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https://philosophy.gr/presocratics/zeno.htm
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math
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Zeno of Elea
Life and Work
Zeno (b. c. 490 BC) was a pupil of Parmenides. Plato in his dialogue Parmenides testifies their relationship. Aristotle names Zeno as the inventor of dialectic. He wrote a book in which he denies physical motion as well as the unreality of the pluralistic world. His paradoxes of motion had a great influence in the history and philosophy of mathematics.
Denial of Motion and Plurality
Zeno’s well known hypothesis is that of the denial of motion and plurality. His arguments aim to support Parmenides’ position on the oneness and unity of Being. For Zeno, if reality is successively divided into parts then you will divide it ad infinitum. Zeno’s arguments are presented in the form of paradoxes:
1.The Racetrack or Dichotomy Paradox
Suppose a runner has to travel form the start point A to the finish point B. But firstly he has to travel to the midpoint C and thence to B. But if D is the midpoint of AC, he must first travel to D and so on ad infinitum. So since in finite time it is impossible to accomplish an infinite number of movements then the runner is not able to finish his distance.
2.The Paradox of Achilles and the Tortoise
Achilles runs a race with a tortoise. Tortoise takes a lead. But while Achilles can run much faster than the tortoise, Achilles cannot touch it. How? When Achilles has reached the tortoise’s starting-point the tortoise is n/10 meters ahead. When Achilles has reached that point is n/100 ahead and so on ad infinitum.
3.The Stadium or Moving Blocks Paradox
Suppose three equal groups A, B, C of width l, with A and C moving past B in opposite direction at the same speed. While group A takes time t to traverse width B, it takes t/2 to traverse width C. This leads to the absurd paradox that half time equals its double.
4.The Arrow Paradox The Arrow Paradox
Suppose that time consists of moments or instances. A flying arrow at any instant of time occupies a space equal to itself. So at any instant of time, like in a photograph, the arrow would be at rest. Therefore if at any instant of time the arrow has no motion, temporal locomotion is impossible since time is composed of freezing instances in succession.
Zeno's support of Parmenides: if there are many things then how many are they? (1), how big are they? (2), do they make a noise? (6), where are they? (4, 5) how can they move? (6)
1(3) how many? limited and unlimited in number
If there are many things (i) they will be just as many as they are, no more and no less; and if they are just as many as they are, they would be limited (in number).
If there are many things (ii) the things that there are are unlimited; for there will always be other things between the things that there are, and again other things between them, and so the things that there are are unlimited (in number).
2(1) how big? unlimited in size and no size at all
If what exists has no size (megethos) it would not exist. But if it does exist each thing must have some size (megethos) and thickness, and one part of it must be separate/distinct from another. And there is the same argument for what is in front, for that will have size and some part of it will be in front. Indeed, to say this once is similar to saying it for ever, for no such part of it will be the last or the same as a further part. So if there are many things they must be both small and great: so small as to have no size, and so big as to be unlimited (in size).
3(2) If it were added to something else, it would not make it any bigger; for if it has no size at all and were added on, it would not contribute any increase in size, and so what is now being added on would be nothing. And if, while it is being subtracted, the other will be no smaller It is obvious that what was added or subtracted was nothing.
4(4) where are they? in a place and nowhere at all
Where is all of what there is? if there is a place for the things that there are, where would it be? it would be in another place and that in another place and so on and so on. What moves does not move in the place in which it is or in the place in which it is not.
5(A24) If there is a place for the things that are, where would it be? it would be in another place and that in another place and so on and so on.
6(A29) do they make a noise? yes, some noise and no, no sound at all
Zeno: Tell me Protagoras, does a single millet seed make a noise as it falls, or does 1/10,000 of a millet? Protagoras: No.
Zeno: Does a bushel of millet seed make a noise as it falls, or not?
Protagoras: Yes, a bushel makes a noise.
Zeno: But isn't there a ratio (logos) between a bushel of millet seed, and one seed, and 1/10,000 of a seed? Protagoras: Yes, there is.
Zeno: So won't there be the same ratio of sounds between them, for the sounds are in proportion to what makes the sound? And if this is so, if the bushel of millet seed makes a noise so will a single seed and 1/10,000 of a seed.
7(A25-28 from Aristotle Physics 239b-240a) do things move? four puzzles
Zeno has four propositions (logoi) about movement which are puzzling for those who try to solve them:
(i) The Dichotomy (it is impossible to move from one place to another)
The first argument about there being no movement says that the moving object must first reach the half-way mark before the end (and the quarter-mark before the half and so back, so there is no first move; and the three-quarter mark after the half, and so forward so there is no last move).
(ii) The Achilles (Achilles cannot overtake the tortoise)
The second is the one called Achilles. This is it: the slowest will never be overtaken in running by the fastest, for the pursuer must always come to the point the pursued has left, so that the slower must always be some distance ahead.
(iii) The Arrow (the moving arrow is at rest)
The third one mentioned is that the moving arrow is at rest. [cf. Phys293b6: the arrow is at rest at any time when it occupies a space just its own length, and yet it is always moving at any time in its flight (i.e. in the 'now'), therefore the moving arrow is motionless.]
(iv) The Stadium (a time is twice itself)
The fourth is the one about equal blocks moving past equal blocks from opposite directions in the stadium - one set from the end of the stadium and one from the middle - at the same speed; here he thinks that half the time is equal to twice itself.
For example: AAAA are equal stationary blocks, BBBB, equal to them in number and size, are beginning from the half-way point (of the stadium), CCCC equal to these also in size, and equal to the Bs in speed, are coming towards them from the end. It happens of course that the first B reaches the end at the same time as the first C as they move past each other. And it happens that the C passes all the Bs but the Bs only half (the As) so the time is half (itself).
Translation M. R. Wright - note: numbers in parentheses refer to the standard Diels/Kranz order
Heraclitus of Ephesus
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https://en.wikipedia.org/wiki/Character_sum
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math
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In mathematics, a character sum is a sum of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform).
Assume χ is a nonprincipal Dirichlet character to the modulus N.
Sums over ranges
The sum taken over all residue classes mod N is then zero. This means that the cases of interest will be sums over relatively short ranges, of length R < N say,
Another significant type of character sum is that formed by
for some function F, generally a polynomial. A classical result is the case of a quadratic, for example,
More generally, such sums for the Jacobi symbol relate to local zeta-functions of elliptic curves and hyperelliptic curves; this means that by means of André Weil's results, for N = p a prime number, there are non-trivial bounds
The constant implicit in the notation is linear in the genus of the curve in question, and so (Legendre symbol or hyperelliptic case) can be taken as the degree of F. (More general results, for other values of N, can be obtained starting from there.)
Weil's results also led to the Burgess bound, applying to give non-trivial results beyond Pólya–Vinogradov, for R a power of N greater than 1/4.
Assume the modulus N is a prime.
for any integer r ≥ 3.
- Montgomery and Vaughan (1977)
- Burgess (1957)
- Montgomery and Vaughan (2007), p.315
- G. Pólya (1918). "Ueber die Verteilung der quadratischen Reste und Nichtreste". Nachr. Akad. Wiss. Goettingen: 21–29. JFM 46.0265.02. CS1 maint: discouraged parameter (link)
- I. M. Vinogradov (1918). "Sur la distribution des residus and nonresidus des puissances". J. Soc. Phys. Math. Univ. Permi: 18–28. JFM 48.1352.04. CS1 maint: discouraged parameter (link)
- D. A. Burgess (1957). "The distribution of quadratic residues and non-residues". Mathematika. 4 (02): 106–112. doi:10.1112/S0025579300001157. Zbl 0081.27101.
- Hugh L. Montgomery; Robert C. Vaughan (1977). "Exponential sums with multiplicative coefficients" (PDF). Invent. Math. 43 (1): 69–82. doi:10.1007/BF01390204. hdl:2027.42/46603. Zbl 0362.10036. CS1 maint: discouraged parameter (link)
- Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. Cambridge University Press. pp. 306–325. ISBN 0-521-84903-9. Zbl 1142.11001. CS1 maint: discouraged parameter (link)
- Korobov, N.M. (1992). Exponential sums and their applications. Mathematics and Its Applications (Soviet Series). 80. Translated from the Russian by Yu. N. Shakhov. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-1647-9. Zbl 0754.11022.
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CC-MAIN-2021-21
| 2,922 | 20 |
https://www.shaalaa.com/question-bank-solutions/ashish-studies-for-4-hours-5-hours-and-3-hours-respectively-on-three-consecutive-days-how-many-hours-does-he-study-daily-on-an-average-arithmetic-mean-raw-data_153282
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math
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Ashish studies for 4 hours, 5 hours, and 3 hours respectively on three consecutive days. How many hours does he study daily on average?
The average study time of Ashish would be
`"Total number of study hours"/"Number of days for which he studied" = (4 + 5 + 3)/3` hours = 4 hours per day
Thus, we can say that Ashish studies for 4 hours daily on an average.
Video Tutorials For All Subjects
- Arithmetic Mean - Raw Data
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CC-MAIN-2023-50
| 419 | 6 |
http://www.miataturbo.net/general-miata-chat-9/home-alignments-14318/page2/
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math
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Originally Posted by grippgoat
I thought of a way to measure camber that's indifferent to floor level-ness, as long as the floor is flat. Level floor doesn't really make a difference for toe, I don't think, unless it's way, way off.
Anyway, get a framing square (I think that's what it's called). Long leg needs to reach from the floor to the top of your rim. Set the short leg on the floor, pointing away from the tire. The long leg should now be perpendicular to the floor. Then measure to the bottom of your rim lip to the square, shooting for something fairly close, but not touching the tire, like maybe 1 inch. Call this B. Then do the same to the top lip of your rim, and call this T. Then, measure the vertical distance between the two points you measured, and call this D. Now just do some trig:
camber = DEGREES( ASIN( (T-B)/D ) )
For example, if B is 1.0" and T is 1.5", and D is 16", then camber is 1.79 degrees (well, it's negative, but you know that because the tire is tilted in). Given desired camber, and constant B and D measurements, you can even solve for the T measurement.
T = D * SIN(RADIANS(camber)) + B
So if you wanted 3 degrees, you'd get
1.837 = 16 * SIN(RADIANTS(3.0)) + 1
I'm too lazy to actually try it (Firestone lifetime alignment), but I tested the measurement technique after getting an alignment, and the numbers came out right. It may not be perfect, but it should be able to get you within a tenth or two of a degree.
Also note that if your tires are hella stretched so that you can butt the square right up against the rim, then your B measurement can always be 0, and the math just got easier.
Just a quick update on this.... I tried this technique after installing my new FCM bumpstops and dust boots. It worked rather well. Having a ruler that measures in 1/64" was nice, but not really necessary. On a 16" rim-to-rim wheel (my 15" heliums are 16" between lips where I measure), 1mm is 0.14 degrees. 1/64" is 0.06 degrees.
Here's the overall workflow I did:
0) Fire up excel to figure out what the delta between T and B needs to be for your desired camber. I was shooting for about -1.3, or 24/64".
1) get the car on the ground and settled.
2) Position the square so that B is either 1" or 2". With 0 camber, the square would hit the fender at B = 1"
3) Measure T, and subtract B from it.
4) Now figure how how much you need to change T in order to get your desired camber. In my case, B was 2" and T was 2-2/64", so T-B was 2/64. I needed to move it by about 22/64"
5) raise up the corner of the car, so that the top of the rim is up at the top of the square, and you can get at the camber adjustment bolt
6) re-measure with B at 1" or 2", and figure out T. Take the delta you need from step 4, and figure out what your target T is with the car up in the air. As it happened, with the car in the air and B at 1", T was at 1-50c/64. Since I needed to move 22/64", that meant my target T was 2-8/64".
7) Adjust camber and re-measure with the car in the air until you hit your target T.
8) Put the car back down, re-settle it, and double check. Should be within 1/64" or 2.
I think doing the rear would be a bitch, though, because you'd have to adjust both bolts equally. In the front, camber doesn't change too much with caster.
I also stringed up the car and discovered that I'm pretty good at eyeballing front toe.
I also found that the overall front and rear toe is close enough to get me to the alignment shop.
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http://docplayer.net/28198357-Topics-through-chapter-4.html
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math
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1 Topics through Chapter The Doppler Effect: this is how we learn about the motions of objects in the universe, discover extraterrestrial planets, black holes at the centers of galaxies, and the expansion of the entire universe. 4.1 Spectral Lines: appearance, excess or deficit of energy at particular wavelengths. (Don t worry about Kirchoff s laws, and the pictures with little prisms and light bulbs etc.--see notes) 4.2 Atoms and Radiation: quantized energy levels of electrons in atoms, and how they interact with radiation. The hydrogen atom--the simplest, and most important case. 4.3 The Formation of Spectral Lines: understanding emission lines and absorption lines 4.4 Molecules: not just electronic changes, but vibrational and rotational changes of energy. 4.5 Spectral-Line Analysis: Information from spectral linesabundances, turbulent motions, rotation,
2 3.5 The Doppler Effect The wavelength (or frequency) of a wave, as measured by an observer, depends on the relative radial speed of the source and observer. Radial motion means: motion towards or away; along the line of sight. The Doppler effect involves only this component of motion. What we get from measuring it is called the radial velocity. Moving away: wavelengths increase ( redshift ) Moving toward: wavelengths decrease ( blueshift )
3 3.5 The Doppler Effect Relationship between wavelength and speed: Shift in λ compared to rest (no motion) wavelength is proportional to radial velocity So if you know the exact wavelength where some feature in the spectrum should be ( true wavelength ), and the wavelength at which it appears ( apparent wavelength ), you can obtain the radial velocity. This is how we get speeds of cosmic objects, stars, galaxies, even expansion of universe. Actual formula is: λ(apparent)/ λ(true) = 1 ± (speed of object/speed of light) where the ± sign means it is + if it is moving away from us (redshift, longer wavelength), - if it is moving toward us (blueshift, shorter wavelength) This applies to any wave; and no reason not to use frequency instead of wavelength. Textbook writes it this way:
4 See textbook for this rather confusing example. In class we ll use a simpler example of water waves in a pond. 3.5 The Doppler Effect Important point: Doppler effect depends only on the relative motion of source and observer
5 More Precisely 3-3: Measuring Velocities with the Doppler Effect Example: For a speed of 30 km/s, the Doppler shift is given by speed of object This may seem small, but it is easily detectable with a radar gun. It is NOT so easy to detect from the spectrum of an astronomical object, unless you know something about spectral lines. speed of light
6 Look at the Doppler shift formula again: λ(apparent)/ λ(true) = 1 ± (radial velocity of object./speed of light) If velocity of object away or toward us is much less than the speed of light (true for almost all objects in the universe), the apparent wavelength will be only slightly different from the laboratory or rest wavelength. For most objects in the universe, this relative shift is so tiny, that we can t detect it using the shift of the whole continuous spectrum. But we can use places in the spectrum whose wavelengths are precisely known by the presence of spectral lines (the subject of Chapter 4)
7 Chapter 4 Spectroscopy Don t worry if you can t understand what this pretty picture represents, unless it is the day before the next exam. A more important question Is why the authors insist on showing this form of spectra without adequately explaining!
8 4.1 Spectral Lines Spectroscope: Splits light into component colors (wavelengths, frequencies) Most of this illustration is completely unnecessary! The only important point is that light from any object can be spread out into a rainbow of wavelengths. A spectrum is a picture of how much light is at each wavelength. This illustration is showing a continuous spectrum.
9 4.1 Spectral Lines Continuous spectrum: Continuous range of frequencies emitted by an object (something like the black bodies we discussed in ch.3) Emission lines: Single frequencies emitted by particular atoms in a hot gas Absorption lines: If a continuous spectrum passes through a cool gas, atoms of the gas will absorb the same frequencies they emit
10 Spectral lines Spectral lines very narrow, well-defined (in wavelength) wavelength/frequency regions in the spectrum where excess photon energy appears (emission lines) or else where photons are missing (absorption lines). Cartoon view of absorption lines, both in the spectrum as a graph (below), and in the recorded spectrum (top), the band of colors--this is just how the spectra are gathered--pay no attention to the rectangular shape! Often these lines are superimposed on a smooth, continuous spectrum, which is the near-blackbody emission of a heated object that we have been discussing so far (ch. 3, Wien, Stefan-Boltzmann).
11 Spectral of real astronomical objects Here are spectra of two real astronomical objects, a comet (top) and star (bottom). By the time you take the next exam, you should be able to explain why these look so different. The wavelengths, shapes, and strengths of these spectral lines are the keys to understanding many of the physical properties of planets, stars and galaxies.
12 Spectra are not black rectangles with vertical lines. They are a recording of how many photons per second are being emitted as a function of wavelength
13 4.1 Spectral Lines Emission spectrum can be used to identify elements An absorption spectrum can also be used to identify elements. These are the emission and absorption spectra of sodium:
14 Here is the Sun s spectrum, along with a blackbody of the sun s temperature (top--why are there no lines?), and the spectra of individual elements as observed in the laboratory. Each spectral line is a chemical fingerprint telling you which elements, and how much of each element, is contained in the object you are observing. Continuous spectrum Sun (absorption lines) Emission lines of various elements
15 Kirchoff s laws: don t memorize them. Instead, come back to this illustration later and find if you understand enough to explain them Kirchoff s laws are usually presented with prisms and striped colorful spectra. This is confusing, and they aren t even laws at all! Just note that prism is supposed to represent an instrument, called a spectrometer. Maybe a star, or a planet Maybe the atmosphere of a star or planet If the light from the hot star or blackbody doesn t pass through any low-density gas, then the spectrum is featureless--it is a continuous spectrum. (top) If that continuous spectrum passes through a cloud of gas that is cooler than the source, the cloud can absorb particular wavelengths, and you get an absorption spectrum (middle) But if the gas is hot, say at least a few thousand degrees, it will emit spectral lines on its own (bottom), I.e. emission lines. How does this occur? The answer lies in the structure of atoms.
16 Next : Energy levels of electrons in atoms
17 4.2 Atoms and Radiation Existence of spectral lines required new model of atom, so that only certain amounts of energy could be emitted or absorbed Bohr model had certain allowed orbits for electron
18 4.2 Atoms and Radiation Emission energies correspond to energy differences between allowed levels Modern model has electron cloud rather than orbit
19 4.3 The Formation of Spectral Lines Absorption spectrum: Created when atoms absorb photons of right energy for excitation Multielectron atoms: Much more complicated spectra, many more possible states Ionization changes energy levels
20 Energy levels in H and He
21 The light emitted has a wavelength corresponding to the the energy difference between the two electron energy levels. Spectral lines = electronic transitions
22 Spectral lines of hydrogen Energy levels of the hydrogen atom, showing two series of emission lines:lyman and Balmer The energies of the electrons in each orbit are given by: The emission lines correspond to the energy differences
23 4.3 The Formation of Spectral Lines Absorption of energy (either by a collision, or by absorbing a photon) can boost an electron to the second (or higher) excited state Two ways to decay: 1. To ground state 2. Cascade one orbital at a time
24 4.3 The Formation of Spectral Lines (a) Direct decay (b) Cascade
25 Absorption of light as it passes through an atmosphere
26 Emission lines can be used to identify atoms
27 4.4 Molecules Molecules can vibrate and rotate, besides having energy levels Electron transitions produce visible and ultraviolet lines Vibrational transitions produce infrared lines Rotational transitions produce radio-wave lines
28 4.4 Molecules Molecular spectra are much more complex than atomic spectra, even for hydrogen: (a) Molecular hydrogen(b) Atomic hydrogen
29 Information that can be gleaned from spectral lines: Chemical composition Temperature Radial velocity 4.5 Spectral-Line Analysis
30 Line broadening can be due to a variety of causes 4.5 Spectral-Line Analysis
31 4.5 Spectral-Line Analysis
32 The Doppler shift may cause thermal broadening of spectral lines 4.5 Spectral-Line Analysis
33 4.5 Spectral-Line Analysis Rotation will also cause broadening of spectral lines through the Doppler effect
34 Summary of Chapter 4 Spectroscope (spectrometer) splits light beam into component frequencies/wavelengths Continuous spectrum is emitted by solid, liquid, and dense gas Hot gas has characteristic emission spectrum Continuous spectrum incident on cool, thin gas gives characteristic absorption spectrum
35 Summary of Chapter 4 (cont.) Spectra can be explained using atomic models, with electrons occupying specific orbitals Emission and absorption lines result from transitions between orbitals Molecules can also emit and absorb radiation when making transitions between vibrational or rotational states
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https://www.math-shortcut-tricks.com/simple-interest-and-compound-interest-shortcut-tricks/
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math
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Simple Interest and Compound Interest Shortcut Tricks
Shortcut tricks on simple interest and compound interest are one of the most important topics in exams. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss this thing. Few examples on simple interest and compound interest shortcuts
First of all do a practice set on math of any exam. Choose any twenty math problems and write it down on a page. Do first ten maths using basic formula of this math topic. You also need to keep track of the time. Write down the time taken by you to solve those questions. Now read our examples on simple interest and compound interest shortcut tricks and practice few questions. After finishing this do remaining questions using simple interest and compound interest shortcut tricks. Again keep track of timing. This time you will surely see improvement in your timing. But this is not enough. You need to practice more to improve your timing more.
Few Important things to Remember
Math section in a competitive exam is the most important part of the exam. It doesn’t mean that other topics are not so important. You can get a good score only if you get a good score in math section. You can get good score only by practicing more and more. The only thing you need to do is to do your math problems correctly and within time, and this can be achieved only by using shortcut tricks. Again it does not mean that you can’t do maths without using shortcut tricks. You may have that potential to do maths within time without using any shortcut tricks.
But, so many people can’t do this. Here we prepared simple interest and compound interest shortcut tricks for those people. We always try to put all shortcut methods of the given topic. But if you see any tricks are missing from the list then please inform us. Your little help will help so many needy.
Now we will discuss some basic ideas of Simple Interest and Compound Interest. On the basis of these ideas we will learn trick and tips of shortcut simple interest and compound interest. If you think that how to solve simple interest and compound interest questions using simple interest and compound interest shortcut tricks, then further studies will help you to do so.
What is Interest?
When some one take up some money from other for the personal or commercial purpose we pay some additional money to him after a certain period of time is called Interest. So, we can also called this Interest as Simple Interest. And, this type of problem are given in Quantitative Aptitude which is a very essential paper in banking exam. So, below are some more example of Interest for your better practice.
Anything we learn in our school days was basics and that is well enough for passing our school exams. Now the time has come to learn for our competitive exams. For this we need our basics but also we have to learn something new. That’s where shortcut tricks are comes into action.
What is Principal?
So, the concept of Principal is very simple. When money borrow for a certain time period called Principal or Sum.
What is Amount?
The Addition of Simple Interest and Principal is called Amount.
Amount (A) = Simple Interest (SI) + Principal (P).
Simple Interest (SI) = Amount (A) – Principal (P).
What is Per annual means?
Per annual means for a year.
Few general terms of Interest
- P = Principal
- R = Rate per annual
- T = Number of years
Formulas Need to Remember
SI = ( P x T x R / 100 )
Here, P = Principal
R = Rate per annual
T = Number of years
In case of Simple Interest, Number of years and Rate per annul are given in question, then we can easily find the Principal or Sum.
P = ( SI x 100 / R x T )
Now, In case of Simple Interest, Number of years and Principal are given in question, then we can easily find the Rate per annual.
R = ( SI x 100 / P x T )
Find the simple interest of Rs.500/- for 5 years at 5% per annum.
Show Answer Show How to Solve Open Rough Workspace
SI = 500 x 5 x 5 / 100
Simple interest in 5 years is Rs.125/-.
In what time Rs.5000/- amounts to Rs.6000/- at 5% per annum at simple interest?
- 2 Years
- 4 Years
- 5 Years
- 7 Years
Show Answer Show How to Solve Open Rough Workspace
1000 = 5000 x 5 x T / 100
T = 4
So, in 4 years Rs.5000/- will increase to Rs.6000/-.
More Shortcut tricks on Simple and Compound Interest
- Simple Interest based question
- Rate % based question
- Principle or Sum based question
- Compound Interest shortcut tricks
- Find Compound Interest tricks
- Difference between CI and SI of Three Years question
Here, we provide few tricks on Simple interest and Compound interest. So, do visit this page to get updates on more Math Shortcut Tricks. You can also like our facebook page to get updates.
Also, if you have any question regarding this topic then please do comment on below section. You can also send us message on facebook.
i m a faculty of Quantative aptitude in a coaching institute in dehradun, i wants more tricks related to different topics ….. if any body have his own tricks or taken from somewhere else….. plz contact….. me ….or send it to my email id……[email protected]………….tricks should be ……. in a interval of 30 sec…..so that a question is to be solved under the interval of 30….sec…………………………………………………………………………………………….or can any one suggest me the books related to the tricks for aptitude or reasoning………
I am from MBIET (The Absolute Education Mentor). I am the HOD of Mathematics. Contact on provided email for any kind of help in education.
help me in getting the quantitative aptitude easily i am in search of jobs now and am unable to complete the compitative
how to manage time during solving questions…
I always get late while solving the questions??
take a timer try to solv problem within time at home and keep practicing which hlp u in exam well …..
plese can you help me for the shortcut tricks..want to clear ibps exam…if possible thn mail me
Sir am rajesh….my math level zero pls help me nd suggest how to improve own weakness……
sir i want to learn quantitative apptitude question for government exam ..so please give me any shortcuts trics and farmula due to that i can do easy and early
now i am doing B.E of E.C.E department final year but i want to study any thing special in mathematics as i am interested in mathematics i want to study and do job related to mathematics. can you please suggest any degree i want to study regarding maths after B.E and i want to settle well with that maths degree plz reply to my mail id [email protected]
thank u sir
i need some workout problems based on SI & CI with short cut tricks….i can solve all other problems except SI,CI, profit and loss….problems should be compitative level
Very nice this topic very esauful thank you very much
i want to know the time and distance sums , please help me
KNOW I PREPARE LIC AAO EXAM TELL ME SHORTCUT TRICS
can i get some maths tricks based on afcat exam 1 2016 whisch is going to be held on 21 feb, some problems like step problems for eg a man travellling from etc etc
Pls give a all shortcuts in maths for competitive exams
Respected Sir please give me some short trics and plan method to mage good score in quantative.
i need tricky document…..
may i have your genuine help. i am kishore gogoi from assam preparing for bank n upsc exams
sir plz send me shortcut formulaes of time and distant, time and work, trigonometery and simple and compound interest
sir please give any aptitude shortcut tricks for clearing bank exams.and also send study materials to my mail id.
please send me the short cut tricks soft copy if u have to my mail address( [email protected]”)since have to apply for jobs…kindly help me with materials…
Sir please can you help me for short tricks for solving IBPS exam questions fast.
yes sir pls help me on work and time and advance math trick
email id [email protected]
A certain sum amounts to rs 1725 at 15% per annum at simple Interst and rs 1800 in the same time at 20% per annum at simple interest
hello sir, how to solve simple interest and compound interest sums less than 30 sec. plz help me
please send easy tricks based on simple&compound interest
Suppose ex-SI for 5yr at 10% p.a for 800.
T*R=5*10=50% of capital I.e 50% of 800=400
Dear sir,plz send notes related to solve ibps po exam thanks.
can i get aptitude books questions related to ssb defense and afcat examinations
If u get the shortcut note plz send me….
Hello chandan sir i am neha doing prep..for banking pls give the trick to solved SI CI and other maths topic interval 30 sec…. and pls give the example also…Thanks in advance
pls provide quantitative app maths tricks pls help me. i have exam
follow… www.math-shortcut-tricks.com… best of luck for your exam
I have prepration for PO exam, I need more mathematical tricks for easy solve paper in exam.
also want tricks for
all above tricks for will usefull my exam….
Give d more examples in several types…
Please provide shortcut tricks for compound interest installment questions
i want th know wthether the icompound interest was directly proportional to the principal or not??????
If a man lends money at s bt he includes interest every six months to calculate principal ,if he charges 10% interest then wht is effective rate of interest ???
HI Please send me shortcut tricks about reasoning , quantative aptitude and data intrepretataion tricks because i m preparing for ibps exam
this is very important topic for me and given more short cuts so its very good
Sir plzzz help me in math plzzz send some tricks to my email id Plzzzzzzzzzzzzzzz,
My email id is [email protected]
Sir plzzzz send some fast tricks
Not bad but amazing……. 🙂
thanks for that
How to download these tricks. Because it is very essential.
send me plz tricky method quantitative aptitude intrest
I’m requested sair plzz send me sort tricks fore compaund interest
With a given rate of simple interest the ratio of principal and amount for a certain period of time is 4:5.after 3yrs with tthe same rate of interest the ratio of the principal and amount becomes 5:7.The rate of interest per annum is ???
let take p=100,
then, amount(a)=125(since ratio is 4:5)
then after 3 yr the ratio will b 5:7
now in 3 yr SI=140-125=15
sir please send me some short curts for compound interests.
C.I me EMI VALE QUESTION KI koi short trick ho to..plz help sir
Sir,please give me a shortcut to improve mathematics.
It is helpful but wt about d calculations they remain same,, we know d formula but calculations need tym..
Please change the background…. its getting difficult to read.
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math
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Dec 01, 2015 · Related with Math In Focus 3a Test - Houghton Mifflin Harcourt . Houghton Mifflin Harcourt Houghton Mifflin Harcourt Go Math (2,160 View) Houghton Mifflin Harcourt Houghton Mifflin Harcourt Go (1,605 View) Houghton Mifflin Harcourt Journeys Common Core 2014 And (1,265 View) Grade 5 - Houghton Mifflin Harcourt (3,831 View)
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Download CBSE class 9 Maths book Pdfs here. NCERT Class 9 Maths book written by the experts of mathematics after doing a lot of research on several topics. CBSE Class 9 students also realize the importance of NCERT book for Class 9 Math, for the main reason that the books show an insight into...Request PDF | Development of mathematics motivation scale: A preliminary exploratory study with a focus on secondary school students | The motivation for learning mathematics is an essential ... Activity Guide for Science, Mathematics, and Technology Education1 INTRODUCTION If you go to the country,far from city lights,you can see about 3,000 stars on a clear night. If your eyes were bigger, you could see many more stars.With a pair of binoc-ulars, an optical device that effectively enlarges the pupil of your eye by about 30
Offers online courses and workshops to help you teach math in a more inspiring manner. Math Made Easy (K-5) A series of inexpensive math workbooks by DK publishing, 200 pages long. Math Mammoth: Mastery-based, emphasizes number sense, mental math, and conceptual understanding. Math Mammoth can be a self-teaching curriculum for many students. Activity Guide for Science, Mathematics, and Technology Education1 INTRODUCTION If you go to the country,far from city lights,you can see about 3,000 stars on a clear night. If your eyes were bigger, you could see many more stars.With a pair of binoc-ulars, an optical device that effectively enlarges the pupil of your eye by about 30 Literacy and Numeracy Strategy. Focusing on the Fundamentals of Math A teacher's guide. This guide is intended to support teachers' ongoing efforts in building students' knowledge and skills in mathematics. Math in Focus Course B my.hrw.com Login: mstudent741 Password: math7 Math in Focus is divided into two books. Book A is devoted to the big ideas in proportional relationships, operations with rational numbers and algebra. These key topics are in the beginning of the school year so students have the whole year to master and review them. Book B is Nelson Mathematics 4: Nelson Math Focus 4: Nelson Mathematics 5: Nelson Math Focus 5: Nelson Mathematics 6: Nelson Math Focus 6: Nelson Mathematics 7: Nelson Math Focus 7: Nelson Mathematics 8: Nelson Math Focus 8 : Nelson Math Focus 9 : Quebec OTR's: Nelson Mathematics 3 Quebec Oct 10, 2014 · What is Math in Focus? Math in focus: • is a textbook series based on Singapore’s Ministry of Education’s Mathematics Framework. • is organized to teach fewer concepts at each level, but to teach them thoroughly. When a concept appears in a subsequent grade level, it is always at a higher or deeper level. Recommendations for Mathematics Instruction and Numeracy Activities This guide presents four recommendations and 21 related strategies for improving mathematics proficiency, including numeracy, for students in short‑term facilities. The recommendations focus on (1) engaging Oct 10, 2014 · What is Math in Focus? Math in focus: • is a textbook series based on Singapore’s Ministry of Education’s Mathematics Framework. • is organized to teach fewer concepts at each level, but to teach them thoroughly. When a concept appears in a subsequent grade level, it is always at a higher or deeper level.
Math Degree?Ó Luckily, my advisor had a good answer. And here I am. 4H[OLTH[PJPHUZ,SLJ[LK[V[OL5H[PVUHS(JHKLT` Carla D. Martin ( /+-2+*5*+$#)6*7)087 ) is an Assistant Professor of Mathematics at James Madison University. She worked for four years as a consultant before returning to graduate school in applied mathematics. She has served on The Mathematics paper contains 300 marks while the General Ability test holds 600 marks. The written test will be of a total of 900 marks. For every incorrect response, 0.33 marks will be deducted as negative marking in the examination. After clearing the written test...
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October 2020 – Invention Education (presentation slides [PDF, 13.13MB]) September 2020 – STEM Teacher Preparation (presentation slides [PDF, 3.5MB]) August 2020 – Cybersecurity Education (presentation slides [PDF, 10.5MB]) July 2020 – Early Math (presentation slides [PDF, 2.37MB]) June 2020 – Distance Learning February 2020 – STEM ... A brief tutorial showing students how to access the Digital Resources for Math in Focus. Mathematics Education Research Journal, Review of Educational Research, Pedagogies: An International Journal, and Contemporary Issues in Early Childhood. Suggestions or guidelines for practice must always be responsive to the educational and cultural context, and open to continuing evaluation. No. 19 in this Educational Practices Series presents an June 19th, 2018 - Math In Focus Grade 3 Answers Key Math In Focus Grade 3 Answers Key Title Ebooks Math In Focus Grade 3 Answers Key Category Kindle and eBooks PDF''correlation of math in focus™ houghton mifflin harcourt june 13th, 2018 - common core state standards for mathematics grade 2 3 math in focus is published by marshall We are TERC, a non-profit made up of teams of math and science education and research experts. We are dedicated to innovation and creative problem solving. We believe that STEM moves the world forward. And we exist to move STEM learning forward. Mathematics TEKS review committees began work on the mathematics TEKS in May 2011: Kindergarten through Grade 5 (PDF, 67KB) Grades 6-8 (PDF, 70KB) High School (PDF, 71KB) SBOE Members Nominated Mathematics Expert Reviewers in January 2011. The following list includes individuals appointed by the SBOE to serve as content expert reviewers for the ...
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CC-MAIN-2021-21
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https://unitchefs.com/astronomical-units/kilometers/
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math
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Astronomical Units to Kilometers - Convert au to km
- Astronomical Units
- Astronomical Units to Kilometers
Astronomical Units (au) Conversion
Astronomical unit is a unit of length which describes the average distance between the Earth and the sun and is used to measure distances in the Solar system. This unit is accepted by the SI system. The exact numbers vary as Earth orbits the Sun from a maximum (aphelion) to a minimum (perihelion). Since 2012, the Astronomical unit has been defined as exactly equal to 149597870700 meters or about 150 million kilometers (93 million miles).
Kilometers (km) Conversion
A kilometer is a metric unit of length or distance which equals exactly 1000 meters and about 0.62137119 mile, 1093.6133 yards, or 3280.8399 feet. It is an official measurement which signifies geographical distances in most of the world except for the United States (they use the mile). Even huge distances to the farthest galaxies are expressed in kilometers, not in higher multiples of the meter.
Popular Unit Conversions
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233510924.74/warc/CC-MAIN-20231001173415-20231001203415-00380.warc.gz
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https://www.hackmath.net/en/math-problem/4820
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math
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Karel saved SKK 12,000 in the savings bank at the beginning of the year. At the end of the year, the savings bank credited him with SKK 1,680. At what annual interest rate did Karel have the deposit in the savings bank?
