagonal hexecontahedron are given by the roots of (5)(6)which have approximate values and .The surface area and volume are both given by the roots of 12th-order..
The small rhombicuboctahedron is the 26-faced Archimedean solid consisting of faces . Although this solid is sometimes also called the truncated icosidodecahedron, this name is inappropriate since true truncation would yield rectangular instead of square faces. It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol .The solid may also be called an expanded (or cantellated) cube or octahedron sinceit may be constructed from either of these solids by the process of expansion.A small rhombicuboctahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicuboctahedron"].Its dual polyhedron is the deltoidal icositetrahedron, also called the trapezoidal icositetrahedron. The inradius of the..
The 32-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedDodecahedron"].The dual polyhedron is the triakisicosahedron.To construct the truncated dodecahedron by truncation, note that we want the inradius of the truncated pentagon to correspond with that of the original pentagon, , of unit side length . This means that the side lengths of the decagonal faces in the truncated dodecahedron satisfy(1)giving(2)The length of the corner which is chopped off is therefore given by(3)The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(4)(5)(6)The distances from the center of the solid to the centroids of the triangular and decagonal faces are given by(7)(8)The surface area and volumeare(9)(10)..
The 62-faced Archimedean solid with faces . It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol . The small dodecicosidodecahedron and small rhombidodecahedron are faceted versions.It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicosidodecahedron"].Its dual polyhedron is the deltoidal hexecontahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)It has surface area(4)and volume(5)
The 26-faced Archimedean solid consisting of faces . It is sometimes (improperly) called the truncated cuboctahedron (Ball and Coxeter 1987, p. 143), and is also more properly called the rhombitruncated cuboctahedron. It is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It can be combined with cubes and truncated octahedra into a regular space-filling pattern.The small cubicuboctahedron is a facetedversion of the great rhombicuboctahedron.Its dual is the disdyakis dodecahedron, also called the hexakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)(4)(5)(6)Additional quantities are(7)(8)(9)(10)(11)The distances between the solid center and centroids of the square and octagonal faces are(12)(13)The surface..
The 14-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedCube"].The dual polyhedron of the truncated cube is the small triakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and octagonal faces are(4)(5)The surface area and volumeare(6)(7)
The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron (Holden 1971, p. 55). It is Wenninger dual . It is composed of 30 golden rhombi joined at 32 vertices. It is a zonohedron and one of the five golden isozonohedra.The intersecting edges of the dodecahedron-icosahedron compound form the diagonals of 30 rhombi which comprise the triacontahedron. The cube 5-compound has the 30 facial planes of the rhombic triacontahedron and its interior is a rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987).More specifically, a tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the rhombic triacontahedron (E. Weisstein, Dec. 25-27, 2009).The rhombic triacontahedron is implemented in the WolframLanguage as PolyhedronData["RhombicTriacontahedron"].The short diagonals of the faces..
In general, a triakis tetrahedron is a non-regular dodecahedron that can be constructed as a positive augmentation of a regular tetrahedron. Such a solid is also known as a tristetrahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting dodecahedron is not regular, its faces are all identical. "The" triakis tetrahedron is the dual polyhedron of the truncated tetrahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit edge-length tetrahedron by a pyramid with height .Five tetrahedra of unit edge length (corresponding to a central tetrahedron and its regular augmentation) and one tetrahedron of edge length 5/3 can be inscribed in the vertices of the unit triakis tetrahedron, forming the configurations illustrated above.The triakis tetrahedron formed by taking the dual of a truncated tetrahedron with unit edge..
The (first) rhombic dodecahedron is the dual polyhedron of the cuboctahedron (Holden 1971, p. 55) and Wenninger dual . Its sometimes also called the rhomboidal dodecahedron (Cotton 1990), and the "first" may be included when needed to distinguish it from the Bilinski dodecahedron (Bilinski 1960, Chilton and Coxeter 1963).A rhombic dodecahedron appears in the upper right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The rhombic dodecahedron is implemented in the WolframLanguage as PolyhedronData["RhombicDodecahedron"].The 14 vertices of the rhombic dodecahedron are joined by 12 rhombiof the dimensions shown in the figure below, where(1)(2)(3)(4)(5)The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal "jack" (left figure). Joining..
The disdyakis triacontahedron is the dual polyhedron of the Archimedean great rhombicosidodecahedron . It is also known as the hexakis icosahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can be inscribed in the vertices of a disdyakis triacontahedron (E. Weisstein, Dec. 26-27, 2009).Starting with an Archimedean great rhombicosidodecahedron of unit edge lengths, the edge lengths of the corresponding disdyakis triacontahedron are(1)(2)(3)The corresponding midradius is(4)The surface area and volume are(5)(6)
The 60-faced dual polyhedron of the truncated dodecahedron (Holden 1971, p. 55) and Wenninger dual . Wenninger (1989, p. 46) calls the small triambic icosahedron the triakis octahedron.A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed on the vertices of the triakis icosahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated dodecahedronwith unit edge lengths gives a triakis icosahedron with edge lengths(1)(2)The surface area and volumeare(3)(4)
The disdyakis dodecahedron is the dual polyhedron of the Archimedean great rhombicuboctahedron and Wenninger dual . It is also called the hexakis octahedron (Unkelbach 1940; Holden 1971, p. 55).If the original great rhombicuboctahedronhas unit side lengths, then the resulting dual has edge lengths(1)(2)(3)The inradius is(4)Scaling the disdyakis dodecahedron so that gives a solid with surface area and volume(5)(6)
In general, a tetrakis hexahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of a cube. Such a solid is also known as a tetrahexahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. "The" tetrakis hexahedron is the 24-faced dual polyhedron of the truncated octahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit cube by a pyramid with height 1/4.A cube, octahedron, and stella octangula can all be inscribed in the vertices of the tetrakis hexahedron (E. Weisstein, Dec. 25, 2009).The edge lengths for the tetrakis hexahedron constructed as the dual of the truncatedoctahedron with unit edge lengths are(1)(2)Normalizing so that gives a tetrakis hexahedron with surface area and volume(3)(4)..
The deltoidal icositetrahedron is the 24-faced dual polyhedron of the small rhombicuboctahedron and Wenninger dual . It is also called the trapezoidal icositetrahedron (Holden 1971, p. 55).A deltoidal icositetrahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).A stella octangula, attractive octahedron 4-compound (whose dual is an attractive cube 4-compound), and cube can all be inscribed in a deltoidal icositetrahedron (E. Weisstein, Dec. 24, 2009). Superposing all these solids gives the beautiful compound illustrated above.For a small rhombicuboctahedron withunit edge length, the deltoidal icositetrahedron has edge lengths(1)(2)and inradius(3)Normalizing so the smallest edge has unit edge length gives a deltoidal icositetrahedron with surface area and volume(4)(5)..
The square antiprism is the antiprism with square bases whose dual is the tetragonal trapezohedron. The square antiprism has 10 faces.The square antiprism with unit edge lengths has surfacearea and volume(1)(2)
The deltoidal hexecontahedron is the 60-faced dual polyhedron of the small rhombicosidodecahedron . It is sometimes also called the trapezoidal hexecontahedron (Holden 1971, p. 55), strombic hexecontahedron, or dyakis hexecontahedron (Unkelbach 1940). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can all be inscribed in the vertices of the deltoidal hexecontahedron (E. W. Weisstein, Dec. 24-27, 2009). The resulting compound of all these inscriptable solids is also illustrated above.Starting from a small rhombicosidodecahedron of unit edge length, the edge lengths of the corresponding deltoidal hexecontahedron are(1)(2)The corresponding midradius is(3)The surface area and volume are(4)(5)..
A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).It is implemented in the Wolfram Languageas PolyhedronData["Cuboctahedron"].It is shown above in a number of symmetric projections.The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron..
The snub dodecahedron is an Archimedean solid consisting of 92 faces (80 triangular, 12 pentagonal), 150 edges, and 60 vertices. It is sometimes called the dodecahedron simum (Kepler 1619, Weissbach and Martini 2002) or snub icosidodecahedron. It is a chiral solid, and therefore exists in two enantiomorphous forms, commonly called laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron and Wenninger model . It has Schläfli symbol s and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubDodecahedron"].An attractive dual of the two enantiomers superposed on one another is illustrated above.The dual polyhedron of the snub dodecahedron isthe pentagonal hexecontahedron.It can be constructed by snubification of a dodecahedron of unit edge length with outward offset(1)and twist angle(2)Here, the notation indicates a polynomial root.The inradius..
The pentakis dodecahedron is the 60-faced dual polyhedron of the truncated icosahedron (Holden 1971, p. 55). It is Wenninger dual . It can be constructed by augmentation of a unit edge-length dodecahedron by a pyramid with height .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentakis dodecahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated icosahedronwith unit edge lengths gives a pentakis dodecahedron with edge lengths(1)(2)Normalizing so that , the surface area and volume are(3)(4)
A matrix with elements(1)for , 2, ..., . Hilbert matrices are implemented in the Wolfram Language by HilbertMatrix[m, n]. The figure above shows a plot of the Hilbert matrix with elements colored according to their values.Hilbert matrices whose entries are specified as machine-precision numbers are difficult to invert using numerical techniques.The determinants for the first few values of for , 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (OEIS A005249). The terms of sequence have the closed form(2)(3)(4)where is the Glaisher-Kinkelin constant and is the Barnes G-function. The numerical values are given in the following table.det()1123456The elements of the matrix inverse of the Hilbert matrix are given analytically by(5)(Choi 1983, Richardson 1999).
The truncated icosahedron is the 32-faced Archimedean solid corresponding to the facial arrangement . It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the structure of pure carbon known as buckyballs (a.k.a. fullerenes).The truncated icosahedron is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedIcosahedron"].Several symmetrical projections of the truncated icosahedron are illustrated above.The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances..
Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..
A superellipse is a curve with Cartesian equation(1)first discussed in 1818 by Lamé. A superellipse may be described parametrically by(2)(3)The restriction to is sometimes made.Superellipses with are also known as Lamé curves or Lamé ovals, and the case with is sometimes known as the squircle. By analogy, the superellipse with and might be termed the rectellipse.A range of superellipses are shown above, with special cases , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used with a number of different ratios for various of his projects. For example, he used for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and for his table.curve(squashed) astroid1(squashed) diamond2ellipsePiet Hein's "superellipse"4rectellipseIf is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers..
A hypotrochoid generated by a fixed point on a circle rolling inside a fixed circle. The curves above correspond to values of , 0.2, ..., 1.0.Additional attractive designs such as the one above can also be made by superposing individual spirographs.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features spirographs when discussing Guilloché patterns.
Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawley (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 227 stellations (including the original solid in the count as usual), some of which are illustrated above.The Great Stella stellation software reproduces Messer's 227 fully supported stellations. Using Miller's rules gives 358833098 stellations, 84959 of them reflexible and 358748139 of them chiral.The convex hull of the dodecadodecahedron is an icosidodecahedron and the dual of the icosidodecahedron is the rhombic triacontahedron, so the dual of the dodecadodecahedron (the medial rhombic triacontahedron) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Others include the great rhombic triacontahedron,..
One of the Kepler-Poinsot solids whose dual is the great stellated dodecahedron. It is also uniform polyhedron , Wenninger model , and has Schläfli symbol and Wythoff symbol .The great icosahedron can be constructed from an icosahedron with unit edge lengths by taking the 20 sets of vertices that are mutually spaced by a distance , the golden ratio. The solid therefore consists of 20 equilateral triangles. The symmetry of their arrangement is such that the resulting solid contains 12 pentagrams.The great icosahedron can most easily be constructed by building a "squashed" dodecahedron (top right figure) from the corresponding net (top left). Then, using the net shown in the bottom left figure, build 12 pentagrammic pyramids (bottom middle figure) and affix them into the dimples (bottom right). This method of construction is given in Cundy and Rollett (1989, pp. 98-99). If the edge lengths of the dodecahedron are unity,..
"Spikey" is the logo of Wolfram Research, makers of Mathematica and the Wolfram Language. In its original (Version 1) form, it is an augmented icosahedron with an augmentation height of , not to be confused with the great stellated dodecahedron, which is a distinct solid. This gives it 60 equilateral triangular faces, making it a deltahedron. More elaborate forms of Spikey have been used for subsequent versions of Mathematica. In particular, Spikeys for Version 2 and up are based on a hyperbolic dodecahedron with embellishments rather than an augmented icosahedron (Trott 2007, Weisstein 2009).The "classic" (Version 1) Spikey illustrated above is implemented in theWolfram Language as PolyhedronData["MathematicaPolyhedron"].The skeleton of the classic Spikey is the graph of thetriakis icosahedron.A glyph corresponding to the classic Spikey, illustrated above, is available as the character \[MathematicaIcon]..
The small stellated dodecahedron is the Kepler-Poinsot solids whose dual polyhedron is the great dodecahedron. It is also uniform polyhedron , Wenninger model , and is the first stellation of the dodecahedron (Wenninger 1989). The small stellated dodecahedron has Schläfli symbol and Wythoff symbol . It is concave, and is composed of 12 pentagrammic faces ().The small stellated dodecahedron appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral, Venice (Muraro 1955). It was rediscovered by Kepler (who used th term "urchin") in his work Harmonice Mundi in 1619, and again by Poinsot in 1809.The skeleton of the small stellated dodecahedron is isomorphic to the icosahedralgraph.Schläfli (1901, p. 134) did not recognize the small stellated dodecahedron as a regular solid because it violates the polyhedral formula, instead satisfying(1)where is the number of vertices, the number of edges,..
The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces, . It is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and the Wythoff symbol is . It is an isohedron, and a special case of the general tetrahedron and the isosceles tetrahedron.The regular tetrahedron is implemented in the Wolfram Language as Tetrahedron, and precomputed properties are available as PolyhedronData["Tetrahedron"].The tetrahedron has 7 axes of symmetry: (axes connecting vertices with the centers of the opposite faces) and (the axes connecting the midpoints of opposite sides).There are no other convex polyhedra other than the tetrahedron having four faces.The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring..
The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and Wythoff symbol . The octahedron of unit side length is the antiprism of sides with height . The octahedron is also a square dipyramid with equal edge lengths.The regular octahedron is implemented in the Wolfram Language as Octahedron, and precomputed properties are available as PolyhedronData["Octahedron"].There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using the Pólya enumeration theorem.The dual polyhedron of an octahedron with unit edge lengths is a cube with edge lengths .The illustration..
The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, .The regular icosahedron is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).The regular icosahedron is implemented in the Wolfram Language as Icosahedron, and precomputed properties are available as PolyhedronData["Icosahedron"].Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.Two icosahedra appears as polyhedral "stars"..
The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .The regular dodecahedron is implemented in the Wolfram Language as Dodecahedron, and precomputed properties are available as PolyhedronData["Dodecahedron"].There are 43380 distinct nets for the regular dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the regular dodecahedron can be addressed using the Pólya enumeration theorem.The image above shows an origami regular dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).A..
In general, an icosidodecahedron is a 32-faced polyhedron. "The" icosidodecahedron is the 32-faced Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is also uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["Icosidodecahedron"].Several symmetric projections of the icosidodecahedron are illustrated above. The dual polyhedron is the rhombic triacontahedron. The polyhedron vertices of an icosidodecahedron of polyhedron edge length are , , , , , . The 30 polyhedron vertices of an octahedron 5-compound form an icosidodecahedron (Ball and Coxeter 1987). Faceted versions include the small icosihemidodecahedron and small dodecahemidodecahedron.The icosidodecahedron constructed in origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction..
The 62-faced Archimedean solid with faces . It is also known as the rhombitruncated icosidodecahedron, and is sometimes improperly called the truncated icosidodecahedron (Ball and Coxeter 1987, p. 143), a name which is inappropriate since truncation would yield rectangular instead of square. The great rhombicosidodecahedron is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicosidodecahedron is an equilateral zonohedron and is the Minkowski sum of five cubes.Its dual is the disdyakis triacontahedron, also called the hexakis icosahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The great rhombicosidodecahedron has surface area(4)and volume(5)The great rhombicosidodecahedron constructed by E. K. Herrstrom in origami is illustrated above (Kasahara and Takahama 1987, pp. 46-49)...
Augmentation is the dual operation of truncation which replaces the faces of a polyhedron with pyramids of height (where may be positive, zero, or negative) having the face as the base (Cromwell 1997, p. 124 and 195-197). The operation is sometimes also called accretion, akisation (since it transforms a regular polygon to an -akis polyhedron, i.e., quadruples the number of faces), capping, or cumulation.B. Grünbaum used the terms elevatum and invaginatum for positive-height (outward-pointing) and negative-height (inward-pointing), respectively, pyramids used in augmentation.The term "augmented" is also sometimes used in the more general context of affixing one polyhedral cap over the face of a base solid. An example is the Johnson solid called the augmented truncated cube, for which the affixed shape is a square cupola--not a pyramid.Augmentation is implemented under the misnomer Stellate[poly,..
The Skewes number (or first Skewes number) is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral.Isaac Asimov featured the Skewes number in his science fact article "Skewered!"(1974).In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper boundwas subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966 (Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987), and less than (Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil 2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson left open the possibility that the inequality could fail around , and thus established a large range of violation around (Derbyshire 2004, p. 237). More recent work by Demichel establishes that the first crossover..
An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..
The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..
The Kuen surface is a special case of Enneper'snegative curvature surfaces which can be given parametrically by(1)(2)(3)(4)(5)for , (Reckziegel et al. 1986; Gray et al. 2006, p. 484).The Kuen surface appears on the cover of volume 2, number 1 of La Gaceta de laReal Sociedad Matemática Española (1999).The coefficients of the first fundamental formare(6)(7)(8)the second fundamental form coefficientsare(9)(10)(11)and the surface area element is(12)The Gaussian and meancurvatures are(13)(14)so the Kuen surface has constant negative Gaussian curvature, and the principal curvatures are(15)(16)(Gray 1997, p. 496).
A surface of constant negative curvature obtained by twisting a pseudosphere and given by the parametric equations(1)(2)(3)The above figure corresponds to , , , and .Dini's surface is pictured in the upper right-hand corner of Gray (1997; left figure), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).The coefficients of the first fundamental formare(4)(5)(6)the coefficients of the second fundamentalform are(7)(8)(9)and the area element is(10)The Gaussian and meancurvatures are given by(11)(12)
Togliatti surfaces are quintic surfaces having the maximum possible number of ordinary double points (31).A related surface sometimes known as the dervish can be defined by(1)where(2)(3)(4)(5)(6)(7)(8)and(9)(10)(11)
The Kummer surfaces are a family of quartic surfacesgiven by the algebraic equation(1)where(2), , , and are the tetrahedral coordinates(3)(4)(5)(6)and is a parameter which, in the above plots, is set to .The above plots correspond to (7)(double sphere), 2/3, 1(8)(Roman surface), 2, 3(9)(four planes), and 5. The case corresponds to four real points.The following table gives the number of ordinary double points for various ranges of , corresponding to the preceding illustrations.parameterreal nodescomplex nodes412412160160The Kummer surfaces can be represented parametrically by hyperelliptic theta functions. Most of the Kummer surfaces admit 16 ordinary double points, the maximum possible for a quartic surface. A special case of a Kummer surface is the tetrahedroid.Nordstrand gives the implicit equations as(10)or(11)..
A cubic algebraicsurface given by the equation(1)with the added constraint(2)The implicit equation obtained by taking the plane at infinity as is(3)(Hunt 1996), illustrated above.On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's seal lines) present on a general smooth cubic surface are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called Eckardt points (Fischer 1986ab, Hunt 1996), and the Clebsch diagonal surface is the unique cubic surface containing 10 such points (Hunt 1996).If one of the variables describing Clebsch's diagonal surface is dropped, leaving the equations(4)(5)the equations degenerate into two intersecting planes given by the equation(6)
An algebraic surface with affine equation(1)where is a Chebyshev polynomial of the first kind and is a polynomial defined by(2)where the matrices have dimensions . These represent surfaces in with only ordinary double points as singularities. The first few surfaces are given by (3)(4)(5)The th order such surface has(6)singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870) for , 2, .... For a number of orders , Chmutov surfaces have more ordinary double points than any other known equations of the same degree.Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces(7)where is even and is again a Chebyshev polynomial of the first kind. For example, the surfaces illustrated above have orders 2, 4, and 6 and are given by the equations (8)(9)(10)..
A surface with tetrahedral symmetry which looks likean inflatable chair from the 1970s. It is given by the implicit equationThe surface illustrated above has , , and .
The dodecic surface defined by(1)where(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) and are both invariants of order 12. It was discovered by A. Sarti in 1999.The version with arbitrary and has exactly 600 ordinary points (Endraß), and taking gives the surface with 560 real ordinary points illustrated above.The Sarti surface is invariant under the bipolyhedralgroup.
Cayley's cubic surface is the unique cubic surface having four ordinary double points (Hunt), the maximum possible for cubic surface (Endraß). The Cayley cubic is invariant under the tetrahedral group and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß).If the ordinary double points in projective three-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is(1)(Hunt). Defining "affine" coordinates with plane at infinity and(2)(3)(4)then gives the equation(5)plotted in the left figure above (Hunt). The slightly different form(6)is given by Endraß (2003) which, when rewritten in tetrahedralcoordinates, becomes(7)plotted in the right figure above.The Hessian of the Cayley cubic is given by(8)in homogeneous coordinates , , , and . Taking the plane at infinity..
A heart-shaped surface given by the sextic equation(Taubin 1993, 1994). The figures above show a ray-traced rendering (left) and the cross section (right) of the surface.A slight variation of the same surface is given by(Nordstrand, Kuska 2004).
The quartic surface obtained by replacing the constant in the equation of the Cassini ovals with , obtaining(1)As can be seen by letting to obtain(2)(3)the intersection of the surface with the plane is a circle of radius .The Gaussian curvature of the surface is givenimplicitly by(4)Let a torus of tube radius be cut by a plane perpendicular to the plane of the torus's centroid. Call the distance of this plane from the center of the torus hole , let , and consider the intersection of this plane with the torus as is varied. The resulting curves are Cassini ovals, and the surface having these curves as cross sections is the Cassini surface(5)which has a scaled on the right side instead of .
Endraß surfaces are a pair of octic surfaces which have 168 ordinary double points. This is the maximum number known to exist for an octic surface, although the rigorous upper bound is 174. The equations of the surfaces arewhere is a parameter. All ordinary double points of are real, while 24 of those in are complex. The surfaces were discovered in a five-dimensional family of octics with 112 nodes, and are invariant under the group .The surfaces illustrated above take . The first of these has 144 real ordinary double points, and the second of which has 144 complex ordinary double points, 128 of which are real.
The Barth sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 65. The surface was discovered by W. Barth in 1994, and is given by the implicit equationwhere is the golden ratio.Taking gives the surface in 3-space illustrated above, which retains 50 ordinary double points.Of these, 20 nodes are at the vertices of a regular dodecahedron of side length and circumradius (left figure above), and 30 are at the vertices of a concentric icosidodecahedron and circumradius 1 (right figure).The Barth sextic is invariant under the icosahedralgroup. Under the mapthe surface is the eightfold cover of the Cayley cubic(Endraß 2003).The Barth sextic appeared on the cover of the March 1999 issue of Notices of theAmerican Mathematical Society (Dominici 1999)...
The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..
The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is(1)The total length of the spiral for an -gon with side length is therefore(2)(3)Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The area of this region, illustrated above for -gons of side length , is(4)The shaded triangular polygonal spiral is a rep-4-tile.
The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.It is given by the parametric equations(1)(2)(3)where(4)and is a constant. Plugging in therefore gives(5)(6)(7)It is a special case of a loxodrome.The arc length, curvature,and torsion are all slightly complicated expressions.A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).
A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1999, p. 229). It is for this reason that squirrels chasing one another up and around tree trunks follow helical paths.Helices come in enantiomorphous left- (coils counterclockwise as it "goes away") and right-handed forms (coils clockwise). Standard screws, nuts, and bolts are all right-handed, as are both the helices in a double-stranded molecule of DNA (Gardner 1984, pp. 2-3). Large helical structures in animals (such as horns) usually appear in both mirror-image forms, although the teeth of a male narwhal, usually..
The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...
Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.
An infinite sequence of circles such that every four consecutive circles are mutually tangent, and the circles' radii ..., , ..., , , , , , , ..., , , ..., are in geometric progression with ratiowhere is the golden ratio (Gardner 1979ab). Coxeter (1968) generalized the sequence to spheres.
A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.A spiral tiling using a single piece is illustrated..
The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of the Mathematisches Forschungsinstitut Oberwolfach library building on January 28, 1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured..
A set of magic circles is a numbering of the intersections of the circles such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circles have magic constants 14 and 39 (Madachy 1979). For circles, the constant is , for , 2, ... corresponding to 3, 14, 39, 84, 155, 258, ... (OEIS A027444).Another type of magic circle arranges the number 1, 2, ..., in a number of rings, which each ring containing the same number of elements and corresponding elements being connected with radial lines. One of the numbers (which is subsequently ignored) is placed at the center. In a magic circle arrangement, the rings have equal sums and this sum is also equal to the sum of elements along each diameter (excluding the central number). Three magic circles using the numbers 1 to 33 are illustrated above...
Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.Mathematicians sometimes use the term "combinatorics" to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as "enumeration."The Season 1 episode "Noisy Edge" (2005) of the television crime drama NUMB3RS mentions combinatorics.
Combinatorial matrix theory is a rich branch of mathematics that combines combinatorics, graph theory, and linear algebra. It includes the theory of matrices with prescribed combinatorial properties, including permanents and Latin squares. It also comprises combinatorial proof of classical algebraic theorems such as Cayley-Hamilton theorem.As mentioned in Season 4 episodes 407 "Primacy" and 412 "Power" of the television crime drama NUMB3RS, professor Amita Ramanujan's primary teaching interest is combinatorial matrix theory.
The rook is a chess piece that may move any number of spaces either horizontally or vertically per move. The maximum number of nonattacking rooks that may be placed on an chessboard is . This arrangement is achieved by placing the rooks along the diagonal (Madachy 1979). The total number of ways of placing nonattacking rooks on an board is (Madachy 1979, p. 47). In general, the polynomialwhose coefficients give the numbers of ways nonattacking rooks can be placed on an chessboard is called a rook polynomial.The number of rotationally and reflectively inequivalent ways of placing nonattacking rooks on an board are 1, 2, 7, 23, 115, 694, ... (OEIS A000903; Dudeney 1970, p. 96; Madachy 1979, pp. 46-54).The minimum number of rooks needed to occupy or attack all spaces on an chessboard is 8 (Madachy 1979), arranged in the same orientation as above.Consider an chessboard with the restriction that, for every subset of , a rook may not..
The rook numbers of an board are the number of subsets of size such that no two elements have the same first or second coordinate. In other word, it is the number of ways of placing rooks on a board such that none attack each other (one form of the so-called rooks problem). The rook number is therefore the leading coefficient of the corresponding rook polynomial .For an board, each permutation matrix corresponds to an allowed configuration of rooks. However, the permutation matrices give only a subset of the total number of solutions, which on an board is simply the factorial . This can be seen easily by noting that there are ways to place the first rook in the first column, ways to place the second rook in the second column, ways to place the third rook, ..., and a single way to place the th rook in the last (th) column.The rook numbers of a board determine the rook numbers of the complementary board , written as . This is known as the rook reciprocity theorem...
Various handshaking problems are in circulation, the most common one being the following. In a room of people, how many different handshakes are possible?The answer is . To see this, enumerate the people present, and consider one person at a time. The first person may shake hands with other people. The next person may shake hands with other people, not counting the first person again. Continuing like this gives us a total number ofhandshakes, which is exactly the answer given above.Another popular handshake problem starts out similarly with people at a party. Not being able to shake hands with yourself, and not counting multiple handshakes with the same person, the problem is to show that there will always be two people at the party, who have shaken hands the same number of times.The solution to this problem uses Dirichlet's box principle. If there exists a person at the party, who has shaken hands zero times, then every person at the party has shaken..
The continuous Fourier transform is definedas(1)(2)Now consider generalization to the case of a discrete function, by letting , where , with , ..., . Writing this out gives the discrete Fourier transform as(3)The inverse transform is then(4)Discrete Fourier transforms (DFTs) are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components. There are however a few subtleties in the interpretation of discrete Fourier transforms. In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length. In particular, if are real, then and are related by(5)for , 1, ..., , where denotes the complex conjugate. This means that the component is always real for real data.As a result of the above relation, a periodic function will contain transformed peaks in not one, but two places. This happens because the periods of the input data..
The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted in modern texts, although the notations and (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where and and are related by(1)Let the multivalued function be interpreted to stand..
Jenny's constant is the name given (Munroe 2012) to the positive real constant defined by(1)(2)(OEIS A182369), the first few digits of which are 867-5309, corresponding to the fictitious phone number in the song "867-5309/Jenny" performed by Tommy Tutone in 1982.Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)where is the hard hexagon entropy constant, the first three of which are "better" than the canonical Jenny expression (E. Weisstein, Jul. 12, 2013).
Triskaidekaphobia is the fear of 13, a number commonly associated with bad luck in Western culture. While fear of the number 13 can be traced back to medieval times, the word triskaidekaphobia itself is of recent vintage, having been first coined by Coriat (1911; Simpson and Weiner 1992). It seems to have first appeared in the general media in a Nov. 8, 1953 New York Times article covering discussions of a United Nations committee.This superstition leads some people to fear or avoid anything involving the number 13. In particular, this leads to interesting practices such as the numbering of floors as 1, 2, ..., 11, 12, 14, 15, ... (OEIS A011760; the "elevator sequence"), omitting the number 13, in many high-rise American hotels, the numbering of streets avoiding 13th Avenue, and so on.Apparently, 13 hasn't always been considered unlucky. In fact, it appears that in ancient times, 13 was either considered in a positive light or..
Nice approximations for the golden ratio are given by(1)(2)the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively. An even more amazing approximation uses Catalan's constant and the Feigenbaum constant is given by(3)which is accurate to within (D. Ross, cited in Pegg 2005).A curious (although not particularly useful) approximation due to D. Barron is given by(4)where is Catalan's constant and is the Euler-Mascheroni constant, which is good to two digits.
A curious approximation to the Feigenbaum constant is given by(1)where is Gelfond's constant, which is good to 6 digits to the right of the decimal point.M. Trott (pers. comm., May 6, 2008) noted(2)where is Gauss's constant, which is good to 4 decimal digits, and(3)where is the tetranacci constant, which is good to 3 decimal digits.A strange approximation good to five digits is given by the solution to(4)which is(5)where is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).(6)gives to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).M. Hudson (pers. comm., Nov. 20, 2004) gave(7)(8)(9)which are good to 17, 13, and 9 digits respectively.Stoschek gave the strange approximation(10)which is good to 9 digits.R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations(11)(12)(13)(14)(15)(16)where e is the base of the natural logarithm and..
A number having 666 digits (where 666 is the beastnumber) is called an apocalypse number.The Fibonacci number is the smallest Fibonacci apocalypse number (Livio 2002, p. 108).Apocalypse primes are given by for , 1837, 6409, 7329, 8569, 8967, 9663, ... (OEIS A115983). The smallest apocalypse prime containing the digits 666 is (Rupinski).
Convergents of the pi continued fractions are the simplest approximants to . The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.Two approximations follow from the near-identity function evaluated at and , giving(1)(2)which are good to 2 and 3 digits, respectively.Kochanski's approximation is the rootof(3)given by(4)which is good to 4 digits.Another curious fact is the almost integer(5)which can also be written as(6)(7)Here, is Gelfond's constant. Applying cosine a few more times gives(8)Another approximation involving is given by(9)which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).An apparently interesting near-identity is given by(10)which becomes less surprising when it is noted that 555555 is a repdigit,so the above is..
E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant given by(1)which is good to 6 digits.M. Hudson (pers. comm., Nov. 8, 2004) found the approximations(2)(3)(4)(5)(6)(7)where is the Euler-Mascheroni constant and is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.A curious approximation to is given by(8)where is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).Lima (2009) found the approximation(9)where is Catalan's constant, which is correct to 21 digits.
An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by(1)found by R. Sabey in 2004 (Friedman 2004).Castellanos (1988ab) gives several curious approximations to ,(2)(3)(4)(5)(6)(7)which are good to 6, 7, 9, 10, 12, and 15 digits respectively.E. Pegg Jr. (pers. comm., Mar. 2, 2002), found(8)which is good to 7 digits.J. Lafont (pers. comm., MAy 16, 2008) found(9)where is a harmonic number, which is good to 7 digits.N. Davidson (pers. comm., Sept. 7, 2004) found(10)which is good to 6 digits.D. Barron noticed the curious approximation(11)where is Catalan's constant and is the Euler-Mascheroni constant, which however, is only good to 3 digits.
Approximations to Catalan's constant include(1)(2)(3)(4)(5)(6)(M. Hudson, pers. comm., Nov. 19, 2004), where is the golden ratio, which are good to 4, 5, 6, 6, 7, 7, and 9 digits, respectively.Other approximations include(7)(8)(K. Hammond, pers. comm., Dec. 31, 2005), where is the golden ratio, which are good to 5 and 9 digits, respectively.
A Woodall prime is a Woodall numberthat is prime. The first few Woodall primes are 7, 23, 383, 32212254719, 2833419889721787128217599, ... (OEIS A050918), corresponding to , 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, ... (OEIS A002234). The following table summarizes large known Woodall primes. As of Mar. 2018, all have been checked (PrimeGrid).decimal digitsdate1467763441847Jun. 20072013992606279Aug. 20072367906712818Aug. 200737529481129757Dec. 2007170166025122515Mar. 2018
, sometimes also denoted (Abramowitz and Stegun 1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94; Andrews 1998, p. 1), gives the number of ways of writing the integer as a sum of positive integers, where the order of addends is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena 1990, p. 51). For example, since 4 can be written(1)(2)(3)(4)(5)it follows that . is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].The values of for , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of for , 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).The first few prime values of are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132,..
Let a prime number generated by Euler's prime-generating polynomial be known as an Euler prime. Then the first few Euler primes occur for , 2, ..., 39, 42, 43, 45, ... (OEIS A056561), corresponding to the primes 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, ... (OEIS A005846).As of Feb. 2013, the largest known Euler prime is , which has 398204 decimal digits and was found by D. Broadhurst (https://primes.utm.edu/primes/page.php?id=111195).
