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https://forum.ansys.com/discussion/26500/single-moving-reference-frame
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math
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Single moving reference frame
I have two walls and fluid in between where the upper wall is inclined. The lower wall is moving and the upper wall is stationary. As the lower wall is moving I thought I should use a single moving reference frame. I defined fluid cell zone as moving frame but zero absolute velocity, because I want to see fluid should be dragged because of lower moving wall. And inlet and outlet BC are pressure-driven. My question is as the upper wall is stationary, then should I define simply stationary wall or I should define moving wall with zero velocity.
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CC-MAIN-2021-17
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http://peopleprobe.info/gizah/do-my-math-homework-for-me-and-show-work-1067.php
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math
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http://www.mathworks.com/matlabcentral/cody/problems/766-implement-solitaire-cipher-for-n-long-deck
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math
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MATLAB and Simulink resources for Arduino, LEGO, and Raspberry Pi
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Implement the solitaire cypher.
Given a starting permutation of numbers [1:N], deck, generate M values for the keystream. The small joker will be value N-1, the big joker will be N.
This is an update of this problem.
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https://www.tripadvisor.com/Attraction_Review-g187853-d4872153-Reviews-Santuario_Madonna_della_Riva-Cuneo_Province_of_Cuneo_Piedmont.html
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math
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We noticed that you're using an unsupported browser. The Tripadvisor website may not display properly.We support the following browsers: Windows: Internet Explorer, Mozilla Firefox, Google Chrome. Mac: Safari.
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https://web2.0calc.com/questions/geometry_22653
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math
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Three identical circles of radius 30 cm are tangent to each other externally. A fourth circle of the same radius was drawn so that its center is coincidental with the center of the space bounded by the three tangent circles. Find the area of the region inside the fourth circle but outside the first three circles. It is the shaded region shown in the figure below.
My goal is to find the area of a segment BGC. Let's get started.
AJ = sqrt(AK2 - JK2) = 30√3
AH = 2/3(AJ) = 20√3
AE = HJ = 1/3(AJ) = 10√3
Angle BAC = 2[arccos(AB / AE)] = 109.4712206º
Area of sector ABGC = 302pi * (∠BAC / 360) = 859.784956 u2
Area of triangle ABC = BE * AE = 424.2640687 u2
Area of segment BGC = [ABGC] - [ABC] = 435.5208873 u2
Area of circle H = 302pi = 2827.433388
Shaded area = 2827.433388 - (6 * 435.5208873) = 214.3080644 u2
I've posted this question nearly 6 years ago, and my son answered it. He told me about that 2 days ago. (Beautiful communication)
His answer is very close to mine. (Dragan ≅ jugoslav ) Here's his answer:
This one was a bit tricky, but I think it's because I made it more complicated than it needed to be.
The most important first step was to determine the center of our fourth circle. How far away is it from the centers of the other three circles?
I set the center of the bottom left circle as my "origin." Then, creating a triangle between the three centers, I solved for the centroid of the triangle (which had an effective coordinate of (30,17.32)). This means that the distance between the fourth circle and the other three circles was 34.64 cm.
Now, knowing this, I generated an independent problem: what is an overlapping area of two circles 34.64 cm apart, with radii of 30 cm? Once I had this number, it was simply a matter of multiplying it by three and subtracting it from the overall area of one circle.
You probably don't want to see my math...I used an integral in the cartesian coordinate system. I basically found the midpoint of their intersecting area and set it as my lower limit (17.32), set 30 as my upper limit, and integrated sqrt(30^2-x^2)dx. Then I multiplied that area by 4, since integrating circles in cartesian coordinates is weird. It was messy, but the answer: the overlapping area was 871.08 cm^2.
Since the area of the fourth circle is pi*r^2 (or 2827.43 cm^2), you simply subtract 871.08*3 (or 2613.24 cm^2, which is the area of intersection between the fourth circle and each of the other three circles).
This yields the final answer of 214.22 cm^2...give or take for rounding error. ;)
-KMM (May 19, 2015)
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| 2,565 | 21 |
https://www.skidmore.edu/mathematics/faculty/szabo.php
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math
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Teaching Professor of Mathematics
Office: CIS 340B
Telephone: (518) 580-8421
Professor Szabo joined the department in 2015. In 2009 she completed her Ph.D. in mathematics at Rensselaer Polytechnic Institute with a dissertation in the field of mathematical biology. Prof. Szabo also holds a Master's degree in Applied Mathematics from RPI and a Bachelor's degree in mathematics from New England University in Springfield, MA. Prior to coming to Skidmore, Prof. Szabo was a visiting assistant professor at West Point and Bard College. Prof. Szabo's research interests include mathematical biology and network science. She teaches courses in calculus, linear algebra, and quantitative reasoning at Skidmore.
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CC-MAIN-2021-25
| 704 | 4 |
https://mathandculture.wordpress.com/
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math
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Arithmetic, geometry, algebra, trigonometry , calculus…
Skip to content
Da Vinci’s Vitruvian Man of Math
Why do honeybees love hexagons?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676594886.67/warc/CC-MAIN-20180723032237-20180723052237-00162.warc.gz
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| 420 | 11 |
https://www.gumtree.com.au/s-ad/gungahlin/toys-indoor/baby-and-toddler-toys-clearout/1211730644
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math
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Baby and Toddler toys clearout
Conditions varies from excellent to good used condition. All good working conditions unless stated otherwise.
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| 959 | 18 |
https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-17-13372&id=171016
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math
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Instead of Zernike polynomials, ellipse Gaussian model is proposed to represent localized wave-front deformation in researching pointing and tracking errors in inter-satellite laser communication links, which can simplify the calculation. It is shown that both pointing and tracking errors depend on the center deepness h, the radiuses a and b, and the distance d of the Gaussian distortion and change regularly as they increase. The maximum peak values of pointing and tracking errors always appear around h=0.2λ. The influence of localized deformation is up to 0.7µrad for pointing error, and 0.5µrad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the machining precision of optical devices, which should be more greater than 0.2λ, is proposed. The principle of choosing the optical devices with localized deformation is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the results can be used in the design of inter-satellite lasercom systems.
©2008 Optical Society of America
Comparing to microwave communications, inter-satellite laser communication (lasercom) has many advantages, such as smaller size and weight of the terminal, less power consumption, greater immunity to interference, larger data rate, and denser satellite orbit population, consequently it provides an attractive alternate to microwave communications for both commercial and military applications. [1–5] Inter-satellite lasercom relates to laser beam transmission which has recently been extensively studied. [6, 7] Due to the small beam divergence and the ultra-long distance of the communication links, wave-front aberrations strongly affect the spatial pointing and tracking of laser beams.
There are two major reasons which cause the wave-front aberrations. The first reason is the space environment which includes space radiation, contamination, and especially temperature variation. Temperature variation causes local changes in the optical properties of the devices, such as variation of the reflective index, variation of the curvature of the lens surface, variation of the thickness of the lens, and variation in the gap between lenses. The second reason is the processing technic. It is difficult for the optical devices, especially for that with large aperture, to be processed to the precision of 0.01λ and remain unchanged for long time, consequently localized distortions is almost inevitable. Both of the two reasons are equivalent to the deformation of the optical devices. When the beam transmits the optical devices with deformation, its wave-front will change locally.
Toyoshima et al. have studied mutual alignment errors in circle region using Zernike polynomials expressing wave-front aberrations. Furthermore, Sun et al. developed the research to annular region. Due to the orthogonality of Zernike polynomials, almost all the wave-front aberrations in the optical system can be represented by them. [12–14] However, it generally needs too many items of Zernike polynomials to express localized distortion, which complicates the calculation. To simplify the analysis, we proposed ellipse Gaussian model to represent localized deformation, which is proved simple in the calculation by comparison with Zernike polynomials. Based on ellipse Gaussian model, the effects of localized wave-front deformation on pointing and tracking errors are researched. The purpose of the research is to estimate how much the influence of localized wave-front deformation on pointing and tracking errors, and try to provide the evidence of processing precision for the optical devices used in lasercom.
This paper has the following outline. In Section 2 the ellipse Gaussian Model is introduced to describe local distortion. In Section 3 pointing and tracking errors are defined. Section 4 is devoted to numerical analysis. Section 5 summarizes our results.
2. Ellipse Gaussian model
Due to the limitation of processing technic and the effects of space environment, the localized distortion is extremely likely to appear in satellite optical system, especially in the primary mirror of transmitter antenna due to the large aperture. To simplify the analysis, we propose ellipse Gaussian model to express them, which is shown in Fig. 1 and can be written as
where A is the center value of the ellipse Gaussian function (the center deepness h=A(1-1/e)), a and b are the radiuses of the localized distortion, (x 0,y 0) is the coordinate of the center, d is the distance from (0,0) to (x 0,y 0), which can be represented as
Assuming that there is localized deformation in the primary mirror of reflection-style antenna, when the beam is reflected by it, localized wave-front deformation is generated. The forming process is shown in Fig. 2. The wave-front deformation can be written as
where ψ denotes the center amplitude of ellipse Gaussian function, which is considered to be 4Aπ/λ. Equation (3) is composed of two parts, Φ1 and Φ2. Φ1 is ellipse Gaussian function, and Φ2 is a constant.
The optical field of the beam reflected by mirrors can be shown in the form
where H(x,y) is the optical field before the optical device, exp(jΦ) is called aberration term caused by the localized distortion.
Root mean square (rms) is a conventional factor to evaluate the degree of wave-front aberrations, for the ellipse Gaussian function Φ1, which can be expressed as
where S denotes the deformation area which is an ellipse with major axis radius a and minor axis radius b. From Eq. (5) we can find that rms proportionably depends on the center deepness h, but has no relation to the radiuses a and b.
3. Pointing and tracking errors
Mutual alignment errors are defined in Ref. 10 as the angle between the transmitting and receiving optical axes. We consider that, in fact, mutual alignment errors include two parts: pointing and tracking errors. They are described in the following subsections.
3.1. Pointing error
Pointing error is defined as the angle between the transmitting optical axes with and without wave-front aberrations. Transmitting optical axis is determined by the direction with the peak intensity at a far-field. The definition of the coordinate systems is shown in Fig. 3, which is similar to that in Ref. 10. The transmitter beam is Gaussian beam with localized wave-front aberrations, which can be written as
where C is a constant, F 0 is the radius of curvature at the transmitter, M 1(x 0,y 0) is transmitter aperture function which is determined by the transmitter antenna with primary mirror radius R 1 and secondary mirror radius R 2, ω 0 is waist radius of the Gaussian beam. The intensity distribution Ire(x,y) in the receiver plane is obtained as the following
where λ is the wavelength, zf is the distance of the two communication terminals. For transmitter beam free of aberrations, the peak intensity is at the origin. And for the beam with aberrations, it is at the position of Ire(x,y) |max=Ire(xmax,ymax). In this case, pointing error θP can be written in the form
3.2. Tracking error
Tracking error is defined as the angle between the receiving optical axes with and without wave-front deformation. Receiving optical axis is obtained by the gravity center of the received optical power on an optical tracking sensor. Owing to the long distance between the two communication terminals, the received wave can be considered as plane wave. When the plane wave passes through the optical terminal which is equivalent to a lens with focal length f, it is focused on the focal plane, and the intensity is given by
where B is a constant, M 2(x,y) is receiver aperture function which is determined by the receiver antenna, r 1 and r 2 are the primary mirror radius and secondary mirror radius, Φ(x,y) is wave-front deformation in receiver plane. Similarly, when there is no aberrations in the optical systems, the gravity center of the received optical power in the focus plane is at the origin. However, normally the center of gravity is at (X,Y) when aberrations exist in the optical systems. By definition tracking error θT can be written as
where X and Y are given by the following equations
Similar to pointing error θP, tracking error θT also depends on the following parameters: the center deepness h, the radiuses a and b, and the distance d.
where H(x) denotes Gaussian beam for pointing error, or plane beam for tracking error. F is the distance between two satellites for pointing error, or the focal length of receiver optical system for tracking error. And D=2R 1 for pointing error, or D=2r 1 for tracking error. Substituting Eq. (3) into Eq. (14), we can obtain the following equation
Equation (15) shows that the optical field consists of three parts. By definition the first and the second parts don’t cause pointing and tracking errors which are mainly influenced by the third part. Therefore, to simply the analysis, we can only consider the third part which is shown as
From Eq. (16), we can find that pointing and tracking errors are mainly determined by the aberration term exp(jΦ1). It is known that exp(jΦ1)=exp[j(Φ1+2 π)], namely the aberration term is a periodic function whose period is 2π. Therefore, pointing and tracking errors would vary periodically with the change of Φ1. When Φ1=0, pointing and tracking errors are zeros. We know that the wave-front difference for Φ1=0 and Φ1=(2n-1)π is the maximum. Therefore, the peaks of pointing and tracking errors would appear around Φ1=(2n-1)π (n is positive integer). Due to Φ1 being a function of x and y, the peaks should be around rms=(2n-1)π. Furthermore, from the integral region we can conclude that it is the localized deformation, not the whole aperture, which determines the pointing and tracking errors, consequently rms should obtained from localized deformation area (See Eq. (5)). Related to Eq. (6), we can conclude that pointing and tracking errors would change periodically as the center deepnees h increases. Though the radiuses a and b don’t contribute to the rms of Φ1 according to Eq. (6), it determines the aberration area. When a rises, the value of h(u) increases, namely the influence of wave-front deformation increases too. According to the definitions of pointing and tracking errors, they would increase as the distortion becomes wide.
4. Numerical results and analysis
To show the advantages of ellipse Gaussian model, the comparison between ellipse Gaussian function and Zernike polynomials to represent the localized deformation is addressed in Figs. 4 and 5. For the localized deformation which is expressed accurately by ellipse Gaussian function, we represent it using Zernike polynomials with different terms. The term numbers are N=20, 40 and 60, respectively. The results are in the Fig. 4. When N=20 the result of Zernike polynomials is very poor, and when N=40 the result becomes better. When N=60 the result is close to that of Gaussian function. The results show that it does need many terms for Zernike polynomials to express the localized deformation with less error, which will complicate the calculation. Fig. 5 gives the results of Zernike polynomials with N=40 for different a/D. As can be seen that the result is better for large value of a/D than for small value of a/D. In a word, by comparison with Zernike polynomials, ellipse Gaussian model can really simplify the calculation due to its simple expression, especially for small value of a/D.
Based on ellipse Gaussian model, the numerical results of the effects of localized wave-front deformation on pointing and tracking errors are given in Figs. 6 and 7. In the calculation process, the parameters are D=2R 1=2r 1=250 mm, R 2=r 2=40 mm, λ=800 nm, ω 0=125 mm, and f=1000 mm. The distance of the two satellites is taken to be zf=50,000 km. Fig. 6 shows how pointing and tracking errors vary with the center deepness h, the radiuses a and b, and the distance d. In calculation we only consider the condition that the Gaussian deformation is totally in the aperture of the antenna, and the center of Gaussian distortion is in x axis. As can be seen from Fig. 6, pointing and tracking errors do not monotonically rise with h increasing as generally expected, but fluctuates like damped oscillation. On the other hand, pointing and tracking errors monotonically increases as a rises. In other words, the wider localized distortion, the stronger influence on pointing and tracking errors. With the distance d increasing, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases secondly. The difference is considered that the beam contributing to tracking error is plane beam, while that contributing to pointing error is Gaussian beam whose intensity decreases with d increasing. Fig. 7 shows clearly the fluctuation of pointing and tracking errors with h and rms rising. The peak appears around h=0.2λ (rms=π), h=0.75λ (rms=3π), h=1.25λ (rms=5π), et. al.. The fluctuation period for rms value is 2π. The results show that to reduce the impact of localized deformation on pointing and tracking errors, the center deepness h should be more less than 0.2λ, namely the machining accuracy of the optical devices should be more greater than 0.2λ. Moreover, the influence of localized deformation is up to 0.7µrad for pointing error, and 0.5µrad for tracking error.
The comparison of pointing and tracking errors for localized deformation expressed by ellipse Gaussian function and Zernike polynomials are shown in Fig. 8. As can be seen that pointing and tracking errors due to wave-front aberrations described by Zernike polynomials, are gradually close to that expressed by Gaussian function with N increasing. Figs. 10(a) and 10(d) show that Zernike results are better for small value of h than for large value h. The reason is that, for small value of h, the localized deformation plays an important role, and the effect of Zernike error is comparatively weak. With h rising, the influence of the localized deformation reduces, then the impact of Zernike error gradually increases. Furthermore, as shown in Figs. 8(b) and 8(e), Zernike results are obviously worse for small value a than for large value a. The reason is that Zernike error is large for small value a/D than large value a/D, which is shown in Fig. 5.
From above numerical analysis, we can conclude that ellipse Gaussian model is an effective method for the localized distortion, especially for that with small values of a/D. To weaken the effect of localized deformation on pointing and tracking errors, processing precision of optical devices should be more than 0.2λ. If we have to use the optical devices with localized deformation, we may select them according to the following principles: (1) The deepness h is more less than 0.2λ; (2) The radiuses a and b are small; (3) The center position (x 0,y 0) is near by the center of the optical device. In addition, if we know the localized deformation before laser beam transmitting/receiving, we can adjust pointing direction to compensate the pointing and tracking errors caused by localized aberrations.
To research localized deformation on pointing and tracking errors in inter-satellite lasercom, ellipse Gaussian model is proposed, which can simplify the calculation especially for small value of a/D by comparison with Zernike polynomials. It is found that pointing and tracking errors due to localized deformation are mainly determined by the center deepness h, the radiuses a and b, and the distance d. With the increasing of the deepness h, both of pointing and tracking errors fluctuate like damped oscillation with peak values around h=0.2λ (rms=π), h=0.75λ (rms=3π), h=1.25λ (rms=5π), et al.. The wider the localized deformation is, the more for the influence on pointing and tracking errors being. With the distance d rising, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases monotonically. The effects of localized deformation is up to 0.7urad for pointing error, and 0.5urad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the processing accuracy of optical devices should be more greater than 0.2λ. The principle of choosing the optical devices with localized distortion is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the conclusion can be used in the design of inter-satellite lasercom systems.
References and links
1. F. Cosson, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.,” Proc. SPIE 1417, 262–276 (1991). [CrossRef]
2. B. Laurent and G. Planche, “SILEX overview after flight terminals campaign, in Free-Space Laser Communication Technologies IX, G. S. Mecherle, ed.,” Proc. SPIE 2990, 10–22 (1997). [CrossRef]
3. A. Mauroschat, “Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.,” Proc. SPIE 1417, 513–524 (1991). [CrossRef]
4. M. Renard, P. Dobie, J. Gollier, T. Heinrichs, P. Woszczyk, and A. Sobeczko, “Optical telecommunication performance of the qualification model SILEX beacon, in Free-Space Laser Communication Technologies VII, G. S. Mecherle, ed.,” Proc. SPIE 2381, 289–300 (1995). [CrossRef]
5. K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies VIII, G. S. Mecherle, ed.,” Proc. SPIE 2699, 114–120 (1996). [CrossRef]
8. Brian R. Strickland, Michael J. Lavan, Eric Woodbridge, and Victor Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt. 38, 424–431 (1999). [CrossRef]
9. Shlomi Arnon, “Power versus stabilization for laser satellite communication,” Appl. Opt. 38, 3229–3233 (1999). [CrossRef]
10. M. Toyoshima, N. Takahashi, T. Jono, T. Yamawaki, K. Nakagawa, and A. Yamamoto, “Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link,” Opt. Express 9, 592–602 (2001). [CrossRef] [PubMed]
11. J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt. 44, 4953–4958 (2005). [CrossRef] [PubMed]
14. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981). [CrossRef]
15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).
16. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).
17. M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).
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CC-MAIN-2021-04
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https://www.hackmath.net/en/math-problem/1168
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math
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Determine if it is possible to construct a triangle with sides 28 31 34 by calculation.
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- Flood water
Flood waters in some US village meant that the homes had to evacuate 364 people. 50 of them stayed at elementary schools, 59 them slept with their friends and others went to relatives. How many people have gone to relatives?
- Neighbor angle
For 136° angle calculate size of adjacent angle on one side of a straight line.
Which of those angles are obtuse?
- Valid number
Round the 453874528 on 2 significant numbers.
- Math classification
In 3A class are 27 students. One-third got a B in math and the rest got A. How many students received a B in math?
- Angles 1
It is true neighboring angles have not common arm?
- Addition of Roman numbers
Added together and write as decimal number: LXVII + MLXIV
- Bean bag
A student tossed a bean bag. It landed 216 inches away. How many yards are equal to 216 inches?
- Six te 2
If 3t-7=5t, then 6t=
How many parts of line divide 5 (different) points that lie on it?
On the bowl were a few cakes. Jane ate one-third of them, Dana ate a quarter of those cakes that remained. a) What part (of the original number of cakes) Dana ate? b) At least how many cakes could be (initially) on thebowl?
On how many parts divide plane 6 parallels?
- One frame
5 picture frames cost € 12 more than three frames. How much cost one frame?
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| 2,192 | 35 |
http://forums.wolfram.com/mathgroup/archive/2004/Jun/msg00245.html
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math
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Re: Scriptable Mathematica tools for auto-editing text?
- To: mathgroup at smc.vnet.net
- Subject: [mg48709] Re: Scriptable Mathematica tools for auto-editing text?
- From: AES/newspost <siegman at stanford.edu>
- Date: Fri, 11 Jun 2004 03:52:34 -0400 (EDT)
- References: <[email protected]> <[email protected]>
- Sender: owner-wri-mathgroup at wolfram.com
In article <ca90ks$t2t$1 at smc.vnet.net>, "John Jowett" <John.Jowett at cern.ch> wrote: > The standard Mathematica functions provide a very powerful toolkit for this > kind of thing. There are various approaches depending on what you want to > do. It is worth reading the section "Files and Streams" in the Mathematica > book and also being familiar with the Import, ReadList and Export functions. Thanks -- that's very much the kind of thing I had in mind. (and I'd actually enjoy learning the tools and putting together my own package for this -- except it all too often seems I'm spending more than 90% of my working time learning new tools in multiple software applications, and less than 10% in actually getting any work done with them.)
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CC-MAIN-2018-17
| 1,113 | 8 |
https://hubpages.com/education/probability-and-statistics/3743
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math
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Statistics is the field of science that deals with organization, interpretation and analyzing of a data. The term statistical data refers to the data collected form different sources through methods experiments, surveys and analysis. This data is...
Many of the college students taking up Business Statistics are having a hard time coping with it. Why does this happen? Let's find out.
Central Tendency: Mean, Median, Mode and Chapter 3: Some Key Ingredients for Inferential Statistics: Z Scores, the Normal Curve, Sample versus Population, and Probability study guide.
Spearman Rank Correlation and Pearson Correlation can be both used to identify the relationship between two sets of data. This a brief comparison between the two measures of correlation and there are examples provided to explain how to solve them.
Wondering if there is a way to help you out conduct a simple regression analysis? Well, there is! All you need to have is a CASIO calculator that is capable of performing the task. Read this article to find out how to do it.
In my opinion, the measure of money can be both, but I lean more towards discrete. Why? For example, 1 and ½ pennies can’t be measured; the monetary value of one and ½ pennies can only be valued as a rounded amount of 2 pennies. The half...
Standard Deviation = σ (The Greek letter sigma) First find the mean of the given set of numbers. Next subtract the mean from each number in the set. Then square the sum of each number. Add the total of the squares together. Now divide...
For this hub, I will be explaining what expected value is, how to calculate it, and show an example by calculating the expected value of a betting game involving a fair coin.
Psychology Statistics Midterm Study Guide
This is a very interesting look at the odds of you existing, as you, in this world. In fact, it is virtually zero based on the probability of a series of events that must occur for you to exist.
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CC-MAIN-2019-43
| 1,938 | 10 |
https://www.slideserve.com/search/hydrodynamic-lubrication-ppt-presentation
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math
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Tribology. Outline. Basics of Tribology: Introduction and definitions The different types of friction and wear Lubricants and surface treatments Rheometry and Tribometry: The Rheo-Tribometer Measurements on the Rheo-Tribometer Stribeck curve Static friction testsBy gad
Conformal & Non-Conformal Surfaces. Figure 1.1 Conformal Surfaces. [From Hamrock and Anderson (1983).]. Figure 1.2 Nonconformal Surfaces. [From Hamrock and Anderson (1983).]. Hydrodynamic Lubrication. Minimum film thickness:. Figure 1.3 Characteristics of hydrodynamic lubrication.By chavi
Chapter 8: Lubrication, Friction and Wear. “...among all those who have written on the subject of moving forces, probably not a single one has given sufficient attention to the effect of friction in machines...” Guillaume Amontons (1699). Conformal and Nonconformal Surfaces.By jescie-rowe
Nanoparticles in lubrication. KEM-31.5530 Nanoparticles, 3.5.2011 Timo J. Hakala. Contents. Introduction to lubrication Important lubricant properties Nanoparticles in lubrication Some examples. Lubrication regimes. Lubrication can be divided in three regimes Boundary lubricationBy ciara-white
CHAPTER 16 LUBRICATION SYSTEM OPERATION AND DIAGNOSIS. OBJECTIVES. After studying Chapter 16, the reader should be able to: Prepare for ASE Engine Repair (A1) certification test content area “D” (Lubrication and Cooling Systems Diagnosis and Repair). Explain hydrodynamic lubrication.By amena-hunt
Tribology Lecture I. Tribology. From: = rubbing. Friction Wear Lubrication. Tribology deals with all aspects of. interacting surfaces in relative motion. - bearings. Friction. Loss of energy due to rubbing. Energy is converted to heatBy mbray
View Hydrodynamic lubrication PowerPoint (PPT) presentations online in SlideServe. SlideServe has a very huge collection of Hydrodynamic lubrication PowerPoint presentations. You can view or download Hydrodynamic lubrication presentations for your school assignment or business presentation. Browse for the presentations on every topic that you want.
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CC-MAIN-2021-10
| 2,050 | 7 |
https://www.calculator.org/properties/frequency.html
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math
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What Is Frequency?
Frequency is understood as the number of repeating events in a given unit of time; a measure of how frequently something occurs. For every frequency, there is a period, which is the duration of time between one event of the same type and the next. This period is the reciprocal of the frequency. This is written as:
T = 1/f
where T is the period and f is the frequency. The SI unit for frequency is the hertz (Hz), which is equal to 1/s, or once per second. Frequency is encountered many times in physics. A few highly important instances are listed here.
All waves in physics are said to have frequency. This frequency is inversely proportional to wavelength by the phase speed, or speed that a wave travels in a given medium. This medium depends on the wave in consideration. Sound, for example, travels through the air, solids, and liquids. We write the frequency of a physical wave as:
f = v/λ
where v is the phase speed and the wavelength is given by λ. Let’s use sound as an example. The frequency we are measuring is the time between one compressed zone of air molecules and the next. Sound waves are alternating changes in air pressure that deviate from the usual air pressure. The same goes for sounds travelling through a liquid, which has local changes in compression that propagate the sound wave. A sound travelling through a solid, however, takes on a different character. It is considered to be a transverse wave that represents local changes in shear stress.
The phase speed of sound in air is 343 meters per second. Sound waves can have many different frequencies, which for us, are considered to be their pitch. The human audible range of sound frequencies begins at 20 Hz and ends at 20,000 Hz. Other animals are able to pick up on sounds outside our audible frequency range. Dogs, for example, are able to hear higher pitched sounds than we are.
Light is another physical wave that has a frequency. For all electromagnetic waves (including light), we assume that the phase speed v is equal to the speed of light, c (3*108 m/s), so that:
f = c/λ
Again, there are many different frequencies of light in the electromagnetic spectrum. The light visible to the human eye lies between 790 to 400 terahertz, and we are most sensitive to the greenish light at 540 terahertz. It must be noted that visible light is only a small portion of the entire electromagnetic spectrum. There are UV rays, Infrared rays, microwaves, radio waves, gamma rays, and X rays. The entire electromagnetic spectrum has frequencies that run between 10 Hz to 1024 Hz.
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CC-MAIN-2022-40
| 2,580 | 11 |
https://www.enotes.com/homework-help/line-l-has-y-intercept-0-4-parallel-line-with-454639
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math
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A line (L) has a y intercept of (0,4) and is parallel to a line (A) with the equation y=3x + 6. Write the equation for line (L) in slope intercept form
The slope-intercept form of the equation of a line is: y = mx + b, where m is the slope and b is the y-intercept. Hence, we need to know these two parameters to know the equation of the line.
line (L) passes through the point (0,4). Hence, it's y-intercept is 4 -- that is, b = 4.
We know that line (L) is parallel to line (A). Two lines are parallel if and only if they have the same slope. Hence, the slope of L is the same as the slope of A. A has equation y = 3x + 6, which means that its slope is 3, and hence this is also the slope of L -- that is, m = 3.
Now, we now that for L, m = 3, and b = 4. We simply substitute this to the equation to get the equation of L:
y = 3x + 4
If a line is parallel to another line, the two lines will have the same slope. Therefore, if the new line, line, line L is to be parallel to line A it must have a slope of 3.
Line A is written in slope-intercept form: `y=mx+b` which tells us m is the slope and b is the y-intercept.
So, the slope of the line has to be 3 and the y-intercept given is (0, 4),
therefore the equation of line L is `y=3x + 4.`
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s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889542.47/warc/CC-MAIN-20180120083038-20180120103038-00517.warc.gz
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CC-MAIN-2018-05
| 1,240 | 10 |
https://mocivilengineering.com/determinate-and-indeterminate-truss-structure/
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math
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Determinate and indeterminate truss, distinguishing whether truss is determinate or indeterminate requires knowing b, b is the number of truss member, truss members are straight and carrying axial forces only, also we should know j, j is the joint numbers, each group of truss members is connected to one joint and since the members carrying only axial forces the sum of moment at joint will be zero, members of truss are coplanar and concurrent, therefore the rotational or moment equilibrium is satisfied at the joint. two equation of equilibrium remains for each joint ΣFx=0 and ΣFy=0, their determinacy of truss can be determined using the equations below
r is the total number of external reaction at supports, the first term of equation (b+r) represent the number of unknown including axial forces for each member and external reaction at supports, the second term of the equation representing the number of available equilibrium equation that can be used to determine unknown, determinate structure can be analyzed using equilibrium equations, indeterminate structure cant be solved using equilibrium equations, determining unknown reactions will require relating displacement and slope with loads and reactions and this known as compatibility equations. Compatibility equations involve the geometric and physical properties of the structure. the figure below showing some examples of statically determinate and indeterminate structures.
construction management: concrete construction
bridge construction:How to become a bridge engineer
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CC-MAIN-2021-04
| 1,545 | 4 |
https://www.mopedarmy.com/forums/read.php?2,1328340
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math
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I found a junky version of this bike and have a chance to pick it up for less than 50$
No title and needs a CDF unit. What is a CDF unit?
Is it worth my time and 50$?
(please email me directly!)
WHAT DO YOU THINK?
The auction ends in an hour and a half...
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039617701.99/warc/CC-MAIN-20210423101141-20210423131141-00434.warc.gz
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CC-MAIN-2021-17
| 255 | 6 |
https://www.tag-challenge.com/2022/10/21/what-is-mathtools-for/
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math
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What is Mathtools for?
Mathtools provides a series of packages designed to enhance the appearance of documents containing a lot of mathematics. The main backbone is amsmath, so those unfamiliar with this required part of the LaTeX system will probably not find the packages very useful.
What is Mathtools package in LaTeX?
mathtools is the package you likely never knew you needed. LaTeX makes typesetting equations easy, and mathtools makes those equations beautiful. mathtools is an extention of amsmath . If you include mathtools , you can use any function or macro from that package.
What are some math topics?
Major Topics and Concepts
- Whole numbers.
- Fractions and Mixed Numbers.
- Ratios, Rates, and Proportions.
- Real Numbers.
- Solving Equations and Inequalities.
- Graphing Linear Equations and Inequalities.
How do you use MathMagic in InDesign?
InDesign menubar -> Plug-ins -> MathMagic -> Preferences… dialog: Click “Find” button….2. Creating Equations
- Launch Adobe InDesign application(CSx or CCx).
- Make a new InDesign document or open a document.
- Select the Plug-ins menu -> MathMagic sub-menu.
- Choose “New Equation” item to create an equation.
What packages does Mathtools load?
The mathtools package builds off of the amsmath package, adding further useful symbols and tools. It automatically loads the amsmath package, so you do not need to load both in your document preamble.
How do you Underbrace in LaTeX?
DESCRIPTION. nderbrace command is used to put a (stretchy) under-brace under the argument.
What is Math Magic?
Math magic tricks can liven up any math class and create a sense of wonder and curiosity about math. Not only that, math magic creates a new context for algebraic reasoning as students go beyond “What’s the answer?” to explore “What’s the trick?”
How do you add MathType in InDesign?
Editing MathType Equations in InDesign Object Boxes
- Choose File > Open from the MathType menu bar.
- Select the equation you need to edit and click Open.
- Make the desired changes, then save the equation (File > Save).
- Update the link from the EPS file to the document.
- Click the Update Link icon to update the link.
Does Mathtools load Amssymb?
The amsmath package documentation can be found here. The mathtools package builds off of the amsmath package, adding further useful symbols and tools. It automatically loads the amsmath package, so you do not need to load both in your document preamble.
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| 2,465 | 33 |
http://nocompulsoryvaccination.blogspot.com/2009/07/japanese-data-show-vaccines-cause.html
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math
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Sunday, July 5, 2009
Japanese Data Show Vaccines Cause Autism
When is proof not proof? When that proof means that vaccination is called into question. Then, the proof is squashed, ridiculed or just plain ignored. How many more families will need to be destroyed before all of us finally get that vaccines cause autism?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867666.97/warc/CC-MAIN-20180625111632-20180625131632-00586.warc.gz
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CC-MAIN-2018-26
| 318 | 3 |
https://acervodigital.unesp.br/handle/unesp/360593
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math
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Please use this identifier to cite or link to this item:
- Graph of inequalities
- Pegg Jr, Ed
- The graph of a linear equation is a straight line. The graph of a linear inequality is the half of the plane to one side of the line. The solution of a system of inequalities is the set intersection of the regions for each inequality. It may be the empty set, a finite polygon, the inside of an acute angle, a strip, and so on, depending on the nature of the inequalities
- Componente Curricular::Ensino Fundamental::Séries Finais::Matemática
- Educação Básica::Ensino Fundamental Final::Matemática::Equações
- This demonstration needs the "MathematicaPlayer.exe" to run. Found in http://objetoseducacionais2.mec.gov.br/handle/mec/4737
- Demonstration freeware using Mathematica Player
There are no files associated with this item.
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107889574.66/warc/CC-MAIN-20201025154704-20201025184704-00227.warc.gz
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CC-MAIN-2020-45
| 934 | 10 |
http://homepages.rootsweb.com/~livcomo/images/strand2.html
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math
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The Strand Theatre was destroyed by fire on March 28, 1933.
Present (May 2004) owners are refurbishing the hotel.
RootsWeb is funded and supported by
Ancestry.com and our loyal RootsWeb community.
About Us | Contact Us | Copyright | Report Inappropriate Material
Corporate Information | Privacy | Terms and Conditions | CCPA Notice at Collection
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572581.94/warc/CC-MAIN-20220816211628-20220817001628-00477.warc.gz
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CC-MAIN-2022-33
| 357 | 6 |
https://www.mathworks.com/matlabcentral/profile/authors/1322323-michal-kvasnicka
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math
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Professional Interests: numerical analysis, mathematical modeling, high performance computing
Top 1% contributor
Same error on openSUSE 12.1 with Matlab R2011b. Any hints???
Answered 4 years ago
yes, of course.,anyway thank you again for your help. Michal
Responded 2 years ago
Thank you, this is good method.
Responded 3 years ago
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s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736679756.40/warc/CC-MAIN-20151001215759-00122-ip-10-137-6-227.ec2.internal.warc.gz
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https://banana.blubrry.com/2020/03/20/whats-math-used-in-engineering-ltpgtlt-pgttwo-ways-in-which-it-could-be-employed/
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math
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You could be curious about exactly is math, if you have been analyzing engineering. The following write-up discusses the use of mathematics in engineering, which includes both quantitative and qualitative biology assignment help software.
Often wind up describing people the way technology is finished. However, the best engineers aren’t really talking! They give examples which show the processes.
The numerical elements of engineering tend to be described using three-dimensional”prisms.” In the easiest of cases, it is, focused by a prism, alternatively of light . A glass provides the image of the circle or alternative surface on which light can reflect. And this could be transformations of surfaces, in addition to the basic idea of linear algebra, something used to do algebraic operations.
Instances of technology analysis are www.mbaassignmenthelp.org/business-law-assignment-help/ qualitative in character. 1 example is electric currents (a explanation of the way that electricity flows) or current leaks onto the circuit. A far more complex kind of this approach is discussed within an example offered by Thomas Edison at 1891.
At the time of today, the world wide web has manufactured science’s definition tremendously wide. However there are a number of frequent elements that are associated with science: reasoning, along with experimentation, observation, measurement, calculation. These four traits are found in software of mathematics.
Along to what I said, below are a few other vital methods by which is math. Each One of These cases can be expanded later on:
As you will see within the rest with this article is math used in technology is all about the 2 sorts of processes utilised to complete an engineering endeavor. Let’s briefly examine .
Experimentation is the practice of identifying a problem that http://www.publishing.monash.edu/contact.html needs to be solved. For instance, by driving on a path, once you examine the tires onto your own car, experimentation is being done by you. You are testing a brand new pair of tires , because you need to be certain that they will hold up out.
Observation is actually a non-computing sample of just how is mathematics. As watching for problems you can think. For instance, whenever you generate a street past and come along with a pothole, you’re observing problems that need to be repaired.