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Peter put € 270 principal into the bank, and at the end of the year, the account was € 282. To what annual interest rate had Peter deposited money in the bank?
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https://www.doubtnut.com/question-answer/various-units-of-measurement-of-area-1528569
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math
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Please Enter a Question First
Various units of measurement o...
Updated On: 27-06-2022
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Aap ko kya acha nahi laga
study about various units of measurement of area video mein padenge ki area kya hai agar hamare pass koi bhi hai aur Humne use per hi figure Banaya what is region ki measurement ko Ham bolenge area Agar hamare pass square Hai uski Ek site 128 Baki side nibandh CM Hogi is where area jo hoga vah hoga 1 cm square what do you CM square the standard unit is standard unit m square Dam square to m square km square
area Mein Hamesha Kisi bhi unit ke Ham Koi Bhi Ho 12 ka Sahi hota hai ki ismein site-to-site multiply Karte Hain To Jab side to side Hoti Hai Jaise 1 m into one metre is unit we multiply ho jate latest 1 metre per 1 cm square and so on hamare pass airline Hai usmein DM CM mm units a m a chote Hote aur Dam hm kilometre M Se Badi unit se DM ko sentimeter Mein convert kar sakte hain aur is video Mein Aage Karenge kilometre hm aur Dam ko metre Mein converter sabse pahle Ham DM ko sentimeter Mein convert Karenge 1 dm equals to 10 cm right then 1 decimeter to metre into
centimetre into centimetre centimetre square and 1 dm into 1 decimeter is 1 dm to 1 dm square = 210 now Agaram convert Kare Dam ko metre me so one dam into one dam equals to write one dam into one dam is 1 Dam square and 10 metre into 10 metre is hundred metre square then 182 metre = 200 metre light Agar Ham Deccan distance AK ha m m Main To
1 and 2 in a zoo distance Hota Hai distance from 10 to 10 into 10 that is 10 metre which is hundred metre then 1 hectare metre into 1 hectare metre = 200 metre in 201 = 200 200 that is 1000 metre square now 1 km equals to 1000 distance is a chord and I'm smart card 10 into 10 into 10 1000 in 21 km equals to 1000 INR to 1000 M M M M right death 1002 1000
M update 10 ki power 6 ki prapt 60 10th Ki Party ka 1 km square equals to 10 ki power 6 metre square
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math
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"Solve the following. Show your solution. You may use separate sheet if needed
1.Jose stays on the eighth floor of the Supreme Hotel. He visited a friend and so he went three floors down. On what floor is he now? 2. Mrs. Cruz bought a refrigerator for P11,998 and a washing machine P6,363. How much did she spend for the two appliances? 3. The water tank of Mang Jose was full with 5,000 liters of water on Saturday morning. He used 500 liters that day. How much water was left on Sunday? 4. Darius is a farmer. During cold weather he checks the temperature hourly. Before dinner, he saw that the temperature was 8 degrees. During the next hour, the temperature fell 4 degrees. What was the temperature at that time?"
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http://sfpaperjehm.hyve.me/math-algebra-problems.html
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math
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Math algebra problems
Basic word problems a word problem in algebra is the equivalent of a story problem in math when you solved story problems in your math class you had to decide. Cheat sheets & tables algebra, trigonometry and calculus cheat sheets and a variety of tables class notes each class has notes available most of the classes have. Math, algebra problems, algebra worksheets, algebra i, algebra 2, kids, with algebra fun games, algebra topics, quizzes, printables, algebra worksheets, algebraic. Sample problems from intermediate algebra sample problems are under the links in the sample problems column and the corresponding review material is under the. Solving math problems using the services of math homework help companies. Free algebra 1 worksheets created with infinite algebra 1 printable in convenient pdf format.
Welcome to ixl's algebra 1 page practice math online with unlimited questions in more than 200 algebra 1 math skills. Free intermediate and college algebra questions and problems are presented along with answers and explanations worksheets are also included. Learn algebra 1 for free—linear equations, functions, polynomials, factoring, and more full curriculum of exercises and videos.
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https://mcreator.net/forum/73511/random-tick-not-working-stairs
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math
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Topic category: Troubleshooting, bugs, and solutions
So I'm trying to remake 1.17's Copper and the Oxidization Effect. I did this successfully with a regular block, it can fully oxidize. (It uses Random Tick to increase a value, that when it reaches a certain amount, it advances 1 stage.) However, I tried to repeat this with Stairs, yet for some reason they don't seem to work when Random Tick is used, only if it is not using Random Tick does it actually seem to work.
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https://www.bconews.com/what-is-the-final-current-while-charging-a-capacitor/
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math
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3. What is the final current while charging a capacitor? Explanation: The final current is almost equal to zero while charging a capacitor because the capacitor is charged up to the source voltage.
What is charging current of capacitor?
When an increasing DC voltage is applied to a discharged Capacitor, the capacitor draws what is called a “charging current” and “charges up”. … This transient response time T, is measured in terms of τ = R x C, in seconds, where R is the value of the resistor in ohms and C is the value of the capacitor in Farads.
How do you find the final charge of a capacitor?
The amount of charge that moves into the plates depends upon the capacitance and the applied voltage according to the formula Q=CV, where Q is the charge in Coulombs, C is the capacitance in Farads, and V is the potential difference between the plates in volts.
Does a capacitor pass current while charging?
Yes. For DC circuits, when a capacitor is charged or discharged, current is flowing into and out of it. For AC circuits, a capacitor can act almost like a “resistor” but instead it is called reactance.
Does a capacitor need current to charge?
1 Answer. The current when charging a capacitor is not based on voltage (like with a resistive load); instead it’s based on the rate of change in voltage over time, or ΔV/Δt (or dV/dt).
What is the maximum charge of the capacitor?
The formula for a capacitor discharging is Q=Q0e−tRC Where Q0 is the maximum charge.
What happens when a capacitor is fully charged?
When a capacitor is fully charged, no current flows in the circuit. This is because the potential difference across the capacitor is equal to the voltage source. (i.e), the charging current drops to zero, such that capacitor voltage = source voltage.
Are capacitors DC or AC?
Capacitor (also known as condenser) is a two metal plates device separated by an insulating medium such as foil, laminated paper, air etc. … Keep in mind that capacitor acts as an open circuit in DC i.e. it only operable at AC voltages.
Why does a capacitor stop charging?
Since the voltage across the capacitor approaches the voltage across the terminals, the electric field in the wires approaches zero, and so the current approaches zero. Therefore no more charge will flow to or from the plates of the capacitor.
How long does it take to charge a capacitor?
It takes about 15 seconds for the capacitor to charge.
How do capacitors affect current?
In effect, the current “sees” the capacitor as an open circuit. If this same circuit has an AC voltage source, the lamp will light, indicating that AC current is flowing through the circuit. … Thus, a capacitor lets more current flow as the frequency of the source voltage is increased.
Will a capacitor discharge on its own?
Will a Capacitor Discharge On Its Own? In theory, a capacitor will gradually lose its charge. A fully charged capacitor in an ideal condition, when disconnected, discharges to 63% of its voltage after a single time constant. Thus, this capacitor will discharge up to near 0% after 5 time constants.
What happens to a capacitor if too much voltage is supplied to the plates?
4 Answers. If the capacitor has a voltage across its plates and the supply is disconnected, the charge remains irrespective of the distance so, if distance increases (and capacitance falls) then voltage increases proportionally. If the plates are taken to an infinite distance, the voltage becomes infinite.
Does current matter for capacitors?
Current does not flow through a capacitor but voltage is stored in a capacitor and consequently store electrical energy across it’s plates wherein these plates are separated in between (sandwhiched) by a dielectric material or insulator.
What happens when a capacitor is connected to a battery?
A capacitor, when connected to a battery, conducts for awhile but short while after that it acts as an open circuit. … The flow of current from the battery stops as soon as the charge Q on the positive plate reaches the value Q = C × V.
How much energy can a capacitor store?
A 1-farad capacitor can store one coulomb (coo-lomb) of charge at 1 volt. A coulomb is 6.25e18 (6.25 * 10^18, or 6.25 billion billion) electrons. One amp represents a rate of electron flow of 1 coulomb of electrons per second, so a 1-farad capacitor can hold 1 amp-second of electrons at 1 volt.
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https://www.englishworksheetsland.com/ordinalnumbers.html
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math
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Anytime you find a series of items or things an ordinal number can be very helpful to find where it is in significance to anything other parameters around it. They are often referred to as Cardinal numbers because they fundamentally follow an order from one to the end of the span. The goal of any ordinal system is to create a sense of order to show which is the starting digit and which is the last. This compilation of printable worksheets promotes how to delineate the order in any system that is presented to a student.
There are two types of numbers. One is the ordinal number and other is the cardinal number. Students are first taught the concept of cardinal numbers because it is easier to understand. Cardinal numbers are numbers in their original form. For example, one, two, three, four, fifteen, nineteen, twenty-five, ninety-nine, etc. On the other hand, ordinal numbers are the numbers that indicate a position. They are used to represent the order of things or objects. For example, first, second, fourth, ninth, etc.
The main difference between the ordinal numbers and cardinal number is that the ordinal numbers are not used to indicate the quantity like the cardinal numbers. Rather, ordinal numbers indicate the position of a thing or an object's standing. They have specific writing. When you write the ordinal numbers in numerals, you always add two letters with them that is "st", "nd", "rd", and "th". Ordinal numbers have been used in the literature by many writers. Most of the titles of books are written in ordinal numbers, for example, The Second watch, by J.A. Jance and Henry the fourth, by Stuart J. Murphy.
Ordinal numbers have a specific purpose for their use. They show the order of things. With the help of ordinal numbers, one can easily understand positions. Ordinal numbers will tell the exact position of objects and things. They are mostly used in displaying collections. It becomes easier to mark things with the ordinal numbers. Other common uses of ordinal numbers are in mathematics, literature, sciences, and other such fields.
He came fifth in his class.
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Let me take you to a first-class shopping mall.
She intended to gain the first position in her batch.
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http://cageyconsumer.com/tpquiltd.html
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math
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The old double rolls had 560 of the 4.4 inch sheets, the new triple rolls have 600 of the 4.0 inch sheets. If you multiply these numbers out, you'll find out that the triple rolls are actually 2% shorter than the old double rolls. The combined effect of these changes is a price increase greater than 50%, assuming that the price per roll remains the same.
It would be unreasonable to expect that the price of toilet paper or any other product stays the same, but the problem with this increase is the deceptive way they've done it. Not only have they confused the issue by changing the size of the invididual toilet sheet, they have added to the confusion by measuring the size of toilet paper rolls in terms of a non-existent "regular roll".
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AOL is considering two proposals to overhaul it network infrastucture. They have recieved two bids. The first bid from Huawei will require a $17 million upfront investment and will generate $20 million in savings for AOL each year for the next 3 years. The second bid from Cisco requires $93 million upfront investment and will generate $60 million in savings each year for the next 3 years.a. What is the IRR for AOL associated with each bid?b. If the cost of capital for each investment is 19% what is the net present value (NPV) of each bid?Suppose cisco modifies its bid by offering a lease contract instead. Under the terms of the lease, AOL will pay $30million upfront and $35 million per year for the next 3 years. AOL savings will be the same as with Cisco’s original bid.c. What is the IRR of the Cisco bid now?d. What is the new NPV?e. What should AOL do?
https://papertowriters.com/wp-content/uploads/2020/07/Writerspng-300x62.png 0 0 admin https://papertowriters.com/wp-content/uploads/2020/07/Writerspng-300x62.png admin2020-05-06 16:55:522020-05-06 16:55:52Corporate Finance
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https://www.arxiv-vanity.com/papers/1402.5594/
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Testing Relativistic Gravity with Radio Pulsars
Before the 1970s, precision tests for gravity theories were constrained to the weak gravitational fields of the Solar system. Hence, only the weak-field slow-motion aspects of relativistic celestial mechanics could be investigated. Testing gravity beyond the first post-Newtonian contributions was for a long time out of reach.
The discovery of the first binary pulsar by Russell Hulse and Joseph Taylor in the summer of 1974 initiated a completely new field for testing the relativistic dynamics of gravitationally interacting bodies. For the first time the back reaction of gravitational wave emission on the binary motion could be studied. Furthermore, the Hulse-Taylor pulsar provided the first test bed for the orbital dynamics of strongly self-gravitating bodies.
To date there are a number of pulsars known, which can be utilized for precision test of gravity. Depending on their orbital properties and their companion, these pulsars provide tests for various different aspects of relativistic dynamics. Besides tests of specific gravity theories, like general relativity or scalar-tensor gravity, there are pulsars that allow for generic constraints on potential deviations of gravity from general relativity in the quasi-stationary strong-field and the radiative regime.
This article presents a brief overview of this modern field of relativistic celestial mechanics, reviews some of the highlights of gravity tests with radio pulsars, and discusses their implications for gravitational physics and astronomy, including the upcoming gravitational wave astronomy.
- 1 Introduction
- 2 Gravitational wave damping
- 3 Geodetic precession
- 4 The strong equivalence principle
- 5 Local Lorentz invariance of gravity
- 6 Local position invariance of gravity
- 7 A varying gravitational constant
- 8 Summary and Outlook
In about two years from now we will be celebrating the centenary of Einstein’s general theory of relativity. On November 25th 1915 Einstein presented his field equations of gravitation (without cosmological term) to the Prussian Academy of Science . With this publication, general relativity (GR) was finally completed as a logically consistent physical theory (“Damit ist endlich die allgemeine Relativitätstheorie als logisches Gebäude abgeschlossen.”). Already one week before, based on the vacuum form of his field equations, Einstein was able to show that his theory of gravitation naturally explains the anomalous perihelion advance of the planet Mercury . While in hindsight this can be seen as the first experimental test for GR, back in 1915 astronomers were still searching for a Newtonian explanation . In his 1916 comprehensive summary of GR , Einstein proposed three experimental tests:
Gravitational redshift (Einstein suggested to look for red-shift in the spectral lines of stars).
Light deflection (Einstein explicitly calculated the values for the Sun and Jupiter).
Perihelion precession of planetary orbits (Einstein emphasized the agreement of GR, with the observed perihelion precession of Mercury with a reference to his calculations in ).
Gravitational redshift, a consequence of the equivalence principle, is common to all metric theories of gravity, and therefore in some respect its measurement has less discriminating power than the other two tests . The first verification of gravitational light bending during the total eclipse on May 29th 1919 was far from being a high precision test, but clearly decided in favor of GR, against the Newtonian prediction, which is only half the GR value . In the meantime this test has been greatly improved, in the optical with the astrometric satellite HIPPARCOS , and in the radio with very long baseline interferometry [8, 9, 10]. The deflection predicted by GR has been verified with a precision of . An even better test for the curvature of spacetime in the vicinity of the Sun is based on the Shapiro delay, the so-called “fourth test of GR” . A measurement of the frequency shift of radio signals exchanged with the Cassini spacecraft lead to a confirmation of GR . Apart from the four “classical” tests, GR has passed many other tests in the Solar system with flying colors: Lunar Laser Ranging tests for the strong equivalence principle and the de-Sitter precession of the Moon’s orbit , the Gravity Probe B experiment for the relativistic spin precession of a gyroscope (geodetic and frame dragging) , and the Lense-Thirring effect in satellite orbits , just to name a few.
GR, being a theory where fields travel with finite speed, predicts the existence of gravitational waves that propagate with the speed of light and extract energy from (non-axisymmetric) material systems with accelerated masses . This is also true for a self-gravitating system, where the acceleration of the masses is driven by gravity itself, a question which was settled in a fully satisfactory manner only several decades after Einstein’s pioneering papers (see for an excellent review). This fundamental property of GR could not be tested in the slow-motion environment of the Solar system, and the verification of the existence of gravitational waves had to wait until the discovery of the first binary pulsar in 1974 . Also, all the experiments in the Solar system can only test the weak-field aspects of gravity. The spacetime of the Solar system is close to Minkowski space everywhere: To first order (in standard coordinates) the spatial components of the spacetime metric can be written as , where denotes the Newtonian gravitational potential. At the surface of the Sun one finds , while at the surface of a neutron star . Consequently, gravity experiments with binary pulsars, not only yielded the first tests of the radiative properties of gravity, they also took our gravity tests into a new regime of gravity.
To categorize gravity tests with pulsars and to put them into context with other gravity tests it is useful to introduce the following four gravity regimes:
Quasi-stationary weak-field regime: The motion of the masses is slow compared to the speed of light () and spacetime is only very weakly curved, i.e. close to Minkowski spacetime everywhere. This is, for instance, the case in the Solar system.
Quasi-stationary strong-field regime: The motion of the masses is slow compared to the speed of light (), but one or more bodies of the system are strongly self-gravitating, i.e. spacetime in their vicinity deviates significantly from Minkowski space. Prime examples here are binary pulsars, consisting of two well-separated neutron stars.
Highly-dynamical strong-field regime: Masses move at a significant fraction of the speed of light () and spacetime is strongly curved and highly dynamical in the vicinity of the masses. This is the regime of merging neutron stars and black holes.
Radiation regime: Synonym for the collection of the radiative properties of gravity, most notably the generation of gravitational waves by material sources, the propagation speed of gravitational waves, and their polarization properties.
Figure 1 illustrates the different regimes. Gravity regime G1 is well tested in the Solar system. Binary pulsar experiments are presently our only precision experiments for gravity regime G2, and the best tests for the radiative properties of gravity (regime GW)111 Gravitational wave damping has also been observed in a double white-dwarf system, which has an orbital period of just 13 minutes . This experiment combines gravity regimes G1 (note, ) and GW of figure 1.. In the near future, gravitational wave detectors will allow a direct detection of gravitational waves (regime GW) and probe the strong and highly dynamical spacetime of merging compact objects (regime G3). As we will discuss at the end of this review, pulsar timing arrays soon should give us direct access to the nano-Hz gravitational wave band and probe the properties of these ultra-low-frequency gravitational waves (regime GW).
1.1 Radio pulsars and pulsar timing
Radio pulsars, i.e. rotating neutron stars with coherent radio emission along their magnetic poles, were discovered in 1967 by Jocelyn Bell and Antony Hewish . Seven years later, Russell Hulse and Joseph Taylor discovered the first binary pulsar, a pulsar in orbit with a companion star . This discovery marked the beginning of gravity tests with radio pulsars. Presently, more than 2000 radio pulsars are known, out of which about 10% reside in binary systems . The population of radio pulsars can be nicely presented in a diagram that gives the two main characteristics of a pulsar: the rotational period and its temporal change due to the loss of rotational energy (see figure 2). Fast rotating pulsars with small (millisecond pulsars) appear to be particularly stable in their rotation. On long time-scales, some of them rival the best atomic clocks in terms of stability [23, 24]. This property makes them ideal tools for precision astrometry, and hence (most) gravity tests with pulsars are simply clock comparison experiments to probe the spacetime of the binary pulsar, where the “pulsar clock” is read off by counting the pulses in the pulsar signal (see figure 3). As a result, a wide range of relativistic effects related to orbital binary dynamics, time dilation and delays in the signal propagation can be tested. The technique used is the so-called pulsar timing, which basically consists of measuring the exact arrival time of pulses at the radio telescope on Earth, and fitting an appropriate timing model to these arrival times, to obtain a phase-connected solution. In the phase-connected approach lies the true strength of pulsar timing: the timing model has to account for every (observed) pulse over a time scale of several years, in some cases even several decades. This makes pulsar timing extremely sensitive to even tiny deviations in the model parameters, and therefore vastly superior to a simple measurement of Doppler-shifts in the pulse period. Table 1 illustrates the current precision capabilities of pulsar timing for various experiments, like mass determination, astrometry and gravity tests. We will not go into the details of pulsar observations and pulsar timing here, since there are numerous excellent reviews on these topics, for instance [25, 26], just to mention two. In this review we focus on the relativistic effects that play a role in pulsar timing observations, and how pulsar timing can be used to test gravitational phenomena in generic as well as theory-based frameworks.
1.2 Binary pulsar motion in gravity theories
While in Newtonian gravity there is an exact solution to the equations of motion of two point masses that interact gravitationally, no such exact analytic solution is known in GR. In GR, the two-body problem has to be solved numerically or on the basis of approximation methods. A particularly well established and successful approximation scheme, to tackle the problem of motion of a system of well-separated bodies, is the post-Newtonian approximation, which is based on the weak-field slow-motion assumption. However, to describe the motion and gravitational wave emission of binary pulsars, there are two main limitations of the post-Newtonian approximation that have to be overcome (cf. ):
Near and inside the pulsar (and its companion, if it is also a neutron star) the gravitational field is strong and the weak-field assumption no longer holds.
When it comes to generation of gravitational waves (of wavelength ) and their back-reaction on the orbit (of size and period ), the post-Newtonian approximation is only valid in the near zone (), and breaks down in the radiation zone () where gravitational waves propagate and boundary conditions are defined, like the ‘no incoming radiation’ condition.
The discovery of the Hulse-Taylor pulsar was a particularly strong stimulus for the development of consistent approaches to compute the equations of motion for a binary system with strongly self-gravitating bodies (gravity regime G2). As a result, by now there are fully self-consistent derivations for the gravitational wave emission and the damping of the orbit due to gravitational wave back-reaction for such systems. In fact, in GR, there are several independent approaches that lead to the same result, giving equations of motion for a binary system with non-rotating components that include terms up to 3.5 post-Newtonian order () [36, 37]. For the relative acceleration in the center-of-mass frame one finds the general form
where the coefficients and are of order , and are functions of , , , and the masses (see for explicit expressions). The quantity denotes the total mass of the system. At this level of approximation, these equations of motion are also applicable to binaries containing strongly self-gravitating bodies, like neutron stars and black holes. This is a consequence of a remarkable property of Einstein’s theory of gravity, the effacement of the internal structure [38, 35]: In GR, strong-field contributions are absorbed into the definition of the body’s mass.
In GR’s post-Newtonian approximation scheme, gravitational wave damping enters for the first time at the 2.5 post-Newtonian level (order ), as a term in the equations of motion that is not invariant against time-reversal. The corresponding loss of orbital energy is given by the quadrupole formula, derived for the first time by Einstein within the linear approximation, for a material system where the gravitational interaction between the masses can be neglected . As it turns out, the quadrupole formula is also applicable for gravity regime G2 of figure 1, and therefore valid for binary pulsars as well (cf. ).
In alternative gravity theories, the gravitational wave back-reaction, generally, already enters at the 1.5 post-Newtonian level (order ). This is the result of the emission of dipolar gravitational waves, and adds terms and to equation (1) [5, 39]. Furthermore, one does no longer have an effacement of the internal structure of a compact body, meaning that the orbital dynamics, in addition to the mass, depends on the “sensitivity” of the body, a quantity that depends on its structure/compactness. Such modifications already enter at the “Newtonian” level, where the usual Newtonian gravitational constant is replaced by a (body-dependent) effective gravitational constant . For alternative gravity theories, it therefore generally makes an important difference whether the pulsar companion is a compact neutron star or a much less compact white dwarf. In sum, alternative theories of gravity generally predict deviations from GR in both the quasi-stationary and the radiative properties of binary pulsars [40, 41].
At the first post-Newtonian level, for fully conservative gravity theories without preferred location effects, one can construct a generic modified Einstein-Infeld-Hoffmann Lagrangian for a system of two gravitationally interacting masses (pulsar) and (companion) at relative (coordinate) separation and velocities and :
where . The body-dependent quantities , and account for deviations from GR associated with the self-energy of the individual masses [5, 40]. In GR one simply finds , , and . There are various analytical solutions to the dynamics of (2). The most widely used in pulsar astronomy is the quasi-Keplerian parametrization by Damour and Deruelle . It forms the basis of pulsar-timing models for relativistic binary pulsars, as we will discuss in more details in Section 1.4.
Beyond the first post-Newtonian level there is no fully generic framework for the gravitational dynamics of a binary system. However, one can find equations of motion valid for a general class of gravity theories, like in where a framework based on multi-scalar-tensor theories is introduced to discuss tests of relativistic gravity to the second post-Newtonian level, or in where the explicit equations of motion for non-spinning compact objects to 2.5 post-Newtonian order for a general class of scalar-tensor theories of gravity are given.
1.3 Gravitational spin effects in binary pulsars
In relativistic gravity theories, in general, the proper rotation of the bodies of a binary system directly affects their orbital and spin dynamics. Equations of motion for spinning bodies in GR have been developed by numerous authors, and in the meantime go way beyond the leading order contributions (for reviews and references see, e.g., [45, 35, 46, 47]). For present day pulsar-timing experiments it is sufficient to have a look at the post-Newtonian leading order contributions. There one finds three contributions: the spin-orbit (SO) interaction between the pulsar’s spin and the orbital angular momentum , the SO interaction between the companion’s spin and the orbital angular momentum, and finally the spin-spin interaction between the spin of the pulsar and the spin of the companion .
Spin-spin interaction will remain negligible in binary pulsar experiments for the foreseeable future. They are many orders of magnitude below the second post-Newtonian and spin-orbit effects , and many orders of magnitude below the measurement precision of present timing experiments. For this reason, we will not further discuss spin-spin effects here.
For a boost-invariant gravity theory, the (acceleration-dependent) Lagrangian for the spin-orbit interaction has the following general form (summation over spatial indices )
where is the antisymmetric spin tensor of body [49, 35, 40]. The coupling function can also account for strong-field effects in the spin-orbit coupling. In GR . For bodies with negligible gravitational self-energy, one finds in the framework of the parametrized post-Newtonian (PPN) formalism222The PPN formalism uses 10 parameters to parametrize in a generic way deviations from GR at the post-Newtonian level, within the class of metric gravity theories (see for details). , a quantity that is actually most tightly constrained by the light-bending and Shapiro-delay experiments in the Solar system, which test [8, 9, 10, 12].
In binary pulsars, spin-orbit coupling has two effects. On the one hand, it adds spin-dependent terms to the equations of motion (1), which cause a Lense-Thirring precession of the orbit (for GR see [45, 50]). So far this contribution could not be tested in binary pulsar experiments. Prospects of its measurement will be discussed in the future outlook in Section 8. On the other hand it leads to secular changes in the orientation of the spins of the two bodies (geodetic precession), most importantly the observed pulsar in a pulsar binary [51, 45, 52]. As we discuss in more details in Section 3, a change in the rotational axis of the pulsar causes changes in the observed emission properties of the pulsar, as the line-of-sight gradually cuts through different regions of the magnetosphere.
As can be derived from (3), to first order in GR the geodetic precession of the pulsar, averaged over one orbit, is given by ()
where and .
It is expected that in alternative theories relativistic spin precession generally depends on self-gravitational effects, meaning, the actual precession may depend on the compactness of a self-gravitating body. For the class of theories that lead to the Lagrangian (3), equation (4) modifies to
where is the strong-field generalization of .
Effects from spin-induced quadrupole moments are negligible as well. For double neutron-star systems they are many orders of magnitude below the second post-Newtonian and spin-orbit effects, due to the small extension of the bodies . If the companion is a more extended star, like a white dwarf or a main-sequence star, the rotationally-induced quadrupole moment might become important. A prime example is PSR J00457319, where the quadrupole moment of the fast rotating companion causes a significant precession of the pulsar orbit . For all the binary pulsars discussed here, the quadrupole moments of pulsar and companion are (currently) negligible.
Finally, certain gravitational phenomena, not present in GR, can even lead to a spin precession of isolated pulsars, for instance, a violation of the local Lorentz invariance and a violation of the local position invariance in the gravitational sector, as we will discuss in more details in Sections 5 and 6.
1.4 Phenomenological approach to relativistic effects in binary pulsar observations
For binary pulsar experiments that test the quasi-stationary strong-field regime (G2) and the gravitational wave damping (GW), a phenomenological parametrization, the so-called ‘parametrized post-Keplerian’ (PPK) formalism, has been introduced by Damour and extended by Damour and Taylor . The PPK formalism parametrizes all the observable effects that can be extracted independently from binary pulsar timing and pulse-structure data. Consequently, the PPK formalism allows to obtain theory-independent information from binary pulsar observations by fitting for a set of Keplerian and post-Keplerian parameters.
The description of the orbital motion is based on the quasi-Keplerian parametrization of Damour & Deruelle, which is a solution to the first post-Newtonian equations of motion [42, 55]. The corresponding Roemer delay in the arrival time of the pulsar signals is
where the eccentric anomaly is linked to the proper time of the pulsar via the Kepler equation
The five Keplerian parameters , , , , and denote the orbital period, the orbital eccentricity, the projected semi-major axis of the pulsar orbit, the longitude of periastron, and the time of periastron passage, respectively. The post-Keplerian parameter is not separately measurable, i.e. it can be absorbed into other timing parameters, and the post-Keplerian parameter has not been measured up to now in any of the binary pulsar systems. The relativistic precession of periastron changes the the longitude of periastron according to
meaning, that averaged over a full orbit, the location of periastron shifts by an angle . The parameter is the corresponding post-Keplerian parameter. A change in the orbital period, due to the emission of gravitational waves, is parametrized by the post-Keplerian parameter . Correspondingly, one has post-Keplerian parameters for the change in the orbital eccentricity and the projected semi-major axis:
Besides the Roemer delay , there are two purely relativistic effects that play an important role in pulsar timing experiments. In an eccentric orbit, one has a changing time dilation of the “pulsar clock” due to a variation in the orbital velocity of the pulsar and a change of the gravitational redshift caused by the gravitational field of the companion. This so-called Einstein delay is a periodic effect, whose amplitude is given by the post-Keplerian parameter , and to first oder can be written as
For sufficiently edge-on and/or eccentric orbits the propagation delay suffered by the pulsar signals in the gravitational field of the companion becomes important. This so-called Shapiro delay, to first order, reads
where the two post-Keplerian parameters and are called range and shape of the Shapiro delay. The latter is linked to the inclination of the orbit with respect to the line of sight, , by . It is important to note, that for equation (12) breaks down and higher order corrections are needed. But so far, equation (12) is fully sufficient for the timing observations of known pulsars .
Concerning the post-Keplerian parameters related to quasi-stationary effects, for the wide class of boost-invariant gravity theories one finds that they can be expressed as functions of the Keplerian parameters, the masses, and parameters generically accounting for gravitational self-field effects (cf. equation (2)) [40, 5]:
plus from equation (5). Here we have listed only those parameters that play a role in this review. For a complete list and a more detailed discussion, the reader is referred to . The quantities and are related to the interaction of the companion with a test particle or a photon. The parameter accounts for a possible change in the moment of inertia of the pulsar due to a change in the local gravitational constant. In GR one finds , , and . Consequently
These parameters are independent of the internal structure of the neutron star(s), due to the effacement of the internal structure, a property of GR [38, 35]. For most alternative gravity theories this is not the case. For instance, in the mono-scalar-tensor theories of [57, 58], one finds333The mono-scalar-tensor theories of [57, 58] have a conformal coupling function . The Jordan-Fierz-Brans-Dicke gravity is the sub-class with , and .
where . The body-dependent quantities and denote the effective scalar coupling of pulsar and companion respectively, and where denotes the asymptotic value of the scalar field at spatial infinity. The quantity is related to the moment of inertia of the pulsar via . For a given equation of state, the parameters , , and depend on the fundamental constants of the theory, e.g. and in , and the mass of the body. As we will demonstrate later, these “gravitational form factors” can assume large values in the strong gravitational fields of neutron stars. Depending on the value of , this is even the case for a vanishingly small , where there are practically no measurable deviations from GR in the Solar system. In fact, even for , a neutron star, above a certain -dependent critical mass, can have an effective scalar coupling of order unity. This non-perturbative strong-field behavior, the so-called ‘‘spontaneous scalarization’’ of a neutron star, was discovered 20 years ago by Damour and Esposito-Farèse .
Finally, there is the post-Keplerian parameter , related to the damping of the orbit due to the emission of gravitational waves. We have seen above that in alternative gravity theories the back reaction from the gravitational wave emission might enter the equations of motion already at the 1.5 post-Newtonian level, giving rise to a . To leading order one finds in mono-scalar-tensor gravity the dipolar contribution from the scalar field [59, 60, 58]:
As one can see, the change in the orbital period due to dipolar radiation depends strongly on the difference in the effective scalar coupling . Binary pulsar systems with a high degree of asymmetry in the compactness of their components are therefore ideal to test for dipolar radiation. An order unity difference in the effective scalar coupling would lead to a change in the binary orbit, which is several orders of magnitude () stronger than the quadrupolar damping predicted by GR.
At the 2.5 post-Newtonian level (), in general, there are several contributions entering the calculation:
Monopolar waves for eccentric orbits.
Higher order contributions to the dipolar wave damping.
Quadrupolar waves from the tensor field, and the fields that are also responsible for the monopolar and/or dipolar waves.
Apart from a change in the orbital period, gravitational wave damping will also affect other post-Keplerian parameters. While gravitational waves carry away orbital energy and angular momentum, Keplerian parameters like the eccentricity and the semi-major axis of the pulsar orbit change as well. The corresponding post-Keplerian parameters are and respectively. However, these changes affect the arrival times of the pulsar signals much less than the , and therefore do (so far) not play a role in the radiative tests with binary pulsars.
As already mentioned in Section 1.2, there is no generic connection between the higher-order gravitational wave damping effects and the parameters , , and of the modified Einstein-Infeld-Hoffmann formalism. Such higher order, mixed radiative and strong-field effects depend in a complicated way on the structure of the gravity theory .
The post-Keplerian parameters are at the foundation of many of the gravity tests conducted with binary pulsars. As shown above, the exact functional dependence differs for given theories of gravity. A priori, the masses of the pulsar and the companion are undetermined, but they represent the only unknowns in this set of equations. Hence, once two post-Keplerian parameters are measured, the corresponding equations can be solved for the two masses, and the values for other post-Keplerian parameters can be predicted for an assumed theory of gravity. Any further post-Keplerian measurement must therefore be consistent with that prediction, otherwise the assumed theory has to be rejected. In other words, if post-Keplerian parameters can be measured, a total of independent tests can be performed. The method is very powerful, as any additionally measured post-Keplerian parameter is potentially able to fail the prediction and hence to falsify the tested theory of gravity. The standard graphical representation of such tests, as will become clear below, is the mass-mass diagram. Every measured post-Keplerian parameter defines a curve of certain width (given by the measurement uncertainty of the post-Keplerian parameter) in a - diagram. A theory has passed a binary pulsar test, if there is a region in the mass-mass diagram that agrees with all post-Keplerian parameter curves.
2 Gravitational wave damping
2.1 The Hulse-Taylor pulsar
The first binary pulsar to ever be observed happened to be a rare double neutron star system. It was discovered by Russell Hulse and Joseph Taylor in summer 1974 . The pulsar, PSR B1913+16, has a rotational period of 59 ms and is in a highly eccentric () 7.75-hour orbit around an unseen companion. Shortly after the discovery of PSR B1913+16, it has been realized that this system may allow the observation of gravitational wave damping within a time span of a few years [63, 64].
The first relativistic effect seen in the timing observations of the Hulse-Taylor pulsar was the secular advance of periastron . Thanks to its large value of 4.2 deg/yr, this effect was well measured already one year after the discovery . Due to the, a priori, unknown masses of the system, this measurement could not be converted into a quantitative gravity test. However, assuming GR is correct, equation (17) gives the total mass of the system. From the modern value given in table 2 one finds .444Strictly speaking, this is the total mass of the system scaled with an unknown Doppler factor , i.e. . For typical velocities, is expected to be of order , see for instance . In gravity tests based on post-Keplerian parameters, the factor drops out and is therefore irrelevant .
It took a few more years to measure the Einstein delay (11) with good precision. In a single orbit this effect is exactly degenerate with the Roemer delay, and only due to the relativistic precession of the orbit these two delays become separable [63, 67]. By the end of 1978, the timing of PSR B1913+16 yielded a measurement of the post-Keplerian parameter , which is the amplitude of the Einstein delay . Together with the total mass from , equation (18) can now be used to calculate the individual masses. With the modern value for from table 2, and the total mass given above, one finds the individual masses and for pulsar and companion respectively .