A prime is called a Wolstenholme prime if the central binomial coefficient(1)or equivalently if(2)where is the th Bernoulli number and the congruence is fractional.A prime is a Wolstenholme prime if and only if(3)where the congruence is again fractional.The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to (McIntosh 2004).
A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having , 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, ... (OEIS A016115; De Geest) and the total number of palindromic primes less than 10, , , ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251). Gupta (2009) has computed the numbers of palindromic primes up to .The following table lists palindromic primes in various small bases. OEISbase- palindromic primes2A11769711, 101, 111, 10001, 11111, 1001001, 1101011, ...3A1176982, 111, 212, 12121, 20102, 22122, ...4A1176992, 3, 11, 101, 131, 323, 10001, 11311, 12121, ...5A1177002, 3, 111, 131, 232, 313, 414, 10301, 12121,..
The Euler numbers, also called the secant numbers or zig numbers, are defined for by(1)(2)where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.A different sort of Euler number, the Euler number of a finite complex , is defined by(3)This Euler number is a topological invariant.To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes.Some values of the (secant) Euler numbers are(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(OEIS A000364).The slightly different convention defined by(16)(17)is..
A Wilson prime is a prime satisfyingwhere is the Wilson quotient, or equivalently,The first few Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) showed there are no others less than (McIntosh 2004), a limit that has subsequently been increased to (Costa et al. 2012).
A Wieferich prime is a prime which is a solution to the congruence equation(1)Note the similarity of this expression to the special case of Fermat'slittle theorem(2)which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220), with none other less than (Lehmer 1981, Crandall 1986, Crandall et al. 1997), a limit since increased to (McIntosh 2004) and subsequently to by PrimeGrid as of November 2015.Interestingly, one less than these numbers have suggestive periodic binaryrepresentations(3)(4)(Johnson 1977).If the first case of Fermat's last theorem is false for exponent , then must be a Wieferich prime (Wieferich 1909). If with and relatively prime, then is a Wieferich prime iff also divides . The conjecture that there are no three consecutive powerful numbers implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1996, p. 341; Vardi 1991). In addition, the abc..
In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers.To this day, it is not known if any odd perfect numbers exist, although..
A "weird number" is a number that is abundant (i.e., the sum of proper divisors is greater than the number) without being pseudoperfect (i.e., no subset of the proper divisors sums to the number itself). The pseudoperfect part of the definition means that finding weird numbers is a case of the subset sum problem.Since prime numbers are deficient, prime numbers are not weird. Similarly, since multiples of 6 are pseudoperfect, no weird number is a multiple of 6.The smallest weird number is 70, which has proper divisors 1, 2, 5, 7, 10, 14, and 35. These sum to 74, which is greater that the number itself, so 70 is abundant, and no subset of them sums to 70. In contrast, the smallest abundant number is 12, which has proper divisors 1, 2, 3, 4, and 6. These sum to 16, so 12 is abundant, but the subset sum equals 12, so 12 is not weird.The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...(OEIS A006037).An infinite number of weird..
An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers, and has complexity(1)reducing the exponent over the continued fraction factorization algorithm and quadratic sieve. There are three values of relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbers near a large power,(2)for the "general" case applicable to any odd positive number which is not a power,(3)and for a version using many polynomials (Coppersmith1993),(4)
Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers(1)(2)known as Euclid numbers, where is the th prime and is the primorial.The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (OEIS A006862; Tietze 1965, p. 19).The indices of the first few prime Euclid numbers are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (OEIS A014545), so the first few Euclid primes (commonly known as primorial primes) are 3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239). The largest known Euclid number is , and it is not known if there are an infinite number of prime Euclid numbers (Guy 1994, Ribenboim 1996).The largest factors of for , 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS A002585)...
Elliptic curve primality proving, abbreviated ECPP, is class of algorithms that provide certificates of primality using sophisticated results from the theory of elliptic curves. A detailed description and list of references are given by Atkin and Morain (1990, 1993).Adleman and Huang (1987) designed an independent algorithm using hyperellipticcurves of genus two.ECPP is the fastest known general-purpose primality testing algorithm. ECPP has a running time of . As of 2004, the program PRIMO can certify a 4769-digit prime in approximately 2000 hours of computation (or nearly three months of uninterrupted computation) on a 1 GHz processor using this technique. As of 2009, the largest prime certified using this technique was the 11th Mills' prime (https://primes.utm.edu/primes/page.php?id=77907)which has decimal digits. The proof was performed using a distributed computation that started in September 2005 and ended in June 2006..
A Wagstaff prime is a prime number of the form for a prime number. The first few are given by , 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, and 4031399 (OEIS A000978), with and larger corresponding to probable primes. These values correspond to the primes with indices , 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).The Wagstaff primes are featured in the newMersenne prime conjecture.There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving.A Wagstaff prime can also be interpreted as a repunit prime of base , asif is odd, as it must be for the above number to be prime.Some of the largest known Wagstaff probable primes are summarized in the following..
The numerical value of is given by(OEIS A002392). It was computed to decimal digits by S. Kondo on May 20, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).The starting positions of the first occurrence of , 1, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first..
The elliptic curve factorization method, abbreviated ECM and sometimes also called the Lenstra elliptic curve method, is a factorization algorithm that computes a large multiple of a point on a random elliptic curve modulo the number to be factored . It tends to be faster than the Pollard rho factorization and Pollard p-1 factorization methods.Zimmermann maintains a table of the largest factors found using the ECM. As of Jan. 2009, the largest prime factor found using the ECM had 67 decimal digits. This factor of was found by B. Dodson on Aug. 24, 2006 (Zimmermann).
A pair of numbers and such thatwhere is the unitary divisor function. Hagis (1971) and García (1987) give 82 such pairs. The first few are (114, 126), (1140, 1260), (18018, 22302), (32130, 40446), ... (OEIS A002952 and A002953; Pedersen).On Jan. 30, 2004, Y. Kohmoto discovered the largest known unitary amicable pair, where each member has 317 digits.Kohmoto calls a unitary amicable pair whose members are squareful a proper unitary amicable pair.
The Earls sequence gives the starting position in the decimal digits of (or in general, any constant), not counting digits to the left of the decimal point, at which a string of copies of the number first occurs. The following table gives generalized Earls sequences for various constants, including .constantOEISsequenceApéry's constantA22907410, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ...Catalan's constantA2248192, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ...Champernowne constantA2248961, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ...Copeland-Erdős constantA2248975, 113, 1181, 21670, 263423, 7815547, 35619942, 402720247, 450680638eA2248282, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011Euler-Mascheroni constantA2248265, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186Glaisher-Kinkelin constantA2257637,..
The decimal expansion of the natural logarithmof 2 is given by(OEIS A002162). It was computed to decimal digits by S. Kondo on May 14, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).The starting positions of the first occurrence of , 1, ... in the decimal expansion of are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901), which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of , and 9876543210 occurs starting..
Twin primes are pairs of primes of the form (, ). The term "twin prime" was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 and A006512).All twin primes except (3, 5) are of the form .It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes,(1)converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201)...
The constant e with decimal expansion(OEIS A001113) can be computed to digits of precision in 10 CPU-minutes on modern hardware. was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). was computed to decimal digits by S. Kondo on Jul. 5, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).The starting positions of the first occurrence of in the decimal expansion of (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).The digit sequence 0123456789..
A number is -multiperfect (also called a -multiply perfect number or -pluperfect number) iffor some integer , where is the divisor function. The value of is called the class. The special case corresponds to perfect numbers , which are intimately connected with Mersenne primes (OEIS A000396). The number 120 was long known to be 3-multiply perfect () sinceThe following table gives the first few for , 3, ..., 6.2A0003966, 28, 496, 8128, ...3A005820120, 672, 523776, 459818240, 1476304896, 510011801604A02768730240, 32760, 2178540, 23569920, ...5A04606014182439040, 31998395520, 518666803200, ...6A046061154345556085770649600, 9186050031556349952000, ...Lehmer (1900-1901) proved that has at least three distinct prime factors, has at least four, at least six, at least nine, and at least 14, etc.As of 1911, 251 pluperfect numbers were known (Carmichael and Mason 1911). As of 1929, 334 pluperfect numbers were known, many of them found..
In the 1980s, Samuel Yates defined a titanic prime to be a prime number of at least 1000 decimal digits. The smallest titanic prime is . As of 1990, more than 1400 were known (Ribenboim 1990). By 1995, more than were known, and many tens of thousands are known today. The largest prime number known as of December 2018 is the Mersenne prime , which has a whopping decimal digits.
A double Mersenne number is a number of the formwhere is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime can be prime only for prime , a double Mersenne prime can be prime only for prime , i.e., a Mersenne prime. Double Mersenne numbers are prime for , 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).The next four , , , and have known factors summarized in the following table. The status of all other double Mersenne numbers is unknown, with being the smallest unresolved case. Since this number has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer test to be practical. The only possible method of determining the status of this number is therefore attempting to find small divisors..
Theodorus's constant has decimal expansion(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does,..
Given the Mertens function defined by(1)where is the Möbius function, Stieltjes claimed in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes (1885) claimed that he had a proof of the general result. However, it seems likely that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens (1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim(2)for was "very probable."The Mertens conjecture has important implications, since the truth of any equalityof the form(3)for any fixed (the form of the Mertens conjecture with ) would imply the Riemann hypothesis. In fact, the statement(4)for any is equivalent to the Riemann hypothesis (Derbyshire 2004, p. 251).Mertens (1897) verified the conjecture for , and this was subsequently extended to by..
A Thâbit ibn Kurrah prime, sometimes called a 321-prime, is a Thâbit ibn Kurrah number (i.e., a number of the form for nonnegative integer ) that is prime.The indices for the first few Thâbit ibn Kurrah primes are 0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, ... (OEIS A002235), corresponding to the primes 2, 5, 11, 23, 47, 191, 383, 6143, ... (OEIS A007505). Riesel (1969) extended the search to . A search for larger primes was coordinated by P. Underwood. PrimeGrid has continued that search and has checked values of up to as of Nov. 2015 (PrimeGrid). The table below summarizes the largest known Thâbit ibn Kurrah primes.digitsdiscovererPrimeGrid (Dec. 2005; https://primes.utm.edu/primes/page.php?id=76506)PrimeGrid (Mar. 2007; https://primes.utm.edu/primes/page.php?id=79671)PrimeGrid (Apr. 2008; https://primes.utm.edu/primes/page.php?id=84769)PrimeGrid..
A Mersenne prime is a Mersenne number, i.e., anumber of the formthat is prime. In order for to be prime, must itself be prime. This is true since for composite with factors and , . Therefore, can be written as , which is a binomial number that always has a factor .The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (OEIS A000668) corresponding to indices , 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS A000043).Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers.It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line through the origin to the asymptotic number of Mersenne primes with for the first 51 (known) Mersenne primes gives a best-fit line with , illustrated above. If the line is not restricted to pass through..
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CC-MAIN-2020-45
| 154,131 | 200 |
https://www.brainkart.com/article/Important-Short-Objective-Question-and-Answers--Initial-Value-Problems-for-Ordinary-Differential-Equations_6469/
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math
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2. State the disadvantage of Taylor series method.
In the differential equation f (x, y), = f (x, y), the function f (x, y),may have a complicated algebraical structure. Then the evaluation of higher order derivatives may become tedious. This is the demerit of this method.
3. Write the merits and demerits of the Taylor method of solution.
The method gives a straight forward adaptation of classic to develop the solution as an infinite series. It is a powerful single step method if we are able to find the successive derivatives easily. If f (x.y) involves some complicated algebraic structures then the calculation of higher derivatives becomes tedious and the method fails.This is the major drawback of this method. However the method will be very useful for finding the starting values for powerful methods like Runge - Kutta method, Milne’s method etc.
4.Which is better Taylor’s method or R. K. Method?(or) State the special advantage of Runge-Kutta method over taylor series method
R.K Methods do not require prior calculation of higher derivatives of y(x) ,as the Taylor method does. Since the differential equations using in applications are often complicated, the calculation of derivatives may be difficult.
Also the R.K formulas involve the computation of f (x, y) at various positions, instead of derivatives and this function occurs in the given equation.
5.Compare Runge-Kutta methods and predictor –corrector methods for solution of initial value problem.
1.Runge-methods are self starting,since they do not use information from previously calculated points.
2.As mesne are self starting,an easy change in the step size can be made at any stage. 3.Since these methods require several evaluations of the function f (x, y), they are time consuming.
4.In these methods,it is not possible to get any information about truncation error. Predictor Corrector methods:
1.These methods require information about prior points and so they are not self starting. 2.In these methods it is not possible to get easily a good estimate of the truncation error.
6. What is a Predictor-collector method of solving a differential equation?
Predictor-collector methods are methods which require the values of y at xn,xn-1,xn-2,… for computing the value of y at . x n+1We first use a formula to find the
value of y at . x n+1 and this is known as a predictor formula.The value of y so got is improved or corrected by another formula known as corrector formula.
7. State the third order R.K method algorithm to find the numerical solution of the first order differential equation.
To solve the differential equation y′ f (=x, y) by the third order R.K method, we use the following algorithm.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707948223038.94/warc/CC-MAIN-20240305060427-20240305090427-00617.warc.gz
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CC-MAIN-2024-10
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https://forum.domegaia.com/topic/285/form-design-measurements
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math
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JoyJoy777 last edited by
Hi, I need help to build the forms for a 24' dome. I am a builder and have lots of concrete experience. I needed to build the triangle aircrete pieces at my farm and then truck them to the building site. Can anyone help me with the angles or mathematical equation that I will need to use? Is there a web site available? Thank-you so much for your time!
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| 377 | 2 |
https://publications.mfo.de/handle/mfo/188?locale-attribute=de
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math
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An Explicit Formula for the Dirac Multiplicities on Lens Spaces
MFO Scientific ProgramOWLF 2013
Lauret, Emilio A.
We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $\Gamma \setminus Spin(2m)/Spin(2m-1)$ and exploiting the representation theory of the Spin groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of infinitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945144.17/warc/CC-MAIN-20230323100829-20230323130829-00230.warc.gz
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CC-MAIN-2023-14
| 887 | 4 |
https://www.hyperfinecourse.org/forums/topic/nuclear-properties-44/
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math
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– The three numbers: Z,N,A
– made of protons and neutrons: obey Fermi-Dirac statistics
– different types of decay
– strong/weak nuclear force + electromagnetic force
– deformation defined by the ratio of the difference between the semi-major and the semi-minor axis over the average nuclear radius.
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| 308 | 5 |
https://www.chess2u.com/t10662-brianfish
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math
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Hi everyone, I imagine that many are aware of this new engine, very interesting and good. I have some questions to ask, any insurance someone can answer the forum. 1) It proved necessary to load the book on the brianfish engine, these are together in one folder or not? 2) You can play with end tables, although brianfish this playing with his book ?, I say this because as we know the book brings games middle game and end. 3) A look loaded the book brianfish in the engine, meaning through book path, I have to disable the book that have activated inside the gui, or can I do play an engine with a book and brianfish with yours.? 3) route within book path, the need to handwrite, or I copy and paste? 5) Since I realize, if there is some way that the engine briafish, I take the book.? 6) You can use the brianfish in the sand? 7) This book serves to infinite analysis? 8) the book bears the name of ligh, which means ?, no other book is best for that? I hope someone or together we can go revealing these questions, of course thank you very much.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571847.45/warc/CC-MAIN-20220812230927-20220813020927-00208.warc.gz
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CC-MAIN-2022-33
| 1,049 | 1 |
https://www.shaalaa.com/question-bank-solutions/find-05-decimal-number-multiplied-10-multiplication-decimal-numbers-10-100-1000_17520
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math
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Find:0.5 × 10
We know that when a decimal number is multiplied by 10, 100, 1000, the decimal point in the product is shifted to the right by as many places as there are zeroes. Therefore, these products can be calculated as
0.5 × 10 = 5
Concept: Multiplication of Decimal Numbers by 10, 100 and 1000
Is there an error in this question or solution?
Video Tutorials For All Subjects
- Multiplication of Decimal Numbers by 10, 100 and 1000
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| 438 | 7 |
https://elementarymath.edc.org/mindset/developing-a-division-algorithm-lesson-7/
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math
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Read the Headline Story to the students. Encourage them to think of creative ways to solve the problem.
Suppose that every day someone in your class helps the kindergarteners at the crosswalk. A different student helps each day. How many times will each of you get to help?
There are 180 days in our school year. I divided 180 by 26, the number of students in our class, and got 6 and a remainder of 24. So, 6 turns per student × 26 students uses up 156 days, with 24 days left. On those 24 days, most of us could get a 7th turn.
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| 530 | 3 |
https://findquestionanswer.com/q/what-does-being-a-constant-mean
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math
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Think of something or someone that does not change as constant. In math and science, a constant is a number that is fixed and known, unlike a variable which changes with the context. That idea crosses over to real life. If a friend is a constant in your life, that means they have always been with you and there for you. Constant: A symbol which has a fixed numerical value is called a constant. For example: 2, 5, 0, -3, -7, 2/7, 7/9 etc., are constants. In the expression 5x + 7, the constant term is 7. 1 fixed and invariable; unchanging. 2 continual or continuous; incessant. constant interruptions. 3 resolute in mind, purpose, or affection; loyal. nt ] A quantity that is unknown but assumed to have a fixed value in a specified mathematical context. A theoretical or experimental quantity, condition, or factor that does not vary in specified circumstances. Avogadro's number and Planck's constant are examples of constants. In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1. 5 + y.
In programming, a variable is a value that can change, depending on conditions or on information passed to the program. Typically, a program consists of instruction s that tell the computer what to do and data that the program uses when it is running.
The most common usage in English is that zero is neither positive nor negative. That is "positive" is normally understood to be "strictly positive". In the same way, "greater than" is normally understood to mean "strictly greater than", as in k>j (not k≥j). This is just a matter of definition.
Since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0; any polynomial term or expression of degree zero is a constant.
A fixed value. In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number. Example: in "x + 5 = 9", 5 and 9 are constants. See: Variable.
Difference between Constant and Variables. A constant does not change its value over time. A variable on the other hand changes its value dependent on the equation. Constants usually represent known values in an equation, expression or in line of programming.
The minus sign is actually 'attached' to the constant in a way. Therefore, any time you see a minus to the left of a constant, it belongs to that constant. Again, a minus or negative sign is to the left of the 3, so we get a -3 constant. There is also one just to the left of the 4, so we get a -4 constant.
Constants and Variables. Constant : A symbol having a fixed numerical value is called a constant. OR. The number before an alphabet (variable) is called a constant. Variable : A symbol which takes various numerical values is called a variable.
Synonyms, Antonyms & Associated Words constant(a) Synonyms: permanent, unchanging, unwavering, unshaken, steadfast, stanch, unswerving, loyal, faithful, continuous, incessant, continual, perpetual, uninterrupted.
Word forms: plural constants. 1. adjective [usually ADJECTIVE noun] You use constant to describe something that happens all the time or is always there. Inflation is a constant threat.
Since k is constant (the same for every point), we can find k when given any point by dividing the y-coordinate by the x-coordinate. For example, if y varies directly as x, and y = 6 when x = 2, the constant of variation is k = = 3. Thus, the equation describing this direct variation is y = 3x.
A constraint is something that limits or controls what you can do. Their decision to abandon the trip was made because of financial constraints. Constraint is control over the way you behave which prevents you from doing what you want to do.
Independent Variable Definition. An independent variable is defines as the variable that is changed or controlled in a scientific experiment. It represents the cause or reason for an outcome. Independent variables are the variables that the experimenter changes to test their dependent variable.
What's the adverb for constant? Here's the word you're looking for. constantly. (archaic) With steadfastness; with resolve; in loyalty, faithfully. In a constant manner; occurring continuously; persistently.
dash someone's hopes. Destroy someone's plans, disappoint or disillusion. For example, That fall dashed her hopes of a gold medal. This term uses dash in the sense of “destroy, ” a usage surviving only in this idiom. [
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| 4,672 | 15 |
https://support.strava.com/hc/es-mx/community/posts/208835307-Custom-Unit-Preferences-Miles-Kg-Celsius-?page=6
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math
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Custom Unit Preferences (Miles, Kg, Celsius)
I'd like more control over what units are used in the app. Specifically, I'd like miles, kilograms and degrees Celsius. Miles is the most important to me so at the moment I'm stuck with other units which mean nothing to me (pounds and Fahrenheit).
Can we just get fields in the settings area for each unit of measurement please?
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s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400212039.16/warc/CC-MAIN-20200923175652-20200923205652-00756.warc.gz
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CC-MAIN-2020-40
| 373 | 3 |
https://scholarsmine.mst.edu/biosci_facwork/176/
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math
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We give an exposition of the principles of quantum computing (logic gates, exponential parallelism from polynomial hardware, fast quantum algorithms, quantum error correction, hardware requirements, and experimental milestones). A compact description of the quantum Fourier transform to find the period of a function-the key step in Shor's factoring algorithm-illustrates how parallel state evolution along many classical computational paths produces fast algorithms by constructive interference similar to Bragg reflections in x-ray crystallography. On the hardware side, we present a new method to estimate critical time scales for the operation of a quantum computer. We derive a universal upper bound on the probability of a computation to fail due to decoherence (entanglement of the computer with the environment), as a function of time. The bound is parameter-free, requiring only the interaction between the computer and the environment, and the time-evolving state in the absence of any interaction. For a simple model we find that the bound performs well and decoherence is small when the energy of the computer state is large compared to the interaction energy. This supports a recent estimate of minimum energy requirements for quantum computation.
P. Pfeifer and C. Hou, "Quantum Computing: From Bragg Reflections to Decoherence Estimates," MRS Online Proceedings Library, vol. 746, pp. 265 - 276, Cambridge University Press, Jan 2002.
Keywords and Phrases
Algorithms; Coherent Light; Error Correction; Fiber Bragg Gratings; Fourier Transforms; Mathematical Models; Probability; Reflection; X Ray Crystallography, Bragg Reflections; Decoherence; Quantum Algorithms; Quantum Computing; Quantum Error Correction, Quantum Theory
International Standard Serial Number (ISSN)
Article - Journal
© 2002 Cambridge University Press, All rights reserved.
01 Jan 2002
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| 1,869 | 8 |
https://www.hackmath.net/en/examples/high-school?page_num=30
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math
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Examples for secondary school students - page 30
- Pool 2
The first supply by the pool fill for five hours and the second fill for six hours, drain should be drained for 15 hours. For how many hours the pool is full, when we open both inlet now and outlet open two hours later?
- Simple interest 4
Find the simple interest if 5243 USD at 4.3% for 261 days. Assume a 361-day year.
- Digits A, B, C
For the various digits A, B, C is true: the square root of the BC is equal to the A and sum B+C is equal to A. Calculate A + 2B + 3C. (BC is a two-digit number, not a product).
I have bevel in the ratio 1:6. What is the angle and how do I calculate it?
What percentage vinegar we get if we mix 1 if dm³ eight percent vinegar with 1.5 dm³ six percent vinegar?
Ball was fired at an angle of 35° at initial velocity 437 m/s. Determine the length of the litter. (g = 9.81 m/s2).
What is the slope of the line defined by the equation -2x +3y = -1 ?
- BW-BS balls
Adam has a full box of balls that are large or small, black or white. Ratio of large and small balls is 5:3. Within the large balls the ratio of the black to white is 1:2 and between small balls the ratio of the black to white is 1:8 What is the ratio of.
- Comparing powers
How many times is number 56 larger than number 46?
- Secret number
Determine the secret number n, which reversed decrease by 16.4 if the number increase by 16.4.
After 548 hours decreases the activity of a radioactive substance to 1/9 of the initial value. What is the half-life of the substance?
Parallelogram has sides lengths in the ratio 3: 4 and perimeter 2.8 meters. Determine the lengths of the sides.
- Two boxes-cubes
Two boxes cube with edges a=38 cm and b = 81 cm is to be replaced by one cube-shaped box (same overall volume). How long will be its edge?
- Three-digit numbers
How many three-digit numbers are from the numbers 0 2 4 6 8 (with/without repetition)?
- Cylinder surface, volume
The area of the cylinder surface and the cylinder jacket are in the ratio 3: 5. The height of the cylinder is 5 cm shorter than the radius of the base. Calculate surface area and volume of cylinder.
- Nuts, girl and boys
Milena collected fallen nuts and called a bunch of boys let them share. She took a condition: the first boy takes one nut and tenth of the rest, the second takes 2 nuts and tenth new rest, the third takes 3 nuts and tenth new rest and so on. Thus managed.
- Isosceles IV
In an isosceles triangle ABC is |AC| = |BC| = 13 and |AB| = 10. Calculate the radius of the inscribed (r) and described (R) circle.
- 3d vector component
The vector u = (3.9, u3) and the length of the vector u is 12. What is is u3?
- Golden ratio
Divide line of length 14 cm into two sections that the ratio of shorter to greater is same as ratio of greater section to whole length of the line.
A circuit has an input power of 58 mW. Its output power is 6 mW. What is the loss in decibels?
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| 2,916 | 34 |
https://civilengineering365.com/response-surfaces-for-water-distribution-system-pipe-roughness-calibration-journal-of-water-resources-planning-and-management-vol-148-no-3/
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math
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AbstractWater distribution system (WDS) model calibration research has focused on estimating model input/output parameters and analyzing several uncertainties (e.g., model uncertainty) to improve models with best-fit parameters. Numerous studies have shown that optimization algorithms generally quickly converge to very good parameter solutions. However, the generality and reasoning behind this have not been identified. This paper examines the shape and convexity of WDS response surfaces (i.e., objective function surfaces) and whether the surfaces have single global or multiple local optima. To that end, three networks with different network topologies are evaluated: (1) the Modena network as presented, (2) a modified form of the Modena network, and (3) a real Austrian network. Various conditions were evaluated to consider field measurement error, parameter uncertainty through pipe grouping, and model uncertainty. Results demonstrate that the response surfaces remained smooth and convex even when uncertainties are introduced, but the best parameter solutions are shifted from the true solution. The impact and sensitivities of the uncertainties are evaluated by examining the change in best-fit parameter estimates.
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CC-MAIN-2023-40
| 1,230 | 1 |
http://calendariu.com/p/prime-numbers.html
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math
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List mersenne prime numbers - primenet, List of all known mersenne prime numbers along with the discoverer's name, dates of discovery and the method used to prove its primality.. What prime number? - definition whatis., This definition explains what prime numbers are and how to find them. it also explains how prime numbers are used in computer security and why large prime numbers are. Math games: monkey drive prime numbers, Learn multiples with this fun racing math game with monkey drive prime numbers.
Prime number -- wolfram mathworld, A prime number ( prime integer, simply called "prime" short) positive integer p>1 positive integer divisors 1 p .. http://mathworld.wolfram.com/PrimeNumber.html Murderous maths: prime numbers!, Mathematicians love prime numbers maths prime numbers. , scientists love atoms . http://www.murderousmaths.co.uk/games/primcal.htm Math forum: dr. math faq: prime numbers, What prime number? find prime numbers? ' 'sieve eratosthenes'? decide number prime? ' largest prime?. http://mathforum.org/dr.math/faq/faq.prime.num.html
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https://stats.oarc.ucla.edu/sas/library/sas-libraryrepeated-measures-anova-using-sas-proc-glm/
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math
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This page was developed by the Consulting group of the Department of Statistics and Data Sciences at the University of Texas at Austin. We thank them for permission to distribute it via our web site.
31 July 1997 Usage Note: Stat-40 Copyright 1995-1997, ACITS, The University of Texas at Austin Statistical Services, 475-9372 http://stat.utexas.edu/ Originally available online at: http://ssc.utexas.edu/consulting/answers/sas/sas43.html
This usage note describes how to run a repeated measures analysis of variance (ANOVA), including a between-subjects variable, using the SAS GLM procedure. The document first explains when one should use such a procedure; describes the terminology used; gives a sample research problem; and finally, in a detailed example, shows how to use the SAS GLM procedure.
You should already know how to write a SAS program to read an external data file and run SAS procedures using the data. In addition, you should be familiar with basic ANOVA methods and assumptions.
As with any ANOVA, repeated measures ANOVA tests the equality of means. However, repeated measures ANOVA is used when all members of a random sample are measured under a number of different conditions. As the sample is exposed to each condition in turn, the measurement of the dependent variable is repeated. Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures: the data violate the ANOVA assumption of independence. Keep in mind that some ANOVA designs combine repeated measures factors and nonrepeated factors. If any repeated factor is present, then repeated measures ANOVA should be used.
This approach is used for several reasons. First, some research hypotheses require repeated measures. Longitudinal research, for example, measures each sample member at each of several ages. In this case, age would be a repeated factor. Second, in cases where there is a great deal of variation between sample members, error variance estimates from standard ANOVAs are large. Repeated measures of each sample member provides a way of accounting for this variance, thus reducing error variance. Third, when sample members are difficult to recruit, repeated measures designs are economical because each member is measured under all conditions.
Repeated measures ANOVA can also be used when sample members have been matched according to some important characteristic. Here, matched sets of sample members are generated, with each set having the same number of members and each member of a set being exposed to a different random level of a factor or set of factors. When sample members are matched, measurements across conditions are treated like repeated measures in a repeated measures ANOVA.
For example, suppose that you select a group of depressed subjects, measure their levels of depression, and then match subjects into pairs having similar depression levels. One subject from each matching pair is then given a treatment for depression, and afterwards the level of depression of the entire sample is measured again. ANOVA comparisons between the two groups for this final measure would be most efficient using a repeated measures ANOVA. In this case, each matched pair would be treated as a single sample member.
One should be clear about the difference between a repeated measures design and a simple multivariate design. For both, sample members are measured on several occasions, or trials, but in the repeated measures design, each trial represents the measurement of the same characteristic under a different condition. For example, one can use a repeated measures ANOVA to compare the number of oranges produced by an orange grove at years one, two, and three. The measurement is the number of oranges, and the condition that changes is the year. In contrast, for the multivariate design, each trial represents the measurement of a different characteristic. You should not, for example, use a repeated measures ANOVA to compare the number, weight, and price of oranges produced by a grove of orange trees. The three measurements are number, weight, and price, and these do not represent different conditions, but different qualities. It is generally inappropriate to test for mean differences between such disparate measurements.
Several terms in this document may be unfamiliar. It will be helpful for you to know their meaning.
A sample member is called a subject.
When a dependent variable is measured repeatedly for all sample members across a set of conditions, this set of conditions is called a within-subjects factor. The conditions that constitute this type of factor are called trials.
When a dependent variable is measured on independent groups of sample members, where each group is exposed to a different condition, the set of conditions is called a between-subjects factor. The conditions that constitute this factor type are called groups.
When an analysis has both within-subjects factors and between subjects factors, it is called a repeated measures ANOVA with between-subjects factors.
The remainder of this document uses a detailed example to illustrate repeated measures ANOVA. Suppose that, as a health researcher, you want to examine the impact of dietary habit and exercise on pulse rate. To investigate these issues, you collect a sample of individuals and group them according to their dietary preferences: meat eaters and vegetarians. You then divide each diet category into three groups, randomly assigning each group to one of three types of exercise: aerobic stair climbing, racquetball, and weight training. So far, then, your design has two between-subjects (grouping) factors: dietary preference and exercise type.
Suppose that, in addition to these between-subjects factors, you want to include a single within-subjects factor in the analysis. Each subject’s pulse rate will be measured at three levels of exertion: after warm-up exercises, after jogging, and after running. Thus, intensity (of exertion) is the within-subjects factor in this design. In the SAS syntax that appears below, this factor is labeled Intensity. The order of these three measurements will be randomly assigned for each subject.
Note that all the factors just described can be considered fixed effects. The levels of intensity, diet, and exercise-type were selected because you are interested in those specific categories. In contrast, the levels of a random effect are chosen at random from a population of possible levels. Random effects cannot be appropriately analyzed with the method being described.
Cases with missing values at any trial must be dropped from the analysis. SAS PROC GLM will automatically delete the entire observation if it has any missing data (this is called “listwise deletion”). If an observation is deleted this way, or if the number of group members is uneven in some other way, then the design is unbalanced. Consider an experiment that has one between-subjects grouping factor: dietary preference. Suppose the design incorporated 25 meat eaters and 24 vegetarians. Such a design would be unbalanced, while a design with 25 members in each group would be balanced. Unbalanced designs create special difficulties for the analysis of variance. If you have an unbalanced design, then you should consult an advanced statistics text or see a statistical consultant for more information about this topic.
With any inferential statistical procedure, it is important to state the hypotheses of interest clearly before undertaking any statistical analyses of the data. In this example, then, you have carefully considered your research goals and decided that you are interested in answers to the following questions:
Within-Subjects Main Effect
Does intensity influence pulse rate? (Does mean pulse rate change across the trials for intensity?) This is the test for a within-subjects main effect of intensity.
Between-Subjects Main Effects
Does dietary preference influence pulse rate? (Do vegetarians have different mean pulse rates than meat eaters?) This is the test for a between-subjects main effect of dietary preference.
Does exercise type influence pulse rate? (Are there differences in mean pulse rates between stair climbers, racquetball players, and weight trainers?) This is the test for a between-subjects main effect of exercise type.
Between-Subjects Interaction Effect
Does the influence of exercise type on pulse rate depend on dietary preference? (Does the pattern of differences between mean pulse rates for exercise-type groups change for each dietary-preference group?) This is the test for a between-subjects interaction of exercise type by dietary preference. Keep in mind that other formulations of this interaction are equivalent. This hypothesis can also be expressed as “Does the influence of dietary preference depend on exercise type?”)
Interaction hypotheses can be difficult to understand, so an example may help. You might believe that vegetarian racquetball players have lower pulse rates than all meat eaters and vegetarians weight-lifters and stair-climbers. In other words, you may wonder if something unique in the combination of a vegetarian diet and racquetball exercise produces an unusually low mean pulse rate. This pattern of differences between pulse rates would ignore intensity trials.
Within-Subjects by Between-Subjects Interaction Effects
Does the influence of diet on pulse rate depend upon intensity? (Does the pattern of differences between mean pulse rates for dietary-preference groups change at each intensity trial?) This is the test for a between-subjects by within-subjects interaction of dietary preference by intensity. You might suspect, for example, that the mean pulse rate of meat eaters will increase more than the mean pulse rate of vegetarians as the intensity of exercise changes.