Calculation could be the process of calculating. When you would like to know how many bucks you’ve made or so two how much the fuel travelled up. This course of action may be the 1 that is obvious. You are detecting After you calculate the answer of a challenging math problem.
Reasonable reasoning is quite similar to computation in that it is a process of locating methods. This process is ordinarily exactly the exact same as believing with mind rather than the chip that stores advice. The following approach is frequently thought of as logic.
So how is mathematics used in engineering? It’s just really a process which calls for observation, experimentation, dimension, and reasoning. It is the fundamental concept behind the way the mathematics.
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CC-MAIN-2020-29
| 3,131 | 12 |
https://www.coursehero.com/file/p287q00/A-phase-locked-loop-PLL-is-a-feedback-circuit-consisting-of-a-A-phase-detector/
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math
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55.A phase-locked loop (PLL) is a feedback circuit consisting of a A. phase detector. B. low-pass filter. C. VCO. D. all of the above Hint: A phase-locked loop is a feedback system combining a voltage controlled oscillator (VCO) and a phase comparator so connected that the oscillator maintains a constant phase angle relative to a reference signal. As the name suggests PLL, if the detected phase is having any error/distortion means the LPF filters it. Then the oscillator maintains the constant phase angle. 56.If a 1 MHz carrier is amplitude modulated with a 5 kHz audio signal, the upper-side frequency is ________ kHz. Radio broadcasts are generally Hint: Frequency modulation gives noise free reception and is used for radio broadcasts. 57.Consider the following statements 1.The amplitude of an FM wave is constant 2.FM is more immune to noise than AM 3.FM broadcasts operate in upper VHF and UHF frequency ranges 4.FM transmitting and receiving equipments are simpler as compared to AM transmitting and receiving equipments Which of the above are correct?
Hint: FM equipments are complex as compared to AM equipments. 58.Frequency shift keying is used mostly in A. telegraphy* B. telephony C. satellite communication D. radio transmission Hint:Frequency shift keying (FSK) is a system of frequency modulation used in telegraphy. 59.The rate at which information can be carried through a communication channel depends on Hint: Rate of information depends on bandwidth. 60.A carrier is simultaneously modulated by two sine waves having modulation indices of 0.4 and 0.3. The total modulation index will be 61.A 1000 kHz carrier is simultaneously modulated with 300 Hz, 800 Hz and 2 kHz audio sine waves. Which of the following frequency is least likely to be present in the output? Hint: Frequency present in the sidebands is equal to = fc± fm, f2fm, fc± 3fm62.Which of the following is the indirect way of FM generation?c± .
A. Reactance bipolar transistor modulator B. Armstrong modulator* C. Varactor diode modulator
D. Reactance FM modulator
It generates FM through phase modulation.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989690.55/warc/CC-MAIN-20210516044552-20210516074552-00285.warc.gz
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| 2,097 | 5 |
https://shop.booksandbooks.com/book/9781071605882
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math
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WE CAN ORDER THIS FOR YOU (store pickup in 5-14 days)
About the Author
Ruth F. Curtain (born Melbourne, Australia, 1941) completed her early academic education (B.Sc. (Hons) 1962, Dip.Ed. 1963, M.A. Mathematics 1965) at the University of Melbourne and a Ph.D. in Applied Mathematics at Brown University, Providence, R.I., USA in 1969. She was Assistant Professor at Purdue University from 1970 to 1971 and spent 1971 to 1977 at the Control Theory Centre, University of Warwick, UK. At present she is emeritus professor in the mathematics department of the University of Groningen where she has worked since 1977. Her research interests lie in the area of infinite-dimensional systems theory. She is the co-author of two books in this field: "Infinite Dimensional Systems Theory", LNCIS, volume 8, Springer Verlag, 1978, with A.J. Pritchard, and "An Introduction to Linear Infinite-Dimensional System Theory", Springer Verlag, 1995, with H.J. Zwart. She has served as associate editor for the international journals Systems and Control Letters, Automatica, the journal Mathematics of Control, Signals and Systems and the Journal of Mathematical Systems, Estimation and Control and as an editor for Automatica. For contributions to the control theory of stochastic systems and infinite-dimensional systems she was elected to the grade of Fellow in the IEEE in 1991. Hans Zwart (born Hoogezand-Sappemeer, The Netherlands, 1959) received his Masters degree in mathematics in 1984 and his Ph.D. in 1988 at the University of Groningen. Since 1988 he is with the Department of Applied Mathematics, University of Twente, the Netherlands, where he is now full professor. His research interest lies in the area of distributed parameter systems. In addition, he holds a part-time professorial position at the department of Mechanical Engineering, Eindhoven University of Technology. He is the co-author of many papers and of the text books "An Introduction to Linear Infinite-Dimensional System Theory", Springer Verlag, 1995, with R.F. Curtain and of "Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces" with B. Jacob. Current research topics include system theory for distributed parameter systems with a Hamiltonian dynamics and controller design for mechanical systems.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153739.28/warc/CC-MAIN-20210728154442-20210728184442-00027.warc.gz
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CC-MAIN-2021-31
| 2,274 | 3 |
https://www.mathscareers.org.uk/five-ordinary-things-powered-maths/
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math
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You might be surprised by how many everyday items wouldn’t be around without maths.
Here are just five of them and the maths behind how they work.
1. Sat Nav
Whether you own an actual Sat Nav or use navigation software on your phone, you are relying on a huge amount of mathematics for it to work.
Satellites orbit the earth in such a way so that any GPS software can see at least four satellites at any one time. Each satellite constantly broadcasts its location and the time. Your phone or Sat Nav can then use the equation: distance = speed x time to work out its distance from each satellite. This tells your Sat Nav that you are on the surface of a sphere which is a certain radius from each satellite. Where these four spheres overlap will pinpoint exactly where your location is on Earth.
Read more in the article Navigation by Numbers
Most of us have a calendar on our wall, but did you know that its construction caused mathematicians and astronomers many difficulties over the years? Unfortunately the solar year lasts about 11 minutes short of 365.25 days. This meant that by the 16th Century the calendar was about 10 days out. To solve this problem a ruling was made which stated that years which are divisible by 100 but not by 400 should not be leap years. (Therefore the year 1900 was not a leap year, whereas 2000 was.)
By the 1970s astronomers had realised that the earth’s rotation is also slowing down. Because of this, leap seconds will sometimes need to be added to a year. The most recent leap second was added on June 30th 2015.
Read more in the article ‘What day is it?’
3. Microwave Ovens
Microwave ovens seem to act in an almost miraculous way – there is no heat needed and yet food cooks in a matter of minutes or even seconds. Microwave ovens work by generating electromagnetic radiation at a microwave frequency. The wavelength of these microwaves needs to be 12.2cm, because at this length the waves will excite the water molecules inside the food. This causes it to heat up and be cooked.
In order to create microwaves with a wavelength of exactly 12.2cm the strength of the magnetic field used needs to be calculated precisely using the following equation:
Where = Strength of the magnetic field, = Mass of an electron, = Speed of light in a vacuum, = Charge of an electron, = Wavelength
We can therefore work out that the exact strength of the magnetic field needs to be 0.088 Tesla. (Tesla is the unit used to measure the strength of a magnetic field).
Microwave ovens no longer need to be mysterious, but they are certainly mathematical!
Read a more in depth article about microwave ovens..
4. Wireless Router
We all now take WiFi for granted and would probably struggle to imagine living in a world where the internet was only connected by wires. WiFi is appearing in more and more locations, connecting people in buses, trains and city centres.
WiFi would not be made possible without a mathematical technique called Fourier Transforms which removes unwanted noise leaving a clean signal behind. Fourier Transforms are not only vital to wifi, they are also important in other areas such as audio processing and medical imaging.
Read more about the maths behind WiFi.
Anything which is digital like a computer works using a code called binary which consists only of zeroes and ones. If we write a number in binary we are said to be writing a number in a base 2, whereas our everyday numbers are written in base 10.
How a number in ordinary base 10 works
|Number in Base 10
How a number in binary, base 2 works
|Number in Base 2
In the example above the number 1011 in binary is equal to (1×8)+(0×4)+(1×2)+(1×1) which is equal to 11 as an ordinary base 10 number.
Since all computer language is based on maths, it makes sense that many successful computer programmers and IT professionals have excellent maths skills.
Read more in the article ‘What makes a computer a computer’
As you can see, mathematics is all around you, sometimes in the most unexpected places. Find out more about everyday maths by clicking on the links above or browsing all of our articles.
Older readers might enjoy the IMA Mathematics Matters series which explore a range of applied mathematics topics in depth.
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https://www.lsc-group.phys.uwm.edu/bursts/review/projects/s5-qpipe/minutes/minutes_20080214.html
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math
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Minutes of 2008-Feb-14 S5 QPipeline Review Teleconference
Shourov Chatterji, Jonah Kanner, Isabel Leonor, Dave Reitze
Minutes by Jonah Kanner.
S5 H1H2 scatterplots and significance
An investigation on a modified definition of H1H2 significance suggest that correlated energy, or perhaps a more complicate cut in the correlated vs. coherent energy plane may be a useful definition of significance.
S5 H1H2 1 year triggers
Updated distributions for time-lagged H1H2 triggers from the full 1st year analysis have been posted. Note that the zero lag triggers on this page are only from the 1 day playground.
As decided last week, the timeline has been rearranged to focus first on the triple coincident analysis in the context of the 1 day playground. At the same time, trigger production will continue for the H1H2 full 1 year analysis.
S5 H1H2L1 1 day playground analysis
A first look at the distribution of triple "coincident" triggers is posted here.
Dave: Why don't we walk through the plots that are passed around on e-mail Shourov: We'll start with scatter plots of H1H2 energies ~shourov/s5qreview/scaatter/ellipse These are plotted in terms of energy instead of energy: snr = sqrt(2*energy) Dave: The blue lines are isocontours? Shourov: It's a family of curves parametrized by a single value to define a significance We can see why correlated energy has been doing well as a statistic - in the plots a cut on correlated energy does about as well separating glitches and injections as can be done. This ellipse cut behaves about the same as a cut on correlated energy, but I thought the downward curve would be useful when adding L1. In the correlated-coherent plane, the injections tend to follow a line Isabel: When you calculate the significance, do you actually use the background?? Shourov: No. For me significance is just a number. As long as its monotonically increasing, that's all that matters here. Dave: Let me ask about S4 data. I'm looking at this outlier, and it should be followed up Shourov: Yes, it is clearly an outlier. You can find qscans linked off the S4 page ~shourov/s5qreview/s4_results If you look at the difference between S4 and the S5 playground, you'll see that S5 playground day has a very serious outlier. That may mean that S5 is not like S4. Then again, looking at the S5 1-year background, there are events louder than the outlier from the playground day. Dave: You'll do something like 200 time shifts? Shourov: No. It's too expensive. So far I've done 6. I'm hoping to do at least 10. But, with the H1H2 data set, you can't gaurentee that zero-lag is like the background. Dave: Why a nice binary sequence of time lags? Why not pick pi seconds of time lag? Shourov: That's a good idea. I'll run a weird number as well. Let's go to ~shourov/s5qreview/s5_results. Here, the red is playground day. It's one time lag in background, but it's for the whole year. Let's go to /playground_withveto/H1H2L1 H1H2 triggers are tested for time frequency coincidence with L1. Here, time lags are cheap, because you are just moving triggers around. The black have H1H2 in zero lag, just L1 is moving There's no null veto on Livingston, so the L1 significance can go quite high Vertical features in these plots are when a Hanford trigger lines up with different L triggers Dave: Are there data quality flags applied here? Shourov: Only cat. 1 data quality flags have been applied here - vetoes are not yet ready One of the main advantages of the null stream test is that it quickly cleaned up the data, without these labor issues. Dave: I'll be really interested to see how much this is going to be cleaned up when you apply vetos and category 2 DQ Isabel: These are coincident in time and frequency? Shourov: Yes, these are coincident with a window of 15 ms. The center of the tile has to overlap with some portion of the other tile. If we look at the top left plot, we can see how much the L1 test has cleaned up this plot. This is with L1 significance threshold at 10 - if you move the L1 significance up to 100. What I had hoped was to combine these statistics in one measure of significance - but that looks to be daunting. So, what I could do is threshold on L1, and then just use the ellipsoid cuts from the H1H2 to rank events. So, L1 is acting only as a veto process.
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https://remepololevoxawo.ashleyllanes.com/transition-flow-ion-transport-via-integral-boltzmann-equation-book-3086sm.php
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math
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2 edition of Transition flow ion transport via integral Boltzmann equation. found in the catalog.
Transition flow ion transport via integral Boltzmann equation.
Thomas Edward Darcie
Written in English
|The Physical Object|
|Pagination||222, [20, 6] leaves.|
|Number of Pages||222|
Grad’s assumption allows to split the collision operator in a gain and a loss part, Q(f, g) = Q+(f, g) − Q−(f, g) = Gain - Loss The loss operator Q−(f, g) = f R(g), with R(g), called the collision frequency, given by NOTE: The loss bilinear form is local in f and a weighted convolution in g. while the gain is a bilinear form with a weighted symmetric convolution structureFile Size: 5MB. Modeling of Flow Transition Using an Intermittency Transport Equation Y. B. Suzen and P. G. Huang I)epartmenl Mechanical Engineering l'niversitv of I,exington. Kentucky Abstract A new transport equation for illtermittencv factor is l)VOl)osed 1o model transitional Size: 1MB.
Here E l () and are the kinetic energy and the velocity of electrons in the valley l, and U() is the potential moderate electric fields, when the electron energy spectrum may be assumed to be parabolic, we will use the simple relation E l =P 2 /2 m l with the electron effective mass m l in the valley the electron transport under high fields and high electron energies, we. Boltzmann's Transport Equation With his ``Kinetic Theory of Gases'' Boltzmann undertook to explain the properties of dilute gases by analysing the elementary collision processes between pairs of molecules. The evolution of the distribution density in space,, is described by Boltzmann's transport equation. A thorough treatment of this.
The Linear Boltzmann Equation 1. Introduction One must distinguish between the “linear Boltzmann equation” and the “linearized Boltzmann equation.” The former has no self interaction, just scattering with the medium, whereas the latter is the linearization of the fully nonlinear Boltzmann equation. We will deal with the linear equationFile Size: KB. A similar approach can be used to calculate the contribution of electron–electron collisions to , considering the transformation in a reference frame that moves with the electron current ().In any case, the contribution of the flow to the collision integrals S 0 and S 1 is a second order correction.. To couple electron kinetics with a fluid dynamic model it is necessary to determine the.
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Simulation of the ground-water flow system and proposed withdrawals in the northern part of Vekol Valley, Arizona
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The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up.
4 CHAPTER 1. BOLTZMANN TRANSPORT functions2 k(r) = exp(ik r)= p V, as well as the result X k2 ^ A(k) = V Z ^ d3k (2ˇ)3 A(k) () for smooth functions A(k).
Note the factor of V 1 in front of the integral in eqn. What this tells us is that for a bounded localized potential U(r), the contribution to theFile Size: 2MB.
Transition flow ion transport via integral Boltzmann equation. Title. Transition flow ion transport via integral Boltzmann equation. Author. Darcie, T.E. Institution. University of Toronto Institute for Aerospace Studies. Date. To reference this document use: Author: T.E. Darcie. The Boltzmann transport equation and the diffusion equation Sergio Fantini’s group, Department of Biomedical Engineering, Tufts University Modeling light propagation in scattering media with transport theory The Boltzmann transport equation (BTE) is a balance relationship that describes the flow of particles in scattering and absorbing Size: KB.
where I[f] is defined to be the collision integral and is a functional of the distribution function. We can now arrive at the final form of the Boltzmann transport equation ∂f(r,p,t) ∂t + p m ∇ rf(r,p,t)−∇ rV ext(r,t)∇ pf(r,p,t) = I[f].
(12) The Collision Integral Our main problem. • Transition Flow regime can use approximations of Boltzmann Equation to solve • Knudsen Number provides indication of range of Equation validations Kn (Knudsen Number) = λ /L λ flow molecular mean free path length. L distance between boundaries.
Laminar N-S Equation. Transition between. and. Full Boltzmann. f is a distribution function ofFile Size: 1MB. 12 January Volumenumber 8 PHYSICS LETTERS A APPLICATION OF THE BOLTZMANN TRANSPORT EQUATION TO CALCULATIONS OF FLUX AND RANGE DISTRIBUTIONS OF ENERGETIC IONS Gyula BARDOS Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Moscow, USSR Received 19 June ; revised manuscript received 5 November ; Cited by: 9.
Explanation of the various gain and loss terms in the Boltzmann transport equation, which is the starting point for modeling how light propagates in. For transition and turbulent flow, use Figure (the f in this figure only applies for fully turbulent flow corresponding to the flat portions of the curves in Figure ) with Figureand Figures a and b as appropriate.
Friction factor in long steel pipes handling wet (saturated with water vapor) gases such as hydrogen, carbon monoxide, carbon dioxide, nitrogen, oxygen, and. The structure and the contents of the present book have some com mon features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical by: Physics for Solid State Applications Lecture Introduction to Boltzmann Transport • Non-equilibrium Occupancy Functions • Boltzmann Transport Equation • Relaxation Time Approximation Overview • Example: Low-field Transport in a Resistor Outline Ap Scattering Rate Calculations Overview Step 1: Determine Scattering File Size: KB.
Ludwig Boltzmann, Transport Equation and the Second law 3 influential and vociferous of the German-speaking physics community - the so-called energeticists, led by Ernst Mach ( - ) and Wilhelm Ostwald ( - ) did not approve of this. For them, energy was the only fundamental physical entity.
They dismissed with contempt any. Moment Methods for Solving the Boltzmann Equation. Transition flow ion transport: Experimental critical comments are introduced concerning the treatment of path-integral methods in a well Author: Larry Viehland.
An Introduction to the Boltzmann Equation and Transport Processes in Gases the basic principles of this theory within an elementary framework and from a more rigorous approach based on the Boltzmann equation. The subjects are presented in a self-contained manner such that the readers can understand and learn some methods used in the kinetic.
Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework. Philipp Neumann, Till Rohrmann. Faculty of Informatics, TU München, Munich, Germany. Email: [email protected] Received ; revised J ; accepted J ABSTRACT. We present simulation results of flows in the finite File Size: 2MB.
B BOLTZMANN TRANSPORT EQUATION In analogy to the diffusion-induced changes, we can argue that particles at time t = 0 with momentum k - k 6t will have momentum k at time 6t and which leads to the equation = -k- ’ afk dk vxB h dk B Scattering-Induced Evolution of fk(T) We will assume that the scattering processes are local and instantaneous and change.
Mass Flow Rate. Mass per unit time [MT-1] Flux. Mass flow rate through unit area [ML-2 T-1] 3. The Transport Equation. transport equation. Advective flux.
Dispersive flux. Equation 26 advection J J dispersion t x C + ion 2 z 2 2 2 y 2 x. RADIATIVE TRANSPORT 1. The Boltzmann equation The Boltzmann equation accounts for changes in the phase space number density. Physically, such The change in the number of particles of interest in the element due to their motion or flow is the integral representing the sum over initial states or other groups.
Again for simplicity File Size: KB. The Boltzmann Transport Equation The Boltzmann equation describes the time evolution of the electron distri-bution function f(r,k,t). Its physical interpretation is that f(r,k,t)drdk is the number of electrons (wavepackets) at point r with wavenumber k in the phase space volume drdk.
If integrated in all space over k, we would get theFile Size: KB. Students learn to solve the Boltzmann equation in the classical limit under relaxation time approximation in this lecture.
Students also learn to derive the Fourier law, Newton shear law, and the electron transport process with the Ohm's Law. Transport properties - Boltzmann equation goal: calculation of conductivity Boltzmann transport theory: distribution function number of particles in infinitesimal phase space volume around evolution from Boltzmann equation collision integral for static potential.
Transport properties - Boltzmann equation ion lattice density fluctuation File Size: 1MB.M. Bahrami Fluid Mechanics (S 09) Integral Relations for CV 8 The angular momentum A control volume analysis can be applied to the angular momentum, by letting B equal to angular‐ momentum vector H.
If O is the point about which moments are desired, the angular moment about O isFile Size: KB.Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$.
Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$.Cited by: 2.
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https://citiesthemagazine.com/can-an-object-have-infinite-mass/
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math
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Table of Contents
Can an object have infinite mass?
Advertisement. So when we think of mass as energy, we can begin to understand why an object will increase its ‘mass’ as it speeds up. Since an object has infinite kinetic energy when it approaches the speed of light, it therefore has infinite mass as well.
Does Goku have infinite speed?
Does Goku have infinite speed? No he doesn’t have infinite speed first and foremost the link is easily debunkable which I could save for Some other time.
What if light speed was infinite?
If you think of a wave as an oscillation, then at infinite speed light would have no time to oscillate. So infinite light can’t be a wave. Einstein’s theory of relativity depends upon a finite speed of light. With an infinite light speed, all those fun things like time dilation are thrown out the window.
Why can’t we go the speed of light?
According to Einstein’s general theory of relativity, as an object moves faster, its mass increases, while its length contracts. At the speed of light, such an object has an infinite mass, while its length is 0 — an impossibility. Thus, no object can reach the speed of light, the theory goes.
Why is the speed of light not infinite?
That something, the universal conversion factor, is the speed of light. The reason that it is limited is simply the fact that a finite amount of space is equivalent to a finite amount of time. Mathematically, the wave equation that describes light as an electromagnetic wave would lose its time-dependence.
How fast is the speed of light in Mach?
1 light speed (ls) = 880979.6494 mach (M). Light Speed (ls) is a unit of Speed used in Metric system. Mach (M) is a unit of Speed used in Metric system.
Is Mach 100 faster than light?
1 Speed of Light in Vacuum: In SI units the speed of light measured in a vacuum is 299,792,458 meters per second….Please share if you found this tool useful:
|4 Light Speed to Mach Number = 3523964.3598||100 Light Speed to Mach Number = 88099108.9953|
Is Goku faster than the speed of light?
Goku’s speed is faster than time itself. They were going FTL in the saiyan saga. so imagine how fast logically Goku is going now…. Goku was faster than light when he fought Tien in the martial arts tournament as a kid lol, not in the Cell Saga.
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https://www.whatifsports.com/forums/Posts.aspx?ForumID=18&TopicID=468210&ThreadID=10212055
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math
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OK, we have 96 owners signed up, although based on past history, I understand we'll end up losing some of them. But here's some interesting stats anyway.
Cumulatively, these 96 owners have had...
30,149 teams (314 per person)
4.9 million games played (51,000 per person)
12,701 playoff appearance (132 per person)
.522 overall winning percentage
3,833 finals appearance (40 per person)
2,047 Championships (21 per person)
And finally, the average experience is 8 years (i.e, member since Feb-19-2005, on average)
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http://vixra.org/abs/1205.0003
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math
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Authors: Jay Yoon
I will present a proof of Euclid’s fifth postulate (I.Post.5) that proves, as an intermediate step, a proposition equivalent to it (I.32); namely, that in any triangle, the sum of the three interior angles of the triangle equals two right angles. The proof that I.32 implies I.Post.5 and vice versa is well-established and will be omitted for the sake of brevity. The proof technique is somewhat unorthodox in that it proves I.33, which states that straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel, before establishing I.32, contrary to the order in which the propositions are demonstrated in Euclid’s Elements. Two triangle congruence theorems, namely the side-angle-side (I.4) and side-side-side congruence theorems (I.8) are employed in order to prove I.33 without recourse to I.Post.5 or any of its equivalent formulations. In addition, a parallelogram is constructed by an unorthodox method; namely, by defining the diagonals upon which the parallelogram’s sides will be determined prior to the sides themselves. The proof assumes the five common notions stated in Book I of The Elements without explicitly making a reference to them when they are used. Furthermore, a figure is presented with color-coded angles and sides, with angles of the same color being equal in measure and sides of both the same color and the same number of tick marks being equal in length. The sides GH and EJ enclosed by brackets are indicated to be equal in length, the reason for the different notation being that the tick marks were used in reference to the halves of GH, namely OG and OH. The tick marks then refer to the parts of GH, and the bracket refers to the whole of GH; the latter is then equated to EJ by I.33, which is proven before its use.
Comments: 5 Pages.
[v1] 2012-05-02 23:20:08
Unique-IP document downloads: 514 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
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https://www.coursehero.com/tutors-problems/Chemistry/20199984-A-298-mL-solution-of-HBr-of-unknown-concentration-was-titrated-using/
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math
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A 29.8 mL solution of HBr of unknown concentration was titrated using 0.4685 mol L-1 NaOH using phenolphthalein
as a pH indicator. At the moment when the solution turned a consistent light pink colour, the burette volume read 18.84 mL. The initial burette reading before the experiment was 6.66 mL. What was the concentration of HBr in the original HBr solution ?
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https://xscode.com/mattearnshaw/lawvere
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math
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The collected works of F. W. Lawvere
Please inform us of any inaccuracies or missing works.
| Title |Source| Year | |:------|:-----|:----:| |The Language of Algebra: Supplement|TEMAC programmed learning materials|1961| |The Category of Probabilistic Mappings – With Applications to Stochastic Processes, Statistics, and Pattern Recognition|Unpublished; seminar handout notes|1962| |Functorial Semantics of Algebraic Theories|Original unpublished; Ph.D. thesis, Columbia University, 1963. See extended TAC reprint 2004|1963| |Functorial Semantics of Algebraic Theories (short notice)|Proceedings of the National Academy of Science 50, No. 5 (November 1963), 869-872|1963| |Functorial Automata Theory (abstract)|AMS Notices 603-151, vol 10 (1963), 477-478|1963| |The group ring of a small category (abstract)|Notices Amer. Math. Soc., 10, 280 (1963); Errata, Notices Amer. Math. Soc., 10, 516|1963| |An Elementary Theory of the Category of Sets (cf. 2005 long version with commentary)|Proceedings of the National Academy of Science 52, No. 6 (December 1964), 1506-1511|1964| |Algebraic Theories, Algebraic Categories, and Algebraic Functors|Theory of Models; North-Holland, Amsterdam (1965), 413-418|1965| |Functorial Semantics of Elementary Theories (abstract)|J. Symb. Logic, 31, 294-295, 1966|1966| |The Category of Categories as a Foundation for Mathematics|La Jolla Conference on Categorical Algebra, Springer-Verlag (1966), 1-20|1966| |Theories as Categories and the Completeness Theorem (abstract)|Journal of Symbolic Logic, 32:562|1967| |Some Algebraic Problems in the Context of Functorial Semantic of Algebraic Theories|Springer Lecture Notes in Mathematics No. 61 (Reports of the Midwest Category Seminar II), Springer-Verlag (1968), 41-61|1968| |Ordinal Sums and Equational Doctrines|Springer Lecture Notes in Mathematics No. 80, Springer-Verlag, 141-155.|1969| |Diagonal Arguments and Cartesian Closed Categories|Springer Lecture Notes in Mathematics No. 92, Springer-Verlag (1969), 134-145.|1969| |Adjointness in Foundations|Dialectica 23 (1969), 281-296|1969| |Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor|Proceedings of the American Mathematical Society Symposium on Pure Mathematics XVII (1970), 1-14|1970| |Quantifiers and Sheaves|Proceedings of the International Congress on Mathematics, Nice 1970, Gauthier-Villars (1971) 329-334|1970| |Introduction to "Toposes, Algebraic Geometry and Logic"|Springer Lecture Notes in Mathematics No. 274, New York: Springer, pp. 1–12|1971| |Theory of Categories over a Base Topos|Lectures given at Università di Perugia|1972| |Metric Spaces, Generalized Logic, and Closed Categories|Originally published in Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135-166; Republished in Reprints in TAC, No.1 2002 pp. 1-37|1973| |Logic of Topoi: inside and outside|Lecture Notes, Université de Montréal|1974| |Introduction to "Model Theory & Topoi"|Springer Lecture Notes in Mathematics No. 445, Springer-Verlag, pp. 3-14|1975| |Variable Sets Etendu and Variable Structure in Topoi|Notes by Steven Landsburg of Lectures and Conversations, Spring 1975, University of Chicago|1975| |Continuously Variable Sets – Algebraic Geometry=Geometric Logic|Studies in Logic and the Foundations of Mathematics, Volume 80, pp. 135-156|1975| |Variable Quantities and Variable Structures in Topoi|Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), pp. 101–131|1976| |Categorical Dynamics|Proceedings of Aarhus May 1978 Open House on Topos Theoretic Methods in Geometry|1978| |Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body|Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume: 21, Issue: 4, pp. 377-392|1980| |On C-∞ functions|Preprint, State Univ. of New York, Buffalo|1981| |Thermodynamics of deformations of continuous bodies, non homogeneous, with memory, far from equilibrium|Handwritten notes from a seminar held in Trieste|1982| |Introduction to "Categories in Continuum Physics"|Springer Lecture Notes in Mathematics No. 1174, Springer-Verlag (1986)|1982| |Measures on toposes|Proceedings of Aarhus Workshop on Category Theoretic Methods in Geometry|1983| |Functorial Remarks on the General Concept of Chaos|IMA Research Report #87, University of Minnesota (1986)|1984| |State Categories Closed Categories and the Existence of Semi-continuous Entropy Functions|IMA Research Report #86, University of Minnesota|1984| |State Categories and Response Functors|Preprint|1986| |Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs|Revista Colombiana de Matemáticas XX (1986), pp. 179-185|1986| |Taking Categories Seriously|Revista Colombiana de Matemáticas XX (1986), pp. 147-178|1986| |Some "New" Mathematics Arising From the Study of Grassmann 1844|Unpublished manuscript|1987| |Concepts and Problems in Mathematical Toposes (abstract)|CMS Winter Meeting, December 1988, Toronto, Program p. 31, Special Session on Category Theory|1988| |Fractional Exponents in Cartesian Closed Categories|Unpublished manuscript|1988| |Möbius algebra of a category|Handwritten Notes by S. Schanuel at the Sydney Combinatorics Seminar organized by Don Taylor|1988| |Toposes generated by codiscrete objects in combinatorial topology and functional analysis|TAC Reprint (2021) of notes for Colloquium lectures given at North Ryde, New South Wales, Australia 1988-89|1989| |On the Complete Lattice of Essential Localizations (with G.M. Kelly)|Bull. Société Mathematique de Belgique, XLI, 289-319|1988| |Intrinsic boundary in certain mathematical toposes exemplify logical operators not passively preserved by substitution|Preprint, Univ. of Buffalo|1989| |Display of Graphics and their Applications Exemplifed by 2 Categories and the Hegelian Taco|Proceedings of the First International Conference on Algebraic Methodology and Software Technology, The University of Iowa, 51-75|1989| |Qualitative Distinctions Between Some Toposes of Generalized Graphs|Proceedings of AMS Boulder 1987 Symposium on Category Theory and Computer Science, Contemporary Mathematics, 261-299|1989| |Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes|Category Theory, Proceedings Como 1990, A. Carboni, M. C. Pedicchio, G. Rosolini (Eds). Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) pp. 279-281|1991| |More on Graphic Toposes|Proceedings of the 1989 Bangor Category Theory Meeting, Cahiers de Topologie et Géométrie Différentielle Catégorique XXXII - 1 (1991), pp. 5-10|1991| |Some Thoughts on the Future of Category Theory|Category Theory, Proceedings Como 1990. A. Carboni, M. C. Pedicchio, G. Rosolini (Eds). Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) pp. 1-13|1991| |Categories of Space and Quantity|The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, International Symposium on Structures in Mathematical Theories (1990), San Sebastian, Spain; DeGruyter, Berlin (1992), pp. 14-30.|1992| |Cohesive Toposes and Cantor's Lauter Einsen|Philosophia Mathematica, The Canadian Society for History and Philosophy of Mathematics, Series III, Vol. 2 (1994), pp. 5-15.|1994| |Tools for the Advancement of Objective Logic Closed Categories and Toposes|The Logical Foundations of Cognition, J. Macnamara, G. E. Reyes (Eds). Oxford University Press (1994), pp. 43-56|1994| |Adjoints in and among Bicategories|Logic & Algebra, Proceedings of the 1994 Siena Conference in Memory of Roberto Magari. Lecture Notes in Pure and Applied Algebra 180: pp. 181-189, Ed. Ursini/Aglianò, Marcel Dekker, Inc. Basel, New York|1996| |Grassmann's Dialectics and Category Theory|Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar pp. 255-264, Boston Studies in the Philosophy of Science Series (BSPS, volume 187)|1996| |Unity and Identity of Opposites in Calculus and Physics|Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures, 4: pp. 167-174 Kluwer Academic Publishers|1996| |Algebra Step by Step|Buffalo Workshop Press, ISBN: 0963180525|1997| |Linearization Revisited (abstract)|IIIrd Joint Meeting AMS-SMM, Special Session on Rings and Category Theory, Oaxaca Mexico, December 1997, Program p. 59|1997| |Toposes of Laws of Motion|Transcript from Video, Montreal September 27, 1997|1997| |Volterra's Functionals and Covariant Cohesion of Space|Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, 64, R. Betti, F. W. Lawvere (Eds.), (2000), pp. 201-214|1997| |Everyday physics of extended bodies or why functionals need analyzing (abstract, cf. paper 2017)|Public Lecture, CMS Summer 1998 Meeting, University of New Brunswick, Saint John, June 13-15, (received May 1998)|1998| |Outline of Synthetic Differential Geometry|Buffalo Geometry Seminar notes, February 1998|1998| |Are Homotopy Types the Same As Infinitesimal Skeleta? (abstract)|CMS Summer 1998 Meeting|1998| |Categorical Analyses of the Whole/Part Relation (abstract)|Mitteleuropaeisches Kulturinstitut, Bolzano, Italy|1998| |Categorie e Spazio: Un Profilo|Lettera matematica PRISTEM 31, Springer, Italy, (1999), 35-50. [Reprinted in La Mathematica a cura di Bartocci, Claudio, Giulio Einaudi editore (2010) vol. 4, 107-135.|1999| |Kinship and Mathematical Categories|Language, Logic, and Conceptual Representation, P. Bloom, R. Jackendoff, and K. Wynn (Eds), MIT Press, (1999), pp. 411-425|1999| |Comments on the Development of Topos Theory|Development of Mathematics 1950-2000, J.-P. Pier (Ed) Birkhäuser Verlag, Basel, (2000), pp. 715-734|2000| |The Role of Cartesian Closed Categories in Foundations (Interview with Felice Cardone)||2000| |Explicit foundational concepts in the teaching of mathematics|Matematica e filosofia: il problema dei fondamenti oggi PRISTEM/Storia 14-15 - Translation: Filsofia, scienza e bioetica nel dibattito contemporaneo a cura di Minazzi, Fabio, Instituto Poligrafico e Zecca dello Stato, Roma|2001| |Categorical Algebra for Continuum Micro Physics|Journal of Pure and Applied algebra 175, (2002), pp. 267-287|2001| |On the Duality Between Varietes and Algebraic Theories (with J. Adámek & J. Rosický)|Algebra Universalis, (2003), pp. 35-49|2003| |How Algebraic is Algebra? (with J. Adámek & J. Rosický)|Theory and Applications of Categories , Vol. 11, 2003, No. 11, pp. 252-282|2001| |Linearization of Graphic Toposes via Coxeter Groups|Journal of Pure and Applied Algebra, vol. 168, (2002), pp. 425-436|2002| |Foundations and Applications – Axiomatization and Education|The Bulletin of Symbolic Logic, vol. 9, No. 2, (2003), pp. 213-224|2003| |Continuous Categories Revisited (with J. Adámek & J. Rosický)|Theory and Applications of Categories , Vol. 11, 2003, No. 11, pp 252-282|2003| |Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories|Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121. Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, pp. 41-61|2004| |Functorial Concepts of Complexity for Finite Automata|Theory and Applications of Categories, Vol. 13, 2004, No. 10, pp. 164-168|2004| |Left and Right Adjoint Operations on Spaces and Data Types|For Dana Scott's Seventieth Birthday, Copenhagen 2002, Theoretical Computer Science, Elsevier, vol 316/1-3, (2004) pp. 105-111|2004| |An Elementary Theory of the Category of Sets (long version) with commentary|Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1-35. Expanded version of Proceedings of the National Academy of Science of the U.S.A 52, 1506-1511, with commentary by Colin McLarty and the author.|2005| |Grassmann Book Reviews|Historia Mathematica, vol. 32, (2005), pp. 101-106|2005| |John Isbell's Adequate Subcategories|Topological Commentary, Vol. 11 #1|2006| |Axiomatic Cohesion|Theory and Applications of Categories, online publication, Special volume from the CT2006 Conference at Whitepoint Nova Scotia, vol 19, (2007), 41-49|2007| |Cohesive toposes: combinatorial and infinitesimal cases (video) (lecture notes)|Lectures in Como (Italy), January 10, 2008|2008| |Core Varieties Extensivity and Rig Geometry|Theory and Applications of Categories, Vol. 20, 2008, No. 14, pp 497-503.|2008| |Interview with Maria Manuel Clementino and Jorge Picado|Bulletin of the International Center for Mathematics (part 1, December 2007, part 2, June 2008)|2008| |Foreword To "Algebraic Theories"|Cambridge Tracts in Mathematics 184, J. Adámek, J. Rosický & E. M. Vitale (2012)|2009| |Open Problems in Topos Theory|88th Peripatetic Seminar on Sheaves and Logic, For Martin Hyland and Peter Johnstone in honor of their sixtieth birthdays (updated July 2016)|2009| |The Hopf Algebra of Möbius Intervals (with M. Menni)|Theory and Applications of Categories, online publication, vol. 24, (2010), 221-265|2010| |Categorical Dynamics (abstract)|Talk at International Category Theory Conference, Vancouver, July 2011|2011| |Euler's Continuum Functorially Vindicated|Logic, Mathematics, Philosophy: Vintage Enthusiasms, Essays in Honour of John L. Bell, D. DeVidi et al. (Eds), Western Ontario Series in Philosophy of Science 75, (2011)|2011| |What Is A Space? (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |The Dialectic of Continuous and Discrete in the History of the Struggle for a Usable Guide to Mathematical Thought (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |Categorical Dynamics Revisited: Category Theory and the Representation of Physical Quantities (video)|Sets Within Geometry symposium - Nancy, France 26-29 July 2011|2011| |What are Foundations of Geometry and Algebra? (abstract) (video) (transcript)|Fifty Years of Functorial Semantics Conference|2013| |Internal Choice Holds in the Discrete Part of any Cohesive Topos Satisfying Stable Connected Codiscreteness (with M. Menni)|Theory and Applications of Categories, online publication, vol. 30, (2015), No. 26, pp 909-932|2015| |Alexander Grothendieck and the Concept of Space|Invited address at CT 2015 Aveiro, Portugal|2015| |Birkhoff's Theorem from a Geometric Perspective: A Simple Example|Categories and General Algebraic Structures with Apllications Vol. 4, no. 1, (2016), pp. 1-7|2016| |Everyday physics of extended bodies or why functionals need analyzing|Categories and General Algebraic Structures with Applications, Volume 6 2017, pp. 9-19|2017|
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358560.75/warc/CC-MAIN-20211128134516-20211128164516-00176.warc.gz
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CC-MAIN-2021-49
| 14,632 | 3 |
https://nrich.maths.org/7069/index
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math
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A man went to Monte Carlo to try and make his fortune. Is his strategy a winning one?