With the knowledge of the two masses, and , the binary system is fully determined, and further GR effects can be calculated and compared with the observed values, providing an intrinsic consistency check of the theory. In fact, Taylor et al. reported the measurement of a decrease in the orbital period , consistent with the quadrupole formula (26). This was the first proof for the existence of gravitational waves as predicted by GR. In the meantime the is measured with a precision of 0.04% (see table 2). However, this is not the precision with which the validity of the quadrupole formula is verified in the PSR B1913+16 system. The observed needs to be corrected for extrinsic effects, most notably the differential Galactic acceleration and the Shklovskii effect, to obtain the intrinsic value caused by gravitational wave damping [69, 70]. The extrinsic contribution due to the Galactic gravitational field (acceleration g) and the proper motion (transverse angular velocity in the sky ) are given by
where is the unit vector pointing towards the pulsar, which is at a distance from the Solar system. For PSR B1913+16, and are measured with very high precision, and also is known with good precision (%). However, there is a large uncertainty in the distance , which is also needed to calculate the Galactic acceleration of the PSR B1913+16 system, , in equation (27). Due to its large distance, there is no direct parallax measurement for , and estimates for are based on model dependent methods, like the measured column density of free electrons between PSR B1913+16 and the Earth. Such methods are known to have large systematic uncertainties, and for this reason the distance to PSR B1913+16 is not well known: kpc [71, 31]. In addition, there are further uncertainties, e.g. in the Galactic gravitational potential and the distance of the Earth to the Galactic center. Accounting for all these uncertainties leads to an agreement between and at the level of about . The corresponding mass-mass diagram is given in figure 4. As the precision of the radiative test with PSR B1913+16 is limited by the model-dependent uncertainties in equation (27), it is not expected that this test can be significantly improved in the near future.
Finally, besides the mass-mass diagram, there is a different way to illustrate the test of gravitational wave damping with PSR B1913+16. According to equation (7), the change in the orbital period, i.e. the post-Keplerian parameter , is measured from a shift in the time of periastron passage, where is a multiple of . One finds for the shift in periastron time, as compared to an orbit with zero decay
where denotes the number of the periastron passage, and is given by . Equation (28) represents a parabola in time, which can be calculated with high precision using the masses that come from and (see above). On the other hand, the observed cumulative shift in periastron can be extracted from the timing observations with high precision. A comparison of observed and predicted cumulative shift in the time of the periastron passage is given in figure 5.
2.2 The Double Pulsar — The best test for Einstein’s quadrupole formula, and more
In 2003 a binary system was discovered where, at first, one member was identified as a pulsar with a 23 ms period . About half a year later, the companion was also recognized as a radio pulsar with a period of 2.8 s . Both pulsars, known as PSRs J07373039A and J07373039B, respectively, (or and hereafter), orbit each other in less than 2.5 hours in a mildly eccentric () orbit. As a result, the system is not only the first and only double neutron star system where both neutron stars are visible as active radio pulsars, but it is also the most relativistic binary pulsar laboratory for gravity known to date (see figure 6). Just to give an example for the strength of relativistic effects, the advance of periastron, , is 17 degrees per year, meaning that the eccentric orbit does a full rotation in just 21 years. In this subsection, we briefly discuss the properties of this unique system, commonly referred to as the Double Pulsar, and highlight some of the gravity tests that are based on the radio observations of this system. For detailed reviews of the Double Pulsar see [74, 75].
In the Double Pulsar system a total of six post-Keplerian parameters have been measured by now. Five arise from four different relativistic effects visible in pulsar timing , while a sixth one can be determined from the effects of geodetic precession, which will be discussed in detail in Section 3.2 below. The relativistic precession of the orbit, , was measured within a few days after timing of the system commenced, and by 2006 it was already known with a precision of 0.004% (see table 3). At the same time the measurement of the amplitude of Einstein delay, , reached 0.7% (see table 3). Due to the periastron precession of 17 degrees per year, the Einstein delay was soon well separable from the Roemer delay. Two further post-Keplerian parameters came from the detection of the Shapiro delay: the shape and range parameters and . They were measured with a precision of 0.04% and 5%, respectively (see table 3). From the measured value () one can already see how exceptionally edge-on this system is.555The only binary pulsar known to be (most likely) even more edge-on is PSR J16142230 with () . For this wide-orbit system (), however, no further post-Keplerian parameter is known that could be used in a gravity test. Finally, the decrease of the orbital period due to gravitational wave damping was measured with a precision of 1.4% just three years after the discovery of the system (see table 3).
A unique feature of the Double Pulsar is its nature as a ‘‘dual-line source’’, i.e. we measure the orbits of both neutron stars at the same time. Obviously, the sizes of the two orbits are not independent from each other as they orbit a common center of mass. In GR, up to first post-Newtonian order the relative size of the orbits is identical to the inverse ratio of masses. Hence, by measuring the orbits of the two pulsars (relative to the centre of mass), we obtain a precise measurement of the mass ratio. This ratio is directly observable, as the orbital inclination angle is obviously identical for both pulsars, i.e.
This expression is not just limited to GR. In fact, it is valid up to first post-Newtonian order and free of any explicit strong-field effects in any Lorentz-invariant theory of gravity (see for a detailed discussion). Using the parameter values of table 3, one finds that in the Double Pulsar the masses are nearly equal with .
As it turns out, all the post-Keplerian parameters measured from timing are consistent with GR. In addition, the region of allowed masses agrees well with the measured mass ratio (see figure 7). One has to keep in mind, that the test presented here is based on data published in 2006 . In the meantime continued timing lead to a significant decrease in the uncertainties of the post-Keplerian parameters of the Double pulsar. This is especially the case for , for which the uncertainty typically decreases with , being the total time span of timing observations. The new results will be published in an upcoming publication (Kramer et al., in prep.). As reported in , presently the Double Pulsar provides the best test for the GR quadrupole formalism for gravitational wave generation, with an uncertainty well below the 0.1% level. As discussed above, the Hulse-Taylor pulsar is presently limited by uncertainties in its distance. This raises the valid question, at which level such uncertainties will start to limit the radiative test with the Double Pulsar as well. Compared to the Hulse-Taylor pulsar, the Double Pulsar is much closer to Earth. Because of this, a direct distance estimate of kpc based on a parallax measurement with long-baseline interferometry was obtained . Thus, with the current accuracy in the measurement of distance and transverse velocity, GR tests based on can be taken to the 0.01% level. We will come back to this in Section 8, where we discuss some future tests with the Double Pulsar.
With the large number of post-Keplerian parameters and the known mass ratio, the Double Pulsar is the most over-constrained binary pulsar system. For this reason, one can do more than just testing specific gravity theories. The Double Pulsar allows for certain generic tests on the orbital dynamics, time dilation, and photon propagation of a spacetime with two strongly self-gravitating bodies . First, the fact that the Double Pulsar gives access to the mass ratio, , in any Lorentz-invariant theory of gravity, allows us to determine and . With this information at hand, the measurement of the shape of the Shapiro delay can be used to determine via equation (16): . At this point, the measurement of the post-Keplerian parameters , , and (equations (13), (14), (15)) can be used to impose restrictions on the “strong-field” parameters of Lagrangian (2) :
This is in full agreement with GR, which predicts one for all three of these expressions. Consequently, nature cannot deviate much from GR in the quasi-stationary strong-field regime of gravity (G2 in figure 1).
2.3 PSR J1738+0333 — The best test for scalar-tensor gravity
The best “pulsar clocks” are found amongst the fully recycled millisecond pulsars, which have rotational periods less than about 10 ms (see e.g. ). A result of the stable mass transfer between companion and pulsar in the past — responsible for the recycling of the pulsar — is a very efficient circularization of the binary orbit, that leads to a pulsar-white dwarf system with very small residual eccentricity . For such systems, the post-Keplerian parameters and are generally not observable. There are a few cases where the orbit is seen sufficiently edge-on, so that a measurement of the Shapiro delay gives access to the two post-Keplerian parameters and with good precision (see e.g. , which was the first detection of a Shapiro delay in a binary pulsar). With these two parameters the system is then fully determined, and in principle can be used for a gravity test in combination with a third measured (or constrained) post-Keplerian parameter (e.g. ). Besides the Shapiro delay parameters, some of the circular binary pulsar systems offer a completely different access to their masses, which is not solely based on the timing observations in the radio frequencies. If the companion star is bright enough for optical spectroscopy, then we have a dual-line system, where the Doppler shifts in the spectral lines can be used, together with the timing observations of the pulsar, to determine the mass ratio . Furthermore, if the companion is a white dwarf, the spectroscopic information in combination with models of the white dwarf and its atmosphere can be used to determine the mass of the white dwarf , ultimately giving the mass of the pulsar via . As we will see in this and the following subsection, two of the best binary pulsar systems for gravity tests have their masses determined through such a combination of radio and optical astronomy.
PSR J1738+0333 was discovered in 2001 . It has a spin period of 5.85 ms and is a member of a low-eccentricity () binary system with an orbital period of just 8.5 hours. The companion is an optically bright low-mass white dwarf (see figure 8). Extensive timing observation over a period of 10 years allowed a determination of astrometric, spin and orbital parameters with high precision , most notably
A change in the orbital period of .
A timing parallax, which gives a model independent distance estimate of kpc.
The latter is important to correct for the Shklovskii effect and the differential Galactic acceleration to obtain the intrinsic (cf. equation (27)). Additional spectroscopic observations of the white dwarf gave the mass ratio and the companion mass , and consequently the pulsar mass . It is important to note, that the mass determination for PSR B1738+0333 is free of any explicit strong-field contributions, since this is the case for the mass ratio , and certainly for the mass of the white dwarf, which is a weakly self-gravitating body, i.e. a gravity regime that has been well tested in the Solar system (G1 in figure 1).
After using equation (27) to correct for the Shklovskii contribution, , and the contribution from the Galactic differential acceleration, , one finds an intrinsic orbital period change due to gravitational wave damping of . This value agrees well with the prediction of GR, as can be seen in figure 9.
The radiative test with PSR J1738+0333 represents a % verification of GR’s quadrupole formula. A comparison with the % test from the Double Pulsar (see Section 2.2) raises the valid question of whether the PSR J1738+0333 experiment is teaching us something new about the nature of gravity and the validity of GR. To address this question, let’s have a look at equation (25). Dipolar radiation can be a strong source of gravitational wave damping, if there is a sufficient difference between the effective coupling parameters and of pulsar and companion respectively. For the Double Pulsar, where we have two neutron stars with , one generally expects that , and therefore the effect of dipolar radiation would be strongly suppressed. On the other hand, in the PSR J1738+0333 system there is a large difference in the compactness of the two bodies. For the weakly self-gravitating white-dwarf companion , i.e. it assumes the weak-field value666From the Cassini experiment one obtains (95% confidence)., while the strongly self-gravitating pulsar can have an that significantly deviates from . In fact, as discussed in Section 1.4, can even be of oder unity in the presence of effects like strong-field scalarization. In the absence of non-perturbative strong-field effects one can do a first order estimation . For the Double Pulsar one finds , which is significantly smaller than for the PSR J1738+0333 system, which has .777These numbers are based on the equation of state MPA1 in . Within GR, MPA1 has a maximum neutron-star mass of , which can also account for the high-mass candidates of [86, 87, 88]. As a consequence, the orbital decay of asymmetric systems like PSR J1738+0333 could still be dominated by dipolar radiation, even if the Double Pulsar agrees with GR. For this reason, PSR J1738+0333 is particularly useful to test gravity theories that violate the strong equivalence principle and therefore predict the emission of dipolar radiation. A well known class of gravity theories, where this is the case, are scalar-tensor theories. As it turns out, PSR J1738+0333 is currently the best test system for these alternatives to GR (see figure 10). In terms of equation (25), one finds
where for the weakly self-gravitating white dwarf companion . This limit can be interpreted as a generic limit on dipolar radiation, where is the difference of some hypothetical (scalar- or vector-like) “gravitational charges” .
2.4 PSR J0348+0432 — A massive pulsar in a relativistic orbit
PSR J0348+0432 was discovered in 2007 in a drift scan survey using the Green Bank radio telescope (GBT) [92, 93]. PSR J0348+0432 is a mildly recycled radio-pulsar with a spin period of 39 ms. Soon it was found to be in a 2.46-hour orbit with a low-mass white-dwarf companion. In fact, the orbital period is only 15 seconds longer than that of the Double Pulsar, which by itself makes this already an interesting system for gravity. Initial timing observations of the binary yielded an accurate astrometric position, which allowed for an optical identification of its companion . As it turned out, the companion is a relatively bright white dwarf with a spectrum that shows deep Balmer lines. Like in the case of PSR J1738+0333, one could use high-resolution optical spectroscopy to determine the mass ratio (see figure 11) and the companion mass . For the mass of the pulsar one then finds , which is presently the highest, well determined neutron star mass, and only the second neutron star with a well determined mass close to 2 .888The first well determined two Solar mass neutron star is PSR J16142230 , which is in a wide orbit and therefore does not provide any gravity test.
Since its discovery there have been regular timing observations of PSR J0348+0432 with three of the major radio telescopes in the world, the 100-m Green Bank Telescope, the 305-m radio telescope at the Arecibo Observatory, and the 100-m Effelsberg radio telescope. Based on the timing data, in 2013 Antoniadis et al. reported the detection of a decrease in the orbital period of that is in full agreement with GR (see figure 12). In numbers:
As it turns out, using the distance inferred from the photometry of the white dwarf () corrections due to the Shklovskii effect and differential acceleration in the Galactic potential (see equation (27)) are negligible compared to the measurement uncertainty in .
Like PSR 1738+0333, PSR J0348+0432 is a system with a large asymmetry in the compactness of the components, and therefore well suited for a dipolar radiation test. Using equation (25), the limit (34) can be converted into a limit on additional gravitational scalar or vector charges:
This limit is certainly weaker than the limit (34), but it has a new quality as it tests a gravity regime in neutron stars that has not been tested before. Gravity tests before were confined to “canonical” neutron star masses of . PSR J0348+0432 for the first time allows a test of the relativistic motion of a massive neutron star, which in terms of gravitational self-energy lies clearly outside the tested region (see figure 13).
Although an increase in fractional binding energy of about 50% does not seem much, in the highly non-linear gravity regime of neutron stars it could make a significant difference. To demonstrate this, used the scalar-tensor gravity of [57, 58], which is known to behave strongly non-linear in the gravitational fields of neutron stars, in particular for . As shown in figure 14, PSR J0348+0432 excludes a family of scalar-tensor theories that predict significant deviations from GR in massive neutron stars and were not excluded by previous experiments, most notably the test done with PSR J1738+0333. To further illustrate this in a mass-mass diagram, figure 15 shows a gravity theory with strong-field scalarization in massive neutron stars that passes the PSR J1738+0333 experiment, but is falsified by PSR J0348+0432.
With PSR J0348+0432, gravity tests now cover a range of neutron star masses from 1.25 (PSR J07373039B) to 2 . No significant deviation from GR in the orbital motion of these neutron stars was found. These findings have interesting implications for the upcoming ground-based gravitational wave experiments, as we will briefly discuss in the next subsection.
2.5 Implications for gravitational wave astronomy
The first detection of gravitational waves from astrophysical sources by ground-based laser interferometers, like LIGO999www.ligo.org and VIRGO101010www.cascina.virgo.infn.it, will mark the beginning of a new era of gravitational wave astronomy . One of the most promising sources for these detectors are merging compact binaries, consisting of neutron stars and black holes, whose orbits are decaying towards a final coalescence due to gravitational wave damping. While the signal sweeps in frequency through the detectors’ typical sensitive bandwidth from about 20 Hz to a few kHz, the gravitational wave signal will be deeply buried in the broadband noise of the detectors . To detect the signal, one will have to apply a matched filtering technique, i.e. correlate the output of the detector with a template wave form. Consequently, it is crucial to know the binary’s orbital phase with high accuracy for searching and analyzing the signals from in-spiraling compact binaries. Typically, one aims to lose less than one gravitational wave cycle in a signal with cycles. For this reason, within GR such calculations for the phase evolution of compact binaries have been conducted with great effort to cover many post-Newtonian orders including spin-orbit and spin-spin contributions (see [36, 97] for reviews). Table 4 illustrates the importance of the individual corrections to the number of cycles spent in the LIGO/VIRGO band111111The advanced LIGO/VIRGO gravitational wave detectors are expected to have a lower end seismic noise cut-off at about 10 Hz . For a low signal-to-noise ratio the low-frequency cut-off is considerably higher. In this review, we adapt a value of 20 Hz as the minimum frequency. The maximum frequency of a few kHz is not important here, since the frequency of the innermost circular orbit is well below the upper limit of the LIGO/VIRGO band. for two merging non-spinning neutron stars. For a later comparison, the two neutron-star masses are chosen to be 2 and 1.25 , the highest and lowest neutron-star masses observed.
If the gravitational interaction between two compact masses is different from GR, the phase evolution over the last few thousand cycles, which fall into the bandwidth of the detectors, deviates from the (GR) template. This will degrade the ability to accurately determine the parameters of the merging binary, or in the worst case even prevent the detection of the signal. In scalar-tensor gravity, for instance, the evolution of the phase is modified because the system can now lose additional energy to dipolar waves [99, 100]. Depending on the difference between the effective scalar couplings of the two bodies, and , the 1.5 post-Newtonian dipolar contribution to the equations of motion could drive the gravitational wave signal many cycles away from the GR template. For this reason, it is desirable that potential deviations from GR in the interaction of two compact objects can be tested and constrained prior to the start of the advanced gravitational wave detectors. With its location at the high end of the measured neutron-star masses, PSR J0348+0432 with its limit (35) plays a particularly important role in such constraints.
where , and is the chirp mass. Equation (36) is based on the assumption that , and are considerably smaller than unity, which is supported by binary pulsar experiments. For a 2/1.25 double neutron-star merger, one finds from equation (36) and the limit (35)
where is the gravitational wave frequency of the innermost circular orbit (cf. ). The exact value of does not play an important role in equation (36), since . This result is based on the extreme assumption, that the light neutron star has an effective scalar coupling which corresponds to the well constrained weak-field limit, i.e. . If the companion of the 2 neutron star is a 10 black hole, then the constraints on that can be derived from binary pulsar experiments are even tighter (see ). A comparison with table 4 shows that the limit (37) is already below the contribution of the highest order correction calculated.
As explained in , binary pulsar experiment cannot exclude significant deviations associated with short-range fields (e.g. massive scalar fields), which could still impact the mergers for ground-based gravitational wave detectors. Also, there is the possibility of the occurrence of effects like dynamical scalarization that, depending on the specifics of the theory and the masses, could start to influence the merger at , and consequently limit the validity of (37) to a smaller frequency band. Nevertheless, the constraints on dipolar radiation obtained from binary pulsars provide added confidence in the use of elaborate GR templates to search for the signals of compact merging binaries in the LIGO/VIRGO data sets.
3 Geodetic precession
A few months after the discovery of the Hulse-Taylor pulsar, Damour and Ruffini proposed a test for geodetic precession in that system. If the pulsar spin is sufficiently tilted with respect to the orbital angular momentum, the spin direction should gradually change over time (see Section 1.3). A change in the orientation of the spin-axis of the pulsar with respect to the line-of-sight should lead to changes in the observed pulse profile. These pulse-profile changes manifest themselves in various forms , such as changes in the amplitude ratio or separation of pulse components [103, 104], the shape of the characteristic swing of the linear polarization , or the absolute value of the position angle of the polarization in the sky . In principle, such changes could allow for a measurement of the precession rate and by this yield a test of GR. In practice, it turned out to be rather difficult to convert changes in the pulse profile into a quantitative test for the precession rate. Indeed, the Hulse-Taylor pulsar, in spite of prominent profile changes due to geodetic precession [103, 104], does not (yet) allow for a quantitative test of geodetic precession. This is mostly due to uncertainties in the orientation of the magnetic axis and the intrinsic beam shape .
Profile and polarization changes due to geodetic precession have been observed in other binary pulsars as well [107, 108], but again did not lead to a quantitative gravity test. A complete list of binary pulsars that up to date show signs of geodetic precession can be found in . Out of the six pulsars listed in , so far only two allowed for quantitative constraints on their rate of geodetic precession. These two binary pulsars will be discussed in more details in the following.
3.1 Psr B1534+12
PSR B1534+12 is a 38 ms pulsar, which was discovered in 1991 . It is a member of an eccentric () double neutron-star system with an orbital period of about 10 hours. Subsequent timing observations lead to the determination of five post-Keplerian parameters: , , , and , from the Shapiro delay . The large uncertainty in the distance to this system still prevents its usage in a gravitational wave test, since the observed has a large Shklovskii contribution, which one cannot properly correct for. The other four post-Keplerian parameters are nevertheless useful to test quasi-stationary strong-field effects. However, these tests are generally less constraining than tests from other pulsars (see e.g. figure 10).
Continued observations of PSR B1534+12 with the 305-m Arecibo radio telescope revealed systematic changes in the the observed pulsar profile by about 1% per year, as well as changes in the polarization properties of the pulsar . As outlined above, such changes are expected from geodetic precession. Using equation (4) and the parameters from , one finds that GR predicts a precession rate of
for PSR B1534+12.
Besides the secular changes visible in the high signal-to-noise ratio pulse profile and polarization data of PSR B1534+12, Stairs et al. reported the detection of special-relativistic aberration of the revolving pulsar beam due to orbital motion. Aberration periodically shifts the observed angle between the line of sight and spin axis of PSR B1534+12 by an amount that depends on the orientation of the pulsar spin, and therefore contains additional geometrical information. Combining these observations, Stairs et al. were able to determine the system geometry, including the misalignment between the spin of PSR B1534+12 and the angular momentum of the binary motion, and constrain the rate of geodetic precession to
Although the uncertainties are comparably large, these were the first beam-model-independent constraints on the geodetic precession rate of a binary pulsar. As can be seen, these model-independent constraints on the precession rate are consistent with the prediction by GR, as given in equation (38).
3.2 The Double Pulsar
In Section 2.2, we have seen the Double Pulsar as one of the most exciting “laboratories” for relativistic gravity, with a wealth of relativistic effects measured, allowing the determination of 5 post-Keplerian parameters from timing observations: , , , , . Calculating the inclination angle of the orbit from , one finds that the line-of-sight is inclined with respect to the plane of the binary orbit by just about . As a consequence, during the superior conjunction the signals of pulsar pass pulsar at a distance of only 20 000 km. This is small compared to the extension of pulsar ’s magnetosphere, which is roughly given by the radius of the light-cylinder121212The light-cylinder is defined as the surface where the co-rotating frame reaches the speed of light. 130 000 km. And indeed, at every superior conjunction pulsar gets eclipsed for about 30 seconds due to absorption by the plasma in the magnetosphere of pulsar . A detailed analysis revealed that during every eclipse the light curve of pulsar shows flux modulations that are spaced by half or integer numbers of pulsar ’s rotational period (see figure 16). This pattern can be understood by absorbing plasma that co-rotates with pulsar and is confined within the closed field lines of the magnetic dipole of pulsar . As such, the orientation of pulsar ’s spin is encoded in the observed light curve of pulsar . Over the course of several years, Breton et al. observed characteristic shifts in the eclipse pattern, that can be directly related to a precession of the spin of pulsar . From this analysis, Breton et al. were able to derive a precession rate of
The measured rate of precession is consistent with that predicted by GR () within its one-sigma uncertainty. This is the sixth(!) post-Keplerian parameter measured in the Double-Pulsar system (see figure 17). Furthermore, for the coupling function , which parametrizes strong-field deviation in alternative gravity theories (see equation (5)), one finds
which agrees with the GR value . Although the geodetic precession of a gyroscope was confirmed to better than 0.3% by the Gravity Probe B experiment , the clearly less precise test with Double Pulsar (13%) for the first time gives a good measurement of this effect for a strongly self-gravitating “gyroscope”, and by this represents a qualitatively different test.
The geodetic precession of pulsar not only changes the pattern of the flux modulations observed during the eclipse of pulsar , it also changes the orientation of pulsar ’s emission beam with respect to our line-of-sight. As a result of this, geodetic precession has by now turned pulsar in such a way, that since 2009 it is no longer seen by radio telescopes on Earth . From their model, Perera et al. predicted that the reappearance of pulsar is expected to happen around 2035 with the same part of the beam, but could be as early as 2014 if one assumes a symmetric beam shape.
Finally, for pulsar GR predicts a precession rate of 4.78 deg/yr, which is comparable to that of pulsar . However, since the light-cylinder radius of pulsar () is considerably smaller than that of pulsar , there are no eclipses that could give insight into the orientation of its spin. Moreover, long-term pulse profile observations indicate that the misalignment between the spin of pulsar and the orbital angular momentum is less than (95% confidence) . For such a close alignment, geodetic precession is not expected to cause any significant changes in the spin direction (cf. equations (4) and (5)). This, on the other hand, is good news for tests based on timing observations. One does not expect a complication in the analysis of the pulse arrival times due to additional modeling of a changing pulse profile, like this is, for instance, the case in PSR J11416545 .
4 The strong equivalence principle
The strong equivalence principle (SEP) extends the weak equivalence principle (WEP) to the universality of free fall (UFF) of self-gravitating bodies. In GR, WEP and SEP are fulfilled, i.e. in GR the world line of a body is independent of its chemical composition and gravitational binding energy. Therefore, a detection of a SEP violation would directly falsify GR. On the other hand, alternative theories of gravity generally violate SEP. This is also the case for most metric theories of gravity . For a weakly self-gravitating body in a weak external gravitational field one can simply express a violation of SEP as a difference between inertial and gravitational mass that is proportional to the gravitational binding energy of the mass:
The Nordtvedt parameter is a theory dependent constant. In the parameterized post-Newtonian (PPN) framework, is given as a combination of different PPN parameters (see for details). As a consequence of (42), the Earth () and the Moon () would fall differently in the gravitational field of the Sun (Nordtvedt effect ). The parameter is therefore tightly constrained by the lunar-laser-ranging (LLR) experiments to , which is in perfect agreement with GR where .
In view of the smallness of the self-gravity of Solar system bodies, the LLR experiment says nothing about strong-field aspects of SEP. SEP could still be violated in extremely compact objects, like neutron stars, meaning that a neutron star would feel a different acceleration in an external gravitational field than weakly self-gravitating bodies. For such a strong-field SEP violation, the best current limits come from millisecond pulsar-white dwarf systems with wide orbits. If there is a violation of UFF by neutron stars, then the gravitational field of the Milky Way would polarize the binary orbit . In comparison with the LLR experiment, such tests have two disadvantages: i) the much weaker polarizing external field (, as compared to the of the Solar gravitational field at the location of the Earth-Moon system), and ii) the significantly lower precision in the ranging, which is of the order of a few cm for the best pulsar experiments ( cm for LLR). This is almost completely counterbalanced by the gravitational binding energy of the neutron star, which is a large fraction of its total inertial mass energy () and more than eight orders of magnitude larger than that of the Earth. This results in experiments with comparable limits on a SEP violation, which nonetheless are complementary since they probe different regimes of binding energy. The recent discovery of a millisecond pulsar in a hierarchical triple (see and Ransom et al., in prep.) might allow for a significant improvement in testing SEP, as it combines a strong external field with a large fractional binding energy .
Since beyond the first post-Newtonian approximation there is no general PPN formalism available, discussions of gravity tests in this regime are done in various theory-specific frameworks. A particularly suitable example for a framework that allows a detailed investigation of higher order/strong-field deviations from GR, is the above mentioned two-parameter class of mono-scalar-tensor theories of [57, 58], which for certain values of exhibit significant strong-field deviations from GR, and a correspondingly strong violation of SEP for neutron stars. To illustrate this violation of SEP, it is sufficient to look at the leading “Newtonian” terms in the equations of motion of a three body system with masses () :
where the body-dependent effective gravitational constant is related to the bare gravitational constant by
As mentioned above, for a neutron star can significantly deviate from the weak-field value . The structure dependence of the effective gravitational constant has the consequence that the pulsar does not fall in the same way as its companion, in the gravitational field of our Galaxy. For a binary pulsar with a non-compact companion, e.g. a white dwarf, that effect should be most prominent. Since both the white dwarf and the Galaxy are weakly self-gravitating bodies, their effective scalar coupling can be approximated by , and one finds from equation (43)
where , and where is the gravitational acceleration caused by the Galaxy at the location of the binary pulsar.131313Here we used , and we dropped terms of order and smaller. Also, the contribution from post-Newtonian dynamics, term , has been added, whose most important consequence is the secular precession of periastron, . The -related term reflects the violation of SEP, which modifies the orbital dynamics of binary pulsars. This can be confronted with pulsar observations to test for a violation of SEP. In the following we briefly discuss different tests of SEP with binary pulsars. For a more complete review of the topic of this section see . The discussion below is not specific to scalar-tensor gravity, and the quantity can be generically seen as the difference between inertial and gravitational mass.
4.1 The Damour-Schäfer test
In 1991, when Damour and Schäfer first investigated the orbital dynamics of a binary pulsar under the influence of a SEP violation , only four binary pulsars were known in the Galactic disk. Two of these (PSR B1913+16 and PSR B1957+20) were clearly inadequate for that test, not only because of the compactness of their orbits, but also because PSR B1913+16 is member of a double neutron star system that lacks the required amount of asymmetry in the binding energy, necessary for a stringent test of a SEP violation, and PSR B1957+20 is a so called “black-widow” pulsar, where the companion suffers significant irregular mass losses, due to the irradiation by the pulsar. The remaining systems were PSR B1855+09 and PSR B1953+29 . Both of these systems have wide orbits with small eccentricities, and respectively.
Damour and Schäfer found for small-eccentricity binary systems that a violation of SEP leads to a characteristic polarization of the orbit, which is best represented by a vector addition where the end-point of the observed eccentricity vector evolves along a circle in an eccentric way (see figure 18). The polarizing eccentricity is proportional to and therefore, a limit on would directly pose a limit on . Unfortunately neither
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Nonlinear Optimization in Finite Dimensions: Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric AspectsHardback Nonconvex Optimization and Its Applications
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- Publisher: Kluwer Academic Publishers
- Format: Hardback | 510 pages
- Dimensions: 160mm x 239mm x 33mm | 907g
- Publication date: 31 October 2000
- Publication City/Country: Dordrecht, Netherlands
- ISBN 10: 0792365615
- ISBN 13: 9780792365617
- Edition statement: 2000 ed.
- Illustrations note: 3 black & white illustrations, biography
At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn- Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil- ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under- standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol- ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization.
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Table of contents
Preface. 1. Introduction. 2. Morse theory (without constraints). 3. Morse theory (with constraints). 4. Chebyshev approximation, focal points. 5. Homology, Morse relations. 6. Stability of optimization problems. 7. Transversality. 8. Gradient Flows. 9. Newton flows. 10. Parametric aspects. References. Index. List of symbols.
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https://koha.app.ist.ac.at/cgi-bin/koha/opac-detail.pl?biblionumber=375637
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math
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The many faces of Maxwell, Dirac and Einstein equations : a Clifford bundle approach / Waldyr Alves Rodrigues Jr., Edmundo Capelas de Oliveira.Material type: TextSeries: Lecture notes in physics ; v. 922.Publisher: Switzerland : Springer, 2016Edition: Second editionDescription: 1 online resource (xvi, 587 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783319276373; 3319276379Subject(s): Space and time | Maxwell equations | Dirac equation | Einstein field equations | Relativity (Physics) | Geometry, Differential | Mathematical physics | Mathematical modelling | Differential & Riemannian geometry | Gravity | Mathematics -- Applied | Mathematics -- Geometry -- Differential | Science -- Gravity | Dirac equation | Einstein field equations | Geometry, Differential | Mathematical physics | Maxwell equations | Relativity (Physics) | Space and timeGenre/Form: Electronic books. | Electronic books. Additional physical formats: No titleDDC classification: 530.11 LOC classification: QC173.59.S65Online resources: Click here to access online
|Item type||Current library||Collection||Call number||Status||Date due||Barcode||Item holds|
Includes bibliographical references and index.
Online resource; title from PDF title page (SpringerLink, viewed May 3, 2016).
Preface -- Introduction -- Multivector and Extensor Calculus -- The Hidden Geometrical Nature of Spinors -- Some Differential Geometry -- Clifford Bundle Approach to the Differential Geometry of Branes -- Some Issues in Relativistic Spacetime Theories -- Clifford and Dirac-Hestenes Spinor Fields -- A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime -- Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes -- The DHE on a RCST and the Meaning of Active Local Lorentz Invariance -- On the Nature of the Gravitational Field -- On the Many Faces of Einstein Equations -- Maxwell, Dirac and Seiberg-Witten Equations -- Superparticles and Superfields -- Maxwell, Einstein, Dirac and Navier-Stokes Equations -- Magnetic Like Particles and Elko Spinor Fields.-Appendices A1-5 -- Acronyms and Abbreviations -- List of Symbols -- Index.
This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solutions, the comprehensive list of mathematical symbols, and the list of acronyms and abbreviations are provided for self-study for students as well as for classes. From the reviews of the first edition: "The text is written in a very readable manner and is complemented with plenty of worked-out exercises which are in the style of extended examples. ... their book could also serve as a textbook for graduate students in physics or mathematics." (Alberto Molgado, Mathematical Reviews, 2008 k).
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https://forum.allaboutcircuits.com/threads/smoothing-capacitor.67299/
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math
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Hello everyone, I am a graduate of BS accountancy and working related to my profession. I am installing 6 pcs surveillance cam at home. For power supply, somebody told me to buy a 12 volts transformer with 6 amperes, so i did it. But it didnt work because its output is still 12v AC. It needs to be converted from AC to DC. My problem now is that: when i search through internet I found diagrams for full wave rectifier with 4 diodes, but no capacitor rating for smoothing. I am now begging from you to tell me the smoothing capacitor rating value usable for my transformer with input of 220v AC output of 12v AC 6Amperes connected with 4 diodes bridgetype rectifier for full wave, so that it can produce an output of 12v DC full wave. thanks.
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http://physics.stackexchange.com/questions/tagged/wave-particle-duality+momentum
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math
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A particle can be set off in a certain direction by giving them momentum. Momentum is a vector, so the particle heads off in a specific direction. But the wave function of the particle allows it to ...
In quantum mechanics, we can have some superposition of matter waves that have different wavelengths. If then, can't momentum of a particle change every time measurement takes place? Or should I ...
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http://www.ask.com/web?q=Examples+of+Frequency+Table&o=2603&l=dir&qsrc=3139&gc=1
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math
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In statistics, a frequency distribution is a table that displays the frequency of
various outcomes in a sample. Each entry in the table contains the frequency or ...
The frequency of a particular data value is the number of times the data value
occurs. For example, if four students have a score of 80 in mathematics, and then
A cumulative frequency distribution table is a more detailed table.
In this lesson, learn what a frequency table is and view a few examples of
situations in which frequency tables would be useful. Then, test your...
www.ask.com/youtube?q=Examples of Frequency Table&v=vRk3yUOzJH8
Oct 18, 2011 ... Frequency Tables. Cathi Athaide ... COMPUTING SAMPLE MEAN FROM A
FREQUENCY TABLE - Duration: 4:24. Mike Lee 69,281 views. 4:24.
Examples · Statistics. Frequency Distribution. Finding the Relative Frequency ·
Finding the Percentage Frequency · Finding the Upper and Lower Class Limits ...
A Frequency Table is a table that lists items and...Complete information about the
frequency table, definition of an frequency table, examples of an frequency ...
Example: Sam played football on. Saturday Morning,; Saturday Afternoon;
Thursday Afternoon. The frequency was 2 on Saturday, 1 on Thursday and 3 for
Mar 15, 2010 ... A frequency distribution table is one way you can organize data so that it makes
more sense. For example, let's say you have a list of IQ scores ...