Does the influence of exercise type on pulse rate depend upon intensity? (Does the pattern of differences between mean pulse rates for exercise-type groups change at each intensity trial?) This is the test for a between-subjects by within-subjects interaction of exercise type by intensity.
Does the influence of dietary preference on pulse rate depend upon exercise type and intensity? (Does the pattern of differences between mean pulse rates for dietary-preference groups change for some exercise-type group and for some intensity trial?) This is the test for a between-subjects by within-subjects interaction of dietary preference by exercise type by intensity.
Recall that, for all of the hypotheses specified above, you test the null hypothesis of no differences between population means. In most cases, some difference will occur in the sample between any levels of a factor. However you want to draw conclusions not about the sample, but about the larger population from which it was taken. F ratios and the analysis of variance were developed to enable you to do that. A large F value yields a correspondingly small p value. The p value is the observed significance level, or probability of a Type 1 error: concluding that a difference between population means exists when in fact there is no difference. This type 1 error is also known as alpha error.
You examine the p value to determine if it meets your criterion for an acceptable level of alpha error. You must decide on an alpha level that is acceptable to you before you conduct each analysis. If the p value appearing on the SAS printout is larger than your previously set alpha level, then you fail to reject the null hypothesis. On the other hand, if your p value is smaller than your alpha level, then you reject the null hypothesis. The alpha level you set before you conduct each hypothesis test can be influenced by a number of factors; by convention it is usually set at 0.05.
Throughout this document, all SAS syntax and output appear in Courier font. Words that appear in ALL CAPITALS are keywords that must be typed exactly as shown. Words or numbers to be supplied by the user (such as variable names) are written in lower case (e.g., trial-1). Here is the general syntax for SAS’s GLM procedure:
PROC GLM DATA = sas-dataset-name ; CLASS group-factor-1 group-factor-2 ... group-factor-k ; MODEL trial-1 trial-2 ... trial-k = group-factor-1 ... group-factor-k ; REPEATED repeated-factor-name number-of-trials / PRINTE ; LSMEANS grouping-factor-1 group-factor-2 ... group-factor-k ; RUN;
To make the description of analysis techniques more concrete, a repeated measures ANOVA example on our health research data is now provided. This example includes the SAS syntax necessary to run a repeated measures ANOVA with grouping factors, as well as a brief guide to interpreting the output provided by SAS PROC GLM.
Recall that you have measured the pulse of your subjects at three trials, and these three variables have been entered into a SAS dataset as Pulse1, Pulse2, and Pulse3. Pulse1 is the pulse measurement taken at the warmup exercising trial whereas Pulse3 is the pulse measurement taken after running. The variable Diet denotes dietary preference, with values of 1 signifying meat eaters and 2 signifying vegetarians. Finally, the variable Exertype is the type of exercise assigned to the subjects, with 1 signifying aerobic stairs, 2 signifying racquetball, and 3 signifying weight training.
Here is a subset of the data:
EXERTYPE PULSE1 PULSE2 PULSE3 DIET 1 112 166 215 1 1 111 166 225 1 1 89 132 189 1 1 95 134 186 2 1 66 109 150 2 1 69 119 177 2 2 125 177 241 1 2 85 117 186 1 2 97 137 185 1 2 93 151 217 2 2 77 122 178 2 2 78 119 173 2 3 81 134 205 1 3 88 133 180 1 3 88 157 224 1 3 58 99 131 2 3 85 132 186 2 3 78 110 164 2
To perform a repeated measures ANOVA with grouping factors, one that tests all of the hypotheses described above, use the following SAS PROC GLM statements.
PROC GLM DATA = repeated ; CLASS diet exertype ; MODEL pulse1 pulse2 pulse3 = diet exertype diet*exertype / nouni; REPEATED intensity 3 / PRINTE ; LSMEANS diet exertype diet*exertype ; RUN ;
The keywords PROC GLM are immediately followed by the DATA = option, which tells GLM which SAS dataset the analysis will be performed on — in this case, the dataset named “repeated”. The CLASS statement tells SAS which between-subject variables are grouping (classification) variables.
The MODEL statement contains three variable names (pulse1, pulse2, and pulse3) which appear on the left side of an equals sign. These represent the three response variables which are the three levels of the within-subjects factor, intensity. On the right side of the equals sign are the groups or between-subjects factors: diet is the variable representing dietary preference, exertype represents exercise type, and diet*exertype represents the interaction between diet and exertype in the population from which the data were sampled. If you have no grouping variables, then this side of the equals sign will be blank. The /nouni option tells SAS not to print out univariate tests for each individual dependent variable; these particular univariate tests do not deal with any of the hypotheses mentioned above.
Following the MODEL statement is the REPEATED statement, and intensity is the user-supplied name for the single within-subjects factor of exertion intensity. Since it has three levels or trials (three measurements of each subject’s pulse rate), 3 is specified after the factor name. This statement tells SAS how to interpret the list of response variables. The /PRINTE option requests that SAS print out Mauchly’s test of sphericity (described in more detail below).
The LSMEANS statement requests that SAS print the cell means associated with the main effects for diet and exertype, as well as with the interaction between diet and exertype. These means help you evaluate any patterns in the data.
Finally, the RUN statement tells SAS to run this set of PROC GLM statements.
When SAS executes this PROC GLM command, the first page of output contains descriptive information about the analysis:
Repeated measures analysis with grouping factors Two betw. S"S factors, 1 within w/3 levels General Linear Models Procedure Class Level Information Class Levels Values DIET 2 Meat Eater Vegetarian EXERTYPE 3 Aerobic Stairs Racquetball Weight Training Number of observations in data set = 150
The first set of tests reported by SAS is for the within-subjects effects. When there are more than two levels of a within-subjects factor, PROC GLM prints out two different sets of within-subjects hypothesis tests: one using the multivariate approach, the other using the univariate approach. Generally, both sets of tests yield similar results.
Repeated measures ANOVA carries the standard set of assumptions associated with an ordinary analysis of variance, extended to the matrix case: multivariate normality, homogeneity of covariance matrices, and independence. Repeated measures ANOVA is robust to violations of the first two assumptions. Violations of independence produce a nonnormal distribution of the residuals, which results in invalid F ratios. The most common violations of independence occur when either random selection or random assignment is not used.
In addition to these assumptions, the univariate approach to tests of the within-subject effects requires the assumption of sphericity, which is described in more detail below. When sample sizes are small, the univariate approach can be more powerful, but this is true only when the assumption of a common spherical covariance matrix has been met.
When at least one within-subjects factor has three or more trials, SAS will run Mauchly’s test of sphericity if the /PRINTE option is specified as part of the REPEATED statement. If your within-subject factors fail to meet the assumption of sphericity, then you should either use the multivariate approach or you should adjust the univariate results by using one of the correction factors described below.
The assumption of sphericity is tested using transformed dependent variables. The original variables representing each trial are transformed according to a set of orthogonal contrasts. The choice of transform does not affect the outcome of the test, as long as the transformation matrix is orthonormal.
If you do not specify a set of contrasts, a default set is used. If you choose a nonorthogonal contrast scheme for any within-subjects factor, SAS will orthonormalize the contrast matrix and you will not get the contrasts you ask for. The first transformed variable, T1, is always a constant and is not used in any tests involving covariance matrices. Thus, there will always be one less transformed variable than original variables.
The default contrast scheme is Deviation. For this contrast scheme, each level of the within-subjects factor is compared to the overall mean of all levels. T1 is a constant, T2 represents Pulse1-(Pulse1+Pulse2+Pulse1)/3, while T3 represents Pulse2-(Pulse1+Pulse2+Pulse3)/3. The covariance matrices have the variances of T2 and T3 on the diagonal, and the covariance of these two variables off the diagonal. Keep in mind that you should also have balanced cell sizes across the between-subjects factors in the analysis: there should be equal numbers of subjects in each between-subjects group.
The test of sphericity, when requested, immediately precedes both sets of within-subjects tests. Although the output shows two separate tests of sphericity, the only one of interest is the second test, which is the test of sphericity applied to the common covariance matrix of the transformed within-subject variables. The test for the health research dataset appears below.
Test for Sphericity: Mauchly's Criterion = 0.4069598 Chisquare Approximation = 128.56285 with 2 df Prob > Chisquare = 0.0000 Applied to Orthogonal Components: Test for Sphericity: Mauchly's Criterion = 0.7335312 Chisquare Approximation = 44.313583 with 2 df Prob > Chisquare = 0.0000
Mauchly’s sphericity test examines the form of the common covariance matrix. A spherical matrix has equal variances and covariances equal to zero. The common covariance matrix of the transformed within-subject variables must be spherical, or the F tests and associated p values for the univariate approach to testing within-subjects hypotheses are invalid. If the Chi-square approximation has an associated p value less than your alpha level, the sphericity assumption has been violated. The chi-square approximation for this test is 44.31 with 2 df and an associated probability of less than 0.001. Since this is less than the alpha level of 0.05, we can be confident that the data do not meet the sphericity assumption.
For practical purposes, these issues are important only in helping you decide which output to use, and if the output should be adjusted. If you can use the univariate output, you may have more power to reject the null hypothesis in favor of the alternative hypothesis. However, the univariate approach is appropriate only when the sphericity assumption is not violated. If the sphericity assumption is violated, then in most situations you are better off staying with the multivariate output.
An alternative to using the multivariate approach is to adjust the univariate test degrees of freedom. SAS prints two different correction factors: the Greenhouse-Geisser Epsilon (G-G) and the Huynh-Feldt Epsilon (H-F). Generally, the H-F correction factor is used because the G-G correction factor has been shown to be too conservative: it sometimes fails to detect a true difference between group means. By default, SAS prints the adjusted p values for both the G-G and the H-F epsilon values for each univariate F test involving a within-subjects effect. See the section of this document entitled “Univariate Approach to Within-Subjects Tests” for more information on this topic.
As noted above, the multivariate output is still valid even if the sphericity assumption is not met. SAS prints the multivariate approach to testing the within-subjects factors after Mauchly’s test of sphericity. The first multivariate test of a within-subjects effect is the within-subjects main effect test. It examines changes in pulse rate as a function of intensity. The null hypothesis is that the mean pulse rate does not change across different intensities.
Manova Test Criteria and Exact F Statistics for the Hypothesis of no INTENSIT Effect H = Type III SS&CP Matrix for INTENSIT E = Error SS&CP Matrix S=1 M=0 N=70.5
Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.018601 3772.3 2 143 0.0001 Pillai's Trace 0.981399 3772.3 2 143 0.0001 Hotelling-Lawley Trace 52.7594 3772.3 2 143 0.0001 Roy's Greatest Root 52.7594 3772.3 2 143 0.0001
SAS prints four lines, each reporting a separate multivariate test statistic (Pillais', Hotelling's, Wilks', and Roy's); the Wilk’s test is commonly used. Notice that following the label “E = Error SS&CP Matrix“, there are three values, S, M, and N. These are the degrees of freedom for the multivariate statistics. Statistics such as Wilks’ Lambda are distributed in three dimensions; thus three separate values for degrees of freedom are required to determine a critical value. These multivariate statistics are converted to F values. In some cases, the converted F and its degrees of freedom are approximations. When this is not the case, a note at the bottom of the output states that the statistics are exact.
Since the F ratio for this hypothesis is very large [F(2, 143) = 3772.3, p = .0001], you can confidently reject the null hypothesis and conclude that the pulse rate changes with intensity in the population from which the sample was drawn.
Next SAS tests the hypothesis that dietary preference interacts with intensity.
Manova Test Criteria and Exact F Statistics for the Hypothesis of no INTENSIT*DIET Effect H = Type III SS&CP Matrix for INTENSIT*DIET E = Error SS&CP Matrix S=1 M=0 N=70.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.774461 20.822 2 143 0.0001 Pillai's Trace 0.225539 20.822 2 143 0.0001 Hotelling-Lawley Trace 0.29122 20.822 2 143 0.0001 Roy's Greatest Root 0.29122 20.822 2 143 0.0001
In this instance, the F value associated with these multivariate tests of the interaction is high; therefore, the associated p value is low [F(2, 143) = 20.82, p = .0001]. Like the previous example, then, you can now reject the null hypothesis and conclude that change in mean pulse rate across intensity levels depends upon dietary preference. This finding may complicate the interpretation of the main effects for diet and intensity.
Next, turn your attention to the null hypothesis that exercise type will not interact with intensity to produce different mean pulse rates. Here is the multivariate test of this hypothesis:
Manova Test Criteria and F Approximations for the Hypothesis of no INTENSIT*EXERTYPE Effect H = Type III SS&CP Matrix for INTENSIT*EXERTYPE E = Error SS&CP Matrix S=2 M=-0.5 N=70.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.976386 0.8595 4 286 0.4887 Pillai's Trace 0.023676 0.8626 4 288 0.4868 Hotelling-Lawley Trace 0.024122 0.8563 4 284 0.4906 Roy's Greatest Root 0.021115 1.5203 2 144 0.2221 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact.
By examining the Wilks' value for this test (.976), its associated F value, and p value [F(4, 286) = .859, p = .489], you can conclude that any differences between pulse rate levels do not reliably depend on intensity in conjunction with the type of exercise the subject was assigned, in samples of this size.
Finally, SAS prints a multivariate hypothesis test of the null hypothesis of no exercise-type by diet by intensity interaction:
Manova Test Criteria and F Approximations for the Hypothesis of no INTENSIT*DIET*EXERTYPE Effect H = Type III SS&CP Matrix for INTENSIT*DIET*EXERTYPE E = Error SS&CP Matrix S=2 M=-0.5 N=70.5 Statistic Value F Num DF Den DF Pr > F Wilks' Lambda 0.983058 0.6135 4 286 0.6532 Pillai's Trace 0.017014 0.6178 4 288 0.6502 Hotelling-Lawley Trace 0.017162 0.6092 4 284 0.6563 Roy's Greatest Root 0.009598 0.691 2 144 0.5027
NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact.
Since the F value associated with the Wilks' test [F(4, 286) = .613, p = .653] has a p value greater than 0.05, you cannot conclude that there is an interaction among these variables, and you retain the null hypothesis.
Following the multivariate tests of significance for within-subjects effects, SAS prints tests of the between-subjects effects. There is only one approach to testing these effects.
General Linear Models Procedure Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS F Value Pr > F DIET 1 33024.500000 45.54 0.0001 EXERTYPE 2 449.231111 0.31 0.7341 DIET*EXERTYPE 2 757.960000 0.52 0.5941 Error 144 104435.066667
The line labeled DIET reports the sum of squares, degrees of freedom, and mean square for DIET. This mean square is the F ratio’s numerator for the test of the diet hypothesis. This line also reports the F value and associated p value for the test of the diet hypothesis. In this case, with a p value less than .0001, you have a statistically significant effect (using the alpha criterion of .05 to define “statistical significance”). You can therefore conclude that a statistically significant difference exists between vegetarians and meat eaters on their overall pulse rates. In other words, there is a main effect for diet. The cell means (not shown here) show that meat eaters experience higher pulse rates than vegetarians.
The next line shows the EXERTYPE test. It is nonsignificant: F(2, 144) = .31, p=.7341. Thus, you can conclude that the type of exercise has no statistically significant effect on overall mean pulse rates. Finally, the test of the DIET BY EXERTYPE interaction also shows a nonsignificant result (F(2, 144) = .52, p=.594). This suggests that dietary preferences and type of exercise do not combine to influence the overall average pulse rate. Recall that when an interaction effect is significant, the pattern of cell means must be examined to determine the meaning not only of the interaction, but also the meaning of any main effects involved in the interaction.
Finally, the line labeled Error reports the within-cells sum of squares, degrees of freedom, and mean square. This mean square is the F ratio’s denominator for any between-subjects hypothesis.
It is important to understand that these tests of between-subjects effects are based on the average of the within-subject trials. For example, the pulse rate average of all three trials of pulse rate is computed, and then this mean pulse rate for vegetarians on this index is compared to the mean for meat eaters. As such, these tests yield no information about within-subjects effects. If you expect important differences in pulse rate across trials, then these between-subjects main-effect tests tests may not be meaningful for you.
While each of the within-subject effects have a separate page of multivariate-approach output, the univariate tests are together on a single page in the standard ANOVA table format. Those for the example data appear below:
General Linear Models Procedure Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Source: INTENSIT Adj Pr > F DF Type III SS Mean Square F Value Pr > F G - G H - F 2 768792.018 384396.009 5755.76 0.0001 0.0001 0.0001
Source: INTENSIT*DIET Adj Pr > F DF Type III SS Mean Square F Value Pr > F G - G H - F 2 4181.493 2090.747 31.31 0.0001 0.0001 0.0001
Source: INTENSIT*EXERTYPE Adj Pr > F DF Type III SS Mean Square F Value Pr > F G - G H - F 4 310.929 77.732 1.16 0.3269 0.3251 0.3256
Source: INTENSIT*DIET*EXERTYPE Adj Pr > F DF Type III SS Mean Square F Value Pr > F G - G H - F 4 159.587 39.897 0.60 0.6648 0.6258 0.6329
DF Type III SS Mean Square 288 19233.973 66.785
Greenhouse-Geisser Epsilon = 0.7896 Huynh-Feldt Epsilon = 0.8246
The sphericity assumption was violated for these data, and so these F’s and p values are not valid. With nonspherical data either use the multivariate test results described earlier or correct the univariate tests results. These corrected univariate p values appear under the G - G and H - F headers in the output shown above. Note that in this case, the univariate approach agrees with the multivariate approach that there is a statistically significant within-subjects main effect for intensity, as well as a statistically significant interaction between diet and intensity.
To interpret a significant interaction, examine the cell means and standard deviations (not shown here) produced by the LSMEANS statement. By plotting these cell means (easily done by hand), you realize first that the mean pulse rate increases across trials: this is the within-subject effect. Further, it’s clear that vegetarians have a lower average pulse rate than do meat eaters at every trial: this is the diet main effect.
Then looking closer, you see that this difference is different at each trial. This is the result of the diet by intensity interaction. As the subjects experience more intense exertion, the average pulse rate of the meat eaters increases more than that of the vegetarians. A graph of the cell sample averages shown below illustrates this point.
In this graph, the cell averages are collapsed across the exertype variable with the diet variable defining the two separate lines shown in the graph. This is justified since exertype, and all its interactions, are nonsignificant. The mean pulse rate is displayed on the Y-axis labeled “Pulse Rate”. Exertion intensity defines the X-axis, labeled “Intensity (Trials)”. Recall that this factor is the within-subjects factor. The lower line shows vegetarian subjects’ average pulse rates, and the upper line shows the meat-eating subjects’ average pulse rates, at the three exertion intensities.
The main effect for diet is interpretable in this instance because the interaction is not complex enough to qualify the main effect. Not all interactions are this simple, however. If you are uncertain as to whether you have an interaction which qualifies a main effect, you should see a statistical consultant.
It is clear from the graph that the main effect for intensity is much stronger than that for the interaction. For both diet groups, the mean pulse rate after jogging increased about 40 points beyond the rate after warmup exercises, and increased another (roughly) 50 points after running. The main effect for diet is reflected in the fact that meat-eaters had a mean pulse rate roughly 10 to 20 points higher than that for vegetarians. The interaction shows this difference between meat eaters and vegetarians increases with exertion intensity. Thus you might want to conclude that the effects for intensity and diet are practically as well as statistically significant, while the interaction between these two variables is too small to have any practical significance.
The following references can be helpful in conducting repeated measures analysis of variance in SAS.
SAS/STAT User’s Guide, Version 6, Fourth Edition, Volume1 and Volume 2, Cary NC: SAS Institute Inc., 1989.
DiIorio, Frank C., SAS Applications and Programming: A Gentle Introduction, Belmont CA, Duxbury Press, 1991.
Stevens, James P., Applied Multivariate Statistics for the Social Sciences, Third Edition, Mahway NJ, Lawrence Erlbaum Associates, Inc., 1996.
This page was developed by the Consulting group of the Division of Statistics and Scientific Computing at the University of Texas at Austin. We thank them for permission to distribute it via our web site.
31 July 1997 Usage Note: Stat-40 Copyright 1995-1997, ACITS, The University of Texas at Austin Statistical Services, 475-9372 http://ssc.utexas.edu/ Originally available online at: http://ssc.utexas.edu/consulting/answers/sas/sas43.html
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CC-MAIN-2023-40
| 34,010 | 101 |
http://www.amazon.ca/Mathematicians-Lament-School-Fascinating-Imaginative/dp/1934137170
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math
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Once in a while we read books that we just know are especially important, and that we know we will be thinking and talking about long after reading them. This book is one of them for me.
I am a returning adult student, and I am about to finish my training to become a math teacher. Having gone through my education program, my enthusiasm was just about completely drained, and I've been having trouble remembering why I ever wanted to become a math teacher in the first place. Why would anyone?
Paul Lockhart knows, and his book has reawakened my desire to help students discover the joy of mathematics. His argument is concise, and he makes it forcefully. His book is a joy to read, mainly because his understanding of the subject and his passion for it are clear in every page. He reinforces ideas I already had about how school sucks the life out of math (and all subjects), but he also challenges some of my opinions. I think this will happen with most people who read it.
Once he finishes making his argument about math education in about the first two-thirds of this short book, he devotes the remaining section to describing what he finds wonderful about mathematics itself. This section should make just about anyone want to become either a mathematician or a math teacher.
I want people to read the book for the specifics of his arguments, but I want to discuss one important point that he makes. Many people in math education claim that in order to make math more understandable and interesting to students, we need to show how practical it is and how it is used in everyday life. I've always felt like this idea was wrong, or at least limited in its usefulness in that regard. Well, Lockhart demolishes the idea, essentially claiming that practical uses are simply by-products of math, and that the real excitement and beauty of mathematics is in the abstract, imaginary, and creative world of mathematical ideas that have no specific connection to the everyday. By-products and applications can make math seem boring and secondary to the uses it serves. I agree with him--and much more now after having read his argument.
I honestly think just about everyone should read this book. Of course math teachers should, as should anybody involved in math education in any way. But I think people outside of math education should read it too. The specific mathematical ideas discussed in the book do not require a strong mathematical background, and I can't think of a better book that so concisely conveys the nature of the subject and the way it is viewed and misunderstood in society. I'm still not sure I agree with Lockhart's every point, but I love this book. (And I might agree with his every point after more thought and experience in the classroom.)
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CC-MAIN-2013-20
| 2,763 | 6 |
https://iim-cat-questions-answers.2iim.com/quant/geometry/geometry-triangles/geometry-triangles_31.shtml
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math
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The question is from the topic circles. From the diagram, we have to find out the circumference of the circle. A perpendicular to the diagram of the circle and it's distance is given. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems.
Question 31: What is the circumference of the below circle given that AB is the diameter and XY is perpendicular to AB?
Join AY and let AX = a
Diameter = a + 3
Since AB is the diameter, ∠AYB = 90∘
Since XY is perpendicular to AB, ∠BXY = 90∘
BY = √(32+52) = √34
Consider triangles ABY and BXY
∠ABY is same as ∠XBY and both the triangles are right angled
Therefore they are similar triangles and sides are proportional!
AB/BY = BY/BX = AY/XY
(a+3)/√34 = √34/3
3a + 9 = 34
or a = 25/3
Diameter = 25/3 + 3 = 34/3
Circumference = π * d = 34π/3 cm
The question is "What is the circumference of the below circle given that AB is the diameter and XY is perpendicular to AB?"
Choice C is the correct answer.
CAT® (Common Admission Test) is a registered trademark of the Indian Institutes of Management. This website is not endorsed or approved by IIMs.
2IIM Online CAT Coaching
A Fermat Education Initiative,
58/16, Indira Gandhi Street,
Kaveri Rangan Nagar, Saligramam, Chennai 600 093
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CC-MAIN-2023-40
| 1,279 | 23 |
https://www.urch.com/forums/tags/adding/
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math
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Search the Community
Showing results for tags 'adding'.
My GRE exam is on 20th August, still I am not confident about awa and facing hard times how to improve it. here i have attached 2 essays from my power prep 1 test. It would be so helpful if someone could give me valuable suggestion about how to focus on the topic, improve writing by adding some skills and what points I am missing in the writings. Also If I write essays like this level what score in AWA I am supposed to get? Any help will be greatly appreciated.
hi there in GRE Revised ETS book example said the sequence of numbers a1,a2,a3 ,,,,,,, An is defined by a = 1/n - 1/n+2 for each integer then we have to find the total of adding the first 20 sequence of this ok i know that answer is number B my question is in explaination they said we dont have to add all the 20 sequences , instead its show to add first 2 terms like that (1/1-1/3) + (1/3-1/4) = (1/1+1/2)-(1/3+1/4) i need to know how this equal , i didnt understand why we substitute the signs , and became opposite signs , thanks for help my GRE angels
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100258.29/warc/CC-MAIN-20231130225634-20231201015634-00666.warc.gz
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CC-MAIN-2023-50
| 1,078 | 4 |
https://byjus.com/question-answer/three-brothers-are-sharing-a-birthday-cake-equally-amongst-themselves-if-the-weight-of-one-1/
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math
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The correct option is D 750 g
Since the cake is being equally divided amongst the 3 brothers, so each brother is eating 13rd of the cake.
If the weight of one piece is 250 grams, then the weight of the entire cake would be three times the weight of one piece of cake.
weight of 13rdof the cake = 250 grams
weight of whole cake = 3 x weight of 13rdof the cake
So, the total weight of the cake
= 3 x 250 grams
= 750 grams
Therefore, the total weight of the cake is 750 grams.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00520.warc.gz
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CC-MAIN-2021-49
| 473 | 9 |
https://pratibha.eenadu.net/tenth/lesson/andhrapradesh/english-medium/statistics/1-2-4-218-481-811-704-1966-20040002218
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math
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4 Marks Questions:
1. A frequency distribution of the life times of 400 T.V. picture tubes tested in a tube company is given below. Find the average life of tube.
2. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency ‘f’.
⇒ 18(44 + f) = 752 + 20 f
⇒ 792 + 18 f = 752 + 20 f
⇒ 792 – 752 = 20 f – 18 f
⇒ 2f = 40
⇒ f = 20
3. To find out the concentration of SO2 in the air (in parts per million i.e. ppm), the data was collected for 30 localities in a certain city and is presented below:
∴ The mean concentration of SO2 in the air = 0.099 ppm
4. Thirty women were examined in a hospital by a doctor and their of heart beats per minute were recorded and summarized as shown. Find the mean heart beats per minute for these women, choosing a suitable method.
5. The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify?
Thus, the average number of wickets taken by these 45 bowlers in one-day cricket is 152.89
Note: Even if the class sizes are unequal, and xi are large numerically, we can still apply the step-deviation method by taking ‘h’ to be a suitable divisor of all the di s.
6. A student noted the number of cars passing through a spot on a road for 100 periods; each of 3 minutes, and summarized this in the table given below.
Find the mode of the data.
Here is the maximum class frequency is 20, and the class corresponding to this frequency is 40-50. So, the modal class is 40-50.
7. The median of the following data is 525. Find the values of x and y, if the total frequency is 100.
8. The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:
Find the median length of the leaves.
2 Marks Questions:
1. Following table shows the weight of 12 students is given. Find the mean weight of the students.
2. If the mean of the following distribution is 6, find the value of ‘p’
3. Find the median for the following frequency distribution
Here N = 120 ⇒ = 60
We find that the cumulative frequency just greater than i.e. 60 is 65 and the value of x corresponding to 65 is 5. Therefore Median = 5.
4. Prepare tables to draw less than cumulative frequency curve and greater than cumulative frequency curve for the following frequency distribution (No need to draw graphs)
For Less than cumulative frequency curve
For Greater than cumulative frequency curve
1 Mark Questions:
1. Write the principle to find mean of the grouped data by direct method and explain the terms in the principle.
Here fi = frequencies and xi = observations
2. Write the principle to find mean of the grouped data by deviation method and explain the terms in the principle.
Here A = Assumed Arithmetic Mean, fi = frequencies and di = xi – A (deviations)
3. Write the principle to find mean of the grouped data by step deviation method and explain the terms in the principle.
Here A = Assumed Arithmetic Mean, fi = frequencies and (deviations), h = height of the class.
4. Write the principle to find mode of the grouped data and explain the terms in the principle.
Here l = lower boundary of the modal class, h = size of the modal class
f1 = frequency of the modal class, f0 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class.
5. Write the principle to find median of the grouped data and explain the terms in the principle.
Here l = lower boundary of median class, n = number of observations
cf = cumulative frequency of class preceding the median class
f = frequency of median class, h = class size (size of the median class)
Dr. T.S.V.S. Suryanarayana Murthy
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233510994.61/warc/CC-MAIN-20231002100910-20231002130910-00420.warc.gz
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CC-MAIN-2023-40
| 3,837 | 45 |
http://web.kias.re.kr/sub05/sub05_02_01_01.jsp?seqno=PGN1720190210-0001&nowBlock=0&page=1&subject=&mjrcd=&mjrcd2=all&sdate=20190215&edate=&keyField=&keyWord=&list_url=/sub05/sub05_02_01.jsp&slides=
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math
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|DATE||February 15 (Fri), 2019|
|TITLE||On rationally connected varieties over C_1 fields of characteristic 0|
In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr), but it is still open for the maximal unramified extensions of p-adic fields. I use birational geometry in characteristic 0 to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities.
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CC-MAIN-2019-22
| 789 | 3 |
https://www.studypool.com/discuss/368807/a-what-is-the-y-intercept-of-this-line-and-what-does-it-represent?free
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math
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The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4000 to produce 300 chairs in one day.
b)What is the slope of the line in part (a), and what does it represent?
C = m c + b
Where C is the cost of production and c is the number of chairs produced.
Using the given data
2200 = 100m +b
4000 = 300m +b
Subtract one equation from the other
1800 = 200m
m = $9/chair
This is the slope of the line and shows that it costs $9 to produce each extra chair
Now we can solve for the y intercept (b)
2200 = 100*9 +b
2200 = 900 +b
b = $1300
This means that it costs $1300 to start each day before any chairs are produced
Content will be erased after question is completed.
Enter the email address associated with your account, and we will email you a link to reset your password.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806676.57/warc/CC-MAIN-20171122213945-20171122233945-00765.warc.gz
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CC-MAIN-2017-47
| 840 | 19 |
http://lwn.net/Articles/183267/
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math
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On the safety of Linux random numbers
Posted May 11, 2006 17:21 UTC (Thu) by Ross
In reply to: On the safety of Linux random numbers
Parent article: On the safety of Linux random numbers
I don't see how /dev/random can be called pseudorandom. It uses a fixed algorithm to produce the output, sure, but the amount of output is no greater than the amount of physically random input. It's not just a seemingly-random number sequence with an unknown starting point.
Now /dev/urandom is also more than a pseudorandom number generator, but unlike /dev/random, it doesn't keep track of entropy, so it may dengenerate into a pseudorandom sequence at any time without warning.
to post comments)
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s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368710115542/warc/CC-MAIN-20130516131515-00028-ip-10-60-113-184.ec2.internal.warc.gz
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| 685 | 7 |
https://due.com/terms/what-is-an-amortization-schedule-how-to-calculate-with-formula/
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math
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An amortization schedule is a complete table of periodic loan payments that shows the amount of principal and the amount of interest that comprise each payment until the loan is paid off at the end of its term. It reflects the breakdown of the payment over time, with the interest cost and principal repayment changing with every installment. The formula to calculate amortization is A = P[r(1+r)^n]/[(1+r)^n – 1], where A is the payment amount per period, P is the initial principal (loan amount), r is the interest rate per period, and n is the total number of payments (or periods).
The phonetics of the keyword “What Is an Amortization Schedule? How to Calculate with Formula” are as follows:- What: /wɒt/- Is: /ɪz/- an: /æn/- Amortization: /əˌmɔːtɪˈzeɪʃn/- Schedule: /ˈʃɛdjuːl/ – How: /haʊ/- to: /tuː/- Calculate: /ˈkælkjəleɪt/- with: /wɪð/- Formula: /ˈfɔːrmjʊlə/Please note that phonetic transcriptions may vary slightly based on accents and regional pronunciations. This transcription is mainly based on British English pronunciation.
- Definition of An Amortization Schedule: An amortization schedule reflects the pay-off plan of your loans. It’s a table outlining each payment towards a loan, showing the amount of interest and principal included in each payment. The schedule provides a detailed breakdown of the loan period, showing you how much you’re paying towards the principal and how much is going toward interest over time.
- The Importance of Amortization Schedule: An amortization schedule helps you understand exactly how your loan works. It tells you how much of your payments go towards interest versus principal every month. It provides visibility into how your debt decreases over time and can be a helpful tool if you’re considering prepayments on your loan.
- How to calculate using the Amortization Formula: The formula for an amortization schedule calculation is P = r(PV)/(1 – (1+r)^-n)). Where: P = payment, r = interest rate (per period), PV = loan amount (present value), and n = total number of payments (or periods). To build the schedule, you list each of your scheduled loan payments, then subtract the interest cost for the period (calculated as remaining loan balance multiplied by interest rate) to find out how much of the payment goes towards principal reduction, which then helps determine the remaining balance.
An amortization schedule is crucial in the fields of business and finance as it provides a detailed plan of how a loan will be paid off over time. It indicates the split between the principal amount (the initial amount borrowed) and the interest payments across the lifespan of the loan. This enables borrowers and lenders to understand precisely how much of each payment goes towards reducing the principal versus paying off interest, thus providing a clear view of the financial timeline. Having this clarity is important for strategic planning and determining affordability. Additionally, the use of the amortization formula (A = P[r(1 + r)n]/[(1 + r)n – 1]), where A represents the payment amount per period, P is the principal loan amount, r is the interest rate per period, and n is the number of payments, allows lenders and borrowers to compute possibilities under various scenarios, offering indispensable insights for financial decision-making.