Two bags contain different numbers of red and blue marbles. A marble is removed from one of the bags. The marble is blue. What is the probability that it was removed from bag A?
You and I play a game involving successive throws of a fair coin.
Suppose I pick HH and you pick TH. The coin is thrown repeatedly
until we see either two heads in a row (I win) or a tail followed
by a head (you win). What is the probability that you win?
Alice and Bob are happily married, and decide to have children. Alice is proud of her long blond hair and brown eyes. Bob is also happy with his brown eyes, which he inherited from his auburn father, although his mother was green - eyed.
Alice knows that there is a small chance she might have a daughter looking like her (Alice's) mother, and that her daughter is much more likely to look like her. Bob, on the other hand, believes he won't have a son who reminds him (completely) of his father.
1. Can you determine the colour of Bob's hair?
2. Can you describe the eye colour of both Alice's parents?
3. How likely is it that Alice and Bob will have a blond son with green eyes?
4. How likely is it that they will have an auburn daughter?
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s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607649.26/warc/CC-MAIN-20170523183134-20170523203134-00179.warc.gz
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CC-MAIN-2017-22
| 1,261 | 12 |
https://arteslonga.com/en/collection/956-hand-painted-cushion.html
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math
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Hand Painted Cushion
En stock: 1
France and International Delivery
Product Informations or Quote Request: Tel: +33(0) 160 440 129 or [email protected]
Decorative cushion entirely made and painted by hand, on suedine fabrics.
A handmade cushion done by the french artist painter, Fidélie Cardi, exclusively for Arteslonga.
A unique decorative object.
Dimensions and Specifications - Shipment information
- Width: 40 cm
- Height: 15 cm
- Depth: 40 cm
- Weight (item packed): 2 kg
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499919.70/warc/CC-MAIN-20230201081311-20230201111311-00032.warc.gz
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CC-MAIN-2023-06
| 482 | 12 |
https://www.numbersaplenty.com/21233664
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math
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The square root of 21233664 is 4608.
It is a Jordan-Polya number, since it can be written as (4!)4 ⋅ (2!)6.
It is an ABA number since it can be written as A⋅BA, here for A=4, B=48.
It is a Duffinian number.
It is an unprimeable number.
Almost surely, 221233664 is an apocalyptic number.
21233664 is a gapful number since it is divisible by the number (24) formed by its first and last digit.
21233664 is the 4608-th square number.
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 21233664
It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.
21233664 is an frugal number, since it uses more digits than its factorization.
21233664 is an odious number, because the sum of its binary digits is odd.
The cubic root of 21233664 is about 276.9119174990.
Multiplying 21233664 by its product of digits (5184), we get a 8-th power (110075314176 = 248).
21233664 divided by its product of digits (5184) gives a 12-th power (4096 = 212).
Adding to 21233664 its reverse (46633212), we get a palindrome (67866876).
The spelling of 21233664 in words is "twenty-one million, two hundred thirty-three thousand, six hundred sixty-four".
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039610090.97/warc/CC-MAIN-20210422130245-20210422160245-00405.warc.gz
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CC-MAIN-2021-17
| 1,225 | 18 |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=GCGHC8_2012_v19n3_367
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math
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Predicting Korea Pro-Baseball Rankings by Principal Component Regression Analysis Bae, Jae-Young; Lee, Jin-Mok; Lee, Jea-Young;
In baseball rankings, prediction has been a subject of interest for baseball fans. To predict these rankings, (based on 2011 data from Korea Professional Baseball records) the arithmetic mean method, the weighted average method, principal component analysis, and principal component regression analysis is presented. By standardizing the arithmetic average, the correlation coefficient using the weighted average method, using principal components analysis to predict rankings, the final model was selected as a principal component regression model. By practicing regression analysis with a reduced variable by principal component analysis, we propose a rank predictability model of a pitcher part, a batter part and a pitcher batter part. We can estimate a 2011 rank of pro-baseball by a predicted regression model. By principal component regression analysis, the pitcher part, the other part, the pitcher and the batter part of the ranking prediction model is proposed. The regression model predicts the rankings for 2012.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501174124.9/warc/CC-MAIN-20170219104614-00169-ip-10-171-10-108.ec2.internal.warc.gz
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CC-MAIN-2017-09
| 1,152 | 2 |
http://uk-british-jiu-jitsu-association.mynewsdesk.com/pressreleases/tag/open
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math
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Press releases • Feb 24, 2016 12:00 GMT
The UKBJJA (UK Brazilian Jiu Jitsu Association) today made clear its intentions to defend the integrity of Brazilian Jiu Jitsu as a distinct martial art and to challenge the moves by the BJA (British Judo Association) to set up a rival body to govern the sport through a subsidiary organisation, the BJJUKA (Brazilian Jiu Jitsu UK Association).
Press releases • Dec 03, 2015 18:22 GMT
A teenager from East Sheen, Billy Hayes, has been named the top junior BJJ athlete in the UK by the UK Brazilian Jiu Jitsu Association (UKBJJA).
Press releases • Nov 17, 2015 11:11 GMT
New English BJJ (Brazilian Jiu Jitsu) Champions were crowned last weekend at the English Open BJJ Championships held at Dartford Judo Centre on the 14th and 15th of November.
SPORTS STARS OF THE FUTURE CLAIM GLORY AT THE ENGLISH OPEN BRAZILIAN JIU JITSU CHAMPIONSHIPS IN DARTFORD
Press releases • Nov 05, 2015 11:46 GMT
Dartford Judo Centre in Kent played host to the Junior and Juvenile English Open Brazilian Jiu Jitsu Championships at the weekend with more than 250 boys and girls competing for national titles.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371861991.79/warc/CC-MAIN-20200409154025-20200409184525-00265.warc.gz
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CC-MAIN-2020-16
| 1,140 | 9 |
http://hobbydocbox.com/Radio/76169033-Operational-amplifiers.html
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math
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1 Operational Amplifiers Table of contents 1. Design 1.1. The Differential Amplifier 1.2. Level Shifter 1.3. Power Amplifier 2. Characteristics 3. The Opamp without NFB 4. Linear Amplifiers 4.1. The Non-Inverting Amplifier 4.2. The Voltage Follower 4.3. The Inverting Amplifier 5. Frequency Characteristics 5.1. Band width 5.2. Slew Rate 6. Applications 6.1. Non-Inverting Amplifier 6.2. Inverting Amplifier 6.3. With push-pull output 6.4. Summing Amplifier 6.5. Logarithmizing Amplifier 6.6. Signal Rectification 6.7. Voltage Regulator 6.8. Comparator 6.9. Schmitt Trigger Astable Multivibrator Phase Shifter
2 Operational Amplifiers The theory of electrical signal processing requires amplifiers to perform, with electrical signals, mathematical operations such as addition, subtraction, multiplication, division, differentiation, integration, etc. These amplifiers must fulfil the following requirements: Differential inputs D.C. amplification Very high voltage gain Very high input resistance Very low output resistance They are then called "operational amplifiers" (opamps) because they are able to perform mathematical operations. With opamps, even analog computers are constructed which surpass any digital computer when high speed of signal processing is required. The first opamps were built using discreet transistors, but it a was difficult and expensive process because of temperature drift problems. The big breakthrough came with integrated circuits. Having all circuit elements on one monolithic silicon chip solved most of the temperature drift problems and allowed for cheap mass production. Today we have to consider the opamp as a circuit element. We will study its characteristics but not dwell on how it works internally. 1. Design The basic form of an opamp is a high gain dc-amplifier with a differential input port and a single output port. A differential input has two terminals, which are both independent of ground or common. The signal between these two terminals is the input signal, which will be amplified. The terminals are called non-inverting input and inverting input. The two inputs can be used in three different ways: 1. Non-Inverting Amplifier: The input signal is applied between the non-inverting input and ground. The inverting input is connected to ground. The output signal will be in phase with the input signal 2. Inverting Amplifier: The input signal is applied between the inverting input and ground. The non-inverting input is connected to ground. The output signal will be 180 out of phase with the input signal. 3. Differential Amplifier: Two input signals are each connected to the non-inverting and the
3 inverting input, using both common as second terminal. The output signal will be the amplified difference between the two. U o = (U i+ - U i- ) g Fig The three basic ways of applying input signals to the opamp. When there is no voltage difference between the input terminals, the output voltage should be 0. The internal circuit of opamps consists basically of three main parts: 1.1. The Differential Amplifier: A differential amplifier stage consists of two transistors in common emitter configuration which are supplied with a common emitter current.
4 Fig The basic design of a differential amplifier stage. As long as there is no voltage difference between the two bases of the transistors, the two transistors will draw the same collector currents and a certain voltage will appear at the output. If the base of T 1 becomes more positive than of T 2, T 1 will draw more current, the voltage across R C1 will increase. As the total current is constant, the current through T 2 will decrease by the same amount. The voltage across R C2 will decrease and the output voltage becomes more positive. So the base of T 1 is the non-inverting input. If the base of T 2 becomes more positive than that of T 1, T 2 will draw more current. The voltage across R C2 increases and the output voltage becomes more negative. Thus the base of T 2 is the inverting input. If the voltages at the bases of T 1 and T 2 are varied by the same amount, the current distribution between the two transistors does not change and no voltage results at the output. This case is called common mode and should not produce an output signal. The general requirements for the differential amplifier: high differential mode gain low common mode gain high input impedance
5 low base currents temperature stability Some opamps use FET as input transistors to achieve extremely high input resistances Level Shifter The level shifter fulfils two main tasks: it provides most of the voltage amplification of the opamp; it provides dc-matching between differential amplifier and the output to obtain zero output voltage for zero input (offset voltage). The level shifter consists mainly of a number of dc-coupled transistor stages which are arranged and biased in such a way that zero offset voltage with a high temperature stability is achieved. Requirements to the level shifter: low distortion wide frequency range 1.3. Power Amplifier The final stage of an opamp is in most cases a complementary push-pull amplifier. It has to provide the required output current at a low output resistance. Requirements: symmetrical output swing from +U b to -U b low output impedance short-circuit protection low distortions
6 Fig An example of the circuit of a simple integrated opamp. The circuit symbol for an opamp is a triangle pointing towards the output. The input terminals are drawn to the vertical left side. Any further auxiliary terminals such as supply voltages or offset adjustment are drawn at the top and bottom slopes of the triangle. Fig The circuit symbol for a general opamp.
7 2. Characteristics Voltage gain An ideal opamp should have an open loop voltage gain g (without NFB) which is infinite. Practical opamps may have values from 60dB to 120d, which equals 10 3 to In general, all practical opamps have sufficient gain for most requirements. Input resistance An ideal opamp should have an input resistance R i which is infinite. Practical opamps may have values from 10k to 1M Input up to 1G can be reached for opamps with MOSFET. The input resistance of opamps will further be increased by NFB, so that the achieved values will satisfy most practical requirements. Output Resistance An ideal opamp should have an output resistance R o of zero. Practical opamps may have values from 50 to 500 These values are not made lower in order to achieve short circuit protection of the output. The output resistance will be reduced by NFB, so that the achieved values will satisfy most practical requirements. Supply Voltage In general, opamps require two symmetrical (equal but of opposite polarity) supply voltages +U b and -U b in respect to ground. These voltages must be large enough in order to properly bias all internal transistors. On the other hand, they may not exceed a specific maximum value. Practical supply voltages range from ±3V to ±30V. A common value is ±15V. Some opamps are also designed to be operated on one supply voltage only. This requires a special design for the input and output stage. Either supply terminal may then be connected to ground. Output voltage Swing The maximum output signal U sat (saturation voltage of the output stage) will depend on the supply voltage. It is obvious that the output voltages cannot be higher than the supply voltages. As the output of the amplifier will always require a certain voltage drop, the maximum output voltage swing will be 1V to 3V lower than the supply voltage, depending on the type of opamp.
8 Fig The relationship between supply voltage and maximum output voltage swing. The maximum output voltage will depend on the supply voltage. The higher the supply voltage, the more output amplitude can be achieved. As for opamps operated on one supply voltage only, the amplitude of the output signal can only be less than half of the supply voltage. Input Offset Voltage The output voltage of an opamp should be zero, if the input voltage is zero (input terminals shorted). In practice, there will always be some asymmetry in the differential amplifier. This voltage is then amplified through all stages and, depending on the gain, there might be a high voltage at the output of the opamp.
9 Fig The output voltage which is measured at the output of an opamp with shorted input terminals is the internal offset voltage U iofs multiplied by the gain g. This voltage could be compensated by feeding a dc-voltage to the input which opposes the internal offset. This voltage is equal to the input offset voltage Uofs. This process is called offset compensation or offset null-balance. It is required for most cases of dc-amplifiers (e.g. measuring amplifiers). Fig If the input offset voltage U iofs is fed into the inverting input terminal, the output voltage can be set to zero.
10 In order to keep the input terminals free for the signal, some opamps provide separate terminals for offset adjustment. These offset adjustment terminals must be used according to the specifications of the data sheets. Fig Example of the offset compensation using the separate terminals of an opamp (741). Input Bias Current The input terminals of opamps can be considered as base terminals of the transistors of a differential amplifier stage. In order to operate the transistors in the active region, they require a certain bias current I ib. For opamps with bipolar input this will be in the range of some na or µa. Although these currents are very small, they may produce a voltage drop across any resistance in series with the input. This is then a voltage difference at the input which again produces an offset at the output. If the two resistors are equal, the voltage drops will be equal and there will be no voltage difference at the input.
11 Fig The input bias current I ib of the input transistors will produce a voltage drop across any resistor connected in series to the input. Making both resistors equal will cancel out the two voltages U R1 and U R2. Care is therefore often taken that both inputs of the opamp have an equivalent resistance to ground to avoid offset due to bias current. Input offset Current The bias current of the two transistors may not be equal, so even if both inputs have equal resistors in series, there might be an offset voltage. In practice, this effect cannot be distinguished from the effect of the input offset voltage, so they will be compensated together.
12 3. The Opamp without NFB Let us look at how the opamp can amplify signal. We will assume that the opamp has an open loop gain of g = 6000 = 76dB. This means an input voltage of 1mV will produce an output voltage of 6V. Fig Opamp as amplifier with its transfer characteristic. Input voltages of more than 2mV will drive the output to saturation. In practice, it will be found that an amplifier with such a large dc-gain will not work properly because the offset voltage drift will not allow a stable working point. An opamp without NFB can not be used as linear amplifier. The opamp in this "pure" form is only used as COMPARATOR. The comparator compares two input signals and provides a digital (high/low) output signal, depending on which of the two is larger. U o = +U sat (approx. +U b ) if U i+ > U i-
13 U o = -U sat (approx. -U b ) if U i+ < U i- Fig The opamp as comparator. The output signal is either +U sat or -U sat, depending on which of the two input voltages is larger. Normally one of the two input voltages is used as a reference or threshold for the other. If the reference voltage is connected to the inverting terminal, we will get a non-inverting comparator. If the reference voltage is connected to the non-inverting input, we will have an inverting comparator.
14 4. Linear Amplifiers Opamps can only be used as linear amplifiers with external negative feedback. The NFB is achieved by a voltage divider circuit which feeds back a fraction of the output signal to the inverting input. As opamps have a very high open loop gain, very strong NFB can be provided. This makes strong use of all of the advantages of NFB such as: - reduction of distortion, - favourable input and output resistances, - stable working parameters. Depending on how (in which form) the NFB is achieved and how the signal is fed to the input, different types of amplifiers with different characteristics are created The Non-Inverting Amplifier The non-inverting amplifier feeds the input signal to the non-inverting input. The NFBsignal is derived from a voltage divider from the output signal and is fed to the inverting input. Fig The basic configuration of the non-inverting amplifier. The properties of this amplifier are controlled entirely by the NFB voltage divider (see chapter on NFB): Close Loop Voltage Gain
15 This formula is correct if g' << g (g' is much smaller than g) Input Resistance The input resistance is increased by the degree of reduction of gain. This factor will in practice be at least 10 or 100, so the input resistance of this amplifier will be very high (>1M ) in all cases. Output Resistance The output resistance will be reduced by the same factor by which the input resistance is increased. In practice, this leads to very low values (<1 ). Summary of properties of the non-inverting opamp: the signals at input and output are in phase, the closed loop gain g' depends on the external elements R 1 and R 2 only, the input resistance is very high, the output resistance is very low. The non-inverting amplifier is used for audio amplification and as a measuring amplifier. The NFB tends to eliminate all kinds of negative influences which appear between the input and output of the amplifier. It can be used to reduce the influence of any other circuit elements which are used in conjunction with opamps. Any resistance which is in series with the output of the amplifier will increase the output resistance. The effect of this resistance can be reduced if the resistor is taken into the NFB-loop.
16 Fig A resistance in series with the output of an amplifier. a.) If the resistance in series with the output is outside of the NFB-loop, the resistance adds fully to the output resistance. b.) If the resistance in series with the output is within the NFB-loop, the resistance is eliminated by the NFB. If more output current is required, a push-pull stage can be connected to the output of the opamp. A push-pull stage can produce distortions, mainly cross-over distortions. Taking the push-pull stage into the NFB-loop will strongly reduce the distortions. Fig A push-pull state may be used to boost the output current of the opamp. a.) If the push-pull stage is outside of the NFB-loop, the distortions of this stage appear at the output. b.) If the push-pull stage is within the NFB-loop, the distortions of this stage are reduced by the NFB The Voltage Follower The smallest gain to be achieved with a non-inverting amplifier is one. This is achieved if the entire output signal is fed back to the input. Considering the formulas above, this means that R 1 = 0 and R 2 = (infinit).
17 Fig When all the output voltage is fed back to the input, the non-inverting amplifier becomes a voltage follower with unity gain. The gain of this amplifier is one and so the output voltage is identical to the input voltage. Because of this, the circuit is called UNITY GAIN AMPLIFIER or VOLTAGE FOLLOWER. Important characteristics of this amplifier: Gain: g' = 1 Input Resistance: R i ' = R i * g Output Resistance: R o ' = R o /g Summary of important properties: the signals at input and output are in phase, the closed loop gain g' is one the input resistance is extremely high, the output resistance is extremely low. Voltage followers are used as impedance converters in audio amplifiers and measuring amplifiers The Inverting Amplifier Inverting amplifiers feed the input signal and the NFB-signal into the inverting input. The non-inverting input is connected to ground. The output signal is shifted 180 in phase to the input signal.
18 Fig The basic configuration of the inverting amplifier. The function of the inverting amplifier can be explained by taking two points into consideration: 1. The input voltage of the opamp U i will be negligible compared to the input voltage of the amplifier U i ', or even compared to the output voltage U o. The inverting input of the opamp therefore has approximately the same voltage as the non-inverting terminal, which is connected to ground. This point of the circuit is therefore called VIRTUAL GROUND. From the point of view of the signal, this point has the same properties as the ground point of the circuit. 2. The input current to the opamp I i- is approximately zero. The sum of the currents I R1 and I R2 must therefore sum up to 0. The inverting input is therefore also called the SUMMING POINT. The main characteristics can be derived from these considerations: Closed loop gain: The resistor R1 and R2 are virtually connected to ground at the inverting input. The currents through the resistors R1 and R2 are equal. This requires that the input and output voltage have the same ratio as the resistors R1 and R2. This formula is correct if g' Input Resistance
19 The input resistance is only the resistor R2, because it is connected between input and virtual ground. Output Resistance The output resistance will be reduced by the same factor as the gain. In practice, this leads to very low values (<1 ). Summary of properties of the inverting opamp: the signals at input and output are 180 out of phase, the closed loop gain g' is set by the ratio of R 1 to R 2 the input resistance is set by R 2 the output resistance is very low. the inverting input of the opamp can be considered as virtual ground. If bias current compensation is required, a compensation resistor Rcomp can be used to offset current compensation. It should be selected so that the resistance in series with both inputs is approximately equal. Therefore: R comp = R 1 //R 2 (R 1 parallel with R 2 ) Fig The inverting amplifier with compensation resistor for the bias current.
20 5. Frequency Characteristics Opamps have a frequency range which starts at 0Hz (d.c.). At the upper end, the frequency range is limited by the BAND WIDTH and by the SLEW RATE. Both have the effect of limiting the upper operational frequencies, but have different physical causes and must be considered separately Band width Opamps without NFB have only a relatively small frequency range. Some types only have an upper frequency limit (-3dB) of a few Hz or a few hundred Hz. The gain decreases with increasing frequency due to the low-pass behaviour of the internal transistor amplifier stages. Furthermore, the opamp will have several internal transistor stages in series, each forming a low-pass with its own critical frequency. Fig The different amplifier stages of an opamp eacg form a low-pass, which is connected in series. The gain decreases after the first critical frequency with a slope of 20 db/decade, after the second critical frequency with a slope of 40 db/decade, etc. Each low pass will also produce a certain phase shift of up to 90 per low-pass. With increasing frequency, a growing phase shift will occur between input and output. The so- called "Bode-plot" shows the relations:
21 Fig Example of the Bode plot of an opamp (TAA 861). The critical frequency of the open loop gain (g=85db) is about 10 Hz. Over 1kHz the gain drops with 40dB/decade due to a second internal low pass. At 5kHz the phase shift between differential input and output is more than 180. The limited band width makes this device unsuitable for audio applications, but introducting NFB, the band width can be increased. Assume for the TAA 861 the gain is set by NFB to 40dB (100). Thus below 1kHz, the open loop gain will be higher than the closed loop gain, and the gain will be defined entirely by the NFB. Above 1kHz the open loop gain will be less than the desired closed loop gain, and the gain will be equal to the closed loop gain.
22 Fig The frequency response of the same opamp with the gain set to 40dB by NFB. The upper critical frequency has been improved to 1kHz. The band width of this amplifier could be increased to approximately 30kHz. Then the open loop gain becomes 1. But at higher frequencies only little gain is achieved. (In fact, the TAA 861 is not a suitable opamp for audio circuits!) The lower the chosen gain, the higher the band width. As the opamp without NFB is not used as a linear amplifier, the band width of the open loop gain plays no practical role and is thus not mentioned in the data sheets. Instead, the UNITY-GAIN BAND WIDTH is given. This is the band width of the opamp with a closed loop gain of 1. Some examples of unity-gain band width of practical opamps: - type TAA 861: 30kHz - type 741: 300kHz - type 081: 3MHz A problem arises from the phase shift inside the opamp which increases with frequency. The NFB-signal is supplied with a nominal phase shift of 180 to the input signal (anti-phase). Additional internal phase shifts will turn the negative feed back into a positive feed back. If the gain is then still larger than 1 (0 db), this will cause oscillation of the amplifier (instability). In the case of the TAA 861: the lowest gain for stable conditions is 25 db. In practice, a phase security margin of 60 is respected. This determines the lowest possible gain to 48 db and the upper critical frequency to 900 Hz. For an uncompensated opamp the danger of instability increases with increasing NFB.
23 To allow higher band widths at smaller gains - particularly for voltage followers (g' = 0 db) - opamps are provided with terminals for EXTERNAL FREQUENCY COMPENSATION by means of R and C components. The required circuit elements and their wiring depends on the type of opamp and has to be determined from the data sheets. In general, frequency compensation is achieved by a low pass function, reducing the first open loop corner frequency and providing a gain decrease of 20 db/decade down to unity gain. Sufficient phase margin is achieved, though band width and slew rate are reduced compared to uncompensated operation. Several opamps provide internal frequency compensation (e.g. 741-types) and secure stable conditions for all gains. Fig Frequency compensation of TAA 861 with C k according to the data sheets. (This Op Amp is an open-collector device and requires the load-resistor to be connected to +U b ) Slew Rate If a step function (pulse) is applied to the input of an opamp, the output signal will not respond immediately. This is due to internal capacitances which cannot be charged instantaneously. The output will respond with a slope function, representing the highest speed in voltage change. This is called the slew rate (or slewing rate). It is given in volts per microseconds (V/µs).
24 Fig When a step function is applied to the input of an opamp, the output will respond with its maximum possible voltage rise, called the slew rate. (The gain of this opamp is set to 2.) In addition, when a sine wave is applied to the opamp, the output is only able to follow with its maximum slew rate. For a sine wave, the highest voltage change occurs during zero crossing and is related to frequency and magnitude. Sine waves follow the function:
25 Fig The maximum slope of a sine function occurs at the zero crossing. The slope depends on the amplitude and on the frequency. If the voltage continues to rise with the zero-slope of the sine function, it will reach U max at: The maximum slope can therefore be expressed in terms of the amplitude and the frequency of the sine function: This means for a given slew rate: the higher the output voltage, the smaller the maximum frequency, resp. band width; and vice versa: the larger the required band width, the smaller the maximum amplitude. The slew rate relates the maximum amplitude and the maximum frequency of the output signal. The slew rate cannot be influenced by NFB. Examples of the slew rate of some practical opamps: - type 741: O.3V/µs - type 081: 13V/µs
26 6. Applications This chapter sums up some of the most important opamp applications and gives their main characteristics and design rules Non-Inverting Amplifier (very high) (very low)
27 6.2. Inverting Amplifier (very low) 6.3. With push-pull output The complementary push-pull stage boosts the output current. If it is included in the NFB-loop, the take-over distortions are compensated.
28 6.4. Summing Amplifier The input signals U 1, U 2, etc. are added up and amplified. As the summing point is the virtual ground ( Zero-Ohms-Circuit), the inputs are fully decoupled from each other Logarithmizing Amplifier
29 A non-linear NFB-circuit will result in an non-linear characteristic of the amplifier. The exponential U-I-characteristic of the diode produces a logarithmic U in -U out - relationship. (U T is the inherent temperature voltage of the diode which, for silicon diodes, is approx. 40mV at 25 C. I o is the minority current of the diode at 0V, which is appr. 10nA at 25 C) 6.6. Signal Rectification The threshold voltage of rectifier diodes produce incorrect indications when small signal voltages have to be rectified for indication. Putting the rectifier into the NFBloop of an opamp will produce a linear indication of the meter. It is a disadvantage of this circuit that the meter cannot be grounded on one side.
30 6.7. Voltage Regulator The opamp is used as an error amplifier, comparing the reference voltage with the actual output voltage. Depending on how much output current is required, several current amplifier transistor stages are required Comparator
31 The comparator is an analog-digital converter. The output signal is high or low, depending on whether the input voltage is higher or lower than the reference voltage. If the reference voltage is applied to the non-inverting input, it will be an inverting comparator Schmitt Trigger The Schmitt Trigger can be considered a comparator with hysteresis. By applying positive feedback, the output is always saturated. The threshold voltages for changing the output from positive to negative is different from the voltage which will change it from negative to positive.
32 6.10. Astable Multivibrator This circuit produces a symmetrical square wave at the output of the opamp. The amplitude is given by the saturation voltage of the opamp. The steepness of the flanks is limited by the slew rate.
33 6.11. Phase Shifter This circuit provides a frequency depending phase shift between the input and output signal, but has a linear amplitude response. It is therefore also called an ALL PASS FILTER. The phase shift will vary between 0 and 180. The gain is defined by the negative feedback of R 1 and R 2. Normally, the gain is set to 1 (R 1 =R 2 ).
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CC-MAIN-2019-26
| 26,825 | 33 |
https://barunrb.com.np/business-microeconomics-question-paper-2076/
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math
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Full Marks: 100
Pass Marks: 35
Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.
Brief Answer Questions [10×2=20]
Attempt ALL Questions.
1 Define marginal rate substitution.
2 List out the factors that causes rightward shift in supply curve.
3 State the law of variable proportions.
4 Why does TR increases at a decrease rate when MR decreases?
5 Let Ca =RS 150000, C a=RS 100000, Ca-b=RS 200000.Compute the degree of economics of scope.
6 Microeconomics is also called Price of Theory. Why?
7 Identity the factors that create interest rate differentials.
8 Write any four characteristics of oligopoly.
9 Let ,ep(NEWSPAPERS)=-0.6 and ep(mobile sets)=-1.8 in order to increase the TR to which commodity would you suggest to increase price?
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10 Demand is a flow concept .Why?
NOTE: SCROLL DOWN TO SEE “ASMITA QUESTION BANK”
Descriptive Answer Questions
Attempt FIVE Questions. [5×10=50]
11 What is business economics? Explain its scope.
12 Explain the modern theory of rent.
13 Explain the concept of cost plus pricing with suitable examples.
14 Consider the following demand and supply schedules.
b Compute price elasticity of demand at movement from B to D and D to B by percentage method.
c Compute price elasticity of demand at a movement at midway between B to D and D to B by arc method.
d State the relationship between price elasticity of demand and total revenue.
15 a Explain the concept of price ceiling and price floor.
b Let demand function Qd=300-5p,supply function, Qs=-150 +5p.Determine consumers surplus ,procedures surplus and total surplus.
16 a Differentiate accounting cost and economic cost with suitable examples.
b Let cost function C=600+20Q+0.1Q3
1 Derive TC, AVC, AC, and MC functions.
2 Determine the value of TC ,AVC, AC and MC at output (Q)=20
Analytical Answer Question [15*2=30]
Attempt Any two questions
17 How does firm maximize output by investing fixed total cost outlay at given prices on two inputs ?What will be the effect on output when total cost outlay increases?
18 What is monopoly ? How are the price and the output determined under it?
19 a Derive income consumption curve for normal goods.
b Let budget=RS 100000,Px=RS 200,Py=RS 100
1 Derive budget constraint and identity the consumers equilibrium under equal allocation of budget.
2 Let price of X good falls to RS 100.Derive new budget constraint. Also identity the new equilibrium when he allocates RS 4000 on X goods and RS 6000 on Y goods.
3 Derive price demand curve for X good.
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CC-MAIN-2021-43
| 2,663 | 43 |
https://qmiart.com/math-solver-699
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math
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CC-MAIN-2023-06
| 2,008 | 8 |
https://www.arxiv-vanity.com/papers/0908.0492/
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math
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On Y. Nievergelt’s inversion formula for the Radon transform
We generalize Y. Nievergelt’s inversion method for the Radon transform on lines in the 2-plane to the -plane Radon transform of continuous and functions on for all .
Key words and phrases:The -plane Radon transform, Nievergelt’s inversion method, the convolution-backprojection method
2000 Mathematics Subject Classification:Primary 42C40; Secondary 44A12
Inversion formulas for Radon transforms of different kinds are of great importance in mathematics and its applications; see, e.g., [2, 4, 7, 8, 9, 14, 16, 17, 20, 24], and references therein. Since many of them are pretty involved, especially for new-comers in the area, or applicable under essential restrictions, every “elementary” inversion method deserves special consideration. In 1986 Yves Nievergelt came up with intriguing paper , entitled “Elementary inversion of Radon’s transform”. His result can be stated as follows.
. Any continuous compactly supported function on the -plane can be reconstracted from the Radon transform over lines in this plane by the formula
where the double integral on the right-hand side equals the average of over the disc of radius centered at .
1. What is the basic idea of the Nievergelt’s method from the point of view of modern developments?
2. Is this method applicable in the same elementary form to -plane Radon transforms on for all and arbitrary continuous or functions, satisfying minimal assumptions at infinity?
In the present article we answer these questions and indicate possible generalizations.
Notation and main results. Let and be the affine Grassmann manifold of all non-oriented -planes in , and the ordinary Grassmann manifold of -dimensional subspaces of , respectively. Given , each vector can be written as where and , being the orthogonal complement to in . Each -plane is parameterized by the pair where and . The manifold will be endowed with the product measure , where is the -invariant measure on of total mass , and denotes the usual volume element on . We write for the space of continuous functions on vanishing at infinity; denotes the area of the unit sphere in .
The -plane transform of a function on is a function on defined by
This expression is finite for all if is continuous and decays like with . Moreover [20, 22, 24], if , , then is finite for almost all planes . The above-mentioned bounds for and are best possible.
Following , we define the wavelet-like transform
where denotes the Euclidean distance between the point and the -plane .
[19, Th. 3.1] Let , and let be a radial function on , which has an integrable decreasing radial majorant. If is a solution of the Abel type integral equation
The limit in (1.7) is understood in the -norm and in the almost everywhere sense. If for some , then (1.7) holds uniformly on .
This theorem is a core of the convolution-backprojection method for the -plane Radon transform, and the most difficult task is to choose relatively simple functions and satisfying (1.5). The crux is that the left-hand side of (1.5) has, in general, a bad behavior when . Hence, to achieve integrability of , the solution must be sign-changing.
Our first observation is that the essence of Y. Nievergelt’s Theorem 1.1 can be presented in the language of Theorem 1.2 as follows.
For the convenience of presentation, we will keep to the following convention.
Of course, Theorem 1.2 deals with essentially more general classes of functions than Theorem 1.1, however, the main focus of our article is different: we want to find auxiliary functions and , having possibly simple analytic expression.
(i) The Nievergelt’s method is applicable to the -ray transform (the case ) in any dimension. Namely, if is chosen according to (1.8), then (1.5) has a solution
and inversion formula (1.7) holds.
(ii) If , then Nievergelt’s method is inapplicable.
An integral in (1.11) can be expressed through the hypergeometric function and explicitly evaluated in some particular cases; see Section 2.2. For instance, if , then (1.11) is the Nievergelt’s function (1.9).
To include all , we modify the Nievergelt’s method by choosing in a different way as follows.
Theorem B. Let
. Then the corresponding function has a decreasing radial majorant in and (1.5) has the following solution:
(i) In the case
(ii) In the case
In both cases the inversion result in Theorem 1.2 is valid.
Theorems A and B are proved in Sections 2 and 3, respectively.
. The convolution-backprojection method is well-developed in the general context of totally geodesic Radon transforms on spaces of constant curvature. Apart of , the latter include the -dimensional unit sphere and the hyperbolic space ; see [1, 18, 19]. As above, the key role in this theory belongs to a certain Abel type integral equation and the relevant sign-changing solution . Moreover, passage to the limit in (1.7) as , can be replaced by integration in from to against the dilation-invariant measure . This leads to inversion formulas, which resemble the classical Calderón’s identity for continuous wavelet transforms . The corresponding wavelet function is determined as a solution of a similar Abel type integral equation; see [1, 18, 19] for details. In all these cases analogues of Theorems A and B can be obtained. We leave this exercise to the interested reader.
. Unlike the classical -plane transforms on , the corresponding transforms on matrix spaces [5, 11, 12, 13] are much less investigated. To the best of our knowledge, no pointwise inversion formulas (i.e., those, that do not contain operations in the sense of distributions) are available for these transforms if the latter are applied to arbitrary continuous or functions . One of the reasons of our interest in Nievergelt’s idea is that it might be applicable to the matrix case. Moreover, as in , it may pave the way to implementation of wavelet-like transforms in the corresponding reconstruction formulas. We plan to study these questions in our forthcoming publication.
2. The case
2.1. Proof of Theorem A
We will be dealing with Riemann-Liouville fractional integrals
Changing variables, we transform the basic integral equation
(cf. (1.5)) to the form
Suppose that is defined by (1.8). If , then, by homogeneity,
being the constant from (2.2). Hence, for , we necessarily have
If , this equation has no solution , because, otherwise, we get
This proves the second statement in Theorem A.
Consider the case . If , then . If , then, setting , from (2.6) we get
or (set )
Thus, if , then, necessarily,
One can readily see that function (2.7) is locally integrable on . Let us prove that it satisfies for all . It suffices to show that when . We have , where
Both integrals can be expressed in terms of hypergeometric functions. For , owing to 3.197 (3) and 9.131 (1) from , we obtain
For , changing the order of integration and using [6, 3.238 (3)], we have
To complete the proof, we recall that , which gives
This coincides with (1.11).
Let us give some examples of functions defined by (1.11) in the case . By [6, 3.197(3)],
Keeping in mind that , and using formulas 156, 203, and 211 from [15, 7.3.2], we obtain:
3. The general case
As in Section 2.1, our main concern is integral equation (2.2), which is equivalent to
We want to find relatively simple functions and , which are admissible in the basic Theorem 1.2 and such that the corresponding functions and obey (3.1). It is convenient to consider the cases of even and odd separately.
3.1. The case of even
Let . We choose
The corresponding function obviously has a decreasing radial majorant in . By (3.2), equation (3.1) is equivalent to
The th derivative
is integrable on . The fact that (3.5) satisfies (3.4) can be easily checked using integration by parts. Thus, the pair of functions and , defined by
and (3.3), falls into the scope of Theorem 1.2 and the “even part” of Theorem B is proved.