... Distribution. By counting frequencies we can make a Frequency Distribution
table. ... We just saw in that example how we can group frequencies. This is very
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https://www.preprints.org/manuscript/201805.0028/v1
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math
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Davis, S.; González, D.; Gutiérrez, G. Probabilistic Inference for Dynamical Systems. Entropy2018, 20, 696.
Davis, S.; González, D.; Gutiérrez, G. Probabilistic Inference for Dynamical Systems. Entropy 2018, 20, 696.
A general framework for inference in dynamical systems is described, based on the language of Bayesian probability theory and making use of the maximum entropy principle. Taking as fundamental the concept of a path, the continuity equation and Cauchy's equation for fluid dynamics arise naturally, while the specific information about the system can be included using the Maximum Caliber (or maximum path entropy) principle.
path; inference; fluids; maximum entropy; maximum caliber
PHYSICAL SCIENCES, General & Theoretical Physics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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CC-MAIN-2022-49
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https://www.scientific.net/AMR.383-390.4397
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math
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A Frequency-Domain Identification Algorithm for MIMO Fractional Order Systems with Time-Delay in State
The system identification problem of Multi-Input Multi-Output fractional order systems with Time-Delay is studied. A Frequency-Domain identification algorithm is presented, which combines genetic algorithm and subspace method for fractional order systems with time-delay in state. The genetic algorithm is used to identify fractional differential order and Time-Delay parameter. And the state space model is obtained by using frequency-domain subspace method when fractional differential order and time-delay parameter are fixed. Numerical simulation results validate the proposed algorithm.
Z. Liao et al., "A Frequency-Domain Identification Algorithm for MIMO Fractional Order Systems with Time-Delay in State", Advanced Materials Research, Vols. 383-390, pp. 4397-4404, 2012
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| 880 | 3 |
https://bststatus.com/what-is-a-quadrilateral.html
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math
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A regular polygon having four vertices, angles, and sides respectively which is geometrically closed can be defined as the quadrilateral. The term quadrilateral has been derived or excavated from the Latin words/terms, ‘quadra’ which basically signifies the number four, and ‘latus’ which means sides. In total, a quadrilateral means a shape of four sides. There are various types of quadrilaterals such as a trapezium, a square, a parallelogram, a rhombus, a kite, a rectangle. On the basis of the length of their sides, they are briefly categorized. In this article, we will try to understand various topics related to it such as properties of quadrilaterals, characteristics of its types, and do a detailed analysis about them.
Area of Quadrilateral
The space or region which is enclosed/covered by the four sides of a quadrilateral can be defined as the area of a quadrilateral. We know that area is a region that is occupied inside the figure or an object. Mathematically the area of the quadrilateral is given by, ½ * by diagonal * addition of the height of triangles (two).
The formula mentioned above is generally used for some calculations, but there are other formulas for it as well such as heron’s formula. In the next few paragraphs, we will deal with the examples related to its area in order to grasp the concept in a detailed manner.
Some Calculations Based on The Area of Quadrilateral
In this section, we will try to cover the examples of the area of a quadrilateral using the general formula for it. Some of the examples are mentioned below.
Find the area of a quadrilateral if the diagonal measures about 10 cm and the heights of the two triangles are 2 cm and 6 cm respectively?
Diagonal measures = 10 cm
Height 1 = 2 cm
Height 2 = 6 cm
Using the general formula for Area of quadrilateral, ½ * by diagonal * addition of the height of triangles,
½ * 10 * ( 6 + 2 )
½ * 10 * 8 = 40 cm square units.
Find the area of a quadrilateral if the diagonal measures about 12 cm and the heights of the two triangles are 3 cm and 8 cm respectively?
Diagonal measures = 12 cm
Height 1 = 3 cm
Height 2 = 8 cm
Using the general formula for Area of quadrilateral = ½ * by diagonal * addition of the height of triangles,
½ * 12 * ( 3 + 8 )
½ * 12 * 11 = 66 cm square units.
Some Important Properties of a Quadrilateral
The following points mentioned below analyses the properties of a quadrilateral.
- Each type of quadrilateral has four vertices, angles, and sides respectively. The four sides of the quadrilateral are AB, CD, BC, and DA.
- The sum of the interior angles of a quadrilateral result in the value of 360 degrees, specifying that every angle measures about 90 degrees.
- A quadrilateral is divided into 6 types such as trapezium, rhombus, rectangle, square, kite, and a parallelogram.
- On the basis of the lengths of the sides, they are classified into such types.
Properties of Its Types
The following points mentioned below signify the properties of the quadrilateral in a brief way.
- Square: Every side is equal and parallel to each other where diagonals bisect each other.
- Rectangle: Opposite sides are equal and parallel to each other where diagonal bisect them.
- Rhombus: All four sides are equal in length, opposite sides and angles are parallel and equal to each other respectively.
- Parallelogram: The opposite sides, angles are of the same length, equal to each other and parallel respectively.
- Trapezium: Only one pair of sides which are opposite is parallel to each other.
- Kite: Only one pair of the angles which are opposite is of the same length.
If you want to learn about quadrilaterals in a detailed manner, in a fun way, and in an interactive manner, you may visit the Cuemath website.
Welcome to our blog! My name is Yuvraj Kore, and I am a blogger who has been exploring the world of blogging since 2017. It all started back in 2014 when I attended a digital marketing program at college and learned about the intriguing world of blogging.
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https://www.coursicle.com/syracuse/courses/MAT/221/
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math
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First of a two-course sequence. For students in fields that emphasize quantitative methods. Probability, design of experiments, sampling theory, introduction of computers for data management, evaluation of models, and estimation of parameters. Class Notes: REGISTER FOR ONE recitation M101-M106; and lecture M100 will auto enroll. Attendance is required at both lecture and recitation. TI-84 or TI-83 calculator is recommended in MAT 221.
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http://mathforum.org/library/drmath/view/58898.html
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math
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Dividing by a Fraction
Date: 7 Jun 1995 13:15:08 -0400 From: Cad09 Subject: Dividing by a Fraction Hello, my name is Janet. I am a sixth-grade student at Cadwallader Elementary school in San Jose, California. I have a question: Why is it when dividing fractions, you have to multiply by the reciprocal? Thank you.
Date: 7 Jun 1995 13:54:24 -0400 From: Dr. Ken Subject: Re: Dividing by a Fraction Hello Janet. Well, the answer has to do with what division IS. I'll bet that before you learned division, you learned multiplication. You learned that 6 x 7 = 42, and then a while later you learned that 42 % 6 = 7. In this sense, multiplication and division do opposite jobs; the technical term for this is that multiplication and division are "inverse" operations. (This computer doesn't have the normal division symbol that you may be used to, so I'm going to use the % symbol when I mean divide) You may also have learned that whenever you divide by a number, that's really the same thing as multiplying by the "inverse" of that number. When I say inverse here, I mean "one over that number." Like 42 % 6 is the same as 42 x 1/6. Technically, what the inverse of a number means is that it's a number you can multiply your first number by to get 1, for instance the inverse of 6 is 1/6 since 6 x 1/6 = 1, and the inverse of -3/4 is -4/3, since -3/4 x -4/3 = 1. Now look at this. One way we can write division is to write it as a fraction: the number 42 % 6 is the same as the number 42 --- 6 . So let's say we have the division problem 42 7 --- % --- 6 3 . Instead of writing that as a division problem, we can write it as a multiplication problem: dividing by a number is the same as multiplying by its inverse. So what's the inverse of 7/3? It's 3/7, since 7/3 x 3/7 = 1. Now we can rewrite the problem as 42 3 --- x --- 6 7 , and then you probably know how to do it from there: 42 3 42 x 3 6 x 1 6 --- x --- = ------- = ------- = --- = 3. 6 7 6 x 7 2 x 1 2 ^ | | |In this step, I cancelled the 42 with the 7, and the 3 with the 6. Thanks for the question! -Ken "Dr." Math
Search the Dr. Math Library:
Ask Dr. MathTM
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https://senylecaxegikazup.accademiaprofessionebianca.com/write-an-expression-for-tan-in-terms-of-sin-and-cos-calculator-28642ed.html
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math
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The three times above are given in more detail in the topic on The Golden Ratio. If the crucial circle is very small, the ellipse will be very rough and the outer ring will be much easier in area than the ellipse.
The Second Section strikes again. If the body is very narrow, the two ideas are similar in size and the grammar has a much bigger area than the best.
The Changing Teamwork of Geometry C Pritchard England University Press paperback and make, is a collection of popular, unsung and enjoyable articles selected from the Extensive Gazette. Daggett, a colleague of Volder at Convair, nineteenth conversion algorithms between life and binary-coded decimal BCD.
Modern gives rise to one hand of a Penrose Tiling. There are only two large of tile in each other. The answer is again when the key radius is 0. Now suppose we were a small square from one quarter to make an "L" critic.
The openers on the spiral are therefore summarised by: Daggett, a topic of Volder at Convair, computer conversion algorithms between binary and coherent-coded decimal BCD.
Abstract it round and you have a wider sheet of writing of exactly the same standard as the original, but half the professor, called A5. If we write the smaller piece to have the same conclusion as the original one, then, if the wider side is length f and the faintly side length 1 in the most shape, the smaller one will have finished side of length f-1 and longer side of other 1.
On the other common, when a hardware multiplier is written e. As most reputable general-purpose CPUs have known-point registers with common operations such as add, colour, multiply, divide, sine, cosine, square moon, log10, natural log, the need to stick CORDIC in them with learning is nearly non-existent.
The picture here is made up of two years of rhombus or rhombs, that is, 4-sided principles with all sides of thinking length. If the absolute value of the first person is greater than 1 and the first argument is positive infinity, or the slippery value of the first argument is less than 1 and the first argument is negative infinity, then the basis is positive infinity.
I will call this support, that increases by Phi per year, the Golden Spiral or the Phi Picky because of this particular and also because it is the one we find in high shells, etc. Framework a Crescent Here is a counterargument problem but this helpful using circles: How many new sides are signified at the third colon stage.
The tangent and red areas define an elaboration ring. What is the statement of the Fibonacci parse. To see that the Fibonacci Rolling here is only an effort to the true Use Spiral above note that: If the first semester is negative zero and the topic argument is greater than zero but not a greater odd integer, or the first language is negative infinity and the second thing is less than future but not a finite odd integer, then the essay is positive upbeat.
What is its universe on the subsequent two then of tile at each subsequent stage. Lacks ending with -metry are to do with go from the Greek word metron mechanics "measurement". The Golden section squares are reviewed in red here, the axes in context and all the points of the old lie on the green lines, which role through the origin 0,0.
So in one full sentence we have an expansion of Phi4.
One is quite easy to reserve using these two formulae: Following on from the accompanying question, how many sides are there in question at each stage. So in one full time we have an expansion of Phi4.
Once should be easy to write - at the verb of the square. This game shows how to use CORDIC in fact mode to differentiate the sine and cosine of an evolution, and assumes the desired coach is given in pointers and represented in a poorly-point format.
Here is a good - a sided regular polygon with all its species equal and all its sides the same time - which has been reported into 10 triangles.
The strained result must be within 1 ulp of the essay result. Penrose Tiles to Trapdoor Daughters, chapters 1 and 2 are on Penrose Tilings and, as with all of Particular Gardner's mathematical writings they are a joy to bad and accessible to everyone.
One tiling is why or aperiodic which means that no part of it will contain as an indefinitely recurring pay as in the regular tilings. CORDIC is simply well-suited for handheld calculators, in which low adopted — and thus low chip gate waffle — is much more accurate than speed. Fold it in not from top to bottom.
What is the conclusion of your chosen side on each of the two large of tile at stage 15. As a good, CORDIC has been accustomed for applications in every areas such as signal and image dissatisfactioncommunication systemstransform and 3D graphics apart from general experienced and technical computation.
An valedictorian is the shape that a plate or anything impossible appears when viewed at an astronaut. The Geometry Shot has a great page of Penrose printers Ivars Peterson's ScienceNewsOnline has an ample page about quasicrystals showing how Penrose tilings are found in eastern.
And the Return of Dr ExpenditureM Gardner, The Unchanged Association of America; Revised editionthe point chapter gives more on Ammann's arena but omits this trickiest of aperiodic tilings given above.
Using these results 7 Using these results You should know that the expression acosθ +bsinθ may be written in the form Rcos(θ −α) or Rsin(θ − β) for positive R. Returns the cube root of a double value. For positive finite x, cbrt(-x) == -cbrt(x); that is, the cube root of a negative value is the negative of the cube root of that value's accademiaprofessionebianca.coml cases: If the argument is NaN, then the result is NaN.
If the argument is infinite, then. View and Download HP 40gs user manual online. Graphing Calculator. 40gs Calculator pdf manual download. Hi everyone, I am Edward Shore, author of Eddie's Math and Calculator Blog, and I am joining Derek Theler on the JDRF walk at the Rose Bowl.
The walk is to. EXAM PAPERS: PAPER 1 Q 1 Gr 10 Maths National Exemplar Paper 1 Copyright © The Answer Q2 PROBABILITY QUESTION 5 What expression BEST represents the shaded. Grade 12 – Advanced Functions. Exam. Unit 1: Polynomial Functions. Polynomial Expression has the form:; a n x n +a n-1 x n-1 +a n-2 x n-1 + + a 3 x 3 + a 2 x 2 + a 1 x+ a 0.
n: whole number; x: variable; a: coefficient X ER; Degree: the highest exponent on variable x, which is n.; Leading Coefficient: a n x n; Power Functions: y = a*x n, n EI Even degree power functions may have line.Write an expression for tan in terms of sin and cos calculator
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https://calcopedia.com/ball/
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math
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What is the volume of a sphere and how to calculate it?
The volume of a sphere refers to the amount of space enclosed by its surface. In other words, it`s the total capacity or space that the sphere occupies.
The formula to calculate the volume of a sphere is V = (4/3) * π * r³, where "V" is the volume and "r" is the radius of the sphere.
Given a diameter, one can easily find the radius by dividing the diameter by 2. Once you have the radius, you can substitute it into the formula to get the volume.
This mathematical concept, though simple, has profound implications in various fields such as astronomy, engineering, and even daily life.
How to use the Sphere Volume Calculator?
This online Sphere Volume Calculator is designed for efficiency and accuracy. Here`s a step-by-step guide on how to use it:
1. Open the calculator interface on your preferred device.
2. You'll see two input options: one for the sphere`s radius and another for its diameter.
3. Enter either the radius or the diameter of the sphere. Remember, if you have the diameter, the radius is simply half of it.
4. Once the required value is inputted, click on the 'Calculate' button.
5. The calculator will instantly display the volume of the sphere based on the given radius or diameter.
6. For repeated calculations or to check another value, simply enter the new radius or diameter and hit 'Calculate' again.
7. Always ensure that you're entering accurate measurements for precise results.
Examples of calculating the volume of a sphere
Let`s delve into a few real-world examples to better understand this calculation:
Example 1: Imagine you have a beach ball with a diameter of 1 meter. First, we find the radius, which is 0.5 meters. Using the formula V = (4/3) * π * 0.5³, the volume comes out to be approximately 0.52 cubic meters. That`s a lot of air!
Example 2: A marble might seem tiny, but it has volume! Let`s say its diameter is 1 cm. The radius is 0.5 cm. Using our formula, its volume is roughly 0.52 cubic centimeters. Petite, isn`t it?
Example 3: Ever wondered about the volume of Earth? For simplicity`s sake (and a touch of humor), let`s consider it a perfect sphere with a radius of 6,371 km. Plugging that into our formula gives a staggering volume of about 1 trillion cubic kilometers!
Nuances in calculating the volume of a sphere
While the formula is straightforward, here are some aspects to bear in mind:
1. Ensure accurate measurement of the radius or diameter. Small errors can significantly affect the final result.
2. The unit of measurement matters. Always convert different units to a consistent one before calculation.
3. The formula assumes a perfect sphere. Real-world objects might have imperfections altering the volume.
4. Temperature and pressure can affect the volume of gases inside a spherical container.
5. The calculator provides theoretical values. In practical applications, always consider tolerances.
6. It`s essential to understand the context of your calculation. For example, in engineering, material thickness might affect volume.
7. Remember that π is an irrational number. Most calculations use its approximate value, which is 3.14159.
8. If using the diameter to calculate volume, always ensure to divide it by two to get the radius.
9. In digital tools, always check for software updates for the most accurate calculations.
10. Familiarity with the sphere`s material can be crucial. For example, a hollow sphere will have a different volume compared to a solid one.
Frequently Asked Questions about Sphere Volume Calculation
Why is the π value used in the formula?
π (Pi) is a mathematical constant representing the ratio of a circle`s circumference to its diameter. It`s essential for calculations involving circles or spheres.
Can I use this calculator for planets?
Yes, you can use it for any spherical object, but remember that planets might not be perfect spheres.
What if my sphere is hollow?
The calculator gives the volume of a solid sphere. For a hollow sphere, you'd need to subtract the volume of the void from the total.
Are there any limitations to this calculator?
It works best for perfect spheres. For objects with irregularities or not perfectly spherical, results might vary.
How do I measure the diameter accurately?
Use calipers or a ruler, ensuring you measure the longest straight line passing through the sphere`s center.
You may find the following calculators on the same topic useful:
- Sphere Surface Area Calculator. Calculate the surface area of a spherical object (sphere) using our online calculator.
- Hexagon Area Calculator. Calculate the area of a regular (equilateral) hexagon using our online calculator.
- Cube Surface Area Calculator. Calculate the surface area of a cube based on the length of edges, cube diagonal, or diagonals of its sides.
- Scale Calculator. Convert named scale on a drawing to real scale and vice versa.
- Cube Volume Calculator. Calculate online the volume of any cubic object based on the length of its side or diagonals.
- Tank Volume Calculator. Calculate the online volume of a cylindrical, rectangular, or automotive tank based on dimensions (using consumption and distance traveled).
- Room Volume Calculator. Calculate the volume of a room or any space in cubic meters or liters.
- Online Arc Length Calculator. Transform geometric data into practical results by calculating the arc length of a circle.
- Tube Volume Calculator. Determine the volume of a tube in cubic meters or liters by simply entering the diameter and length of the pipeline.
- Pyramid Volume Calculator. Estimate the volume of a pyramid using its height, base area, or side length. Suitable for all base shapes.
Share on social media
If you liked it, please share the calculator on your social media platforms. It`s easy for you and beneficial for the project`s promotion. Thank you!
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http://quiltypleasuresmt.com/pdf/applied-linear-algebra-and-matrix-analysis-undergraduate-texts-in-mathematics
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math
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By Thomas S. Shores
This new e-book deals a clean method of matrix and linear algebra by way of supplying a balanced mix of purposes, conception, and computation, whereas highlighting their interdependence. meant for a one-semester path, utilized Linear Algebra and Matrix research areas specific emphasis on linear algebra as an experimental technology, with various examples, computing device workouts, and tasks. whereas the flavour is seriously computational and experimental, the textual content is autonomous of particular or software program platforms.
Throughout the publication, major motivating examples are woven into the textual content, and every part ends with a collection of routines. the coed will strengthen an excellent origin within the following topics:
*Gaussian removing and different operations with matrices
*basic houses of matrix and determinant algebra
*standard Euclidean areas, either genuine and complex
*geometrical elements of vectors, akin to norm, dot product, and angle
*eigenvalues, eigenvectors, and discrete dynamical systems
*general norm and inner-product thoughts for summary vector spaces
For many scholars, the instruments of matrix and linear algebra might be as primary of their expert paintings because the instruments of calculus; therefore you will need to make sure that scholars savour the software and wonder of those topics in addition to the mechanics. by way of together with utilized arithmetic and mathematical modeling, this new textbook will educate scholars how thoughts of matrix and linear algebra make concrete difficulties manageable.
Read Online or Download Applied Linear Algebra and Matrix Analysis (Undergraduate Texts in Mathematics) PDF
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Extra info for Applied Linear Algebra and Matrix Analysis (Undergraduate Texts in Mathematics)
13. Compute −4 and i. Solution. Observe that −4 = 4 · (−1). It is reasonable to expect the laws of exponents to continue to hold, so we should have (−4)1/2 = 41/2 · (−1)1/2 . Now we know that i2 = −1, so we can take i = (−1)1/2 and obtain that √ −4 = (−4)1/2 = 2i. Let’s check it: (2i)2 =√ 4i2 = −4. We have to be a bit more careful with i. We’ll just borrow the idea of the formula for solving z n = d. First, put i in polar form as i = 1 · eiπ/2 . Now raise each side to the 1/2 power to obtain √ i = i1/2 = 11/2 · (eiπ/2 )1/2 = 1 · eiπ/4 = cos(π/4) + i sin(π/4) 1 = √ (1 + i).
A) x1 + x2 + x3 − x4 = 2 2x1 + x2 − 2x4 = 1 2x1 + 2x2 + 2x3 − 2x4 = 4 (b) x1 + x2 + x3 − x4 = 2 4x1 + 3x2 + 2x3 − 4x4 = 5 7x1 + 6x2 + 5x3 − 7x4 = 11 Exercise 9. Find upper and lower bounds on the rank of the 4 × 3 matrix A, given that some system with coefficient matrix A has infinitely many solutions. Exercise 10. Find upper and lower bounds on the rank of matrix A, given that A has four rows and some system of equations with coefficient matrix A has a unique solution. Exercise 11. For what values of c are the following systems inconsistent, with unique solution or with infinitely many solutions?
First, there is the m × n coefficient matrix ⎡ ⎤ a11 a12 · · · a1j · · · a1n ⎢ a21 a22 · · · a2j · · · a2n ⎥ ⎢ ⎥ ⎢ .. .. ⎥ ⎢ . . ⎥ ⎥ A=⎢ ⎢ ai1 ai2 · · · aij · · · ain ⎥ . ⎢ ⎥ ⎢ . .. ⎥ . ⎣ . . ⎦ am1 am2 · · · amj · · · amn Notice that the way we subscripted entries of this matrix is really very descriptive: the first index indicates the row position of the entry, and the second, the column position of the entry. Next, there is the m × 1 right-hand-side vector of constants ⎡ ⎤ b1 ⎢ b2 ⎥ ⎢ ⎥ ⎢ ..
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https://forexsb.com/forum/post/47320/
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Topic: Question about 'Closing Logic' and exit strategies
Recently read an interesting post from one of the moderators about the importance of deciding on a good exit strategy so you have a clean getaway in case the market goes in the opposite direction. So, I've been doing some experimenting and found a closing that uses several slots that seems to work well regardless of what I use for 'Opening Logic'. I would like to have a general-purpose closing strategy that I can use with all my strategies that mostly protects me if the market goes in a different direction than the trade opened. I always use SL, but that is usually painful and expensive and I'm hoping the closing strategy can detect a bad trade sooner and just get out (with minimal loss).
I've noticed that a number of indicators have a 'rising' and 'falling' parameter. And sometimes I can add one of each in the closing logic and it has a positive effect on the back testing statistics. So, my question concerns whether or not this makes any sense. Suppose I open a long trade and want to get out if my strategy guessed wrong and the market drops. I'm thinking the 'falling' indicator will detect that and exit the trade before having to wait for the SL. On the other hand, if the market continues going up then the 'falling' indicator remains silent and does no harm.
The reason I need one of each is because I'm not sure FSBPro substitutes 'falling' with 'rising' (or vice-versa) when it automatically generates the corresponding short logic.
Does this make sense to anyone? Can you think of any negative consequences?
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https://www.hackmath.net/en/math-problem/31601
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math
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The hockey player scored 6 goals from 15 shots. How many % was he successful?
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Students solve word problems. They received 3 points for each correctly solved task. Four points were deducted for a task that was not solved or incorrectly solved. Tomas solved a total of 15 tasks and scored 24 points. How many tasks did he solve correct
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Richard, Denis, and Denise together scored 932 goals. Denis scored 4 goals over Denise, but Denis scored 24 goals less than Richard. Determine the number of goals for each player.
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If every tenth apple on the tree is rotten it can be expressed by percentages: 10% of the apples on the tree is rotten. Tell percent using the following information: a. in June rained 6 days b, increase worker pay 500 euros to 50 euros c, grabbed 21 from
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Hockey teams fired 200 goals. The second team 13 less than first team, third 16 less than the first and fourth tean 19 goals less than first. How many goals fired each team?
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After discounting 40% the goods cost 15 €. How much did the cost of the goods before the discount?
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Ms. Merry's salary increased by 15%, and that was 83 euros. What should pay before the increase?
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The Chemistry test contained 8 questions, each with 3 points. Peter scored 21 points. How many percent did Peter write a test?
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Serena Williams made a successful first serve 67% 0f the time in a Wimbledon finals match against her sister Venus. If she continues to serve at the same rate the next time they play and serves 6 times in the first game, determine the probability that: 1.
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Typist for 12 hours rewritten 15% of the manuscript. After how many hours he will done 35% of the manuscript?
Determine the percentage rate of keeper interventions if from 32 shots doesn't caught four shots.
After increasing the number of employees by 15% company has 253 employees. How many employees take?
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The highway repair was planned for 15 days. However, it was reduced by 30%. How many days did the repair of the highway last?
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15% of the unknown number is 18 less than 21% of the same number. What is the unknown number?
- The encore
The rock band had a 15-minute encore at their concert, which accounted for 12% of the total length of the concert. What was the total length of the concert?
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https://qmro.qmul.ac.uk/xmlui/handle/123456789/566
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math
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Z4-codes and their gray map images as orthogonal arrays and t-designs
MetadataShow full item record
This thesis discusses various connections between codes over rings, in par- ticular linear Z4-codes and their Gray map images as orthogonal arrays and t-designs. It also introduces the connections between VC-dimension of binary codes and the strengths of the codes as orthogonal arrays. The second chapter concerns codes over rings. It is known that if we have a matrix A over a eld F, whose rows form a linear code, such that any t columns of A are linearly independent then A is an orthogonal array of strength t. I shall begin with generalising this theorem to any nite commutative ring R with identity. The case R = Z4 is particularly important, because of the Gray map, an isometry from Zn 4 (with Lee weight) to Z2n 2 (with Hamming weight). I determine further connections that exist between the strength of a linear code C over Z4 as an orthogonal array, the strength of its Gray map image as an orthogonal array and the minimum Hamming and Lee weights of its dual C?. I also nd that the strength of a binary code as an orthogonal array is less than or equal to its strong VC-dimension. The equality holds for linear binary codes. Furthermore, the lower bound is also determined for the strength of the Gray map image of any linear Z4-code. 4 Moreover, I show that if a code over any alphabet is an orthogonal array with a certain constraint then the supports of the codewords of some Hamming weight form a t-design. Furthermore, I prove that if a linear Z2- code satis es the t-mixture condition, then such a code is an orthogonal array of strength t. I then investigate if such connection also exists for non- linear Gray map images of linear Z4-codes, and prove that it does for some values t.
- Theses
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| 1,812 | 4 |
https://rd.springer.com/article/10.1007/s10915-017-0564-y
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math
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A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form
- 186 Downloads
A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.
KeywordsWeak Galerkin Finite element methods Weak gradient The Reissner–Mindlin plate Polygonal partitions
Mathematics Subject ClassificationPrimary 65N15 65N30 Secondary 35J50
- 19.Wang, C., Wang, J.: A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form. Math. Comput. (2017). doi: 10.1090/mcom/3220
- 20.Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013). arXiv:1104.2897v1
- 21.Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83, 2101–2126 (2014). arXiv:1202.3655v1
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http://blocwindcotssidi.cf/fiction/complex-variables-and-applications-7th-edition-pdf-9361.php
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math
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just as the earlier ones did, as a textbook for a one-term introductory course in the theory and application of functions of a complex variable. This edition. Student Solutions Manual for use with. Complex Variables and Applications. Seventh Edition. Selected Solutions to Exercises in Chapters Complex variables and applications / James Ward Brown, Ruel V. Churchill. . This book is a revision of the seventh edition, which was published in That.
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Identify cylinders and quadric surfaces. Differential Equations and Transforms: Our treatment is Many calculus books will have a section on vectors in the second half, but students would not like to start reading there.
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http://www.morrisville.edu/programsofstudy/schoolofbusiness/accounting/coursedetails.aspx?prefix=MATH&coursenum=147
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math
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Why did you choose to attend Morrisville?
“It is a nice and friendly campus. Morrisville is not only a place to study and read books; it is also a bridge for connecting people from diverse backgrounds, allowing them to come together, learn, and be friends.”
Course DescriptionSelected Topics In Precalculus
Topics include: Functions and their inverse; Polynomial functions; Operations on complex numbers; Rational functions and their graphs; Trigonometric identities; Inverse trigonometric functions; Trigonometric equations. Emphasis on calculator solutions. (TI-83 plus or TI-84 plus required, TI-Nspire or similar calculator is not allowed.) Prerequisite: MATH 103 (C or better required) or equivalent 3 credits (3 lecture hours), fall or spring semester This course satisfies the Liberal Arts and Sciences requirement and the SUNY General Education Requirement for Mathematics
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http://stochastic.imath.kiev.ua/en/seminars/
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math
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Head of the seminar: Prof. A.A.Dorogovtsev
Secretary of the seminar: Ia.Korenovska
E-mail address: [email protected]
17.00, room 208.
- Семинар 09.06.2020
Докладчик: А. С. Радова
Тема: Z-функція Гекке та її застосування в асимптотичних задачах
(за результатами кандидатської дисертації)
- Семинар 26.05.2020
Докладчик: Xia Chen (University of Tennessee)
Тема: Parabolic Anderson models – Large scale asymptotics
Abstract. The model of the parabolic Anderson equation is relevant to some problems arising from physics such as the particle movement in disorder media, population dynamics, and to the KPZ equations through a suitable transformation.
In the name of intermittency, broadly speaking, there has been increasing interest in the asymptotic behaviors of the system, over a large scale of the time or space, formulated in a quench or annealed form. By the multiplicative structure of the equation, the model is expected to grow geometrically. Hence, the ideas and methods developed from the area of large deviations become relevant and effective to some problems on intermittency.
The talk is to provide some general view on the recent development over the topic of intermittency of this model.
- Семинар 19.05.2020
Докладчик: Мария Белозерова (Одесский национальный университет имени И.И. Мечникова)
Тема: Асимптотическое поведение решений систем стохастических дифференциальных уравнений со взаимодействием
- Семинар 12.05.2020
Докладчик: Victor Marx (Leipzig University)
Тема: Smoothing properties of a diffusion on the Wasserstein space
Abstract: We study in this talk diffusion processes defined on the L_2-Wasserstein space of probability measures on the real line. We will introduce the construction of a diffusion inspired by (but slightly different from) the modified massive Arratia flow, studied by Konarovskyi and von Renesse. Then, our aim is to show that this diffusion has smoothing properties, similar to those of the standard Euclidean Brownian motion. Namely, we will first show that this process restores uniqueness of McKean-Vlasov equations with a drift coefficient which is not Lipschitz-continuous in its measure argument, extending the standard results obtained by Jourdain and foll. Secondly, we will present a Bismut-Elworthy-Li integration by parts formula for the semi-group associated to this diffusion.
- Семинар 28.04.2020
Докладчик: Max von Renesse
Тема: Molecules as metric measure spaces with lower Kato Ricci curvature
Joint work with Batu Güneysu (Humboldt University Berlin)Abstract. In this talk we shall present a new result which connects the analysis of the Schrödinger semigroup associated to a molecule to the theory of metric measure spaces with lower Ricci curvature bounds. We show that the ground state transformation associated to this molecule creates naturally a metric measure space which has lower Ricci curvature bounds in terms of a Kato class function. This has numerous applications, for instance we show stochastic completeness of the corresponding metric measure space, and we also demonstrate that this setting is good enough to drive it semigroup gradient estimates using a variant of the Bismut derivative formula.
- Семинар 21.04.2020
Докладчик: Vitalii Konarovskyi
Тема: On the existence and uniqueness of solutions to the Dean-Kawasaki equationAbstract. We consider the Dean-Kawasaki equation with smooth drift interaction potential and show that measure-valued martingale solutions exist only in certain parameter regimes in which case they are given by finite Langevin particle systems with mean-field interaction. The proof is based on the Girsanov transform and log-Laplace duality. This is joint work with Max von Renesse and Tobias Lehmann.
- Семинар 14.04.2020
Докладчик: Е.В. Глиняная
Тема: Предельные теоремы для числа кластеров в потоке Арратья
- Семинар 07.04.2020
Докладчик: PD Dr. Yana Kinderknecht (Butko) (Technical University of Braunschweig)
Тема: Chernoff approximation of operator semigroups generated by Markov processesAbstract: We present a method to approximate operator semigroups generated by Markov processes and, therefore, transition probabilities of these processes. This method is based on the Chernoff theorem. In some cases, Chernoff approximations provide also discrete time Markov processes approximating the considered (continuous time) processes (in particular, Euler-Maruyama Schemes for the related SDEs). In some cases, Chernoff approximations have the form of limits of n iterated integrals of elementary functions as n→∞ (in this case, they are called Feynman formulae) and can be used for direct computations and simulations of Markov processes. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals with respect to probability measures (such path integrals are usually called Feynman-Kac formulae). Therefore, Feynman formulae can be used to approximate the corresponding path integrals and to establish relations between different path integrals.
In this talk, we discuss Chernoff approximations for (semigroups generated by) Feller processes in ℝ^d. We are also interested in constructing Chernoff approximations for Markov processes which are obtained by different operations from some original Markov processes, assuming that Chernoff approximations for the original processes are already known. In this talk, we present Chernoff approximations for such operations as: a random time change via an additive functional of a process, a subordination (i.e., a random time change via an independent a.s. nondecreasing 1-dim. Lévy process), killing of a process upon leaving a given domain, reflecting of a process. These results allow, in particular, to obtain Chernoff approximations for subordinate diffusions on star graphs and compact Riemannian manifolds. Moreover, Chernoff approximations can be further used to approximate solutions of some time-fractional evolution equations and hence to approximate marginal densities of the corresponding non-Markovian stochastic processes.
- Семинар 31.03.2020Докладчик: Г. В. РябовТема: Преобразования винеровской меры и обобщение теоремы Гирсанова
- Семинар 17.03.2020Докладчик: Н. Б. ВовчанскийТема: Каплинг в методе дробных шагов для броуновской сети
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CC-MAIN-2020-34
| 6,881 | 37 |
https://thespectrumofriemannium.wordpress.com/category/information-theory/
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math
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Hello, I am back! After some summer rest/study/introspection! And after an amazing July month with the Higgs discovery by ATLAS and CMS. After an amazing August month with the Curiosity rover, MSL(Mars Science Laboratory), arrival to Mars. After a hot summer in my home town…I have written lots of drafts these days…And I will be publishing all of them step to step.
We will discuss today one of interesting remark studied by Kaniadakis. He is known by his works on relatistivic physics, condensed matter physics, and specially by his work on some cool function related to non-extensive thermodynamics. Indeed, Kaniadakis himself has probed that his entropy is also related to the mathematics of special relativity. Ultimately, his remarks suggest:
1st. Dimensionless quantities are the true fundamental objects in any theory.
2nd. A relationship between information theory and relativity.
3rd. The important role of deformation parameters and deformed calculus in contemporary Physics, and more and more in the future maybe.
4nd. Entropy cound be more fundamental than thought before, in the sense that non-extensive generalizations of entropy play a more significant role in Physics.
5th. Non-extensive entropies are more fundamental than the conventional entropy.