An amortization schedule is essentially a complete table of periodic loan payments, showing the amount of principal and the amount of interest that make up each payment until the loan is paid off at the end of its term. Its purpose is to break down each payment into what amounts are going towards interest and what amounts are going towards the principal of the loan. This is valuable for the borrower, as it allows them to understand and visualize how each payment affects the total loan balance, and for the lender, to keep track of the outstanding balance over time.To calculate an amortization schedule, you’ll use the formula: A=P(1+(r/n))^(nt), where A is the total loan amount, P is the principal loan amount, r is the annual interest rate (in decimal form), n is the number of times that interest is compounded per year, and t is the loan term in years. The formula helps break down each payment into two parts: total interest paid over the life of the loan, and the total cost of the loan (interest plus principal). By using this formula, borrowers can have a clearer understanding of their financial obligation, and plan and manage their finances more effectively.
1. Home Mortgage: One of the most common applications of an amortization schedule is in a home mortgage. For example, if you take out a 30-year mortgage for $200,000 at 4% annual interest rate, an amortization schedule will outline your monthly payments for the next 360 months (30 years). It will break down every payment into principal and interest. In the early years, a bulk of your payment goes towards interest, while towards the end, it predominantly goes towards principal. The schedule allows you to see when you would have paid off half the principal, full principal, and how much interest you’re paying over the life of the loan.2. Car Loan: Another real-world application is in auto financing. Let’s say you purchase a new car for $25,000. You put down a 20% down payment ($5,000) and finance the rest ($20,000) at an annual interest rate of 5% for a period of 5 years. An amortization schedule will detail each of your monthly payments of around $377 over these 60 months. Similar to the mortgage example, it’ll highlight how much of it is going towards paying down the principal of your loan, and how much is going toward interest.3. Business Loans: Lastly, business loans also use amortization schedules. If a business borrows $50,000 to buy new equipment at an interest rate of 6% with a loan term of 7 years, an amortization schedule will lay out each of the 84 monthly payments, detailing the proportion that goes towards the principal and interest. This is extremely beneficial for the company’s financial planning; it can make accurate budget forecasts knowing exactly how much it has to pay back each month, as well as seeing the tax-deductible interest component.In all these examples, you can calculate an amortization schedule using the formula:A = P [r(1 + r)^n] / [(1 + r)^n – 1]where:A = monthly paymentP = principal loan amountr = monthly interest rate (annual rate/12)n = number of payments (months).You can use this formula manually to get the monthly payment, and then create a schedule of payments for each period, showing the split between principal and interest.
Frequently Asked Questions(FAQ)
What is an Amortization Schedule?
An Amortization Schedule is a complete table of periodic loan payments, showing the amount of principal and the amount of interest that comprise each payment until the loan is paid off at the end of its term.
How is an Amortization Schedule used in Finance and Business?
An Amortization Schedule is used in Finance and Business for outlining the repayment schedule of a loan. Businesses can use it to plan future payments and assess the impact of interest rates on loan repayment.
What are the key components of an Amortization Schedule?
The key components of an Amortization Schedule include the loan amount or principal, the periodic interest rate, the number of payments per year, the total number of payments, and the payment amount per period.
How to calculate using the Amortization Formula?
The basic formula to calculate an amortization schedule is: A = P[r(1+r)^n]/[(1+r)^n – 1], where:A is the payment amount per periodP is the initial principal (loan amount)r is the interest rate per periodn is the total number of payments (or periods)
What is an example of an Amortization Schedule calculation?
Given: P=$100,000, r=5% annual rate or (0.05/12)=0.004167 per month, n=30 years or 360 months.By plugging into the formula: A = [$100,000(0.004167(1+0.004167) ^ 360]/[(1+0.004167)^360 – 1] = $536.82/month.
Does the portion paid towards the principal increase over time in an Amortization Schedule?
Yes, in an Amortization Schedule, the early payments primarily cover interest costs. As the schedule progresses, a larger proportion of the payments goes toward paying down the principal.
Can the Amortization Schedule change over time?
Generally, an Amortization Schedule is fixed if the loan agreement terms remain the same. However, it may change if there is a loan refinance, modification of payment terms or prepayment of the loan.
Where can I find or create an Amortization Schedule?
You can create an Amortization Schedule using spreadsheet software like Excel or Google Sheets. You can also find online calculators or financial software that will generate an Amortization Schedule.
Related Finance Terms
- Principal Amount: The initial total amount of loan that is borrowed from a lender. It does not include any interest charges.
- Interest Rate: The percentage of the loan amount that a lender charges for borrowing money. These charges are added on top of the principal amount.
- Amortization Period: The total length of time, usually described in years or months, that a borrower agrees to clear the loan with the lender.
- Loan Payment: The regular amounts made by a borrower to the lender. Each payment is usually a mix of principal repayment and interest charges.
- Outstanding Balance: The total amount of money still owed by the borrower at any given time during the loan period. It decreases with each payment as per the amortization schedule.
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| 9,579 | 31 |
https://www.asiapapermarkets.com/fibria-recorded-1q-net-income-of-r-329-million-net-revenue-decreased/
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math
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Fibria recorded 1Q net income of R$ 329 million; net revenue decreased
(Brazil, May 09, 2017) Fibria recorded net income of R$ 329 million ($102.8 million) in 1Q 2017, versus a net loss of R$ 92 million ($28.75 million) in 4Q 2016 and net income of R$ 978 million ($305.6 million) in 1Q 2016.
Fibria’s net revenue totalled R$ 2,074 million ($648 million) in 1Q 2017, 18% less than in 4Q 2016, due to the reduction in sales volume, as previously explained. Compared to 1Q 2016, net revenue fell by 13%, as a result of the 25% drop in the average net price in reais, in turn caused by the 19% devaluation of the average dollar against the real and a reduction of 7% in the price in dollars, offsetting the 15% increase in sales volume.
Adjusted EBITDA came to R$ 644 million ($201 million) in 1Q 2017, with a margin of 37%, excluding the Klabin effect, 20% lower than in 4Q 2016, due to lower sales volume and the depreciation of the dollar against the real, partially offset by the 4% increase in the average net price of pulp in dollars. The 49% year-on-year decline was a result of the lower average net price in reais, due to the 19% depreciation of the average dollar against the real, a reduction in the price in dollars and increase in cash COGS, mainly due to the greater effect of scheduled downtime for maintenance.
Fibria is a Brazilian forestry company and the world’s leading eucalyptus pulp producer. (Source: Fibria)
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CC-MAIN-2023-40
| 1,433 | 5 |
http://www.gogeometry.com/school-college/2/p1182-two-squares-sum-areas-metric-relations.htm
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math
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< PREVIOUS PROBLEM |
NEXT PROBLEM >
The figure shows the squares ABCD
and EFGH. S1, S2, S3, and S4, are the
areas of the shaded quadrilaterals.Prove that:
(1) AE2 + CG2 = BF2 + DH2; (2) S1 + S3 = S2 + S4.
Puzzle of problem 1182.
Search | Geometry
Problems Art Gallery |
Area of a square Last updated:
Mar 20, 2020
Post or view a solution to the problem 1182
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https://23equations.com/docs/About.html
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math
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The first aim of the app is to help students to learn the physics equations they needed for the new science GCSE. Without the equations at their fingertips, they are at a huge disadvantage for in Physics exams.
The second aim of the app is to allow the students to practice using the equations in exam style questions and progress independently by giving support and feedback at every stage.
The models used to generate questions "know" how to answer the question so they know what needs to be done at each stage. The models can generate a similar question and show how to answer it as an example. They can understand different units and give useful feedback to answers. Once a question is answered, if the student is unsure, they can see a model answer and when they have learned from mistakes, they can try a similar question to check they have understood.
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CC-MAIN-2023-06
| 858 | 3 |
https://www.assignmentexpert.com/homework-answers/physics/mechanics-relativity/question-36236
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math
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A block A of mass 5 kg is placed on a rough table. The
coefficient of static and kinetic friction between
surfaces of block and table be 0.4 and 0.3 respectively.
If the force F exerted on the block is 10 N (g = 10 ms–2)
the force of friction between block and table is
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| 271 | 5 |
https://buy-essay.com/answered-essay-sam-a-calendar-year-taxpayer-purchased-an-annuity-contract-for-3600-that-would-pay-him-120-a-month-beginning-on/
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math
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Sam, a calendar year taxpayer, purchased an annuity contract for $3,600 that would pay him $120 a month beginning on January 1, 2016. His expected return under the contract based on his life expectancy is $10,800. Assuming Sam received a total of $1,440 in payments during 2016, how much of this annuity income is included in Sam’s gross income for 2016, using the general rule?
|Can someone show how to get annuity income by using general rule above? Thanks.|
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CC-MAIN-2023-14
| 462 | 2 |
https://slrfc.org/a-time-series-graph-is-useful-for-which-of-the-following-purposes/
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math
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1. What is a Timeplot?
A timeplot (sometimes called a time series graph) displays worths against time. They are equivalent to Cartesian plane x-y graphs, however while an x-y graph have the right to plot a variety of “x” variables (for example, height, weight, age), timeplots deserve to only display time on the x-axis. Unfavor pie charts and bar charts, these plots perform not have actually categories. Timeplots are excellent for showing exactly how information alters over time. For example, this type of chart would occupational well if you were sampling information at random times.
You are watching: A time series graph is useful for which of the following purposes?
Time Series Analysis
The goal of time series evaluation is to find fads in the information and also use the information for predictions. For example, if your information is influenced by previous information, one method to version that actions is with the AR procedure.
The adhering to graph mirrors a physics-connected timeplot through the place vs. time for two spark tapes pulled via a spark timer at various constant speeds.
See more: Why Did Americans Enjoy Escapist Films In The 1930S? The Culture Of The 1930S Flashcards
Many real-life data sets aren’t stationary. If you’ve got a real-life data set, in most situations you won’t have the ability to run any kind of procedures on the data collection directly, and also you won’t have the ability to make advantageous predictions from it. One solution is to make the design stationary by transcreating it. A stationary data collection will certainly not endure a change in circulation form as soon as there’s a change in time; Basic properties of the distribution favor the mean, variance and also covariance remain constant. This renders the design better at predictions. After you’ve made predictions, the transformations are reversed so that the brand-new model predicts the behavior of the original time series.
Some models can’t be easily transformed—choose models via seasonality, which refers to consistent, regular fluctuations in time series data. These have the right to periodically be damaged dvery own right into smaller pieces (a process dubbed stratification) and individually transcreated. Another means to resolve seasonality is to subtract the mean worth of the routine function from the data.
Chatfield, C, 1995, The evaluation of Time Series, 4th edition. Chapman & Hall
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| 2,441 | 10 |
https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/projective-algebra-and-the-calculus-of-relations1/FB3399D3BC851CB518D34F571603FF51
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math
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In there were given postulates for an abstract “projective algebra” which, in the words of the authors, represented a “modest beginning for a study of logic with quantifiers from a boolean point of view”. In , D. Monk observed that the study initiated in was an initial step in the development of algebraic versions of logic from which have evolved the cylindric and polyadic algebras.
Several years prior to the publication of , J. C. C. McKinsey presented a set of postulates for the calculus of relations. Following the publication of , McKinsey showed that every projective algebra is isomorphic to a subalgebra of a complete atomic projective algebra and thus, in view of the representation given in , every projective algebra is isomorphic to a projective algebra of subsets of a direct product, that is, to an algebra of relations.
Of course there has since followed an extensive development of projective algebra resulting in the multidimensional cylindric algebras . However, what appears to have been overlooked is the correspondence between the Everett–Ulam axiomatization and that of McKinsey.
It is the purpose of this paper to demonstrate the above, that is, we show that given a calculus of relations as defined by McKinsey it is possible to introduce projections and a partial product so that this algebra is a projective algebra and conversely, for a certain class of projective algebras it is possible to define a multiplication so that the resulting algebra is McKinsey's calculus of relations.
Email your librarian or administrator to recommend adding this journal to your organisation's collection.
* Views captured on Cambridge Core between September 2016 - 30th April 2017. This data will be updated every 24 hours.
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| 1,752 | 6 |
http://www.itextbook.cn/f/book/bookDetail?bookId=dc179c0b867d4fc1ad9fe70d2bb0b1e5
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math
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University of Cambridge
University of Oxford
National Library of China
University of Chicago
From economics and business to the biological sciences to physics and engineering, professionals successfully use the powerful mathematical tool of optimal control to make management and strategy decisions. Optimal Control Applied to Biological Models thoroughly develops the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models.
Focusing on mathematical concepts, the book first examines the most basic problem for continuous time ordinary differential equations (ODEs) before discussing more complicated problems, such as variations of the initial conditions, imposed bounds on the control, multiple states and controls, linear dependence on the control, and free terminal time. In addition, the authors introduce the optimal control of discrete systems and of partial differential equations (PDEs).
Featuring a user-friendly interface, the book contains fourteen interactive sections of various applications, including immunology and epidemic disease models, management decisions in harvesting, and resource allocation models. It also develops the underlying numerical methods of the applications and includes the MATLAB® codes on which the applications are based.
Requiring only basic knowledge of multivariable calculus, simple ODEs, and mathematical models, this text shows how to adjust controls in biological systems in order to achieve proper outcomes.
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https://ask.okorder.com/questions/how-many-speed-grades-are-the-tires-in-the-car-what-are-they-like_659856.html
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math
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How many speed grades are the tires in the car? What are they like?
The car tire "speed level" is a maximum speed corresponding to the tire supported by the figures, from the structure diagram of automobile tires, we can see that the upper left corner of the first mark mark refers to a car tire speed level, find the car tire speed level in the tires position, you can it is easy to see the tire speed rating level
The speed rating indicates the maximum speed that the automobile tires are carrying under specified conditions. The main letters, A to Z, represent speed levels between 4.8 km / h to 300 km / h.
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| 610 | 3 |
https://www.bionicturtle.com/forum/threads/ong-1999-unexpected-loss-derivation.23532/
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math
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Does anyone have the derivation for the Unexpected Loss formula? It's formula 5.5 from Schroeck but the derivation is only available in Ong, Michael, Internal Credit Risk Models (Risk Books, 1999).
HI @Kosuke Yamaji It is based on the variance of the product of two independent random variables, see https://en.wikipedia.org/wiki/Produ...f_the_product_of_independent_random_variables
You can see that σ^2(XY) = σ^2(X)*σ^2(Y) + σ^2(X)*μ^2(Y) + σ^2(Y)*μ^2(X). So the assumption here is independence between PD and LGD such that σ^2(PD*LGD) = σ^2(PD)*σ^2(LGD) + σ^2(PD)*μ^2(LGD) + σ^2(LGD)*μ^2(PD). But as a Bernoulli μ^2(PD) = PD^2 and μ^2(LGD) = LGD^2 so we have really this version of the variance of a product:
σ^2(PD*LGD) = σ^2(PD)*σ^2(LGD) + σ^2(PD)*LGD^2 + σ^2(LGD)*PD^2
= ( σ^2(PD)*σ^2(LGD) + σ^2(LGD)*PD^2 ) + σ^2(PD)*LGD^2
= ( σ^2(LGD)*[σ^2(PD) + PD^2] ) + σ^2(PD)*LGD^2, because σ^2(PD) = PD*(1-PD):
= ( σ^2(LGD)*[PD*(1-PD) + PD^2] ) + σ^2(PD)*LGD^2
= ( σ^2(LGD)*[PD - PD^2 + PD^2] ) + σ^2(PD)*LGD^2
= σ^2(LGD)*PD + σ^2(PD)*LGD^2; i.e., since this is the variance, we can see that UL is the standard deviation of the product PD*LGD. I hope that's helpful.
Hat tip to @MarekH who actually showed me (because I previously assumed it was more difficult) at https://www.bionicturtle.com/forum/threads/the-origin-of-ongs-unexpected-loss-ul.1792/post-78444
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| 1,393 | 10 |
http://www.ccl.net/chemistry/resources/messages/2000/06/19.005-dir/index.html
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math
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CCL: molecular chain dimensions
- From: John <jparm "at@at" umich.edu>
- Subject: CCL: molecular chain dimensions
- Date: Mon, 19 Jun 2000 13:48:40 -0400
I'm conducting research on biological nanoscale composites. I have a
situation where I know the connectivity of a molecular chain (sulfated
fucose polymer), and would like to
calculate the physical dimensions of the chain (average diameter for all
of the various conformations) as well as, say, how many molecular chains
may be bundled together to make a nano-scale fiber of known diameter.
I was hoping that someone could recommend some references so that I
could learn how to do this. Also I understand that there are computer
programs that do these types of calculations, what would be an
appropriate one? Thank you.
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| 773 | 14 |
http://www.primidi.com/calculation
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math
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A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.
The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition or calculating the chance of a successful relationship between two people.
Statistical estimations of the likely election results from opinion polls also involve algorithmic calculations, but provide results made up of ranges of possibilities rather than exact answers.
To calculate means to ascertain by computing. The English word derives from the Latin calculus, which originally meant a small stone in the gall-bladder (from Latin calx). It also meant a pebble used for calculating, or a small stone used as a counter in an abacus (Latin abacus, Greek abax). The abacus was an instrument used by Greeks and Romans for arithmetic calculations, preceding the slide-rule and the electronic calculator, and consisted of perforated pebbles sliding on an iron bars.
Read more about Calculation: Comparison To Computation
Other articles related to "calculation":
... In English, calculation involves numbers and the word usually connotes a simple process, but computation may be done by applying specific rules, with ... Calculation is a prerequisite for computation ... The difference in the meaning of calculation and computation appears to originate from the late medieval period ...
... Calculation of square roots of irrational numbers was not an easy task in the third century with counting rods ... happy with this result because he had acquired the same result with the calculation for a 1536-gon, obtaining the area of a 3072-gon ... After all calculation of square roots was not a simple task with rod calculus ...
2 ... In a multiplicative time-series model, the seasonal component is expressed in terms of ratio and percentage as Seasonal effect = (T*S*C*I)/( T*C*I)*100 = Y/(T*C*I )*100 However in practice the detrending of time-series is done to arrive at S*C*I ...
Famous quotes containing the word calculation:
“Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.”
—Henri-Frédéric Amiel (18211881)
“To my thinking boomed the Professor, begging the question as usual, the greatest triumph of the human mind was the calculation of Neptune from the observed vagaries of the orbit of Uranus.
And yours, said the P.B.”
—Samuel Beckett (19061989)
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| 2,539 | 15 |
https://papers.nips.cc/paper_files/paper/1997/hash/489d0396e6826eb0c1e611d82ca8b215-Abstract.html
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math
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Michael Lewicki, Terrence J. Sejnowski
We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data. Over(cid:173) complete bases allow for better approximation of the underlying statistical density. Using a Laplacian prior on the basis coefficients removes redundancy and leads to representations that are sparse and are a nonlinear function of the data. This can be viewed as a generalization of the technique of independent component anal(cid:173) ysis and provides a method for blind source separation of fewer mixtures than sources. We demonstrate the utility of overcom(cid:173) plete representations on natural speech and show that compared to the traditional Fourier basis the inferred representations poten(cid:173) tially have much greater coding efficiency.
A traditional way to represent real-values signals is with Fourier or wavelet bases. A disadvantage of these bases, however, is that they are not specialized for any particular dataset. Principal component analysis (PCA) provides one means for finding an basis that is adapted for a dataset, but the basis vectors are restricted to be orthogonal. An extension of PCA called independent component analysis (Jutten and Herault, 1991; Comon et al., 1991; Bell and Sejnowski, 1995) allows the learning of non-orthogonal bases. All of these bases are complete in the sense that they span the input space, but they are limited in terms of how well they can approximate the dataset's statistical density.
Representations that are overcomplete, i. e. more basis vectors than input variables, can provide a better representation, because the basis vectors can be specialized for
Learning Nonlinear Overcomplete Representations for Efficient Coding
a larger variety of features present in the entire ensemble of data. A criticism of overcomplete representations is that they are redundant, i.e. a given data point may have many possible representations, but this redundancy is removed by the prior probability of the basis coefficients which specifies the probability of the alternative representations.
Most of the overcomplete bases used in the literature are fixed in the sense that they are not adapted to the structure in the data. Recently Olshausen and Field (1996) presented an algorithm that allows an overcomplete basis to be learned. This algorithm relied on an approximation to the desired probabilistic objective that had several drawbacks, including tendency to breakdown in the case of low noise levels and when learning bases with higher degrees of overcompleteness. In this paper, we present an improved approximation to the desired probabilistic objective and show that this leads to a simple and robust algorithm for learning optimal overcomplete bases.
Inferring the representation
The data, X 1 :L ' are modeled with an overcomplete linear basis plus additive noise:
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http://daassignmentiuce.hyve.me/graphing-and-writing-inequalities.html
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math
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Graphing and writing inequalities
This purchase includes: - guided notes on inequality vocabulary with practice on writing inequalities - independent practice on graphing inequalities. Section 31 writing and graphing inequalities 105 work with a partner use 8 to 10 pieces of spaghetti break one piece of spaghetti into three parts that can be used to. Concept 11 write and graph linear inequalities concept 11: writing & graphing inequalities pre score 5 = level 4 deadline: (c) level 2 1 watch the video (level 2. Section 81 writing and graphing inequalities 313 work with a partner use 8 to 10 pieces of spaghetti break one piece of spaghetti into three parts that can be used to. Free practice questions for algebra 1 - writing inequalities includes full solutions and score reporting. 328 chapter 8 inequalities state standards ma6a32 s 81 writing and graphing inequalities how can you use a number line to represent solutions of an inequality.
Fit an algebraic two-variable inequality to its appropriate graph. Inequalities what is an inequality an inequality is a statement that two quantities are not equal welcome today, we will be learning about inequality time. Fun math practice improve your skills with free problems in 'graph inequalities' and thousands of other practice lessons.
Solving and graphing inequalities worksheets linear inequality worksheets contain graphing inequalities, writing inequality from the graph, solving one-step, two. Khan academy is a nonprofit with the mission of providing a free create number line graphs of inequalities learn for free about math, art, computer. Fun math practice improve your skills with free problems in 'write inequalities from graphs' and thousands of other practice lessons.
- Activating activity: this is an excellent activity to give students additional practice solving linear inequalities and graphing number lines this.
- 324 chapter 7 equations and inequalities 75 writing and graphing inequalities how can you use a number line to represent solutions of an inequality.
3-1 graphing and writing inequalities lesson describe the solutions of x 2 6 choose different values for x all rights reserved #opyright©by. Writing, solving, and graphing inequalities in one variable learning objective · solve algebraic inequalities in one variable using a combination of the. Graphing and writing inequalities short answer 1 wright the solution of inequality in words k–2 –7 2 wright the solution of inequality in words 7l.
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| 2,490 | 7 |
https://www.physicsforums.com/threads/quick-torque-and-static-equilibrium.271315/
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math
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1. The problem statement, all variables and given/known data A uniform beam of length 1.0m and mass 10 kg is attached to a wall by a cable (the cable makes a 30 degree angle with the beam). The beam is free to pivot at the point where it attaches to the wall. What is the tension in the cable? 2. Relevant equations F(net) = F(beam) - F(cable) = 0 torque = [F(beam) x d(beam)]+[F(cable) x d(cable)] 3. The attempt at a solution I think these equations are correct. I was able to figure out the torque of the beam (10kg x 9.8m/s2 x 1.0m) but I am having trouble figuring out what to do with the cable. How and where do I use the angle? Do I use it to find the distance of the cable itself? or its distance up the wall?
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| 717 | 1 |
https://mail.dafyomi.co.il/discuss_daf.php?gid=4&sid=20&daf=94&n=3
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math
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Does anyone offer an explanation for the astronomical descriptions that do not seem to fit with the known facts of our universe today?
Avraham Grant, Baltimore, USA
AKEIDAH SHEMOS 37 P. 10
I am not aware of any treatises on the subject, although it is certainly a fascinating candidate for a comprehensive discussion. For lack of other sources, I will share with you my own understanding of the astronomical Agados on these Dapim, in the hope that it will inspire someone to do a more thorough work on the subject.
There are four main points on this Daf that require clarification in light of current scientific understanding (in order of appearance):
(a) Is the world really 6,000 Parsa'os wide? How are we to understand the various proofs that the earth is larger than 6,000 Parsa'os? And who is correct?
(b) What is meant by "the "thickness of the Raki'a"?
(c) What is the meaning of the argument whether the "Galgal" is stiff and the "Mazalos" are moving, or v.v.? And what does Rashi mean when he says that the Mazalos "just pass the sun on to the next Mazal, and return to their place"? And how are we to understand the proof for the Chachamim and the refutation that the Gemara offers?
(d) What is meaning of the argument whether the sun goes under or over the world at night? And how does the sun heat the subterranean springs at night?
(e) What is the meaning of the "four paths" that the sun takes over the months of the year?
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| 1,436 | 10 |
https://www.payscale.com/research/HK/Job=Associate_-_Investment_Banking/Salary/3a6a3c9e/Mid-Career
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math
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The average salary for a Mid-Career Associate - Investment Banking is HK$610,000.
Is Associate - Investment Banking your job title? Get a personalized salary report!
HK$407k - HK$830k
HK$55k - HK$165k
HK$459k - HK$914k
Your Market Worth Over Time
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Job Description for Associate - Investment Banking
Associates in investment banking are usually recent MBA graduates. They may also be internal staff, who were promoted after three or more years working experience as financial analysts. They perform extensive investment valuation analyses. They must be consumer-focused. They must have direct investment banking experience. Some employers require that associates maintain series 6 and/or 7 certifications, issued by the Financial Industry Regulatory Authority. They must be knowledgeable about …Read more
Associate - Investment Banking Tasks
- Manage client portfolios, evaluating profitability and conducting financial transactions.
- Attend meetings with clients and prepare presentations providing an overview of clients' investment portfolios.
- Analyze investment strategies, identify potential investments, and conduct risk assessments.
Associate - Investment Banking Job Listings
Years of Experience
This data is based on 7 survey responses.
Related Job Salaries
HK$206k - HK$834k
HK$25k - HK$533k
HK$506k - HK$1m
HK$61k - HK$783k
HK$329k - HK$700k
HK$240k - HK$540k
HK$171k - HK$605k
HK$843k - HK$2m
HK$589k - HK$2m
HK$39k - HK$442k
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| 1,528 | 27 |
https://projecteuclid.org/proceedings/advanced-studies-in-pure-mathematics/The-Role-of-Metrics-in-the-Theory-of-Partial-Differential/Chapter/Lower-bound-for-the-lifespan-of-solutions-to-the-generalized/10.2969/aspm/08510303
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math
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This paper is a survey article to review the Cauchy problem for the generalized KdV equation with low-degree of nonlinearity. In , the local well-posedness for the equation is established in an appropriate class under a non-degenerate condition for the initial data. In this paper, we give a lower bound estimate for the lifespan of the solution under the condition as a consequence of the contraction principle. The lifespan depends on the size and a quantity corresponding to the distance from zero of the initial data.
Digital Object Identifier: 10.2969/aspm/08510303
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| 570 | 2 |
http://www.writework.com/essay/income-distribution
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math
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Economics have long sought to understand the links between economic growth and income distribution. According to Kuznets' hypothesis: A country's income distribution would first widen, then stabilise, and finally narrow with the economy development.
However, if the necessary result of the economy development is the income distribution's worsening, what's the value of the growth? It is meaningless for most people become poorer and only a small number of rich people become richer as a result of growth. Efforts must be made to avoid the worsening of income distribution with the economy growth.
Lucy enough, recent research has shown that there are no systematic relationship between growth and the worsening of income distribution and it's possible to achieve both growth and equal income distribution by effective policies.
This paper will focus on the following areas:
Lorenz curve and Gini coefficient.
The Kuznets Hypothesis
Empirical evidence for the inverted-U hypothesis:
Policy can pursue both objectives
Inequality is a matter of income distribution--that is, how the total income of a country is shared out among individuals and families. Inequality can be observed between different income groups and different countries, particularly, the advanced and less advanced of the world.
2. Lorenz curve and Gini coefficient.
In order to measure inequality, two common methods are adopted: Lorenz curve and Gini coefficient.
2.1 Lorenz Curves
Lorenz curves are an effective way of showing inequality of income within and between countries. The curve illustrates the actual relationship between the percentage of income recipients and the percentage of income that they actually receive. The cumulative percentage of population is plotted along the horizontal axis. The cumulative percentage of income is plotted along the vertical axis(as shown in the following diagram).
The 45 degree line is called the line of absolute equality. It...
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http://www.storyworks.in/blog/?tag=airtel
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What did you like most about the workshop ?
# Structure storytelling. Importance of story as a leadership tool.
# Simple techniques of storytelling. The learning can be used in life apart from office.
# The impact of a narrative. Makes the communication effective.
# Power of stories, bring clarity and influence. I have struggled at times getting all the above with facts.
# Simple format; message sticks!
# The content of the workshop and the seamless flow of the two days made it very interesting.
# Variety in instructions that kept me engaged.
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http://www.osti.gov/eprints/topicpages/documents/record/274/2527282.html
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math
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Summary: Math 845, Spring 2009, Topics in Mathematical Physics
Instructor: Abdelmalek Abdesselam, [email protected]. Phone: 434-924-
4926. Office: KER 227.
Location and times: MWF from 10 am to 10:50 am, in KER 326.
Office hours: MW from 1pm to 2pm and by appointment.
Prerequisites: Math 731 or instructor permission. Recommended but not re-
quired: Math 732, Math 745.
Topics: The main object of study in this course are functional integrals, i.e., mea-
sures on spaces of functions or distributions (Sobolev spaces, Schwartz spaces)
which arise in modern statistical mechanics and quantum field theory. Such
measures also arise as invariant measures for some partial differential equations,
i.e., infinite-dimensional dynamical systems. The goal of this course is to provide
you with the essential tools needed in order to tackle research problems in the
area. These are Feynman diagrams, the cluster and Mayer expansions, pertur-
bative renormalization and the rigorous theory of the renormalization group. A
typical example of measure one would like to study is that of a random function
q : Rd
R with a probability density distribution of the form exp(-V (q))Dq.
Here Dq should be thought of as the Lebesgue measure on the function space to
which q belongs. V (q) is the integral dx over Rd
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https://questions.llc/questions/387481/what-mass-of-chloral-hydrate-c2h3cl3o2-would-contain-1-0-grams-of-chlorine
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math
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What mass of chloral hydrate (C2H3Cl3O2) would contain 1.0 grams of chlorine?
Do you want 1 g Cl2 or 1 g of the Cl atom?
I expect you want 1 g chlorine atom (not Cl2).
moles Cl = 1/atomic mass Cl = 1/35.5 = 0.0282 moles Cl.
Since chloral hydrate contains 3 Cl atoms/mole of the hydrate you will want 1/3 x (0.0282) = ?? moles hydrate.
g hydrate = moles hydrate x molar mass hydrate.
If I have interpreted the problem the wrong way, just adjust the numbers above to fit.
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http://pellwillfigalus.gq/biography/electric-machinery-fundamentals-chapman-4th-edition-pdf-17667.php
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math
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IN Libmry of Co n ~ress Gltalo~in~-in-l'ublic:ltion Data Chapman. Stephen J. Electric machinery fundamentals / Stephen Chapman. - 4th ed. p. em. Includes. Stephen J. Electric machinery fundamentals / Stephen Chapman. - 4th ed. p. em Stephen J. Chapman received a B.S. in Electrical Engineering from Louisiana. instructor's manual to accompany chapman electric machinery fundamentals fourth edition stephen chapman bae systems australia instructor's manual to.
|Language:||English, German, Hindi|
|ePub File Size:||17.69 MB|
|PDF File Size:||11.84 MB|
|Distribution:||Free* [*Register to download]|
Instructor's Manual to accompany. Chapman. Electric Machinery Fundamentals. Fourth Edition. Stephen J. Chapman. BAE SYSTEMS Australia. Stephen J. Electric machinery fundamentals / Stephen Chapman. - 4th ed. Electric Machinery Fundamentals, 4th Edition (McGraw-Hill Series in Electrical and Computer pellwillfigalus.gq Oh Crap! Potty Training. Electric Machinery Fundamentals 4th ED (Chapman) - Ebook download as PDF File .pdf) or view presentation slides online. Electric Machinery Fundamentals.
Segment cd The velocity of the wire is tangential to the path of rotation, while B points to the right. The quantity v x B points into the page, which is the same direction as segment cd. Segment da same as segment bc, v x B is perpendicular to l. To determine the magnitude and direction of the torque, examine the phasors below: The force on each segment of the loop is given by: Segment ab The direction of the current is into the page, while the magnetic field B points to the right.
Segment bc The direction of the current is in the plane of the page, while the magnetic field B points to the right. Segment cd The direction of the current is out of the page, while the magnetic field B points to the right. Segment da The direction of the current is in the plane of the page, while the magnetic field B points to the right. An alternative way to express the torque equation can be done which clearly relates the behaviour of the single loop to the behaviour of larger ac machines.
Examine the phasors below: If the current in the loop is as shown, that current will generate a magnetic flux density Bloop with the direction shown. The magnitude of Bloop is: The area of the loop A is 2rl and substituting these two equations into the torque equation earlier yields: The Rotating Magnetic Field Before we have looked at how if two magnetic fields are present in a machine, then a torque will be created which will tend to line up the two magnetic fields.
If one magnetic field is produced by the stator of an ac machine and the other by the rotor, then a torque will be induced in the rotor which will cause the rotor to turn and align itself with the stator magnetic field.
How do we make the stator magnetic field to rotate? It is a 2-pole winding one north and one south. Assume currents in the 3 coils are: It produces the magnetic field intensity: Refer again to the stator in Figure 4. Assume that we represent the direction of the magnetic field densities in the form of: We know that: The Relationship between Electrical Frequency and the Speed of Magnetic Field Rotation The figure above shows that the rotating magnetic field in this stator can be represented as a north pole the flux leaves the stator and a south pole flux enters the stator.
These magnetic poles complete one mechanical rotation around the stator surface for each electrical cycle of the applied current. The mechanical speed of rotation of the magnetic field in revolutions per second is equal to electric frequency in hertz: In this winding, a pole moves only halfway around the stator surface in one electrical cycle.
To prove this, phases B and C are switched: However, the flux in a real machine does not follow these assumptions, since there is a ferromagnetic rotor in the centre of the machine with a small air gap between the rotor and the stator. The rotor can be cylindrical a nonsalient-pole , or it can have pole faces projecting out from its surface b salient pole. The reluctance of the air gap in this machine is much higher than the reluctances of either the rotor or the stator, so the flux density vector B takes the shortest possible path across the air gap and jumps perpendicularly between the rotor and the stator.