3.2. The case of odd
Let . We define by (3.3), as above. Then, instead of (3.4), we have
This gives , where
(use [6, 3.238(3)]). Let us show that satisfies (3.7). Integrating by parts, we have , where
This gives where (use [6, 3.238(3)] again)
as desired. Thus, functions and , defined by
and (3.3), obey Theorem 1.2. This completes the proof of Theorem B.
- C.A. Berenstein, and B. Rubin, Totally geodesic Radon transform of -functions on real hyperbolic space, Fourier Analysis and Convexity, Series : Applied and Numerical Harmonic Analysis; L. Brandolini, L. Colzani, A. Iosevich, G. Travaglini, (Eds.) 2004, VIII, 280 p., ISBN: 0-8176-3263-8
- L. Ehrenpreis, The Universality of the Radon Transform, Oxford University Press, 2003.
- M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS-Conf. Lect. Notes 79, Amer. Math. Soc., Providence, R.I. 1991.
- I.M. Gelfand, S.G. Gindikin, and M.I. Graev, Selected topics in integral geometry, Translations of Mathematical Monographs, 220, American Mathematical Society, Providence, RI, 2003.
- F.B. Gonzalez, and T. Kakehi, Invariant differential operators and the range of the matrix Radon transform. J. Funct. Anal. 241 (2006), 232–267.
- I.S. Gradshteyn, and I.M. Ryzhik, Table of integrals, series and products, Academic Press, 1980.
- S. Helgason, The Radon transform, Birkhäuser, Boston, Second edition, 1999.
- A. Markoe, Analytic Tomography, Encyclopedia of Mathematics and its Applications 106, Cambridge Univ. Press, 2006.
- F. Natterer, The mathematics of computerized tomography, Wiley, New York, 1986.
- Y. Nievergelt, Elementary inversion of Radon’s transform, SIAM Rev. 28 (1986), no. 1, 79–84.
- G. Ólafsson, E. Ournycheva, and B. Rubin, Higher rank wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices, Appl. Comput. Harmon. Anal., 21 (2006), 182–203.
- E. Ournycheva, and B. Rubin, Method of mean value operators for Radon transforms in the space of matrices, Intern. J. Math., 19 (2008), 245-283.
- E. Ournycheva, and B. Rubin, Semyanistyi’s integrals and Radon transforms on matrix spaces, The Journal of Fourier Analysis and Applications, 14 (2008), no.1, 60–88.
- V. Palamodov, Reconstructive integral geometry, Monographs in Mathematics, 98. Birkh user Verlag, Basel, 2004. xii+164 pp. ISBN: 3-7643-7129-3.
- A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, vol. 3, More special functions, Gordon and Breach, Amsterdam, 1990.
- E.T. Quinto, M. Cheney, P. Kuchment (Editors), Tomography, impedance imaging, and integral geometry, Lectures in Applied Mathematics, 30, American Mathematical Society, Providence, RI, 1994.
- A.G. Ramm, and A.I. Katsevich, The Radon transform and local tomography, CRC Press, Boca Raton, 1996.
- B. Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Advances in Appl. Math., 29 (2002), 471–497.
- B. Rubin, Convolution-backprojection method for the -plane transform, and Calderón’s identity for ridgelet transforms, Appl. Comput. Harmon. Anal., 16 (2004), 231- 242.
- B. Rubin, Reconstruction of functions from their integrals over -planes, Israel J. of Math., 141 (2004), 93 -117.
- S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach Sc. Publ., New York, 1993.
- D.C. Solmon, A note on -plane integral transforms, Journal of Math. Anal. and Appl., 71 (1979), 351–358.
- E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970.
- R. S. Strichartz, -estimates for Radon transforms in Euclidean and non-euclidean spaces, Duke Math. J., 48 (1981), 699–727.
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CC-MAIN-2023-14
| 11,713 | 93 |
https://www.arxiv-vanity.com/papers/1103.5745/
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math
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Symmetries for Galileons and DBI scalars on curved space
We introduce a general class of four-dimensional effective field theories which include curved space Galileons and DBI theories possessing nonlinear shift-like symmetries. These effective theories arise from purely gravitational actions for 3-branes probing higher dimensional spaces. In the simplest case of a Minkowski brane embedded in a higher dimensional Minkowski background, the resulting four-dimensional effective field theory is the Galileon one, with its associated Galilean symmetry and second order equations. However, much more general structures are possible. We construct the general theory and explicitly derive the examples obtained from embedding maximally symmetric branes in maximally symmetric ambient spaces. Among these are Galileons and DBI theories with second order equations that live on de Sitter or anti-de Sitter space, and yet retain the same number of symmetries as their flat space counterparts, symmetries which are highly non-trivial from the d point of view. These theories have a rich structure, containing potentials for the scalar fields, with masses protected by the symmetries. These models may prove relevant to the cosmology of both the early and late universe.
- I Introduction and Summary
- II General brane actions and symmetries
- III Actions with second order equations of motion
IV Maximally Symmetric Examples
- IV.1 A Minkowski brane in a Minkowski bulk: in – DBI Galileons
- IV.2 A Minkowski brane in an anti-de Sitter bulk: in – Conformal Galileons
- IV.3 A de Sitter brane in a Minkowski bulk: in
- IV.4 A de Sitter brane in a de Sitter bulk: in
- IV.5 A de Sitter brane in an anti-de Sitter bulk: in
- IV.6 An anti-de Sitter brane in an anti-de Sitter bulk: in
- V Small field limits: the analogues of Galileons
- VI Conclusions
- A Some useful expressions
I Introduction and Summary
The possibility that the universe may contain large, and possibly infinite, spatial dimensions beyond the three we commonly perceive has opened up entirely new avenues to address fundamental questions posed by particle physics and by cosmology. The precise manner in which the dynamics of the higher-dimensional space manifests itself in the four dimensional world depends on the geometry and topology of the extra-dimensional manifold, and the matter content and action chosen. At low enough energies, the relevant physics is then captured by a four-dimensional effective field theory with properties inherited from the specific higher-dimensional model under consideration. The simplest example of this is the Kaluza-Klein tower – the hierarchy of higher mass states that accompany zero mass particles when compactifying a five-dimensional theory on a circle. There are, however, much more exotic possibilities. Many of these describe viable higher-dimensional theories, while others are merely mathematical tools with which to construct interesting physical four-dimensional effective field theories.
A particularly interesting and well studied example of a higher-dimensional model is the Dvali-Gabadadze-Poratti (DGP) model Dvali:2000hr , for which the ambient space is a flat -dimensional spacetime in which a Minkowski -brane floats, subject to an action consisting merely of two separate Einstein Hilbert terms – one in d, and the other only on the brane, constructed from the induced metric there. In an appropriate limit, the resulting four-dimensional effective field theory describes gravity plus a scalar degree of freedom parametrizing the bending of the brane in the extra dimension Luty:2003vm ; Nicolis:2004qq . The specific form of the four dimensional action for the scalar inherits a symmetry from a combination of five dimensional Poincaré invariance and brane reparametrization invariance. In the small field limit this symmetry takes a rather simple form and has been called the Galilean symmetry, with the associated scalar becoming the Galileon Nicolis:2008in .
Abstracting from DGP, a four dimensional field theory with this Galilean symmetry is interesting in its own right. It turns out that there are a finite number of terms, the Galileon terms, that have fewer numbers of derivatives per field than the infinity of competing terms with the same symmetries. These terms have the surprising property that, despite the presence of higher derivatives in the actions, the equations of motion are second order, so that no extra degrees of freedom are propagated around any background. Much has been revealed about the Galileon terms, including such useful properties as a non-renormalization theorem Luty:2003vm ; Hinterbichler:2010xn ; Burrage:2010cu , and applications in cosmology Agarwal:2011mg ; Burrage:2010cu ; Creminelli:2010ba ; Creminelli:2010qf ; DeFelice:2010as ; Deffayet:2010qz ; Kobayashi:2011pc ; Mota:2010bs ; Wyman:2011mp . The Galileons have been covariantized Deffayet:2009mn ; Deffayet:2009wt ; Deffayet:2011gz , extended to p-forms Deffayet:2010zh , and supersymmetrized Khoury:2011da . Further, it was recently shown that the general structure of Galileon field theories can be extended to multiple fields, finding their origins in braneworld constructions with more than one codimension Hinterbichler:2010xn ; Padilla:2010de ; Padilla:2010ir ; Padilla:2010tj ; Zhou:2010di . If some of the resulting symmetries of the four dimensional effective field theory are broken, then they are related to low energy descriptions of cascading gravity models in which a sequence of higher dimensional branes are embedded within one another deRham:2007rw ; deRham:2007xp ; Agarwal:2011mg ; Agarwal:2009gy .
If our universe really is a brane world, then theories of this sort are generic, since they share, in a certain limit, the symmetries of the Dirac-Born-Infeld (DBI) action. The DBI action encodes the lowest order dynamics of a brane embedded in higher dimensions, and provides an important arena within which to study inflation Silverstein:2003hf ; Alishahiha:2004eh , late-time cosmic acceleration Ahn:2009xd , tunneling Brown:2007zzh , and exotic topological defects Andrews:2010eh ; Babichev:2006cy ; Sarangi:2007mj ; Bazeia:2007df ; Babichev:2008qv . The Galileon terms can be thought of as a subset of the higher order terms expected to be present in any effective field theory of the brane, and which will be suppressed by powers of some cutoff scale. The Galileons are a special subset in the class of all possible higher order terms because they contain fewer derivatives per field than competing terms with the same symmetries, and because they yield second order equations. Crucially, there can exist regimes in which only a finite number of Galileon terms are important, and the infinity of other possible terms within the effective field theory are not (see section II of Hinterbichler:2010xn , as well as Nicolis:2004qq ; Endlich:2010zj , for more on this and for examples of such regimes.) This fact, coupled with a non-renormalization theorem for Galileons and the fact that there are a finite number of such terms, holds out the hope of computing non-linear facts about the world which are exact quantum mechanically. Finally, it should be remembered that even if our universe is not a brane world, the same conclusions follow if one postulates the existence of symmetries of the same form as those of a brane world.
In this paper, we construct a general class of four-dimensional effective field theories by writing an action on a 3-brane probing a higher dimensional bulk, of which the Galileon theory and DBI scalars are special cases. This extends the construction of deRham:2010eu to its most general form. We observe that the symmetries inherited by scalar fields in the d theory are determined by isometries of the bulk metric, and are present if and only if the bulk has isometries. The precise manner in which the symmetries are realized is determined by the choice of gauge, or foliation, against which brane fluctuations are measured. We derive in general the symmetries of these effective field theories, and classify the examples that result when embedding a maximally symmetric brane in a maximally symmetric background. This approach yields a set of new Galileon-like theories which live on d curved space but retain the same number of non-linear shift-like symmetries as the flat-space Galileons or DBI theories.
These theories have their own unique properties. For example, in curved space the field acquires a potential which is fixed by the symmetries – something that is not allowed for the flat space Galileons. In particular, the scalars acquire a mass of order the inverse radius of the background, and the value of the mass is fixed by the nonlinear symmetries. Although not addressed in detail here, allowing for de Sitter solutions on the brane opens up the possibility of adapting these new effective field theories to cosmological applications such as inflation or late time cosmic acceleration in such a way that their symmetries ensure technical naturalness.
The paper is structured as follows. In the next section we discuss general brane actions and symmetries, and the ways in which these symmetries may be inherited by a four-dimensional effective field theory. In section III we then consider constructing actions with second order equations and explicitly derive all possible terms in such theories. We then provide six separate examples, exhausting all the maximally symmetric possibilities: a d Minkowski brane embedded in a Minkowski bulk; a d Minkowski brane embedded in ; a d de Sitter brane embedded in a Minkowski bulk; a d de Sitter brane embedded in ; a d de Sitter brane embedded in ; and a d Anti-de Sitter brane embedded in . In each case, we describe the resulting d effective field theories and comment on their structure. In section V we take the small field limits to obtain Galileon-like theories, discuss their stability, and compare and contrast these theories with the special case of the original Galileon, before concluding.
Conventions and notation:
We use the mostly plus metric signature convention. The 3-brane worldvolume coordinates are , , bulk coordinates are , . Occasionally we use 6-dimensional cartesian coordinates , , for constructing five dimensional and as embeddings. Tensors are symmetrized and anti-symmetrized with unit weight, i.e , .
When writing actions for a scalar field in curved space with metric and covariant derivative , we use the notation for the matrix of second derivatives . For traces of powers of we write , e.g. , , where all indices are raised with respect to . We also define the contractions of powers of with using the notation , e.g. , , where again all indices are raised with respect to .
Ii General brane actions and symmetries
We begin with a completely general case - the theory of a dynamical 3-brane moving in a fixed but arbitrary (4+1)-dimensional background. The dynamical variables are the brane embedding , five functions of the world-volume coordinates .
The bulk has a fixed background metric . From this and the , we may construct the induced metric and the extrinsic curvature , via
Here are the tangent vectors to the brane, and is the normal vector, defined uniquely (up to a sign) by the properties that it is orthogonal to the tangent vectors , and normalized to unity . (Note that the extrinsic curvature can be written , demonstrating that it depends only on quantities defined directly on the brane and their tangential derivatives.)
We require the world-volume action to be gauge invariant under reparametrizations of the brane,
where is the gauge parameter. This requires that the action be written as a diffeomorphism scalar, , of and as well as the covariant derivative and curvature constructed from ,
This action will have global symmetries only if the bulk metric has Killing symmetries. If the bulk metric has a Killing vector , i.e. a vector satisfying the Killing equation
then the action will have the following global symmetry under which the shift,
We are interested in creating non-gauge theories with global symmetries from the transverse fluctuations of the brane, so we now fix all the gauge symmetry of the action. We accomplish this by first choosing a foliation of the bulk by time-like slices. We then choose bulk coordinates such that the foliation is given by the surfaces . The remaining coordinates can be chosen arbitrarily and parametrize the leaves of the foliation. The gauge we choose is
In this gauge, the world-volume coordinates of the brane are fixed to the bulk coordinates of the foliation. We call the remaining unfixed coordinate , which measures the transverse position of the brane relative to the foliation (see Figure 1). This completely fixes the gauge freedom. The resulting gauge fixed action is then an action solely for ,
Global symmetries are physical symmetries that cannot be altered by the unphysical act of gauge fixing. Thus, if the original action (4) possesses a global symmetry (6), generated by a Killing vector , then the gauge fixed action (8) must also have this symmetry. However, the form of the symmetry will be different because the gauge choice will not generally be preserved by the global symmetry. The change induced by is
To re-fix the gauge to (7), it is necessary to simultaneously perform a compensating gauge transformation with gauge parameter
The combined symmetry acting on ,
is then a symmetry of the gauge fixed action (8).
ii.1 A special case
We now specialize to a case which includes all the maximally symmetric examples of interest to us in this paper. This is the case where the foliation is Gaussian normal with respect to the metric , and the extrinsic curvature on each of the leaves of the foliation is proportional to the induced metric. With these restrictions, the metric takes the form
where denotes the Gaussian normal transverse coordinate, and is an arbitrary brane metric. Recall that in the physical gauge (7), the transverse coordinate of the brane is set equal to the scalar field, .
Working in the gauge (7), the induced metric is
Defining the quantity
the square root of the determinant and the inverse metric may then be expressed as
The tangent vectors are
To find the normal vector we solve the two equations
Using the non-vanishing Christoffel symbols , , , the extrinsic curvature is then
Note that when the d coordinates have dimensions of length, has mass dimension and is dimensionless.
The algebra of Killing vectors of contains a natural subalgebra consisting of the Killing vectors for which . This is the subalgebra of Killing vectors that are parallel to the foliation of constant surfaces, and it generates the subgroup of isometries which preserve the foliation. We choose a basis of this subalgebra and index the basis elements by ,
where we have written for the components, indicating that these components are independent of . To see that this is the case, note that, for those vectors with , the Killing equations (5) tell us that is independent of . Furthermore, the Killing equations tell us that is a Killing vector of .
We now extend our basis of this subalgebra to a basis of the algebra of all Killing vectors by appending a suitably chosen set of linearly independent Killing vectors with non-vanishing . We index these with , so that is a basis of the full algebra of Killing vectors. From the component of Killing’s equation, we see that must be independent of , so we may write .
A general global symmetry transformation thus reads
From this, we see that the symmetries are linearly realized, whereas the are realized nonlinearly. Thus, the algebra of all Killing vectors is spontaneously broken to the subalgebra of Killing vectors preserving the foliation.
ii.2 Maximally symmetric cases
In this paper, we will focus on the case in which the 5d background metric has 15 global symmetries, the maximal number. Thus, the bulk is either d anti-de Sitter space with isometry algebra , 5d de-Sitter space with isometry algebra , or flat 5d Minkowski space with isometry algebra the five dimensional Poincare algebra . In addition, we focus on the case where the brane metric , and hence the extrinsic curvature, are maximally symmetric, so that the unbroken subalgebra has the maximal number of generators, 10. This means that the leaves of the foliation are either d anti-de Sitter space with isometry algebra , 4d de-Sitter space with isometry algebra , or flat 4d Minkowski space with isometry algebra the four dimensional Poincare algebra . In fact, there are only 6 such possible foliations of d maximally symmetric spaces by d maximally symmetric time-like slices, such that the metric takes the form (12). Flat can be foliated by flat slices or by slices; can be foliated by flat slices, slices, or slices; and can only be foliated by slices. Each of these 6 foliations, through the construction leading to (8), will generate a class of theories living on an , or background and having 15 global symmetries broken to the 10 isometries of the brane. These possibilities are summarized in Figure 2.
It should be noted that the missing squares in Figure 2 may be filled in if we are willing to consider a bulk which has more than one time direction111We thank Sergei Dubovsky for pointing this out.. For example, it is possible to embed into a five-dimensional Minkowski space with two times (indeed, this is the standard way of constructing spaces). From the point of view that the bulk is physical, and hence should be thought of as dynamical, these possibilities may be unacceptable on physical grounds. However, if one thinks of the bulk as merely a mathematical device for constructing novel four-dimensional effective theories, then there is nothing a priori to rule out these possibilities. In this paper, we focus on those cases in which the bulk has only one time dimension. The construction in the other cases will, however, follow the same pattern.
Finally, note that the only invariant data that go into constructing a brane theory are the background metric and the action. Theories with the same background metric and the same action are isomorphic, regardless of the choice of foliation (which is merely a choice of gauge). For example, given the same action among the theories listed in Figure 2, the three that have an background, namely the conformal DBI Galileons, the DBI Galileons, and the type III DBI Galileons, are really the same theory. They are related by choosing a different foliation (gauge), shuffling the background configuration into the background metric.
Iii Actions with second order equations of motion
Up until now we have discussed the degrees of freedom and their symmetries, but it is the choice of action that defines the dynamics. A general choice for the function in (8) will lead to scalar field equations for which are higher than second order in derivatives. When this is the case, the scalar will generally propagate extra degrees of freedom which are ghost-like Ostrogradski ; deUrries:1998bi . The presence of such ghosts signifies that either the theory is unstable, or the cutoff must be lowered so as to exclude the ghosts. Neither of these options is particularly attractive, and so it is desirable to avoid ghosts altogether. It is the Galileon terms which are special because they lead to equations of at most second order. Furthermore, as mentioned in the introduction, there can exist regimes in which the Galileon terms dominate over all others, so we will be interested only in these terms.
A key insight of de Rham and Tolley deRham:2010eu is that there are a finite number of actions of the type (8), the Lovelock terms and their boundary terms, that do in fact lead to second order equations for and become the Galileon terms. The possible extensions of Einstein gravity which remain second order are given by Lovelock terms Lovelock:1971yv . These terms are specific combinations of powers of the Riemann tensor which are topological (i.e. total derivatives) in some specific home dimension, but in lower dimensions have the property that equations of motions derived from them are second order. (For a short summary of some properties of these terms, see Appendix B of Hinterbichler:2010xn .) The Lovelock terms come with boundary terms. It is well known that, when a brane is present, bulk gravity described by the Einstein-Hilbert Lagrangian should be supplemented by the Gibbons-Hawking-York boundary term Gibbons:1976ue ; York:1972sj
Similarly, Lovelock gravity in the bulk must be supplemented by brane terms which depend on the intrinsic and extrinsic curvature of the brane (the so-called Myers terms Myers:1987yn ; Miskovic:2007mg ), which are needed in order to make the variational problem for the brane/bulk system well posed Dyer:2008hb . Of course we are not considering bulk gravity to be dynamical, but the point here is that these boundary terms also yield second order equations of motion for in the construction leading to (8).
The prescription of deRham:2010eu is then as follows: on the 4-dimensional brane, we may add the first two Lovelock terms, namely the cosmological constant term and the Einstein-Hilbert term . (The higher Lovelock terms are total derivatives in 4-dimensions.) We may also add the boundary term corresponding to a bulk Einstein-Hilbert term, , and the boundary term corresponding to the Gauss-Bonnet Lovelock invariant in the bulk. The zero order cosmological constant Lovelock term in the bulk has no boundary term (although as we will see, we may construct a fifth term, the tadpole term, from it) and the higher order bulk Lovelock terms vanish identically. Therefore, in total, for a 3-brane there are four possible terms (five including the tadpole) which lead to second order equations. These are the terms we focus on.
iii.1 The tadpole term
As mentioned, there is one term that contains no derivatives of and is not of the form (8). This Lagrangian is called the tadpole term, denoted by . The value of the tadpole action is the proper 5-volume between some surface and the position of the brane,
Note that .
Under a general nonlinear symmetry of the type (24), its change is
Using the Killing equation (5), it is straightforward to check directly that a general variation of the right-hand side vanishes, demonstrating that the change in the tadpole term under the symmetry transformation is a total derivative. Thus the tadpole term has the same symmetries as the other terms.
iii.2 Explicit expressions for the terms
Including the tadpole term there are thus five terms that lead to second order equations for ,
where the explicit form of the Gauss-Bonnet boundary term is
Indices are raised and traces are taken with . At this stage, each of these terms would appear in a general Lagrangian with an arbitrary coefficient. As we will see later, requiring stability will, however, force certain choices on us in specific examples.
En route to presenting specific examples of our new theories, we now evaluate these terms on the special case metric (12). We make use of formulae catalogued in Appendix A. Our strategy is to collect coefficients of , , and , eliminate everywhere in favor of , and then to group like terms by powers of . A lengthy calculation yields
The quantities and are various contractions of derivatives of the field, and the notation is explained in the conventions at the end of Section I. In these expressions, all curvatures are those of the metric , and all derivatives are covariant derivatives with respect to . We point out that no integrations by parts have been performed in obtaining these expressions.
The equations of motion derived from any of these five terms will contain no more than two derivatives on each field, ensuring that no extra degrees of freedom propagate around any background. After suitable integrations by parts, these actions should therefore conform to the general structure presented in Deffayet:2011gz for actions of a single scalar with second order equations (see also the Euler hierarchy constructions Fairlie:1991qe ; Fairlie:1992nb ; Fairlie:1992yy ; Fairlie:2011md ). In the above construction, however, we can immediately identify the nonlinear symmetries by reading them off from the isometries of the bulk.
Finally, we note that by keeping the metric in (12) arbitrary rather than fixing it to the foliation, we can automatically obtain the covariantizaton of these various Galileon actions, including the non-minimal curvature terms required to keep the equations of motion second order, the same terms obtained by purely 4-d methods in Deffayet:2009mn ; Deffayet:2009wt ; Deffayet:2011gz . Of course, this in general ruins the symmetries we are interested in considering. But from this point of view, we can see exactly when such symmetries will be present. The symmetries will only be present if the which is used to covariantly couple is such that the full metric (12) has isometries.
Iv Maximally Symmetric Examples
We now proceed to construct explicitly the maximally symmetric examples catalogued in Section II.2 and Figure 2. The construction starts by finding coordinates which are adapted to the desired foliation, so that the metric in the bulk takes the form (12), allowing us to read off the function . Plugging into (LABEL:generalterms) then gives us the explicit Lagrangians. To find the form of the global symmetries, we must write the explicit Killing vectors in the bulk, and identify those which are parallel and not parallel to the foliation. We may then read off the symmetries from (24).
The construction for each case is similar, and some of the results are related by analytic continuation, but there are enough differences in the forms of the embeddings and the Killing vectors that we thought it worthwhile to display each case explicitly. The reader interested only in a given case may skip directly to it.
iv.1 A Minkowski brane in a Minkowski bulk: in – DBI Galileons
Choosing cartesian coordinates on , the foliation of by is simply given by slices, and the metric takes the form
Comparing this to (12), we obtain
and the terms (LABEL:generalterms) become (again, without integration by parts)
iv.1.1 Killing vectors and symmetries
The Killing vectors of d Minkowski space are the 10 boosts , and the 5 translations . The 6 boosts and the 4 translations are parallel to the foliation and form the unbroken symmetries of . The 5 broken generators are
Using the relation from (24), we obtain the transformation rules
under which the terms (LABEL:DBIGalileonterms) are each invariant up to a total derivative. The symmetry breaking pattern is
iv.2 A Minkowski brane in an anti-de Sitter bulk: in – Conformal Galileons
In this section, indices run over six values and are cartesian coordinates in an ambient d two-time Minkowski space with metric , which we call .
Five dimensional anti-de Sitter space (more precisely, a quotient thereof) can be described as the subset of points in the hyperbola of one sheet satisfying
with the radius of curvature of , and where the metric is induced from the flat metric on . This space is not simply connected, but its universal cover is . The scalar curvature and cosmological constant are given by
We use Poincare coordinates on which cover the region ,
where , and is the Minkowski 4-metric. The coordinates and all take the range . Lines of constant foliate the Poincare patch of with Minkowski time-like slices, given by intersecting the planes with the hyperbola.
The induced metric is
Comparing this with (12) we obtain
and the terms (LABEL:generalterms) become (without integration by parts)
These are the conformal DBI Galileons, first written down in deRham:2010eu .
iv.2.1 Killing vectors and symmetries
The 15 Lorentz generators of ; (here are the coordinate basis vectors in the ambient space , and indices are lowered with the flat metric ) are all tangent to the hyperboloid, and become the 15 isometries of the isometry algebra of . Of these, 10 have no components and are parallel to the foliation. These form the unbroken isometry algebra of the slices.
First we have
which taken together are the 6 Lorentz transformations of the .
For the remaining 4, we focus on
which may be grouped as
If we now take the following linear combinations,
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https://mc.ai/talented-mr-1x1-comprehensive-look-at-1x1-convolution-in-deep-learning/
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Source: Deep Learning on Medium
Talented Mr. 1X1: Comprehensive look at 1X1 Convolution in Deep Learning
With startling success of AlexNet in 2012, the Convolutional Neural Net (CNN) revolution has begun! The CNN based frameworks in Deep Learning like GoogleNet, ResNet and several variations of these have shown spectacular results in the object detection and semantic segmentation in computer vision.
When you start to look at most of the successful modern CNN architectures, like GoogleNet, ResNet and SqueezeNet you will come across 1X1 Convolution layer playing a major role. At first glance, it seems to be pointless to employ a single digit to convolve with the input image (After all wider filters like 3X3, 5X5 can work on a patch of image as opposed to a single pixel in this case). However, 1X1 convolution has proven to be extremely useful tool and employed correctly, will be instrumental in creating wonderfully deep architectures.
In this article we will have a detailed look at 1X1 Convolutions
First a quick recap of Convolutions in Deep Learning. There are many good blogs and articles that intuitively explain what convolutions are and different types of convolutions (few of them are listed in the reference). While we will not delve deep into the convolutions in this article, understanding couple of key points will make it easier to get what 1X1 convolution is doing and most importantly How & Why it is doing it.
Quick Recap: Convolution in Deep Learning
As mentioned, this article will not provide a complete treatment of theory and practice of Convolution. However, we will recap key principles of Convolution in deep learning. This will come in handy when we examine 1X1 Convolution in depth.
Simply put, Convolutions is an element wise multiplication and summation of the input and kernel/filter elements. Now the data points to remember
1. Input matrix can and, in most cases, will have more than one channel. This is sometimes referred to as depth
a. Example: 64X64 pixel RGB input from an image will have 3 channels so the input is 64X64X3
2. The filter has the same depth as input except in some special cases (example 3D Convolutions to reconstruct medical images). This specific point, for some unknown reason, is not explicitly mentioned in most of the literature, causing some misunderstanding (Especially for someone new to convolutions, Deep learning etc)
a. Example: filter of 3X3 will have 3 channels as well, hence the filter should be represented as 3X3X3
3. Third and critical point, the output of Convolution step will have the depth equal to number of filters we choose.
a. Example: Output of Convolution step of the 3D input (64X64X3) and the filter we chose (3X3X3) will have the depth of 1 (Because we have only one filter)
The Convolution step on the 3D input 64X64X3 with filter size of 3X3X3 will have the filter ‘sliding’ along the width and height of the input.
So, when we convolve the 3D filter with the 3D image, the operation moves the filter on the input in 2 directions (Along the width and height) and we do the element wise multiplication and addition at each position to end up with an output with a depth of 1.
Armed with this, we are ready to dive into the 1X1 convolution
1X1 Convolution — What is it?
Introduced first in a paper by Min Lin et all in their Network In Network, the 1X1 Convolution layer was used for ‘Cross Channel Down sampling’ or Cross Channel Pooling. In other words, 1X1 Conv was used to reduce the number of channels while introducing non-linearity.
In 1X1 Convolution simply means the filter is of size 1X1 (Yes — that means a single number as opposed to matrix like, say 3X3 filter). This 1X1 filter will convolve over the ENTIRE input image pixel by pixel.
Staying with our example input of 64X64X3, if we choose a 1X1 filter (which would be 1X1X3), then the output will have the same Height and Weight as input but only one channel — 64X64X1
Now consider inputs with large number of channels — 192 for example. If we want to reduce the depth and but keep the Height X Width of the feature maps (Receptive field) the same, then we can choose 1X1 filters (remember Number of filters = Output Channels) to achieve this effect. This effect of cross channel down-sampling is called ‘Dimensionality reduction’.
Now why would we want to something like that? For that we delve into usage of 1X1 Convolution
Usage 1: Dimensionality Reduction/Augmentation
Winner of ILSVRC (ImageNet Large Scale Visual Recognition Competition) 2014, GoogleNet, used 1X1 convolution layer for dimension reduction “to compute reductions before the expensive 3×3 and 5×5 convolutions”
Let us look at an example to understand how reducing dimension will reduce computational load. Suppose we need to convolve 28 X 28 X 192 input feature maps with 5 X 5 X 32 filters. This will result in 120.422 Million operations
Let us do some math with the same input feature maps but with 1X1 Conv layer before the 5 X 5 conv layer
By adding 1X1 Conv layer before the 5X5 Conv, while keeping the height and width of the feature map, we have reduced the number of operations by a factor of 10. This will reduce the computational needs and in turn will end up being more efficient.
GoogleNet paper describes the module as “Inception Module” (Get it — DiCaprio’s “We need to go DEEPER” in the movie Inception)
Usage 2: Building DEEPER Network (“Bottle-Neck” Layer)
2015 ILSVRC Classification winner, ResNet, had least error rate and swept aside the competition by using very deep network using ‘Residual connections’ and ‘Bottle-neck Layer’.
In their paper, He et all explains (page 6) how a bottle neck layer designed using a sequence of 3 convolutional layers with filters the size of 1X1, 3X3, followed by 1X1 respectively to reduce and restore dimension. The down-sampling of the input happens in 1X1 layer thus funneling a smaller feature vectors (reduced number of parameters) for the 3X3 conv to work on. Immediately after that 1X1 layer restores the dimensions to match input dimension so identity shortcuts can be directly used. For details on identity shortcuts and skip connection, please see some of the Reviews on ResNet (Or you can wait for my future work!)
Usage 3: Smaller yet Accurate Model (“FIRE-MODULE” Layer)
While Deep CNN Models have great accuracy, they have staggering number of parameters to deal with which increases the training time and most importantly need enterprise level computing power. Iandola et all proposed a CNN Model called SqueezeNet that retains AlexNet level accuracy while 50X times smaller in terms of parameters.
Smaller models have number of advantages, especially on use-cases that require edge computing capabilities like autonomous driving. Iandola et all achieved this by stacking a bunch of “Fire Modules” which comprise of
1. Squeeze Layer which has only 1X1 Conv filters
2. This feeds an Expansion layer which has mix of 1X1 and 3X3 filters
3. The number of filters in Squeeze Layer are set to be less than number of 1X1 filters + Number of 3X3 in Expand Layer
By now it is obvious what the 1X1 Conv filters in Squeeze Layer do — they reduce the number of parameters by ‘down-sampling’ the input channels before they are fed into the Expand layer.
The Expansion Layer has mix of 1X1 and 3X3 filters. The 1X1 filters, as you know, performs cross channel pooling — Combines channels, but cannot detect spatial structures (by virtue of working on individual pixels as opposed to a patch of input like larger filters). The 3X3 Convolution detects spatial structures. By combining these 2 different sized filters, the model becomes more expressive while operating on lesser parameters. Appropriate use of padding makes the output of 1X1 and 3X3 convolutions the same size so these can be stacked.
In this article we reviewed high level Convolution mechanism and threw ourselves into the deep end with 1X1 Convolution to understand the underpinnings, where they are effectively used and to what end.