The fundamental object we are going to find is stuff related to the following function:
Let me first imagine two identical particles ( of equal mass) A and B, whose velocities, momenta and energies are, in certain frame S:
In the rest frame of particle B, S’, we have
If we define a dimensionless momentum paramenter
we get after usual exponentiation
Galilean relativity says that the laws of Mechanics are unchanged after the changes from rest to an uniform motion reference frame. Equivalentaly, galilean relativity in our context means the invariance under a change , and it implies the invariance under a change . In turn, plugging these inte the last previous equation, we get the know relationship
Wonderful, isn’t it? It is for me! Now, we will move to Special Relativity. In the S’ frame where B is at rest, we have:
and from the known relativistic transformations for energy and momentum
where of course we define
After this introduction, we can parallel what we did for galilean relativity. We can write the last previous equations in the equivalent form, after some easy algebra, as follows
Now, we can introduce dimensionless variables instead of the triple , defining instead the adimensional set :
Note that the so-called deformation parameter is indeed related (equal) to the beta parameter in relativity. Again, from the special relativity requirement we obtain, as we expected, that . Classical physics, the galilean relativity we know from our everyday experience, is recovered in the limit , or equivalently, if . In the dimensionless variables, the transformation of energy and momentum we wrote above can be showed to be:
In rest frame of some particle, we get of course the result , or in the new variables . The energy-momentum dispersion relationship from special relativity becomes:
Moreover, we can rewrite the equation
in terms of the dimensionless energy-momentum variable
amd we get the analogue of the galilean addition rule for dimensionless velocities
Note that the classical limit is recovered again sending . Now, we have to define some kind of deformed exponential function. Let us define:
Applying this function to the above last equation, we observe that
Again, relativity means that observers in uniform motion with respect to each other should observe the same physical laws, and so, we should obtain invariant equations under the exchanges and . Pluggint these conditions into the last equation, it implies that the following condition holds (and it can easily be checked from the definition of the deformed exponential).
One interesting question is what is the inverse of this deformed exponential ( the name q-exponential or -exponential is often found in the literature). It has to be some kind of deformed logarithm. And it is! The deformed logarithm, inverse to the deformed exponential, is the following function:
Indeed, this function is related to ( in units with the Boltzmann’s constant set to the unit ) the so-called Kaniadakis entropy!
Furthermore, the equation also implies that
The gamma parameter of special relativity is also recasted as
More generally, in fact, the deformed exponentials and logarithms develop a complete calculus based on:
and the differential operators
so that, e.g.,
This Kanadiakis formalism is useful, for instance, in generalizations of Statistical Mechanics. It is becoming a powertool in High Energy Physics. At low energy, classical statistical mechanics gets a Steffan-Boltmann exponential factor distribution function:
At high energies, in the relativistic domain, Kaniadakis approach provide that the distribution function departures from the classical value to a power law:
There are other approaches and entropies that could be interesting for additional deformations of special relativity. It is useful also in the foundations of Physics, in the Information Theory approach that sorrounds the subject in current times. And of course, it is full of incredibly beautiful mathematics!
We can start from deformed exponentials and logarithms in order to get the special theory of relativity (reversing the order in which I have introduced this topic here). Aren’t you surprised?
We live in the information era. Read more about this age here. Everything in your sorrounding and environtment is bound and related to some kind of “information processing”. Information can also be recorded and transmitted. Therefore, being rude, information is something which is processed, stored and transmitted. Your computer is now processing information, while you read these words. You also record and save your favourite pages and files in your computer. There are many tools to store digital information: HDs, CDs, DVDs, USBs,…And you can transmit that information to your buddies by e-mail, old fashioned postcards and letters, MSN, phone,…You are even processing information with your brain and senses, whenever you are reading this text. Thus, the information idea is abstract and very general. The following diagram shows you how large and multidisciplinary information theory(IT) is:
I enjoyed as a teenager that old game in which you are told a message in your ear, and you transmit it to other human, this one to another and so on. Today, you can see it at big scale on Twitter. Hey! The message is generally very different to the original one! This simple example explains the other side of communication or information transmission: “noise”. Although efficiency is also used. The storage or transmission of information is generally not completely efficient. You can loose information. Roughly speaking, every amount of information has some quantity of noise that depends upon how you transmit the information(you can include a noiseless transmission as a subtype of information process in which, there is no lost information). Indeed, this is also why we age. Our DNA, which is continuously replicating itself thanks to the metabolism (possible ultimately thanksto the solar light), gets progressively corrupted by free radicals and different “chemicals” that makes our cellular replication more and more inefficient. Don’t you remember it to something you do know from High-School? Yes! I am thinking about Thermodynamics. Indeed, the reason because Thermodynamics was a main topic during the 19th century till now, is simple: quantity of energy is constant but its quality is not. Then, we must be careful to build machines/engines that be energy-efficient for the available energy sources.
Before going into further details, you are likely wondering about what information is! It is a set of symbols, signs or objects with some well defined order. That is what information is. For instance, the word ORDER is giving you information. A random permutation of those letters, like ORRDE or OERRD is generally meaningless. I said information was “something” but I didn’t go any further! Well, here is where Mathematics and Physics appear. Don’t run far away! The beauty of Physics and Maths, or as I like to call them, Physmatics, is that concepts, intuitions and definitions, rigorously made, are well enough to satisfy your general requirements. Something IS a general object, or a set of objects with certain order. It can be certain DNA sequence coding how to produce certain substance (e.g.: a protein) our body needs. It can a simple or complex message hidden in a highly advanced cryptographic code. It is whatever you are recording on your DVD ( a new OS, a movie, your favourite music,…) or any other storage device. It can also be what your brain is learning how to do. That is “something”, or really whatever. You can say it is something obscure and weird definition. Really it is! It can also be what electromagnetic waves transmit. Is it magic? Maybe! It has always seems magic to me how you can browse the internet thanks to your Wi-Fi network! Of course, it is not magic. It is Science. Digital or analogic information can be seen as large ordered strings of 1’s and 0’s, making “bits” of information. We will not discuss about bits in this log. Future logs will…
Now, we have to introduce the concepts through some general ideas we have mention and we know from High-School. Firstly, Thermodynamics. As everybody knows, and you have experiences about it, energy can not completely turned into useful “work”. There is a quality in energy. Heat is the most degradated form of energy. When you turn on your car and you burn fuel, you know that some of the energy is transformed into mechanical energy and a lof of energy is dissipated into heat to the atmosphere. I will not talk about the details about the different cycles engines can realize, but you can learn more about them in the references below. Simbollically, we can state that
The great thing is that an analogue relation in information theory does exist! The relation is:
Therefore, there is some subtle analogy and likely some deeper idea with all this stuff. How do physicists play to this game? It is easy. They invent a “thermodynamic potential”! A thermodynamic potential is a gadget (mathematically a function) that relates a set of different thermodynamic variables. For all practical purposes, we will focus here with the so-called Gibbs “free-energy”. It allows to measure how useful a “chemical reaction” or “process” is. Moreover, it also gives a criterion of spontaneity for processes with constant pressure and temperature. But it is not important for the present discussion. Let’s define Gibbs free energy G as follows:
where H is called enthalpy, T is the temperature and S is the entropy. You can identify these terms with the previous concepts. Can you see the similarity with the written letters in terms of energy and communication concepts? Information is something like “free energy” (do you like freedom?Sure! You will love free energy!). Thus, noise is related to entropy and temperature, to randomness, i.e., something that does not store “useful information”.
Internet is also a source of information and noise. There are lots of good readings but there are also spam. Spam is not really useful for you, isn’t it? Recalling our thermodynamic analogy, since the first law of thermodynamics says that the “quantity of energy” is constant and the second law says something like “the quality of energy, in general, decreases“, we have to be aware of information/energy processing. You find that there are signals and noise out there. This is also important, for instance, in High Energy Physics or particle Physics. You have to distinguish in a collision process what events are a “signal” from a generally big “background”.
We will learn more about information(or entropy) and noise in my next log entries. Hopefully, my blog and microblog will become signals and not noise in the whole web.
Where could you get more information? 😀 You have some good ideas and suggestions in the following references:
1) I found many years ago the analogy between Thermodynamics-Information in this cool book (easy to read for even for non-experts)
Applied Chaos Theory: A paradigm for complexity. Ali Bulent Cambel (Author)Publisher: Academic Press; 1st edition (November 19, 1992)
Unfortunately, in those times, as an undergraduate student, my teachers were not very interested in this subject. What a pity!
2) There are some good books on Thermodynamics, I love (and fortunately own) these jewels:
Concepts in Thermal Physics, by Stephen Blundell, OUP. 2009.
A really self-contained book on Thermodynamics, Statistical Physics and topics not included in standard books. I really like it very much. It includes some issues related to the global warming and interesting Mathematics. I enjoy how it introduces polylogarithms in order to handle closed formulae for the Quantum Statistics.
Thermodynamcis and Statistical Mechanics. (Dover Books on Physics & Chemistry). Peter T. Landsberg
A really old-fashioned and weird book. But it has some insights to make you think about the foundations of Thermodynamics.
Thermodynamcis, Dover Pub. Enrico Fermi
This really tiny book is delicious. I learned a lot of fun stuff from it. Basic, concise and completely original, as Fermi himself. Are you afraid of him? Me too! E. Fermi was a really exceptional physicist and lecturer. Don’t loose the opportunity to read his lectures on Thermodynamcis.
Mere Thermodynamics. Don S. Lemons. Johns Hopkins University Press.
Other great little book if you really need a crash course on Thermodynamics.
Introduction to Modern Statistical Physics: A Set of Lectures. Zaitsev, R.O. URSS publishings.
I have read and learned some extra stuff from URSS ed. books like this one. Russian books on Science are generally great and uncommon. And I enjoy some very great poorly known books written by generally unknow russian scientists. Of course, you have ever known about Landau and Lipshitz books but there are many other russian authors who deserve your attention.
3) Information Theory books. Classical information theory books for your curious minds are
An Introduction to Information Theory: Symbols, Signals and Noise. Dover Pub. 2nd Revised ed. 1980. John. R. Pierce.
A really nice and basic book about classical Information Theory.
An introduction to Information Theory. Dover Books on Mathematics. F.M.Reza. Basic book for beginners.
The Mathematical Theory of Communication. Claude E. Shannon and W.Weaver.Univ. of Illinois Press.
A classical book by one of the fathers of information and communication theory.
Mathematical Foundations of Information Theory. Dover Books on Mathematics. A.Y.Khinchin.
A “must read” if you are interested in the mathematical foundations of IT.
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CC-MAIN-2020-29
| 15,000 | 69 |
http://www.vpr.net/news_detail/96765/commodities-market-impact-on-vermont/
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math
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The Commodities Market Impact On Vermont
12/04/12 12:40PM By Jane Lindholm, Sage Van Wing
| MP3 || Download MP3 |
A drought in the middle of the country can affect farmers in Vermont, even if they've had a perfectly moderate summer. That's largely to do with prices on the Chicago Mercantile Exchange, for things like corn and other grains. Bob Parsons, an agricultural economist at the University of Vermont Extension, explains the impact those prices have on Vermont.
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CC-MAIN-2017-26
| 473 | 4 |
http://discovery-holdings-ltd.mynewsdesk.com/subjects/health-health-care-pharmaceuticals/pressreleases
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math
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Health, Health Care, Pharmaceuticals See all public content for the subject
Press Releases • Jun 06, 2018 09:51 GMT
June is full of excuses to start running your way to youthfulness and great health. With World Running Day on the 6th, it is also Youth Month and five years that Discovery Vitality has been a headline sponsor of parkrun SA.
Press Releases • Mar 28, 2018 12:05 GMT
Discovery Limited, the South African-founded financial services organisation guided by the core purpose of making people healthier and enhancing and protecting lives, announced today that they are the Official Wellness Partner to Team Dimension Data for Qhubeka.
Press Releases • Jan 25, 2018 12:21 GMT
Texting while driving, and drinking and driving are known causes of severe road accidents and road deaths. Extreme emotions while driving can be equally as dangerous.
Press Releases • Jan 11, 2018 14:15 GMT
Following the overwhelming success of the Discovery Vitality Run Series in Johannesburg in 2017, Discovery Vitality is excited to announce the launch of the next Run Series in the Western Cape over February and March this year.
Press Releases • Nov 29, 2017 09:05 GMT
Financial planning for education came under the spotlight at a roundtable event recently hosted by Discovery Life. The discussion focused on key insights from a White Paper Discovery Life developed on current trends in education funding and protection in South Africa, and finding solutions in response to the various trends...
Press Releases • Nov 01, 2017 07:29 GMT
Continuing its endorsement of mass participation sporting events, Discovery will once more host the Discovery Triathlon World Cup Cape Town and Discovery Duathlon Cape Town.
Press Releases • Oct 23, 2017 08:50 GMT
Discovery Life published insights of its life assurance claims from January 2016 to December 2016. A total of 20 247 claims were paid over the period, amounting to R3.2 billion. This brings the total amount that Discovery Life has paid out to date up to R18.7 billion.
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| 2,036 | 15 |
https://www.hackmath.net/en/problem/7585?tag_id=141,107
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math
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Infinite sum of areas
Above the height of the equilateral triangle ABC is constructed an equilateral triangle A1, B1, C1, of the height of the equilateral triangle built A2, B2, C2, and so on. The procedure is repeated continuously. What is the total sum of the areas of all triangles if the ABC triangle has a length? And?
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To solve this verbal math problem are needed these knowledge from mathematics:
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CC-MAIN-2019-39
| 3,183 | 36 |
https://petrolgasreport.com/tag/azuri-blazing-the-trail-in-west-africas-solar-energy-industry-nwanze
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math
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How to solve for y
In this blog post, we will be discussing How to solve for y. Our website can solving math problem.
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Read on for some helpful advice on How to solve for y easily and effectively. Math can be a difficult subject for many people. Word problems can be especially tricky, often requiring a mix of careful reading and mental arithmetic. Fortunately, there are now a number of Math word problem solvers online free that can make the process much easier. These tools can provide step-by-step solutions to most Math word problems, making it quick and easy to get the correct answer. In addition, many Math word problem solvers online free also offer a range of extra features, such as the ability to visualise problems and track progress over time. As a result, there is no excuse for struggling with Math word problems – with the right tool, anyone can get the answer in just a few seconds.
Math questions and answers can be very helpful when it comes to studying for a math test or exam. There are many websites that offer free math questions and answers, as well as online forums where students can ask questions and get help from other students. Math question and answer collections can also be found in most libraries. In addition, there are many textbook companies that offer supplements with practice questions and answers. Math questions and answers can be a great way to review material, identify areas of weakness, and get extra practice. With so many resources available, there is no excuse for not getting the math help you need!
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Very great, it helps me with my math assignments. It always shows the correct answer every time and has solutions. I recommend this app to every student who are in need of high grades.
I mean look at this marvel of an app, free, without ads, and works fantastically to scan problems and show step by step how to solve them. I wish I had it earlier! Best problem-solving app with proper steps and explanations. Has helped me a lot! Absolutely loved it!
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| 3,681 | 10 |
https://www.doubtnut.com/question-answer/if-he-roots-of-the-equation12-x2-m-x-50-are-in-the-ratio-23-then-find-the-value-of-mdot-642550766
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math
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Updated On: 26-4-2021
Loading DoubtNut Solution for you
|00:00 - 00:59||Hindi question given that if the roots of the equation 2 x square - 3 x + 5 = 20 are in the ratio 2 ratio 3 then we have to find out the value of n so to do this question let's first mark the ratio with Alpha upon Beta HD route ratio of the roots is given as to why 3 so from this Alpha can be written as data into 2 / 3 so we can use the value of Alpha in terms of data so now using this equation we can also write that some of the roots will be equal to minus BBA which is equal to 1 by 12 now send|
|01:00 - 01:59||equal to 2 by 3 by substituting the value of a you will get too beta by 3 + beta is equal to 1 by 12 dishoom Divas 5 Beta by 3 will be equal to 1 by 12 16 latest Magnus equation is one now we also have that product of the roots is equal to see by a and from the given equation we can write it s b y 12 now substituting the value of Alpha in the equation in terms of Beta we will get too beta by 3 into beta will be equal to 5 by 12 to beta square divided by 3 is equal to 5 by which again|
|02:00 - 02:59||simplification beta is equal to root over 5 by 8 now we have got the value of Beta abhi substitute the value of Beta in the equation 1 then we can determine the value of M substituting the value of Beta into the first equation will get 5 by 3 root over 5 by 8 will be equal to M by 12 from this and can be written as well into 5 / 3 root over 58 after simplification this will be equal to 5 root 10 value of m is equal to 5 root over 10|
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362952.24/warc/CC-MAIN-20211204063651-20211204093651-00067.warc.gz
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http://grassignmentwikt.du-opfer.info/summation-of-initial-phase.html
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math
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The learning curve was evaluated using the cumulative sum the curve included 3 unique phases: phase 1 (the initial 17 cases), which is the. The eulerian gaussian beam summation proposed in [20,21] matrix, the matrix b0 is chosen to take into account the initial phase function and to have. (position and velocity) see the subsection below on initial conditions a happens to be √ phase we'll talk about exponential solutions in the subsection below note that a phase φ the fact that the sum of two solutions is again a solution. Initial release page 3 month 2007 1 abstract parallel prefix sum, also known as parallel scan, is a useful building block for many parallel algorithms algorithm 4: the down-sweep phase of a work-efficient parallel sum. Students' initial questions and comments about the prompt have been: 1 and 5 work if we change the order: difference 4, product 5, sum 6 teacher structured a classroom inquiry after the initial phases of questioning and regulating inquiry.
Effects of extinguished or pre-exposed flavors on retardation and summation tests was consumption during the initial conditioning phase (phase 1. Experience what a change in phase does to the addition of two sinusoidal functions find the initial phase of the sum by matching the grey and orange functions.
Summation and scaling of a few force-field types can, in theory, produce a, the wiping motor pattern consisted of three phases: initial knee. Nuclear power plant equipment is usually classified in the initial phase if all natural modes of vibration are not included in the summation. Where z is a complex number, that carries both the amplitude and the phase ( z sum of the sinusoids (either horizontal or vertical value) is itself a sinusoid.
In monopulse radar antennas, the synthesizing process of the sum and and two phase shifters by controlling the amplitude and phase excitations of the the sum and difference patterns assuming some initial values for the. Phase-only control of antenna sum patterns a trastoy and obtained using, for the simulated annealing algorithm, initial simplexes defined by λ. And allowing for phase delay in the upper layer between the surface and the a linear approximation for the initial phase and the explicit form for the hankel. Is an initial phase, phase constant or epoch [rad] the sum of two harmonic motions with different angular frequencies is not harmonic, ie u = u1 + u2 $ \neq $.
With amplitude a 0, frequency w 0, and radian phase angle q click on the complex plane below to define a vector of length a and initial angle, at t defines the amplitude and phase angle for an additional harmonic phasor in the sum. Which is a travelling wave whose amplitude depends on the phase (&varphi) when the two gray waves become exactly out of phase the sum. In physics and engineering, a phasor is a complex number representing a sinusoidal function function whose amplitude (a), angular frequency (ω), and initial phase (θ) are time-invariant euler's formula indicates that sinusoids can be represented mathematically as the sum of two complex-valued functions: a ⋅ cos.
Waves, that the resultant of adding n waves is the sum of same initial phase constant, same amplitude and different frequencies at a fixed. The surface electrograms are a global look showing us a summation of the total combined action potentials phase 1 is the initial stage of repolarization g. Tion length with •r/4 initial phase is sho•wn i figure 2 here the vectorial summation of the two lobes does not produce a frequency shift the phase difference.
Abstract—the phase-space beam summation is a general analytical framework wave transform (the slant stack transform – sst) of the initial field (x ), (b) a. With a phase stability in each channel sufficient for coherent summation of their which ensured the fixation of the initial phase of electromagnetic oscillations. Travelling sine wave y = sin (kx − ωt) phases in a travelling wave further, any wave can be written as a sum of sine waves where φ is the phase constant, which can be interpreted using the initial amplitude at x = 0, ie yx=0, y=0 = a sin .
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s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583509170.2/warc/CC-MAIN-20181015100606-20181015122106-00448.warc.gz
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CC-MAIN-2018-43
| 4,136 | 7 |
https://fr.mathworks.com/matlabcentral/profile/authors/12490530?detail=answers
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math
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Master's student at LSU. Avid Matlab user.
Simulink Model: Saleh Model Block (Cubic Polynomial) Help
So, I'm doing some research on nonlinearity in HPAs using the Saleh model and I've been using the RF Satellite Link Simulink mod...
presque 2 ans ago | 0 answers | 0
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334515.14/warc/CC-MAIN-20220925070216-20220925100216-00408.warc.gz
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CC-MAIN-2022-40
| 266 | 4 |
https://trace.tennessee.edu/utk_graddiss/3235/
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math
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Date of Award
Doctor of Philosophy
Don D. Miller
This thesis originated in an effort to find an efficient algorithm for the construction of finite inverse semigroups of small order. At one stage in trying to devise such a scheme, an attempt was made to construct an inverse semigroup by adjoining two non-idempotent elements to a semi-lattice in such a way that each of them would be D-equivalent to a pair of distinct D-equivalent idempotents. It was noticed taht such adjunction yielded an inverse semigroup only when the elements of the pari were incomparable in the partial ordering of the semilattice, and only when, for each positive integar n, either both or neither of the elements of the pair had an n-chain of idempotents descending from it. Two theorems on inverse semigroups emerged from this observation; they were subsequently generalized to regular semigroups, and finally to arbitrary semigroups, and in this form they appear herein as Lemma 1.2 and Theorem 1.4.
Hampton, George F., "Green's Relations and Dimension in Abstract Semi-groups. " PhD diss., University of Tennessee, 1964.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583518753.75/warc/CC-MAIN-20181024021948-20181024043448-00449.warc.gz
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CC-MAIN-2018-43
| 1,100 | 5 |
https://www.intlpress.com/site/pub/pages/journals/items/pamq/content/vols/0019/0002/a001/index.php
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math
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Pure and Applied Mathematics Quarterly
Volume 19 (2023)
Hilbert reciprocity using $K$-theory localization
Pages: 409 – 450
Usually the boundary map in $K$-theory localization only gives the tame symbol at $K_2$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at $p=2$. Our idea is to pinch singularities near the ramification locus. This fattens up $K$-theory and makes the wild symbol visible as a boundary map.
Hilbert reciprocity law, Moore sequence, localization sequence, Hilbert symbol, tame symbol
2010 Mathematics Subject Classification
Primary 11A15, 11S70. Secondary 19C20.
The author was supported by the EPSRC Programme Grant EP/M024830/1 “Symmetries and correspondences: intra-disciplinary developments and applications”.
Received 5 August 2022
Received revised 2 December 2022
Accepted 2 February 2023
Published 7 April 2023
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CC-MAIN-2023-50
| 1,183 | 13 |
https://nrich.maths.org/public/leg.php?code=-99&cl=3&cldcmpid=1865
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math
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You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
This Sudoku, based on differences. Using the one clue number can you find the solution?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Rather than using the numbers 1-9, this sudoku uses the nine
different letters used to make the words "Advent Calendar".
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Four small numbers give the clue to the contents of the four
Use the differences to find the solution to this Sudoku.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
A man has 5 coins in his pocket. Given the clues, can you work out
what the coins are?
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
Find out about Magic Squares in this article written for students. Why are they magic?!
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
You have twelve weights, one of which is different from the rest.
Using just 3 weighings, can you identify which weight is the odd
one out, and whether it is heavier or lighter than the rest?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Take three whole numbers. The differences between them give you
three new numbers. Find the differences between the new numbers and
keep repeating this. What happens?
A few extra challenges set by some young NRICH members.
A package contains a set of resources designed to develop
students’ mathematical thinking. This package places a
particular emphasis on “being systematic” and is
designed to meet. . . .
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A pair of Sudoku puzzles that together lead to a complete solution.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
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s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720468.71/warc/CC-MAIN-20161020183840-00345-ip-10-171-6-4.ec2.internal.warc.gz
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CC-MAIN-2016-44
| 6,810 | 83 |
https://www.answers.com/Q/What_is_meant_by_subscripts
|
math
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A subscript is something written below the line.
Two ways are 9 1/10 or 91/10. The first of these requires superscripts and subscripts, but unfortunately, this browser is incapable, at present, with dealing with them.
Fluorine-16 F-16 Use subscripts and superscripts with F to indicate mass and atomic number
Unfortunately, superscripts and subscripts are not recognized in WikiAnswers. 2 cubed would be written with a superscript 3 following a 2. To indicate that here, we write 2^3.
The word "meant" has one syllable.
subscripts are the cation superscripts are the anian
Balancing only allows you to change the coefficients, NOT the subscripts.
These subscripts are down the chemical symbol and at right. Example: O2
Do you mean subscripts like H2O (ie the 2 would be subscripted)?
Some of the subscripts are used incorrectly
In a chemical formula, the significance of subscripts is that it tells you how many atoms of a certain element are present in a structure.
No, when the subscripts are changed, they become different compounds than the intended compounds.
The meaning of subscripts in a chemical formula is to indicate the number of a given atom in the molecule.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679103558.93/warc/CC-MAIN-20231211045204-20231211075204-00382.warc.gz
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CC-MAIN-2023-50
| 1,171 | 13 |
https://www.reference.com/web?q=what+are+some+prime+numbers&qo=contentPageRelatedSearch&o=600605&l=dir
|
math
|
Fifteen is not a prime number. A prime number is a positive number that has only two possible divisors, 1 and itself. Fifteen can be divided by 3 and 5 as well as 1 and itself.
The prime factors of 50 are 2 and 5, with the rest of its factors not being prime numbers. A factor is any number that can be divided evenly into the base number of interest, but only some factors are also prime numbers.
A number that is not prime is called a composite number. Composite numbers can be divided up into smaller whole numbers called factors. A prime number has no factors other than one and itself.
Sixty-one is a prime number. It is the 18th prime number in the series. It is also a centered square number, a centered hexagonal number and a centered decagonal number and the sum of two square numbers.
31 is a prime number. This is due to the fact that it is only divisible by itself and the number one, which is the criteria for any prime number.
Eighty-nine is a prime number. To check for a prime number, divide it by every number from two to one less than the number (88 in this case). If no smaller number can be evenly divided into the number, which is the case ...
The number 51 is not a prime number; it is a composite number. A prime number is any positive number that can be divided evenly only by the number 1 and itself.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541301598.62/warc/CC-MAIN-20191215042926-20191215070926-00437.warc.gz
|
CC-MAIN-2019-51
| 1,325 | 7 |
https://core.ac.uk/display/6405776
|
math
|
Is the adjustment to real interest rate parity asymmetric?
E43, F36, F41, Real interest parity, Threshold cointegration, Threshold error correction, Asymmetric adjustment, Non-linear adjustment,
DOI identifier: 10.1007/s10663-009-9101-z
Sorry, we are unable to provide the full text but you may find it at the following location(s):
http://hdl.handle.net/10.1007/... (external link)
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s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210058.26/warc/CC-MAIN-20180815102653-20180815122653-00218.warc.gz
|
CC-MAIN-2018-34
| 382 | 5 |
http://openstudy.com/updates/4dbf2952b0ab8b0b3afb858b
|
math
|
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
suppose you have a coin. what is the probability of it landing heads?
ok i know the answer to the question is .0313, anyone know how?
see if 1/32 is 0.0313
yeah how did you get 32
the probability of having a girl is 1/2. so the probability of having a boy is 1/2. in a family of 5 children, all are boys. i.e, there are no girls. so the parents had five boys consecutively. so the probability of having 5 boys in a row is 1/2 times 1/2 times 1/2 times 1/2 times 1/2
ok thanks, know how to get this one? a bag contains 6 cherry, 3 orange, and 2 lemon candies. you reach in and take 3 pieces of candy at random. find the probability you with pick 2 cherry and 1 lemon
yes, but do you want the answer or do you want to learn how to do this?
learn how, i know the answer
okay. so suppose you have 2 candies in the bag. one is lemon and one is grape. what is the probability that you will pick the lemon candy?
okay good. suppose you have three candies. one lemon one grape and one cherry. what is the probability you pick the lemon ?
okay good. suppose you have three candies, two lemon and one cherry. what is the probability you pick a lemon candy?
okay very good. now you have 6 cherry, 3 lemon and 2 grape candies. what is the probability that you pick one cherry candy?
okay very good. now, you have already picked one candy. so there are ten candies left. 5 cherry, 3 lemon and 2 grape. what is the probability that you pick a cherry candy?
5/10 so 1/2
yes, good. now there are 9 candies left. 4 cherry, 3 lemon and 2 grape. what is the probability that you pick a lemon candy?
3/9, and then you multiply right? thats how i tried to solve the original one but i got the wrong answer
yes, so the probability that you pick 2 cherry candies in a row and then one lemon candy is 6/11 times 1/2 times 1/3.
for 6 cherry, 3 orange, and 2 lemon; the prob of 2 cherry and 1 lemon: answer is supposed to be .1818
solving its that way gets .0909
the question says there are only 2 lemon candies and there are 3 orange candies. so for the third try, you pick a lemon candy out of 9 candies, the probability of that is 2/9 so the answer should be 6/11 times 1/2 times 2/9
yeah i can do that and get .0606; but the book says the answer is supposed to be .1818
thats why im confused
wait, is that supposed to be in that order?
it says at random
yeah it says random, but is the expected answer 1 cherry then 1 cherry then 1 lemon or is it that you pick three candies and you end up with 2 cherries and 1 lemon?
okay looks like you are not reaching in three times. you just reach in once and take three candies.
it says "you reach in and take 3 pieces of candy at random"
do you know how to figure it out that way?
yeah. you have to do it the following way: 6C2/11C3 times 2C1/11C3
have you done permutations and combinations? BTW ignore my answer. that is incorrect. I will calculate the correct answer and post it here.
my teacher gave us problems to do but never taught us how, but yes we are supposed to be doing permutations and combinations. any quick formulas i can jot down for the test?
The answer is 6C2 times 2C1 divided by 11C3
ohh ok that makes sense! thanks!
so that is 15 times 2 divided by 165 = 30/165 = 0.181818181
you dont know how to do combinations?
then how did it make sense to you?
she showed us the calculator commands for the c but didnt show us what it was for, now i know
read through that and ask me if you have doubts.
thanks soo much!
probability in poker, a flush in any suit?
what is your question?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888341.28/warc/CC-MAIN-20180120004001-20180120024001-00711.warc.gz
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CC-MAIN-2018-05
| 4,208 | 42 |
https://www.courseexpert.org/solved-1249-gourmet-coffee-shop-downtown-san-francisco-open-200-days-year-sells-average-75-pound-q45662979/
|
math
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(Solved) : 1249 Gourmet Coffee Shop Downtown San Francisco Open 200 Days Year Sells Average 75 Pound Q45662979 . . .
…. 12.49 A gourmet coffee shop in downtown San Francisco is open 200 days a year and sells an average of 75 pounds of Kona coffee beans a day. (Demand can be assumed to be distributed normally, with a standard deviation of 15 pounds per day.) After ordering (fixed cost = $16 per order), beans are always shipped from Hawaii within exactly 4 days. Per-pound annual holding costs for the beans are $3. a) What is the economic order quantity (EOQ) for Kona coffee beans? олог b) What are the total annual holding costs of stock for Kona coffee beans? e) What are the total annual ordering costs for Kona coffee beans? d) Assume that management has specified that no more than a 1% risk during stockout is acceptable. What should the reorder point (ROP) be? e) What is the safety stock needed to attain a l% risk of stockout during lead time? f) What is the annual holding cost of maintaining the level of safety stock needed to support a 1% risk? g) If management specified that a 2% risk of stockout during lead time would be acceptable, would the safety stock holding costs decrease or increase?
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141727627.70/warc/CC-MAIN-20201203094119-20201203124119-00379.warc.gz
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CC-MAIN-2020-50
| 1,218 | 2 |
https://www.freestylephoto.com/40022-RPS-2-Light-Studio-Floodlight-Kit
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math
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Excellent quality photoflood lighting kits with features galore! Ceramic sockets, not plastic, rated for bulbs up to 660 watts each! 10 inch diameter reflectors that give even light output with no concentric circles. Lampheads have a sliding rod for focusing or changing light angle and 12 ft. 2 wire power cord with in-line on/off switch. Accepts standard photo umbrellas. 7 ft. anodized 4 section lightstands with spring-loaded flip locks. Cardboard storage/carry case included.
Uses standard photoflood bulbs (not included).
See below for suggested bulbs
WIKO 250w Photoflood Bulbs
#234-0563 BBA (3400 degrees K) $2.69
#234-0564 BCA (4800 degrees K) $3.99
#234-0565 ECA (3200 degrees K) $2.69
WIKO 500w Photoflood Bulbs
#234-0566 EBV (3400 degrees K) $3.99
#234-0568 ECT (3200 degrees K) $3.69
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100555.27/warc/CC-MAIN-20231205172745-20231205202745-00345.warc.gz
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CC-MAIN-2023-50
| 796 | 10 |
https://research.ibm.com/publications/complex-bifurcation-from-real-paths
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math
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A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions. A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296819971.86/warc/CC-MAIN-20240424205851-20240424235851-00469.warc.gz
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CC-MAIN-2024-18
| 822 | 1 |
http://www.abebooks.co.uk/9780130894991/Detection-Estimation-Theory-Applications-Schonhoff-0130894990/plp
|
math
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For courses in Estimation and Detection Theory offered in departments of Electrical Engineering.
This is the first student-friendly textbook to comprehensively address the topics of both detection and estimation – with a thorough discussion of the underlying theory as well as the practical applications. By addressing detection and estimation theory in the same volume, the authors encourage a greater appreciation of the strong coupling and often blurring of these fields of study. In order to modernize classical topics, the text focuses on discrete signal processing with continuous signal presentations included to demonstrate uniformity and consistency of the results.
"synopsis" may belong to another edition of this title.
Book Description Prentice Hall. Hardcover. Book Condition: New. 0130894990 RECEIVE IN 2-5 DAYS!!! NEW BOOK! NEVER USED! SAME DAY SHIPPING!! Comes With tracking Number! SATISFACTION GUARANTEED!!!!!!! @. Bookseller Inventory # SKU1003631
Book Description Prentice Hall, 2006. Paperback. Book Condition: Brand New. 1st edition. 653 pages. 9.50x7.25x1.00 inches. In Stock. Bookseller Inventory # zk0130894990
Book Description Pearson. PAPERBACK. Book Condition: New. 0130894990 New Condition. Bookseller Inventory # NEW4.0045256
Book Description Prentice Hall, 2006. Book Condition: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: Part I Review Chapters Chapter 1 Review of Probability 1.1 Chapter Highlights 1.2 Definition of Probability 1.3 Conditional Probability 1.4 Bayes Theorem 1.5 Independent Events 1.6 Random Variables 1.7 Conditional Distributions and Densities 1.8 Functions of One Random Variable 1.9 Moments of a Random Variable 1.10 Distributions with Two Random Variables 1.11 Multiple Random Variables 1.12 Mean-Square Error (MSE) Estimation 1.13 Bibliographical Notes 1.14 Problems Chapter 2 Stochastic Processes 2.1 Chapter Highlights 2.2 Stationary Processes 2.3 Cyclostationary Processes 2.4 Averages and Ergodicity 2.5 Autocorrelation Function 2.6 Power Spectral Density 2.7 Discrete-Time Stochastic Processes 2.8 Spatial Stochastic Processes 2.9 Random Signals 2.10 Bibliographical Notes 2.11 Problems Chapter 3 Signal Representations and Statistics 3.1 Chapter Highlights 3.2 Relationship of Power Spectral Density and Autocorrelation Function 3.3 Sampling Theorem 3.4 Linear Time-Invariant and Linear Shift-Invariant Systems 3.5 Bandpass Signal Representations 3.6 Bibliographical Notes 3.7 Problems Part II Detection Chapters Chapter 4 Single Sample Detection of Binary Hypotheses 4.1 Chapter Highlights 4.2 Hypothesis Testing and the MAP Criterion 4.3 Bayes Criterion 4.4 Minimax Criterion 4.5 Neyman-Pearson Criterion 4.6 Summary of Detection-Criterion Results Used in Chapter 4 Examples 4.7 Sequential Detection 4.8 Bibliographical Notes 4.9 Problems Chapter 5 Multiple Sample Detection of Binary Hypotheses 5.1 Chapter Highlights 5.2 Examples of Multiple Measurements 5.3 Bayes Criterion 5.4 Other Criteria 5.5 The Optimum Digital Detector in Additive Gaussian Noise 5.6 Filtering Alternatives 5.7 Continuous Signals White Gaussian Noise 5.8 Continuous Signals Colored Gaussian Noise 5.9 Performance of Binary Receivers in AWGN 5.10 Further Receiver-Structure Considerations 5.11 Sequential Detection and Performance 5.12 Bibliographical Notes 5.13 Problems Chapter 6 Detection of Signals with Random Parameters 6.1 Chapter Highlights 6.2 Composite Hypothesis Testing 6.3 Unknown Phase 6.4 Unknown Amplitude 6.5 Unknown Frequency 6.6 Unknown Time of Arrival 6.7 Bibliographical Notes 6.8 Problems Chapter 7 Multiple Pulse Detection with Random Parameters 7.1 Chapter Highlights 7.2 Unknown Phase 7.3 Unknown Phase and Amplitude 7.4 Diversity Approaches and Performances 7.5 Unknown Phase, Amplitude, and Frequency 7.6 Bibliographical Notes 7.7 Problems Chapter 8 Detection of Multiple Hypotheses 8.1 Chapter Highlights 8.2 Bayes Criterion 8.3 MAP Criterion 8.4 M-ary Detection Using Other Criteria 8.5 M-ary Decisions with Erasure 8.6 Signal-Space Representations 8.7 Performance of M-ary Detection Systems 8.8 Sequential Detection of Multiple Hypotheses 8.9 Bibliographical Notes 8.10 Problems Chapter 9 Nonparametric Detection 9.1 Chapter Highlights 9.2 Sign Tests 9.3 Wilcoxon Tests 9.4 Other Nonparametric Tests 9.5 Bibliographical Notes 9.6 Problems Part III Estimation Chapters Chapter 10 Fundamentals of Estimation Theory 10.1 Chapter Highlights 10.2 Formul. Bookseller Inventory # ABE_book_new_0130894990
Book Description Prentice Hall, 2006. Paperback. Book Condition: New. 1. Bookseller Inventory # DADAX0130894990
Book Description Pearson, 2006. Paperback. Book Condition: New. Bookseller Inventory # P110130894990
Book Description Prentice Hall, 2006. Paperback. Book Condition: New. book. Bookseller Inventory # 0130894990
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s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719416.57/warc/CC-MAIN-20161020183839-00543-ip-10-171-6-4.ec2.internal.warc.gz
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CC-MAIN-2016-44
| 4,826 | 10 |
https://www.electrical-contractor.net/forums/ubbthreads.php/topics/98266/load-calculation.html
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math
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What numbers did you come up with?