To produce a sinusoidal voltage in a machine like this, the magnitude of the flux density vector B must vary in a sinusoidal manner along the surface of the air gap.
The flux density will vary sinusoidally only if the magnetizing intensity H and mmf varies in a sinusoidal manner along the surface of the air gap. A cylindrical rotor with sinusoidally varying air-gap flux density The mmf or H or B as a function of angle in the air gap To achieve a sinusoidal variation of mmf along the surface of the air gap is to distribute the turns of the winding that produces the mmf in closely spaced slots around the surface of the machine and to vary the number of conductors in each slots in a sinusoidal manner.
The number of conductors in each slot is indicated in the diagram. The mmf distribution resulting from the winding, compared to an ideal transformer. The distribution of conductors produces a close approximation to a sinusoidal distribution of mmf. The more slots there are and the more closely spaced the slots are, the better this approximation becomes. In practice, it is not possible to distribute windings exactly as in the nC equation above, since there are only a finite number of slots in a real machine and since only integral numbers of conductors can be included in each slot.
Induced Voltage in AC Machines The induced voltage in a Coil on a Two-Pole Stator Previously, discussions were made related to induced 3 phase currents producing a rotating magnetic field. Now, lets look into the fact that a rotating magnetic field may produce voltages in the stator. The Figures below show a rotating rotor with a sinusoidally distributed magnetic field in the centre of a stationary coil. Assume that the magnetic of the flux density vector B in the air gap between the rotor and the stator varies sinusoidally with mechanical angle, while the direction of B is always radially outward.
The magnitude of the flux density vector B at a point around the rotor is given by: In this case, the wire is stationary and the magnetic field is moving, so the equation for induced voltage does not directly apply. The total voltage induced in the coil will be the sum of the voltages induced in each of its four sides.
Electric Machinery Fundamentals 4th Edition (Stephen J Chapman) Part 2.pdf
These are determined as follows: Segment bc The voltage is zero, since the vector quantity v x B is perpendicular to l. Segment da The voltage is zero, since the vector quantity v x B is perpendicular to l. Therefore total induced voltage: This derivation goes through the induced voltage in the stator when there is a rotating magnetic field produced by the rotor. The Induced Voltage in a 3-Phase Set of Coils If the stator now has 3 sets of different windings as such that the stator voltage induced due to the rotating magnetic field produced by the rotor will have a phase difference of o, the induced voltages at each phase will be as follows: Induced Torque in an AC Machines In ac machines under normal operating conditions, there are 2 magnetic fields present - a magnetic field from the rotor circuit and another magnetic field from the stator circuit.
The interaction of these two magnetic fields produces the torque in the machine, just as 2 permanent magnets near each other will experience a torque, which causes them to line up.
A simplified ac machine with a sinusoidal stator flux distribution and a single coil of wire mounted in the rotor.
Its resultant direction may be found in the diagram above. How much torque is produced in the rotor of this simplified ac machine? This is done by analyzing the force and torque on each of the two conductors separately: Since the total magnetic field density will be the summation of the BS and BR, hence: Synchronous Generator Construction 2. The Speed of Rotation of a Synchronous Generator 3. The Equivalent Circuit of a Synchronous Generator 5.
The Phasor Diagram of a Synchronous Generator 6. Power and Torque in Synchronous Generator 7. Measuring Synchronous Generator Model Parameters 8. Parallel operation of AC Generators - The conditions required for paralleling - The general procedure for paralleling generators - Frequency-power and Voltage-Reactive Power characteristics of a synchronous generator. Synchronous Generator Construction A DC current is applied to the rotor winding, which then produces a rotor magnetic field.
The rotor is then turned by a prime mover eg. Steam, water etc. This rotating magnetic field induces a 3-phase set of voltages within the stator windings of the generator. Generally a synchronous generator must have at least 2 components: Salient Pole b. Non Salient Pole b Stator Windings or Armature Windings The rotor of a synchronous generator is a large electromagnet and the magnetic poles on the rotor can either be salient or non salient construction.
Non-salient pole rotors are normally used for rotors with 2 or 4 poles rotor, while salient pole rotors are used for 4 or more poles rotor. Non-salient rotor for a synchronous machine Salient rotor A dc current must be supplied to the field circuit on the rotor. Since the rotor is rotating, a special arrangement is required to get the dc power to its field windings. The common ways are: One end of the dc rotor winding is tied to each of the 2 slip rings on the shaft of the synchronous machine, and a stationary brush rides on each slip ring.
If the positive end of a dc voltage source is connected to one brush and the negative end is connected to the other, then the same dc voltage will be applied to the field winding at all times regardless of the angular position or speed of the rotor. Some problems with slip rings and brushes: Small synchronous machines — use slip rings and brushes. Larger machines — brushless exciters are used to supply the dc field current. A brushless exciter is a small ac generator with its field circuit mounted on the stator and its armature circuit mounted on the rotor shaft.
The 3-phase output of the exciter generator is rectified to direct current by a 3-phase rectifier circuit also mounted on the shaft of the generator, and is then fed to the main dc field circuit. By controlling the small dc field current of the exciter generator located on the stator , we can adjust the field current on the main machine without slip rings and brushes.
Since no mechanical contacts occur between the rotor and stator, a brushless exciter requires less maintenance. A brushless exciter circuit: A small 3-phase current is rectified and used to supply the field circuit of the exciter, which is located on the stator. The output of the armature circuit of the exciter on the rotor is then rectified and used to supply the field current of the main machine.
To make the excitation of a generator completely independent of any external power sources, a small pilot exciter can be used. A pilot exciter is a small ac generator with permanent magnets mounted on the rotor shaft and a 3-phase winding on the stator.
It produces the power for the field circuit of the exciter, which in turn controls the field circuit of the main machine. If a pilot exciter is included on the generator shaft, then no external electric power is required. The permanent magnets of the pilot exciter produce the field current of the exciter, which in turn produces the field current of the main machine.
Even though machines with brushless exciters do not need slip rings and brushes, they still include the slip rings and brushes so that an auxiliary source of dc field current is available in emergencies. The Speed of Rotation of a Synchronous Generator Synchronous generators are by definition synchronous, meaning that the electrical frequency produced is locked in or synchronized with the mechanical rate of rotation of the generator.
Hence, the rate of rotation of the magnetic field in the machine is related to the stator electrical frequency by: The Internal Generated Voltage of a Synchronous Generator Voltage induced is dependent upon flux and speed of rotation, hence from what we have learnt so far, the induced voltage can be found as follows: The Equivalent Circuit of a Synchronous Generator The voltage EA is the internal generated voltage produced in one phase of a synchronous generator.
If the machine is not connected to a load no armature current flowing , the terminal voltage will be equivalent to the voltage induced at the stator coils. This is due to the fact that there are no current flow in the stator coils hence no losses. These differences are due to: We will explore factors a, b, and c and derive a machine model from them.
The effect of salient pole rotor shape will be ignored, and all machines in this chapter are assumed to have nonsalient or cylindrical rotors. Armature Reaction When the rotor is spun, a voltage EA is induced in the stator windings.
If a load is attached to the terminals of the generator, a current flows. But a 3-phase stator current flow will produce a magnetic field of its own.
This stator magnetic field will distorts the original rotor magnetic field, changing the resulting phase voltage. This effect is called armature reaction because the armature stator current affects the magnetic field, which produced it in the first place. Refer to the diagrams below, showing a two-pole rotor spinning inside a 3-phase stator. This stator magnetic field BS and its direction are given by the right-hand rule.
The stator field produces a voltage of its own called Estat. If X is a constant of proportionality, then the armature reaction voltage can be expressed as: In series with RF is an adjustable resistor Radj which controls the flow of the field current. The rest of the equivalent circuit consists of the models for each phase. If the 3 phases are connected in Y or I, the terminal voltage may be found as follows: If it is not balanced, a more in-depth technique is required.
The per-phase equivalent circuit: The phasor diagrams are as follows: Unity power factor Lagging power factor Leading power factor For a given phase voltage and armature current, a larger internal voltage EA is needed for lagging loads than for leading loads.
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Alternatively, for a given field current and magnitude of load current, the terminal voltage is lower for lagging loads and higher for leading loads. Power and Torque in Synchronous Generators A generator converts mechanical energy into electrical energy, hence the input power will be a mechanical prime mover, e. Regardless of the type of prime mover, the rotor velocity must remain constant to maintain a stable system frequency.
The power-flow diagram for a synchronous generator is shown: Stray losses, friction and windage losses, core loss Converted power: Copper losses Output: This gives a phasor diagram as shown: Based upon the simplified phasor diagram: Measuring Synchronous Generator Model Parameters There are basically 3 types of relationship which needs to be found for a synchronous generator: This plot is called open-circuit characteristic OCC of a generator.
With this characteristic, it is possible to find the internal generated voltage of the generator for any given field current. Open-circuit characteristic OCC of a generator At first the curve is almost perfectly linear, until some saturation is observed at high field currents. The unsaturated iron in the frame of the synchronous machine has a reluctance several thousand times lower than the air-gap reluctance, so at first almost all the mmf is across the air-gap, and the resulting flux increase is linear.
When the iron finally saturates, the reluctance of the iron increases dramatically, and the flux increases much more slowly with an increase in mmf.
The linear portion of an OCC is called the air-gap line of the characteristic. During the short circuit analysis, the net magnetic field is very small, hence the core is not saturated, hence the reason why the relationship is linear.
SCC is essentially a straight line. To understand why this characteristic is a straight line, look at the equivalent circuit below when the terminals are short circuited. When the terminals are short circuited, the armature current IA is: Since the net magnetic field is small, the machine is unsaturated and the SCC is linear. Therefore, an approximate method for determining the synchronous reactance XS at a given field current is: Find XS by applying the equation above.
Problem with this method: EA is taken from the OCC whereby the core would be partially saturated for large field currents while IA is taken from the SCC where the core is not saturated at all field currents. Hence the value of XS is only an approximate. Hence to gain better accuracy, the test should be done at low field currents which looks at the linear region of the OCC test. To find out on the resistive element of the machine, it can simply be found by applying a DC voltage to the machine terminals with the rotor stationary.
Value obtained in this test RA may increase the XS accuracy. Short Circuit Ratio Definition: Ratio of the field current required for the rated voltage at open circuit to the field current required for rated armature current at short circuit. Example A kVA, V, 50Hz, Y-connected synchronous generator with a rated field current of 5A was tested, and the following data were taken: When a dc voltage of 10V was applied to two of the terminals, a current of 25A was measured.
Find the values of the armature resistance and the approximate synchronous reactance in ohms that would be used in the generator model at the rated conditions. The synchronous generator operating alone The behaviour of a synchronous generator under load varies greatly depending on the power factor of the load and on whether the generator is operating alone or in parallel with other synchronous generator.
The next discussion, we shall disregard RA and rotor flux is assumed to be constant unless it is stated that the field current is changed. Also, the speed of the generator will be assumed constant, and all terminal characteristics are drawn assuming constant speed.
Load increase: An increase of load is an increase in real and reactive power drawn from the generator. Such a load increase increases the load current drawn from the generator. If EA is constant, what actually varies with a changing load?? Initially lagging load: This change may be seen in the phasor diagram. The effect of an increase in generator loads at constant power factor upon its terminal voltage — lagging power factor.
The effect of an increase in generator loads at constant power factor upon its terminal voltage — unity power factor. Initially leading load: The effect of an increase in generator loads at constant power factor upon its terminal voltage — leading power factor.
However, in practical it is best to keep the output voltage of a generator to be constant, hence EA has to be controlled which can be done by controlling the field current IF. This generator has a synchronous reactance of 0. At full load, the machine supplies A at 0. Under full-load conditions, the friction and windage losses are 40kW, and the core losses aree 30kW.
Ignore any field circuit losses. How much power is supplied to the generator by the prime mover? How much field current would be required to keep VT at V? Its full-load armature current is 60A at 0. This generator has friction and windage losses of 1. Since the armature resistance is being ignored, assume that the I2R losses are negligible. The field current has been adjusted so that the terminal voltage is V at no load. It is loaded with the rated current at 0. It is loaded with the rated current at 1.
How large is the induced countertorque? Conditions required for Paralleling The figure below shows a synchronous generator G1 supplying power to a load, with another generator G2 about to be paralleled with G1 by closing switch S1.
Chapman Electric Machinery Fundamentals Fourth Edition Solutions Manual
What conditions must be met before the switch can be closed and the 2 generators connected? If the switch is closed arbitrarily at some moment, the generators are liable to be severely damaged, and the load may lose power.
If the voltages are not exactly the same in each conductor being tied together, there will be a very large current flow when the switch is closed. To avoid this problem, each of the three phases must have exactly the same voltage magnitude and phase angle as the conductor to which it is connected.
Thus, paralleling 2 or more generators must be done carefully as to avoid generator or other system component damage. Conditions are as follows: If the generators were connected in this manner, there would be no problem with phase a, but huge currents would flow in phases b and c, damaging both machines.
This is done so that the phase angles of the incoming machine will change slowly with respect to the phase angles of the running system. General Procedure for Paralleling Generators Suppose that generator G2 is to be connected to the running system as shown below: Using Voltmeters, the field current of the oncoming generator should be adjusted until its terminal voltage is equal to the line voltage of the running system.
Check and verify phase sequence to be identical to the system phase sequence. There are 2 methods to do this: Alternately connect a small induction motor to the terminals of each of the 2 generators. If the motor rotates in the same direction each time, then the phase sequence is the same for both generators.
If the motor rotates in opposite directions, then the phase sequences differ, and 2 of the conductors on the incoming generator must be reversed. Another way is using the 3 light bulb method, where the bulbs are stretched across the open terminals of the switch connecting the generator to the system as shown in the figure above. As the phase changes between the 2 systems, the light bulbs first get bright large phase difference and then get dim small phase difference.
If all 3 bulbs get bright and dark together, then the systems have the same phase sequence. If the bulbs brighten in succession, then the systems have the opposite phase sequence, and one of the sequences must be reversed. Using a Synchroscope — a meter that measures the difference in phase angles it does not check phase sequences only phase angles. Check and verify generator frequency to be slightly higher than the system frequency.
This is done by watching a frequency meter until the frequencies are close and then by observing changes in phase between the systems. Once the frequencies are nearly equal, the voltages in the 2 systems will change phase with respect to each other very slowly. The phase changes are observed, and when the phase angles are equal, the switch connecting the 2 systems is shut.
All prime movers tend to behave in a similar fashion — as the power drawn from them increases, the speed at which they turn decreases.
The decrease in speed is in general non linear, but some form of governor mechanism is usually included to make the decrease in speed linear with an increase in power demand. Whatever governor mechanism is present on a prime mover, it will always be adjusted to provide a slight drooping characteristic with increasing load.
The speed droop SD of a prime mover is defined as: Most governors have some type of set point adjustment to allow the no-load speed of the turbine to be varied. A typical speed vs.
Since mechanical speed is related to the electrical frequency and electrical frequency is related with the output power, hence we will obtain the following equation: Example Figure below shows a generator supplying a load. A second load is to be connected in parallel with the first one.
The generator has a no-load frequency of Load 1 consumes a real power of kW at 0. Operation of Generators in Parallel with Large Power Systems Changes in one generator in large power systems may not have any effect on the system. A large power system may be represented as an infinite bus system.
An infinite bus is a power system so large that its voltage and frequency do not vary regardless of how much real and reactive power is drawn from or supplied to it. The power-frequency characteristic and the reactive power-voltage characteristic are shown below: We shall consider the action or changes done to the generator and its effect to the system. When a generator is connected in parallel with another generator or a large system, the frequency and terminal voltage of all the machines must be the same, since their output conductors are tied together.
Thus, their real power-frequency and reactive power-voltage characteristics can be plotted back to back, with a common vertical axis.
Such a sketch is called a house diagram, as shown below: A synchronous generator operating in parallel with an infinite bus The frequency-power diagram house diagram for a synchronous generator in parallel with an infinite bus.
This is shown here: Suppose the generator had been paralleled to the line but instead of being at a slightly higher frequency than the running system, it was at a slightly lower frequency. In this case, when paralleling is completed, the resulting situation is as shown here: At this frequency, the power supplied by the generator is actually negative. Assume that the generator is already connected, what effects of governor control and field current control has to the generator?
Governor Control Effects: In theory, if the governor set points is increased, the no load frequency will also increase the droop graph will shift up.
Since in an infinite bus system frequency does not change, the overall effect is to increase the generator output power another way to explain that it would look as if the generator is loaded up further.
Hence the output current will increase. As the governor set points are further increased the no-load frequency increases and the power supplied by the generator increases.
If the governor is set as such that it exceeds the load requirement, the excess power will flow back to the infinite bus system. The infinite bus, by definition, can supply or consume any amount of power without a change in frequency, so the extra power is consumed. Field Current Control Effects: Increasing the governor set point will increase power but will cause the generator to absorb some reactive power. The question is now, how do we supply reactive power Q into the system instead of absorbing it?
This can be done by adjusting the field current of the generator. Power into the generator must remain constant when IF is changed so that power out of the generator must also remain constant. Now, the prime mover of a synchronous generator has a fixed-torque speed characteristic for any given governor setting.
This curve changes only when the governor set points are changed. Since the generator is tied to an infinite bus, its speed cannot change. In other words, increasing the field current in a synchronous generator operating in parallel with an infinite bus increases the reactive power output of the generator. Hence, for a generator operating in parallel with an infinite bus: Note that these effects are only applicable for generators in a large system only. Operation of Generators in Parallel with Other Generators of the Same Size When a single generator operated alone, the real and reactive powers supplied by the generators are fixed, constrained to be equal to the power demanded by the load, and the frequency and terminal voltage were varied by the governor set points and the field current.
When a generator is operating in parallel with an infinite bus, the frequency and terminal voltage were constrained to be constant by the infinite bys, and the real and reactive powers were varied by the governor set points and the field current. What happens when a synchronous generator is connected in parallel not with an infinite bus, but rather with another generator of the same size?
What will be the effect of changing governor set points and field currents? The system is as shown here: In this system, the basic constraint is that the sum of the real and reactive powers supplied by the two generators must equal the P and Q demanded by the load.
The system frequency is not constrained to be constant, and neither is the power of a given generator constrained to be constant. The power-frequency diagram for such a system immediately after G2 has been paralleled to the line is shown below: As a result, the power-frequency curve of G2 shifts upward as shown here: The total power supplied to the load must not change. At the original frequency f1, the power supplied by G1 and G2 will now be larger than the load demand, so the system cannot continue to operate at the same frequency as before.
In fact, there is only one frequency at which the sum of the powers out of the two generators is equal to Pload. That frequency f2 is higher than the original system operating frequency. At that frequency, G2 supplies more power than before, and G1 supplies less power than before.
Thus, when 2 generators are operating together, an increase in governor set points on one of them 1. What happens if the field current of G2 is increased? The resulting behaviour is analogous to the real-power situation as shown below: When 2 generators are operating together and the field current of G2 is increased, 1.
The system terminal voltage is increased. The reactive power Q supplied by that generator is increased, while the reactive power supplied by the other generator is decreased.
Example shows how this can be done. Example G1 has a no-load frequency of The 2 generators are supplying a real load totalling 2. The resulting system power-frequency or house diagram is shown below. What would the new system frequency be, and how much power would G1 and G2 supply now? When 2 generators of similar size are operating in parallel, a change in the governor set points of one of them changes both the system freq and the power sharing between them.
How can the power sharing of the power system be adjusted independently of the system frequency, and vice versa? Therefore, to adjust power sharing without changing the system frequency, increase the governor set points of one generator and simultaneously decrease the governor set points of the other generator. Shifting power sharing without affecting Shifting system frequency without affecting system frequency power sharing Reactive power and terminal voltage adjustment work in an analogous fashion.
To shift the reactive power sharing without changing VT, simultaneously increase the field current on one generator and decrease the field current on the other. Shifting reactive power sharing without Shifting terminal voltage without affecting affecting terminal voltage reactive power sharing It is very important that any synchronous generator intended to operate in parallel with other machines have a drooping frequency-power characteristic.
If two generators have flat or nearly flat characteristics, then the power sharing between them can vary widely with only the tiniest changes in no-load speed. This problem is illustrated below: Rated frequency will depend upon the system at which the generator is connected. Voltage Ratings: Generated voltage is dependent upon flux, speed of rotation and mechanical constants.
However, there is a ceiling limit of flux level since it is dependent upon the generator material. Hence voltage ratings may give a rough idea on its maximum flux level possible and also maximum voltage to before the winding insulation breaks down. Apparent Power and Power Factor Ratings Constraints for electrical machines generally dependent upon mechanical strength mechanical torque on the shaft of the machine and also its winding insulation limits heating of its windings.
For a generator, there are 2 different windings that has to be protected which are: The heating effect of the stator copper losses is given by: And since we can find the maximum field current and the maximum EA possible, we may be able to determine the lowest PF changes possible for the generator to operate at rated apparent power.
Figure below shows the phasor diagram of a synchronous generator with the rated voltage and armature current. The current can assume many different angles as shown. Notice that for some possible current angles the required EA exceeds EA,max. If the generator were operated at the rated armature current and these power factors, the field winding would burn up. It is possible to operate the generator at a lower more lagging power factor than the rated value, but only by cutting back on the kVA supplied by the generator.
Synchronous Generator Capability Curves. Based upon these limits, there is a need to plot the capability of the synchronous generator. This is so that it can be shown graphically the limits of the generator. The capability curve can be derived back from the voltage phasor of the synchronous generator.
Assume that a voltage phasor as shown, operating at lagging power factor and its rated value: Note that the capability curve of the must represent power limits of the generator, hence there is a need to convert the voltage phasor into power phasor.
The powers are given by: The length corresponding to EA on the power diagram is: The final capability curve is shown below: Any point that lies within both circles is a safe operating point for the generator. It has a synchronous reactance of 1 ohm per phase. Assume that this generator is connected to a steam turbine capable of supplying up to 45kW.
The friction and windage losses are 1. Why or why not? Synchronous Motor In general, a synchronous motor is very similar to a synchronous generator with a difference of function only. Steady State Operations A synchronous motor are usually applied to instances where the load would require a constant speed. Hence for a synchronous motor, its torque speed characteristic is constant speed as the induced torque increases. Since, v ind? If load exceeds the pullout torque, the rotor will slow down.
Due to the interaction between the stator and rotor magnetic field, there would be a torque surge produced as such there would be a loss of synchronism which is known as slipping poles.
Also based upon the above equation, maximum induced torque can be achieved by increasing Ea hence increasing the field current. Effect of load changes Assumption: A synchronous generator operating with a load connected to it.
The field current setting are unchanged.
Varying load would in fact slow the machine down a bit hence increasing the torque angle. Due to an increase to the torque angle, more torque is induced hence spinning the synchronous machine to synchronous speed again. The overall effect is that the synchronous motor phasor diagram would have a bigger torque angle f. In terms of the term Ea, since If is set not to change, hence the magnitude of Ea should not change as shown in the phasor diagram fig.
Since the angle of d changes, the armature current magnitude and angle would also change to compensate to the increase of power as shown in the phasor diagram fig.
Effect of field current changes on a synchronous motor Assumption: The synchronous generator is rotating at synchronous speed with a load connected to it. The load remains unchanged. As the field current is increased, Ea should increase. Unfortunately, there are constraints set to the machine as such that the power requirement is unchanged.
Therefore since P is has to remain constant, it imposes a limit at which Ia and jXsIa as such that Ea tends to slide across a horizontal limit as shown in figure Ia will react to the changes in Ea as such that its angle changes from a leading power factor to a lagging power factor or vice versa. This gives a possibility to utilise the synchronous motor as a power factor correction tool since varying magnetic field would change the motor from leading to lagging or vice versa.
This characteristic can also be represented in the V curves as shown in figure Synchronous motor as a power factor correction Varying the field current would change to amount of reactive power injected or absorbed by the motor. Hence if a synchronous motor is incorporated nearby a load which require reactive power, the synchronous motor may be operated to inject reactive power hence maintaining stability and lowering high current flow in the transmission line.
Starting Synchronous Motors Problem with starting a synchronous motor is the initial production of torque which would vary as the stator magnetic field sweeps the rotor. As a result, the motor will vibrate and could overheat refer to figure for diagram explanations. There are 3 different starting methods available: Stator magnetic field speed reduction The idea is to let the stator magnetic field to rotate slow enough as such that the rotor has time to lock on to the stator magnetic field.
This method used to be impractical due to problems in reducing stator magnetic field. Now, due to power electronics technology, frequency reduction is possible hence makes it a more viable solution. Using a prime mover This is a very straightforward method. Motor Starting using Amortisseur windings This is the most popular way to start an induction motor. Amortisseur windings are a special kind of windings which is shorted at each ends. Its concept is near similar to an induction motor hence in depth explanation can be obtained in the text book page The final effect of this starting method is that the rotor will spin at near synchronous speed.
Note that the rotor will never reach synchronous speed unless during that time, the field windings are switched on hence will enable the rotor to lock on to the stator magnetic field. Induction Motor Construction 2. The Equivalent Circuit of an Induction Motor. Powers and Torque in Induction Motor. Starting Induction Motors 8. No dc field current is required to run the machine. Induction Motor Construction There are basically 2 types of rotor construction: Wound rotor are known to be more expensive due to its maintenance cost to upkeep the slip rings, carbon brushes and also rotor windings.
Cutaway diagram of a wound rotor induction motor. Basic Induction Motor Concepts The Development of Induced Torque in an Induction Motor When current flows in the stator, it will produce a magnetic field in stator as such that Bs stator magnetic field will rotate at a speed: This rotating magnetic field Bs passes over the rotor bars and induces a voltage in them.
The voltage induced in the rotor is given by: And this rotor current will produce a magnetic field at the rotor, Br. Hence the interaction between both magnetic field would give torque: An induction motor can thus speed up to near synchronous speed but it can never reach synchronous speed.
This can be easily termed as slip speed: Apart from that we can describe this relative motion by using the concept of slip: But unlike a transformer, the secondary frequency may not be the same as in the primary. If the rotor is locked cannot move , the rotor would have the same frequency as the stator refer to transformer concept. Another way to look at it is to see that when the rotor is locked, rotor speed drops to zero, hence by default, slip is 1. But as the rotor starts to rotate, the rotor frequency would reduce, and when the rotor turns at synchronous speed, the frequency on the rotor will be zero.
Example 7. The Equivalent Circuit of an Induction Motor An induction motor relies for its operation on the induction of voltages and currents in its rotor circuit from the stator circuit transformer action. This induction is essentially a transformer operation, hence the equivalent circuit of an induction motor is similar to the equivalent circuit of a transformer.
The Transformer Model of an Induction Motor A transformer per-phase equivalent circuit, representing the operation of an induction motor is shown below: The transformer model or an induction motor, with rotor and stator connected by an ideal transformer of turns ratio aeff. As in any transformer, there is certain resistance and self-inductance in the primary stator windings, which must be represented in the equivalent circuit of the machine.
They are - R1 - stator resistance and X1 — stator leakage reactance Also, like any transformer with an iron core, the flux in the machine is related to the integral of the applied voltage E1. The curve of mmf vs flux magnetization curve for this machine is compared to a similar curve for a transformer, as shown below: This is because there must be an air gap in an induction motor, which greatly increases the reluctance of the flux path and thus reduces the coupling between primary and secondary windings.
The higher reluctance caused by the air gap means that a higher magnetizing current is required to obtain a given flux level. Therefore, the magnetizing reactance Xm in the equivalent circuit will have a much smaller value than it would in a transformer. The primary internal stator voltage is E1 is coupled to the secondary ER by an ideal transformer with an effective turns ratio aeff. The turns ratio for a wound rotor is basically the ratio of the conductors per phase on the stator to the conductors per phase on the rotor.
It is rather difficult to see aeff clearly in the cage rotor because there are no distinct windings on the cage rotor. ER in the rotor produces current flow in the shorted rotor or secondary circuit of the machine. The primary impedances and the magnetization current of the induction motor are very similar to the corresponding components in a transformer equivalent circuit.
The Rotor Circuit Model When the voltage is applied to the stator windings, a voltage is induced in the rotor windings. In general, the greater the relative motion between the rotor and the stator magnetic fields, the greater the resulting rotor voltage and rotor frequency. The largest relative motion occurs when the rotor is stationary, called the locked-rotor or blocked-rotor condition, so the largest voltage and rotor frequency are induced in the rotor at that condition.
The smallest voltage and frequency occur when the rotor moves at the same speed as the stator magnetic field, resulting in no relative motion. The magnitude and frequency of the voltage induced in the rotor at any speed between these extremes is directly proportional to the slip of the rotor.
Therefore, if the magnitude of the induced rotor voltage at locked-rotor conditions is called ER0, the magnitude of the induced voltage at any slip will be given by: The rotor resistance RR is a constant, independent of slip, while the rotor reactance is affected in a more complicated way by slip.
The reactance of an induction motor rotor depends on the inductance of the rotor and the frequency of the voltage and current in the rotor. With a rotor inductance of LR, the rotor reactance is: The resulting rotor equivalent circuit is as shown: The rotor circuit model of an induction motor. The rotor current flow is: The rotor circuit model with all the frequency slip effects concentrated in resistor RR.
Based upon the equation above, at low slips, it can be seen that the rotor resistance is much much bigger in magnitude as compared to XR0. At high slips, XR0 will be larger as compared to the rotor resistance. The Final Equivalent Circuit To produce the final per-phase equivalent circuit for an induction motor, it is necessary to refer the rotor part of the model over to the stator side.
In an ordinary transformer, the voltages, currents and impedances on the secondary side can be referred to the primary by means of the turns ratio of the transformer. Power and Torque in Induction Motor Losses and Power-Flow diagram An induction motor can be basically described as a rotating transformer. Its input is a 3 phase system of voltages and currents.
The secondary windings in an induction motor the rotor are shorted out, so no electrical output exists from normal induction motors.
Instead, the output is mechanical. The relationship between the input electric power and the output mechanical power of this motor is shown below: The input power to an induction motor Pin is in the form of 3-phase electric voltages and currents. Then, some amount of power is lost as hysteresis and eddy currents in the stator Pcore. The power remaining at this point is transferred to the rotor of the machine across the air gap between the stator and rotor.
This power is called the air gap power PAG of the machine. After the power is transferred to the rotor, some of it is lost as I2R losses the rotor copper loss PRCL , and the rest is converted from electrical to mechanical form Pconv. The remaining power is the output of the motor Pout. The core losses do not always appear in the power-flow diagram at the point shown in the figure above.
Because of the nature of the core losses, where they are accounted for in the machine is somewhat arbitrary. The core losses of an induction motor come partially from the stator circuit and partially from the rotor circuit.
Since an induction motor normally operates at a speed near synchronous speed, the relative motion of the magnetic fields over the rotor surface is quite slow, and the rotor core losses are very tiny compared to the stator core losses. Since the largest fraction of the core losses comes from the stator circuit, all the core losses are lumped together at that point on the diagram. These losses are represented in the induction motor equivalent circuit by the resistor RC or the conductance GC.
If core losses are just given by a number X watts instead of as a circuit element, they are often lumped together with the mechanical losses and subtracted at the point on the diagram where the mechanical losses are located. The higher the speed of an induction motor, the higher the friction, windage, and stray losses. On the other hand, the higher the speed of the motor up to nsync , the lower its core losses.
Therefore, these three categories of losses are sometimes lumped together and called rotational losses. The total rotational losses of a motor are often considered to be constant with changing speed, since the component losses change in opposite directions with a change in speed. The stator copper losses are 2kW, and the rotor copper losses are W.
The friction and windage losses are W, the core losses are W, and the stray losses are negligible. The input current to a phase of the motor is: The stator copper losses in the 3 phases are: Thus, the air-gap power: The power converted, which is called developed mechanical power is given as: This torque differs from the torque actually available at the terminals of the motor by an amount equal to the friction and windage torques in the machine.
Hence, the developed torque is: Hence it may be useful to separate the rotor copper loss element since rotor resistance are both used for calculating rotor copper loss and also the output power.
The core loss is lumped in with the rotational losses. For a rotor slip of 2. Induced Torque from a Physical Standpoint The magnetic fields in an induction The magnetic fields in an induction motor under light loads motor under heavy loads No-load Condition Assume that the induction rotor is already rotating at no load conditions, hence its rotating speed is near to synchronous speed. The net magnetic field Bnet is produced by the magnetization current IM.
The magnitude of IM and Bnet is directly proportional to voltage E1. If E1 is constant, then Bnet is constant. In an actual machine, E1 varies as the load changes due to the stator impedances R1 and X1 which cause varying volt drops with varying loads.
However, the volt drop at R1 and X1 is so small, that E1 is assumed to remain constant throughout. At no-load, the rotor slip is very small, and so the relative motion between rotor and magnetic field is very small, and the rotor frequency is also very small.
Since the relative motion is small, the voltage ER induced in the bars of the rotor is very small, and the resulting current flow IR is also very small. Since the rotor frequency is small, the reactance of the rotor is nearly zero, and the max rotor current IR is almost in phase with the rotor voltage ER.
The rotor current produces a small magnetic field BR at an angle slightly greater than 90 degrees behind Bnet. The stator current must be quite large even at no-load since it must supply most of Bnet. The induced torque which is keeping the rotor running, is given by: Since the rotor speed is slower, there is now more relative motion between rotor and stator magnetic fields.
Greater relative motion means a stronger rotor voltage ER which in turn produces a larger rotor current IR. With large rotor current, the rotor magnetic field BR also increases. However, the angle between rotor current and BR changes as well. Therefore, the rotor current now lags further behind the rotor voltage, and the rotor magnetic field shifts with the current.
This effect is known as pullout torque. Modelling the torque-speed characteristics of an induction motor Looking at the induction motor characteristics, a summary on the behaviour of torque: Current flow will increase as slip increase reduction in velocity b The net magnetic field density will remain constant since it is proportional to E1 refer to equivalent induction motor equivalent circuit.
Since E1 is assumed to be constant, hence Bnet will assume to be constant. Adding the characteristics of all there elements would give the torque speed characteristics of an induction motor. Graphical development of an induction motor torque-speed characteristics The Derivation of the Induction Motor Induced-Torque Equation Previously we looked into the creation of the induced torque graph, now we would like to derive the Torque speed equation based upon the power flow diagram of an induction motor.