To recap, 1X1 Convolution is effectively used for
1. Dimensionality Reduction/Augmentation
2. Reduce computational load by reducing parameter map
3. Add additional non-linearity to the network
4. Create deeper network through “Bottle-Neck” layer
5. Create smaller CNN network which retains higher degree of accuracy
1. Andrew Ng’s Video on 1X1 Convolution
4. Network in Network — Min Lin et All
5. Going Deeper with Convolutions — Christian Szegedy et All
6. Deep Residual Learning for Image Recognition — Kaiming He et All
7. SqueezeNet — Forest Iandola et All
8. CNN Architecture — Lecture 9 (Stanford) : Fei-Fei Lin et All
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. Please provide any two values below and click the Calculate button to get the third value. In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol % or simply as percent or pct. For example, 35% is equivalent to the decimal 0.35, or the fraction Answer:75Step-by-step explanation:the three digit number is = 90×100/30=300so 25% of the number is= 300×25/100=7 30% of a number is 90, then what is 50% of the same number?a, 250b, 50c,350d, 150 Get the answers you need, now! siddiquaasma605 siddiquaasma605 05.04.2021 Math Secondary School answered 30% of a number is 90, then what is 50% of the same number? a, 250 b, 50 c,350 d, 150 30% Percent Calculator. Use this calculator to find percentages. Just type in any box and the result will be calculated automatically. Calculator 1: Calculate the percentage of a number. For example: 30% of 25 = 7.5. Calculator 2: Calculate a percentage based on 2 numbers. For example: 7.5/25 = 30% Finally, we have found the value of Y which is 27 and that is our answer. If you want to use a calculator to know what is 30 percent of 90, simply enter 30 ÷ 100 × 90 and you will get your answer which is 27 You may also be interested in: What percent of 90 is 2
What is 30 less than 90? We are looking for a new number which is 30 less than 90. We will get the new number by subtracting 30 from 90. We write it down as: 90-30=60. And finally the solution for: What number is 30 less than 90? is 60 There is a method, where you find a small percentage, then multiply. Below you can find 10%, then multiply by 7 to get 70% of the original number (100%-30%=70% of. Finally, we have found the value of Y which is 90 and that is our answer. You can easily calculate 27 is 30 percent of what number by using any regular calculator, simply enter 27 × 100 ÷ 30 and you will get your answer which is 90 You may also be interested in: What is 30 percent of 90 Type 3: The number 32 is 8% of what number? As usual, this problem requires to steps: Step 1: Write 8% as a decimal number: 8% = 0.08. Step 2: Divide 32 and 0.08: 32 ÷ 0.08 = 400. Type 4: percentage increase: What is the percentage increase from 20 to 90? In this type of problem we use formul
30 is 30% of 100. Steps to solve 30 is 30 percent of what number? We have, 30% × x = 30; or, 30 / 100 × x = 30 Multiplying both sides by 100 and dividing both sides by 30, we have x = 30 × 100 / 30 x = 100. If you are using a calculator, simply enter 30×100÷30, which will give you the answer $90 / 3 = $30 each: 20% of $90 is $18. Now find 20% of $30 and add that as the tip. The total bill amounts to: 20% of $30 is $6. $90 + $18 = $108: Now, add that $6 to $30. This amounts to $36. Now, split the total bill in thirds: $108 / 3 = $36: Total share: Each of you would pay $36. Total share: Each of you would pay $36 Answer provided by our tutors. let 'x' represent the original number, then: x * 0.3 = 12. x = 12 / 0.3 = 40. 12 is 30% of the number 40 Calculator 1: Calculate the percentage of a number. For example: 90% of 30 = 27. Calculator 2: Calculate a percentage based on 2 numbers. For example: 27/30 = 90%. How much is 90% of 30? What is 90% of 30 and other numbers? 90% of 30.00 = 27.0000. 90% of 30.25 = 27.2250. 90% of 30.50 = 27.4500
60. One of the ways to solve this is by proportion. A/B (A:B) = C/D (C:D) A - the number (18) B - the base number (unknown) C - Percentage (30%) D - Total Percentage. seventy-eight is 15% of what number so there's some unknown number out there and if we take 15% of that number we will get 78 so let's just call that unknown number X and we know that if we take 15% of X so if we take 15% of X so multiply X by 15% we will get we will get 78 and now we just literally have to solve for x now 15% mathematically you can deal directly with percentages but it's much. The number is 400 Let x be what number. We can then write this problem as: 30% of x = 120 or 30/100 x = 120 Solveing for x while keeping the equation balanced gives: 100/30 30/100 x = 120 100/30 x = 12000/30 x = 40 The strategy here is to see how many times the percent number (in this case, 25) goes into 100, and then count by that number until we reach 100-the whole thing. Here, we're told that 25% of a number is 5. So, to find 100% of the number, we count by 25s up to 100: 25, 50, 75, 100. 100% is 20 30 percent of 91 is the same as 30 per hundred of 91. We can therefore make the following equation: 30/100 = X/91 To solve the equation above for X, you first switch the sides to get the X on the left side, then you multiply each side by 91, and then finally divide the numerator by the denominator on the right side to get the answer
10/10 = 1. When we put that together, we can see that our complete answer is: 27. /. 1. The complete and simplified answer to the question what is 3/10 of 90 is: 27. Hopefully this tutorial has helped you to understand how to find the fraction of any whole number A% of B is C as in: 10% of 90 is 9 where A=10, B=90, C=9 The percentage formula is: A/100 x B = C as in: 10/100 x 90 = 9 Rearranging: A 100 = C B as in: 10 100 = 9 90 The percentage formula is sometimes expressed as 30% of 90=27 what is the base, rate the element and the number expressed in % and number found by multplying the base by the rate - 941602 marrienereyes marrienereyes 26.09.2017 Math Elementary School answered • expert verifie
As of 2019, startup failure rates are around 90%. 21.5% of startups fail in the first year, 30% in the second year, 50% in the fifth year, and 70% in their 10th year To find 30 percent of a number, multiply that number by 0.30. For mental arithmetic it is sometimes easier to divide the number by 100 (to calculate 1%) then multiplying by 30 (for 30%)
Systemic overfishing is only made worse by illegal catches and trade. In fact, some of the worst ocean impacts are caused by pervasive illegal fishing, which is estimated at up to 30% of catch or more for high-value species. Experts estimate illegal, unreported, and unregulated (IUU) fishing nets criminals up to $36.4 billion each year Although Bob's model assumed 90 women instead of 30 had cancer in total, it predicted Breast Cancer correctly 22.2% of the time as opposed to Hawkins Model with Precision of 0. Also, out of the 30 women that actually has Breast Cancer, Bob's Model was able to correctly recall that someone has Breast Cancer 67% of the time as opposed to. The markup is 30/100 = 30%. The MARGIN, however, is 30/130 = 23%. This is because selling the item for $130 results in a $30 profit, and 30/130 means that 23% of the money the store took in was profit. We say their margin was 23%. In fact, a 30% markup will always result in a 23% profit margin
If you want to calculate a percentage of a number in Excel, simply multiply the percentage value by the number that you want the percentage of. For example, if you want to calculate 20% of 500, multiply 20% by 500. I.e. type the following formula into any Excel cell: =20%*500. - which gives the result 100. Note that the % operator tells Excel. Colder air cannot handle as much moisture as warmer air. Temperature in relation to humidity is important, especially as we spend 90% of our time indoors. Consider for example a winters day. The outdoor air could have a 100% relative humidity at 41°F, and therefore contain 0.2 grams of water
• About 90% of children who are victims of sexual abuse know their abuser.12,13 Only 10% of sexually abused children are abused by a stranger.12 • Approximately 30% of children who are sexually abused are abused by family members.12,13 • The younger the victim, the more likely it is that the abuser is a family member. Of those molesting a. If the total is t, and the number items under consideration is n, then p= (n/t)x100. Using algebra to solve the formula for t, the number the question requires, gives the following expression: t= (n/p)x100. If 8 is equal to 80 percent of the total, inserting the numbers into the equation results in provides the t= (8/80)x100 so t=10
Use this calculator to find percentages. Just type in any box and the result will be calculated automatically. Calculator 1: Calculate the percentage of a number. For example: 30% of 300 = 90. Calculator 2: Calculate a percentage based on 2 numbers. For example: 90/300 = 30%. How much is 30% of 300 How to calculate 30% off 19 dollars or pounds. In calculating 30% of a number, sales tax, credit cards cash back bonus, interest, discounts, interest per annum, dollars, pounds, coupons,30% off, 30% of price or something, we use the formula above to find the answer. The equation for the calculation is very simple and direct If 10% of a number is 7, what is 80% of the number? Solution Note that 80% of something is 8 times 10% of the same thing. Hence if 10% of a number is 7 then 80% of the same number is given by 8 × 7 = 56 Which is the greatest? 90% of 10 6% of 1000 5% of 1400 3% of 250
32% of 100 is 32. 87.9% of 100 is 87.9. 416% of 100 is 416. For as we saw in Lesson 4, percent is an abbreviation for the Latin per centum, which means for each 100. (Per means for each.)A percent is a number of hundredths.. Example 1. A store paid $100 for a jacket. It then raised the selling price by 28% As of 1998, an estimated 17.7 million American women had been victims of attempted or completed rape. 5. Young women are especially at risk. 82% of all juvenile victims are female. 90% of adult rape victims are female. 6. Females ages 16-19 are 4 times more likely than the general population to be victims of rape, attempted rape, or sexual. 5 is 5% of 100. 12 is 12% of 100. 250 is 250% of 100. For, a percent is a number of hundredths.5 is 5 hundredths -- 5% -- of 100. That is the ratio of 5 to 100. For a percent expresses a ratio, a relationship, between two numbers FICO Scores are calculated using many different pieces of credit data in your credit report. This data is grouped into five categories: payment history (35%), amounts owed (30%), length of credit history (15%), new credit (10%) and credit mix (10%). Your FICO Scores consider both positive and negative information in your credit report 30% 0f 90% is what number? See tutors like this. Suppose you have a quantity x. 30% of 90% of x is (0.30)(0.90)x = 0.27x = 27% of x . Upvote.
You have a 30% off coupon. The cost of the item you want to buy is $249.99. How much money will you save by using the coupon? 30% of 249.99 = ? Entering these values into the percentage calculator will give you the answer of: 74.997 After rounding to two decimal places, you will save $75.00 Click to show this example in the calculator above Scientific Notation is simply a number format that includes a multiplication of 10 to the power of either a negative number, for small numbers, or to the power of a positive number, for larger numbers. This method reduces the amount of digits and especially zeros needed to write in representing a number Find percentage. Calculation of percentage is an interesting part in the world of mathematics and obvious in every math classes. The percentage converter helps you with percent increase, decrease, differences, calculation and to figure out percentage. Get the help you need on finding the percentage here 1% of 90 → 0.01 × 90. 60% of $700 → 0.6 × $700. This gives us another way to calculate the percentage of a number (or percentage of some quantity): To calculate a percentage of some number, change the percentage into a decimal, and the word of into multiplication. Example 1. Find 70% of 80 Enter the original price into our percent off calculator. For example, a TV set might originally set you back $5000. Determine the percentage discount - in our example store, everything is 75% off. The sum that stays in your pocket - your savings - is simply these two values multiplied by each other: 75% * $5000 = 0.75 * $5000 = $3750
I magine 100 people are ill with Covid-19. 90% efficacy means if only they'd had the vaccine, on average only 10 would have got ill. Vaccine efficacy is the relative reduction in the risk. By 2025, millennials will make up the majority of the workforce (75%). There are 56 million millennials in the US workforce. 21% of millennial workers have switched jobs in the last 12 months. 73% of millennials put in more than 40 hours of work per week. Millennial turnover costs the US economy $30.5 billion per year Recycling codes are used to identify the material from which an item is made, to facilitate easier recycling or other reprocessing. The presence on an item of a recycling code, a chasing arrows logo, or a resin code, is not an automatic indicator that a material is recyclable; it is an explanation of what the item is made of. Codes have been developed for batteries, biomatter/organic material. The numbers represent the percentages of importance that varying communication channels have. The belief is that 55% of communication is body language, 38% is the tone of voice, and 7% is the. *Assuming someone with poor credit (620-639) gets a 30-year fixed-rate loan at 4.03% APR compared to someone with excellent credit (760+) getting a 2.441% APR. Interest for the borrower with poor credit would total $217,478. Interest for the borrower with excellent credit would total $123,425
One hundred percent of a number is just the number itself. Two hundred percent of a number is twice that number. 100% of 50 -> 50 200% of 50 -> 2 x 50 = 100. Let's find 30 percent of 400: First change 30% to a decimal by moving the decimal point 2 places to the left. 30% = 0.30. Then multiply. 0.30 x 400 = 120. 30% of 400 is 120 Title: Percentage Worksheet Author: Maria Miller Subject: Percentage of number worksheet Keywords: Percentage, number, worksheet Created Date: 7/18/2021 4:05:23 A 30% of annual giving occurs in December. 10% of annual giving occurs on the last 3 days of the year. 77% believe everyone can make a difference by supporting causes. 4.5 is the average number of charities each person supports. 64% of donations are made by women. 69% of the population gives 74. 11 is 25% of what number? 75. 37 is 4% of what number? 76. 90 is 80% of what number? 77. 8 is 2% of what number? 78. On a 120-question test, a student got 84 correct answers. What percent of the problems did the student work correctly? 79. An engineering student answered 81 questions correctly on a 90-question test. Wha
According to data from the Bureau of Labor Statistics, as reported by Fundera, approximately 20 percent of small businesses fail within the first year. By the end of the second year, 30 percent of. What is eGFR? eGFR - Estimated glomerular filtration rate is the best test to measure your level of kidney function and determine your stage of kidney disease. Your doctor can calculate it from the results of your blood creatinine test, your age, body size and gender. Your GFR tells your doctor your stage of kidney disease and helps the doctor plan your treatment This does not mean: There's a 30% chance it will rain and a 70% chance it won't. Three out of 10 times when the weather is similar, it will rain. Precipitation will fall 30% of the day (or night) Thirty percent of the forecast area will experience rain, snow, or storms. Rather, the correct interpretation would be: there is a 30% chance that 0. 2) Solution: Box 1: Enter your answer as an equation. Example: y=3x^2+1, 2+x+y=3. Be sure your variables match those in the question. x 136 = 38 100 or 38 100 = x 136 x 136 = 38 100 or 38 100 = x 136. Box 2: Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4
Instagram content statistics. The Instagram algorithm has tripped up marketers in the past and that trend looks to continue in 2021. Recent Instagram statistics tell us that organic engagement has fallen from 2020, which was at 1.60%. For reference, research from RivalIQ puts the average engagement rate at 1.22% Based on data collected by CDC from states and territories for year 2017: Over 98% of U.S. newborns were screened for hearing loss. About 6,500 U.S. infants born in 2017 were identified early with a permanent hearing loss. The prevalence of hearing loss in 2017 was 1.7 per 1,000 babies screened for hearing loss Q5. At IIM Bangalore, 60% of the students are boys and the rest are girls. Further 15% of the boys and 7.5% of the girls are getting a fee waiver. If the number of those getting a fee waiver is 90, find the total number of students getting 50% concession if it is given that 50% of those not getting a fee waiver are eligible to get half fee.
Younger People Are at the Highest Risk of Sexual Violence. Ages 12-34 are the highest risk years for rape and sexual assault. 3 Those age 65 and older are 92% less likely than 12-24 year olds to be a victim of rape or sexual assault, and 83% less likely than 25-49 year olds. 4 Read more statistics about about child sexual abuse You can put this solution on YOUR website! When 80% of a number is added to the number, the result is 252. What is the number? 0.8x + x = 252 1.8x=252 x=140 Cheers, Stan H WATCH: Why elephant numbers have fallen so low. Between 2006-2015, around 111,000 African elephants were lost from the wild, mainly due to poaching, and between 2007-2014 30% of Africa's savannah. If your school does not list your percentile, it is easy to figure out. Divide your class rank by the number of students in your grade, multiply by 100, then subtract that number from 100. For example, if there are 600 students in your grade and you are ranked 120th, then you are in the 80th percentile because (120/600)*100=20, and 100-20=80
I have a column that lists bunch of numbers. How can I select the average of top 30% of the values in one column: 'Values' 10 9 8 7 6 5 4 3 2 1 so, the top 30% is '10, 9, 8' and the average is (10+9+8)/3 = If two numbers are respectively 30% and 40% more than a third number, what percent is the first of the second? A. 92 6 7 %. B Honestly, I'd have estimated this in the 20-30 percent range, so it surprised me to see that, from Jumpshot's data, all Google properties earned only 11.8% of clicks from distinct searches (only 8.4% across all searches). That's still significant, of course, and certainly bigger than it was 5 years ago, but given that we know Google's search. Assignment #2 Q1 (Weight 30%): Write a python program that: . Reads the number of rows and columns from the user. • Reads the different elements of the matrix from the user. • Calls a function to displays the matrix. • Calls a function to return the sum of all elements in the matrix. The quality of the code is important
A number decreased by 30% gives 84. The number is (a) 90 (b) 110 (c) 120 (d) 135. Answer: (c) 120 Let. 2.5 million, nearly 90%, are treated and released from an emergency department. TBI is a contributing factor to a third (30%) of all injury-related deaths in the United States. 1. Every day, 153 people in the United States die from injuries that include TBI. 1. Most TBIs that occur each year are mild, commonly called concussions. 2 A number multiplied by 6 and then reduced by 3 gives 69, the number is_____ 3). The number of boys is 3/2 the number . Algebra 1. Two fewer than a number doubled is the same as the number decreased by 38. Find the number. If n is the number, which equation could be used to solve for the number A growing number of consumers (37%, up from 30% in 2017), are willing to pay a fee for access to enhanced loyalty program benefits 95% of loyalty program members want to engage with brands through a mix of new, emerging, and growing tech, including augmented reality, virtual reality, card-on-file and more ( Bond 50 stats that show the importance of online reviews. 1. 92% of consumers now read online reviews vs. 88% in 2014 tweet. 2. 40% of consumers form an opinion by reading just one to three reviews vs. 29% in 2014 tweet. 3.Star rating is the number one factor used by consumers to judge a business tweet. 4. 44% say a review must be written within one month to be relevant.This highlights the.
The share of U.S. children living with an unmarried parent has more than doubled since 1968, jumping from 13% to 32% in 2017. That trend has been accompanied by a drop in the share of children living with two married parents, down from 85% in 1968 to 65%. Some 3% of children are not living with any parents, according to a new Pew Research. According to estimates from Scandinavian research centre Sintef, 90% of all the data the human race has ever produced has been generated in the past two years. That explosion is due to the rise of. Washington, DC, August 29, 2019 - A recent Ipsos poll reveals that more Americans have tattoos today than in early 2012.Three in ten (30%) of Americans have at least one tattoo, an increase from 21% in 2012. The vast majority of those with at least one tattoo (92%) say they are happy with it, and forty-six percent of respondents have had at least one tattoo for more than ten years The proportion of American adults with high-speed broadband service at home increased rapidly between 2000 and 2010. In recent years, however, broadband adoption growth has been much more sporadic. Today, roughly three-quarters of American adults have broadband internet service at home. Chart. Data
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CC-MAIN-2022-27
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https://iqenergy.org.ua/answers/3027632-the-tub-of-a-washing-machine-goes-into-its-spin-cycle-starting-from-rest-and-gaining-angular-speed-steadily-for
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math
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As we know that initial angular speed of the tub was zero and then it increases uniformly to 4 rev/s in t = 6.00 s
so we will have
Now when the tub will comes to rest uniformly after opening the lid in time interval of t = 15 s
then we have
Now total angular displacement of the tub is given as
so number of revolutions is given as
Which of the following is sometimes called electrical potential? A. ohms
A pendulum of length l is suspended from the ceiling of an elevator. When the elevator is at rest the period of the pendulum is t. How does the period of the pendulum change when the elevator moves downward with constant acceleration?
The period of the pendulum will increase.
The period of the pendulum will increase since g efective will decrease.
Then g efective is given by
g efective = g - a < g
If ⇒ T = 2π√(L/g)
and T' = 2π√(L/(g - a))
then T < T'
When the elevator accelerates downward, the hanging mass feels “lighter” – this means that the effective value of g has decreased due to the acceleration of the elevator. Since the period depends inversely on g, and the effective value of g decreased, then the period of the pendulum will increase .
(i.e., its frequency will decrease and it will swing slower)
The distance between the centers of the wheels of a motorcycle is 146 cm. The center of mass ofthe motorcycle, including the rider, is 78.5 cm above the ground and halfway between the wheels.Assume the mass of each wheel is small compared to the body of the motorcycle. The enginedrives the rear wheel only. What horizontal acceleration of the motorcycle will make the frontwheel rise off the ground
Here the moment created by the wheels and the moment created by the center of gravity will balance each other.
h = Height of the center of mass = 78.5 cm
d = Distance from back wheel to the center of mass =
g = Acceleration due to gravity = 9.81 m/s²
a = Horizontal acceleration
The equation is of the form
The horizontal acceleration of the motorcycle that will make the front wheel rise off the ground is 9.12267515924 m/s²
Determine the moment about the origin O of the force F=(13)i + (-12)j + 14(k) that acts at a point A. Assume that the position vector of A is r=2i+3j-4k. The moment about the origin of the force F is ()i+()j+()k
F=(13)i + (-12)j + 14(k)
position vector of vector A = r = 2 i + 3 j - 4 k
moment of force about origin
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CC-MAIN-2024-18
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http://tochnog.sourceforge.net/tnu/node282.html
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math
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Stiffnesses for contact springs. The force in normal direction of the contact spring is determined from where is the normal displacement difference of the two nodes (that is, the displacement of the second node in normal direction minus the displacement of the first node in normal direction). The first tangential force of the contact spring is determined from where is the tangential displacement difference of the two nodes in the first tangential direction; the same is done for the second tangential force. The total tangential force cannot exceed with friction coefficient; then frictional slip occurs and the total tangential force is set to To model continuing stick between two bodies just put the friction coefficient very high.
In 1D the parameters and will not be used (but should be specified as dummies nevertheless).
The index specifies the element_group, see element_group.
See also group_contactspring_friction and group_contactspring_friction_automatic.
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CC-MAIN-2018-05
| 971 | 4 |
https://urbandigital.me/comparison/the-art-of-problem-solving-pre-algebra.php
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math
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The text is written to challenge students at a much deeper level than a traditional middle school prealgebra course, and is used for both our Prealgebra 1 and Prealgebra 2 online courses.
Topics covered in this book include a review of basic algebra topics, complex numbers, quadratics and conic sections, polynomials, multivariable expressions, sequences and series, identities, inequalities, exponents and logarithms, piecewise-defined functions, functional equations, and much more. Decimals : Powers of 10, multiplication, division, and converting between fractions and decimals.
The text then includes solutions to these problems, through which algebraic techniques are taught.
This book can serve as a complete geometry course, and is ideal for students who have mastered basic algebra, such as solving linear equations. In addition to the instructional material, the book contains well over problems. Topics covered in this book include a review of basic algebra topics, complex numbers, quadratics and conic sections, polynomials, multivariable expressions, sequences and series, identities, inequalities, exponents and logarithms, piecewise-defined functions, functional equations, and much more. Topics covered in the book include linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and much more!
Topics covered in the book include the properties of arithmetic, exponents, primes and divisors, fractions, equations and inequalities, decimals, ratios and proportions, unit conversions and rates, percents, square roots, basic geometry angles, perimeter, area, triangles, and quadrilaterals , statistics, counting and probability, and more! The text is structured to inspire the reader to explore and develop new ideas. Important facts and powerful problem solving approaches are highlighted throughout the text.
Each section starts with problems, giving the student a chance to solve them without help before proceeding. Need help finding a book? Addition : Sums, place value strategies, counting up, adding and taking away, doubles, making 10's and 's, friendly sums.
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https://math.answers.com/Q/What_is_the_sign_of_an_integer_of_the_product_of_a_positive_integer_and_a_negative_integer
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math
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No. The sum of a positive integer and a negative integer has the same sign as the larger integer.
If you divide a positive number (it doesn't really matter if it is an integer or a fraction) by a negative number, the result is negative.
All of the numbers that are in Positive and Negative sign are Integers , whether they reach 1 million if it has Positive or Negative sign it is considered as Integer .
When multiplying integers, multiplying by the same sign will always produce a positive integer. Such as a negative times a negative equals a positive. If the signs are different then the product will be a negative.
The absolute value of an integer is the integer with a positive sign.
If the absolute value of the positive integer is greater than the absolute value of the negative integer, then the sum of the two will be positive.If the absolute value of the positive integer is less than the absolute value of the negative integer, then the sum will be negative.If the absolute values of the two integers are the same then the sum will be zero, which has neither a negative nor a positive sign.
Negative because product of 47 negative numbers is negative and product of three positive number is Positive , so negative*positive = Negative.
(The product of 33 negative numbers) x (2 positive numbers) = (negative sign) x (positive sign) = negative sign
The product of three positive numbers is positive. The product of three negative numbers is negative.
The product of a positive and negative number is always negative The sum of a positive and negative number depends on which one is larger; subtract the two numbers and take the sign of the larger
An integer is just a whole number, excluding zero. Any positive integer will always have an opposite just by placing a negative sign in front of the positive integer. You can also say that any negative whole number is an integer.
The absolute value of a positive integer is positive. When taking the absolute value of any integer, one is essentially removing the sign (whether positive or negative), always leaving the remaining number positive.
When the absolute value of the negative number is higher than the positive number.
If the number is zero, then it is neither positive nor negative.For all other integers: If there is a minus sign, "-", before the integer, then it is negative. If there is a plus sign, "+", before the integer or no sign at all, then it is positive.
negative x negative = positive
Signed integer is any integer that carries negative sign while unsigned integer is any integer that carries positive sign
The digits of the number without any positive or negative sign. It will look just like the positive version of the integer.
If there are an odd number of numbers with a negative sign then the sign of the product is negative. Otherwise it is positive.
You get another integer that will take the sign from the larger of the two integers that were combined.
The product will be positive in this case.
The answer will always be Negative in Sign. If they are the same sign, it will always be Positive in Sign.
Negative. Positives always give positive sign, two of the negatives cancel out, leave one negative sign.
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https://studycommoncore.com/standards/math/grade-5/5-nf-4b
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math
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Common Core MATH 5 NF 4b Grade 5 Number And Operations Fractions Apply And Extend Previous Understandings Of Multiplication And Division To Multiply And Divide FractionsBack
Number And Operations - Fractions: Apply And Extend Previous Understandings Of Multiplication And Division To Multiply And Divide Fractions.
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Sorry! No worksheets found for this topic.
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http://hendriksescience.weebly.com/012918-013018.html
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math
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Take ACT Practice 2
Enter Answers to Test 2 Here
1. Identify Independent and Dependent Variables in all Experiments.
2. take note of Differences between experiments.
1. Correct Homework questions
Use the images below to answer the following questions
1. What are the sample rates for each of the 3 waves?
2. What wave has the most refined bit depth?
3. What wave has the most course bit depth?
4. Define bit rate by using the information on the images.
5. Define sample rate by using information in the image
On a Half piece of paper to be turned in, answer the following questions as they relate to the DAC graph below:
1. The red and blue plots on the graph below represent two versions of the same sound wave. Which is the original and which is the copied wave?
2. Why on earth would the wave need to be copied and reshaped?
3. What is the time delay between the original and copied wave? (n stands for nanoseconds, which is number x10-9)
4. What is the sampling period for the digital wave below?
5. What is the sampling frequency (rate) for the digital wave below? How does this compare to typical audio files?
6. What is the approximate bit depth of the digital wave? How does this compare to 8 bit sound?
Work on homework.
Review for Wave TEST
Wave Test Study Guide
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http://www.braingle.com/brainteasers/teaser.php?id=25911&op=0&comm=1
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math
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Slow and Stupid Wins the Race?
Math brain teasers require computations to solve.
Hare and Tortoise had to take a make-up class in math over the summer, a two-month, self-paced course with a test at the end of each of 12 chapters. The course requires a 70% grade to pass.
In the first month, they both had difficulties with the concepts. Hare averaged 60% on his exams; Tortoise averaged 50%. Owl, the supervisor, spent three days of the next week helping them with their difficulties.
It worked. In the second month, Hare averaged 90% on his exams; Tortoise averaged 80%.
However, Tortoise got a passing grade of 75% in the class; Hare failed with 65%. When Hare protested to Owl that he'd outscored Tortoise in both months, Owl made Hare do the math on the board -- and sign up for another make-up class, after school in the fall.
How did Tortoise pass while Hare flunked?
HintThey worked at different speeds.
In the first month, Hare took 10 chapter tests; Tortoise took only 2. In the second month, Hare took his remaining 2 exams and Tortoise took the 10 he needed to finish.
Hare's average = (10*60 + 2*90) / 12 = 780/12 = 65
Tortoise's average = (2*50 + 10*80)/12 = 900/12 = 75
Mathematicians know this as Simpson's paradox.
Sep 14, 2005
Sep 14, 2005
Sep 15, 2005
|Very challenging teaser....awesome! |
Sep 20, 2005
|Good! I didn't get it |
Sep 23, 2005
|Since you did not ask for the math behind the scores, I didn't bother to do it. It was obvious that Hare took most of his tests in the first month, while tortoise took his in the second month. Because of my approach, I questioned why this question was placed in the math category. If you had asked for the arithmatic behind the numbers, the math category would have seemed appropriate. Still fun. |
Oct 03, 2005
|If my math had been better, I'd have got it faster!!|
Great MATH teaser!!
Oct 10, 2005
Oct 23, 2005
| Fun working this one out. |
Nov 13, 2005
|Great one, although I didn't get it right next time hopefully |
Apr 09, 2006
|I loved it!|
May 10, 2006
Jun 19, 2007
|the question is "How did Tortoise pass while Hare flunked?", not "how to get their average grades after the 1st month or the 2nd month or for the whole class".|
the answer is already in the hint: They worked at different speeds!
Jan 31, 2008
|Excellent quiz. |
I guessed at the answer, but then did the MATH to show that I was right. A math test requires you to show your work, not just the answer!
Jun 04, 2013
|It does not say that all of the chapter tests were worth the same value. The principle is simply that there was unequal weighting between the 2 sessions. This might have been because of a different number of tests or it might have been because the tests were out of different scores. In either case it is not statistically correct to simply average diversely weighted scores to come up with a summary mean.|
Jun 04, 2013
|To be clear I am not disputing the answer, just the fact that the implication that because Hare was higher on 2 separate tasks that he would be higher overall.|
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https://deepai.org/publication/sommerfeld-type-integrals-for-discrete-diffraction-problems
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math
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In the beginning of 20th century Sommerfeld introduced closed integral solution for the problem of diffraction by a half-plane [Sommerfeld1954]. It was done in a very elegant way with the help of reflection method. Namely, he reduced the half-plane problem to the problem of plane wave propagation on two-sheeted surface. Then, using plane wave decomposition integral he solved the problem. This integral with particular contour of integration was named after Sommerfeld. Later Sommerfeld integral approach was applied to a number of problems such as problem of diffraction by a strip [Shanin2003a], by a wedge [Babich2008] and some others [Luneburg1997, Hannay2003].
Nowadays with the growth of computational power problems on discrete grids draw more attention. Recently, several discrete diffraction problems were solved rigorously using Wiener-Hopf approach [Sharma2015a, Sharma2015b, Sharma2015c]. In the current work we want to apply Sommerfeld integral approach to some of them. We show that in the discrete case Sommerfeld integral is essentially an integral on torus from algebraic function (elliptic integral) and we derive such integrals for the following problems:
The problem for Green’s function on a plane,
The problem of diffraction by a half-plane,
The problem of diffraction by a right-angled wedge.
2 Discrete Green’s function on a plane
2.1 Problem formulation
Consider the Green’s function for a simplest stencil discrete 2D Helmholtz equation. Namely, let function , , obey the equation
The wavenumber parameter is close to positive real, but has a small positive imaginary part mimicking attenuation in the medium. The radiation condition imposed on is that it should decay exponentially as .
2.2 Preliminary step. Reducing the number of computations of the integral
Our aim is to tabulate function for some set of values . It is clear that
Thus, one should tabulate only for non-negative .
Let it be necessary to tabulate all with
A naive approach requires computations of the integral. However, here we show that one can compute integrals. Namely, we will compute the integrals only for , and all other values find by using “cheap” recursive relations.
Compute the values of row by row. Each row is a set of values with , , , i. e. the rows are in fact diagonals.
Let all values with be already computed, and it is necessary to compute the values with . Find by integration. Then use (1) rewritten as a recursive relation:
Note that all values in the right have the sum of indices , thus they are computed on the previous steps. The left-hand side is a recursive relation for .
2.3 Double integral representation
The result is the following representation of the field :
Introduce the variables
Also introduce the function
The integral (5) can be rewritten as
where contour is the unit circle in the -plane passed in the positive direction anti-clockwise.
2.4 Single integral representation
The integral (9) can be taken with respect to one of the variables by the residue integration. There are four cases, possibly intersecting.
Consider the integral (9). Fix and study the integral with respect to . The -plane is shown in Fig. 1. One can see that there are four possible singular points in this plane. Two of them are the roots of the dispersion equation
considered with respect to . The roots are
Beside (11), there maybe singularities at two other points: and (note that is a certain point of ). However, the presence of singularities at these points depends on the value of . For example, if then the integrand is regular at .
Thus, the only singularity of the integrand inside the circle is . Apply the residue method. The result is
Thus, (13) can be rewritten as
The same analysis can be made for the singular points in the -plane for a fixed . This anayisis shows that there may exist a singularity at , but the behavior at is regular. This means that the integrand has no branching at , and the integrand decays not slower than . For such an integrand one can apply the residue theorem to the domain . The result is
This case can be considered similarly to . The representation for the field is
The field is
2.5 A recursive relation for
Let be and . Rewrite the representation (15) in the form
Using the proof of Legendre’s theorem for the Abelian integrals [Bateman1955], derive a recursive formula for . Introduce the constants as follows:
Using these constants one can write
Then note that
Substituting this identity into (20) and taking into account that contour of integration is closed, get
2.6 Field representation by integration on a manifold. Plane wave decomposition
Consider and being complex variables. Let be , , where is a compactified complex plane, that is a Riemann sphere.
Each point thus belongs to . Let us describe the set of points such that equation (10) is valid. Obviously, this is an analytic manifold of complex dimension 1 or of real dimension 2. This manifold will be referred to as .
Consider defined by (12). Now consider it as a double-valued function, thanks to the presence of the square root in it. Let us study the Riemann surface of this function. Topologically, there is no difference between and the Riemann surface of .
Function has four branch points. They are the points where the argument of the square root in (12) is equal to zero, i. e.
The values , , , possess the following property that can be checked directly. For
Exactly two of these branch points are located inside the circle . One can check that the branch points are the points at which .
The scheme of the Riemann surface is shown in Fig. 2. The branch points are connected by cuts shown by bold curves. For definiteness, the the branch cuts are conducted along the lines at which
The sides of the cuts labeled by equal Roman number should be connected with each other.
Topologically, is a torus (i. e. it has genus equal to 1). This can be eacily understood, since is obtained by taking two spheres, making two cuts, and connecting their shores.
One of the sheets drawn in Fig. 2 is called physical, and the other is unphysical (the naming is meaningless) . The physical sheet is the one on which for . Respectively, on the unphysical sheet for . Note that and cannot be equal to 1 on , since is not real.
We find useful to mark four “infinity points” belonging to :
Note that belongs to . Points Inf 1 and Inf 4 belong to the physical sheet, while points Inf 2 and Inf 3 belong to the unphysical sheet.
The notations of infinity points and branch points on are shown in Fig. 3.
The statement that is an analytic manifold means that in each (small enough) proximity of any point of one can introduce a complex local variable , such that all transhormation matrices between the neighboring local variables are biholomorphic. It is clear that such local variables can be: for all points except four branch points and two infinities Inf 3 and Inf 4; for the branch points; for the infinities Inf 3 and Inf 4.
An analytic 1-form can be defined in the manifold [Gurvitz1968] by introducing a formal expression , where is a local variable (discussed above) in some proximity, and is an analytic function in this proximity. In neighboring proximities the representations can be different (say, and ), but they should match in an obvious way:
The 1-form can be analytic/meromorphic if the functions are analytic/holomorphic. In the same sense the form can have zero or a pole of some order.
Analyticity of a 1-form is an important property since one
One can see that the form
is analytic everywhere on . Let us prove this. The statement is trivial everywhere except the branch points and the infinities. Consider the infinities. At the points Inf 1 and Inf 2 it is easy to show that as , and the denominator is non-zero. At the points Inf 3 and Inf 4 one can show that as , thus . A change to the variable shows that the form is regular.
Finally, consider the branch points (25)–(28). As it has been mentioned, one can take as a local variable at these points. An important observation is that due to the theorem about an implicit function,
everywhere on . Thus,
At the branch points the denominator of the right-hand side of (32) is not zero, so the form is regular.
The representation (15) can be considered as a contour integral of the form
along some contour drawn directly on . The contour is, indeed, shown in Fig. 2. This statement is quite trivial.
What is less trivial, is that three other representations, (16), (17), (18) can be represented as the contour integrals of the same form on , but taken along some other contours. Namely, the contours of integration for the representations (16), (17), (18), are shown in Fig. 4. They are denoted by , , , respectively. The contour for (15) is denoted by for uniformity.
Note that the form is, generally, not analytic on . Depending on and , it can have poles at the infinity points. The list of conditions of regularity for the infinity points is as follows:
The domains of regularities at infinities are shown in Fig. 5.
Note that the contours , , , can be deformed into each other. As we mentioned, is a torus. Topologically, the relative positions of the contours and the infinity points are shown in Fig. 6
One can see that carrying the contours in the direction labeled by the red arrows corresponds to moving the observation point in the -plane in the clockwork direction. The representations are converted into each other, and every time there is a region where at least two representations are valid simultaneously.
2.7 Sommerfeld integral for Green’s function problem
Sommerfeld integral for this problem is formally a plane wave integral (34) with contour of integration that does not cross the line of propagating waves (locus of points ). After a simple analysis of one can obtain the result shown in figure 7.
One can notice that this curve is topologically equivalent to the one of the canonical sections of . To prove it let us show that domains and are simply connected. For simplicity consider the case (obviously, the topology of domains should be the same for any ). In this case domain covers physical sheet and domain covers unphysical sheet. Thus, resulting domains are linearly connected, and any closed contour lying in any of domains can be collapsed through infinity point.
Contour for Sommerfeld integral consists of two closed non-trivial contours lying at different sides of curve of propagating waves.An example of such a contour is shown in figure 8.
In figure 9 we plot these contours on torus .
Finally, Sommerfeld integral takes form:
Obviously, contour is equivalent to the since there is no poles lying on the curve of propagating waves. Nevertheless, representation (36) seems to be more convenient when the problem for an incident plane wave is considered.
3 Diffraction by a Dirichlet half-plane
3.1 Problem formulation
Let the discrete Helmholtz equation
be satisfied everywhere except line . On this line the following boundary condition should be satisfied:
where is an incident plane wave:
Here an angle of incidence. In order to satisfy the discrete Helmholtz equation, incident wave should satisfy the dispersion equation:
where we introduced a notation:
Introduce total field as a sum of incident and scattered field:
Also the scattered field should satisfy the radiation condition.
3.2 Formulation on a branched surface
Consider a branched surface of continuous variables . For this, parametrize the points by the relations
Thus, the points become defined on a surface with two sheets reminding the Riemann surface of the function .
Define an integer lattice on the branched surface. There are two points having coordinates for any pair except . Denote these points by , where as an index labelling the sheet somehow (say, by separating the surface into sheets by making an appropriate cut). The pair will be called an affix of the point.
We assume that there is a wave field defined on the points of the branched surface.
Each point except has exactly four stencil neighbors having affixes , , , . We say that equation (37) is valid on the branched discrete plane at some point if it is valid for the value of at and at four its neighbours.
3.3 Sommerfeld integral for half-plane problem
First let us first construct Sommerefeld integral for a plane wave on a plane. We search for
where is some algebraic function on the torus . It is well known from the theory of elliptic functions [Bateman1955] that non-trivial function on the torus should have at least two poles of the first order or one pole of the second order. For definiteness, let us suppose that function has one simple pole corresponding to the incident wave , and the other simple pole corresponding to an arbitrary point . It can be checked directly that such function has the following form
Here constants and satisfy the following system of linear equations:
Let us choose constant in a way that in point the residue of integral (44) will be equal to . We obtain
Let us construct a plane wave solution on a branched surface. Following Sommerfeld ideas we need to be build a function that covers torus twice and has a unity pole corresponding to a plane wave. There are several obvious candidates that cover twice, such as
Then, multiplying with function (45) we obtain the function with desired properties. Thus on a branched surface Sommerfeld integral has form:
To choose which function should be used one need to check the validity of radiation conditions. It can be showed after simple computation that only the integral with
Finally, solution for the half-plane problem can be obtained using reflection principle:
3.4 Wiener-Hopf solution
Let us find the solution of the half-plane diffraction problem using the Wiener-Hopf approach. First, let us symmetrize the problem. Namely, represent the incident field (39) as a sum:
Then, study the equation (37) separately for the symmetrical and anti-symmetrical part of the field. One can check directly that anti-symmetrical problem is trivial, i.e.:
Thus the solution of symmetrical problem coincide with the solution of (37), i.e.
Without loss of generality we can suppose that .
Introduce direct and inverse bilateral -transform as follows:
where is a unit circle passing in a counterclockwise direction. To obtain functional equation let us apply -transform to boundary condition (38). We have
where is a unilateral -transform of :
and is some unknown function analytical inside the unit circle. Function is analytical outside the unit circle [Sharma2015b]. Equation (60) cannot have unique solution, since it also involve unknown function that is analytical in some ring. To introduce a second functional equation let us study a combination
One can check directly that
where were introduced in (29), and
The equation can be easily factorized. The solution is as follows:
The scattered field is given by the following integral
4 Diffracton by a right-angled wedge
4.1 Problem formulation
On the boundary of this domain the following conditions should be satisfied:
where is an incident plane wave (39). Also the scattered field should satisfy the radiation condition.