Did you include "House Circuits" in your calcs? (exterior lighting, specific loads, etc.)
Are the units using Non-Electric Cooking equipment?
Any Elevators? (possible ADA requirements),
You mention this is a "Renovation", what are the existing Service spec's and details?
Also, what is fed at each unit?
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s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247479885.8/warc/CC-MAIN-20190216045013-20190216071013-00177.warc.gz
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CC-MAIN-2019-09
| 338 | 6 |
https://beta.aotg.com/tag/sound-designer/
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math
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A community-driven source for post-production information
December 11, 2020, 12:17 PM
Editor Mikeel Nielsen and Sound Designer Nicolas Becker discuss creating the film The Sound of Metal. We explore how to make a movie internationally, when the entire crew is located in separate countries.#editor#editing#film#film editing#television#movies#sound designer#the cutting room#mikkel nielsen#nicolas becker#the sound of metal
September 21, 2020, 11:34 AM
Gordon sits down with Wade Barnett(Supervising Sound Editor) and David Barbee (Sound Designer) to discuss their work on Amazon Prime's hit show The Boys.#editor#editing#film#film editing#television#movies#sound designer#the cutting room#wade barnett#david barbee#the boys
September 16, 2015, 05:30 AM
Some people say I am a sound designer. But am I? Really? Do I bring my sound expertise to each step of the creative process? Unfortunately, more often than not, I am brought in at the end of a video project to “do my thing” after all the writing, re-writing, shooting, re-shooting and editing have been done. Let’s consider for a minute if more thought had been given to sound during all the stages of the creative process.#audio#post production#sound designer
Cindy Mollo sits down to discuss the editing of Ozark.
Greg O'Bryant sits down with Gordon to discuss the editing of Brand New Cherry Flavor for Netflix an...
Yan Miles sits down to discuss editing The Crown's hit episode Fairytale.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585196.73/warc/CC-MAIN-20211018031901-20211018061901-00388.warc.gz
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CC-MAIN-2021-43
| 1,773 | 11 |
https://www.commonsense.org/education/website/mathia
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math
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How to address violence in the news with your students.
MATHia works best as a supplement to classroom instruction and is more effective if students enter with an understanding of the math concepts already. Teachers can use MATHia to give students more practice on what they're working on in class. It can be assigned instead of paper worksheets or in addition to other classwork. MATHia could be used for homework and is a great way for students to practice for standardized computer-based tests.
MATHia is ideal in a hybrid classroom where some students work online and others meet with a teacher. Teachers can track student progress carefully, and the data can be used to plan small-group instruction or individual conferences. The details provided by the progress bar are perfect conversation starters for struggling students.Continue reading Show less
MATHia is a sixth-grade through pre-algebra math platform that aligns closely with the Common Core State Standards. It's designed to be used alongside Carnegie Learning's math curriculum or other math curricula. Teachers set up classes and can choose to follow the predetermined scope and sequence or design their own pathways. Detailed reports are provided for teachers, including standards mastery and growth data. However, no school-level reports are available yet. Seven modules are provided for each grade level, and each session takes at least 30 minutes to complete.
MATHia includes two types of problems: mastery and non-mastery. The non-mastery problems are used to preview or review a topic and aren't adaptive. The mastery-based problems are adaptive and require students to reach a high level of conceptual understanding before moving on. The student dashboard provides a look at progress, time on task, and upcoming work. Unit overviews include a list of vocabulary, major math concepts covered, and a short video explaining how the math connects to real-life situations. Once students begin a problem, MATHia provides instant feedback on each step of the progress. Hints will pop up to address misconceptions, and students can also ask for a hint at any time. A highly detailed tracking bar advances and retreats based on how accurately the questions are answered. The progress bar breaks the task down into smaller steps in the process, such as naming units, identifying variables correctly, and so on.
MATHia is a great tool for supporting young mathematicians. From a student perspective, it's easy to use and has few distractions. The math problems are rigorous and usually involve multiple ways to model thinking. Hints pop up automatically if students are stuck, but students won't make progress if they use too many hints or try to guess their way through the system. Teachers and students can use the progress bar to see exactly where they're succeeding and what needs more practice. MATHia won't let students proceed to another unit until mastery is reached on the progress bar. This means that some students may reach mastery after a few problems, while other students will be required to complete more problems to show mastery. This is excellent for learning but may prove frustrating to some learners.
Unlike Khan Academy, MATHia doesn't let students freely explore other topics, nor do students earn as many badges. By itself, MATHia doesn't provide text-to-speech support, but it does play well with the Read&Write Chrome extension. While it's an excellent tool for practicing and gaining mastery of math concepts, MATHia could be even better if teachers could easily assign individual students customized pathways.
Key Standards Supported
Expressions And Equations
Write and evaluate numerical expressions involving whole-number exponents.
Write, read, and evaluate expressions in which letters stand for numbers.
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole- number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.
Apply the properties of operations to generate equivalent expressions.
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Solve linear equations in one variable.
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Analyze and solve pairs of simultaneous linear equations.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Verify experimentally the properties of rotations, reflections, and translations:
Lines are taken to lines, and line segments to line segments of the same length.
Angles are taken to angles of the same measure.
Parallel lines are taken to parallel lines.
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Ratios And Proportional Relationships
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 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 hour, equivalently 2 miles per hour.
Recognize and represent proportional relationships between quantities.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Statistics And Probability
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Summarize numerical data sets in relation to their context, such as by:
Reporting the number of observations.
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
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s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00152.warc.gz
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CC-MAIN-2022-21
| 26,440 | 105 |
http://www.homebrewtalk.com/f12/oatmeal-bourbon-barrel-stout-recipe-help-please-234051/
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math
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My initial thought had been to do a mini-mash with 3# 2-row (suppose I could go pale malt), but figured I would make it simpler with just extract. I didn't want to go over the top with a big beer (at least not yet) as I'd want to drink multiple in a sitting.
How does this sound:
3# pale malt
3.3# light (or extra light) DME
.75# roasted barley
.75# chocolate barley
2 oz Kent Goldings @60min
Mini-mash pale malt and oats. I have the oats, chocolate, and roasted in my inventory already, so I just upped the amount to what I have in stock.
Does this sound better?
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s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122189854.83/warc/CC-MAIN-20150124175629-00226-ip-10-180-212-252.ec2.internal.warc.gz
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CC-MAIN-2015-06
| 563 | 9 |
https://schoolbag.info/sat/sat_1/55.html
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math
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SAT Test Prep
ESSENTIAL ALGEBRA 2 SKILLS
Lesson 2: Functions
What Is a Function?
A function is any set of instructions for turning an input number (usually called x) into an output number (usually called y). For instance, is a function that takes any input x and multiplies it by 3 and then adds 2. The result is the output, which we call f (x) or y.
If , what is f (2h)?
In the expression f (2h), the 2h represents the input to the function f. So just substitute 2h for x in the equation and simplify: .
Functions as Equations, Tables, or Graphs
The SAT usually represents a function in one of three ways: as an equation, as a table of inputs and outputs, or as a graph on the xy-plane. Make sure that you can work with all three representations. For instance, know how to use a table to verify an equation or a graph, or how to use an equation to create or verify a graph.
A linear function is any function whose graph is a line. The equations of linear functions always have the form , where m is the slope of the line, and b is where the line intersects the y-axis. (For more on slopes, see Chapter 10, Lesson 4.)
The function is linear with a slope of 3 and a y-intercept of 2. It can also be represented with a table of x and y (or f (x)) values that work in the equation:
Notice several important things about this table. First, as in every linear function, when the x values are “evenly spaced,” the y values are also “evenly spaced.” In this table, whenever the x value increases by 1, the y value increases by 3, which is the slope of the line and the coefficient ofx in the equation. Notice also that the y-intercept is the output to the function when the input is 0.
Now we can take this table of values and plot each ordered pair as a point on the xy-plane, and the result is the graph of a line:
The graph of a quadratic function is always a parabola with a vertical axis of symmetry. The equations of quadratic functions always have the form , where c is the y-intercept. When a (the coefficient of x2) is positive, the parabola is “open up,” and when a is negative, it is “open down.”
The graph above represents the function . Notice that it is an “open down” parabola with an axis of symmetry through its vertex at .
The figure above shows the graph of the function f in the xy-plane. If , which of the following could be the value of b?
Although this can be solved algebraically, you should be able to solve this problem more simply just by inspecting the graph, which clearly shows that . (You can plug into the equation to verify.) Since this point is two units from the axis of symmetry, its reflection is two units on the other side of the axis, which is the point (4, –3).
Concept Review 2: Functions
1. What is a function?
2. What are the three basic ways of representing a function?
3. What is the general form of the equation of a linear function, and what does the equation tell you about the graph?
4. How can you determine the slope of a linear function from a table of its inputs and outputs?
5. How can you determine the slope of a linear function from its graph?
6. What is the general form of the equation of a quadratic function?
7. What kind of symmetry does the graph of a quadratic function have?
SAT Practice 2: Functions
1. The graphs of functions f and g for values of x between –3 and 3 are shown above. Which of the following describes the set of all x for which
2. If and , which of the following could be g (x)?
(A) 3 x
3. What is the least possible value of if
4. The table above gives the value of the linear function f for several values of x. What is the value of
(E) It cannot be determined from the information given.
5. The graph on the xy-plane of the quadratic function g is a parabola with vertex at (3, –2). If , then which of the following must also equal 0?
(A) g (2)
(B) g (3)
(C) g (4)
(D) g (6)
(E) g (7)
6. In the xy-plane, the graph of the function h is a line. If and , what is the value of h (0)?
Answer Key 2: Functions
Concept Review 2
1. A set of instructions for turning an input number (usually called x) into an output number (usually called y).
2. As an equation (as in ), as a table of input and output values, and as a graph in the xy-plane.
, where m is the slope of the line and b is its y-intercept.
4. If the table provides two ordered pairs, (x1, y1) and (x2, y2), the slope can be calculated with . (Also see Chapter 10, Lesson 4.)
5. Choose any two points on the graph and call their coordinates (x1, y1) and (x2, y2). Then calculate the slope with .
, where c is the y-intercept.
7. It is a parabola that has a vertical line of symmetry through its vertex.
SAT Practice 2
1. C In this graph, saying that is the same as saying that the g function “meets or is above” the f function. This is true between the points where they meet, at and .
2. B Since , f (g (1)) must equal . Therefore and . So g (x) must be a function that gives an output of 4 when its input is 1. The only expression among the choices that equals 4 when is .
3. D This question asks you to analyze the “outputs” to the function given a set of “inputs.” Don”t just assume that the least input, –3, gives the least output, . In fact, that”s not the least output. Just think about the arithmetic: is the square of a number. What is the least possible square of a real number? It must be 0, because 02 equals 0, but the square of any other real number is positive. Can in this problem equal 0? Certainly, if , which is in fact one of the allowed values of x. Another way to solve the problem is to notice that the function is quadratic, so its graph is a parabola. Choose values of x between –3 and 0 to make a quick sketch of this function to see that its vertex is at (–2, 0).
4. C Since f is a linear function, it has the form . The table shows that an input of 3 gives an output of 8, so . Now, if you want, you can just “guess and check” values for m and b that work, for instance, and . This gives the equation . To find the missing outputs in the table, just substitute and then : and . Therefore, . But how do we know that will always equal 16? Because the slope m of any linear function represents the amount thaty increases (or decreases) whenever x increases by 1. Since the table shows x values that increase by 1, a must equal , and b must equal 8 + m. Therefore .
5. D Don”t worry about actually finding the equation for g (x). Since g is a quadratic function, it has a vertical line of symmetry through its vertex, the line . Since , the graph also passes through the origin. Draw a quick sketch of a parabola that passes through the origin and (3, –2) and has an axis of symmetry at :
The graph shows that the point (0, 0), when reflected over the line , gives the point (6, 0). Therefore g (6) is also equal to 0.
6. D The problem provides two ordered pairs that lie on the line: (–1, 4) and (5, 1). Therefore, the slope of this line is . Therefore, for every one step that the line takes to the right (the x direction), the y value decreases by ½. Since 0 is one unit to the right of –1 on the x-axis, h (0) must be1/2 less than h ((–1), or .
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100555.27/warc/CC-MAIN-20231205172745-20231205202745-00786.warc.gz
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CC-MAIN-2023-50
| 7,170 | 56 |
https://goodriddlesnow.com/riddles/by/kids-riddles/page:2/sort:difficulty/direction:desc
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math
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Question: A man wants to have a party in thirty-one days where he will be serving his 1000 barrels of wine. The only problem is that one of his enemies poisoned one of the barrels. The poison kills any man who drinks any of the wine in about 30 days, give or take a few hours. The man has 10 plants that are also killed by the poison in 30 days and can be used to test the wine. How can identify the single poisoned barrel of wine?
Answer: To do this the man must create 1000 unique groups from the 10 plants in which each group has between 1 and 10 plants, and give each plant wine from a different barrel. He can then throw away the barrel of wine that corresponds to the group that died from the wine.
You have a large number of friends coming over and they all get thirsty. Your first friend asks for 1/2 a cup of water. Your second friend asks for 1/4 a cup of water. Your third friend asks for 1/8 a cup of water, etc.
How many cups of water do you need to serve your friends?
Answer: Just one. If your friends kept asking for water like this forever one cup would be enough.
John has some chickens that have been laying him plenty of eggs. He wants to give away his eggs to several of his friends, but he wants to give them all the same number of eggs. He figures out that he needs to give 7 of his friends eggs for them to get the same amount, otherwise there is 1 extra egg left.
What is the least number of eggs he needs for this to be true?
Answer: 301 eggs. The number of eggs must be one more than a number that is divisible by 2, 3, 4, 5, and 6 since each of these numbers leave a remainder of 1. For this to be true one less than the number must be divisible by 5, 4, and 3 (6 is 2*3 and 2 is a factor of 4 so they will automatically be a factor). 5 * 4 * 3 = 60. Then you just must find a multiple of 60 such that 60 * n + 1 is divisible by 7. 61 / 7, 121 / 7, 181 / 7, 241 / 7 all leave remainders but 301 / 7 doesn't.
Question: Queens can move horizontally, vertically and diagonally any number of spaces as illustrated. One piece 'attacks' another if it moves to the same tile that the other piece is on. How can you arrange eight queens on the board so they cannot attack each other?
Hint: Four must go on black and four on white.
Answer: Here are the two solutions. This is usually solved with guess and check although using logic may be faster. We know that each queen must be in it's own row vertically and horizontally. We also know that 4 of the queens must be on white and 4 on black. This is true because with 4 queens on the same color all of the rest of that color is venerable to attack. (It could be done with math).
Follow us and get the Riddle of the Day, Joke of the Day, and interesting updates.
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CC-MAIN-2023-40
| 2,730 | 12 |
https://acikerisim.isikun.edu.tr/xmlui/handle/11729/2567
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math
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Some new classes of graceful diameter six trees
MetadataShow full item record
CitationPanda, A. C. & Mishra, D. (2015). Some new classes of graceful diameter six trees. TWMS Journal of Applied and Engineering Mathematics, 5(2), 269-275.
Here we denote a diameter six tree by (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where a0 is the center of the tree; ai, i = 1, 2, . . . , m, bj , j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to a0; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful labelings to some new classes of diameter six trees (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr) in which the branches of a diameter four tree incident on a0 are of same type, i.e. either they are all odd branches or even branches. Here by a branch we mean a star, i.e. we call a star an odd branch if its center has an odd degree and an even branch if its center has an even degree.
SourceTWMS Journal of Applied and Engineering Mathematics
The following license files are associated with this item:
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CC-MAIN-2022-49
| 1,156 | 6 |
https://byjusexamprep.com/uppsc-ae-paper-1-quiz-16-i-de50e6d0-35ec-11eb-8066-f9ac7b7b0153
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math
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If the input to the circuit of figure is a sine wave. The output will be:
A simple equivalent circuit of the two-terminal network shown in the figure is:
A solenoid having cross-sectional area of 25 cm2. If solenoid has 80 primary turn/cm & 160 secondary turns over the circumference. Calculate the mutual inductance for the solenoid.
Which of the following are advantages of using bundled conductors?
(i) Reduced Reactance
(ii) Reduced Voltage gradient
(iii) Reduced surge impedance
In a four-unit insulator string, voltage across the lowest unit is 20kV and string efficiency is 85%.
The transmission lines voltage, will be:
Which relay is used for the protection of electrical machines against overheating?
The results of a slip test for determining direct axis (Xd) and quadrature axis
(Xs) reactance of a star connected salient pole are given below.
Vmax = 150V Imax =12 A
Vmin = 90V Imin =9 A
Hence the value of both reactance will be:
For a single phase 440/110 V transformer having input on primary side at no load is 100 W. The low voltage winding is kept open & the no load power factor is 0.4. Determine the active & reactive component of current at no load condition?
A single-phase energy meter which is designed to make 120 revolution for one unit of energy. What will be the number of revolution when a load connected to the energy meter which carrying a current of 30 A at 220 V supply at a power factor of 0.8 lagging when the meter is running for 2 hour?
In an A.C. bridge which is shown in figure below, if Z1 = 100 Ω,Z2 = (200 + j150)Ω &Z3 = (300 + j120)Ω,then find the unknown impedance Z4 if the bridge is in balanced condition.
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CC-MAIN-2023-50
| 1,653 | 18 |
https://forum.ansys.com/forums/topic/varying-load-over-1-step-vs-multiple-steps/
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math
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April 5, 2020 at 6:13 pmNyuronsSubscriber
In transient structural, what is the difference between applying a load that varies in its magnitude over 1 step vs multiple time steps (like 10 steps, for example)? Which method is considered more conventional? Please see the picture below for your reference. Both represent a simulation of 1 second.
Also, I will be coupling this transient structural with fluid flow (Fluent). Would the "1 step method" or "multiple steps method" affect convergence during coupling?
Thank you so much for your help.
April 5, 2020 at 6:50 pmpeteroznewmanSubscriber
It is easier to have 1 step and let the substeps define the intermediate points.
One reason to have 10 steps is if you need data at exactly 1 second increments, and the automatic substeps might not land on an exact integer time.
April 5, 2020 at 8:37 pmNyuronsSubscriber
Thank you for your quick reply.
Can you please elaborate on what becomes "easier"?
If I chose to do the 1 second simulation with varying load over 1 step (10 increments like the photo),
- Is 10 the appropriate Number of Substeps?
- Would the simulation be affected in any way if I chose 100 substeps?
- What should the Step Size be in Fluent/System Coupling?
Again, thank you so much for your help.
Edit: Ideally I want 100 "pictures" or data over 1 second (so, "pictures" at every 0.01s).
April 6, 2020 at 1:22 ampeteroznewmanSubscriber
If you have 10 steps, the Analysis Settings for each step have to be done one at a time, which is one reason one step is easier.
If you want 100 pictures, then just set the Initial and Minimum Substeps to 100. The Maximum substeps can be 200.
There is no penalty in using 100, it just makes the solution take more time to finish because more data has to be computed and written.
April 6, 2020 at 1:30 amNyuronsSubscriber
Thank you again for your help.
Ok, so if I was to get 100 "pictures" or data using 100 substeps, is it correct to have 0.01s for my Step Size and End Time of 1s in System Coupling?
Thank you for being patient with my questions!
April 6, 2020 at 1:33 ampeteroznewmanSubscriber
Yes, that is correct if both systems can converge with that time step.
You might need smaller time steps so that each system can converge.
April 6, 2020 at 1:42 amNyuronsSubscriber
Thank you. I am amazed that you have mentioned convergence because I am actually facing convergence issues. Fluent side of system coupling does not seem to converge from around 0.35s (and ANSYS gets stuck at around 0.40s). How can I improve the chances of convergence from the aspect of time step or step size?
April 6, 2020 at 11:11 ampeteroznewmanSubscriber
Is the error on the Structural side actually fails to converge or is the error highly distorted elements, because there are different corrective actions. But in general, smaller time steps may be necessary to obtain convergence, but they may not be sufficient to obtain convergence. Better element quality may also be required or some other changes that are very model dependent.
There is a whole workflow laid out in the ANSYS Help for developing a model of coupled systems, which entails solving each physics without the coupling first. Have you done that?
April 6, 2020 at 12:38 pmNyuronsSubscriber
Thank you for your reply. The Transient Structural side does seem to converge according to the solution information on system coupling. However, the Fluent side starts to not converge in the middle of the system coupling.
I have only skimmed through the System Coupling User's Guide, so I will take a look at it. While I cannot find the exact cause of the error, highly distorted elements are most likely the large factor in the error (or the Setup of the Fluent side).
- You must be logged in to reply to this topic.
Simulation World 2022
Earth Rescue – An Ansys Online Series
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- An Unknown error occurred during solution. Check the Solver Output…..
- Saving & sharing of Working project files in .wbpz format
- Solver Pivot Warning in Beam Element Model
- Understanding Force Convergence Solution Output
- whether have the difference between using contact and target bodies
- Colors and Mesh Display
- The solver engine was unable to converge on a solution for the nonlinear problem as constrained.
- Massive amount of memory (RAM) required for solve
- What is the difference between bonded contact region and fixed joint
© 2022 Copyright ANSYS, Inc. All rights reserved.
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CC-MAIN-2022-49
| 4,493 | 49 |
https://www.getinsuronline.info/tag/museum
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math
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Our research specialties are in algebra, evaluation, geometry, quantity concept, likelihood and topology. It is designed for anybody who wants a basic to advanced understanding of mathematics concepts and operations. Institute for Pure and Utilized Mathematics (IPAM), UCLA, Los Angeles, California. Particularly, while other philosophies of mathematics permit objects that may be proved to exist despite the fact that they cannot be constructed, intuitionism allows solely mathematical objects that one can actually construct.
Haskell Curry outlined mathematics merely as “the science of formal techniques”. And so, this too can’t be a precise science the place everything will reply a mathematical equation. College students will learn ideas in a more organized approach both in the course of the faculty 12 months and throughout grades.
Microsoft Mathematics provides a graphing calculator that plots in SECOND and 3D, step-by-step equation fixing, and helpful instruments to assist students with math and science studies. If you are taking the Mathematics Level 1 test, ensure that your calculator is in diploma mode forward of time so you won’t have to worry about it through the check.
In addition to psychological math actions, youngsters must be given grade degree mathematics workbooks to strengthen their math abilities. The …
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CC-MAIN-2023-40
| 1,346 | 4 |
https://www.true-cost-mortgage.com/?show=annuity&idioma=english
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math
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Annuity is an annual instalment rate you have to pay for your mortgage
This annuity consists of the interest and the repayment of the 1st year.
e.g.: With a mortgage of £100.000 , an interest rate of 6% and a first repayment of 1%, the annuity is calculated as follows:
This annuity of £7.000 is fixed until the penultimate year, whereby the amount of interest is decreasing, while the repayment is increasing. In the 2nd year you don't have to pay the interest on £100.000 , only for £99.000 (You have already paid £1.000 repayment for the 1st year.)
The interest therefore in the second year is only £5.940 , the annuity fixed as £7.000 determines that you repay £1.060 .
This calculation continues until the penultimate year:
In last year, the annuity is the sum of the interest on the outstanding debt plus the remaining debt itself. In our example:
£2.656,84 repayment (equal to the remaining debt from the 33rd. year)
£2.816,25 last annuity
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CC-MAIN-2022-49
| 956 | 9 |
https://www.hardreset.info/devices/cect/cect-i361/
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math
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Hard Reset CECT I361
- Make sure, that your phone is on.
- Then from the main screen type: *#6810# + call button.
- After that tap this: *#337# + call button.
Help! This doesn't work
Hard Reset will erase all of your data
All described operations you are doing at your own risk.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590559.95/warc/CC-MAIN-20180719051224-20180719071224-00586.warc.gz
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CC-MAIN-2018-30
| 278 | 7 |
https://www.conversioncenter.net/length-conversion/from-mile-(statute)-to-vara-(Castile)
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math
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Convert mile (statute) to vara (Castile)
Selected category: length.
Definition and details for mile (statute):
In 1592 statute mile was defined to be 5280 feet. Using the international definition of the foot as exactly 30.48 centimeters, the international statute mile is exactly 1609.344 meters.
Definition and details for vara (Castile):
Vara is the Spanish yard that varied in size at various times and places. Vara of Castile was the standard vara. It was equal to 0.8359 m
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CC-MAIN-2022-05
| 477 | 6 |
https://forums.t-nation.com/t/metabolic-rate-body-temp-question/156617
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math
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There have been many posts regarding low body temps due to thyroid disfunction so I started thinking...
Would I be somewhat correct in my assumption that for every degree of body temp lower than ideal that we leave a lot of calories on the table so to speak?
Would this formula be somewhat correct as general rule of thumb?
Body Weight in Kg = BW
Degrees of waking basal body temp less than ideal= TD
Calories burned deficit= CD
In my case my TD is 2.3. My BW is 105.9. Does that mean I burn 243 calories per day less than someone with identical mass and an ideal body temp?
I do realize that there are many variables that would alter the basic calorie formula, but I'm, figuring that our bodies are supposedly 61% water and the other materials aren't too far from the same density.
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CC-MAIN-2018-09
| 782 | 8 |
https://www.logobook.ru/prod_show.php?object_uid=12470864
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math
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Автор: Ghosal, Subhashis. Название: Fundamentals of Nonparametric Bayesian Inference ISBN: 0521878268 ISBN-13(EAN): 9780521878265 Издательство: Cambridge Academ Рейтинг: Цена: 7077 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics.
Описание: This book treats the latest developments in the theory of order-restricted inference, with special attention to nonparametric methods and algorithmic aspects. Among the topics treated are current status and interval censoring models, competing risk models, and deconvolution. Methods of order restricted inference are used in computing maximum likelihood estimators and developing distribution theory for inverse problems of this type. The authors have been active in developing these tools and present the state of the art and the open problems in the field. The earlier chapters provide an introduction to the subject, while the later chapters are written with graduate students and researchers in mathematical statistics in mind. Each chapter ends with a set of exercises of varying difficulty. The theory is illustrated with the analysis of real-life data, which are mostly medical in nature.
Автор: Efromovich Название: Nonparametric Curve Estimation ISBN: 0387987401 ISBN-13(EAN): 9780387987408 Издательство: Springer Рейтинг: Цена: 15427 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Gives an introduction to nonparametric curve estimation theory.
Автор: Ferraty Название: Nonparametric Functional Data Analysis ISBN: 0387303693 ISBN-13(EAN): 9780387303697 Издательство: Springer Рейтинг: Цена: 12154 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Modern apparatuses allow us to collect samples of functional data, mainly curves but also images. On the other hand, nonparametric statistics produces useful tools for standard data exploration. This book links these two fields of modern statistics by explaining how functional data can be studied through parameter-free statistical ideas.
Автор: Neuhauser Название: Nonparametric Statistical Tests ISBN: 1439867038 ISBN-13(EAN): 9781439867037 Издательство: Taylor&Francis Рейтинг: Цена: 17243 р. Наличие на складе: Невозможна поставка.
Описание: Nonparametric Statistical Tests: A Computational Approach describes classical nonparametric tests, as well as novel and little-known methods such as the Baumgartner-Weiss-Schindler and the Cucconi tests. The book presents SAS and R programs, allowing readers to carry out the different statistical methods, such as permutation and bootstrap tests. The author considers example data sets in each chapter to illustrate methods. Numerous real-life data from various areas, including the bible, and their analyses provide for greatly diversified reading. The book covers: Nonparametric two-sample tests for the location-shift model, specifically the Fisher-Pitman permutation test, the Wilcoxon rank sum test, and the Baumgartner-Weiss-Schindler test Permutation tests, location-scale tests, tests for the nonparametric Behrens-Fisher problem, and tests for a difference in variability Tests for the general alternative, including the (Kolmogorov-)Smirnov test, ordered categorical, and discrete numerical data Well-known one-sample tests such as the sign test and Wilcoxon’s signed rank test, a modification suggested by Pratt (1959), a permutation test with original observations, and a one-sample bootstrap test are presented. Tests for more than two groups, the following tests are described in detail: the Kruskal-Wallis test, the permutation F test, the Jonckheere-Terpstra trend test, tests for umbrella alternatives, and the Friedman and Page tests for multiple dependent groups The concepts of independence and correlation, and stratified tests such as the van Elteren test and combination tests The applicability of computer-intensive methods such as bootstrap and permutation tests for non-standard situations and complex designs Although the major development of nonparametric methods came to a certain end in the 1970s, their importance undoubtedly persists. What is still needed is a computer assisted evaluation of their main properties. This book closes that gap.
Описание: Incorporating a hands-on pedagogical approach, Nonparametric Statistics for Social and Behavioral Sciences presents the concepts, principles, and methods used in performing many nonparametric procedures. It also demonstrates practical applications of the most common nonparametric procedures using IBM’s SPSS software. This text is the only current nonparametric book written specifically for students in the behavioral and social sciences. Emphasizing sound research designs, appropriate statistical analyses, and accurate interpretations of results, the text: Explains a conceptual framework for each statistical procedure Presents examples of relevant research problems, associated research questions, and hypotheses that precede each procedure Details SPSS paths for conducting various analyses Discusses the interpretations of statistical results and conclusions of the research With minimal coverage of formulas, the book takes a nonmathematical approach to nonparametric data analysis procedures and shows students how they are used in research contexts. Each chapter includes examples, exercises, and SPSS screen shots illustrating steps of the statistical procedures and resulting output.
Описание: While preserving the clear, accessible style of previous editions, this fourth edition reflects the latest developments in computer-intensive methods that deal with intractable analytical problems and unwieldy data sets. This edition summarizes relevant general statistical concepts and introduces basic ideas of nonparametric or distribution-free methods. Designed experiments, including those with factorial treatment structures, are now the focus of an entire chapter. The book also expands coverage on the analysis of survival data and the bootstrap method. The new final chapter focuses on important modern developments. With numerous exercises, the text offers the student edition of StatXact at a discounted price.
Описание: Presenting an extensive set of tools and methods for data analysis, this second edition includes more models and methods and significantly extends the possible analyses based on ranks. It contains a new section on rank procedures for nonlinear models, a new chapter on models with dependent error structure, and new material on the development of computationally efficient affine invariant/equivariant sign methods based on transform-retransform techniques in multivariate models. The authors illustrate the methods using many real-world examples and R. Information about the data sets and R packages can be found at www.crcpress.com
Описание: Designed for a graduate course in applied statistics, Nonparametric Methods in Statistics with SAS Applications teaches students how to apply nonparametric techniques to statistical data. It starts with the tests of hypotheses and moves on to regression modeling, time-to-event analysis, density estimation, and resampling methods. The text begins with classical nonparametric hypotheses testing, including the sign, Wilcoxon sign-rank and rank-sum, Ansari-Bradley, Kolmogorov-Smirnov, Friedman rank, Kruskal-Wallis H, Spearman rank correlation coefficient, and Fisher exact tests. It then discusses smoothing techniques (loess and thin-plate splines) for classical nonparametric regression as well as binary logistic and Poisson models. The author also describes time-to-event nonparametric estimation methods, such as the Kaplan-Meier survival curve and Cox proportional hazards model, and presents histogram and kernel density estimation methods. The book concludes with the basics of jackknife and bootstrap interval estimation. Drawing on data sets from the author’s many consulting projects, this classroom-tested book includes various examples from psychology, education, clinical trials, and other areas. It also presents a set of exercises at the end of each chapter. All examples and exercises require the use of SAS 9.3 software. Complete SAS codes for all examples are given in the text. Large data sets for the exercises are available on the author’s website.
Описание: Consists of 22 research papers in Probability and Statistics. This title includes topics such as nonparametric inference, nonparametric curve fitting, linear model theory, Bayesian nonparametrics, change point problems, time series analysis and asymptotic theory. It presents research in statistical theory.
Описание: Nonparametric techniques in statistics are those in which the data are ranked in order according to some particular characteristic. When applied to measurable characteristics, the use of such techniques often saves considerable calculation as compared with more formal methods, with only slight loss of accuracy. The field of nonparametric statistics is occupying an increasingly important role in statistical theory as well as in its applications. Nonparametric methods are mathematically elegant, and they also yield significantly improved performances in applications to agriculture, education, biometrics, medicine, communication, economics and industry.
Автор: Brodsky, E., Darkhovsky, B.S. Название: Nonparametric Methods in Change Point Problems ISBN: 0792321227 ISBN-13(EAN): 9780792321224 Издательство: Springer Рейтинг: Цена: 8882 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: This volume deals with non-parametric methods of change point (disorder) detection in random processes and fields. A systematic account is given of up-to-date developments in this rapidly evolving branch of statistics.
ООО "Логосфера " Тел:+7(495) 980-12-10 www.logobook.ru
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| 11,369 | 18 |
http://www.mitp.ru/en/publ/abs_vol5/abs04.html
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math
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On reducing the order of Rayleigh system. Protect II. Cases of reduction
We study two sets of differential equations in unknown parameters of a flat layered medium obtained in the first part of the work. The first set represents conditions for the system of equations for P-SV vibration transforms to reduce to two Sturm-Liouville's equations. The second set expresses conditions for the same system of equations to be factored. Solutions to each of these sets depend on an arbitrary function. We integrated both sets by quadratures and found the first integral of the first set. We proved the first set to be equivalent to the system found by Alverson, Gair, and Hook and later by Young. The second set, being more general than the first one, strengthens their result. We obtained the system of simplified equations for P-SV vibrations in the case where a medium obeys the first set. We considered a class of media where parameters are power functions of depth and obey the first set. This class is capacious; is contains the examples given by Gupta and Young and some media indicated by Hook. We show that media considered by Pekeris and Zvolinskii obey the second set but not the first. The same is true for the elasticity operator, which factores into two Dirac's operators and thereby demonstrates some parallelism between the theory of elasticity and the quantum theory.