Hence there is a need to derive PAG. By definition, air gap power is the power transferred from the stator to the rotor via the air gap in the induction machine. Based upon the induction motor equivalent circuit, the air gap power may be defined as: The easiest way is via the construction of the Thevenin equivalent circuit. Since Pconv may be derived as follows: Based upon the maximum power transfer theorem, maximum power transfer will be achieved when the magnitude of source impedance matches the load impedance.
Since the source impedance is as follows: By adding more resistance to the machine impedances, we can vary: At what speed and slip does it occur? What is the new starting torque?
S decrease, PAG increase, and efficiency increase. Use a wound rotor induction motor and insert extra resistance into the rotor during starting, and then removed for better efficiency during normal operations. But, wound rotor induction motors are more expensive, need more maintanence etc. Solution - utilising leakage reactance — to obtain the desired curve as shown below A torque-speed characteristic curve combining high- resistance effects at low speeds high slip with low resistance effects at high speed low slip.
Thus, if the bars of a cage rotor are placed near the surface of the rotor, they will have small leakage flux and X2 will be small. Deep-Bar and Double-Cage rotor design How can a variable rotor resistance be produced to combine the high starting torque and low starting current of Class D, with the low normal operating slip and high efficiency of class A?? Deeper in the bar, the leakage inductance is higher.
The impedances of all parts of the bar are approx equal, so current flows through all the parts of the bar equally. The resulting large cross sectional area makes the rotor resistance quite small, resulting in good efficiency at low slips. Since the effective cross section is lower, the rotor resistance is higher.
Thus, the starting torque is relatively higher and the starting current is relatively lower than in a class A design. It is similar to the deep-bar rotor, except that the difference between low-slip and high-slip operation is even more exaggerated. Hence, high starting torque. However, at normal operating speeds, both bars are effective, and the resistance is almost as low as in a deep-bar rotor.
They are low starting torque machines. Starting Induction Motors An induction motor has the ability to start directly, however direct starting of an induction motor is not advised due to high starting currents, which will be explained later.
In order to know the starting current, we should be able to calculate the starting power required by the induction motor. The Code Letter designated to each induction motor, which can be seen in figure , may represent this. The starting code may be obtained from the motor nameplate. For a wound rotor type induction motor, this problem may be solved by incorporating resistor banks at the rotor terminal during starting to reduce current flow and as the rotor picks up speed, the resistor banks are taken out.
Reducing the starting terminal voltage will also reduce the rated starting power hence reducing starting current. One way to achieve this is by using a step down transformer during the starting sequence and stepping up the transformer ratio as the machine spins faster refer figure below.
A magnetic motor starter of this sort has several built in protective features: There is a finite delay before the 1TD contacts close. During that time, the motor speeds up, and the starting current drops. And finally 3TD contacts close, and the entire starting resistor is out of the circuit. Speed Control of Induction Motor Induction motors are not good machines for applications requiring considerable speed control.
Varying slip may be achieved by varying rotor resistance or varying the terminal voltage. Consider one phase winding in a stator. By changing the current flow in one portion of the stator windings as such that it is similar to the current flow in the opposite portion of the stator will automatically generate an extra pair of poles. In terms of torque, the maximum torque magnitude would generally be maintained. This method will enable speed changes in terms of 2: Multiple stator windings have extra sets of windings that may be switched in or out to obtain the required number of poles.
Unfortunately this would an expensive alternative. Read More Posted on March 8, Read More Posted on May 16, Workshop Instructors: Would you like to know more about how to use the formatting features in Microsoft Word? Research Commons staff will help you with your questions about the nuts and bolts of formatting: This includes a special section about copyright and your thesis. Graduate students at any stage of the writing process are welcome, but some prior knowledge of Microsoft Word is recommended.
To keep up-to-date with all of the workshops, consults, and events subscribe to the UBC Library Research Commons monthly newsletter. Workshop Facilitators: Amir and David Workshop Description: This workshop provides hands-on experience with NVivo for literature reviews, focusing on coding, advanced queries, and visualization methods.
It is recommended to take this workshop after completing NVivo Parts 1 and 2. If possible, bring articles that you plan to use for your own research.
This workshop is offered as part of our Literature Reviews workshop series. Searching and Keeping Track -Literature Reviews: Analyzing with NVivo -Literature Reviews:Now, connect a load to the terminals of the machine, and a current will flow in its armature windings. Research Commons staff will help you with your questions about the nuts and bolts of formatting: You've reached the end of this preview.
In other words, the concept of magnetic permeability corresponds to the ability of the material to permit the flow of magnetic flux through it. These are determined as follows: The transformer has Np turns of wire on its primary side and Ns turns of wire on its secondary sides.
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http://relativity.li/en/epstein2/read/i0_en/i1_en/
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math
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I1 The Precession of the Perihelion of Mercury
|Should a single, isolated planet orbit the sun then it must do so according to Kepler and Newton in an exact ellipse. Newton already recognized that this is not the case in the solar system because the planets affect each other gravitationally. An exact solution to the ‘three-body problem’ evaded even great people like Poincaré (whose attempt at a solution deeply penetrated into the territory known today as ‘chaos theory’). Today, iterative numeric methods can calculate the orbits of all the planets with high precision for a long time into the future. The apsis, the straight line through the aphelion (furthest point from sun) and perihelion (closest point to sun) of the orbit precedes very slowly under the influence of the outer planets, in the same direction in which the planets orbit. This results in a rosette-like path, where the effect as shown in the diagram is greatly exaggerated. By the way, these numerical simulations have also shown that the solar system will remain stable even over very long periods [36-315ff].|
There is, however, a small difference between the calculated values for the precession of the perihelion within Newtonian physics and those measured observationally. The following table shows the numerical values in the units ‘arc seconds per century’. The fuzziness of the values can be read from the column ‘difference’:
|Planet||Newtonian Value||Observed Value||Difference||Prediction GTR|
|Mercury mmmm||mmmm 532.08 mmmm||mmmm 575.19 mmmm||mm 43.11 ± 0.45 mm||mmmm 43.03 mmmm|
|Venus||13.2||21.6||8.4 ± 4.8||8.6|
|Earth||1165||1170||5 ± 1.2||3.8|
The difference between the calculated and the measured value, especially in the case of Mercury was so big that it demanded an explanation. The French astronomer Urbain Le Verrier, who predicted in 1845 the existence and location of the new planet Neptune based on the interference of the planet Uranus, postulated in 1859 the existence of another planet Vulcan, whose orbit was closer to the sun than Mercury’s.
The GTR explains precisely this difference between the Newtonian theory and observation. Einstein was overjoyed when he calculated near the end of 1915 that his new theory predicted an addition of just 43 arc seconds per century to the precession of the perihelion of Mercury! He derived the following formula:
where Δφ is the extra rotation per orbit in radians; RS is the Schwarzschild radius of the sun; a is the length of the semi-major axis of the orbit; and ε is the eccentricity of the ellipse.
The effect decreases with increasing distance from the sun and is also greater with highly elliptical orbits than with circular orbits. Therefore Mercury was an ideal candidate. The small eccentricity of the orbit of Venus not only weakens the effect but also makes it difficult to observe the precession. The values in the last column of the table can be obtained from Einstein's formula, if the result is multiplied by the number of revolutions in 100 years and then converted from radians to arc seconds (See problem 1).
That this effect must occur is made nearly self-evident by Epstein’s ‘barrel’ region [15-166]:
In the first drawing space is flat and the planet moves in its ellipse, however, for Epstein in an unconventional direction (one always looks from the north to the ecliptic). That is the situation according to Newton.
Now we cut the plane along the apsis. We make the cut from the aphelion to the sun.
As discussed in section H6 a cone should now arise with its tip in the sun. To this end we must push the edges on both sides of the incision over each other (thus crafts one a cone!). This forces an advance (precession) of the aphelion in the direction of the planet’s orbit!
Amazingly, it is even possible to determine the magnitude of the phenomenon from Epstein’s paper model. Almost with no computation we come surprisingly close to the results of the formula, whose derivation forced Einstein to tussle with elliptic integrals.
The red curve shows the cross-section of Epstein's ‘space bump’ (problem 5 in I10 deals with the analysis of this function). The central mass sits at the origin, and with increasing distance x from the origin the spatial curvature decreases. If our planet has an average distance a from the central mass, we can then approximate the space bump with a local cone. The appropriate angle of inclination φ between the surface line of the cone and the plane through the center of the central mass can be easily determined for the planet at point a. It is cos (φ) = Δx(r, ∞) / Δx(r, r) = 1 - α / a according to G4. If the cone has a surface line of length 1 then the base circle measures a radius of (1 - α / a). We now cut the cone along a surface line and press it flat:
How big is the angle β of the missing sector?
β / (2π) is equal to the ratio of the 'missing' arc length to the circumference, i.e.,
β / (2π) = [2π - 2π • (1 - α / a)] / (2π) = 1 - (1 - α / a) = α / a
We thus obtain the expression for β
β= (2π) • α / a = π • 2 • α / a = π • RS / a
Reform the cone and you will find either a circle or an ellipse offset by about the size of this angle β, that is, β specifies the amount of precession per orbit of the apsis.
We only obtain about one third of the correct value (compare with the formula above in this section). This should not concern us since we have only taken the influence of space curvature into account, and that also using very modest means. We are, in any case, within a correct order of magnitude.
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http://www.av8n.com/irro/richiami_e.html
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math
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We are going to examine the properties of a plane flow the velocity of which satisfies the two conditions
The above conditions follow from the hypothesis of irrotational flow and from the conservation of mass for an incompressible fluid, respectively. In addition to the above continuity equation, the well-known Navier-Stokes momentum equation is available for the solution of the velocity and pressure fields.
where d/dt is the material derivative, represents body forces (due to gravity) and and are fluid density and kinematic viscosity, respectively.
In the following we will analyze how the irrotationality of the flow contributes to the determination of the flow field from the continuity equation alone. The solution involves the definition of a velocity potential the Laplacian (pictured above) of which must vanish. Once the velocity field is known, the pressure field is obtained from the Navier-Stokes equation, which for an irrotational flow takes a simple formulation known as the Bernoulli theorem.
This chapter is divided into the following sections
It is suggested to browse the sections in the above order, at least for first time visitors.
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| 1,158 | 6 |
https://www.onlinemathlearning.com/multiplying-dividing-negative-numbers.html
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math
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Multiplying and Dividing Positive and Negative Numbers
More Lessons for GMAT Math
Videos that will explain how to multiply and divide positive and negative numbers.
Multiplying Positive and Negative Numbers
Multiplying and Dividing Integers (Positive and Negative Numbers)
Multiplying and dividing negative numbers
Understand and Learn the Rules of Positive and Negative Numbers
Explains how to add, subtract, multiply and divide positive and negative numbers...also known as the integer rules.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other 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://ciqujezagynyxyn.southindiatrails.com/correlation-and-regression-book-3627ee.php
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math
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2 edition of Correlation and regression. found in the catalog.
Correlation and regression.
K. E. Selkirk
|Series||Rediguides -- 32|
You could try the combination of Cohen and Cohens Applied Multiple Regression/Correlation Analysis and John Mardens free online book/notes on multivariate analysis, Multivariate - Old School. The first book covers multiple regression in an applied sense very well, while the second is good on multivariate theory, and many skips many of the. Correlation and linear regression each explore the relationship between two quantitative variables. Both are very common analyses. Correlation determines if one variable varies systematically as another variable changes. It does not specify that one variable is the dependent variable and the other is the independent variable.
State the three assumptions that are the basis for the Simple Linear Regression Model. The Simple Linear Regression Model is summarized by the equation \[y=\beta _1x+\beta _0+\varepsilon\] Identify the deterministic part and the random part. Is the number \(\beta _1\) in the equation \(y=\beta _1x+\beta _0\) a statistic or a population parameter? Topic 3: Correlation and Regression September 1 and 6, In this section, we shall take a careful look at the nature of linear relationships found in the data used to construct a scatterplot. The first of these, correlation, examines this relationship in a symmetric manner. The second, regression.
Methods of correlation and regression can be used in order to analyze the extent and the nature of relationships between different variables. Correlation analysis is used to understand the nature of relationships between two individual variables. For example, if we aim to study the impact of foreign. In Lesson 11 we examined relationships between two categorical variables with the chi-square test of independence. In this lesson, we will examine the relationships between two quantitative variables with correlation and simple linear regression. Quantitative variables have numerical values with magnitudes that can be placed in a meaningful were first introduced to correlation and.
Basic mathematical concepts.
time element in criminal prosecutions in Wisconsin.
Our Changing Democracy (Command 6348)
The New York times
Right of franchise.
By the Lower House of Assembly, December 1, 1757.
Projects from pine
Tax planning for groups of companies
Molecular modelling of the elastic behaviour of polymer chains in networks
Linear Regression & Correlation. If you are looking for a short beginners guide packed with visual examples, this book is for you. Linear Regression is a way of simplifying a group of data into a single equation. For instance, we all know Moore’s law: that the number of transistors on a computer chip doubles every two years/5(38).
This book Correlation and Regression is an outcome of authors long teaching experience of the subject. This book present a thorough treatment of what is required for the students of B.A/, of all Indian Universities.
It includes fundamental concepts, illustrated examples and application to various problems. These illustrative examples have been selected carefully on such topic and. on Correlation and Regression Analysis covers a variety topics of how to investigate the strength, direction and effect of a relationship between variables by collecting measurements and using appropriate statistical analysis.
Also this textbook intends to practice data of labor force surveyFile Size: 1MB. Bobko has achieved his objective of making the topics of correlation and regression accessible to students For someone looking for a very clearly written treatment of applied correlation and regression, this book would be an excellent choice."--Paul E.
Spector, University of South Florida. "This book provides one of the clearest treatments of correlations and Correlation and regression. book of any statistics book I have seen Bobko has achieved his objective of making the topics of correlation and regression accessible to studentsCited by: Introduction to Correlation and Regression Analysis.
In this section we will first discuss correlation analysis, which is used to quantify the association between two continuous variables (e.g., between an independent and a dependent variable or between two independent variables). In Correlation and Regression Analysis: A Historian's Guide Thomas J.
Archdeacon provides historians with a practical introduction to the use of correlation and regression analysis.
The book concentrates on the kinds of analysis that form the broad range of statistical methods used in Correlation and regression.
book social sciences. It enables historians to understand and to evaluate critically the quantitative analyses. On the other end, Regression analysis, predicts the value of the dependent variable based on the known value of the independent variable, assuming that average mathematical relationship between two or more variables.
The difference between correlation and regression is one of the commonly asked questions in interviews. Ch 08 - Correlation and Regression - 4.
These videos provide overviews of these tests, instructions for carrying out the pretest checklist, running the tests, and inter-preting the results using the data sets Ch 08 - Example 01 - Correlation and Regression - and Ch 08 - Example 02 - Correlation and Regression - Chapter 4 Covariance, Regression, and Correlation “Co-relation or correlation of structure” is a phrase much used in biology, and not least in that branch of it which refers to heredity, and the idea is even more frequently present than the phrase; but I am not aware of any previous attempt to define it clearly, to trace its mode of.
The points given below, explains the difference between correlation and regression in detail: A statistical measure which determines the co-relationship or association of two quantities is known as Correlation.
Regression describes how an independent variable is numerically related to the dependent variable. David Nettleton, in Commercial Data Mining, Correlation. The Pearson correlation method is the most common method to use for numerical variables; it assigns a value between − 1 and 1, where 0 is no correlation, 1 is total positive correlation, and − 1 is total negative correlation.
This is interpreted as follows: a correlation value of between two variables would indicate that a. The topic of how to properly do multiple regression and test for interactions can be quite complex and is not covered here.
Here we just fit a model with x, z, and the interaction between the two. To model interactions between x and z, a x:z term must be added. Linear Regression & Correlation. If you are looking for a short beginners guide packed with visual examples, this book is for you. Linear Regression is a way of 4/5.
For n> 10, the Spearman rank correlation coefficient can be tested for significance using the t test given earlier. The regression equation. Correlation describes the strength of an association between two variables, and is completely symmetrical, the correlation between A and B is the same as the correlation between B and A.
"This book provides one of the clearest treatments of correlations and regression of any statistics book I have seen Bobko has achieved his objective of making the topics of correlation and regression accessible to students For someone looking for a very clearly written treatment of applied correlation and regression, this book would be an excellent choice."--Paul E.
One of the most common goals of statistical research is to find links between variables. Using correlation, regression, and two-way tables, you can use data to answer questions like these: Which lifestyle behaviors increase or decrease the risk of cancer.
What are the number of side effects associated with this new drug. Can I lower [ ]. It is important to recognize that regression analysis is fundamentally different from ascertaining the correlations among different variables. Correlation determines the strength of the relationship between variables, while regression attempts to describe that relationship between these variables in more detail.
The linear regression model (LRM)File Size: KB. Correlation and Regression, Second Edition, provides students with an accessible textbook on statistical theories in correlation and regression.
Taking an ap. Prelude to Linear Regression and Correlation In this chapter, you will be studying the simplest form of regression, "linear regression" with one independent variable (x).
This involves data that fits a line in two dimensions. You will also study correlation which measures how strong the relationship is.
Linear Equations. With a package that includes regression and basic time series procedures, it's relatively easy to use an iterative procedure to determine adjusted regression coefficient estimates and their standard errors.
Remember, the purpose is to adjust "ordinary" regression estimates for the fact that the residuals have an ARIMA structure.(i) Calculate the equation of the least squares regression line of y on x, writing your answer in the form y a + lox. (ii) Draw the regression line on your scatter diagram.
The mathematics teacher needs to arrive at school no later than am. (5 marks) (l mark) The number of minutes by which the mathematics teacher arrives early at school, when. Correlation and regression 1.
Correlation(Pearson & spearman) &Linear Regression 2. Correlation Semantically, Correlation means Co-together and Relation. Statistical correlation is a statistical technique which tells us if two variables are related. 3.
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https://www.aakash.ac.in/important-concepts/maths/supplementary-and-complementary-angles
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math
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If the sum of linear angles at a common vertex is 180˚, then the angles are supplementary. Even if two right angles are added, equal to 180˚, the pair is known as a supplementary pair of angles.
If one angle out of a pair of linear angles is x, then the other angle is given by 180-x. This linearity remains the same for all the pairs of supplementary angles.
This property is also valid for trigonometric functions like-
From the given image, we can say that the pair of angles BOA and AOC are supplementary angles because their sum is equal to 180˚.
We can also find the other angle if only one of these angles is given. For example, consider only 60 degrees was given, we can find the other angle by 180-60 = 120˚.
When the sum of two angles is equal to 90˚, then the angles are said to be complementary angles. So even if you add two angles and form a right angle, the two angles are known to be complementary.
Complementary angle phenomenon is valid in trigonometry as well. The ratios are given as-
In the above image, we can see that AOD and DOB are complementary angles because their sum is equal to 90˚.
Also, in this image, angles POQ and ABC can be called complementary angles because their sum adds up to 90 degrees.
Find the values of angles P and Q, if angle P and angle Q are supplementary angles such that angle P = 2x+10 and angle Q is 6x-46
We know, the sum of angles of a supplementary pair is equal to 180˚.
Therefore, ∠P + ∠Q = 180˚
(2x + 10) + (6x - 46) = 180˚
8x - 36 =180
8x = 216
x = 27
Therefore, ∠P = 2(27) + 10 = 64˚ and ∠Q = 6(27) - 46 =116˚.
Given that two angles are supplementary in nature. The value of the larger angle is 5 degrees more than 4 times the measure of the smaller angle. Find out the value of a larger angle in degrees.
We need to consider the two supplementary angles as x (larger) and y (smaller).
From the information above, x = 4y+5
We know the sum of angles is 180˚ if they are supplementary.
x + y = 180˚
(4y + 5) + y = 180˚
5y + 5 = 180˚
5y = 175˚ = 35˚
Therefore, the larger angle x = 4(35) + 5 = 145˚
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https://pdfcoffee.com/curves-pdf-free.html
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math
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CHAPTER 3 CURVES Section I. SIMPLE HORIZONTAL CURVES CURVE POINTS TYPES OF HORIZONTAL CURVES By studying TM 5-232, th
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CURVES Section I. SIMPLE HORIZONTAL CURVES CURVE POINTS
TYPES OF HORIZONTAL CURVES
By studying TM 5-232, the surveyor learns to locate points using angles and distances. In construction surveying, the surveyor must often establish the line of a curve for road layout or some other construction.
A curve may be simple, compound, reverse, or spiral (figure 3-l). Compound and reverse curves are treated as a combination of two or more simple curves, whereas the spiral curve is based on a varying radius.
The surveyor can establish curves of short radius, usually less than one tape length, by holding one end of the tape at the center of the circle and swinging the tape in an arc, marking as many points as desired.
Simple The simple curve is an arc of a circle. It is the most commonly used. The radius of the circle determines the “sharpness” or “flatness” of the curve. The larger the radius, the “flatter” the curve.
As the radius and length of curve increases, the tape becomes impractical, and the surveyor must use other methods. Measured angles and straight line distances are usually picked to locate selected points, known as stations, on the circumference of the arc.
Compound Surveyors often have to use a compound curve because of the terrain. This curve normally consists of two simple curves curving in the same direction and joined together.
FM 5-233 Reverse A reverse curve consists of two simple curves joined together but curving in opposite directions. For safety reasons, the surveyor should not use this curve unless absolutely necessary.
Spiral The spiral is a curve with varying radius used on railroads and somemodern highways. It provides a transition from the tangent to a simple curve or between simple curves in a compound curve.
STATIONING On route surveys, the surveyor numbers the stations forward from the beginning of the project. For example, 0+00 indicates the beginning of the project. The 15+52.96 would indicate a point 1,552,96 feet from the beginning. A full station is 100 feet or 30 meters, making 15+00 and 16+00 full stations. A plus station indicates a point between full stations. (15+52.96 is a plus station.) When using the metric system, the surveyor does not use the plus system of numbering stations. The station number simply becomes the distance from the beginning of the project.
ELEMENTS OF A SIMPLE CURVE Figure 3-2 shows the elements of a simple curve. They are described as follows, and their abbreviations are given in parentheses. Point of Intersection (PI) The point of intersection marks the point where the back and forward tangents
intersect. The surveyor indicates it one of the stations on the preliminary traverse. Intersecting Angle (I) The intersecting angle is the deflection angle at the PI. The surveyor either computes its value from the preliminary traverse station angles or measures it in the field. Radius (R) The radius is the radius of the circle of which the curve is an arc. Point of Curvature (PC) The point of curvature is the point where the circular curve begins. The back tangent is tangent to the curve at this point. Point of Tangency (PT) The point of tangency is the end of the curve. The forward tangent is tangent to the curve at this point.
FM 5-233 Length of Curve (L) The length of curve is the distance from the PC to the PT measured along the curve.
Long Chord (LC) The long chord is the chord from the PC to the PT.
Tangent Distance (T) The tangent distance is the distance along the tangents from the PI to the PC or PT. These distances are equal on a simple curve.
External Distance (E) The external distance is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at the PI.
Central Angle The central angle is the angle formed by two radii drawn from the center of the circle (0) to the PC and PT. The central angle is equal in value to the I angle.
Middle Ordinate (M) The middle ordinate is the distance from the midpoint of the curve to the midpoint of the long chord. The extension of the middle ordinate bisects the central angle.
FM 5-233 Degree of Curve (D) The degree of curve defines the “sharpness” or “flatness” of the curve (figure 3-3). There are two definitions commonly in use for degree of curve, the arc definition and the chord definition.
As the degree of curve increases, the radius decreases. It should be noted that for a given intersecting angle or central angle, when using the arc definition, all the elements of the curve are inversely proportioned to the degree of curve. This definition is primarily used by civilian engineers in highway construction. 0
English system. Substituting D = 1 and length of arc = 100 feet, we obtain—
R = 36,000 divided by 6.283185308 R = 5,729.58 ft
Metric system. In the metric system, using a 30.48-meter length of arc and substituting D = 1°, we obtain—
Arc definition. The arc definition states that the degree of curve (D) is the angle formed by two radii drawn from the center of the circle (point O, figure 3-3) to the ends of an arc 100 feet or 30.48 meters long. In this definition, the degree of curve and radius are inversely proportional using the following formula:
R = 10,972.8 divided by 6.283185308 R = 1,746.38 m
Chord definition. The chord definition states that the degree of curve is the angle formed by two radii drawn from the center of the circle (point O, figure 3-3) to the ends of a chord 100 feet or 30.48 meters long. The radius is computed by the following formula:
FM 5-233 The radius and the degree of curve are not inversely proportional even though, as in the arc definition, the larger the degree of curve the “sharper” the curve and the shorter the radius. The chord definition is used primarily on railroads in civilian practice and for both roads and railroads by the military.
Substituting D = 1° and given Sin ½ 1 = 0.0087265335, solve for R as follows:
English system. Substituting D = 1 and given Sin ½ 1 = 0.0087265355. 50 R = 50ft or Sin ½ D 0.0087265355 R = 5,729.65 ft Metric system. Using a chord 30.48 meters long, the surveyor computes R by the formula R=
15.24 m 0.0087265355
Chords On curves with long radii, it is impractical to stake the curve by locating the center of the circle and swinging the arc with a tape. The surveyor lays these curves out by staking the ends of a series of chords (figure 3-4). Since the ends of the chords lie on the circumference of the curve, the surveyor defines the arc in the field. The length of the chords varies with the degree of curve. To reduce the discrepancy between the arc distance and chord distance, the surveyor uses the following chord lengths:
SIMPLE CURVE FORMULAS The following formulas are used in the computation of a simple curve. All of the formulas, except those noted, apply to both the arc and chord definitions.
M= R (l-COs ½ I) LC = 2 R (Sin½ I) In the following formulas, C equals the chord length and d equals the deflection angle. All the formulas are exact for the arc definition and approximate for the chord definition.
This formula gives an answer in degrees.
,3048 in the metric system. The answer will be in minutes.
SOLUTION OF A SIMPLE CURVE
L is the distance around the arc for the arc definition, or the distance along the chords for the chord definition.
To solve a simple curve, the surveyor must know three elements. The first two are the PI station value and the I angle. The third is the degree of curve, which is given in the project specifications or computed using one of the elements limited by the terrain (see section II). The surveyor normally determines the PI and I angle on the preliminary traverse for the road. This may also be done by triangulation when the PI is inaccessible. Chord Definition The six-place natural trigonometric functions from table A-1 were used in the example. When a calculator is used to obtain the trigonometric functions, the results may vary slightly. Assume that the following is known: PI = 18+00, I = 45, and D = 15°.
Chord Definition (Feet) arc definitions are not exact for the chord definition. However, when a one-minute instrument is used to stake the curve, the surveyor may use them for either definition. The deflection angles are— d
d std = 0.3 x 25 x 15° = 112.5’ or 1°52.5’ d1 = 0.3X 8.67X 15° = 0°39.015’ d2 = 0.3 x 16.33 x 15° = 73.485’ or 1°13.485’ The number of full chords is computed by subtracting the first plus station divisible by the chord length from the last plus station divisible by the chord length and dividing the difference by the standard (std) chord length. Thus, we have (19+25 - 16+50)-25 equals 11 full chords. Since there are 11 chords of 25 feet, the sum of the deflection angles for 25foot chords is 11 x 1°52.5’ = 20°37.5’. Chords. Since the degree of curve is 15 degrees, the chord length is 25 feet. The surveyor customarily places the first stake after the PC at a plus station divisible by the chord length. The surveyor stakes the centerline of the road at intervals of 10,25,50 or 100 feet between curves. Thus, the level party is not confused when profile levels are run on the centerline. The first stake after the PC for this curve will be at station 16+50. Therefore, the first chord length or subchord is 8.67 feet. Similarly, there will be a subchord at the end of the curve from station 19+25 to the PT. This subchord will be 16,33 feet. The surveyor designates the subchord at the beginning, C1 , an d at the end, C2 (figure 3-2). Deflection Angles. After the subchords have been determined, the surveyor computes the deflection angles using the formulas on page 3-6. Technically, the formulas for the
The sum of d1, d2, and the deflections for the full chords is— d1 = 0°39.015’ d2 = 1°13.485’ d std = 20°37.500’ Total 22°30.000’ The surveyor should note that the total of the deflection angles is equal to one half of the I angle. If the total deflection does not equal one half of I, a mistake has been made in the calculations. After the total deflection has been decided, the surveyor determines the angles for each station on the curve. In this step, they are rounded off to the smallest reading of the instrument to be used in the field. For this problem, the surveyor must assume that a one-minute instrument is to be used. The curve station deflection angles are listed on page 3-8.
Special Cases. The curve that was just solved had an I angle and degree of curve whose values were whole degrees. When the I angle and degree of curve consist of degrees and minutes, the procedure in solving the curve does not change, but the surveyor must take care in substituting these values into the formulas for length and deflection angles. For example, if I = 42° 15’ and D = 5° 37’, the
surveyor must change the minutes in each angle to a decimal part of a degree, or D = 42.25000°, I = 5.61667°. To obtain the required accuracy, the surveyor should convert values to five decimal places. An alternate method for computing the length is to convert the I angle and degree of curve to
FM 5-233 minutes; thus, 42° 15’ = 2,535 minutes and 5° 37’ = 337 minutes. Substituting into the length formula gives
correction of 0.83 feet is obtained for the tangent distance and for the external distance, 0.29 feet.
L = 2,535 x 100 = 752.23 feet. 337
The surveyor adds the corrections to the tangent distance and external distance obtained from table A-5. This gives a tangent distance of 293.94 feet and an external distance of 99.79 feet for the chord definition.
This method gives an exact result. If the surveyor converts the minutes to a decimal part of a degree to the nearest five places, the same result is obtained. Since the total of the deflection angles should be one half of the I angle, a problem arises when the I angle contains an odd number of minutes and the instrument used is a oneminute instrument. Since the surveyor normally stakes the PT prior to running the curve, the total deflection will be a check on the PT. Therefore, the surveyor should compute to the nearest 0.5 degree. If the total deflection checks to the nearest minute in the field, it can be considered correct. Curve Tables The surveyor can simplify the computation of simple curves by using tables. Table A-5 lists long chords, middle ordinates, externals, and tangents for a l-degree curve with a radius of 5,730 feet for various angles of intersection. Table A-6 lists the tangent, external distance corrections (chord definition) for various angles of intersection and degrees of curve. Arc Definition. Since the degree of curve by arc definition is inversely proportional to the other functions of the curve, the values for a one-degree curve are divided by the degree of curve to obtain the element desired. For example, table A-5 lists the tangent distance and external distance for an I angle of 75 degrees to be 4,396.7 feet and 1,492,5 feet, respectively. Dividing by 15 degrees, the degree of curve, the surveyor obtains a tangent distance of 293.11 feet and an external distance of 99.50 feet. Chord Definition. To convert these values to the chord definition, the surveyor uses the values in table A-5. From table A-6, a
After the tangent and external distances are extracted from the tables, the surveyor computes the remainder of the curve.
COMPARISON OF ARC AND CHORD DEFINITIONS Misunderstandings occur between surveyors in the field concerning the arc and chord definitions. It must be remembered that one definition is no better than the other. Different Elements Two different circles are involved in comparing two curves with the same degree of curve. The difference is that one is computed by the arc definition and the other by the chord definition. Since the two curves have different radii, the other elements are also different. 5,730-Foot Definition Some engineers prefer to use a value of 5,730 feet for the radius of a l-degree curve, and the arc definition formulas. When compared with the pure arc method using 5,729.58, the 5,730 method produces discrepancies of less than one part in 10,000 parts. This is much better than the accuracy of the measurements made in the field and is acceptable in all but the most extreme cases. Table A-5 is based on this definition.
CURVE LAYOUT The following is the procedure to lay out a curve using a one-minute instrument with a horizontal circle that reads to the right. The values are the same as those used to demonstrate the solution of a simple curve (pages 3-6 through 3-8).
FM 5-233 Setting PC and PT With the instrument at the PI, the instrumentman sights on the preceding PI and keeps the head tapeman on line while the tangent distance is measured. A stake is set on line and marked to show the PC and its station value. The instrumentman now points the instrument on the forward PI, and the tangent distance is measured to set and mark a stake for the PT. Laying Out Curve from PC The procedure for laying out a curve from the PC is described as follows. Note that the procedure varies depending on whether the road curves to the left or to the right. Road Curves to Right. The instrument is set up at the PC with the horizontal circle at 0°00’ on the PI. (1) The angle to the PT is measured if the PT can be seen. This angle will equal one half of the I angle if the PC and PT are located properly. (2)Without touching the lower motion, the first deflection angle, d1 (0° 39’), is set on the horizontal circle. The instrumentman keeps the head tapeman on line while the first subchord distance, C1 (8.67 feet), is measured from the PC to set and mark station 16+50. (3) The instrumentman now sets the second deflection angle, d1 + dstd (2° 32’), on the horizontal circle. The tapemen measure the standard chord (25 feet) from the previously set station (16+50) while the instrument man keeps the head tapeman on line to set station 16+75. (4) The succeeding stations are staked out in the same manner. If the work is done correctly, the last deflection angle will point on the PT, and the last distance will be the subchord length, C2 (16.33 feet), to the PT.
Road Curves to Left. As in the procedures noted, the instrument occupies the PC and is set at 0°00’ pointing on the PI. (1) The angle is measured to the PT, if possible, and subtracted from 360 degrees. The result will equal one half the I angle if the PC and PT are positioned properly. (2) The first deflection, dl (0° 39’), is subtracted from 360 degrees, and the remainder is set on the horizontal circle. The first subchord, Cl (8.67 feet), is measured from the PC, and a stake is set on line and marked for station 16+50. (3)The remaining stations are set by continuing to subtract their deflection angles from 360 degrees and setting the results on the horizontal circles. The chord distances are measured from the previously set station. (4)The last station set before the PT should be C2 (16.33 feet from the PT), and its deflection should equal the angle measured in (1) above plus the last deflection, d2 (1° 14’). Laying Out Curve from Intermediate Setup When it is impossible to stake the entire curve from the PC, the surveyor must use an adaptation of the above procedure. (1) Stake out as many stations from the PC as possible. (2) Move the instrument forward to any station on the curve. (3) Pick another station already in place, and set the deflection angle for that station on the horizontal circle. Sight that station with the instruments telescope in the reverse position. (4)Plunge the telescope, and set the remaining stations as if the instrument was set over the PC.