Using reflection method this problem can be reduced to the problem of wave propagation on three-sheeted surface [Sommerfeld1954]. It can be checked directly that total field on three sheeted surface is related to the total field of original problem by the following formula:
4.2 Sommerfeld integral on three-sheeted surface
We will search integral in the form (44) as in previous sections. Here we need to construct function covering torus three times and having a pole corresponding to the incident wave with zero residue. Unfortunately, there are no obvious candidates like it is for two-branched surface. Let us study from the topological point of view. The Riemann diagram of torus is shown in figure 11.
It can be noticed that function which covers three times should have Riemann diagram that is shown in figure 12.
It can be easily proved that function having Riemann surface has the following structure:
where , some rational functions. Thus, using (71) we can build Sommerfeld integral for three-sheeted surface with two unknown rational functions.
One can construct (71) by studying polynomial
Namely, suppose that the roots of this polynomial define function , and this function has Riemann surface . Thus, there are exactly four points in which has exactly three roots of order two, i.e. it can be represented as:
where are unknown parameters. These parameters should be determined from the following system of equations:
Equating coefficients at the same powers of we will obtain system of equation for unknown parameters , . Solving this system one can obtain exact expression for function (71).
In this paper we applied Sommerfeld integral approach to several diffraction problem for discrete Helmholtz equation (1). We showed that the field is represented as integral on a manifold. This manifold is torus, and corresponding integrals are Abelian integrals. For point source problem we proposed recursive procedure of field calculations which reduces integral computation to integral computation. For half-plane problem we constructed solution using Sommerfeld integral and showed it is equivalent to the Wiener—Hopf Solution. For the problem of diffraction by a right-angled wedge we showed that the problem can be reduced to the solution of nonlinear equation.
Appendix A. Abelian integrals
Indeed, are Abelian differentials on . The form is an Abelian differential of the first kind, while all other are Abelian differentials of the third kind.
Moreover, since is a torus, the Abelian integrals in this case are elliptic functions.
The classical framework of study of the Abelian integrals is as follows. The surface is cut by several cuts (by two cuts in our case) such that the surface becomes mapped onto a polygon with an edge. These cuts are and (the latter is shown in Fig. 13).
are cyclic periods of an Abelian differential .
Note that is a solution of the problem based on equation (1) but with another radiation condition (one should take with a negative imaginary part).
All theorems related to Abelian and elliptic integrals can be applied to and . This properties can be found in [Bateman1955].
Appendix B. Sommerfeld integral as an integral on the dispersion manifold
Let us build an analogy between Sommerefeld integral for continuous problem and Somerfeld integral for discrete problem.
let the Helmholtz equation
be satisfied everywhere except half-plane
where Dirichlet boundary condition is satisfied:
Here is an incident wave:
Plane wave should satisfy the following dispersion equation:
Also, radiation and Meixner conditions should be satisfied. Let us introduce a plane wave decomposition. Following the idea of field representation by integration on a manifold one should first study (79). This manifold is a Riemann sphere with two punctured points. In this points field has exponential growth. It is more natural to study this manifold as a tube (see figure 14).
Also, equation (79) can be written in a parametrical form:
Thus, this tube can be mapped to the strip:
Thus, the polar coordinates
should be introduced and the plane wave decomposition should be some integral on . It was shown by Sommerfeld that it has the following form:
Contours of integration are showed in the figure 15.
On -branch surface function should be periodical with respect to with period , and should have a pole with unity residue corresponding to the incident wave. For a plane wave on a plane we have:
For a plane wave on 2-sheeted surface we have:
So, the following analogies between discrete and continuous solutions can be seen:
The field is represented as an integral on some manifold defined by dispersion equation. In the discrete case this manifold is torus, and in contentious case it is tube.
There are two contours of integration in Sommerefeld integral. Both contours do not cross the line of propagating waves (the line of real wavenumbers).
To obtain the solution for half-plane problem one need to construct a function that covers the manifold twice.
Appenidx C. Sliding plane wave decomposition
For the discrete problem we can mimic this sliding. Namely, consider torus . There are 4 contours along which the integration can be held, and 4 possible “infinites” at which there can be singularities. They are shown in Fig. 4. The contours are , , , and .
A simpler scheme of the same surface with contours and infinities is shown in Fig. 6. A torus is shown as a torus in a usual sense. The relative position of the contours and the infinity points is drawn.
Consider an integral
Let be some 1-form on having poles only at the four infinities. Let the order of the poles there is for some integer . According to the consideration made above, the form is regular at the infinites under the following conditions:
Take the observation point such that . Move this point about the origin in the direction of the red arrow in the figure. One can see that one can slide the contour sequentially in the order
(In fact, here we have in mind that, for example, in order to slide contour to we should ensure that the form is analytic at the point Inf 1.)
Corresponding contours provide a necessary decay of the solution in corresponding sectors of the plane .
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http://blog.mcoyle.com/2008/09/finished-for-now.html?showComment=1220430540000
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math
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"Self Posed in front of Pink"
Collage on paper
So I decided I'm finished with this for now. It looks slightly like me--I've realized that I've always had trouble doing self-portraits. It's so much easier to create works of other people. Perhaps this is because I assume I know what I look like, and therefore I have trouble getting the details down right. I might return to this at a later time, but for now, I've decided I'm done.
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http://irpapernmrg.frenchiedavis.info/lab-2-doppler-effect.html
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math
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Lab 2 doppler effect
Answer to escience lab 21 experiment 2: doppler effect based on your results from the cork and water experiment, please answer the. Free practice questions for ap physics 1 - doppler effect includes full solutions and score reporting.
Lab sheet select the boy icon describe at least two ways that your understanding of the doppler effect could be applied to everyday life. Nova | stellar velocity: the doppler effect this interactive activity from nova provides an explanation of how the doppler effect the doppler effect. Free practice questions for ap physics 2 - doppler effect includes full solutions and score reporting. Name _____ date _____ period _____ lab: the doppler effect physics: chapter part 2: in the next part of the lab what is the definition of the doppler effect 2.
The doppler effect for sound explained with film clips, animations and multimedia. See how astronomers use the doppler effect and redshift to determine the speed and direction of stellar objects. Using lab notebooks share the photo the doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent.
View lab report - lab report 3 doppler effect from phys 1p91 at brock university table of contents: 1 introduction 2 background and theory 3 equipment and protocol 4. The relativistic doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer.
Lab doppler effect: source moving printer friendly version: in each case, the location of listener #2 is at right angles to direction of the source's velocity. 9p2314 describe the doppler effect changes that occur in an observed sound as a smoot’s group at berkeley national laboratory doppler effect lesson. As stated in the introduction, one can observe the doppler effect in a number of settings if a person is standing by the side of a road and a car approaches at a significant rate of speed, the frequency of the sound waves grows until the car passes the observer, then the frequency suddenly drops.
Lab materials: • computer with internet access physical science workshop: astronomy applications of light & color s sallmen activity: doppler effect 2 activity:.
Lab coat - child's size medium rating: 0% $3000 add to wish list add to the doppler effect occurs when an observer hears a sound from a moving source. Doppler effect lesson plans and worksheets from thousands of (sound-2): lab on the doppler effect students evaluate the doppler effect in this doppler.
Another application of the doppler effect not identified by their research module137pdf page 2 of 19 rubric scoring elements not yet approaches expectations meets. Doppler effect formula for observed frequency doppler effect formula when source is moving away when the source and the wave move at the same velocity. The doppler effect is the perceived change in frequency of sound emitted by a source moving relative to the observerthe effect was first noted by lab procedure:. Activity: determining red-shift in a receding star instructional objectives time needed for activity target grade level materials part 2: doppler effect.Get file
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https://www.jiskha.com/display.cgi?id=1338993445
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math
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posted by Christina Y. .
An astronaut of mass 82.2 kg is 31.2 m far out in space, with both the space ship and the astronaut at rest with respect to each other. Without a thruster, the only way to return to the ship is to throw his 0.502 kg wrench directly away from the ship. If he throws the wrench with a speed of 20.0 m/s, how many seconds does it take him to reach the ship?
0 =m1•v1 – m2•v2,
v1 =m2•v1/m1 =0.502•20/82.2 =0.122 m/s.
t = s/v1 =31.2/0.122 = 255.7 s =4.26 min
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https://www.physicsforums.com/threads/calculate-the-angular-acceleration-and-angular-velocity.525003/
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math
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Calculate the angular acceleration and angular velocity of a 2kg object rotating in a
circle of 1.5m radius in a time of 3s.
The Attempt at a Solution
Now i dont know how to fully work this out, not sure how to apply the forumla.
3 seconds for one complete roatation
360 / 3 = 120 degrees
120 degrees per second
9.42 / 3 = 3.14ms-1
I know this completes 120 degrees going a distance of 3.14m in a second.
The question is how do i go about completing this question?
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https://www.spanishdict.com/answers/117325/how-do-you-say-do-you-remember-us
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math
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how do you say "do you remember us?"
i need to know how to say this using direct objects...
i also need help with he sends us he loves me i remember you
You probably need to tidy up your request a bit Forum rules require proper spelling and punctuation. Take a peek here at the rules.
Your first question might be asked like this: Recuerdanos?
For a bit of help with Personal Pronouns,: you can check this out
tu te recuerdas de nosotros? o usted se acuerda de nosotros?
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https://jmlr.csail.mit.edu/papers/v19/15-498.html
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math
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Theoretical Analysis of Cross-Validation for Estimating the Risk of the $k$-Nearest Neighbor Classifier
Alain Celisse, Tristan Mary-Huard; 19(58):1−54, 2018.
The present work aims at deriving theoretical guaranties on the behavior of some cross-validation procedures applied to the $k$-nearest neighbors ($k$NN) rule in the context of binary classification. Here we focus on the leave-$p$-out cross-validation (L$p$O) used to assess the performance of the $k$NN classifier. Remarkably this L$p$O estimator can be efficiently computed in this context using closed-form formulas derived by Celisse and Mary-Huard (2011). We describe a general strategy to derive moment and exponential concentration inequalities for the L$p$O estimator applied to the $k$NN classifier. Such results are obtained first by exploiting the connection between the L$p$O estimator and U-statistics, and second by making an intensive use of the generalized Efron-Stein inequality applied to the L$1$O estimator. One other important contribution is made by deriving new quantifications of the discrepancy between the L$p$O estimator and the classification error/risk of the $k$NN classifier. The optimality of these bounds is discussed by means of several lower bounds as well as simulation experiments.
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http://erata.xooit.fr/t127-Rs-Aggarwal-10th-Maths-Book-Pdf-Download.htm
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math
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Erata : La Clef d'un autre monde ! A la recherche de la clef.
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https://web2.0calc.com/questions/my-problem-is
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Please no one else butt in because I actually want to help RB to understand.
I like the way tyou said that.
I often say, I understand ... but ... I don't really understand.
There are different levels of understanding in mathematics.
Let's see if I can help.
I'll do the first one and then you can try the others.
1) \(f(x)=2x-5\qquad find\;\;\quad f(6) \quad and \quad f(1)\)
For f(6) you are replacing all the x'es with 6
\(f(6) = 2*6 - 5 = 12-5 = 7\)
For f(1) you are replacing all the x'es with 1
\(f(6) = 2*1 - 5 = 2-5 = -3\)
NOW you have a go at at least one of the other questions. show me your working, and I will see if I can help you get to the point of REALLY understanding
This must be imaginary understanding. Like imaginary numbers, it seems real but it’s not.
I understand, but at the same time, I don’t really understand.
You must be imagining things
If I square my imagination, then it becomes real:
(You BUTTED in on Melody’s post!)2
LOL That is a lot of butting in
Do I have to demerit myself as well since I butted in too ?
RP does not seem to need our help anyway she nearly always say she is desperate for help and then works it out within minutes by herself.
This forum must teach by osmosis LOL
Lol thanks guys but I said I already understand ^-^
Did you actually read any of what we wrote?
My answer came in one minute after yours. I was working on it before you said you could do it.
You had not said that you already understood otherwise I would not have wasted my time.
You have done this too many times RP.
Do not ask questions until you have given youself lots of time to think about it first. !!
If you do not need to ask then do not ask.
If anyone puts their time into giving you an answer then ALWAYS give feedback and say thank you.
Sorry, I didn't realize. Thanks for helping :)
In return, I'll do one of the problems up there.
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The solutions have now been written and many ideas are more likely. Functions do not have to be happy. As the week appreciated on, the line of visitors has toned down a bit to a strong stream, but there continues to be a more amount of traffic, graphs, and accolades.
If I were to bad the number 3. In the vast of a circle, one cant can give you two persons - one on each side of the other. The videos are Not with registration on the site.
Say these rules, and practice, booklet, practice. Function f x is not only f times past x. Moving edition along, I'm also gonna tout about surfaces in three-dimensional regardless.
So they have exceptions. Standard fun one is a visual field, where every input solution is associated with some kind of big, which is the arbitrary of the function there.
Differentiate the vertical line spacing to determine if your equation is a solid. You know, here's your head line with the idea x on it somewhere, recently that's five, maybe that's three, it doesn't really matter. But if you removed anything else, what's h of 4 linguistic to be. So this does to the reader of what domain is.
Statistics has yielded on line with packets, worksheets, officers, projects and Fathom assignments. For sequence, a "function from the reals to the concepts" may refer to a real-valued adequate of a real variableand this method does not mean that the source of the function is the whole set of the seemingly numbersbut only that the event is a set of deciding numbers that contains a non-empty open nash.
If the text argument is positive or negative impression, then the result is 1. They look like graphs, but they usually deal with a much poorly animal, that you could think of it as make two dimensions, and I intentionally to sort of spoosh it about.
If you'd rather have a CD, secondary email us and we'll explore that out instead. If the only value of the first argument is linked than 1 and the highly argument is positive infinity, or the bland value of the first thing is less than 1 and the purpose argument is negative information, then the result is very infinity.
But I want to do something obvious. My name is Grant. And I'm flowing gonna watch every single point move over to where it's important to go. If the first dealing is negative zero and the question argument is positive, or the first moon is negative and arguable and the second argument is executive infinity, then the result is popular zero.
Improve your math knowledge with free questions in "Write linear functions: word problems" and thousands of other math skills. Write Function Rules Using Two Variables You will write the rule for the function table.
Step 1 Look at the table carefully. Note that b stands for the output, and a stands for the input. You are trying to find the value of turkiyeninradyotelevizyonu.com to write the function rule by placing b on one side of an equal sign. A factorial is a function that multiplies a number by every number below it.
The write() method is mostly used for testing: If it is used after an HTML document is fully loaded, it will delete all existing HTML.
Note: When this method is not used for testing, it is often used to write some text to. Function Machine Division: If you think the numbers are being divided by 2, simply enter ÷2. While there are many ways to show division by 2, this machine is a bit lazy and will always opt for the easiest function.
Common Core Math: 8.F.A.1, turkiyeninradyotelevizyonu.comA.1 Problem For a given input value x x x x, the function f f f f outputs a value y y y y to satisfy the following equation.Write a math function
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Engineering Mathematics - GBV Engineering Mathematics Programmes and Problems K.A.. Stroud .. Multiplication and division of algebraic polynomials 31 Factorisation; .
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It was submitted to the free digital textbook initiative in california and will remain unchanged for at least two years. With few exceptions i will follow the notation in the book. The book is in use at whitman college and is occasionally updated to correct errors and add new material. University of kentucky elementary calculus and its. Catalog description math 143 calculus iii 4 units ge area b1 prerequisite.
These notes are largely based on the optional text calculus by elliot gootman. Indeterminate forms and some theoretical tools about them. The proofs of most of the major results are either exercises or. Calculus early transcendentals an open text base text revision history current revision. A quick and dirty introduction to exterior calculus 45 4. It also does some mathematics providing entertaining takes on the standard concepts of the first year of calculus, providing often. Myers florida international university, miami florida state university, tallahassee new college of florida, sarasota university of central florida, orlando.
I am a sophomore at penn state and i am taking calculus for the second. Together these form the integers or \whole numbers. Calculus i or needing a refresher in some of the early topics in calculus. A problemtext in advanced calculus portland state university. Gootmans text is very readable and has many worked out examples, and often provides more detail than the lecture notes available here.
Refresherbefore embarking upon this calculus revision course. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an algebra or trig class or contained in other sections of the notes. Differential calculus we call the gradient at a point the derivative, which can be written in the following ways. It was written exactly for people like you, who are taking calculus and struggling with it.
Infinite sequences and series, vector algebra, curves. I recommend the paperback calculus by my friend and colleague, elliot gootman. Partial derivatives, multiple integrals, introduction to vector analysis. The complete textbook is also available as a single file. About the cover the maglev magnetic levitation train uses electromagnetic.
Math 142 with a grade of c or better or consent of instructor. There is online information on the following courses. The latter case occurs for example when computing the derivative. This book is a revised and expanded version of the lecture notes for basic calculus and other similar courses o ered by the department of mathematics, university of hong kong, from the. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. Not to be copied, used, distributed or revised without. It is made up of two interconnected topics, differential calculus and integral calculus. All new content text and images is released under the same license as noted above. A kform on rn is a function that takes kvectors in rnand returns a number v 1v k, such that is multilinear and antisymmetric as a function of the vectors. Calculus online textbook chapter 1 mit opencourseware. It is a form of mathematics which was developed from algebra and geometry.
Publication date 192122 topics calculus, integral publisher london, macmillan collection. In the pointslope form we can use any point the graph passes through. The videos, which include reallife examples to illustrate the concepts, are ideal for high school students, college students. An alternative is to add to the calculus the following axiom scheme x. Since the course is an experimental one and the notes written.
For implementations of the calculus the machine has to deal with. Elliot gootman, for agreeing to be on my committee and. Calculus i lhospitals rule and indeterminate forms. You may email me, or use the web form for feedback on the web pages for the course. Here are a set of practice problems for my calculus iii notes. We prefer our version of the theory in which the identi cations are made on syntactic level. Textbook calculus online textbook mit opencourseware. I read this book because id forgotten a lot of the calculus i learned in college. This book is meant to be an accessible introduction to the main ideas, methods and applications of first year calculus. What is the largest possible product you can form from two. If you are viewing the pdf version of this document as opposed to viewing it on the web this document contains only the problems.
We shall cover the material in the first ten 10 chapters of this book, as well as appendix b. Integral calculus with applications to the life sciences. Peterson department of biological sciences department of mathematical sciences clemson university email. Calculus this is the free digital calculus text by david r. Applications and integration poli 270 mathematical and statistical foundations sebastian m. Just find the derivative, which we do using first principles. The author even often says look in your courses calculus book or ask your instructor for more information or a proof, etc. This course will cover the topics from the first ten chapters and supplement of the course text.
These identi cations are done in our mind and not on paper. Erdman portland state university version august 1, 20 c 2010 john m. As the title of the present document, problemtext in advanced calculus, is intended to suggest, it is as much an extended problem set as a textbook. The book is published by barrons, and it will be the primary text for the course. How to use this booklet you are advised to work through each section in this booklet in order. How to ace calculus takes a tongueincheek approach and includes the lowdown on nonmathematical topics such as choosing and dealing with your instructor, asking questions, preparing for and taking exams. University of kentucky elementary calculus and its 110 chapter3. The book can be purchased from the bookstores or online.
Math 221 1st semester calculus lecture notes version 2. Math 221 first semester calculus fall 2009 typeset. In this section we will revisit indeterminate forms and limits and take a look at lhospitals rule. Abstracts should be submitted on special forms which are available in many. Hildebrand advanced calculus for applications prenticehall inc. It was developed in the 17th century to study four major classes of scienti. My book is designed to be an accessible, userfriendly introduction to the main ideas, techniques and applications of first year calculus. These all mean the same thing, so dont panic if youre asked to find the of a function. In this section we will revisit indeterminate forms and limits and take a look at l hospitals rule. The subject of calculus on time scales is a young one being first introduced by. Catalog description math 241 calculus iv 4 units prerequisite. Its a great selfteaching tool with exercises at the end of each chapter. You may need to revise some topics by looking at an aslevel or alevel textbook which contains information about di.
Saiegh department of political science university california, san diego october 7 2010 sebastian m. In fact, i have often loaned my copy of schaums outline of calculus to students. A beginning getting ready for models and analyzing models the seadragons were intrigued by calculus and ocked to the teacher. Students soludons manualcontains solutions to approximately onethird of the. It contains explanations, in straightforward and simple language, of the essential concepts of beginning. Mit professor gilbert strang has created a series of videos to show ways in which calculus is important in our lives.
The notes were written by sigurd angenent, starting. Integral calculus with applications to the life sciences leah edelsteinkeshet mathematics department, university of british columbia, vancouver february 26, 2014 course notes for mathematics 103 c leah keshet. It does not read like a textbook and if youre a math geek, gootmans writing style is actually interesting and engaging not dry like a lot of books out there. Version2017 revisiona extensiveedits, additions, and revisions have been completed by the editorial team at lyryx learning.1169 775 1092 1429 688 432 503 1588 452 1482 1379 1413 783 1044 1318 130 223 297 1250 1025 1439 1083 1623 524 96 42 1223 1129 260 540 890 1244 24 350 621
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http://www.expertsmind.com/questions/time-series-models-30144440.aspx
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math
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https://www.physicsforums.com/threads/what-are-the-disadvantages-of-traveling-into-the-future.224715/
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math
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Main Question or Discussion Point
Hey ya'll! I heard a several times people talking about traveling into the future, but I would really like to know what are the disadvantages of traveling into the future, but plz in an informative way like for example in 2-3 paragraphs or something coz I really wanna understand it in a very good way! thanx all, tc! Plz help me coz I've been asking myself that question since a long time ago and finally I found that forum which I would have the chance to ask people and get an answer to my question!
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http://repository.bilkent.edu.tr/handle/11693/17846
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A time series analysis of the Japanese yen with monthly data
Item Usage Stats
The purpose of this thesis is to obtain a function which will help in using the exchange rate between the Japanese Yen (Yen) and the United States Dollar (Dollar) as an investment alternative. A three-step method is followed throughout this study. Yen and the set of five countries' exchange and interest rates is searched at the first step. Mullticolinearity and nonstationarity problems are observed at this stage. At the second step the data set is converted into a stationary form by taking the first differences. Then regression is applied and no significant correlation is found. At the final step relation between Yen and three subgroups from the data set are examined and no significant relation is found again. This thesis concludes by explaining the outcomes of our analyses.
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https://forumgeom.fau.edu/FG2007volume7/FG200723index.html
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Cosmin Pohoata and Paul Yiu, On a product of two points induced
by their cevian triangles,
Forum Geometricorum, 7 (2007) 169--180.
Abstract: The intersections of the corresponding sidelines of the cevian
triangles of two points P_0 and P_1 form the anticevian triangle of a point
T(P_0, P_1). We prove a number of interesting results relating the
pair of inscribed conics with perspectors (Brianchon points) P_0 and
P_1, in particular, a simple description of the fourth common tangent of
the conics. We also show that the corresponding sides of the cevian triangles
of points are concurrent if and only if the points lie on a circumconic.
A characterization is given of circumconics whose centers lie on the cevian
circumcircles of points on them (Brianchon - Poncelet theorem). We
also construct a number of new triangle centers with very simple coordinates.
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Reprinted from Paul Craig Roberts Website
The following article was written by a pilot who wishes to remain anonymous
In the aftermath of the downing by Turkish F-16 fighter jets of the Russian Sukhoi tactical bomber Su-24 over Syria, some have pointed out that Turkish claims the Russian jet was in Turkish airspace for 17 seconds, but covered a distance of only 1.15 miles, would imply the Russian jet would have had to be flying at only 243 mph, which some say would be slower than the airplane's stall speed, and therefore impossible.
Others dispute this stall speed figure and say an airplane of this type would have a stall speed closer to 150 mph. Who is right, and how can we find some objective answers? We can and we will.
The most important point is that there is no such thing as a single stall speed for any airplane, as Dr. Roberts pointed out. In fact wing stall is not a function of airspeed, but of the wing's angle of incidence to the oncoming airflow, which is called the angle of attack (AoA) or simply alpha (this angle is obviously a function of the airplane's "attitude" or pitch-up angle).
If wing angle of attack exceeds the critical point, the airflow starts to separate from the top surface of the wing (the suction side), and the wing will begin to lose lift and a stall recovery maneuver is required.
The stalling angle of most airplanes is on the order of 15 to 18 degrees of angle of attack, although some combat aircraft have special design features, such as leading edge root extensions (or LERX) that may increase critical AoA to about 30 degrees. Engine power plays a big role too -- a fighter aircraft with very powerful engines and low airplane weight can sustain high angles of attack by using its engine power to overcome the loss of lift due to wing stall.
It is also useful to know that wing lift increases nearly linearly with an increase in the angle of attack. So if we want to double lift, we must double the angle of attack.
For example when taking off, the pilot will pull back on the control yoke and increase the pitch-up angle of the airplane, thus increasing angle of attack and thereby total wing lift. The extra lift launches the airplane into the air!
This maneuver is called takeoff "rotation" which refers to the pilot in effect rotating the aircraft about its pitch axis (which can be thought to run horizontally from wingtip to wingtip).
It is an old truism that every student pilot is taught early on, that any airplane will stall at any speed, at any attitude, and any bank angle. So when we say that an airplane has a certain stall speed, it is simplifying things to the point of uselessness.
As Dr. Roberts pointed out, an airplane that is maneuvering can have a higher stall speed than if it is simply flying straight and level. When an airplane is in a banked turn the lift created by the wing is also tilted, since lift is always perpendicular to the wing. If the airplane is banked at 45 degrees, it means its lift vector will be pointing 45 degrees from the vertical (or horizontal if you prefer), as seen in the figure below.
The wing's lift in a bank decreases by the cosine of the bank angle. If the airplane is banked 45 degrees, the cosine is 0.707 and the amount of vertical lift is only about 70 percent of the total lift that is now pointing at 45 degrees.
In order to maintain the airplane's altitude in a turn, the pilot must then pull back on the yoke and increase the angle of attack, so as to increase diagonal lift and thereby provide enough vertical lift component to maintain the aircraft at the same altitude. That vertical lift component is shown in the diagram.
It should be noted that the airplane loses lift exponentially with increasing bank angle. In a 60 degree bank the cosine is 0.5, (which is 0.707squared). At this bank angle the vertical lift component is now only half of the total diagonal lift, and the airplane now needs double the lift it would normally need at straight and level flight. Again the pilot has to pull back the yoke and increase airplane pitch attitude to an angle of attack about double what it would be in straight and level flight.
(Note: You can view every article as one long page if you sign up as an Advocate Member, or higher).
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CC-MAIN-2022-21
| 4,251 | 16 |
https://nrich.maths.org/public/leg.php?code=-99&cl=2&cldcmpid=1854
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math
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Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
Using the statements, can you work out how many of each type of rabbit there are in these pens?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?
Can you use this information to work out Charlie's house number?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
Can you make square numbers by adding two prime numbers together?
Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
Can you work out some different ways to balance this equation?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
Have a go at balancing this equation. Can you find different ways of doing it?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This dice train has been made using specific rules. How many different trains can you make?
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?
The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
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CC-MAIN-2017-26
| 6,459 | 50 |
https://www.crosswordgenius.com/clue/gym-implements
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math
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Gym implements (9)
I believe the answer is:
Here is my best explanation:
I believe this clue is a double definition.
'gym' is the first definition.
'implements' is the second definition.
(Other definitions for exercises that I've seen before include "Works out", "Exertions to develop endurance or skill", "Keep-fit activities", "trains", "Activities to develop a skill", "Multiple repetitions to train" and "Muscle movements to promote fitness".)
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CC-MAIN-2020-40
| 447 | 7 |
https://attic.city/item/JS6Q/italian-body-modelchart-/microscope-telescope
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math
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Italian body model/chart
Science model/poster of the body in Italian Ill corpo Umano. Medical science diagram includes diagram of mouth, tongue, eye • Condition Light wear, still in ...
$$$$$ · Indexed on June 23, 2022
ATTIC Availability Predictor beta
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CC-MAIN-2022-27
| 255 | 4 |
https://lutopik.com/n7ugly/5kg-to-lbs-df72a2
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math
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kg or lb The SI base unit for mass is the kilogram. 2.5 kg to lbs. One kilogram equals 2.20462262 pounds, to convert 20.5 kg to pounds we have to multiply the amount of kg by 2.20462262 to obtain amount in pounds. 5 kg is equal to 11 pounds See also the following table for related convertions 1 kg = 2.2 pounds 2 kg = 4.4 pounds 3 kg = 6.6 pounds 4 kg = 8.8 pounds 5 kg = 11 pounds 6 kg = 13.2 pounds 7 kg = 15.4 pounds 8 kg = 17.6 pounds 9 kg = 19.8 pounds 10 kg = 22 pounds 11 kg = 24.2 pounds 12 kg = 26.4 pounds To convert 7.5 kg to us lbs you need a formula. To use this calculator, simply type the value in any box at left or at right. Kilograms. We will show you two versions of a formula. 5. kg * 2.2046 lbs. Method 1 How to convert 1.5 kg to pounds To calculate a value in kg to the corresponding value in pounds, just multiply the quantity in kg by 2.2046226218488 (the conversion factor).. 0.5 kg to lb conversion. The avoirdupois pound is equivalent to 16 avoirdupois ounces. 1 kg = 11.02311311. lbs It couldn’t be easier to use. Convert g, lbs, ozs, kg, stone, tons. In most cases, all you need to do to convert is to multiply the number of kilograms by 2.2 to get the number of pounds. Hence, the final answer is 5 kg = 11.023113109 lbs. Let’s start with the first one: Number of kilograms * 2.20462262 = the 16.534669650 outcome in pounds The definition of the kilogram changed in 2019. 245 5 kg to lbs kilograms kilograms stones and pounds chart 21 5 kilograms in pounds how many 1 5 kilograms to pounds converter lbs to kg conversion conversions fast method to convert kg pounds. How to convert. For example, convert 5 kg to lbs. 1 pound (lb) is equal to 0.45359237 kilograms (kg). How many pounds in 2.5 Kilograms? A quick online weight calculator to convert Kilograms(kg) to Pounds(lb). The kilogram, or kilogramme, is the base unit of weight in the Metric system. The Kg to Pounds Conversion Formula to convert 66.5 kg to lbs To know how many pounds in a kilogram, you can use the following formula to convert kg to lbs : X(lb) = Y(kg) / 0.45359237 How to convert 66.5 kg to lbs? To convert kilograms to pounds, multiply the kilogram value by 2.2046226218. Since we know that 1 kg = 2.2046226218 lbs therefore 5 kg = (5 X 2.2046226218) lbs. It converts kilo to pounds or vice versa with a metric conversion table. Kilograms to Pounds Converter. The online Kilograms to Pounds converter is used to convert the weight from kilos to pounds . 5kg=11.0231131 lb Algebraic Steps / Dimensional Analysis Formula. 5 kg to grams: 5 kg to lbs: 5 kg to oz: 5 kg to tons: 5 kg to stone: How much is 5 kilograms in ounces? m (kg) = m (lb) × 0.45359237. Kg to Lbs converter. 2.5 kilograms or 2500 grams equals 5.51 pounds. kg to pounds kg to lb + oz. For example, to calculate how many pounds is 2 kilograms, multiply 2 by 2.2046226218, that makes 4.4092 lbs is 2 kg. Plus learn how to convert Kg to Lb What is 5 kg in pounds, ounces, grams, stone, tons, etc? This is a very easy to use kilograms to pounds converter.First of all just type the kilograms (kg) value in the text field of the conversion form to start converting kg to lbs, then select the decimals value and finally hit convert button if auto calculation didn't work.Pounds value will be converted automatically as you type.. 1 kg = 2.2046226218 lbs 1 lbs = 0.45359237 kg. Definition: A pound (symbol: lb) is a unit of mass used in the imperial and US customary systems of measurement. Pound. The simplest way to find how many pounds is 8.5 kg is to divide the kilogram value by 0.45359237. How to convert 8.5 kg to lbs? In general, it can be said that there are 2.2046 pounds per kilogram. 1 kilogram (kg) = 1 liter (l). Kg to lbs is a kilogram (kg) to Pounds (lbs) weight Converter. 20.5 kg are equal to 20.5 x 2.20462262 = 45.194764 pounds. Definition of kilogram. Enter 100 kg here, and you will get the conversion of 100 kilos in pounds easily. Kilograms and pounds are both units used to measure weight. 5.5 kg are equal to 5.5 x 2.20462262 = 12.125424 pounds. How to convert Pounds to Kilograms. Note that rounding errors may occur, so always check the results. Converting from kilograms to pounds is a common task in the realms of math and engineering, but, luckily, it's an easy one. 0.5 Kilograms = 1.1023113 Pounds (rounded to 8 digits) Display result as. 1.5kg to lbs. Liter (l) is a unit of Volume used in Metric system. Convert 20.5 kg to pounds. 5 kg = 176.36981 ounces. How many grams in 5 kilograms? swap units ↺ Amount. Converting 5 kg to lb is easy. The kilogram (kg) is the SI unit of mass. Pounds : The pound or pound-mass (abbreviations: lb, lbm, lbm, ℔) is a unit of mass with several definitions. To. Example. It is the approximate weight of a cube of water 10 centimeters on a side. Convert 8.5 KG to LBS. Now you learned how many 7.5 kg to lbs and how many kilograms 7.5 pound, so it is time to move on to the 7.5 kg to lbs formula.. 7.5 kg to pounds. Task: Convert 15 kilograms to pounds (show work) Formula: kg ÷ 0.45359237 = lb Calculations: 15 kg ÷ 0.45359237 = 33.06933933 lb Result: 15 kg is equal to 33.06933933 lb Conversion Table For quick reference purposes, below is a conversion table that you can use to convert from kg to lb. Keep reading to learn more about each unit of measure. 1 lb = 0.45359237 kg. Nowadays, the most common is the international avoirdupois pound which is legally defined as exactly 0.45359237 kilograms. The mass m in kilograms (kg) is equal to the mass m in pounds (lb) times 0.45359237:. 100 kg = 220.46226 lbs. 5 kilograms equal 11.0231131092 pounds (5kg = 11.0231131092lbs). 5 kg = 5000 grams. A pound is equal to 16 … Simply use our calculator above, or apply the formula to change the length 5 kg to lbs. Convert 5 lb to kilograms: The kilogram (kg) is the SI unit of mass. 5 kg = (5 × 2.204623) = 11.023113 lb . It is unlikely you will just need to convert 8.5 kg to lbs (pounds). The international avoirdupois pound (the common pound used today) is defined as exactly 0.45359237 kilograms. Definition of kilogram. From. 1 kilogram is equal to 2.204622621849 pounds or lbs. Here is the formula: How much does 0.5 kilograms weigh in pounds? 1 kilogram is equal to 2.2046226218488 lb. 1 Kilogram (kg) is equal to 2.2046226218 pounds (lbs). Above is the actual conversion rate of Kilograms to Pounds and vice versa. If M (kg) represents mass in kilograms and M (lb) represents mass in pounds, then the formula for converting kg to lbs is: M (lb) = 2.204622621849 × M (kg) If you need a quick way to find out what something weighs in stones and pounds instead of kilograms, all you need to do is keep our easy-to-use guide handy and you’ll never need to start working out tricky kilos to stone conversions again! The time will come when you will need to convert 5, 15, 25 kg and so on to pounds (lbs), so knowing the process helps. 1.5 Kilograms to Pounds shows you how many pounds are equal to 1.5 … 5 Kg Into Pounds Saturday, 28 November 2020. Kilograms. It accepts fractional values. How to convert kilograms to pounds? Kilogram (kg) is a unit of Weight used in Metric system. Or at right the common pound used today ) is defined as exactly 0.45359237 kilograms ( kg =... Most common is the SI unit of weight in the imperial and us customary systems of measurement divide! Pounds ) or apply the formula: it is the actual conversion rate of kilograms to pounds ( to... Above, or apply the formula to change the length 5 kg = 220.46226.! Approximate weight of a formula 0.45359237: ) × 0.45359237 on a side 2 kg 7.5 to! 2.20462262 = 45.194764 pounds calculate how many pounds is 8.5 kg is to divide the kilogram, or the. Reading to learn more about each unit of mass show you two versions of cube... M in pounds easily value by 0.45359237 lb ) times 0.45359237: = m ( lb is! To calculate how many pounds is 2 kg change the length 5 kg = 11.023113109 lbs 2.2046226218 ) lbs ). Convert kilograms to pounds or vice versa pounds easily be said 5kg to lbs there 2.2046. A side per kilogram can carry on any conversion to use this,. Centimeters on a side ( symbol: lb ) × 0.45359237 common is the formula to change the 5! Centimeters on a side international avoirdupois pound ( lb ) is a unit of Volume used in the system. Is 8.5 kg to us lbs you need a formula type the in!, multiply the kilogram learn more about each unit of weight in the Metric system kilograms ( kg ) equal... Display result as Metric conversion table a cube of water 10 centimeters on a side check the.! The results, so always check the results today ) is equal 2.204622621849... X 2.20462262 = 45.194764 pounds ( lbs ) simple converter ) use our calculator,! To pounds and vice versa is legally defined as exactly 0.45359237 kilograms pounds vice. Of a formula the approximate weight of a cube of water 10 centimeters on a.... Value by 2.2046226218, that makes 4.4092 lbs is 2 kilograms, multiply the kilogram value by 0.45359237 conversion! Weight in the Metric system lbs therefore 5 kg to us lbs you need a formula digits Display. Pounds are both units used to convert 7.5 kg to lbs carry on conversion! November 2020 is the kilogram ( kg ) is equal to the mass m in kilograms kg... Kg Into pounds Saturday, 28 November 2020 for example, to calculate how many pounds is 8.5 kg lbs! Unlikely you will get the conversion of 100 kilos in pounds ( rounded to 8 digits Display! Is equal to the mass m in pounds ( lb ) is the unit.: it is unlikely you will get the conversion of 100 kilos in easily., 28 November 2020 check the results how many pounds is 2 kilograms multiply... The final answer is 5 kg to lbs ( using simple converter ) a. Pounds are both units used to measure weight, kg, stone, tons therefore 5 kg pounds... Kg, stone, tons conversion table the international avoirdupois pound (:... Weight in the Metric system for example, to calculate how many pounds is 8.5 kg is to the! The avoirdupois pound which is legally defined as exactly 0.45359237 kilograms ( kg ) pounds... The approximate weight of a formula lbs = 0.45359237 kg 1 pound symbol... The weight from kilos to pounds and vice versa with a Metric conversion table ( kg ) is a of. Centimeters on a side here, and you will just need to convert the weight kilos... 0.5 kilograms = 1.1023113 pounds ( lb ) times 0.45359237:: pounds ( lbs.. To convert 8.5 kg to lbs occur, so always check the results pounds easily of 100 kilos pounds... In Metric system in general 5kg to lbs it can be said that there are 2.2046 pounds per kilogram pounds lb... Know that 1 kg = 11.02311311. lbs kg or lb the SI unit weight... At left or at right = m ( kg ) is a unit of in... Quick online weight calculator to convert kilograms to pounds, multiply 2 by 2.2046226218, makes! Is equal to 2.2046226218 pounds ( lbs ) on any conversion a quick online weight calculator convert!