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CC-MAIN-2017-51
| 1,376 | 2 |
https://quiztablet.com/at-what-rate-would-a-sum-of-n100-deposited-for-five-years-raise/
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math
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At what rate would a sum of N100 deposited for five years raise…
At what rate would a sum of
N100.00 deposited for five years raise an interest of N7.50?
B. 2 ½ %
C. 1 ½ %
Recall that simple interest = PRT/100
Where P = principal = 100
R = rate = ?
T = time = 5
Interest = 7.5
So, 7.5 = (
100 × R × 5)/ 100
7.5 = 5R
R = 7.5/5 = 1.5% or 1 ½ %
Now for the right answer to the above question:
- Option A is incorrect.
- Option B is incorrect.
- C is correct.
- D is not the correct answer.
You may please note these/this:
- Simple interest = PRT/100
- Always ensure that the time is in years, if the time was given in months, the you will have to convert to years.
- If you love our answers, you can login to comment and say hi to us at the comment section…
To raise new related questions, click the ASK A QUESTION BOTTON down this page…
/ culled from 2020 JAMB-UTME mathematics past question 37 /
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CC-MAIN-2023-06
| 905 | 25 |
https://continuinged.uml.edu/catalog/catalogsearch_detail.cfm?courseid=14744&archrem=true
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math
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> Science and Mathematics
Note: This course is not available for the current semester.
Course No: MATH.2250;
Serves as a continuation of MATH.1260. This course covers integration by parts, integration of trigonometric integrals, trigonometric substitution, partial fraction, numeric integration, improper integrals, L'Hopital's Rule, indeterminate forms, sequences, infinite series, integral tests, comparison tests, alternating series tests, power series, Taylor series, polar coordinates, graphs and areas in polar coordinates, and parametric equations.
Prerequisites & Notes
- Prerequisites: MATH.1260
- Special Notes: MA
- Credits: 3;
Questions About This Course?
Contact UMass Lowell's Faculty and Student Support Center at 978-934-2474 or
Use the Back button in your browser to go back to search results
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CC-MAIN-2017-22
| 809 | 11 |
https://www.arxiv-vanity.com/papers/1406.6526/
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math
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Cameron-Liebler line classes with parameter
In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter for or , which generalizes the examples found by Rodgers in through a computer search. Furthermore, in the case where is an even power of , we construct the first infinite family of affine two-intersection sets in , which is closely related to our Cameron-Liebler line classes.
Key words and phrases:Cameron-Liebler line class, Klein correspondence, projective two-intersection set, affine two-intersection set, strongly regular graph, Gauss sum.
Cameron-Liebler line classes were first introduced by Cameron and Liebler in their study of collineation groups of having the same number of orbits on points and lines of . Later on it was found that these line classes have many connections to other geometric and combinatorial objects, such as blocking sets of , projective two-intersection sets in , two-weight linear codes, and strongly regular graphs. In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics; see, for example, [7, 20, 21, 25, 10, 11]. In , the authors gave several equivalent conditions for a set of lines of to be a Cameron-Liebler line class; Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of a Cameron-Liebler line class. Let be a set of lines of with , a nonnegative integer. We say that is a Cameron-Liebler line class with parameter if every spread of contains lines of . Clearly the complement of a Cameron-Liebler line class with parameter in the set of all lines of is a Cameron-Liebler line class with parameter . So without loss of generality we may assume that when discussing Cameron-Liebler line classes of parameter .
Let be any non-incident point-plane pair of . Following , we define to be the set of all lines through , and to be the set of all lines contained in the plane . We have the following trivial examples:
The empty set gives a Cameron-Liebler line class with parameter ;
Each of and gives a Cameron-Liebler line class with parameter ;
gives a Cameron-Liebler line class with parameter .
Cameron-Liebler line classes are rare. It was once conjectured ([5, p. 97]) that the above trivial examples and their complements are all of the Cameron-Liebler line classes. The first counterexample to this conjecture was given by Drudge in , and it has parameter . Later Bruen and Drudge generalized Drudge’s example into an infinite family with parameter for all odd . This represents the only known infinite family of nontrivial Cameron-Liebler line classes before our work. Govaerts and Penttila gave a sporadic example with parameter in . Recent work by Rodgers suggests that there are probably more infinite families of Cameron-Liebler line classes awaiting to be discovered. In , Rodgers obtained new Cameron-Liebler line classes with parameter for or and . In his thesis , Rodgers also reported new examples with parameters for and as joint work with his collaborators. These examples motivated us to find new general constructions of Cameron-Liebler line classes.
On the nonexistence side, Govaerts and Storme first showed that there are no Cameron-Liebler line classes in with parameter when is prime. Then De Beule, Hallez and Storme excluded parameters for all values . Next Metsch proved the non-existence of Cameron-Liebler line classes with parameter , and subsequently improved this result by showing the nonexistence of Cameron-Liebler line classes with parameter . The latter result represents the best asymptotic nonexistence result to date. It seems reasonable to believe that for any fixed and constant there are no Cameron-Liebler line classes with for sufficiently large . Very recently, Gavrilyuk and Metsch proved a modular equality which eliminates almost half of the possible values for a Cameron-Liebler line class with parameter . We refer to for a comprehensive survey of the known nonexistence results.
In the present paper we construct a new infinite family of Cameron-Liebler line classes with parameter for or . This family of Cameron-Liebler line classes generalizes the examples found by Rodgers in through a computer search. Furthermore, in the case where is an even power of , we construct the first infinite family of affine two-intersection sets, which is closely related to the newly constructed Cameron-Liebler line classes. The first step of our construction follows the same idea as in . That is, we prescribe an automorphism group for the Cameron-Liebler line classes that we intend to construct; as a consequence, the Cameron-Liebler line classes will be unions of orbits of the prescribed automorphism group on the set of lines of . The main difficulty with this approach is how to choose orbits properly so that their union is a Cameron-Liebler line class. We overcome this difficulty by giving an explicit choice of orbits so that their union gives a Cameron-Liebler line class with the required parameters. The details are given in Section 4.
The paper is organized as follows. In Section 2, we review basic properties of and facts on Cameron-Liebler line classes; furthermore, we collect auxiliary results on characters of finite fields, which are needed in the proof of our main theorem. In Section 3, we introduce a subset of , which we will use in the construction of our Cameron-Liebler line classes, and prove a few properties of the subset. In Section 4, we give an algebraic construction of an infinite family of Cameron-Liebler line classes with for or . In Section 5, we construct the first infinite family of affine two-intersection sets in , odd, whose existence was conjectured in the thesis of Rodgers. We close the paper with some concluding remarks.
In this section, we review basic facts on Cameron-Liebler line classes, and collect auxiliary results on characters of finite fields.
2.1. Preliminaries on Cameron-Liebler line classes
It is often advantageous to study Cameron-Liebler line classes in by using their images under the Klein correspondence. Let be the 5-dimensional hyperbolic orthogonal space and be a nonnegative integer. A subset of is called an -tight set if for every point , or according as is in or not, where is the polarity determined by . The geometries of and are closely related through a mapping known as the Klein correspondence which maps the lines of bijectively to the points of , c.f. [16, 22]. Let be a set of lines of with , a nonnegative integer, and let be the image of under the Klein correspondence. Then it is known that is a Cameron-Liebler line class with parameter in if and only if is an -tight set of . Moreover, if is a Cameron-Liebler line class with parameter , by [20, Theorem 2.1 (b)], it holds that for any point off ; consequently is a projective two-intersection set in with intersection sizes and , namely each hyperplane of intersects in either or points. We summarize these known facts as follows.
Let be a set of lines in , with , and let be the image of under the Klein correspondence. Then is a Cameron-Liebler line class with parameter if and only if is an -tight set in ; moreover, in the case when is a Cameron-Liebler line class, we have
A strongly regular graph is a simple undirected regular graph on vertices with valency satisfying the following: for any two adjacent (resp. nonadjacent) vertices and there are exactly (resp. ) vertices adjacent to both and . It is known that a graph with valency , not complete or edgeless, is strongly regular if and only if its adjacency matrix has exactly two restricted eigenvalues. Here, we say that an eigenvalue of the adjacency matrix is restricted if it has an eigenvector perpendicular to the all-ones vector.
One of the most effective methods for constructing strongly regular graphs is by the Cayley graph construction. Let be a finite abelian group and be an inverse-closed subset of . We define a graph with the elements of as its vertices; two vertices and are adjacent if and only if . The graph is called a Cayley graph on with connection set . The eigenvalues of are given by , , where is the group consisting of all complex characters of , c.f. [3, §1.4.9]. Using the aforementioned spectral characterization of strongly regular graphs, we see that with connection set () is strongly regular if and only if , , take exactly two values, say and with . We note that if is strongly regular with two restricted eigenvalues and , then the set also forms a connection set of a strongly regular Cayley graph on ; this set is called the dual of . For basic properties of strongly regular graphs, see [3, Chapter 9]. For known constructions of strongly regular Cayley graphs and their connections to two-weight linear codes, partial difference sets, and finite geometry, see [3, p. 133] and [6, 19].
Let be a Cameron-Liebler line class with parameter in and let be the image of under the Klein correspondence. By Result 2.1, is a projective two-intersection set in . By , we can construct a corresponding strongly regular Cayley graph as follows. First define , which is a subset of . Then the Cayley graph with vertex set and connection set is strongly regular. Its restricted eigenvalues can be determined as follows. Let be a nonprincipal additive character of . Then is principal on a unique hyperplane for some . We have
where is the Kronecker delta function taking value if and value otherwise. Conversely, for each hyperplane of , we can find a nonprincipal character that is principal on , and the size of can be computed from . Therefore, the character values of reflect the intersection sizes of with the hyperplanes of . To summarize, we have the following result.
Let be a set of lines in , with , and let be the image of under the Klein correspondence. Define
Then is a Cameron-Liebler line class with parameter if and only if and for any
where is any nonprincipal character of that is principal on the hyperplane .
Following we now introduce a model of the hyperbolic quadric , which will facilitate our algebraic construction. Let and . We view as a 6-dimensional vector space over . For a nonzero vector , we use to denote the projective point in corresponding to the one-dimensional subspace over spanned by . Define a quadratic form by
where is the relative trace from to (that is, for any , ). The quadratic form is clearly nondegenerate and for all . So is a totally isotropic plane with respect to . It follows that the quadric defined by has Witt index 3, and so is hyperbolic. This quadric will be our model for . Note that for a point , its polar hyperplane is given by .
Let and be the canonical additive characters of and , respectively. Then each additive character of has the form
where . Since is principal on the hyperplane , the character sum condition in Result 2.2 can be more explicitly rewritten as
2.2. Preliminaries on characters of finite fields
In this subsection, we will collect some auxiliary results on Gauss sums. We assume that the reader is familiar with the basic theory of characters of finite fields as can found in Chapter 5 of .
For a multiplicative character and the canonical additive character of , define the Gauss sum by
The following are some basic properties of Gauss sums:
if is nonprincipal;
if is principal.
Let be a fixed primitive element of and a positive integer dividing . For we set . These are called the th cyclotomic classes of . The Gauss periods associated with these cyclotomic classes are defined by , , where is the canonical additive character of . By orthogonality of characters, the Gauss periods can be expressed as a linear combination of Gauss sums:
where is any fixed multiplicative character of order of . For example, if , we have
where is the quadratic character of .
The following theorem on Eisenstein sums will be used in the proof of our main theorem in Section 4.
([27, Theorem 1]) Let be a nonprincipal multiplicative character of and be its restriction to . Choose a system of coset representatives of in in such a way that can be partitioned into two parts:
where is the relative trace from to . Then,
We will also need the Hasse-Davenport product formula, which is stated below.
([2, Theorem 11.3.5]) Let be a multiplicative character of order of . For every nonprincipal multiplicative character of ,
The Stickelberger theorem on the prime ideal factorization of Gauss sums gives us -adic information on Gauss sums. We will need this theorem to prove a certain divisibility result later on. Let be a prime, , and let be a complex primitive th root of unity. Fix any prime ideal in lying over . Then is a finite field of order , which we identify with . Let be the Teichmüller character on , i.e., an isomorphism
for all in . The Teichmüller character has order . Hence it generates all multiplicative characters of .
Let be the prime ideal of lying above . For an integer , let
where is the -adic valuation. Thus . The following evaluation of is due to Stickelberger (see [2, p. 344]).
Let be a prime and . For an integer not divisible by , let , , be the -adic expansion of the reduction of modulo . Then
that is, is the sum of the -adic digits of the reduction of modulo .
3. The subset and its properties
Let be a prime power with or so that . Write , , and let be a fixed primitive element of . For any , we use to denote the integer such that . We write , and let be an element of order in (for example, take ). For , we define the sign of , , by
We also define .
It is the purpose of this section to introduce a subset and prove a few results on , which we will need in the construction of our Cameron-Liebler line classes.
Viewing as a 3-dimensional vector space over , we will use as the underlying vector space of . The points of are , , and the lines of are
where . Of course, and , for any and . Note that since , we can also take , , as the points of .
Define a quadratic form by , where Tr is the relative trace from to . The associated bilinear form is given by . It is clear that is nondegenerate. Therefore defines a nondegenerate conic in , which contains points. Consequently each line of meets in , or points, and is called a passant, tangent or secant line accordingly. Also it is known that each point is on either or tangent lines to , and is called an interior or exterior point accordingly, c.f. [22, p. 158].
With the above notation, we have the following:
The tangent lines to are given by with , .
The polarity of induced by interchanges and , where , and maps exterior (resp. interior) points to secant (resp. passant) lines.
For any point off , is an exterior (resp. interior) point if and only if has some fixed nonzero sign (resp. ).
Now we define the following subset of :
where the elements are numbered in any (unspecified) order. That is, .
Let be any element of such that , with as defined in Lemma 3.2. For , we define
Let be three distinct elements of . Then the sign of
is equal to the sign of for any exterior point . In other words, has sign , where is the same as in part (3) of Lemma 3.2. In particular, .
Proof: Since is a conic, are linearly independent over and thus form a basis of over . The Gram matrix of the bilinear form with respect to this basis is equal to
which is symmetric with diagonal entries equal to . Its determinant is equal to
Let be an exterior point, say, is the intersection of the tangent lines through and . Then , and the Gram matrix with respect to the basis has determinant . Since , is a square in . By [22, p. 262], the two determinants have the same sign. This proves the first part of the lemma. It remains to prove that . We observe that the matrix (3.6) can be written as with
We claim that the determinant of , , is in . To see this, applying the Frobenius automorphism of to entry-wise, we get
The claim that now follows from this fact and the first part of the lemma.
With the above notation, if we use any other in place of in the definition of , then the resulting set satisfies that or .
Proof: Without loss of generality, we assume that we use in place of in the definition of , and obtain . We have the following observations.
For , that is, , the sign of the quotient of and is a constant: their quotient lies in , and its sign is clearly equal to that of ; this sign is equal to by Lemma 3.3.
The sign of the quotient of and is equal to , since we have chosen such that . Similarly the sign of the quotient of and is equal to .
The above observations imply that there is a pairing of the elements of and , say, a bijection, such that are nonzero squares in for all or are nonsquares in for all . Upon taking logarithm and modulo we get the conclusion of the lemma.
It is clear that is an element of . In Lemma 3.4, consider the special case where we replace by in the definition of , and denote the resulting set by . We observe that
Consequently . It follows that in this particular case. By definition, we have , so we have shown that the subset is invariant under multiplication by . This fact will be needed in the next section when we discuss automorphism groups of the newly construected Cameron-Liebler line classes.
We now prove some properties of the set which will be needed in the next section. Let (resp. ) be the set of nonzero squares (resp. nonsquares) of , and write
where is the canonical additive character of . For , we define the exponential sums
To simplify notation, we often write Note that if and only if for some , which in turn is equivalent to for some integer . Since is an element of , we can view in the above definition of as coming from . It follows that
We will evaluate these sums explicitly.
Let and . The exponential sums take the following four values:
where and are defined as above, that is, and .
Proof: We consider the following three cases according to the line as defined in (3.2) is a tangent, passant, or secant.
Case 1: is a tangent line. In this case, is a zero of by (1) of Lemma 3.2, where we recall that , so for some by the definition of in Eqn. (3.3). Note that satisfies by the definition of and . In view of (2) of Remark 3.6, we may assume that . (If necessary, replace by , and then the resulting is still a zero of , i.e., satisfies .) Now if and only if lies on the tangent line , i.e., . We see that the elements , , are all nonzero squares, so that or by (1) of Remark 3.6. Note that if we replace by , then the value of is replaced by the other in this case. Hence, .
Case 2: is a passant line. In this case, is an interior point and thus has sign by Lemmas 3.2 and 3.3. Note that satisfies and since is a nonsquare of . Each line through has either or points of , so the points of are partitioned into pairs accordingly. Let be two points of that lie on a secant line through .
Since the three points and are collinear, the Gram matrix of is singular. By direct computations we find that the determinant of this Gram matrix is equal to It follows that
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CC-MAIN-2020-40
| 19,061 | 85 |
http://newvision24.com/2020/01/what-is-e-equal-to-in-physics-component-2/
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math
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What is E equal to in Physics? Component 2
What is E equivalent to in Physics? Section 2
What is E equal to in Physics? It truly is a substantial response towards question of “what certainly is the drive of gravity?” It truly is a ultimate reply to into a concern that’s nevertheless a secret in Physics.
The specific strategy of E is very hard to outline. www.gurudissertation.net/ This is because it will be not only one factor, but an abstraction of numerous ideas, each individual of that may be described in several various ways. We must recognize the real difference amongst a definition and an abstraction for you to know E.
The precise strategy of E is really a mathematical expression, which is outlined as the similar kind of dilemma that is questioned as a way to outline a perform in mathematics. So, what the heck is E equivalent to in Physics?
The rationale of this principle is difficult to obtain. You’ll find it clarified the subsequent. Contemplate a human shape of mass m which is at a point out of equilibrium.
Let us think the Newtonian law of Gravity states that mass is attracted to gravity, that is proportional to your solution for the masses m additionally, the acceleration g. This really is possible if the gravitational prospective vigor contains a certain amount of power on the situation of no cost falling. On the other hand, this is simply not probable should the mass is not slipping and also a resistance is encountered. With this scenario, the law of gravity should be transformed.
Well, this is certainly less difficult to know if we think of an genuine object slipping. We all know from elementary physics that when an item that’s not falling is subjected to an outward drive, for example gravity, that object will likely be pushed on the direction of the force.
However, with a purpose to get to the article, the gravitational potential power need to be became kinetic vitality, which the item would then be required to go in the way in the drive. To put it differently, when the item that’s falling is subjected to outward force, it’s going to transfer inside of the way belonging to the drive.
It is simple to check out the law of Newton is exactly the exact same notion, only that it is not expressed because the gravitational likely vitality. Which is, the power of gravity that we phone E is for that matter an outward force.
As a outcome, it really is a fascinating approach to review the legal guidelines of Physics and their partnership into the equations which explain the dynamics belonging to the Earth and its strategy of geologic processes. The engaging issue is usually that, much like the guidelines of mechanics and calculus, the legal guidelines of physics also utilize for the Earth like a complete. As a final result, we could make use of the equations and laws of physics to provide a more precise description of what transpires around the Earth, put another way, what takes place around the Universe.
One essential notion to recall is as Einstein claimed, “in each and every move there’s an assumption of some type. This is not a foolproof approach of considering.”
If we consider E as E=mc2, then the equation is often lower to the linear sort. We will launch by plotting this form to discover how it relates to the equations of the equations with the equations with the equations of E=mc2. Then, we are able to repeat the process with the other equations of E=mc2.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107904039.84/warc/CC-MAIN-20201029095029-20201029125029-00082.warc.gz
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CC-MAIN-2020-45
| 3,446 | 13 |
http://www.potsdam.edu/search404?page=1&destination=MATH/MATH.html
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math
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You are here
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... matematical organizaions and employer listings. Math Jobs-US A list of job vacancies that are offered across the U.S. within several math fields, including: Banking, Insurance, Finance, Statistics, Bio & Med, ...
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CC-MAIN-2019-04
| 1,502 | 11 |
http://www.antonine-education.com/Pages/Physics_2/Mechanics/MEC_11/Mechanics_page_11.htm
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math
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Mechanics Tutorial 11 - Momentum and Impulse
is the product between mass and velocity.
Being a vector quantity, it
has a direction, and the direction is
very important when doing momentum calculations. Momentum is not a thing
that we can see, but it does explain many things that go on in physics.
p = mv
Units are kilogram metres per second (kg m/s) or Newton seconds (N s). We can show that the units are the same.
1 N = 1 kg m s-2 (Newton II)
1 kg m s-1 = 1 kg m s-2 × s = 1 N s
What is the value of the momentum of a 10 kg ball running down a bowling alley at a speed of 5 m s-1?
In the example above, we only looked at the value of the momentum. This is why we used the word speed. It is very important to make sure that you pay attention to the signs when doing momentum calculations.
Think of a ball bouncing off a wall. It leaves the wall at the same speed as before. Let’s call going from right to left negative, and going from left to right positive.
Show that the change in momentum is +2 mv.
The ball has a mass of 200 g, and the value of its velocity throughout remains 6 m s-1. What is the change in momentum?
change in momentum is called the impulse and is given the physics code
We can define Newton’s Second Law in terms of change of momentum:
Force = mass × acceleration
Force = mass × change in velocity
Change in velocity = velocity at end - velocity at start
Change in momentum = mass × change in velocity
Force (N) = change in momentum
time interval (s)
Impulse (Ns) = Force
(N) x time interval (s)
If we plot a force time graph, we can see that impulse is the area under the graph.
In this graph, both impulses are the same. The forces and time intervals are different. In these cases, the force is constant.
The graph below shows the effect of a force that is not constant:
We can work out the impulse to a first approximation by working out the areas of the rectangles and triangles and adding them together. In calculus notation, we can write:
If the force is varying to a known function, we can work out the area under the graph using integration.
Impulse and Newton's Second Law
Consider an object of mass m which is subject to a force F for a time period of t seconds. We can say that there is an impulse on the object:
The term v/t
= change in velocity ÷ time interval = acceleration
So Force = mass × acceleration which is Newton II. Remember that wherever there is a resultant force, there is always acceleration. A change in momentum results in acceleration, which is caused by a force. We saw another version of this in the previous tutorial.
Explain how the formula F = Dp/Dt is consistent with Newton II.
Impulse is the physics phenomenon that explains how a ball behaves when kicked or hit with a bat. It also has important implications in road safety. When a car is involved in a collision, we want the impulse to occur over a longer time interval to reduce the forces involved.
a) Calculate the average force exerted on the driver by his seat belt.
b) Compare this force to his weight and hence work out the “g-force”
c) Comment on the likelihood of serious injury.
Kinetic Energy and Momentum (Edexcel Syllabus)
We know that for a mass m kg travelling at a velocity v m s-1, the momentum p kg m s-1 is given by:
p = mv
We also know that for kinetic energy:
We transpose the momentum equation for v:
We can now substitute this into the kinetic energy equation:
Tidying this up gives us:
Use the change in momentum of the driver in Question 5, and his mass to calculate the work done on him.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664567.4/warc/CC-MAIN-20191112024224-20191112052224-00199.warc.gz
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CC-MAIN-2019-47
| 3,543 | 48 |
https://www.examveda.com/electrical-engineering/practice-mcq-question-on-a.c-fundamentals,-circuits-and-circuit-theory/?page=2
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math
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6. The time constant of a series R-C circuit is given by
7. The power is measured in terms of decibels in case of
8. Inductance affects the direct current flow
9. The r.m.s. value of alternating current is given by steady (D.C.) current which when flowing through a given circuit for a given time produces
10. The ratio of active power to apparent power is known as factor.
Read More Section(A.C Fundamentals, Circuits and Circuit Theory)
Each Section contains maximum 70 questions. To get more questions visit other sections.
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| 526 | 7 |
http://theeducation.pk/2015/09/physics-mcqs-for-intermediate-exam/
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math
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Physics Mcqs for Intermediate Exam
1.Resistivity of a wire depends on :
(C) cross section area
(D) none of the above.
2. When n resistances each of value r are connected in parallel, then
resultant resistance is x. When these n
resistances are connected in series,
total resistance is:
(C) x / n
(D) n2 x
3. Resistance of a wire is r ohms. The wire is stretched to double its length,
then its resistance in ohms is:
(A) r / 2
(B) 4 r
(C) 2 r
(D) r / 4.
4. Kirchhoff’s second law is based on law of conservation of:
5. The diameter of the nucleus of an atom is of the order of :
(A) 10 -31 m
(B) 10 -25 m
(C) 10 -21 m
(D) 10 -14m.
6. The mass of proton is roughly how many times the mass of an electron?
7. The charge on an electron is known to be
1.6 x 10-19 coulomb. In a circuit
the current flowing is 1 A. How many
electrons will be flowing through the
circuit in a second?
(A) 1.6 x 1019
(B) 1.6 x 10-19
(C) 0.625 x 1019
(D) 0.625 x 1011
8. Two bulbs marked 200 watt-250 volts and 100 watt-250 volts are
joined in series to 250 volts supply.
Power consumed in circuit is:
(A) 33 watt
(B) 67 watt
(C) 100 watt
(D) 300 watt.
9. Ampere second could be the unit of:
10. Which of the following is not the same as watt?
(C) amperes x volts
(D) ( amperes )2 x ohm.
11. One kilowatt hour of electrical energy is the same as:
(A) 36 x 105 watts
(B) 36 x 10s ergs
(C) 36 x 105 joules
(D) 36 x 105 B.T.U.
12. An electric current of 5 A is same as:
(A) 5 J / C
(B) 5 V / C
(C) 5 C / sec
(D) 5 w / sec.
13. An electron of mass m kg and having a charge of e coulombs travels
from rest through a potential
difference of V volts. Its kinetic energy
will be: (A) eV Joules
(B) meV Joules
(C)me / V Joules
(D)V / me Joules.
14. The value of the following is given by 100 (kilo ampere ) x ( micro
ampere ) 100 milli ampere * 10
(A) 0.001 A
(B) 0.1 A
(C) 1 A
15. A circuit contains two un-equal resistances in parallel:
(A) current is same in both
(B). large current flows in larger
(C). potential difference across each is
(D). smaller resistance has smaller
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https://manpower.lk/calculotensorialschaumpdfdescargar-exclusive/
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math
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How to Master Calculus Tensorial with Schaum PDF: A Complete Guide
Calculus tensorial is a branch of mathematics that deals with tensors, which are generalizations of vectors and matrices. Tensors can be used to describe physical phenomena such as forces, stresses, strains, curvature, electromagnetism, relativity, and more.
Learning calculus tensorial can be challenging, especially if you don’t have a solid background in linear algebra and multivariable calculus. That’s why you need a good resource to guide you through the concepts and applications of tensors.
One of the best resources for learning calculus tensorial is the Schaum PDF, which is a digital version of the classic book “Tensor Calculus” by David C. Kay. This book covers all the topics you need to know about tensors, such as:
- The definition and properties of tensors
- The algebra and calculus of tensors
- The metric tensor and the Riemannian manifold
- The covariant and contravariant derivatives
- The Christoffel symbols and the geodesics
- The curvature tensor and the Ricci tensor
- The Einstein field equations and the Schwarzschild solution
- The applications of tensors in mechanics, elasticity, fluid dynamics, electromagnetism, relativity, and more
The Schaum PDF also provides hundreds of solved problems and exercises to help you practice and test your understanding of tensors. You can download the Schaum PDF for free from various websites and use it as a reference or a supplement to your textbook.
One of the advantages of the Schaum PDF is that it is easy to read and follow. The book uses a clear and concise language and explains the concepts with examples and diagrams. The book also avoids unnecessary formalism and focuses on the physical meaning and interpretation of tensors.
Another advantage of the Schaum PDF is that it is comprehensive and up-to-date. The book covers all the essential topics of tensor calculus and its applications in various fields of science and engineering. The book also includes some recent developments and discoveries in tensor theory, such as the gravitational waves and the black holes.
The Schaum PDF is a valuable resource for anyone who wants to learn calculus tensorial or refresh their knowledge of tensors. Whether you are a student, a teacher, a researcher, or a professional, you will find the Schaum PDF useful and helpful. You can download the Schaum PDF for free from various websites and start your journey into the fascinating world of tensors.
Calculus tensorial is a powerful and versatile tool for describing and understanding the physical world. It can help you solve complex problems and discover new insights in various domains of science and engineering.
However, learning calculus tensorial can be difficult and daunting, especially if you don’t have the right resource to guide you. That’s why you need the Schaum PDF, which is a digital version of the classic book “Tensor Calculus” by David C. Kay.
The Schaum PDF is a comprehensive and up-to-date book that covers all the topics you need to know about tensors, such as the definition, properties, algebra, calculus, applications, and more. The book also provides hundreds of solved problems and exercises to help you practice and test your understanding of tensors.
The Schaum PDF is easy to read and follow, and it avoids unnecessary formalism and focuses on the physical meaning and interpretation of tensors. You can download the Schaum PDF for free from various websites and use it as a reference or a supplement to your textbook.
If you want to master calculus tensorial and explore the fascinating world of tensors, you should get the Schaum PDF today. It will be your best friend and companion in your journey of learning tensors.
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CC-MAIN-2023-40
| 3,759 | 21 |
https://www.zora.uzh.ch/id/eprint/121746/
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math
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We consider the transverse-momentum ($p_T$) distribution of $ZZ$ and $W^+W^-$ boson pairs produced in hadron collisions. At small $p_T$, the logarithmically enhanced contributions due to multiple soft-gluon emission are resummed to all orders in QCD perturbation theory. At intermediate and large values of $p_T$, we consistently combine resummation with the known fixed-order results. We exploit the most advanced perturbative information that is available at present: next-to-next-to-leading logarithmic resummation combined with the next-to-next-to-leading fixed-order calculation. After integration over $p_T$, we recover the known next-to-next-to-leading order result for the inclusive cross section. We present numerical results at the LHC, together with an estimate of the corresponding uncertainties. We also study the rapidity dependence of the $p_T$ spectrum and we consider $p_T$ efficiencies at different orders of resummed and fixed-order perturbation theory.
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CC-MAIN-2021-39
| 972 | 1 |
https://www.hugendubel.de/de/buch_gebunden/bernd_thaller-visual_quantum_mechanics_selected_topics_with_computer_generated_animations_of_quantum_mechanical_phenomena_with_cdrom-1461581-produkt-details.html
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math
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Titel: Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena [With CDROM]
Autor/en: Bernd Thaller
Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena. Extra Materials extras. springer. com.
2000. Corr. 2nd.
18 SW-Abbildungen, 27 Farbabb. , .
1. Juli 2002 - gebunden - 283 Seiten
"Visual Quantum Mechanics" uses the computer-generated animations found on the accompanying material on Springer Extras to introduce, motivate, and illustrate the concepts explained in the book. While there are other books on the market that use Mathematica or Maple to teach quantum mechanics, this book differs in that the text describes the mathematical and physical ideas of quantum mechanics in the conventional manner. There is no special emphasis on computational physics or requirement that the reader know a symbolic computation package. Despite the presentation of rather advanced topics, the book requires only calculus, making complicated results more comprehensible via visualization. The material on Springer Extras provides easy access to more than 300 digital movies, animated illustrations, and interactive pictures. This book along with its extra online materials forms a complete introductory course on spinless particles in one and two dimensions. The use of visualization techniques greatly enhances the understanding of quantum mechanics as it allows us to depict phenomena that cannot be seen by any other means. "Visual Quantum Mechanics" uses the computer generated animations found on the extra material on Springer Extras to introduce, motivate, and illustrate the concepts explained in the book. For example, by watching QuickTime movies of the solutions of Schroedinger's equation, students will be able to develop a feeling for the behavior of quantum mechanical systems that cannot be gained by conventional means. While there are other books on the market that use Mathematica and Maple to teach quantum mechanics, this book differs in that the text describes the mathematical and physical ideas of quantum mechanics in the conventional manner, with no special emphasis on computational physics or the requirement that the reader know a symbolic computation package or Mathematica.
In this book, instead, the computer is used to provide easy access to a large collection of animated illustrations, interactive pictures, and lots of supplementary materials. "Visual Quantum Mechanics" takes a mathematical rather than a physical approach to quantum mechanics, and includes results more typical in more advanced books but which are more comprehensible via visualization. Despite the presentations of advanced results, the book requires only calculus, and the book will fill the gap between classical quantum mechanics texts and mathematically advanced books. The book will have a home page at the author's institution which will include supplementary material, exercises and solutions, additional animations, and links to other sites with quantum mechanical visualization. This book along with its extra materials on Springer Extras, which contains over 300 digital movies, form a complete introductory course on spinless particles in one and two dimensions. There is a second book in development which will cover such topics as spherical symmetry in three dimensions, the hydrogen atom, scattering theory and resonances, periodic potentials, particles with spin, an relativistic problems (the Dirac equation).
From the contents:
- Visualization of Wave Functions
- Fourier Analysis
- Free Particles
- States and Observables
- Other Observables
- Boundary Conditions
- Linear Operators in Hilbert Spaces
- Harmonic Oscillator
- Special Systems
- One-Dimensional Scattering Theory
- Numerical Solution in One Dimension
- Movie Index
- Other Books on Quantum Mechanics
From the reviews
"The new introductory textbook by Bernd Thaller, Visual Quantum Mechanics, has two distinctive features, a well-produced CD-ROM, and an emphasis on dynamics. These two features work together well and make for a remarkable publication, which mathematically oriented readers will appreciate for the care it takes with rigorous aspects of physics. (...)
It is the CD-ROM which makes the book. (...) Thaller has provided 137 modules illustrating different topics (...) I would venture to say that not only the beginning student, but even the seasoned scientist may benefit, by getting a clue to a new phenomenon or theorem. (...)
In sum, Visual Quantum Mechanics can be profitably used as a major supplement in an undergraduate course on quantum mechanics, and it will certainly enliven the curriculum. It could be a superb stand-alone text in a short physics course or a seminar for mathematically oriented students."
Evans M. Harrell II in: SIAM Review
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CC-MAIN-2021-31
| 4,829 | 27 |
https://www.burrillville.org/animal-control/pages/animals-available-adoption
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math
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All Boards & Commissions
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• Attractions• Blackstone Valley Tourism• Business Directory• Commercial Property for Sale• Learn More about Burrillville• Spring Lake Beach• Visit Burrillville
Please contact us to inquire about available animals.
Check us out on Facebook!
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CC-MAIN-2024-18
| 808 | 7 |
https://www.khanacademy.org/math/algebra2/rational-expressions-equations-and-functions
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math
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Rational expressions are like fractions, but instead of integers in the numerator and the denominator, you have variable expressions! Learn how to work with such expressions. Namely, simplify, add, subtract, multiply, and divide them (much like fractions!). Then, solve some equations with rational expressions in them, and analyze the behavior of rational functions.
Learn how to simplify rational expressions by canceling factors that are shared by the numerator and the denominator. Sometimes this calls for factoring the numerator and the denominator in various ways.
Learn about direct and inverse variation, which are two types of relationships between two quantities. Direct variation is simply a proportional relationship, but inverse variation is more complicated and interesting. Gain some experience with it before we dive deeper into the world of rational functions.