FM 5-233 Laying Out Curve from PT If a setup on the curve has been made and it is still impossible to set all the remaining stations due to some obstruction, the surveyor can “back in” the remainder of the curve from the PT. Although this procedure has been set up as a method to avoid obstructions, it is widely used for laying out curves. When using the “backing in method,” the surveyor sets approximately one half the curve stations from the PC and the remainder from the PT. With this method, any error in the curve is in its center where it is less noticeable. Road Curves to Right. Occupy the PT, and sight the PI with one half of the I angle on the horizontal circle. The instrument is now oriented so that if the PC is sighted, the instrument will read 0°00’. The remaining stations can be set by using their deflections and chord distances from the PC or in their reverse order from the PT. Road Curves to Left. Occupy the PT and sight the PI with 360 degrees minus one half of the I angle on the horizontal circle. The instrument should read 0° 00’ if the PC is sighted. Set the remaining stations by using their deflections and chord distances as if computed from the PC or by computing the deflections in reverse order from the PT.
CHORD CORRECTIONS Frequently, the surveyor must lay out curves more precisely than is possible by using the chord lengths previously described. To eliminate the discrepancy between chord and arc lengths, the chords must be corrected using the values taken from the nomography in table A-11. This gives the corrections to be applied if the curve was computed by the arc definition. Table A-10 gives the corrections to be applied if the curve was computed by the chord definition. The surveyor should recall that the length of a curve computed by the chord definition was the length along the chords. Figure 3-5 illustrates the example given in table A-9. The chord distance from station 18+00 to station 19+00 is 100 feet. The nominal length of the subchords is 50 feet.
INTERMEDIATE STAKE If the surveyor desires to place a stake at station 18+50, a correction must be applied to the chords, since the distance from 18+00 through 18+50 to 19+00 is greater than the chord from 18+00 to 19+00. Therefore, a correction must be applied to the subchords to keep station 19+00 100 feet from 18+00. In figure 3-5, if the chord length is nominally 50 feet, then the correction is 0.19 feet. The chord distance from 18+00 to 18+50 and 18+50 to 19+00 would be 50.19.
Section II. OBSTACLES TO CURVE LOCATION TERRAIN RESTRICTIONS To solve a simple curve, the surveyor must know three parts. Normally, these will be the PI, I angle, and degree of curve. Sometimes, however, the terrain features limit the size of various elements of the curve. If this happens, the surveyor must determine the degree of curve from the limiting factor. Inaccessible PI Under certain conditions, it may be impossible or impractical to occupy the PI. In this case, the surveyor locates the curve elements by using the following steps (figure 3-6).
(1) Mark two intervisible points A and B, one on each of the tangents, so that line AB (a random line connecting the tangents) will clear the obstruction. (2) Measure angles a and b by setting up at both A and B. (3) Measure the distance AB. (4) Compute inaccessible distances AV and BV as follows: I=a+b
FM 5-233 (5) Determine the tangent distance from the PI to the PC on the basis of the degree of curve or other given limiting factor. (6) Locate the PC at a distance T minus AV from the point A and the PT at distance T minus BV from point B. (7) Proceed with the curve computation and layout. Inaccessible PC When the PC is inaccessible, as illustrated in figure 3-7, and both the PI and PT are set and readily accessible, the surveyor must establish the location of an offset station at the PC. (1)Place the instrument on the PT and back the curve in as far as possible.
(2) Select one of the stations (for example,
“P”) on the curve, so that a line PQ, parallel to the tangent line AV, will clear the obstacle at the PC.
(3) Compute and record the length of line PW
so that point W is on the tangent line AV and line PW is perpendicular to the tangent. The length of line PW = R (l - Cos dp), where dp is that portion of the central angle subtended by AP and equal to two times the deflection angle of P. (4) Establish point W on the tangent line by setting the instrument at the PI and laying off angle V (V = 180° - I). This sights the instrument along the tangent
FM 5-233 AV. Swing a tape using the computed length of line PW and the line of sight to set point W. (5) Measure and record the length of line VW along the tangent. (6) Place the instrument at point P. Backsight point W and lay off a 90-degree angle to sight along line PQ, parallel to AV. (7) Measure along this line of sight to a point Q beyond the obstacle. Set point Q, and record the distance PQ. (8) Place the instrument at point Q, backsight P, and lay off a 90-degree angle to sight along line QS. Measure, along this line of sight, a distance QS equals PW, and set point S. Note that the station number of point S = PI - (line VW + line PQ).
(9) Set an offset PC at point Y by measuring from point Q toward point P a distance equal to the station of the PC minus station S. To set the PC after the obstacle has been removed, place the instrument at point Y, backsight point Q, lay off a 90-degree angle and a distance from Y to the PC equal to line PW and QS. Carefully set reference points for points Q, S, Y, and W to insure points are available to set the PC after clearing and construction have begun. Inaccessible PT When the PT is inaccessible, as illustrated in figure 3-8, and both the PI and PC are readily accessible, the surveyor must establish an
FM 5-233 offset station at the PT using the method for inaccessible PC with the following exceptions. (1) Letter the curve so that point A is at the PT instead of the PC (see figure 3-8). (2)Lay the curve in as far as possible from the PC instead of the PT. (3) Angle dp is the angle at the center of the curve between point P and the PT, which is equal to two times the difference between the deflection at P and one half of I. Follow the steps for inaccessible PC to set lines PQ and QS. Note that the station at point S equals the computed station value of PT plus YQ. (4)Use station S to number the stations of the alignment ahead.
Obstacle on Curve Some curves have obstacles large enough to interfere with the line of sight and taping. Normally, only a few stations are affected. The surveyor should not waste too much time on preliminary work. Figure 3-9 illustrates a method of bypassing an obstacle on a curve. (1) Set the instrument over the PC with the 0 horizontal circle at 0 00’, and sight on the PI. Check I/2 from the PI to the PT, if possible. (2)Set as many stations on the curve as possible before the obstacle, point b. (3) Set the instrument over the PT with the plates at the value of I/2. Sight on the PI.
FM 5-233 (4)Back in as many stations as possible beyond the obstacle, point e.
(2) Measure line y, the distance from the PI to the fixed point.
(5) After the obstacle is removed, the obstructed stations c and d can be set.
(3) Compute angles c, b, and a in triangle COP. c = 90 - (d + I/2)
CURVE THROUGH FIXED POINT Because of topographic features or other obstacles, the surveyor may find it necessary to determine the radius of a curve which will pass through or avoid a fixed point and connect two given tangents. This may be accomplished as follows (figure 3-10): (1)Given the PI and the I angle from the preliminary traverse, place the instrument on the PI and measure angle d, so that angle d is the angle between the fixed point and the tangent line that lies on the same side of the curve as the fixed point.
To find angle b, first solve for angle e Sin e = Sin c Cos I/2 Angle b = 180°- angle e a = 180° - (b + c) (4)Compute the radius of the desired curve using the formula
FM 5-233 (5) Compute the degree of curve to five decimal places, using the following formulas: (arc method) D = 5,729.58 ft/R D = 1,746.385 meters/R (chord method) Sin D = 2 (50 feet/R) Sin D = 2 (15.24 meters/R) (6) Compute the remaining elements of the curve and the deflection angles, and stake the curve.
LIMITING FACTORS In some cases, the surveyor may have to use elements other than the radius as the limiting factor in determining the size of the curve. These are usually the tangent T, external E, or middle ordinate M. When any limiting factor is given, it will usually be presented in the form of T equals some value x, x. In any case, the first step is to determine the radius using one of the following formulas: Given: Tangent; then R = T/(Tan ½I) External; then R = E/[(l/Cos ½I) - 1] Middle Ordinate; then R = M/(l - Cos ½I)
The surveyor next determines D. If the limiting factor is presented in the form T equals some value x, the surveyor must compute D, hold to five decimal places, and compute the remainder of the curve. If the then D is limiting factor is presented as rounded down to the nearest ½ degree. For example, if E 50 feet, the surveyor would round down to the nearest ½ degree, recompute E, and compute the rest of the curve data using the rounded value of D, The new value of E will be equal to or greater than 50 feet. If the limiting factor is the D is rounded is to the nearest ½ degree. For example, if M 45 feet, then D would be rounded up to the nearest ½ degree, M would be recomputed, and the rest of the curve data computed using the rounded value of D. The new value of M will be equal to or less than 45 feet. The surveyor may also use the values from table B-5 to compute the value of D. This is done by dividing the tabulated value of tangent, external, or middle ordinate for a l-degree curve by the given value of the limiting factor. For example, given a limiting tangent T 45 feet and I = 20°20’, the T for a l-degree curve from table B-5 is 1,027.6 and D = 1,027.6/45.00 = 22.836°. Rounded up to the nearest half degree, D = 23°. Use this rounded value to recompute D, T and the rest of the curve data.
Section III. COMPOUND AND REVERSE CURVES COMPOUND CURVES A compound curve is two or more simple curves which have different centers, bend in the same direction, lie on the same side of their common tangent, and connect to form a continuous arc. The point where the two curves connect (namely, the point at which the PT of the first curve equals the PC of the second curve) is referred to as the point of compound curvature (PCC).
Since their tangent lengths vary, compound curves fit the topography much better than simple curves. These curves easily adapt to mountainous terrain or areas cut by large, winding rivers. However, since compound curves are more hazardous than simple curves, they should never be used where a simple curve will do.
FM 5-233 Compound Curve Data The computation of compound curves presents two basic problems. The first is where the compound curve is to be laid out between two successive PIs on the preliminary traverse. The second is where the curve is to be laid in between two successive tangents on the preliminary traverse. (See figure 3-11.) Compound Curve between Successive PIs. The calculations and procedure for laying out a compound curve between successive PIs are outlined in the following steps. This procedure is illustrated in figure 3-11a. (1) Determine the PI of the first curve at point A from field data or previous computations.
the tangent for the second curve must be held exact, the value of D2 must be carried to five decimal places. (9) Compare D1 and D2. They should not differ by more than 3 degrees, If they vary by more than 3 degrees, the surveyor should consider changing the configuration of the curve. (l0) If the two Ds are acceptable, then compute the remaining data and deflection angles for the first curve. (11) Compute the PI of the second curve. Since the PCC is at the same station as the PT of the first curve, then PI2 = PT1 + T2.
(2) Obtain I1, I2, and distance AB from the field data.
(12) Compute the remaining data and deflection angles for the second curve, and lay in the curves.
(3) Determine the value of D1 , the D for the first curve. This may be computed from a limiting factor based on a scaled value from the road plan or furnished by the project engineer.
Compound Curve between Successive Tangents. The following steps explain the laying out of a compound curve between successive tangents. This procedure is illustrated in figure 3-llb.
(4) Compute R1, the radius of the first curve as shown on pages 3-6 through 3-8.
(1) Determine the PI and I angle from the field data and/or previous computations.
(5) Compute T1, the tangent of the first curve.
(2) Determine the value of I1 and distance AB. The surveyor may do this by field measurements or by scaling the distance and angle from the plan and profile sheet.
T1 = R1 (Tan ½ I) (6) Compute T2, the tangent of the second curve. T2= AB - T1 (7) Compute R2, the radius of the second curve. R2
= T2 Tan ½ I
(8) Compute D2 for the second curve. Since
(3) Compute angle C. C = 180 - I (4) Compute I2. I2=180-(I l+C) (5) Compute line AC. AC = AB Sin I2 Sin C
FM 5-233 ( 6 ) Compute line BC. BC = AB Sin I1 Sin C (7) Compute the station of PI1. PI1 = PI - AC (8) Determine D1 and compute R1 and T1 for the first curve as described on pages 3-6 through 3-8. (9) Compute T2 and R2 as described on pages 3-6 through 3-8. (l0) Compute D2 according to the formulas on pages 3-6 through 3-8. (11) Compute the station at PC. PC1 = PI - (AC + T1) (12) Compute the remaining curve data and deflection angles for the first curve. (13) Compute PI2.
Compound Curve between Successive Tangents. Place the instrument at the PI and sight along the back tangent. (1) Lay out a distance AC from the PI along the back tangent, and set PI1. (2) Continue along the back tangent from PI2 a distance T1, and set PC1. (3) Sight along the forward tangent with the instrument still at the PI. (4) Lay out a distance BC from the PI along the forward tangent, and set PI2. (5) Continue along the forward tangent from PI a distance T2, and set PT2. (6) Check the location of PI1 and PI2 by either measuring the distance between the two PIs and comparing the measured distance to the computed length of line AB, or by placing the instrument at PI1, sighting the PI, and laying off I1. The resulting line-of-sight should intercept PI2. (7) Stake the curves as outlined on pages 3-10
(14) Compute the remaining curve data and deflection angles for the second curve, and stake out the curves. Staking Compound Curves Care must be taken when staking a curve in the field. Two procedures for staking compound curves are described. Compound Curve between Successive PIs. Stake the first curve as described on pages 3-10 and 3-11. (1) Verify the PCC and PT2 by placing the instrument on the PCC, sighting on PI2, and laying off I2/2. The resulting line-ofsight should intercept PT2. (2) Stake the second curve in the same manner as the first.
A reverse curve is composed of two or more simple curves turning in opposite directions. Their points of intersection lie on opposite ends of a common tangent, and the PT of the first curve is coincident with the PC of the second. This point is called the point of reverse curvature (PRC). Reverse curves are useful when laying out such things as pipelines, flumes, and levees. The surveyor may also use them on low-speed roads and railroads. They cannot be used on high-speed roads or railroads since they cannot be properly superelevated at the PRC. They are sometimes used on canals, but only with extreme caution, since they make the
FM 5-233 canal difficult to navigate and contribute to erosion. Reverse Curve Data The computation of reverse curves presents three basic problems. The first is where the reverse curve is to be laid out between two successive PIs. (See figure 3-12.) In this case, the surveyor performs the computations in exactly the same manner as a compound curve between successive PIs. The second is where the curve is to be laid out so it connects two parallel tangents (figure 3-13). The third problem is where the reverse curve is to be laid out so that it connects diverging tangents (figure 3-14).
FM 5-233 Connecting Parallel Tangents Figure 3-13 illustrates a reverse curve connecting two parallel tangents. The PC and PT are located as follows. (1) Measure p, the perpendicular distance between tangents. (2)Locate the PRC and measure m1 and m2. (If conditions permit, the PRC can be at the midpoint between the two tangents. This will reduce computation, since both arcs will be identical.) (3) Determine R1. (4) Compute I1.
(5) R 2,I 2,andL 2 are determined in the same way as R1, I1, and L1. If the PRC is to be the midpoint, the values for arc 2 will be the same as for arc 1. (6) Stake each of the arcs the same as a simple curve. If necessary, the surveyor can easily determine other curve components. For example, the surveyor needs a reverse curve to connect two parallel tangents. No obstructions exist so it can be made up of two equal arcs. The degree of curve for both must be 5°. The surveyor measures the distance p and finds it to be 225.00 feet. m 1 = m 2 a n d L1 = L 2 R 1=
R2 a n d I 1 = I2
FM 5-233 R1 and R2, which are computed from the specified degree of curve for each arc. (1) Measure I at the PI. (2) Measure Ts to locate the PT as the point where the curve is to join the forward tangent. In some cases, the PT position will be specified, but Ts must still be measured for the computations. (7) The PC and PT are located by measuring off L1 and L2. Connecting Diverging Tangents The connection of two diverging tangents by a reverse curve is illustrated in figure 3-14. Due to possible obstruction or topographic consideration, one simple curve could not be used between the tangents. The PT has been moved back beyond the PI. However, the I angle still exists as in a simple curve. The controlling dimensions in this curve are the distance Ts to locate the PT and the values of
(3) Perform the following calculations: Determine R1 and R2. If practical, have R1 equal R2. Angle s = 180-(90+I)=90-I m = Ts (Tan I) L = Ts Cos I angle e = I1 (by similar triangles)
FM 5-233 angle f = I1 (by similar triangles) therefore, I2 = I + I1
degrees. The PT location is specified and the Ts is measured as 550 feet.
n = (R2 - m) Sin e p = (R2 - m) Cos e Determine g by establishing the value of I 1.
Knowing Cos I1, determine Sin I1.
(4) Measure TL from the PI to locate the PC. (5) Stake arc 1 to PRC from PC. (6) Set instrument at the PT and verify the PRC (invert the telescope, sight on PI, plunge, and turn angle I2/2). (7) Stake arc 2 to the PRC from PT. For example, in figure 3-14, a reverse curve is to connect two diverging tangents with both arcs having a 5-degree curve. The surveyor locates the PI and measures the I angle as 41
The PC is located by measuring TL. The curve is staked using 5-degree curve computations.
Section IV. TRANSITION SPIRALS SPIRAL CURVES In engineering construction, the surveyor often inserts a transition curve, also known as a spiral curve, between a circular curve and the tangent to that curve. The spiral is a curve of varying radius used to gradually increase the curvature of a road or railroad. Spiral curves are used primarily to reduce skidding and steering difficulties by gradual transition between straight-line and turning motion, and/or to provide a method for adequately superelevating curves.
The spiral curve is designed to provide for a gradual superelevation of the outer pavement edge of the road to counteract the centrifugal force of vehicles as they pass. The best spiral curve is one in which the superelevation increases uniformly with the length of the spiral from the TS or the point where the spiral curve leaves the tangent. The curvature of a spiral must increase uniformly from its beginning to its end. At
FM 5-233 the beginning, where it leaves the tangent, its curvature is zero; at the end, where it joins the circular curve, it has the same degree of curvature as the circular curve it intercepts. Theory of A.R.E.A. 10-Chord Spiral The spiral of the American Railway Engineering Association, known as the A.R.E.A. spiral, retains nearly all the characteristics of the cubic spiral. In the cubic spiral, the lengths have been considered as measured along the spiral curve itself, but measurements in the field must be taken by chords. Recognizing this fact, in the A.R.E.A. spiral the length of spiral is measured by 10
equal chords, so that the theoretical curve is brought into harmony with field practice. This 10-chord spiral closely approximates the cubic spiral. Basically, the two curves degrees. coincide up to the point where The exact formulas for this A.R.E.A. 10chord spiral, when does not exeed 45 degrees, are given on pages 3-27 and 3-28. Spiral Elements Figures 3-15 and 3-16 show the notations applied to elements of a simple circular curve with spirals connecting it to the tangents. TS = the point of change from tangent to spiral
FM 5-233 SC = the point of change from spiral to circular curve
L = the length of the spiral in feet from the TS to any given point on the spiral
CS = the point of change from circular curve to spiral
Ls = the length of the spiral in feet from the TS to the SC, measured in 10 equal chords
ST = the point of change from spiral to tangent
o = the ordinate of the offsetted PC; the distance between the tangent and a parallel tangent to the offsetted curve
SS = the point of change from one spiral to another (not shown in figure 3-15 or figure 3-16)
r = the radius of the osculating circle at any given point of the spiral
The symbols PC and PT, TS and ST, and SC and CS become transposed when the direction of stationing is changed.
R = the radius of the central circular curve s = the length of the spiral in stations from the TS to any given point
a = the angle between the tangent at the TS and the chord from the TS to any point on the spiral
S = the length of the spiral in stations from the TS to the SC
A = the angle between the tangent at the TS and the chord from the TS to the SC
u = the distance on the tangent from the TS to the intersection with a tangent through any given point on the spiral
b = the angle at any point on the spiral between the tangent at that point and the chord from the TS B = the angle at the SC between the chord from the TS and the tangent at the SC c = the chord from any point on the spiral to the TS C = the chord from the TS to the SC d = the degree of curve at any point on the spiral D = the degree of curve of the circular arc f = the angle between any chord of the spiral (calculated when necessary) and the tangent through the TS I = the angle of the deflection between initial and final tangents; the total central angle of the circular curve and spirals k = the increase in degree of curve per station on the spiral
U = the distance on the tangent from the TS to the intersection with a tangent through the SC; the longer spiral tangent v = the distance on the tangent through any given point from that point to the intersection with the tangent through the TS V = the distance on the tangent through the SC from the SC to the intersection with the tangent through the TS; the shorter spiral tangent x = the tangent distance from the TS to any point on the spiral X = the tangent distance from the TS to the SC y = the tangent offset of any point on the spiral Y = the tangent offset of the SC Z = the tangent distance from the TS to the offsetted PC (Z = X/2, approximately)
FM 5-233 Ts = the tangent distance of the spiraled curve; distance from TS to PI, the point of intersection of tangents Es = the external distance of the offsetted curve
Spiral Formulas The following formulas are for the exact determination of the functions of the 10does chord spiral when the central angle not exceed 45 degrees. These are suitable for the compilation of tables and for accurate fieldwork.
FM 5-233 any degree of curvature and design speed is obtained from the the relationship Ls = 3 1.6V /R, in which Ls is the minimum spiral length in feet, V is the design speed in miles per hour, and R is the radius of curvature of the simple curve. This equation is not mathematically exact but an approximation based on years of observation and road tests. Table 3-1 is compiled from the above equation for multiples of 50 feet. When spirals are inserted between the3 arcs of a compound curve, use Ls = 1.6V /Ra. Ra represents the radius of a curve of a degree equal to the difference in degrees of curvature of the circular arcs. Railroads Spirals applied to railroad layout must be long enough to permit an increase in superelevation not exceeding 1 ¼ inches per second for the maximum speed of train operation. The minimum length is determined from the equation Ls = 1.17 EV. E is the full theoretical superelevation of the curve in inches, V is the speed in miles per hour, and Ls is the spiral length in feet. Empirical Formulas For use in the field, the following formulas are sufficiently accurate for practical purposes when does not exceed 15 degrees. a= A=
(degrees) (degrees) 2
a = 10 ks (minutes) 2
S = 10 kS (minutes) Spiral Lengths Different factors must be taken into account when calculating spiral lengths for highway and railroad layout. Highways. Spirals applied to highway layout must be long enough to permit the effects of centrifugal force to be adequately compensated for by proper superelevation. The minimum transition spiral length for
This length of spiral provides the best riding conditions by maintaining the desired relationship between the amount of superelevation and the degree of curvature. The degree of curvature increases uniformly throughout the length of the spiral. The same equation is used to compute the length of a spiral between the arcs of a compound curve. In such a case, E is the difference between the superelevations of the two circular arcs.
SPIRAL CALCULATIONS Spiral elements are readily computed from the formulas given on pages 3-25 and 3-26. To use these formulas, certain data must be known. These data are normally obtained from location plans or by field measurements. The following computations are for a spiral when D, V, PI station, and I are known. D = 4° I = 24°10’
Determining L s (1) Assuming that this is a highway spiral, use either the equation on page 3-28 or table 3-1.
(2) From page 3-28,
(2) From table 3-1, when D = 4° and V = 60 mph, the value for Ls is 250 feet. Determining
FM 5-233 (2) I = 24° 10’= 24.16667° A=5” D=4° (3) L, = 24.16667- 10 x 100 = 354.17 ft 4 Determining Chord Length (1) Chord length =Ls 10
(2) Chord length=250 ft 10 Determining Station Values With the data above, the curve points are calculated as follows:
(l) Z = X - (R Sin A) (2) From table A-9 we see that X = .999243 x Ls X = .999243 x 250 X = 249.81 ft R = 1,432.69 ft
Station PI Station TS Station TS
Sin 5° = 0.08716
(3) Z = 249.81- (1,432.69X 0.08716) Station CS Z = 124.94 ft Determining T s
(l) Ts= (R + o) Tan(½ I) + Z (2) From the previous steps, R = 1,432.69 feet, o = 1.81 feet, and Z = 124.94 feet. 0
(3) Tan 1 .Tan 24° 10’= Tan 12 05'=0.21408 2 2 (4) Ts = (1,432.69 + 1.81) (0.21408) + 124.94
= 42 + 61.70 = -4 + 32.04 = 38+ 29.66 +2 + 50.()() = 40 + 79.66 +3 + 54.17 = 44+ 33.83 +2 + 50.()() = 46+ 83,83
= Ts = L, = La = Ls
Determining Deflection Angles One of the principal characteristics of the spiral is that the deflection angles vary as the square of the distance along the curve. 2 a—... . L 2 Ls A From this equation, the following relationships are obtained: 2
Ts = 432.04 ft Determining Length of the Circular Arc (La)
a 1 =(1) 2 A, a2 = 4 a1 , a3 = 9a,=16a 1 ,...a 9 = (l0) 81a1, and a10 = 100a1 = A. The deflection angles to the various points on the spiral from the TS or ST are a1, a2, a3 . . . a9 and a10. Using these relationships, the deflection angles for the spirals and the circular arc are
FM 5-233 computed for the example spiral curve. Page 3-27 states that
SPIRAL CURVE LAYOUT The following is the procedure to lay out a spiral curve, using a one-minute instrument with a horizontal circle that reads to the right. Figure 3-17 illustrates this procedure.
Setting TS and ST With the instrument at the PI, the instrumentman sights along the back tangent and keeps the head tapeman on line while the tangent distance (Ts) is measured. A stake is
set on line and marked to show the TS and its station value. The instrumentman now sights along the forward tangent to measure and set the ST.
(3) The remaining circular arc stations are set by subtracting their deflection angles from 360 degrees and measuring the corresponding chord distance from the previously set station.
Laying Out First Spiral from TS to SC Set up the instrument at the TS, pointing on the PI, with 0°00’ on the horizontal circle.
Laying Out Second Spiral from ST to CS Set up the instrument at the ST, pointing on the PI, with 0°00’ on the horizontal circle.
(1) Check the angle to the ST, if possible. The angle should equal one half of the I angle if the TS and ST are located properly.
(1) Check the angle to the CS. The angle should equal 1° 40’ if the CS is located properly.
(2) The first deflection (a 1 / 0 01’) is subtracted from 360 degrees, and the remainder is set on the horizontal circle. Measure the standard spiral chord length (25 feet) from the TS, and set the first spiral station (38 + 54.66) on line. (3) The remaining spiral stations are set by subtracting their deflection angles from 360 degrees and measuring 25 feet from the previously set station. Laying Out Circular Arc from SC to CS Set up the instrument at the SC with a value of A minus A (5° 00’- 1°40’ = 3° 20’) on the horizontal circle. Sight the TS with the instrument telescope in the reverse position. (1) Plunge the telescope. Rotate the telescope until 0°00’ is read on the horizontal circle. The instrument is now sighted along the tangent to the circular arc at the SC. (2) The first deflection (dl /0° 24’) is subtracted from 360 degrees, and the remainder is set on the horizontal circle. The first subchord (c1 / 20.34 feet) is measured from the SC, and a stake is set on line and marked for station 41+00.
(2) Set the spiral stations using their deflection angles in reverse order and the standard spiral chord length (25 feet). Correct any error encountered by adjusting the circular arc chords from the SC to the CS. Intermediate Setup When the instrument must be moved to an intermediate point on the spiral, the deflection angles computed from the TS cannot be used for the remainder of the spiral. In this respect, a spiral differs from a circular curve. Calculating Deflection Angles Following are the procedures for calculating the deflection angles and staking the spiral. Example: D = 4° Ls = 250 ft (for highways) V =60 mph I = 24°10’ Point 5 = intermediate point (1) Calculate the deflection angles for the first five 0points These angles are: a1 = 0° 0 01’, a2 =0 04’, a3 =0 09’, a4 =0° 16’, and a5 = 0°25’. (2) The deflection angles for points 6, 7,8,9, and 10, with the instrument at point 5, are
FM 5-233 calculated with the use of table 3-2. Table 3-2 is read as follows: with the instrument at any point, coefficients are obtained which, when multiplied by a1, give the deflection angles to the other points of the spiral. Therefore, with the instrument at point 5, the coefficients for points 6,7,8,9, and 10 are 16, 34, 54, 76, and 100, respectively. Multiply these coefficients by a1 to obtain the deflection angles. These angles are a6 = 16a1 =0° 16’, a7 = 34a1 =0034’, a8 = 54a1 = 0°54’, a9 = 76a1 = 1°16’, and a10 = 100a1 = 0 1 40’. (3) Table 3-2 is also used to orient the instrument over point 5 with a backsight
on the TS. The angular value from point 5 to point zero (TS) equals the coefficient from table 3-2 times a1. This angle equals 50a1= 0° 50’. Staking. Stake the first five points according to the procedure shown on page 3-33. Check point 5 by repetition to insure accuracy. Set up the instrument over point 5. Set the horizontal circle at the angular value determined above. With the telescope inverted, sight on the TS (point zero). Plunge the telescope, and stake the remainder of the curve (points 6, 7, 8, 9, and 10) by subtracting the deflection angles from 360 degrees.
FM 5-233 Field Notes for Spirals. Figure 3-18 shows a typical page of data recorded for the layout
of a spiral. The data were obtained from the calculations shown on page 3-31.
Section V. VERTICAL CURVES FUNCTION AND TYPES COMPUTATIONS When two grade lines intersect, there is a vertical change of direction. To insure safe and comfortable travel, the surveyor rounds off the intersection by inserting a vertical parabolic curve. The parabolic curve provides a gradual direction change from one grade to the next. A vertical curve connecting a descending grade with an ascending grade, or with one descending less sharply, is called a sag or invert curve. An ascending grade followed by a descending grade, or one ascending less sharply, is joined by a summit or overt curve.
In order to achieve a smooth change of direction when laying out vertical curves, the grade must be brought up through a series of elevations. The surveyor normally determines elevation for vertical curves for the beginning (point of vertical curvature or PVC), the end (point of vertical tangency or PVT), and all full stations. At times, the surveyor may desire additional points, but this will depend on construction requirements. Length of Curve The elevations are vertical offsets to the tangent (straightline design grade)
FM 5-233 elevations. Grades G1 and G2 are given as percentages of rise for 100 feet of horizontal distance. The surveyor identifies grades as plus or minus, depending on whether they are ascending or descending in the direction of the survey. The length of the vertical curve (L) is the horizontal distance (in 100-foot stations) from PVC to PVT. Usually, the curve extends ½ L stations on each side of the point of vertical intersection (PVI) and is most conveniently divided into full station increments.
is the terrain. The rougher the terrain, the smaller the station interval. The second consideration is to select an interval which will place a station at the center of the curve with the same number of stations on both sides of the curve. For example, a 300-foot curve could not be staked at 100-foot intervals but could be staked at 10-, 25-, 30-, 50-, or 75-foot intervals. The surveyor often uses the same intervals as those recommended for horizontal curves, that is 10, 25, 50, and 100 feet.
A sag curve is illustrated in figure 3-20. The surveyor can derive the curve data as follows (with BV and CV being the grade lines to be connected).
Since the PVI is the only fixed station, the next step is to compute the station value of the PVC, PVT, and all stations on the curve.
Determine values of G1 and G2, the original grades. To arrive at the minimum curve length (L) in stations, divide the algebraic difference of G1 and G2 (AG) by the rate of change (r), which is normally included in the design criteria. When the rate of change (r) is not given, use the following formulas to compute L: (Summit Curve)
PVC = PVI - L/2 PVT = PVI + L/2 Other stations are determined by starting at the PVI, adding the SI, and continuing until the PVT is reached. Tangent Elevations Compute tangent elevations PVC, PVT, and all stations along the curve. Since the PVI is the fixed point on the tangents, the surveyor computes the station elevations as follows: Elev PVC = Elev PVI + (-1 x L/2 x G1) Elev PVT = Elev PVI + (L/2 x G2) The surveyor may find the elevation of the stations along the back tangent as follows:
If L does not come out to a whole number of stations from this formula, it is usually extended to the nearest whole number. Note that this reduces the rate of change. Thus, L = 4.8 stations would be extended to 5 stations, and the value of r computed from r = These formulas are for road design only. The surveyor must use different formulas for railroad and airfield design.
Elev of sta = Elev of PVC + (distance from the PVC x G1).
Station Interval Once the length of curve is determined, the surveyor selects an appropriate station interval (SI). The first factor to be considered
Vertical Maximum The parabola bisects a line joining the PVI and the midpoint of the chord drawn between the PVC and PVT. In figure 3-19, line VE =
The elevation of the stations along the forward tangent is found as follows: Elev of sta = Elev of PVI + (distance from the PVI x G2)
DE and is referred to as the vertical maximum (Vm). The value of Vm is computed as follows: (L= length in 100-foot stations. In a 600-foot curve, L = 6.) Vm = L/8 (G2 - G1) or
Station Elevation. Next, the surveyor computes the elevation of the road grade at each of the stations along the curve. The elevation of the curve at any station is equal to the tangent elevation at that station plus or minus the vertical offset for that station, The sign of the offset depends upon the sign of Vm (plus for a sag curve and minus for a summit curve).
Vertical Offset = (Distance) x Vm
First and Second Differences. As a final step, the surveyor determines the values of the first and second differences. The first differences are the differences in elevation between successive stations along the curve, namely, the elevation of the second station minus the elevation of the first station, the elevation of the third station minus the elevation of the second, and so on. The second differences are the differences between the differences in elevation (the first differences), and they are computed in the same sequence as the first differences.
A parabolic curve presents a mirror image. This means that the second half of the curve is identical to the first half, and the offsets are the same for both sides of the curve.
The surveyor must take great care to observe and record the algebraic sign of both the first and second differences. The second differences provide a check on the rate of change
In practice, the surveyor should compute the value of Vm using both formulas, since working both provides a check on the Vm, the elevation of the PVC, and the elevation of the PVT. Vertical Offset. The value of the vertical offset is the distance between the tangent line and the road grade. This value varies as the square of the distance from the PVC or PVT and is computed using the formula: 2
FM 5-233 per station along the curve and a check on the computations. The second differences should all be equal. However, they may vary by one or two in the last decimal place due to rounding off in the computations. When this happens, they should form a pattern. If they vary too much and/or do not form a pattern, the surveyor has made an error in the computation. Example: A vertical curve connects grade lines G1 and G2 (figure 3-19). The maximum allowable slope (r) is 2.5 percent. Grades G1 and G2 are found to be -10 and +5.