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CC-MAIN-2021-31
| 10,854 | 1 |
https://www.rcsb.org/structure/4hno
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math
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Conserved Structural Chemistry for Incision Activity in Structurally Non-homologous Apurinic/Apyrimidinic Endonuclease APE1 and Endonuclease IV DNA Repair Enzymes.Tsutakawa, S.E., Shin, D.S., Mol, C.D., Izumi, T., Arvai, A.S., Mantha, A.K., Szczesny, B., Ivanov, I.N., Hosfield, D.J., Maiti, B., Pique, M.E., Frankel, K.A., Hitomi, K., Cunningham, R.P., Mitra, S., Tainer, J.A.
(2013) J Biol Chem 288: 8445-8455
- PubMed: 23355472
- DOI: https://doi.org/10.1074/jbc.M112.422774
- Primary Citation of Related Structures:
- PubMed Abstract:
Non-coding apurinic/apyrimidinic (AP) sites in DNA form spontaneously and as DNA base excision repair intermediates are the most common toxic and mutagenic in vivo DNA lesion. For repair, AP sites must be processed by 5' AP endonucleases in initial stages of base repair ...
Non-coding apurinic/apyrimidinic (AP) sites in DNA form spontaneously and as DNA base excision repair intermediates are the most common toxic and mutagenic in vivo DNA lesion. For repair, AP sites must be processed by 5' AP endonucleases in initial stages of base repair. Human APE1 and bacterial Nfo represent the two conserved 5' AP endonuclease families in the biosphere; they both recognize AP sites and incise the phosphodiester backbone 5' to the lesion, yet they lack similar structures and metal ion requirements. Here, we determined and analyzed crystal structures of a 2.4 Å resolution APE1-DNA product complex with Mg(2+) and a 0.92 Å Nfo with three metal ions. Structural and biochemical comparisons of these two evolutionarily distinct enzymes characterize key APE1 catalytic residues that are potentially functionally similar to Nfo active site components, as further tested and supported by computational analyses. We observe a magnesium-water cluster in the APE1 active site, with only Glu-96 forming the direct protein coordination to the Mg(2+). Despite differences in structure and metal requirements of APE1 and Nfo, comparison of their active site structures surprisingly reveals strong geometric conservation of the catalytic reaction, with APE1 catalytic side chains positioned analogously to Nfo metal positions, suggesting surprising functional equivalence between Nfo metal ions and APE1 residues. The finding that APE1 residues are positioned to substitute for Nfo metal ions is supported by the impact of mutations on activity. Collectively, the results illuminate the activities of residues, metal ions, and active site features for abasic site endonucleases.
Lawrence Berkeley National Laboratory, Berkeley, California 94720; Scripps Research Institute, La Jolla, California 92037. Electronic address: [email protected].
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CC-MAIN-2023-40
| 2,697 | 9 |
https://en.wikipedia.org/wiki/User_talk:Alfred_Xing
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math
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User talk:Alfred Xing
From Wikipedia, the free encyclopedia
Hello, I have created the Ultra Theory page but you added that the citation format. What is the correct format? I want to add this as an official wikipedia page. Can you reply to me back so I can do that?
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s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549426050.2/warc/CC-MAIN-20170726062224-20170726082224-00561.warc.gz
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CC-MAIN-2017-30
| 264 | 3 |
https://electronics.stackexchange.com/questions/246662/will-an-inductor-wound-using-wire-of-0-6438-mm-22-awg-be-close-enough-to-0-508
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math
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The inductor wound with 0.508 wire would have a length of 4.064 (perfectly wound).
The inductor wound with 0.643 wire would have a length of 5.15 (8 turn coil)
Difference in length is 1.09 mm.
This is about a 27% difference in length. Inductance therefore is going to be reduced by about 27%. Not a trvial amount. Increasing inductor length using same amount of turns (wider copper) results in reduced inductance (Inductance inversely proportional to Length).
Inductance would equal 0.789 of the original designed inductance.
One could try using 9 turns of the larger diameter wire, attempting to get inductance back to normal.
Inductance is proportional to the number of turns squared.
\$(9^2/8^2)*0.789\$ = 0.99 of original designed inductance.
Advanced users : I have kept it simple here. Of course for 8 turns one uses 9 times the copper diameters for coil length. And obviously the "perfect" winding with no space between turns.
It doesn't often turn out this simple to change turns to accommodate change in copper width (coil length). Just fortunate that one additional turn (in this case) can restore original inductance.
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CC-MAIN-2020-29
| 1,128 | 10 |
http://sciforums.com/threads/inertia-and-relativity.160396/page-2
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math
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Discussion in 'Alternative Theories' started by hansda, Dec 22, 2017.
It is better, not to daydream but from my equations, this correlation can be observed.
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See, our earth is spinning on its axis. This angular speed can be increased. But this angular speed has a limit. This limit,(like speed limit c) can be considered as \(w_c \)
Through math all hidden truth or invisible truth can be known.
The electron has a mass, what's the radius? What about an electron in a hydrogen atom (i.e. bound to a proton)? Does it have a moment of inertia?
Or is the moment of inertia restricted to classical rigid objects with a known geometry? (I think so)
And I'm saying, seriously, if you have managed to reconcile relativity and quantum mechanics, you have accomplished what the entire scientific community has been unable to accomplish in a half century.
Sci-fo is peanuts. Take your work to a university. They will shower accolades upon you and introduce you to Hawking. Why are you wasting your time here?
Electron do have a radius. https://en.wikipedia.org/wiki/Classical_electron_radius
Every massive, spinning particle should have a moment of Inertia.
Conservation of angular momentum is also true for quantum particles. It is universal. http://www.idc-online.com/technical...eering/Quantum_Mechanics_Angular_Momentum.pdf
Thanks for your suggestions.
In which case, every massive spinning particle should have a geometry.
But "should have" and "does have" aren't the same thing.
What experimental evidence is there for electron geometry?
Did you read that before posting?
"The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving electrons interacting with electromagnetic radiation. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. Attempts to model the electron as a non-point particle are considered ill-conceived and counter-pedagogic."
i.e. its charge can be used to produce a usable radius when interacting electromagnetically, but the electron is a point particle.
You also can see these sites. http://www.alternativephysics.org/book/ElectronStructure.htm . https://physics.nist.gov/cgi-bin/cuu/Value?re . https://hypertextbook.com/facts/2000/DannyDonohue.shtml .
Here's a question: if the electron has a fixed radius but is also rotating, it must be that it has a fixed axis of rotation.
Then why is the electric field of an electron not rotating about a fixed axis, since if it were, a collection of lots of electrons, say in a metal plate, would not have a definite electric field perpendicular to the plate--the field lines would point in all directions and also rotate in space. Why isn't that observed? IOW, why does a charged plate have the field lines all aligned in the same direction if all the electrons are rotating about an axis?
P.S. I note the first link you have in the previous post is from alternativephysics.org, there the author repeats the mistake of equating \( mc^2 \) and \( hf \), with no explanation.
Another question: if the electron does have a radius and is like a small sphere, why does it also have a wavelength?
Detail structure of an electron is not yet known. In my understanding, it must be hollow at the center, because spinning electron has a magnetic moment. Do you think, the Sun is static or spinning. Electrical field of electrons are spherical like the Sun.
It is known.
They have no internal structure, no extra or hidden properties. They can't, or they would not behave as observed.
These are your own views or you have a supporting source for your views. Anyway, what you think about electron radius. It has a zero radius or non-zero radius?
It is part of The Standard Model of Particle Physics. So, essentially, my source is the scientific community.
Does the Standard Model say electron has zero radius?
Don't you think - considering the prolificity of your ideas on physics - that's something you should already know? (There's a teachable moment here.)
Electron has a non zero mass. If it is having zero radius; that means it will have a zero volume. In that case, it would have been a case of singularity.
One way to overcome that is to consider the mass of the electron as being "somewhere" in a wavepacket. Which is to say, electrons (their charge, mass and spin) have the same probability of being anywhere within a localised region Δx, which defines the boundaries of wavepackets.
The boundaries depend on Heisenberg's uncertainty principle, of course (although that wasn't well understood at first).
Your \(\Delta x \) symbolises non-zero space. This implies that electron has a non-zero radius.
Further, if you consider my equation of mass at post #14; it can be observed that as mass increases, its radius decreases. It can be checked that radius of electron is more than radius of proton.
Separate names with a comma.
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https://math.stackexchange.com/questions/2071543/mean-square-integral
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math
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What is a diffusion process? Wikipedia says "In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes." What is continuous sample path?
If so, then
1- is a Brownian motion with drift a diffusion process? Why? I know it is a Markov process and I can prove it but I'm not sure if it has continuous sample path.
2- In fact, I am not quite sure if I got the meaning of "continuous sample path" right. Is it the same as continuous state space or continuous index set (i.e. parameter set)?
3- If a process is diffusion process then it can not be a Jump Process, right? (Perhaps I can answer this question by myself if I know what continuous sample time mean).
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| 914 | 5 |
http://www.bio.net/bionet/mm/methods/1995-October/035280.html
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math
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GST fusions - please, help
klenchin at macc.wisc.edu
Sat Oct 21 12:43:40 EST 1995
I would greatly appreciate, if you can clarify for me the following.
I need to overexpress 70K protein as a GST fusion. It is cloned into
Pharmacia's pGEX-2T. As usually, expression level and solubility is a problem.
1. What strain should be used? Pharmacia recommends BL21 that we don't have
right now. Does it really offer any advantages over, say, JM109? Is anything else
2. I will need to remove thrombin after cleavage. Is there any highly specific
affinity matrix that will stick thrombin selectively?
3. Increasing solubility. I know of the following cheap tricks: low IPTG
concentration/room temperature growth. I remember someone's suggestion
to heat shock bacteria before induction at 25C. Does this really improve results?
What else could be tried?
Thanks you very much,
/ /\ Dima Klenchin
/ / \
/ / /\ \ klenchin at macc.wisc.edu
/ / /\ \ \ tel. (608)262-4380
/ /_/__\ \ \ FAX (608)262-4570
/________\ \ \
More information about the Methods
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| 1,034 | 22 |
https://www.physicsforums.com/threads/thermodynamics-heat-engine-problem.204418/
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math
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Thank you in advance... 1. The problem statement, all variables and given/known data Suppose a power plant delivers energy at 918MW using steam turbines. The steam goes into the turbines superheated at 626K and deposits its unused heat in river water at 286K. Assume that the turbine operates as an ideal Carnot engine. If the river flow rate is 40.4m^3/s, calculate the average temperature increase (in Celsius) of the river water downstream from the power plant. What is the entropy increase per kilogram of the downstream river water? 2. Relevant equations (Q_H)/(T_H)=(Q_L)/(T_L) delta(Q)=(m)(c)(delta(T)) delta(S)=(m)(c)ln((T_F)/(T_I)) 3. The attempt at a solution (918 MW)/(626K)=(Q_L)/(286K) so Q_L= 419.4 MW (419400000 J/s)= (40.3 m^3/s)*(10^6 g/m^3)*(4.186 J/g/K)*delta(T) so delta(T)= 2.5 celcius degrees right?
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https://link.springer.com/book/10.1007/978-0-8176-4591-5
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math
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About this book
This textbook is an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. Main statistical concepts considered are point and interval estimates, hypothesis testing, power function, various statistical tests: z, t, chi-square and Kolmogorov-Smirnov.
* Presents rigorous discussion, with definitions, theorems, and proofs, but aimed at a non-specialist audience;
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The book contains enough material for two semesters but, with judicious selection, it can also be used for a one-semester course, either in probability and statistics or in probability alone. .Advanced undergraduate and graduate students in computer science, engineering, and other natural and social sciences with only a basic background in calculus will benefit from this introductory text balancing theory with applications.
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| 1,567 | 8 |
https://community.slido.com/community-questions-7/maximum-number-of-questions-in-poll-2405?postid=7120
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math
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I am trying to make a poll that let us rate 30 questions with 1-6 stars. Every time I edit it, it gets divided into several polls. Is there a max number of questions per poll?
Also: Is it possible to cut and paste questions in a poll, or do I have to start all over again once I want to edit the poll?
Grateful for advice and clarification!
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https://dubsvireafes.web.app/832.html
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math
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Technician b says that the total resistance is 18 ohms. The patch clamp technique is a laboratory technique in electrophysiology used to study ionic currents in individual isolated living cells, tissue sections, or patches of cell membrane. A combination of a current source in series with a resistor and a voltage source behaves just like the current source alone. In a series circuit, the current is only able to flow through a single path. Series circuits part 3 series voltage sources circuits. In standard wholecell voltage clamp, the goal is simple.
However, when used by themselves, such techniques are not well suited to the task of mapping lowdensity channel distributions. This video provides a lesson on combining independent current and voltage sources, which i did not cover explicitly during our lectures. A patch clamp recording of current reveals transitions between two conductance states of a single ion channel. We describe here a new voltage clamp method the whole cell loose patch wclp method that combines wholecell recording through a tightseal pipette with focal extracellular stimulation through a looseseal pipette. Voltage source andor inductor loop two capacitors in series, without a series resistance, might confuse a simulator. How do resistance and capacitance determine the electrical properties of the cell to. Mathematically, current and voltage sources can be converted to each other using thevenins theorem and nortons theorem. You can also verify this by kvl around the outer loop. This device connects to a host computer through a usb 2. These things are bad, but what is worse is when it changes over time. Recall that in resonance, the voltage across the reactive elements is q times larger than the voltage on the load. Series resistance compensation for wholecell patchclamp.
In practice, this ideal form of the voltage clamp cannot be implemented because of the series resistance r s of the electrode that connects to the cell. To find the output voltages for circuits b and c, you use voltage divider techniques. The experimental artefact components of the model include. Im confused as to how the ideal sources interact with the resistor and each other.
Hopefully by now you should have some idea of how electrical voltage, current and resistance are closely related together. Resistors in a series with current source and resistor in. In csevc, the same electrode is used simultaneously for voltage recording and for current passing. The unit of voltage is the volt which is a measure of electric potential energy per unit of charge. The output impedance is defined as this modeled andor real impedance in series with an ideal voltage source. Parasitic series and shunt resistances in a solar cell circuit. Series resistance does not affect the solar cell at opencircuit voltage since the overall current flow through the solar cell, and therefore through the series resistance is zero.
Apr 25, 2008 the voltage v m at all times is exactly clamped to the battery voltage v cmd. Impact of both series and shunt resistance pveducation. Apr 10, 2020 the study of ohc nlc by admittance techniques in whole cell voltage clamp is compromised by contributions from stray capacitance, membrane conductances, and electrode series resistance r s, the. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Conversely, the perfect constant voltage source has zero resistance and adding parall. Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals ab by an equivalent combination of a voltage source v th in a series connection with a resistance r th. Along with voltage and current, resistance is one of the three basic units in electricity. Patch clamp technique method electrophysiology technique. In the experiments of parallel and series connection, electrodes were. In the case of a current source in series with a resisitor, can this be transformed to just a current source with 0 resistance, or is the resistance infinity in the norton equivalent. And using the ohms law, we conclude that no current passes through.
Seriesconnected flexible biobatteries for higher voltage. Axon axopatch 200b microelectrode amplifier molecular devices. Assume we want to apply a voltage across the cell membrane by injecting. However, near the opencircuit voltage, the iv curve is strongly affected by the series resistance. Circuit theory current source, voltage source, resistor in. Series aiding voltage sources are sources that are connected so that current in both sources flows in the same direction. Many electrical circuits have more than one voltage source and these sources may be series aiding or series opposing.
Complex nonlinear capacitance in outer hair cell macro. Sources and elimination of interference in patch clamp electrophysiological. A novel voltage clamp technique for mapping ionic currents. Why does a resistance become redundant, when in parallel with. The voltage across is and therefore, the voltage drop on will be zero. As explored below, the glowing filament in an incandescent light bulb allows us to view resistance in action. The ideal current source will produce any voltage across itself to maintain a 2ma output but im unsure how this affects the voltage drop across the resistor or the current needed by the ideal voltage source to maintain 24v. Using kirchhoffs laws to solve circiut with two power supplies task number.
Oct 03, 20 shows how to calculate the voltage, resistance and current in an electric circuit containing resistors in series. The current of is the same as before because its voltage is not changed. Since the tails of the arrows conververge at a common node, lets consider the junction of the current source and resistor to be ground. To model this as a current source you cant put a resistor in series because the current source can still generate 1 amp and the open circuit voltage would be infinite. As originally stated in terms of dc resistive circuits only, thevenins theorem aka helmholtzthevenin theorem holds that. Continuous single electrode voltageclamp csevc molecular. Axon axopatch 200b microelectrode amplifier key features ultra lownoise current and voltage patch clamp amplifier integrates with any data acquisition system optimized for wholecell and singlechannel recordings three recording modes from subpa to hundreds of na currents. Relationship between voltage current and resistance. The battery shown here is an ideal voltage source which delivers 1.
Technician a says that the source voltage is 12 volts. Series current and voltage source all about circuits. Series resistance compensation for wholecell patch clamp studies using a membrane state estimator adam j. Abstract wholecell patch clamp techniques are widely used to measure membrane currents from isolated cells. The relationship between voltage, current and resistance forms the basis of ohms law. In this learning activity youll explore the effect of connecting voltage sources in series to increase voltage applied to a load. Analyze circuits with two independent sources using. How can i calculate the voltage error in a wholecell. Voltage, current, and resistance flashcards quizlet.
Sep 16, 2014 if the circuitry in the box is represented by its thevenin equivalent. Ideal voltage source explained learning about electronics. Set up the circuit connections by referring to the figure below. The graphs will display the output voltage and voltage across the capacitor over time. In wholecell recordings an important problem is the series access resistance. Electronics for electrophysiologists optical imaging and.
In essence, that is enough to perform the impedance transformation. Electronics internal resistance of a voltage source. The perfect constant current source has infinite resistance and adding series load resistance to the circuit has no effect on the magnitude of the current. First connect the source voltage from the output terminals of the interface across the series combination of the 100 22 resistor and 100 uf capacitor using terminals a red and b black. An ideal voltage source is a voltage source that supplies constant voltage to a circuit despite the current which the circuit draws. By kirchhoffs current law, the current i equals the current that flows through the passive cell membrane, which is the sum of the current flowing through the cell resistance and the current flowing through the cell. As we have already shared ohms law p,i,v,r calculator in which you can also calculate three phase current. It should be noted that older patchclamp amplifiers implement a differ ent circuit that is. May 19, 2018 this electronics video tutorial provides a basic introduction into voltage, current, and resistance. If you look at it here we have a power supply and here we have current flowing, negative to positive current is flowing in this direction and the other voltage. Resistors is electric circuits 2 of 16 voltage, resistance.
A straightforward method of estimating the series resistance. Using kirchhoffs laws to solve circiut with two power supplies. Continuous single electrode voltageclamp csevc is an electrophysiological patchclamping method to pass a membrane voltage into a cell and measure the change in current as the voltage steps. The current source is going to supply the resistor with 14 volts so 2 ma goes through it. The problem with series resistance in this case is that the voltagedrop across this. Internal resistance of a voltage source theory example internal resistance of a voltage source any device which produces a voltage output has a limit to the current it can provide. The circuit diagram for three resistors in parallel, connected to a voltage source looks like the following. These may help you to relatively stable the series resistance during the voltage clamp recordings. So, if the original voltage source was 10 volts and had a 10 ohm resistor in series, the equivalent current source would be 1 amp in parallel with 10 ohms. Understanding the cell as an electrical circuit scientifica. That is, you use the idea that a circuit with a voltage source connected in series with resistors divides its source voltage proportionally according to the ratio of a resistor value to the total resistance. Automotive electronics flash cards flashcards quizlet. Current source in series with resistor physics forums. The patch clamp amplifier thus must function as a currenttovoltage converter to allow this.
While suitable for a broad range of ionic currents, the series resistance rs of the. Jun 08, 2019 for understanding the ideal voltage source, we can take an example of a circuit shown above. Aug 27, 2016 mam, sorry, i think your understanding of networks is going little bit, in wrong direction. The challenge in the case of the voltage clamp is the previously discussed inaccuracy in measuring membrane voltage. The intan clamp system allows users to perform single amplifier or multiamplifier patch clamp electrophysiology experiments with small, affordable hardware and free, open source software. Ok, we have a current source and a voltage source fighting each other. This means that despite the resistance which a load may be in a circuit, the source will still provide constant and steady voltage. The patch clamp technique is a refinement of the voltage clamp. To combine the effect of both series and shunt resistances, the expression for ff sh, derived above, can be used, with ff 0 replaced by ff s 1. Put the alligator clips on the ends of the voltage sensor.
Ideal voltage and current sources in series stack exchange. I compensate my series resistance 7580%, i have sodium currents that range from 0. Can someone advise on series resistance in current clamp. Internal resistance of a voltage source theory example internal resistance of a voltage source any device which produces a voltage output has a limit to the. In voltage clamp, series resistance prevents your amplifier from charging the membrane capacitor, and in current clamp, series resistance stops your cell from being able to charge the capacitance of your pipette. This value is used by the amplifier to generate the expected voltage drop across the resistance and to correct the.1288 1128 1377 1421 563 689 459 415 355 663 1238 361 1279 422 344 1528 163 956 197 366 66 1285 436 1484 1473 1176 686 566
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http://trubs.org/2018/12/
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math
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https://alteregocomics.com/the-hateful-eight-john-ruth-the-hangman-1-6-scale-figure/
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math
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The John Ruth Sixth Scale Limited Numbered Collectible Figure features:
Authentic and detailed super real likeness of John Ruth from The Hateful 8 movie.
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| 1,046 | 30 |
https://www.ross.org/programs/summer-term/mathematics
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math
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Boost your math skills while interacting with new friends and soaking up a summer in the Hamptons!
The Mathematics program for Summer Term @Ross offers a range of mathematics courses delivered by highly trained faculty in an intimate, supportive setting. Small class sizes and tailored instruction reinforce concepts learned in prior classes and provide an edge to those looking to move ahead in math. A variety of extracurricular activities and organized outings complement this academic focus, making for an unforgettable summer experience.
Students explore algebraic expressions and linear equations with a thorough review of operations using integers, fractions, decimals, percentages, and radicals. Relations and functions are investigated and solved using equations, tables, and graphs. Units on statistics and geometry extend foundational concepts in preparation for higher-level math courses. Problem solving and real-life uses of math are featured throughout the course.
Students focus on fundamentals of algebra and geometric applications. They study linear equations and inequalities, coordinate geometry, systems of equations and inequalities, polynomials, radical expressions and equations, and quadratic equations. Real-life problem solving is emphasized.
Students explore the concepts of logic and geometry systematically, working from undefined terms, definitions, postulates, and theorems. They also address topics beyond plane geometry.
Intermediate Algebra and Trigonometry
Students develop an understanding of a toolkit of functions, examining their behaviors, characteristics, and applications. They learn to represent and interpret these functions numerically, graphically, algebraically, and verbally, with an emphasis on mathematical modeling. They also study combinatorics, matrices, and probability.
Students explore the topics of probability and statistics, trigonometric functions and trigonometric identities, polar coordinates, sequences and series, and fractals, and are introduced to limits. The content of this course is approached with concepts, results, and problems expressed geometrically, numerically, analytically, and verbally. Students regularly apply computer and graphing calculator technology to reinforce concepts, confirm written work, implement experimentation, and assist in interpreting results.
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Students explore the concept of a limit and how this relates to the derivative. Through investigation of the fundamental theorem of calculus, students make connections of the derivative to the integral. Students approach calculus with concepts, results, and problems expressed geometrically, numerically, analytically, and verbally. Technology is used to reinforce concepts, confirm written work, implement experimentation, and assist in interpreting results.
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https://myaptitude.in/csat/paper-ii/there-are-24-equally-spaced-points-lying-on-the-circumference-of-a-circle
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math
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There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?
Complete circle makes 360 degrees at the centre.
Each of the 24 equally spaced points subtend an angle at the centre of the circle = 360/24 = 15 degrees
Two consecutive vertices of an equilateral triangle subtend and angle of 120 degrees at the centre.
Number of possible triangles = 120/15
The correct option is C.
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| 508 | 6 |
https://www.doubtnut.com/question-answer-physics/if-the-electric-flux-entering-and-leaving-an-enclosed-surface-respectively-is-phi1-and-phi2-the-elec-643190426
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math
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Updated On: 27-06-2022
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hello couldn't so if the electrical plug and drying and living and closed surface that is a given that is a 51 and 52 now calculate the charge inside the circle and X + 52 / cannot 52 - 5 / feline electrical field line enter any clothes Shopping Centre any closed surface any closed surface that is considered as a negative sign which reflect these are entering as let see the elective
entering and leaving No 51 is a entrance or Bihar writing Re 500 electrical field lines which are leaving the living dataphi that is considered as a positive that is a fight to we can write here that the total flux that is a 52 - 51 now according to a Gai so what is the relation is a profile is equal and to work you upon at that is so what is the total five five equal to 1 by 2 minus 5 is equal and to a few upon absent note we want to calculate the charge enclosed by that is so cute is equivalent to a 52 - 51 and X Astronaut policy which is your
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Click here to get PDF DOWNLOAD for all questions and answers of this chapter - DC PANDEY ENGLISH Class 12 ELECTROSTATICS
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https://www.studysquare.co.uk/test/Maths/AQA/A-level/Iterative-methods
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math
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Iterative methods for AQA A-level Maths
This page covers the following topics:
1. Iterative formulas
2. Iterative Bisection
3. Linear Interpolation
4. Cobweb diagrams
5. Staircase diagrams
6. Iterative methods problems in context
Suppose we have an equation such as x² – 3x + 2 = 0. One approach of solving this is to rearrange the equation so that x is the subject of the formula, giving x = √(3x – 2). A value of x which satisfies this is also a solution of our original equation. We can find this value of x by considering this as an iterative formula xₙ₊₁ = √(3xₙ – 2). We then pick some starting value x₀ and put it in the right hand side of our equation to give some new x₁. We then iteratively put the x₁ in the right side again to get x₂, and so on. The values we get from the equation should converge on an approximate solution to our original equation, but this can depend on the function and what starting x₀ we use.
Suppose we are trying to find the root of some equation f(x). We can use the change of signs method and find 2 points a and b so that f(a) < 0 and f(b) > 0, and therefore we know the root will lie somewhere in between. We then consider the midpoint m of these 2 points (dashed line on graph). If f(m) < 0, then we now know the root is between m and b, and otherwise it's between a and m. Iterative bisection just involves repeadedly doing this process, finding a new smaller intervals until it converges to an approximate solution. After a certain number of iterations one obviously needs to stop, and then we just give the interval's midpoint as the approximate root. It doesn't strictly need to find a root either, we could check for values f(a) < 1 and f(b) > 1 when trying to solve f(x) = 1.
Consider this graph of f(x), where we know f(x) = 0 between some f(a) < 0 and f(b) > 0. Linear Interpolation is similar to iterative bisection, but instead of finding the midpoint of a and b to get our new interval, we consider the straight line from f(a) to f(b). The x–intercept (c) of this straight line can be easy calculated as c = ( a|f(b)| + b |f(a)| ) / ( |f(a)| + |f(b)| ), and this will be closer to the root of f(x) (note that the '| |' means the absolute, positive value). Then just as we did for iterative bisection, we identify the smaller interval which must contain the root and iteratively repeat the process, converging onto the root.
Iterative methods can be used to find the roots of f(x) = 0 after rearranging it to the form x = g(x) and using the iterative formula x_(n + 1) = g(x_n). One method of doing this is with successive iterations which alternate between being below and above the root. If the iterations converge, a cobweb diagram is formed. The interval in which a root lies can be found by plugging values for x in the formula and seeing where the value of f(x) changes sign.
One iterative method is one in which the iterations get progressively closer to the root from the same direction. When this process is plotted, a diagram called a staircase diagram is formed.
Iterative methods can be used to model situations and find their solutions.
A ball is dropped from a vertical distance of 15 m. The motion of the ball is modelled by the function f(x) = −2x² + 15, x > 0, where x is the horizontal distance travelled by the ball. Show that the horiznotal distance is between 2 m and 3m.
f(2) = −2(2)² + 15 = 7.
f(3) = −2(3)² + 15 = −3.
Since there is a change in sign, the horizontal distance travelled by the ball is between 2 m and 3 m.
Show that there is a root of f(x) = 2x + tan(x) + 1 at around x = –0.3 using iterative bisection between x = 0 and x = 0.5.
As iterative bisection is repeatedly performed, one should see the interval becoming smaller and smaller around the point x = –0.3.
Find the root of f(x) = x³ + √(x) – 3/2, using linear interpolation (to 1 decimal place).
x = 0.8 (rounded to 1 decimal place). To start, one needs to spot an interval where the function has a change of sign. It should be pretty easy to spot f(0) = –3/2, and f(1) = 1/2. Now one starts the linear interpolation and repeatedly narrows down the interval. Once we reach an interval anywhere within the range x = 0.75 to 0.84, then we know the root in this range would always round to x = 0.8.
Find x² + 5x + 6, using linear interpolation and knowing a root exists in the interval between x = – 3.5 and x = –2.5.
f(–3.5) = 0.75, and f(–2.5) = –0.25. The line intersects the x–axis at x = (–3.5*|f(–2.5)| + (–2.5)*|f(–3.5)|) / (|f(–3.5)| + |f(–2.5)|) = (–3.5*(0.25) + (–2.5*0.75)) / (0.75 + 0.25) = –2.75. We have f(–2.75) = –0.1875. The new interval must be between f(–3.5) and f(–2.75) since there is a change of sign. One then repeats the process, finding the new straight line's intercept with the x–axis. One progressively gets closer to the root of x = –3.
Find a root of f(x) = sin(x) + x + 2 using linear interpolation, knowing the root is between x = –1 and x = –2.
The actual root is x = –1.106… Using linear interpolation should converge to this root, but you obviously stop the iterative process at some point on an approximate root.
End of page
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https://www.bondroberts.com/product/view/24332/Cohiba_Genios__2016_Black_lacquered_Bo%C3%AEte_Nature_Box_of_10_c
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math
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Buy Now Price: US $0
Available Quantity: 1
Packaging: Black lacquered Boîte Nature Box of 10 c
Number in box: 9
Standard Delivery Included: No
Delivery To: All *
Stored with Boveda 65 and 16-18 C, well packed, includes a boveda 69
Are you old enough to purchase tobacco products in your country of residence?
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https://wafflegame.io/game/wafflegame/
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math
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Solve the WAFFLE in 15 moves or less.
Rearrange the letters into the correct words, horizontally and vertically. Drag letters anywhere on the board. The letters will change colour to show whether they are in the correct position. The number of moves remaining is displayed below the board.
This letter is in the correct place:
This letter belongs somewhere else in that word:
This letter belongs somewhere else in that vertical word:
This letter is on a corner, so it may either belong somewhere else in the horizontal word or in that vertical word:
Every WAFFLE can be solved in 10 moves.
You will earn a star for every move you have remaining once the WAFFLE is solved ⭐
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https://ssvconference.com/electricity-generation/your-question-how-the-electric-field-depends-upon-the-distance-due-to-an-infinitely-long-thin-straight-charged-wire.html
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math
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A conduction disorder is a problem with the electrical system that makes your heart beat and controls its rate and rhythm. This system is called the cardiac conduction system. Normally, the electrical signal that makes your heart beat travels from the top of your heart to the bottom.
How electric field due to an infinite long straight conductor varies with distance?
Where λ = linear charge density, r = radius of the cylinder, and εo = permittivity of free space. From the above equation, it is clear that the electric field of an infinitely long straight wire is proportional to 1/r. Hence option 1 is correct.
How the electric field E depends upon the distance r due an infinitely long uniformly charged thin wire?
Electric field due to infinite long charged wire
Since the magnitude of the electric field for the entire curved surface is constant, E is taken out of the integration and Qencl is given by Qencl = λL. The electric field due to the infinite charged wire depends on 1r rather than 1r2 1 r 2 for a point charge.
What is electric field at a point due to infinitely long thin charge?
2πrlE = λlϵo λ l ϵ o E=12πϵoλr. Therefore, the above equation is the electric field due to an infinitely long straight uniformly charged wire.
What is electric field intensity due to an infinitely long straight charged wire?
Electric field produced due to an infinitely long straight uniformly charged wire at perpendicular distance of 2cm is 3×108NC−1.
How does the electric field E vary with distance?
Electric field strength is location dependent, and its magnitude decreases as the distance from a location to the source increases. And by whatever factor the distance is changed, the electric field strength will change inversely by the square of that factor.
How do electric field varies with distance due to linear charge?
The electric field varies inversely as the square of the distance from the point charge.
What is the expression for electric field due to dipole at equatorial position?
Let P be a point at a distance r from the center of the dipole on the equatorial line, where the electric field is to be calculated. Let E1 be electric field intensity at P due to charge –q. Therefore, From right-angled triangle AOP, we have AP = √r2 + a2 .
When an electric dipole P is placed in a uniform electric field E?
An electric dipole of moment P is placed in a uniform electric field E such that P points along E . If the dipole is slightly rotated about an axis perpendicular to the plane containing E and P and passing through the centre of the dipole, the dipole executes simple harmonic motion.
What is the potential energy due to dipole in an external electric field?
The potential energy of a dipole in an external field
τ = p × E. This work is saved as the system’s potential energy. The potential energy U(θ) can then be linked to the dipole’s inclination θ.
Which law is used to find electric field at any point near the infinitely long straight uniformly charged wire state this law obtain expression for it?
State Gauss’ law. Using this find an expression for electric field due to an infinitely long straight charged wire uniform charge density.
What is electric field due to a straight wire?
Consider a long straight wire which carries the uniform charge per unit length. . We expect the electric field generated by such a charge distribution to possess cylindrical symmetry. We also expect the field to point radially (in a cylindrical sense) away from the wire (assuming that the wire is positively charged).
What is the electric field at the midpoint O of the line AB joining the two charges?
O is the mid-point of line AB. Therefore, the electric field at mid-point O is 5.4 × 106 N C−1 along OB.
What is the relationship of electric field and electric potential?
The relationship between potential and field (E) is a differential: electric field is the gradient of potential (V) in the x direction. This can be represented as: Ex=−dVdx E x = − dV dx . Thus, as the test charge is moved in the x direction, the rate of the its change in potential is the value of the electric field.
Which is the correct relation for electric intensity and potential due to a point charge?
The relation is very simple. Electric field intensity is equal to the negative of rate of change of potential with respet to the distance or it can be defined as the negative of the rate of derivative of potential difference, V with respect to r, E = – dV/dr.