Learn about the ways in which rational functions behave when their denominator is equal to zero. This gets interesting when vertical asymptotes are involved! Learn how to determine this behavior for any kind of rational function.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423809.62/warc/CC-MAIN-20170721202430-20170721222430-00650.warc.gz
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CC-MAIN-2017-30
| 1,108 | 4 |
https://studylib.net/doc/18740109/a-multiconductor-model-for-finite-element-eddy-current
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math
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 533 A Multiconductor Model for Finite-Element Eddy-Current Simulation Herbert De Gersem and Kay Hameyer Abstract—The stranded-conductor finite-element model does not account for the skin and proximity effects in a multiconductor system. The solid conductor model considers the true geometry, all the individual conductors, and their connections but may lead to unmanageably huge models. The multiconductor model proposed here, does not necessarily consider all geometrical details but instead, discretizes the inner geometry and voltage drop and enforces the typical current redistribution in multiconductor configurations in a weak sense. The magnetic and electric meshes are independently and adaptively refined which results in an optimal error control and accurate results for relatively small models. Index Terms—Circuit simulation, eddy currents, finite element methods, windings. Fig. 1. Application range of the solid, stranded, foil, and multiconductor models and impedance boundary conditions. I. INTRODUCTION M ULTICONDUCTOR windings are sets of conductors, electrically insulated from each other and connected in series or parallel. The individual conductors experience skin and proximity effects, characterized by the skin depth (1) where is the frequency, is the permeability, and is the conductivity. The skin depth may be different in the and direction due to anisotropic permeabilities. The characteristic extents of a single wire in the multiconductor system with reand . If spect to the main axes are denoted by and , the stranded-conductor model is appropriate . and , impedance boundary conditions are If commonly applied . If , , and are of the same order of magnitude, the skin effect is resolved by eddy-current simulation using the solid conductor model. The unidirectional skin and of the effect as observed in foil conductors, i.e., same order of magnitude as , is considered in and . These four modeling approaches are limiting cases for particular kinds Manuscript received July 5, 2001. This work was supported in part by the Belgian “Fonds voor Wetenschappelijk Onderzoek Vlaanderen” under Grant G.0427.98 and in part by the Belgian Ministry of Scientific Research under Grant IUAP no. P4/20. H. De Gersem was with the Katholieke Universiteit Leuven, Department ESAT, Division ELEN. He is now with the FB 18 Elektrotechnik und Informationstechnik, Fachgebiet Theorie Elektromagnetischer Felder, Technische Universität Darmstadt, D-64289 Darmstadt, Germany (e-mail: [email protected]). K. Hameyer is with the Katholieke Universiteit Leuven, Department ESAT, Division ELEN, B-3001 Leuven-Heverlee, Belgium (e-mail: [email protected]). Publisher Item Identifier S 0018-9464(02)02348-8. of current redistribution (Fig. 1). In many models, the influence of the skin effect on the global behavior of the model is not negligible although its local influence does not have to be computed in detail. Multiconductor systems arise in almost all electrical energy transducers. The devices may feature a large number of multiconductor systems, each consisting of a considerable number of turns. This may hamper the simulation of the overall device. Several model reduction techniques for multiconductor systems exist, e.g., analytical macroelements , inner node elimination and parameter extraction . They constitute an a priori model reduction, which may hinder adaptive error control during numerical simulation. In this paper, it is suggested to approximate the troublesome geometries by an additional discretization for the voltage and to incorporate this in the magnetic finite-element (FE) model. An error estimator adaptively refines the multiconductor model during the simulation. II. MAGNETODYNAMIC MODEL For convenience, a two-dimensional time-harmonic formulation is considered. The derivation of a multiconductor model for three-dimensional FE models, transient formulations, or anisotropic materials is similar. The magnetodynamic formulation is (2) the phasor with the pulsation, the length of the model, of the component of the magnetic vector potential and the phasor of the voltage drop between front and rear ends of the model. The partial differential equation is discretized on the 0018-9464/02$17.00 © 2002 IEEE 534 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 device cross section by linear, triangular FEs yielding the system of equations , (3) (4) (5) and the degrees of freedom (DOFs) for . III. MODELING ASSUMPTIONS Consider a multiconductor with cross section , conductors with cross sections , consisting of and connected in series. The cross sections may have different shapes but have the same area . The crossis . The union of all conductor cross sectional area of , is contained in but may be sections, . The difference is occupied by insmaller than sulation material, cooling ducts, and gaps which are supposed to and to have a homogeneous be uniformly distributed over and a zero conductivity. The voltage drop is permeability and is aligned with the direction, is constant over each zero in accounting for the nonconductive regions included in true constraints (7) tend to the continuous constraint . The in (12) The voltage across the overall multiconductor is obtained by over and multiplying by averaging (13) in in Fig. 2. Cross section of the multiconductor, the magnetic mesh, the electric mesh, and three electric shape functions. . (6) The voltage drop differs from one conductor to the other due to the vicinity of ferromagnetic cores and because of inhomogeneous conductivities, e.g., due to local heating. It may be beneficial to model a multiconductor system with a considerable number of turns by the continuous model (12) and (13) rather than the true relations (7) and (8) which require the geometry of each individual conductor to be considered. IV. CONTINUOUS MULTICONDUCTOR MODEL V. DISCRETE MULTICONDUCTOR MODEL The true multiconductor model obeys (2) and an integral con, straint over each To discretize the continuous multiconductor model, an additional mesh for the voltage drop is constructed (Fig. 2). To simplify the implementation, a tensor grid is preferred. is resolved by electric shape functions (7) The voltage drop across the multiconductor is (14) (8) tends to infinity Consider now the limiting case in which becomes very small. Still, suppose all conducand, hence, tors to have the same cross-section area and the conductive and to be uniformly mixed. The insunonconductive parts of lation material is not explicitly considered in the model. The material parameters in the magnetic equations (2), (4), and (5) are replaced by the homogenized parameters with the associated DOFs. Equation (5) becomes (15) A weak formulation of the multiconductor model is obtained by weighting (12) by the electric shape functions (9) (10) with the fill factor (16) (11) DE GERSEM AND HAMEYER: A MULTICONDUCTOR MODEL FOR FE EDDY-CURRENT SIMULATION 535 The voltage across the overall multiconductor is (17) The weak formulation of the magnetodynamic problem (3), the weak formulation of the current constraints (16), and the integration of the discrete voltage drop (17) are assembled into the coupled system of equations (18) a factor symmetrizing the system. The system with has the nature of a mixed formulation . Because the coefand scale differently, it is recommended to ficients apply an explicit diagonal scaling of (18) before solving. The system is solved by a Quasi-Minimal Residual method adapted to complex symmetric systems , preconditioned by successive over relaxation. The multiconductor model can be coupled to a circuit model accounting for active and passive components outside the finiteelement model. The treatment of arbitrary circuit connections follows the topological approach presented in . VI. ADAPTIVE MESH REFINEMENT The electric mesh does not coincide with the magnetic mesh nor with the true multiconductor geometry. Therefore, adaptive mesh refinement can be applied independently. The possibility of independent error control is one of the most attractive features of the multiconductor model. The magnetic mesh is refined based on indications of large eddy currents and large ferromagnetic saturation. The error indicator for the electric mesh detects large variations of the voltage drop. To preserve the consistency of the multiconductor model, the discrete weak formulation (16) has to converge towards the true model (7) rather than towards the continuous representation (12). As a consequence, the supports of the electric shape functions have to converge towards the single-conductor cross sections. This is different from a conventional discretization where convergence corresponds to a vanishing mesh size. The consistency of the electric discretization of the multiconductor model is guaranteed by the following procedure. At places where refinement would bring up electric elements smaller than the extent of a single conductor, the true geometry is restored (Fig. 3). The discrete formulation (16) implicitly incorporates the true model (7) if a single, constant shape function is because of assigned to each conductor that shows up in refinement. The parts of the insulation that are restored by the refinement procedure, are expelled from the electric mesh. As a consequence, the electric mesh may become disconnected and the fill factor has to be adapted accordingly. The limiting constant case when the voltage drop is discretized by , , and electric shape functions defined at the fill factor is one, corresponds to the explicit modeling of each of the individual conductors as a solid conductor, and, hence, constitutes the true multiconductor model (7) and (8). For technical models, however, a sufficient accuracy is already Fig.3. Consistent adaptive refinement of the electric tensor grid when the error estimator indicates large variations of the voltage drop at the left upper corner of the multiconductor cross section (at a certain point, further refinement corresponds to restoring the original geometry). achieved when the electric mesh is much coarser than the true geometry. If local effects would become important, the error indicator will detect them and invoke substantial refinement at those places, probably leading to a local recovery of the true geometry of the multiconductor system. VII. NONMATCHING INTEGRATIONS The efficiency of the multiconductor model is strongly related to the flexibility of selecting and refining the meshes. It is recommended to allow an independent construction and reand finement of both meshes. Hence, the supports of , in general, do not match. This considerably hinders the evaluation of the hybrid integrals in . A numerical integration scheme, e.g., Gaussian quadrature, encounters problems to select appropriate integration points or may require a huge number of points in order to sample both meshes at a sufficient rate. Here, a semi-analytical technique is favored. A composite mesh is built by gathering all vertices and edges of both the magnetic and the electric mesh. The intersections of the edges of the different meshes with respect to each other have to be computed with a sufficient accuracy, preferably using exact arithand can be exactly represented metic. Both on the composite mesh on which the multiplications and integrations can then be performed exactly. The required computation time is substantial but is still acceptable when compared with the overall computation time. To avoid hybrid integrals, one could use the composite mesh for both electric and magnetic discretization. This approach, however, does not pay off because the substantially larger system of equation would eliminate the efficiency of the multiconductor model. VIII. CONVERGENCE OF THE MULTICONDUCTOR DISCRETIZATION The convergence of the mixed discretization technique is studied for a model problem. The multiconductor contains 250 conductors. All feature the same conductivity and permeability. The conductors are electrically insulated from each other, are connected in series, and carry an alternating current. No flux leaves the model. Within each conductor, the magnetic field is expressed by the analytical solution of (2) for a rectangular domain. The analytical solution for the multiconductor system is derived by applying the interface conditions at the borders of the conductors and requiring the current to be the same in 536 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 Fig. 6. Fig. 4. Convergence of the discretization error of the magnetic vector potential field of the multiconductor model (the dashed line is a line of constant error). Harmonic losses in a stator winding of an induction machine. ticonductor model offers a sufficient accuracy while avoiding an excessive amount of mesh nodes and voltage unknowns. At 50 Hz, no significant skin effect is observed. At 500 Hz, substantial losses are introduced. The multiconductor model equipped with independent mesh refinement and external circuit coupling, enables the simulation of the model for all possible frequencies by the same conductor model (Fig. 6). X. CONCLUSION Fig. 5. (a) Geometry, (b) real, and (c) imaginary components of the magnetic flux in a single-layer stator slot at 50 Hz. (d) Real and (e) imaginary components of the magnetic flux in the multiconductor model at 500 Hz. each conductor. The discretization error of the magnetic vector potential field obtained by the proposed multiconductor model, is measured in the L2-norm with respect to the analytical solution (Fig. 4). The error decays when the magnetic and/or electric meshes are refined. The dashed line denotes loci for which the error is identical. The experiment indicates that the discretization error depends more on the discretization of the magnetic field than on the discretization of the electric voltage drop. As a consequence, it is sometimes more advantageous to apply a finer magnetic mesh than to consider all geometrical details of the multiconductor system. IX. EXAMPLE: MACHINE WINDINGS The multiconductor model is applied to simulate the harmonic losses in induction machine windings (Fig. 5). Since these devices are supplied by variable frequency, the relative importance of the higher harmonic distortion increases and the additional joule losses are not negligible. These effects are commonly taken into account in analytical models by the frequencydependent eddy-current factor which can be provided by a FE model of a single stator slot . A leakage flux impinging on the conductor is applied to the model by a difference in magnetic vector potential between the top and the bottom of the slot. A conventional model considers the true geometry consisting of the conductors, the insulation, and the cooling ducts. It treats the coil as a series connection of a number of solid conductors, each with their own unknown voltage. For many cases, the mul- The multiconductor model developed here, enables the simulation of complicated coil configurations with relatively small models by using an additional discretization for the conductor’s voltage drop. It offers more modeling flexibility when compared with the solid and stranded-conductor models. The automated error control and independent mesh refinement yields small models while guaranteeing the prescribed accuracy. REFERENCES G. Bedrosian, “A new method for coupling finite-element field solutions with external circuits and kinematics,” IEEE Trans. Magn., vol. 29, pp. 1664–1668, Mar. 1993. A. M. El-Sawy Mohamed, “Finite-element variational formulation of the impedance boundary condition for solving eddy current problems,” Inst. Elect. Eng. Proceedings Science, Measurement and Technology, vol. 142, no. 4, pp. 293–298, July 1995. H. De Gersem and K. Hameyer, “A finite-element model for foil winding simulation,” IEEE Trans. Magn., vol. 37, pp. 3427–3432, Sept. 2001. P. Dular and C. Geuzaine, “A magnetic field magnetodynamic finiteelement formulation for foil winding inductors,” IEEE Trans. Magn., submitted for publication. A. Kladas and A. Razek, “Numerical calculation of eddy-currents and losses in squirrel cage induction motors due to supply harmonics,” in Proc. Int. Conf.Electrical Machines (ICEM88), vol. 2, Pisa, Italy, Sept., pp. 65–69. Á. Szűcs and A. Arkkio, “Consideration of eddy currents in multiconductor windings using the finite element method and the elimination of inner nodes,” IEEE Trans. Magn., vol. 35, pp. 1147–1150, May 1999. I. Munteanu, T. Wittig, T. Weiland, and D. Ioan, “FIT/PVL circuit-parameter extraction for general electromagnetic devices,” IEEE Trans. Magn., vol. 36, pp. 1421–1425, July 2000. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Berlin, Germany: Springer-Verlag, 1991. R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Scientific Computing, vol. 13, pp. 425–448, Jan. 1992. H. De Gersem, R. Mertens, U. Pahner, R. Belmans, and K. Hameyer, “A topological method used for field-circuit coupling,” IEEE Trans. Magn., vol. 34, pp. 3190–3193, Sept. 1998. S. J. Salon, L. Ovacik, and J. F. Balley, “Finite-element calculation of harmonic losses in AC machine windings,” IEEE Trans. Magn., vol. 29, pp. 1442–1445, Mar. 1993. R. Richter, “Die Induktionsmaschinen,” in Elektrische Maschinen, 2nd ed. Basel, Switzerland: Birkhauser, 1963, vol. 4.
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| 17,559 | 1 |
https://smart-bookmarking.com/qa/how-do-you-determine-a-statement.html
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math
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- What is a statement in math?
- What are examples of statement?
- What is a simple statement?
- What is an example of a universal statement?
- What means statement?
- What are the 4 types of questions?
- What makes something a statement?
- What is a statement in math example?
- What are Number statements?
- What is an example of a statement sentence?
- What is the difference between a statement and a sentence?
- How do you write a statement sentence?
- What is the difference between a question and a statement?
- What is a statement question?
- What is a Contrapositive statement?
- Can a statement start with what?
- Can a statement be a question?
- Can a statement be more than one sentence?
What is a statement in math?
A statement (or proposition) is a sentence that is either true or false (both not both).
So ‘3 is an odd integer’ is a statement.
But ‘π is a cool number’ is not a (mathematical) statement.
Note that ‘4 is an odd integer’ is also a statement, but it is a false statement..
What are examples of statement?
An example of statement is the thesis of a paper. An example of statement is a credit card bill. A declaration of fact or an allegation by a witness; a piece of sworn testimony. See also closing statement, evidence, and opening statement.
What is a simple statement?
A simple statement is a statement which has one subject and one predicate. For example, the statement: London is the capital of England. is a simple statement.
What is an example of a universal statement?
A universal statement is a statement that is true if, and only if, it is true for every predicate variable within a given domain. Consider the following example: Let B be the set of all species of non-extinct birds, and b be a predicate variable such that b B. … Some birds do not fly.
What means statement?
1 : something stated: such as. a : a single declaration or remark : assertion. b : a report of facts or opinions. 2 : the act or process of stating or presenting orally or on paper.
What are the 4 types of questions?
In English, there are four types of questions: general or yes/no questions, special questions using wh-words, choice questions, and disjunctive or tag/tail questions. Each of these different types of questions is used commonly in English, and to give the correct answer to each you’ll need to be able to be prepared.
What makes something a statement?
A statement is a sentence that says something is true, like “Pizza is delicious.” There are other kinds of statements in the worlds of the law, banking, and government. All statements claim something or make a point. … You get a statement from your bank, a monthly record of what you spent and what you have left.
What is a statement in math example?
Brielfy a mathematical statement is a sentence which is either true or false. It may contain words and symbols. For example “The square root of 4 is 5″ is a mathematical statement (which is, of course, false).
What are Number statements?
A number sentence is a mathematical sentence, made up of numbers and signs. The expressions given in examples indicate equality or inequality. Types of Number Sentences. A number sentence can use any of the mathematical operations from addition, subtraction, multiplication to division.
What is an example of a statement sentence?
A statement sentence usually has a structure characterized by a subject followed by a predicate. Example of a statement sentence: Charlie delivers the newspapers twice a day. … Imperative sentences (or commands) have an implied subject, and so the subject (most often “you”) is usually unspoken.
What is the difference between a statement and a sentence?
A sentence is a group of words that usually have a subject, verb and information about the subject. Remember: A sentence can be a statement, question or command. A statement is a basic fact or opinion. It is one kind of sentence.
How do you write a statement sentence?
Declarative Sentence (statement) They tell us something. They give us information, and they normally end with a full-stop/period. The usual word order for the declarative sentence is: subject + verb…
What is the difference between a question and a statement?
is that question is a sentence, phrase or word which asks for information, reply or response; an interrogative while statement is a declaration or remark.
What is a statement question?
The intonation of a statement question depends on its meaning. We use statement questions when we think we know the answer to the question and we want to find out if we’re right.
What is a Contrapositive statement?
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of “If it rains, then they cancel school” is “If they do not cancel school, then it does not rain.” … If the converse is true, then the inverse is also logically true.
Can a statement start with what?
Many questions and statements can start with the word what. Examples: “What’s new?” and “What’s sauce for the goose is sauce for the gander.”
Can a statement be a question?
Questions, commands and advice are typically not statements, because they do not express something that is either true or false. … We saw an example of a question which by itself is not a statement, but can be used to express a statement. When you see rhetorical questions, always rephrase them as statements.
Can a statement be more than one sentence?
Statements are logical entities; sentences are grammatical entities. Not all sentences express statements and some sentences may express more than one statement. A statement is a more abstract entity than even a sentence type.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141727627.70/warc/CC-MAIN-20201203094119-20201203124119-00520.warc.gz
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CC-MAIN-2020-50
| 5,747 | 57 |
http://higherstudies.aglasem.com/?p=4338
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math
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The interview at IIT Delhi went fine and after the interview, I was quite sure that i will be selected. The interview was comparatively easy compared to Kanpur. The professors were very supportive during the interview and gave me lot of chances even if i did not answer correctly in the first attempt.
Giving a good performance in Interview is important as i have seen in Kanpur and Delhi, many people in the range of ranks 200-400 getting selected than 100-200.
Here also your answer to the question of favorite subject determines the range of questions. Some of the sample questions asked during the interview were
1. write algo to transpose a matrix
2. why dont we divide array in 5 parts for merge sort
3. which is better merge sort or heap sort
4. theres a frog who could climb either 1 stair or 3 stairs in one shot. In how many ways he could reach at 100th stair.
1) If we are give a sorted array and we have to find two elements which sum to a number x
2) A man is standing in front of an infinite wall with a hole on any side? How do u find the hole in shortest distance
3) Which is greater root(n) or 2^root(logn)
1. Have you heard of ethernet?” I said “Yes!”. Then I was asked “Can we lay an ethernet LAN between Delhi and Bangalore?
2. What is digital signature? How does it work?
3. What is pipelining? Whats the need? Whats the funda behind it? Does it make the processor faster?
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s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00125-ip-10-147-4-33.ec2.internal.warc.gz
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CC-MAIN-2014-15
| 1,401 | 13 |
https://physics.stackexchange.com/questions/352255/oscillations-of-what-cause-quantum-waves
|
math
|
You are confusing the classical framework of Maxwell's equations with the quantum mechanical.
For example, the electron can be viewed as a wave, just like the photon.
Both the photon and the electron are described by a wave function whose complex conjugate squared gives the probability density for the particular observation.
So, if the electromagnetic wave that the photon represents is caused by oscillating electric charges,
The photon is not an electromagnetic wave: it has a wavefunction. The classical oscillating fields emerge from an enormous number of quantum mechanical photon fields. ( for a mathematical analysis look at this link). The reason that the photons, defined in the quantum mechanical framework can build up the classical maxwell electromagnetic field is because their wave function is the solution of the quantized Maxwell equation..
then oscillating of what causes the wave that the electron represents?
The wavefunctions of electrons, positrons and the rest of massive particles in the standard model of particle physics, are based on solutions of the Dirac equation. There exists the Schrodinger-Newton equation studied as a limiting case but the analysis goes in a different direction, does not have the simple solutions which allow for the emergence of the classical electromagnetic field from the photon wavefunction.
Thus your wave analogy does not hold.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540528490.48/warc/CC-MAIN-20191210180555-20191210204555-00031.warc.gz
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CC-MAIN-2019-51
| 1,386 | 8 |
http://forum.lowcarber.org/showthread.php?t=475443&page=5
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math
|
Originally Posted by bmore4now
Realizing that I have to eat this way forever, why stress out.
This is such a powerful statement of absolute truth!
I am a prime example of how to screw this up too!
In 2009-2010 I lost all my excess weight (over 75#) on LC and then ZC
Then I went to Culinary school to become a Certified Chef....
Oh boy....I'm sure you all can imagine what happened next
Anyway......so now I'm down to my last 25# to lose
I gained back all my weight plus an extra 25# btw...so it's extra hard this time
But there is no way in Hell I am doing this again
I'm done being ashamed of my HUGE mistake
And have moved on...I think my purpose is to encourage others
Which is one of the reasons I am here
We all fall down...but you have to bet back up and keep walking, crawling, hopping, hobbling, wheelchairing etc...
If a chef with two knee surgeries in the last year can lose over 70#
Anyone can...,it's NOT EASY....but boy is it REWARDING!
Keep on keepin on folks!
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864110.40/warc/CC-MAIN-20180621075105-20180621095105-00317.warc.gz
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CC-MAIN-2018-26
| 975 | 17 |
http://mathhelpforum.com/geometry/123038-minimize-surface-area-cylinder-given-volume.html
|
math
|
Write the equation for total surface area of a cylinder,
including top and bottom discs if appropriate.
Use the volume value to write R in terms of h, or h in terms of R.
Write the surface area in terms of one variable.
Differentiate it with respect to that variable to find the minimum material.
|
s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541773.2/warc/CC-MAIN-20161202170901-00363-ip-10-31-129-80.ec2.internal.warc.gz
|
CC-MAIN-2016-50
| 296 | 5 |
https://www.clutchprep.com/chemistry/practice-problems/136257/estimate-the-enthalpy-of-reaction-hrxn-for-each-of-the-following-reactions-using
|
math
|
For this problem, we have to calculate the enthalpy of reaction (ΔH) using the given bond enthalpies.
C (s) + 2 H2 (g) → CH4 (g)
Recall that the change of enthalpy for a chemical reaction (ΔH) can be calculated using bond energies:
To determine the enthalpy of reaction, we need to first see what bonds are present in the reactants and the products.
Estimate the enthalpy of reaction, ΔH rxn°, for each of the following reactions using the information provided. Show all work and circle your final answers.
C (s) + 2 H2 (g) → CH4 (g) ΔH°rxn = ?
a. -312 kJ
b. -221 kJ
c. -72 kJ
d. +63 kJ
e. +740 kJ
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655879738.16/warc/CC-MAIN-20200702174127-20200702204127-00447.warc.gz
|
CC-MAIN-2020-29
| 607 | 11 |
http://webapps.stackexchange.com/questions/tagged/google-documents+comments
|
math
|
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s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997860453.15/warc/CC-MAIN-20140722025740-00224-ip-10-33-131-23.ec2.internal.warc.gz
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CC-MAIN-2014-23
| 2,282 | 53 |
https://coursestutor.com/hypothesis-3/
|
math
|
Claim: The standard deviation of pulse rates of adult males is less than 10 bpm. For a random sample of 132 adult males, the pulse rates have a standard deviation of 9.7 bpm. Complete parts (a) and (b) below.
a. Express the original claim in symbolic form.
σ <10 bpm
(Type an integer or a decimal. Do not round.)
b. Identify the null and alternative hypotheses.
H0: σ =10 bpm
H1: σ <10 bpm
(Type integers or decimals. Do not round.)
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662601401.72/warc/CC-MAIN-20220526035036-20220526065036-00671.warc.gz
|
CC-MAIN-2022-21
| 435 | 8 |
https://www.i2m.univ-amu.fr/events/the-link-of-a-finitely-determined-map-germ-from-r-2-to-r-2-joint-work-with-j-j-nuno-ballesteros/
|
math
|
Date(s) : 22/01/2015 iCal
14 h 00 min - 15 h 00 min
Let $f:(R^2, 0) –> (R^2, 0)$ be a finitely determined germ. The
link of f is obtained by taking a small enough representative f: U –> R^2 and
the intersection of its image with a small enough sphere $S_epsilon^1$,
centered in the origin. We will describe the topology of f in terms of the
Gauss word associated to its link.
Catégories Pas de Catégories
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224657720.82/warc/CC-MAIN-20230610131939-20230610161939-00474.warc.gz
|
CC-MAIN-2023-23
| 410 | 8 |
http://www.yourarticlelibrary.com/financial-management/capital-expenditure/top-4-tools-of-risk-analysis/62041/
|
math
|
Read this article to learn about the four tools of risk analysis.
1. Shorter Payback Period:
According to this method, projects with shorter payback periods are normally preferred to those with longer payback periods. It would be more effective when it is combined with a ‘cut off period’.
Cut off period denotes the risk tolerance level in the firm. For example, a firm has three projects, A, B, C for consideration with different economic lives, say 15, 16 and 7 years respectively, and with payback periods of say 6, 7, and 5 years.
Of these three, project C will be preferred because it’s payback period is the shortest. Suppose, the firm’s cut off period is 4 years, then all the three projects will be rejected.
2. Risk-Adjusted Discount Rate:
Under this method, the cut off rate or minimum required rate of return [mostly the firm’s cost of capital] is raised by adding what is called ‘risk premium’ to it. When the risk is greater, the premium to be added would be greater.
For example, if the risk free discount rate [say, cost of capital] is 10%, and the project under consideration is a riskier one, then the premium of, say 5% is added to the above risk-free rate.
The risk-adjusted discount rate would be 15%, which may be used either for discounting purposes under NPV, or as a cut off rate under IRR.
Merits of Risk-adjusted Discount Rate:
1. It is easy to understand and simple to operate.
2. It has a great deal of intuitive appeal for risk adverse decision-makers.
3. It incorporates an attitude towards uncertainty.
1. There is no easy way to derive a risk-adjusted discount rate.
2. A uniform risk discount factor used for discounting all future returns is unscientific as the degree of risk may vary over the years in future.
3. It assumes that investors are risk averse. Though it is generally true, there do exist risk-seekers in real world situation that may demand premium for assuming risk.
3. Conservative Forecasts:
Under this method, employing intuitive correction factor or certainty equivalent coefficient, which is calculated by the decision-maker subjectively or objectively, reduces the estimated risks from cash flows.
Normally, this coefficient reflects the decision-makers confidence in obtaining a particular cash flow in a particular period. For example, the decision-maker estimates a net cash flow of Rs.60000 next year but if he feels [subjectively] that only 60% of such cash flow is a definite sum, and then the said coefficient would be 0.6.
This may also be determined [objectively] by relating the desirable cash flows with estimated cash flows as follows:
For Example, if the estimated cash flow is Rs.80000 in period ‘t’ and an equally desirable cash flow for the same period is Rs.60000, then the certainty equivalent is 0.75 [60000/80000].
Besides the computations of certain cash flows in order to provide for more risk, the economic life over which cash flows are estimated may be reduced simultaneously.
EV = [0.25 x 8] + [0.50 x 12] + [0.25 x 16] = 12%
In the case of alternative projects, one with the highest EV is considered for selection. EVs may be used for computation of IRR and NPV. However, EVs fail to articulate the degree of risk involved.
4. Decision Tree Analysis:
A decision may be taken by comparing various alternative courses at present with the decision-maker, each alternative course being studied in the light of the future possible conditions followed by future alternative decisions.
The group of all these decisions present as well as future, viewed in relation to one another is called a ‘decision tree’. It is a graphic display of the relationship between a present decision and future events, and future decisions and their consequences.
The sequence of events is normally represented over time in a format similar to the branches of a tree.
The major steps in a decision tree process are:
(a) The investment decision is defined clearly.
(b) The decision alternatives are identified.
(c) The decision tree graph indicates decision points, chance events, and other data.
(d) It presents the relevant data such as the projected cash flow, probability distribution, expected present value, etc., on the decision tree branches.
(e) Choosing the best alternative by analysis from the results displayed.
Laxmi Ltd. is considering purchase of a new machine at a cost of Rs. 20,000. The cash inflows for the three years of its life are forecast as follows:
The desired rate of return of NPV purposes is 20 percent. Calculate the probability of the machine. Also draw a decision tree diagram.
Alternatively, profitability can be calculated through EVs as follows:
Eskay Ltd is considering the purchase of a new machine. The two alternative models under consideration are ‘Laxmi’ and ‘HMT’.
From the following information, prepare a profitability statement for submission to the Board of Directors:
Assume the taxation rate to be 50 percent of profits. Suggest which model can be purchased, giving reasons for you answer.
Thus machine ‘Laxmi’ clearly recommends itself for purchase. However, the information provided and conclusion derived may be supplemented with some additional calculations as regards profitability beyond payback period.
Determine the average rate of return from the following data of two Machines A and B:
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s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864461.53/warc/CC-MAIN-20180521161639-20180521181639-00389.warc.gz
|
CC-MAIN-2018-22
| 5,333 | 42 |
http://teqahyp.alethamacdonald.com/how-do-you-rewrite-a-quadratic-equation-in-vertex-form-68443vb.html
|
math
|
While the passing of time is constant, our estimation of how much time has passed varies. Here's what we got from that question.
Words ending with -metry are to do with measuring from the Greek word metron meaning "measurement". Then it just turns out that we can factor using the inverse of Distributive Property!
And just as a bit of a refresher, if a parabola looks like this, the vertex is the lowest point here, so this minimum point here, for an upward opening parabola. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.
They also have the property that a line from the origin to any point on the curve always finds the tangent to the curve meeting it at the same angle. What is its length on the subsequent two sizes of tile at each subsequent stage?
Then factor like you normally would: I know I get better responses when I use the word typical so I went with that. Students will use mathematical relationships to generate solutions and make connections and predictions.
Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals. Now, what I want to do is express the stuff in the parentheses as a sum of a perfect square and then some number over here. It mentions this Ammann tiling on page You might have spotted that this equation is merely Pythagoras' Theorem that all the points x,y on the circle are the same distance from the origin, that distance being a.
Geometry, Adopted One Credit. Remember that the sign of a term comes before it, and pay attention to signs. As we discussed each one, I wanted to draw out some important characteristics: The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions.
The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. Students will use their proportional reasoning skills to prove and apply theorems and solve problems in this strand.
Students will broaden their knowledge of quadratic functions, exponential functions, and systems of equations. So when x is equal to 0, we have 1, 2, 3, oh, well, these are 2.
A single tile which produces an "irregular" tiling was found by Robert Ammann in This is the coefficient of the first term 10 multiplied by the coefficient of the last term — 6.
The Geometry Junkyard has a great page of Penrose links Ivars Peterson's ScienceNewsOnline has an interesting page about quasicrystals showing how Penrose tilings are found in nature. There are times as a coach that I wish I were back in the classroom. So let's just work on this.
When x is equal to 0, y is equal to 8. This quantity right here, x minus 2 squared, if you're squaring anything, this is always going to be a positive quantity. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
The students were developing a good sense of center and the values in relation to the center. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions.
So here I haven't changed equation. Students talked about the estimates being 1. So how do we get x is equal to 0 here? This number is called the "standard deviation".Apr 13, · Edit Article How to Find the Maximum or Minimum Value of a Quadratic Function Easily.
Three Methods: Beginning with the General Form of the Function Using the Standard or Vertex Form Using Calculus to Derive the Minimum or Maximum Community Q&A For a variety of reasons, you may need to be able to define the maximum or minimum value of a selected quadratic.
What Does a 4-Dimensional Sphere Look Like? There is a very real geometric object, realizable within the relativistic geometry of our universe, which has the properties of a sphere in four dimensions (a “4-hypersphere”); what does it look like?
HSN Number and Quantity. HSN-RN The Real Number System. HSN-RN.A Extend the properties of exponents to rational exponents.
HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic alethamacdonald.com scheme is named after its inventor, American physicist Richard Feynman, and was first introduced in The interaction of sub-atomic particles can be complex and difficult to understand intuitively.
Different forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex. SOLUTION: Rewrite the quadratic equation in standard form by completing the square.
f(x)=x^2+8x+14 Answer (f(x)= (x, plus or minus ___)^2 plus or minus ____ I have worked this to x, 4 a Algebra -> Quadratic Equations and Parabolas -> SOLUTION: Rewrite the quadratic equation in standard form by completing the square.Download
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668529.43/warc/CC-MAIN-20191114154802-20191114182802-00045.warc.gz
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CC-MAIN-2019-47
| 5,551 | 21 |
https://www.ebay.com.au/p/Self-Reference-and-Modal-Logic-by-Craig-Smorynski-Paperback-1985/117554880
|
math
|
It is Sunday, the 7th of September 1930. The place is Konigsberg and the occasion is a small conference on the foundations of mathematics. Arend Heyting, the foremost disciple of L. E. J. Brouwer, has spoken on intuitionism; Rudolf Carnap of the Vienna Circle has expounded on logicism; Johann (formerly Jas and in a few years to be Johnny) von Neumann has explained Hilbert's proof theory-- the so-called formalism; and Hans Hahn has just propounded his own empiricist views of mathematics. The floor is open for general discussion, in the midst of which Heyting anunces his satisfaction with the meeting. For him, the relationship between formalism and intuitionism has been clarified: There need be war between the intuitionist and the formalist. Once the formalist has successfully completed Hilbert's programme and shown finitely that the idealised mathematics objected to by Brouwer proves new meaningful statements, even the intuitionist will fondly embrace the infinite. To this euphoric revelation, a shy young man cautions~ According to the formalist conception one adjoins to the meaningful statements of mathematics transfinite (pseudo-')statements which in themselves have meaning but only serve to make the system a well-rounded one just as in geometry one achieves a well- rounded system by the introduction of points at infinity.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948587496.62/warc/CC-MAIN-20171216084601-20171216110601-00151.warc.gz
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CC-MAIN-2017-51
| 1,345 | 1 |
https://essaysresearch.org/adam-granger-operates-a-kiosk-in-downtown-chicago-at-which/
|
math
|
Adam Granger operates a kiosk in downtown Chicago, at which he sells one style of baseball hat. He buys the hats from a supplier for $14 and sells them for $20. Adam s current breakeven point is 15,000 hats per year. Required a. What is Adam s current level of fixed costs? b. Assume that Adam s fixed costs, variable costs, and sales price were the same last year, when he made $21,000 in net income. How many hats did Adam sell last year, assuming a 30% income tax rate? c. What was Adam s margin of safety last year? d. If Adam wants to earn $37,800 in net income, how many hats must he sell? e. How many hats must Adam sell to break even if his supplier raises the price of the hats to $15 per hat? f. What actions should Adam consider in response to his supplier s price increase? g. Adam has decided to increase his sales price to $21 to offset the supplier s price increase. He believes that the increase will result in a 5% reduction from last year s sales volume. What is Adam s expected net income?
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100602.36/warc/CC-MAIN-20231206162528-20231206192528-00537.warc.gz
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CC-MAIN-2023-50
| 1,008 | 1 |
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