The vertical offsets for each station are computed as in figure 3-20. The first and second differences are determined as a check. Figure 3-21 illustrates the solution of a summit curve with offsets for 50-foot intervals. High and Low Points The surveyor uses the high or low point of a vertical curve to determine the direction and amount of runoff, in the case of summit curves, and to locate the low point for drainage. When the tangent grades are equal, the high or low point will be at the center of the curve. When the tangent grades are both plus, the low point is at the PVC and the high point at the PVT. When both tangent grades are minus, the high point is at the PVC and the low point at the PVT. When unequal plus and
FM 5-233 minus tangent grades are encountered, the high or low point will fall on the side of the curve that has the flatter gradient. Horizontal Distance. The surveyor determines the distance (x, expressed in stations) between the PVC or PVT and the high or low point by the following formula:
Example: From the curve in figure 3-21, G1 = + 3.2%, G2 = - 1.6% L = 4 (400). Since G2 is the flatter gradient, the high point will fall between the PVI and the PVT.
G is the flatter of the two gradients and L is the number of curve stations. Vertical Distance. The surveyor computes the difference in elevation (y) between the PVC or PVT and the high or low point by the formula
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CC-MAIN-2023-06
| 58,921 | 201 |
http://www.dubai-forever.com/recruitment-agencies-in-dubai.html
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math
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There are over 5000 Recruitment Agencies in Dubai. Whew! That's quite a few by far.
Do all of them deliver results? Well, NO. They don't. Some deliver spectacular results. While most others do not.
Now, which ones should YOU apply to? Yeah, that's right. The Best of the Best. You DO want spectacular results, don't you?
Recruitment Dubai is HUGE, since 80% of residents in Dubai/ UAE are non-Emiratis, who come searching for a Lucrative Career & a Good Life from 200+ countries world-wide.
There are many top 10 lists on the Web. It's been our PASSION to provide a TRUE list of the TOP 20 Employment Agencies.
We've left no stone un-turned to make this list as genuine as possible. We rated agencies on the following criteria:
and many many more...
Here is the list of the TOP 20 Recruitment Agencies in Dubai.
Researched, compiled and published by our team of experts, with real feedback from job-seekers like you. And also, from Human Resources Managers from a cross-section of Corporates and Industries.
Have listed the registration links to make it easy for you. Click/ Copy-paste the link into a new browser window...
NOTE: A 5 rating is Excellent, 4 is Very Good, 3 is Good, 2 is Okay and 1 is Bad.
# 1 Recruitment Agencies in Dubai by far is: Bayt
Although not a recruiting agency, it is the leading job search network, hence make sure you CLICK the banner below and Register with all your details. I highly recommend it!
# 2 Enrollment Consultants in Dubai is: Work Circle
Once again this is an Important step in your job search. Click The Banner Below To Search For the Best Collection of Jobs in Dubai & UAE. I highly recommend this too...
The # 3 Staffing
Agencies in Dubai is: Stanton Chase
# 4 Recruitment Dubai Consultants is: Edge Executive
# 5 Employment Consultants in Dubai UAE is: First Select International
# 6 Job Recruitment Agency in Dubai is: Clarendon Parker (Middle-East)
# 7 Dubai Recruitment Agency is: ANOC Management Consultants
# 8 Dubai UAE Commissioning Agent is: Inspire Selection
# 9 Job Placement Agency in Dubai is: Purple Square
# 10 Dubai Recruitment Agency is: Engage Selection Dubai
The # 11 Recruitment
Agencies in Dubai is: Al Madina Agencies & Services
# 12 Dubai UAE Recruitment Agent is:
# 13 Recruitment Agencies in Dubai is: Adecco ME
# 14 Recruitment Dubai Consultants is: Cobalt Recruitment
# 15 Job Recruitment Agencies in Dubai is: Nadia Gulf
# 16 Job Recruitment Agency in Dubai is: ND & Associates
# 17 Employment Consultants in Dubai is: Apple Search
# 18 Recruitment Dubai Consultants is: Barclay Simpson
# 19 Dubai Recruiting Agency is: BAC Middle East
# 20 Dubai UAE Enlistment Agent is: Mind Field Resources
The above list is updated frequently (twice a month), so please return again for the HOTTEST TOP 10 / 20.
The above information will provide all the answers to the following queries:
Search This Site Using The Search Box Below:
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s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818685850.32/warc/CC-MAIN-20170919145852-20170919165852-00640.warc.gz
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CC-MAIN-2017-39
| 2,895 | 38 |
https://www.justanswer.com/multiple-problems/6uiia-show-work-a-work-progress-inbventory-account.html
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math
|
Show all work:
a) The work in progress inbventory account…
Show all work: a) The work...
Show all work:Submitted: 9 years ago.Category: Multiple Problems
a) The work in progress inbventory account of a manufacturing company shows a balance of 3,0000.00 at the end of an accounting period. The job-cost sheets of the two incomplete jobs show charges of 500.00 and 300.00 for direct materials, and charges of 400.00 and 600.00 for direct labor. From this info, it appears that the company is using predetermined overhead rate as a percentage of direct labor costs. What percentage is the rate?
B) The break-even point in dollar sales for Rice Company is 480,000.00 and the company's contribution margin ratio is 40 percent. If Rice Companydesires a profit of 84,0000.00, how much would sales have to total.
C)Williams Companys direct labor cost is 25 percent of its conversion cost. If the manufacturing overhead for the last period was 45,000 and the direct material cost was 25,000, how much is the direct labor cost?
D)Grading Company's cash and cash equivalents consist of cash and marketable securities. Last year teh companys cash count decreased by 16,000 ands its marketable securities account increased by 22,0000.00 Cash provided by operating activities was 20,000. Based on this info, was the net cash flow from investing activities onm the statement of cash flows a net increase or decrease? by how much?
E) Gladstone Footwear Cooperations flexible budget cost formula for supplies, a variable cost, is $2.82 per unit of output. The companys flexible budget performance report for last month showed an 8,140.00 unFAVORABLE SPENDING VARIANCE FOR SUPPLIES. During that month, 21,250 units were produced. Budgeted activity for the month had been 20,900 units. What is the actual cost per unit for indirect materials?
F)Lyons Company consists of 2 divisions, A and B. Lyons company reported a contribution margin of 60,000.00 for Division A, ansd had a contribution margin ratio of 30 percent in Division B, when sales in B were 240,000.00. Net operating income for the company was 22,000.00 and traceable fixed expenses were 45,000.00. How much were the Lyons Companys common fixed expenses?
G)Atlantic company produces a single product. For the most recent year, the companys net operating income computed by the absorption costing method was 7,800.00, and its net operating income computed by the variable costing method was 10,500.00. The companys unit product cost was 15.00 under variable costing and 24.00 under absorption costing. If the ending inventory consisted of 1460.00 unitsd, how many units must have been in the beginning inventory?
H)Black Company uses the weighted-average method in its process costing system. The companys ending work in process inventory consists of 6,000 units, 75 percent complete with respect to materials and 50 percent complete with respect to labor and overhead. If the total dollar value of the inventory is 80,000 and the cost per equivalent unit for labor and overhead is 6.00, what is the copst per equivalent unit for materials.
I)At overland company, maintenance cost is exclusively a variable cost that varies directly with machine-hours. The performance report for July showed that actual maintenace costs totaled 11,315 and that the associated rate variance was 146.00 unfavorable. If the 7,300 machine0-hours were actually worked during July, what is the budgeted maintenance cost per machine hour.
J)Teh cost of goods sold in a retail store totaled 650,000. Fixed selling and administrative expenses totaled 115,000 and variable selling and administrative expenses totaled 420,000. If teh stoers contribution margin totaled 590,000, how much were the sales.
K) Denny Coorperation is considering replacing a technologically obsolete machine with a new state of teh art numerically controlled machine. The new machine would cost 600,000 and would have a 10 hyear useful life. The new machine would have no salvage value. The new machine would cost 20,000 per year to operate and maintain, but would save 125,000 per year in labor and other costs. The old machine can be sold now for scrap for 50,000. What percentage is the simple rate of return on the new machine rounded to the nearest tenth percent?(ignore taxes in this problem)
L)Lounsberry inc regularly uses material O55P and currently has in stock 375 liters of the material, for wich it paid 2,700 several weeks ago. If this were to be sold as is on the open market as surplus material, it would fetch 6.35 per liter. New stocks of the material can be purchased on the market for 7.20 per liter, but it must be purchased in lots of 1,000 liters. Youve been asked to determine the relevent costs of 900 liters of the material to be used in a job for a customer. What is the relevant cost of the 900 liters of material O55P?
M)Harwichport company has a current ratio of 3.0 and an acid-test ratio of 2.8. Current assets equal 210,000, wich 5,000 consists of prepaid expenses. The remainder of current assets consists of cash, accounts recievable, marketable securiotis, and inventory. What is the amount of the companys inventory?
N) Tolla Company is estimating the following sales for the first six months of next year:
January-350,000, Feb-300,000, March-320,000, April 410,000, May-450,000, and June 470,000. Sale at tolla are normally collected as 70 percent in the month of sale, 25 percent in the month following the sale, and teh remaining 5 percent being uncollectable. Also, customers paying in the month of sale are given a 2 percent discount. Based on this info, how much cash should the company expect to collect during the month of april.
O)T cooperation has provided the following data from its activity based costing system:
Assembly- total cost-704,880-total activity 44,000 machine hours
Processing orders-totalcost 91,428-total activity 1900 orders
Inspection-total cost 117,546-total activity-1950 inspection-hours
The company makes 360 units of product P23F a year, rewuiring a total of 725 machine hours, 85 orders, and 45 inspection hours per year. The products direct materials cost is 42.30 per unit and its direct labor cost is 14.55 per unit. The product sells for 132.10 per unit. According to the activity based costing systrem, what is the product margin for the product P23F?
p)Williams company direct labor cost is 30 percent of its conversion cost. If the manufacturing overhead for the last period was 59,500 and the direct materials cost was 37,000. what is the direct labor cost?
Q)In a recent period 13,000 units were produced, and there was a favorable labor efficiency variance of 23,000. If 40,000 labor hours were worked and the standard wage rate was 13.00 per labor hour, what would the standard hours allowed per unit of output be?
R)The balance in White companys work in proxcess inventory account was 15,000 on august 1 and 18,000 on aug. 31. The company incurred 30,000 in direct labor cost during AUgust and requistioned 25,000 in raw materials. If the sum of the debits to the masnufacturing overhead account total 28,000 for the month, and if the sum of the credits totaled 30,000 then was Finished Goods debited or credited? by how much?
S)A company has provided the following data:
Sales-4000 units, sales price-80.00 per unit, Variable cost-50.00 per unit, Fixed cost 30,000.00.
If the dollar contribution margin unit is increased by 10 percent, total fixed cost is decreased by 15 percent and all other factors remain the same, will net operating income increase or decrease? by how much?
t) For the current year, Paxman Company incurred 175,000 in actual manufacturing overhead cost. The manufacturing overhead account showed that the overhead was overapplied in teh amount of 9,000 for the year. If the predetermined overhead rate was 8.00 per direct labor hour, how many hours were worked during the year?
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s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662539131.21/warc/CC-MAIN-20220521143241-20220521173241-00645.warc.gz
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CC-MAIN-2022-21
| 7,879 | 31 |
https://books.google.gr/books?id=2JA7AAAAcAAJ&pg=PA49&focus=viewport&vq=Note&dq=editions:UOMDLPabq7928_0001_001&lr=&hl=el&output=html_text
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math
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« ΠροηγούμενηΣυνέχεια »
Всок. Prop.13. Cor.1. A plane surface may be prolonged to any extent in a plane. Prop.13. Cor.2. If a straight line in a plane be prolonged, its prolongation lies
wholly in that same plane. Prop. 13. Nom.
If a given straight line in a plane be turned in that plane about one of its extremities which re
B mains at rest, till the straight line is returned to the situation from which it set out, the plane figure described by such straight line is called a circle, and its boundary the circumference. The point in which the extremity of the straight line remains at rest, is called the centre of the circle. Any straight line drawn from the centre of a circle to the circumference, is called a radius of the circle; and any straight line drawn through the centre and terminated both ways by the circumference, is called a diameter of the circle.
When a circle is said to be described about the centre A with the radius
AB, the meaning is, that it is described by the revolution of the
given straight line AB about the extremity A. Prop.13. Cor.3. A circle may be described about any centre and with any
radius. Prop.13. Cor. 4.
All the radii of the same circle are equal. And circles that
have equal radii, are equal. Prop.13. Cor.5.
A straight line from the centre of a circle to a point outside,
coincides with the circumference only in a point. Prop. 14.
Any three points are in the same plane. [That is to say,
one plane may be made to pass through them all.] Prop. 14. Cor.1.
Any three points which are not in the same straight line being joined, the straight lines which are the sides of the three-sided
figure that is formed lie all in one plane. Prop.14. Cor. 2.
Any two straight lines which proceed from the same point, lie
wholly in one plane. Prop.14. Cor.3. If three points in one plane (which are not in the same straight
line) are made to coincide with three points in another plane; the planes shall coincide throughout, to any extent to which they may be prolonged.
(The above Recapitulation contains the principal matters
likely to be referred to. But should reference be made to any thing that is not found in it, recourse is to be had to the Intercalary Book.)
BOOK continued from page 6.
which are not in one and the same straight line, and surfaces
called curved. XXXIV. Straight lines which proceed from the same point but
do not afterwards coincide, are said to be divergent.
XXXV. If through two divergent straight lines of unlimited • Interc.14. length a plane be* made or supposed to pass, and another straight Cor. 2.
line of unlimited length be turned about the point from which the two divergent straight lines proceed, continuing ever in the same plane with them, and so travel from the place of one to the place of the other ; such travelling straight line is called the
radius vectus. XXXVI. The plane surface (of unlimited extent in some direc
tions but limited in others) passed over by the radius vectus in
travelling from one of the divergent straight lines to the other, See Note.
is called the angle between them.
-C E of the straight lines between which they lie. Thus the angle between the straight lines BC and BA, is greater than that between BC and BD, or that between BD and BA; and is in fact equal to their sum. Also if the radius vectus instead of moving by the shortest road from BC to BA, should go round by the contrary way, the plane surface so passed over is likewise an angle. Such an angle may be called circuitous; and the other where
the radius vectus goes by the nearest road, direct. When several direct angles are at one point B, any one of them is ex
pressed by three letters, of which the letter that is at the vertex of the angle, (that is, at the point from which the straight lines that make the angle, proceed], is put in the middle, and one of the remaining letters is somewhere upon one of those straight lines, and the other upon the other. Thus the angle between the straight lines BA and BC, is named the angle ABC, or CBA ; that between BA and BD, is na ed the angle ABD, or DBA; and that between Figures in which a number of sides is specified or intimated, are
BD and BC, is named the angle DBC, or CBD. But if there be only one such angle at a point, it may be named from a letter placed at that point; as the angle at E, or more briefly still, the
angle E. If the angle intended is the circuitous one, it must be expressed by the
use of the term, or something equivalent. But whenever the contrary
is not expressed, it is always the direct angle that is meant. XXXVII. When a straight line standing on
another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle. And the straight line which stands on the other is called a perpendicular to it;
and is also said to be at right angles to it. XXXVIII. An angle greater than a right angle,
is called obtuse. XXXIX. An angle less than a right angle, is called
acute. XL. Angles greater or less than right angles, are called by the
common title of oblique. XLI. A straight line joining the ex
tremities of any portion of the circumference of a circle, is called a chord.
в А B The same name is applied to a straight line joining the extremities of any series of straight lines that approaches to the like form. The points A, B, where the chord meets the ends of the series, are called the cusps ; and the
angles A and B, the angles at the cusps. XLII. Figures which are bounded by straight lines, are called rectilinear. Linear figures of all kinds are understood to lie wholly in one plane,
when the contrary is not expressed. XLIII. Of rectilinear figures, such as are contained by three straight lines, are called triangles. XLIV. Those contained by four straight lines, are called quad
rilateral. XLV. Those contained by more than four, are called polygons.
always understood to be rectilinear, when the contrary is not expressed.
XLVI. Of triangles, such as have two sides equal,
are called isoskeles. XLVII. A triangle which has all its three sides equal,
belied a quienes which has all its three sides equal, A
Hence all equilateral triangles are at the same time isoskeles ; but
isoskeles triangles are not all equilateral. XLVIII. A triangle which has a right angle, is called right-angled. The side opposite to the right angle is called the hypotenuse.
The other two sides are sometimes called the base and perpendicular. XLIX. A triangle which has an obtuse angle, is
called obtuse-angled. L. A triangle which has all its angles acute, is called
acule-angled. LI. A triangle which has all its angles oblique, is called
oblique-angled. LII. The nomenclature of the various kinds of quadrilateral
figures cannot with propriety be given, till these figures have been shown to be capable of possessing certain properties from which their distinctions are derived. It is therefore to be found in the places where such properties are demonstrated. (See the Nomenclature at the end of Propositions XXVIII A, XXXIII,
and XXXIV bis, of the First Book. LIII. In any quadrilateral figure, a straight line joining two of thc opposite angular points is called a diagonal. For brevity, quadrilateral figures may be named by the letters at two
of their opposite angles, when no obscurity arises therefrom. LIV. Of polygons, such as have five, six, seven, eight, nine,
ten, eleven, twelve, and fifteen sides respectively, are called a pentagon, hexagon, heptagon, oktagon, enneagon, dekagon, hendekagon, dodekagon, pendekagon. For polygons with other numbers of sides, names might probably
be found, or be framed from the Greek; but they are not in
SCHOLIUM.—Henceforward all lines, angles, and figures linear or superficial, whether single or formed by the junction of many, will be understood to lie wholly in one plane, viz. the plane of the paper on which they are represented; when the contrary is not expressed.
See Note. PROBLEM.—To describe an equilateral triangle upon a given
Let AB be the given straight line. It is required to describe an equilateral triangle upon it.
About the centre A, with the radius AB, de-
scribe* the circle BCD; and about the centre B, (DA B E)
crossed by it. From a point in which the circles meet (as for +INTERC. 9. instance C) drawt the straight lines CA, CB, to the points A and Cor.
B. ABC shall be an equilateral triangle.
Because the point A is the centre of the circle BCD, and C INTERC13. and B are points in the circumference, AC ist equal to AB. And Cor. 4.
because the point B is the centre of the circle ACE, and C and A are points in the circumference, BC is equal to AB. But it has been shown that AC is equal to AB; therefore AC and BC
are each of them equal to AB. And things which are equal to the *INTERC. 1.
same, are* equal to one another; therefore AC is equal to BC. Wherefore AC, BC, AB are equal to one another, and the triangle ABC is equilateral; and it is described upon the given straight line AB. Which was to be done.
And by parity of reasoning, the like may be done in every other instance.
SCHOLIUM. - It has not yet been proved that the place where the two circles cross one another is only a point. There might, therefore, for all that has yet been proved, be more equilateral triangles than one, describable on the same side of AB. Which if it were possible (though it will hereafter be shown that it is not), would in no way affect the accuracy of the assertion that it has been shown how to construct an equilateral triangle upon AB.Referred back to, in the Scholium at the end of Prop. VII of the First Book.
PROPOSITION II. See Note. PROBLEM.–From a point assigned, to draw a straight line equal
to a given straight line.
First Case. Let A be the point assigned, and BC the given
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CC-MAIN-2021-49
| 9,782 | 80 |
https://blog.oureducation.in/calorific-value-computation/
|
math
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Calorific Value Computation Based on Ultimate Analysis data
DETERMINING HEATING VALUES OF FUELS:-
The heating values of a fuel can be determined either from chemical analysis or by burning a sample in a calorimeter.
In the former method the calculation is based on an ultimate analysis. It reduces the fuel to its elementary constituents of carbon, hydrogen, oxygen etc.
The proximate analysis determines the percentages of moisture and carbon.
The ultimate analysis when resolves the fuel into its elementary constituents doesn’t reveal how they combine in the fuel. It is given on both moist and dry fuel basis.
When an analysis is given on a moist fuel it is converted to dry basis by dividing the percentage of various constituents by one minus the percentage of moisture.
CALCULATION FROM AN ULTIMATE ANALYSIS
The most commonly used formula is the DULONG’S FORMULA.
Heat units in B.t.u per pound of dry fuel=
Where C is carbon,
H is hydrogen,
O is oxygen and
S is sulphur
Heating value per pound of dry coal=
= 14765 B.t.u
This method gives a satisfactory results. It may be made on a moist or dry basis. The method of converting from a moist to a dry basis is same as that of ultimate analysis.
To compute the efficiency of a boiler the heating value is determined with the help of a fuel calorimeter. In this apparatus the fuel is burned and the heat generated is absorbed by water.
The calorimeter which gives the best results is M.PIERRE.MAHLER.
If the result is not correct we use the PFAUNDLER’S METHOD.
It can be expressed as
Where C= correction in degree centigrade
N= number of intervals over which the correction is made
R= initial radiation
R’= final radiation
T= average temperature for the initial radiation
T’=average temperature for the final radiation
T”=average temperature over period of combustion
EXAMPLE 1. ASSUME A BLAST FURNACE GAS, THE ANALYSIS OF WHICH IN PERCENTAGE BY WEIGHT IS OXYGEN=2.7, CARBON MONOXIDE=19.5, CARBON DIOXIDE=18.7, NITROGEN= 59.1.
Sol:- Here the only combustible gas is the carbon monoxide and the heat value will be
0.195 x 4450 = 867.75 B.t.u per pound
The total volume of air required to burn one pound of this gas will be
0.195 x 30.6 = 5.967 cubic feet
EXAMPLE 2. ASSUME A NATURAL GAS THE ANALYSIS OF WHICH IN PERCENTAGE BY VOLUME IS OXYGEN= 0.40, CARBON MONOXIDE= 0.95, CARBON DIOXIDE= 0.34, OLEFIANT GAS(C2H4)= 0.66, ETHANE(C2H6)=3.55, MARSH GAS(CH4)= 72.15 AND HYDROGEN= 21.95.
Sol:- All except oxygen and carbon dioxide are combustible and the heat per cubic foot will be
From CO= 0.0095 x 347= 3.30
C2H4= 0.0066 x 1675= 11.05
C2H6= 0.0355 x 1862= 66.10
CH4= 0.7215 x 1050= 757.58
H= 0.2195 x 349= 76.61
B.t.u per cubic foot= 414.64
The total air required for combustion of one cubic foot of the gas will be
CO= 0.0095 x 2.39= 0.02
C2H4= 0.0066 x 14.33= 0.09
C2H6= 0.0355 x 16.74= 0.59
CH4= 0.7215 x 9.57= 6.90
H= 0.2195 x 2.41= 0.53
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499911.86/warc/CC-MAIN-20230201045500-20230201075500-00637.warc.gz
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CC-MAIN-2023-06
| 2,904 | 47 |
https://math.answers.com/Q/How_do_you_add_mixed_proper_and_improper_fractions
|
math
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Mixed numbers can be converted to improper fractions. Improper fractions can be added the same way proper fractions are.
Change them into mixed numbers and add the integers and fractions together ensuring that the fractions have a common denominator.
You need to learn how to convert from improper fractions to mixed fractions and vice versa; however, for most other operations, such as compare, add, subtract, multiply, and divide them, it really doesn't make much difference whether the fraction is proper or improper.
Convert them to improper fractions with common denominators and proceed.
Convert them to improper fractions, find a common denominator and proceed.
To find the sum of two mixed numbers, turn the mixed numbers into improper fractions (multiply the base with the denominator and add the numerator), then add the two fractions. To add the two fractions, find the LCD (lowest common denominator) and add the two numerators, but leave the denominators the same.
Express the mixed fraction as an improper fraction and then proceed as you would with ordinary fractions. If the answer is an improper fraction, then remember to convert to a mixed fraction.
First convert the mixed numbers into "top heavy (or "improper) fractions". Now multiply each of the improper fractions by each other - this makes the denominators the same. Now you can add both the fractions together (and cancel down if necessary).
Proper fractions are less than 1. Improper fractions are greater than 1. For a proper fraction to become an improper fraction, you would have to add a quantity that would make it greater than 1.
Convert them to improper fractions with common denominators and proceed with the adding and subtracting.
A mixed number can be converted into an improper fraction. Mixed numbers as improper fractions can be divided just like any other fraction. To convert a mixed number to an improper fraction multiply the whole number by the denominator and add the original numerator to give the new numerator and put this over the original denominator.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945315.31/warc/CC-MAIN-20230325033306-20230325063306-00369.warc.gz
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CC-MAIN-2023-14
| 2,053 | 11 |
https://ctan.org/pkg/rmthm
|
math
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rmthm – Use a roman font for theorem statements
Modifies the theorem environment of LaTeX 2.09 to cause the statement of a theorem to appear in \rm.
LaTeX 2e users should use a more general package (such as ntheorem) to support this requirement.
Maybe you are interested in the following packages as well.
- autobreak: Simple line breaking of long formulae
- shadethm: Theorem environments that are shaded
- newproof: Make commands to define proofs
- amsthm: Typesetting theorems (AMS style)
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s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578532050.7/warc/CC-MAIN-20190421180010-20190421202010-00256.warc.gz
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CC-MAIN-2019-18
| 493 | 8 |
http://lkml.iu.edu/hypermail/linux/kernel/0001.2/0598.html
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math
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Also sprach Andi Kleen:
} > And, btw, since now the number of users of vmalloc() in
} > performance-critical paths is growing, perhaps it is worth to come back to
} > the patch I submitted for review ages ago that adds vmlist_lock rw
} > spinlock and makes vmalloc() SMP-safe?
} I think vmalloc in every poll is totally out of question. The patch
} needs to be fixed to either allocate only once per process (ugly, eats
} lots of memory, needs infrastructure to free on memory pressure), or better
} rework the poll interface to support a linked list of poll blocks.
Maybe if vmalloc was the "thing to use" for allocations and have it call
kmalloc/gfp whenever necessary? Could be a big hassle, though.
Anyway, I tried implementing the linked-list version of the poll(). It
was wrought with problems. The code looked fine (I submitted an alternate
patch about it awhile ago...It should be in the archives), but the
poll()ing would simply poll over and over in some cases...
-- || Bill Wendling [email protected]
- To unsubscribe from this list: send the line "unsubscribe linux-kernel" in the body of a message to [email protected] Please read the FAQ at http://www.tux.org/lkml/
This archive was generated by hypermail 2b29 : Sun Jan 23 2000 - 21:00:18 EST
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s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690591.29/warc/CC-MAIN-20170925092813-20170925112813-00362.warc.gz
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CC-MAIN-2017-39
| 1,270 | 18 |
https://app.seesaw.me/activities/z0npps/lesson-23-subtracting-mixed-numbers-with-regrouping
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math
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1. Use the draw tool to rewrite your equations vertically, and show all of your thinking as you simplify the expressions.
2. Use the to explain each step needed to simplify ONE of the expressions.
Students will edit this template:
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573849.97/warc/CC-MAIN-20220819222115-20220820012115-00228.warc.gz
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CC-MAIN-2022-33
| 230 | 3 |
https://www.mountainbuzz.com/forums/f11/ww-kayak-sales-55062.html
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math
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WW kayak sales?
Can anyone clue me in on how many WW kayaks are sold each year? How many are sold in each category, freestyle, rive running play boat, river running/creek boat and creek boat? What's the pecking order in terms of company sales, obviously JK is 1st(do they really have 50+% of the market?) but who is 2nd, 3rd, ect.? Thanks
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s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986659097.10/warc/CC-MAIN-20191015131723-20191015155223-00538.warc.gz
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CC-MAIN-2019-43
| 338 | 2 |
https://www.skillsworkshop.org/maths?q=maths&f%5B0%5D=%3A48&f%5B1%5D=%3A50&f%5B2%5D=%3A164&f%5B3%5D=%3A952&f%5B4%5D=%3A2185&f%5B5%5D=solr_subjects_maths_numeracy%3A2017
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math
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Challenging set of questions based on an infographic from a recent TV Licensing annual review. Aimed at L1-2 but a few questions are Entry 2/3 - there are also a couple of "stretch questions" making a bridge to GCSE. Two main topics are covered: working with very large numbers and percentages. Extracting data, equivalents and ordering decimals are also touched upon.
FM Straightforward problem(s) with more than 1 step
FM Complex multi-step problem(s)
FM E3.21 Extract information from lists, tables, diagrams, charts; create frequency tables
FM L2.1 Read, write, order & compare positive & negative numbers of any size
FM L2.4 Identify & know the equivalence between fractions, decimals & percentages
FM L2.6 Calculate percentage change (any size increase & decrease), & original value after % change
FM L2.13 Calculate amounts of money, compound interest, percentage increases, decreases & discounts inc. tax & budgeting
This is a set of worksheets in a MS Excel workbook that deals with fractions.
- Sheet one: a set of pictures (pie charts) that shows fractions from halves to tenths
- Sheet two: equivalent fractions with two pie charts, learners can input fractions and see if they are equivalent by looking at the shape of the pie charts
- Sheet three: starting to look at fractions being equivalent to decimals with two pie charts one for fractions and one for decimals
Adult Numeracy N2/L2.2
Adult Numeracy N2/L1.3
GCSE N10 (Work interchangeably with terminating decimals & their corresponding fractions)
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s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662522270.37/warc/CC-MAIN-20220518115411-20220518145411-00254.warc.gz
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CC-MAIN-2022-21
| 1,515 | 15 |
http://web.pdx.edu/~wamserc/CH331F97/E2ans.htm
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math
|
1. (15 points) Write a complete IUPAC name for each of the following compounds, including designation of stereochemistry if it is specifically shown:
2. (15 points) Complete each of the following reactions by adding the missing part: either the starting compound, the necessary reagents and conditions, or the final major product. Indicate stereochemistry if it is specific.
(the second isomer may be more likely, since the first isomer would be expected to give a mix of two tertiary alcohols as products)
3. (10 points) The structure of D-fructose is shown below in a Fischer projection.
How many stereocenters are in this molecule ?
How many stereoisomers of D-fructose are there ?
8 (including D-fructose)
How many of these stereoisomers are meso compounds ?
How many of these stereoisomers are diastereomers of D-fructose ?
What is the relationship of D-fructose to the other stereoisomers (the ones that aren't diasteromers) ?
one is D-fructose itself and one is its enantiomer
4. (15 points) The pKa of CH3OH2+ is - 2.2 .
Write out the specific reaction that this refers to.
Write the Ka in terms of concentrations.
Knowing this pKa value, predict the preferred direction of the following equilibria:
Forward, since HCl is a stronger acid than CH3OH2+ .
Backward, since CH3OH2+ is a stronger acid than H3O+ .
5. (15 points) Write all the steps in the complete mechanism for the addition of HBr to 1-methylcyclohexene. Show electron-pushing arrows. Predict which is likely to be the rate-determining step.
The first step is most likely rate-determining, since it requires breaking two bonds and only making one. The second step is only one new bond making.
6. (15 points) Addition of HCl to 3-methyl-1-butene gives the expected Markovnikov product, but also gives some 2-chloro-2-methylbutane.
Explain by writing a complete mechanism that shows how both products could be formed.
Following Markovnikov's Rule, a 2° carbocation is initially formed. However, there is an adjacent 3° carbon, and the cation can be shifted there if a hydrogen atom (with 2 electrons) is shifted over to the 2° carbon. This is favorable since 3° carbocations are more stable than 2°.
The H shift will be more apparent if you don't use line structures (i.e., write out all atoms and bonds).
7. (15 points) Hydroxide ion could react with acetaldehyde (shown below) in two different ways.
Clearly show electron-pushing arrows to illustrate what happens in each reaction.
Describe each reaction type (i.e. , what has happened?)
(A) - acid-base reaction
(B) - addition reaction
Assume both reactions are favorable, and B is more favorable than A, yet A is a faster reaction than B. Draw an approximate potential energy diagram that would be consistent with this. Your diagram should show both reactions starting from the same place.
Reaction B is more exothermic than reaction A, yet A must have a lower energy of activation than B in order to be faster. Note that this necessarily requires that the potential energy diagrams must cross.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267868876.81/warc/CC-MAIN-20180625185510-20180625205510-00334.warc.gz
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CC-MAIN-2018-26
| 3,021 | 30 |
https://chartreusemodern.com/the-unit-circle/
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math
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The unit circle is a superb way to remember your trig values. It is something that we’re going to start talking about in Geometry, we’re going to talk about it in Algebra too, and you’re going to talk about it in pre-calc and probably a little bit of Calculus as well. To put it differently, it is nothing more than a circle with a bunch of Special Right Triangles. It is an essential math concept that every student needs to understand. So the secret to using the unit circle is to try to remember your radius will be and you can always drop an altitude or whether you’re down here raise an altitude so that you create a perfect triangle. In other words, it shows you all the angles that exist. Complete the unit circle to see whether you know all the crucial pieces.
Quiz yourself as to the information for a particular point, then click the point to see whether you’re correct. You will be supplied the quiz once we’ve got an opportunity to review the homework from this lesson. In almost no time whatsoever, you’ll be prepared for any upcoming unit circle quiz.
Be forewarned, everything in the majority of calculus classes will be completed in radians! The last step is for the students to discover what the x and y values are going to be on the unit circle. So, degrees are simpler to utilize in everyday work, but radians are far better for mathematics.
Remember the way the signals of angles do the job. You can accomplish this with more angles also. Therefore, if you’re able to address these angles you’ll be in a position to deal with the majority of the others. You would have to be in a position to measure an angle of 43 degrees from north to be able to stick to these directions correctly. Thus, to provide an extremely basic definition of angle, an individual could say it is a quantity that specifies direction in space.
An excellent illustration is offered by the formula for the duration of a circular arc. You’ll realize this in the subsequent examples. The exact same goes, obviously, for cosine and for tangent. Now, it’s a truth of arithmetic that there is not any number with denominator 0. In case you have any suggestions for improvement, please don’t be afraid to share them with me! The next step is simple using that which we’ve memorized, we can readily fix this issue. Among the problems with the majority of trig classes is they have a tendency to concentrate on right triangle trig and do everything concerning degrees.
Click each point as you go to determine if you’re right. The purpose of the unit circle is the fact that it makes other components of the mathematics simpler and neater. At this time you may be worried.
There aren’t lots of people we’d visit the other side of a circle for, but pi is among them. For instance, the below graph indicates the 45-45-90 degree Right Triangle in all four Quadrants. With these tricks in mind, the practice of the way to bear in mind the unit circle gets so much simpler! A good comprehension of the unit circle makes trigonometry much simpler to comprehend. You find the importance of this fact when you manage the trig functions for these angles.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314667.60/warc/CC-MAIN-20190819052133-20190819074133-00361.warc.gz
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CC-MAIN-2019-35
| 3,160 | 7 |
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