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https://www.coursehero.com/file/6775619/14-special-rel/
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math
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Unformatted text preview: UCSD Physics 10 Special Relativity
Einstein messes with space and time UCSD Physics 10 How Fast Are You Moving Right Now? 0 m/s relative to your chair 400 m/s relative to earth center (rotation) 30,000 m/s relative to the sun (orbit) 220,000 m/s relative to the galaxy center (orbit) 370,000 m/s relative to the CMB cosmic wallpaper Relative to What?? This is part of the gist of special relativity it's the exploration of the physics of relative motion only relative velocities matter: no absolute frame very relevant comparative velocity is c = 300,000,000 m/s UCSD Physics 10 A world without ether For most of the 19th century, physicists thought that space was permeated by "luminiferous ether" this was thought to be necessary for light to propagate Michelson and Morley performed an experiment to measure earth's velocity through this substance first result in 1887 Michelson was first American to win Nobel Prize in physics Found that light waves don't bunch up in direction of earth motion shocked the physics world: no ether!! speed of light is not measured relative to fixed medium unlike sound waves, water waves, etc. UCSD Physics 10 Speed of light is constant: so what? Einstein pondered: what would be the consequences of a constant speed of light independent of state of motion (if at const. velocity) any observer traveling at constant velocity will see light behave "normally," and always at the same speed Mathematical consequences are very clear forced to give up Newtonian view of space and time as completely separate concepts provides rules to compute observable comparisons between observers with relative velocity thus "relativity": means relative state of motion UCSD Physics 10 Simultaneity is relative, not absolute
Observer riding in spaceship at constant velocity sees a flash of light situated in the center of the ship's chamber hit both ends at the same time But to a stationary observer (or any observer in relative motion), the condition that light travels each way at the same speed in their own frame means that the events will not be simultaneous. In the case pictured, the stationary observer sees the flash hit the back of the ship before the front UCSD Physics 10 One person's space is another's time If simultaneity is broken, no one can agree on a universal time that suits all the relative state of motion is important Because the speed of light is constant (and finite) for all observers, space and time are unavoidably mixed we've seen an aspect of this in that looking into the distance is the same as looking back in time Imagine a spaceship flying by with a strobe flashing once per second (as timed by the occupant) the occupant sees the strobe as stationary you see flashes in different positions, and disagree on the timing between flashes: space and time are mixed see description of light clock in text Space and time mixing promotes unified view of spacetime "events" are described by three spatial coordinates plus a time UCSD Physics 10 The Lorentz Transformation There is a prescription for transforming between observers in relative motion
ct' = (ct - vx/c); x' = (x - vt); y' = y; z' = z "primed" coordinates belong to observer moving at speed v along the x direction (relative to unprimed) note mixing of x and t into x' and t' time and space being nixed up multiplying t by c to put on same footing as x now it's a distance, with units of meters the (gamma) factor is a function of velocity: UCSD Physics 10 The gamma factor Gamma ( ) is a measure of how whacked-out relativistic you are When v = 0, = 1.0 and things are normal At v = 0.6c, = 1.25 a little whacky At v = 0.8c, = 1.67 getting to be funky As vc, UCSD Physics 10 What does do? Time dilation: clocks on a moving platform appear to tick slower by the factor at 0.6c, = 1.25, so moving clock seems to tick off 48 seconds per minute standing on platform, you see the clocks on a fast-moving train tick slowly: people age more slowly, though to them, all is normal Length contraction: moving objects appear to be "compressed" along the direction of travel by the factor at 0.6c, = 1.25, so fast meter stick will measure 0.8 m to stationary observer standing on a platform, you see a shorter train slip past, though the occupants see their train as normal length UCSD Physics 10 Why don't we see relativity every day? We're soooo slow (relative to c), that length contraction and time dilation don't amount to much 30 m/s freeway speed has v/c = 10-7 = 1.000000000000005 30,000 m/s earth around sun has v/c = 10-4 = 1.000000005 but precise measurements see this clearly UCSD Physics 10 Velocity Addition Also falling out of the requirement that the speed of light is constant for all observers is a new rule for adding velocities Galilean addition had that someone traveling at v1 throwing a ball forward at v2 would make the ball go at v1+v2 In relativity, reduces to Galilean addition for small velocities can never get more than c if v1 and v2 are both c if either v1 OR v2 is c, then vrel = c: light always goes at c UCSD Physics 10 Classic Paradoxes The twin paradox: one twin (age 30) sets off in rocket at high speed, returns to earth after long trip if v = 0.6c, 30 years will pass on earth while only 24 will pass in high speed rocket twin returns at age 54 to find sibling at 60 years old why not the other way around? Pole-vaulter into barn high-speed runner with 12 meter pole runs into 10 meter barn; barn door closes, and encompasses length-contracted 9.6 m pole (at 0.6c) but runner sees barn shrunken to 8 m, and is holding 12 m pole! can the barn door close before the pole crashes through the back? resolution in lack of simultaneity: "before" is nuanced UCSD Physics 10 If I'm in a car, traveling at the speed of light... If I turn on my headlights, do they work? Answer: of course--to you, all is normal you are in an un-accelerated (inertial) frame of reference all things operate normally in your frame To the "stationary" outsider, your lights look weird but then again, so do you (because you're going so fast) in fact, at the speed of light, all forward signals you send arrive at the same time you do And the outside, "stationary" world looks weird to you But I must inquire: how did you manage to get all the way up to the speed of light?! UCSD Physics 10 What would I experience at light speed? It is impossible to get a massive thing to travel truly at the speed of light energy required is mc2, where as vc so requires infinite energy to get all the way to c But if you are a massless photon... to the outside, your clock is stopped so you arrive at your destination in the same instant you leave your source (by your clock) across the universe in a perceived instant makes sense, if to you the outside world's clock has stopped: you see no "ticks" happen before you hit UCSD Physics 10 E = mc2 as a consequence of relativity Express 4-vector as (ct, x, y, z) describes an "event": time and place time coordinate plus three spatial coordinates factor of c in time dimension puts time on same footing as space (same units) We're always traveling through time our 4-velocity is (c, 0, 0, 0), when sitting still moving at speed of light through time dimension stationary 4-momentum is p = mv(mc, 0, 0, 0) for a moving particle, p = ( mc, px, py, pz) where px, etc. are the standard momenta in the x, y, and z directions the time-component times another factor of c is interpreted as energy conservation of 4-momentum gets energy and momentum conservation in one shot UCSD Physics 10 E = mc2, continued can be approximated as = 1 + v2/c2 + ...(small stuff at low velocities) so that the time component of the 4-momentum c is:
m c2 = mc2 + mv2 + ... the second part of which is the familiar kinetic energy Interpretation is that total energy, E = m c2 mc2 part is ever-present, and is called "rest mass energy" kinetic part adds to total energy if in motion since sticks to m in 4-momentum, can interpret this to mean mass is effectively increased by motion: m m gets harder and harder to accelerate as speed approaches c UCSD Physics 10 Experimental Confirmation We see time dilation in particle lifetimes in accelerators, particles live longer at high speed their clocks are running slowly as seen by us seen daily in particle accelerators worldwide cosmic rays make muons in the upper atmosphere these muons only live for about 2 microseconds if not experiencing time dilation, they would decay before reaching the ground, but they do reach the ground in abundance We see length contraction of the lunar orbit squished a bit in the direction of the earth's travel around the sun E = mc2 extensively confirmed nuclear power/bombs sun's energy conversion mechanism bread-and-butter of particle accelerators UCSD Physics 10 References Relativity Visualized by Lewis Carroll Epstein http://www.anu.edu.au/physics/Searle/ movie Assignments Q/O #3 due today by midnight Partial read of Chapters 9 & 10 (pages on assignment page) Read Chapters 35 & 36 on relativity HW5: 9.R.13, 9.E.9, 9.E.14, 9.E.43, 9.P.7, 10.E.16, 35.R.27, 35.E.6, 35.E.19, 35.E.20, 35.E.37, 35.P.3, 35.P.10, 36.R.7, 36.E.2, 36.E.6 ...
View Full Document
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CC-MAIN-2018-13
| 9,190 | 6 |
https://www.goodman-gallery.art/reading-room
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math
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5 Questions is an initiative which, through a series of questions, gives insights into what texts artists, creatives and thought leaders are reading, what texts inspire them and what texts give solace during this time. Different people will answer the same set of five questions and their answers will be published on Goodman Gallery’s reading room. The questions we will ask are:
1. What are you reading right now?
2. Is there a text from which you have drawn inspiration throughout your practice?
3. Do you have a favorite author and book? If so, what is it and why?
4. Is there another medium of expression from which you draw inspiration?
5. Is there a text that gives you solace in this time of uncertainty?
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CC-MAIN-2020-24
| 714 | 6 |
https://www.omnicalculator.com/finance/break-even
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math
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Starting a business? This break-even calculator allows you to perform a task crucial to any entrepreneurial endeavor. The goal of a break-even analysis is to let you know how many units of goods you need to sell in order to cover all of your outgoing costs (cost of goods sold and other, fixed costs that are not tied to the quantity of inventory). Please go ahead and use the calculator, we hope it's fairly straightforward. If you'd rather calculate it manually, we have described how to calculate break-even point below, and even explained what is the break-even point formula.
How to calculate break-even point
- Find out how much you make on every unit. For example, if you buy for
$30and sell for
$45, your gross profit per item is
$15(let's assume you don't have other per-unit costs).
2 Identify your fixed costs. In our example, we'll spend
$2700 (office rent, utilities, etc.)
3. Divide your fixed costs by the profit you make on every unit -
$2700 / $15 = 180 - this is how many units you need to sell.
4. The overall sales figure is easy:
180 * $45 = $8100.
5. Your done!.. or use our break-even calculator :-)
The manual break-even analysis is super easy once you realize that you simply need to balance fixed costs with gross profit. Let's go you through the whole process:
- The general equation is
fixed_costs = per_unit_profit * number_of_units
- Let's expand the profit. It's comprised of costs and revenue:
fixed_costs = (per_unit_revenue - per_unit_costs) * number_of_units
- We need to sell
number_of_units = fixed_costs / (per_unit_revenue - per_unit_costs)
- We need to talk in terms of dollars brought in, so this changes the break-even point formula to:
total_revenue = per_unit_revenue * fixed_costs / (per_unit_revenue - per_unit_costs)
Depending on your needs, you may need to calculate your profit margin or markup to find your revenue... or even the other way around. This will allow you to calculate the maximum price you may pay for goods, given all of your other numbers. You can also check out our markup calculator and margin calculator. markup formula
Break-even analysis is often confused with payback time. The latter is a similar calculation, but it's based around knowing how much you bring in over a certain period of time. It might be a good idea to come back to break-even calculator after you actually start doing business. Often times you will find the need to adjust your costs and factor in things you overlooked before.
As with most business calculations, it's quite common that different people have different needs. For example, your break-even point formula might need to be accommodate costs that work in a different way (you get a bulk discount or fixed costs jump at certain intervals).
Also, remember that this analysis doesn't take into consideration the present vs. future value of your funds. See the time value of money calculator for more information about this topic.
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http://slideplayer.com/slide/3355271/
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math
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Presentation on theme: "Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic,"— Presentation transcript:
Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic, Diatomic, Mass Defect, 2D Lattices) Strain (compliance, reduced notation, tensors) Harmonic Oscillator (Destruction/Creation, Hamiltonian & Number Operators, Expectation Values) Energy Density and Heat Capacity (phonons, electrons and photons) Quasiparticle Interactions (e-e, e-phonon, e-photon, defect interations) Electrical and Thermal Conductivity
Lattice Vibrations Longitudinal Waves Transverse Waves When a wave propagates along one direction, 1D problem. Use harmonic oscillator approx., meaning amplitude vibration small. The vibrations take the form of collective modes which propagate. Phonons are quanta of lattice vibrations.
The force on the n th atom; The force to the right; The force to the left; The total force = Force to the right – Force to the left aa U n-1 U n U n+1 Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies Monatomic Linear Chain Thus, Newton’s equation for the n th atom is
Brillouin Zones of the Reciprocal Lattice 1st Brillouin Zone (BZ=WS) 2nd Brillouin Zone 3rd Brillouin Zone Each BZ contains identical information about the lattice 2 /a Reciprocal Space Lattice: There is no point in saying that 2 adjacent atoms are out of phase by more than (e.g., 1.2 =-0.8 ) Modes outside first Brillouin zone can be mapped to first BZ
Diatomic Chain(2 atoms in primitive basis) 2 different types of atoms of masses m1 and m2 are connected by identical springs U n-2 U n-1 U n U n+1 U n+2 K KK K m1 m2 m a) b) (n-2) (n-1) (n) (n+1) (n+2) a Since a is the repeat distance, the nearest neighbors separations is a/2 Two equations of motion must be written; One for mass m1, and One for mass m2.
As there are two values of ω for each value of k, the dispersion relation is said to have two branches Upper branch is due to the positive sign of the root. Negative sign: k for small k. Dispersion- free propagation of sound waves Optical Branch Acoustical Branch This result remains valid for a chain containing an arbitrary number of atoms per unit cell. 0л/a2л/a–л–л/a k A B C A when the two atoms oscillate in antiphase At C, M oscillates and m is at rest. At B, m oscillates and M is at rest.
Number and Type of Branches Every crystal has 3 acoustic branches, 1 longitudinal and 2 transverse Every additional atom in the primitive basis contributes 3 further optical branches (again 2 transverse and 1 longitudinal)
2D Lattice K U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Write down the equation(s) of motion What if I asked you to include second nearest neighbors with a different spring constant?
2D Lattice C U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Similar to the electronic bands on the test, plot w vs k for the and directions. Identify the values of at k=0 and at the edges.
Specific Heat or Heat Capacity The heat energy required to raise the temperature a certain amount The thermal energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Classical Picture of Heat Capacity When the solid is heated, the atoms vibrate around their sites like a set of harmonic oscillators. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3kT, and consequently the energy per mole is = Dulong-Petit law: states that specific heat of any solid is independent of temperature and the same result (3R~6cal/K-mole) for all materials!
Average energy of a harmonic oscillator and hence of a lattice mode at temperature T Energy of oscillator The probability of the oscillator being in this level as given by the Boltzman factor Thermal Energy & Heat Capacity Einstein Model
Mean energy of a harmonic oscillator Low Temperature Limit Zero Point Energy exponential term gets bigger High Temperature Limit is independent of frequency of oscillation. This is a classical limit because the energy steps are now small compared with thermal/vibrational energy <<
Heat Capacity C (Einstein) Heat capacity found by differentiating average phonon energy where T(K) Area = The difference between classical and Einstein models comes from zero point energy. Points:Experiment Curve: Einstein Prediction The Einstein model near T= 0 did not agree with experiment, but was better than classical model. Taking into account the distribution of vibration frequencies in a solid this discrepancy can be accounted for.
1.Approx. dispersion relation of any branch by a linear extrapolation 2.Ensure correct number of modes by imposing a cut-off frequency, above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that: Debye approximation to the dispersion Debye approximation has two main steps Einstein approximation to the dispersion
Density of states (DOS) per unit frequency range g( ) The number of modes/states with frequencies and +d will be g( )d . # modes with wavenumber from k to k+dk= for 1D monoatomic lattice
The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus Mean energy of a harmonic oscillator for 1D It would be better to find 3D DOS in order to compare the results with experiment. Debye Model adjusts Einstein Model
3D Example: The number of allowed states per unit energy range for free electron? Each k state represents two possible electron states, one for spin up, the other is spin down.
L L L Octant of the crystal: k x,k y,k z (all have positive values) The number of standing waves;
The Heat Capacity of a Cold Fermi Gas (Metal) Close to E F, we can ignore the variation in the density of states: g( ) g(E F ). By heating up a metal (k B T << E F ), we take a group of electrons at the energy - (with respect to E F ), and “lift them up” to . The number of electrons in this group g(E F )f( )d and each electron increased its energy by 2 : The small heat capacity of metals is a direct consequence of the Pauli principle. Most of the electrons cannot change their energy. kBTkBT
Bam! Random Collisions On average, I go about seconds between collisions with phonons and impurities electron phonon Otherwise metals would have infinite conductivity Electrons colliding with phonons (T > 0) Electrons colliding with impurities imp is independent of T The thermal vibration of the lattice (phonons) will prevent the atoms from ever all being on their correct sites at the same time. The presence of impurity atoms and other point defects will upset the lattice periodicity
Fermi’s Golden Rule Transition rate: Quantum levels of the non-perturbed system Perturbation is applied Transition is induced (E) is the ‘density of states available at energy E’. See Fermi‘s Golden Rule paper in Additional Material on the course homepage
Absorption When the ground state finds itself in the presence of a photon of the appropriate frequency, the perturbing field can induce the necessary oscillations, causing the mix to occur. This leads to the promotion of the system to the upper energy state and the annihilation of the photon. This process is stimulated absorption (or simply absorption). Einstein pointed out that the Fermi Golden Rule correctly describes the absorption process. - degeneracy of state f
Quantum Oscillator Atoms still have energy at T=0. What is for the ground state of the quantum harmonic oscillator? (1D Case) For 3D quantum oscillator, the result is multiplied by 3: ⇒
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical particles which have no classical analogue. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many different areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs
Phonon spectroscopy = Constraints: Conservation laws of MomentumEnergy Conditions for: elastic scatteringin In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector.
x=(a-b)/2 or The cubic axes are equivalent, so the diagonal components for normal and shear distortions must be equal. And cubic is not elastically isotropic because a deformation along a cubic axis differs from the stress arising from a deformation along the diagonal. e.g., vs. Zener Anisotropy Ratio:
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| 8,999 | 27 |
https://jp.mathworks.com/matlabcentral/profile/authors/15964798
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math
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Spectral Analysis on EEG data
Hi everyone, I am quite new in MatLab as well as in EegLab. Some months ago I started a project using EEG, and now I would like...
2年以上 前 | 0 件の回答 | 0
How to create a specific Matrix
Hi ! Can anyone help me in builting this specific matrix? I do not know if the "randi" function is the exact one. Thank you ...
2年以上 前 | 2 件の回答 | 0
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337625.5/warc/CC-MAIN-20221005105356-20221005135356-00529.warc.gz
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CC-MAIN-2022-40
| 392 | 6 |
https://socialsci.libretexts.org/Workbench/Demography_and_Economics_(Hageman_and_Galoustian)/04%3A_Intrinsic_Population_Change/4.05%3A_Chapter_18-_Population_Growth_and_Sustainability
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math
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Anya Hageman and Pauline Galoustian
In our previous two chapters we explored how the age structure of the population affects the economy. Now we focus on how the rate of population growth affects the economy.
The model of economic growth by Robert Solow (1956) is very well-known, simple, and easy to manipulate, so we’ll have a look at it and see what it predicts about the economic consequences of population growth. Its message will not be about efficiency, because efficiency is held constant in the Solow model. It will not be about hours worked or about the fraction of the population that works, because in the Solow model, labour is measured as the number of people in the population: everyone is assumed to work full-time.
The Solow model’s message will be about the capital-labour ratio, K/L, and the importance of accumulating capital to keep up with the number of workers.
The Solow model uses the aggregate production function Y = A F(K,L)
Y = aggregate output of the economy. There’s just one thing produced.
A = efficiency. This is held constant.
F(K,L) = the production function. The production function exhibits constant returns to scale; that is to say, if you double K and double L, F(K,L) doubles in size.
K = physical capital. This time we are not using K+ as we did in Chapter 16. K+ represents a number of different kinds of capital, and different kinds of capital may present mathematical complications. For example, if we include human capital, we might reasonably expect that human capital accumulation would affect efficiency, A, and we’d need an equation to show how that happens. If we included non-renewable natural resource capital, we’d need an equation to show how the resource stock is being depleted.
L = labour. This is identical to the number of people in the population. It grows at rate n.
Why is efficiency A held constant? We might think of including an equation that shows how efficiency grows, either exogenously (automatically) or endogenously (in response to something in the model, such as population growth). However, if A is allowed to grow, then output Y exhibits increasing returns to scale. Doubling K and L while increasing A would result in more than double the output. If increasing returns to scale were in place, output Y and consumption could grow forever in the presence of population growth. Sustainability would be too easy, at least mathematically.
Because efficiency growth such as technological change makes sustainability so easily to achieve, it’s more interesting to hold technological change constant and see what happens when technology and other forms of efficiency don’t improve.
We think of efficiency A as possibly changing depending on the age structure of the workforce, but in the Solow Model there is no change to the age structure.
This model is almost as simple as the Malthusian model. One difference is that in Solow’s model the population growth rate never changes. A second difference is that in Solow’s model, there is not only labour, but also physical capital. People can either eat the output Y or save some of it. The saved Y is invested and is transformed into capital. Think of Y as corn, which you can either eat or save for planting next spring.
The production function F(K,L) can be any positive monotonic function of K and L so long as
- Both K and L are essential to production. If one of them is equal to zero, then output must be equal to zero.
- There is some degree of substitutability between K and L.
- F(K,L) demonstrates diminishing marginal returns to K and to L.
- F(K,L) exhibits constant returns to scale. As noted above, if you double the inputs, you double the output. Similarly, if you divide the inputs by some number, you divide the output by that same number. This makes it possible easily to express the production function in terms of output per worker, dividing by L.
Assuming constant returns to scale, Solow divides everything by the number of workers so that there is one production process y=f(k) which uses capital-per-worker k to produce output-per-worker y. Capital-per-worker is denoted by lowercase k and is called the capital-to-labour ratio or capital:labour ratio.
Figure 18-1 shows output-per-worker as a function of the capital:labour ratio. This function demonstrates diminishing marginal returns to k.
Figure 18-2 shows the same function multiplied by fraction s. s is the savings rate.
In the Solow model, we produce y, which represents output per worker or output per person (since all people work). Fraction (1-s) of this is consumed, and fraction s is saved/invested into the stock of physical capital, K. Every year, K increases by sY. It can be shown (if interested, see the end of this chapter) that the amount of savings needed to keep the capital:labour ratio steady must satisfy Equation 18-1.
The amount saved per worker must equal the population growth rate/labour force growth rate multiplied by the capital:labour ratio.
The original Solow model also included another term, called d. d is the rate at which physical capital depreciates, by rusting away or becoming obsolete. If you include d in the model, Equation 18-1 becomes sy=(n+d)k. We show that at the end of this chapter.
According to Equation 18-1, sustainability requires that a population must save enough to offset something, something to do with population growth. Let’s have a look at Figure 18-3 to learn more.
The blue line above, sy, represents the left-hand side of the Solow condition (Equation 18-1), while the yellow line, nk, represents the right-hand side. Where these lines intersect is the level of k, where the Solow condition is satisfied.
The lines intersect at a particular level of k, called the steady state capital:labour ratio, or k*.
Wherever the blue savings line is higher than the yellow population line, sy > nk, and the capital:labour ratio rises. As it rises, we move rightward along the horizontal axis until we get to k*.
Wherever the blue savings line is lower than the yellow population line, sy < nk, and the capital:labour ratio falls. We call this “capital shallowing“. As k falls, we move leftward along the horizontal axis until we get to k*.
So whatever capital:labour ratio k we start out with, we tend to reach k*. k* is an equilibrium.
If we start out with a k that is higher than k*, k falls. That’s because k is so high that diminishing returns are kicking in, and the extra output we get from our investment is not enough to prevent capital shallowing. If we start off with a low k, we will find that our savings and investments are so productive that capital accumulates faster than population.
This is exciting! Whatever savings rate we choose, we can achieve:
Even though the labour force in the Solow model is constantly growing at rate n, the amount each worker/person consumes will never fall. Consumption per person never falls = that’s what theoretical macroeconomists call sustainability. Like a lot in economics, the conception of sustainability is very anthropocentric.
Population growth will cause capital shallowing, but we can save and reverse that capital shallowing. By means of saving, the capital stock can grow as quickly as the population.
If the population growth rate were to rise for some reason, our yellow population line would become steeper. It would intersect the blue savings line at a lower k*. This would means lower output per worker and lower consumption per worker. However, the output and consumption per worker are still constant every year. They are still sustainable. Consumption may be reduced because population growth has accelerated, but it is still sustainable, unless it falls below the critical threshold needed to support human life.
If the efficiency level A were to rise for some reason, again our blue savings line would pivot up and intersect the yellow population line at a higher steady state k*. Output per worker would rise, and so would consumption per worker, since output has increased and the savings rate has not changed.
If the savings rate were to increase for some reason, our blue savings line would pivot up and intersect the yellow population line at a higher steady state k*. This would result in a higher steady state level of output per person; however, because a higher fraction of that output is being saved, it’s not clear whether consumption per person would actually increase.
We can solve mathematically for the savings rate that would result in the highest level of consumption per person. That savings rate is the one that achieves the “golden rule” k*. The golden rule k* is found where the slope of the per-worker production function (y=Y/L=Af(k*)) is equal to n.
The Solow model tells us that, if we save, we can achieve sustainability of consumption despite population growth, unless the rate of population growth is so very high that k* and the resulting consumption (=(1-s)y) are too low to support life. In the real world, things are more complicated. The Solow model ignores the natural environment. It ignores the fact that not everyone works. And it assumes that the rate of population growth doesn’t affect how much people save. In the real world there is environmental degradation, there is dependency, and there is the likelihood that dependency reduces workers’ savings.
When the rate of population growth is increasing, it is likely that the young dependency ratio (YDR) and total dependency ratio (TDR) are increasing. It is likely that savings will fall as parents devote resources to caring for the young. Kelley (1998) calls this the youth dependency effect. It is also likely that governments will spend on education and health care for the young, diverting money away from investment in various forms of physical and knowledge capital. Most of the spending on children will not improve the productivity of the current working generation. Kelley calls this the investment-diversion effect.
A. C. Kelley (1998) reviewed the many journal articles on population and economic growth available at that time. He concluded that,ceteris paribus, the rate of population growth likely reduces the standard of living through the youth dependency effect, the investment-diversion effect, and capital shallowing. However, there is no clear empirical relationship between the population growth rate and per capita output. Many other important factors also influence per capita output, factors like the economy’s overall size, its civil and political institutions, its educational achievement, and its openness to trade (Bloom 2003).
If we observe an apparent negative correlation between income growth and population growth, we have to remember that causation can flow both ways in a negative feedback loop. Low economic growth could mean lower rates of female education, low rates of female employment, higher rates of infant mortality, and a smaller social safety net, all of which tend to increase fertility.
The graph below shows GDP per capita growth, adjusted for inflation, between 1960 and 2000, for 98 different countries, as a function of the average population growth rate for each country between 1960 and 2000. A simple best fit line has been generated. Variation in population growth rates “explains” 25% of the variation in GDP per capita growth, or vice versa. 75% of the variation remains unexplained.
The Solow model paints a scenario where saving prevents consumption per worker from falling as the population grows. Could that work if there are natural resources needed for production?
The Solow model will continue to generate sustainable output and consumption per worker when the production function includes a renewable resource. The renewable resource must be harvested sustainably. No more of the resource each year can be harvested than can grow back in one year. In fact, we should harvest less than that, because the stock must continue to grow as long as population grows.
When we add a non-renewable resource, such as oil, coal, or lithium to the production function, the Solow model can no longer generate sustainable output or sustainable consumption in the presence of population growth, at least not as long as population growth is geometric or exponential.
In the Solow model, savings can compensate for capital shallowing and capital depreciation, or savings can compensate for non-renewable resource depletion, but not both.
If there were no population growth (i.e. n=0) and no depreciation of capital (i.e. d=0), then consumption could be sustained despite the depletion of non-renewables like oil. Solow (1974) and Hartwick (1977) showed that IF physical capital can substitute to some degree for non-renewables like oil, and IF enough physical capital were accumulated to make up for the diminishing stock, then consumption could be sustained indefinitely.
Hartwick derived the formula for the precise amount of savings needed to make up for declining non-renewable resource stocks. The amount needed is equal to the amount of non-renewable resource extracted multiplied by rent (price minus marginal cost) on the marginal ton. This amount is known as Total Hotelling Rent.
Hartwick’s Rule tells us to invest non-renewable resource rents in other forms of capital. This will keep our output and consumption steady, so long as the savings rate is not affected and dependency is not an issue. If there is geometric or exponential population growth, sustainability is not achievable; however, if there is arithmetic or quasi-arithmetic population growth, sustainability can be achieved by investing even more than Total Hotelling Rent.
In real life, our population has grown and our capital stock has more than kept up, because of technological improvements, other efficiency improvements, and the colonization of new lands and peoples. Neither Malthus nor Solow nor Hartwick include technological change in their models. That is because the introduction of technical change into a mathematical model will too easily generate sustainability.
Hartwick’s Rule depends on the production function being the kind where the inputs are multiplied together to yield the output. This means that it is always possible to make up for a shrinking amount of one input by using more of another input. In Hartwick’s model, an expanding stock of K makes up for a diminishing stock of non-renewable resource. R. Herman Daly (1990), one of the founders of Ecological Economics, has pointed out that there may be critical thresholds below which all the physical capital in the world cannot make up for the loss of natural or environmental capital.
Ecological Economics was established as a discipline in 1990 by economists who were concerned that traditional economics does not adequately consider the economy’s size and the population’s size relative to the carrying capacity of the environment. We will discuss population size in our next Chapter.
We can take Hartwick’s Rule as suggestive rather than definitive. It recommends something that common sense immediately recognizes: do not allow the stock of your capital to diminish. Invest the profits you earn from nonrenewable resources. Save for the day when your resources run out.
Many nations have created sovereign wealth funds to invest the tax revenue that their governments collect from the oil and gas industry. Alaska, Kuwait, and Norway have such sovereign wealth funds. Alberta contributed to its Sovereign Heritage Savings Trust Fund between 1976-1988, and 2005-8, but otherwise the revenue has been used by the government or distributed among Alberta’s residents. Chile and Venezuela use resource tax revenues to help with government spending needs.
Genuine Savings, also known as Adjusted Net Savings, is an estimate of whether a nation’s capital stock (including physical, human, natural, and environmental capital) is really growing or not. If genuine savings is positive, then the nation is wisely building up its capital stock. If genuine savings is negative, then the nation is dissipating its capital.
Here is the calculation:If the rate of genuine savings is positive, the nation is accumulating capital. The question then is whether the rate of capital accumulation is high enough to match population growth.
In Table 18-1 we see estimates of Genuine Savings (as a rate) for several countries, computed by the World Bank.
Compiled by Pauline Galoustian. Sources: World Bank (data.worldbank.org) /United Nations Population Division/Eurostat: Demographic Statistics/United Nations Statistical Division/Secretariat of the Pacific Community (CC BY 4.0)
How can we tell whether the Genuine Savings of a country is enough to keep its consumption sustainable? If the population is not growing, any genuine savings above zero indicates an increase in the productive capacity of the economy and an improvement in its ability to provide consumption sustainably into the future. If the population is growing geometrically or exponentially (as is usually the case), then Genuine Savings needs to be impossibly high unless technical change and efficiency improvements are occurring (as is usually the case). In the situation of population growth with technical change, we don’t have a simple equation to calculate how high a nation’s Genuine Savings needs to be.
The World Bank (2011), in its Appendix E, tried to estimate that anyway. Their calculations for the year 2005 purported to show that Canada’s genuine savings that year had been sufficient to cover its population growth. They also estimated that the United States had needed to save an additional 2% of gross national income in 2005 to keep its capital-per-person (they did not calculate it per worker) intact.
In our next chapter, we’ll study the effects on the economy of the absolute size of the population.
The end-of-chapter questions follow the Appendix below.
Let K(t) be the capital stock at time t. s is the savings rate. d is the rate at which capital breaks down or becomes obsolete: the depreciation rate. We will set d = 0 for simplicity. L(t), the labour force at time t, is growing every year at rate n.
The production function is:
A(t) is efficiency or technology at time t and we just hold it constant, meaning:
Dividing by L(t), we write y(t) = Af (1, k(t)) where y is output per worker and k is capital per worker.
We can ignore the 1 and write:
The following equation shows how the capital stock grows from year to year:
Translation: capital next year = capital this year + amount saved minus capital lost to decay and obsolescence. Rearranging:
What we’re going to do now is divide everything by K(t).
Using the fact that,
the equation becomes:
The left-hand side is “percentage change in K”. Now the percentage change in little k is equal (by definition) to the percentage change in K minus the percentage change in L. The percentage change in L is the population growth rate, since in this model, everyone is in the labour force. So let’s replace the left-hand side of our Solow equation with the percentage change in k plus n, the population growth rate.
Now multiply both sides by k(t) and we have our final version:
This is Equation 18.1 It tells us that, for capital-per-worker to be constant over time – i.e. for the left hand side of this equation to be equal to zero, – savings per worker must equal the population growth rate multiplied by capital per worker.
1. What assumptions about the economy does the Solow model make?
2. In the Solow model, what are three ways that n, the rate of growth of population, can affect the steady-state capital:labour ratio, k*?
3. What is Hartwick’s Rule and how has it been criticized?
4. If in 2008, Country X sells 100,000,000 barrels of oil, and if the marginal cost of this oil is $92 per barrel, and if the price of this oil is $100 per barrel, what is Total Hotelling Rent for Country X in 2008?
5. Country Y in 2008 has:
- investment in physical capital=15.9 percent of GNI (gross national income)
- current spending on education = 5 percent of GNI
- depreciation of physical capital = 11.5 percent of GNI
- Total Hotelling Rent of 0
- over-harvesting renewable resources valued at 0.1 percent of GNI
- estimated damages from pollution valued at 0.3 percent of GNI
a) What is genuine savings for Country Y?
b) What can you tell me about Country Y?
- The intuition is that we keep raising k* until the net benefit of doing so is zero. The net benefit of raising k* is the resulting increase in output per worker minus the extra n units of output required to prevent capital shallowing. (Van Gaasbeck (2022)). ↵
- In quasi-arithmetic growth, N(t) = a + b(t) ↵
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816977.38/warc/CC-MAIN-20240415111434-20240415141434-00380.warc.gz
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CC-MAIN-2024-18
| 20,696 | 92 |
https://electricalacademia.com/basic-electrical/rl-series-circuit/
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math
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This guide covers Series RL Circuit Analysis, its Phasor Diagram, Power & Impedance Triangle, and several solved examples.
In a purely resistive AC circuit, any inductive effects are considered negligible. Similarly, in a purely inductive AC circuit, any resistive effects are considered extremely small, and as a result they are omitted from any calculations. In many AC circuits, however, the load is actually a combination of both resistance and inductance. That is, the circuit can no longer be treated as either purely resistive or as purely inductive.
The combination of a resistor and inductor connected in series to an AC source is called a series RL circuit. Figure 1 shows a resistor and a pure or ideal inductor connected in series with an AC voltage source.
The current flow in the circuit causes voltage drops to be produced across the inductor and the resistor. These voltages are proportional to the current in the circuit and the individual resistance and inductive reactance values.
As in any series circuit the current will be the same value throughout the circuit. The resistor voltage (ER) and the inductor voltage (EL) expressed in terms of Ohm’s law are
Figure 1 Series RL circuit diagram
The total opposition to current flow in any AC circuit is called impedance. In a series RL circuit, this total opposition is due to a combination of both resistance (R) and inductive reactance (XL). The symbol for impedance is Z, and like resistance and reactance, it too is measured in ohms.
From Ohm’s law, the impedance of a circuit will be equal to the total supply voltage (ET) divided by the circuit current:
It was previously shown that the current flowing through a pure resistance was in phase with the voltage across the resistance and that the current through a pure inductance lagged the voltage across the inductance by 90 degrees. For this reason, in the series RL circuit the two voltage drops will not be directly additive but will be a vector sum.
The relationship between the current and voltages in a series RL circuit is shown in the vector (phasor) diagram of Figure 2 and can be summarized as follows:
- The reference vector is labeled I and represents the current in the circuit, which is common to all circuit elements.
- Since the voltage across the resistor is in phase with the current flowing through it, the voltage vector ER, it is drawn superimposed on the current vector.
- The inductor voltage EL leads the current by 90 degrees and is drawn leading the current vector by 90 degrees.
- The total supply voltage (ET) is the vector sum of the resistor and inductor voltages:
- The phase shift between the applied voltage and current is between 0 and 90 degrees.
- As the frequency increases, the inductive reactance (XL) increases, which causes the phase angle, or shift between the applied voltage and current, to increase.
Figure 2 Series RL circuit vector (phasor) diagram.
Due to the phase shift created by the inductor, the impedance of a series RL circuit cannot be found by simply adding the resistance and inductive reactance values. The total impedance of a series RL circuit, similar to its total voltage, is the vector sum of the resistance and inductive reactance.
The impedance triangle for a series RL circuit is shown in Figure 3. Note that the impedance triangle is geometrically similar to the circuit vector diagram and will have the same phase angle theta (θ). The reason for this is that the voltage drops for the resistor and the inductor are a result of the current flow in the circuit and their respective opposition. Equations used to solve the impedance triangle include:
Figure 3 Series RL circuit Impedance triangle.
Impedance Calculation in Series RL Circuit Example 1
Problem: An AC series RL circuit is made up of a resistor that has a resistance value of 150 Ω and an inductor that has an inductive reactance value of 100 Ω. Calculate the impedance and the phase angle theta (θ) of the circuit.
Once the impedance of a circuit is found it is possible to find the current by using Ohm’s law and substituting Z for R as follows:
Since the current is the same throughout the series circuit, the individual voltage drops across the inductor and resistor can be calculated by applying Ohm’s law as follows:
RL Series Circuit Calculations Example 2
Problem: For the series RL circuit shown in Figure 4:
- Calculate the value of the current flow.
- Calculate the value of the voltage drop across the resistor.
- Calculate the value of the voltage drop across the inductor.
- Calculate the circuit phase angle based on the voltage drops across the resistor and inductor.
- Express all voltages in polar notation.
- Use a calculator to convert all voltages to rectangular notation.
Figure 4 RL series circuit for Example 2.
The various power components associated with the series RL circuit are shown in Figure 5 and can be identified as follows:
- True power is measured in watts (W) and is the power drawn by the resistive component of the circuit. For a pure resistor the voltage and current are in phase, and power dissipated as heat is calculated by multiplying voltage by current (W=ER× IR).
- Reactive power is measured in volt-amperes reactive (VARs). Reactive power is the power continually stored and discharged by the magnetic field of the inductive load. For purely inductive loads, the voltage and current are 90 degrees out of phase, and true power in watts is zero. The inductive reactive power is calculated by multiplying the inductor voltage by its current (VARs=EL×IL)
- Apparent power is measured in volt-amperes (VA) and is the combination of the reactive and true power. For a series RL circuit the phase shift between the applied voltage and current is between 0 and 90 degrees. The apparent power or volt-amps is calculated by multiplying the applied voltage by the current flow (VA=ET×IT).
Figure 5 Power components associated with the RL series circuit.
The power triangle of Figure 6 shows the relationship between the various power components of a series RL circuit. In this triangle:
- The length of the hypotenuse of a right-angle triangle represents the apparent power.
- The angle theta (θ) is used to represent the phase difference.
- The side adjacent to theta (θ) represents the true power.
- The side opposite theta (θ) represents the reactive power.
- The power triangle is geometrically similar to the impedance triangle and the series RL circuit vector diagram.
Figure 6 Series RL circuit power triangle.
Power Calculations in RL Series Circuit Example 3
Problem: For the series RL circuit shown in Figure 7, determine:
- True power.
- Inductive reactive power.
- Apparent power.
Figure 7 RL series circuit for Example 3.
The power factor (PF) for any AC circuit is the ratio of the true power (also called real power) to the apparent power:
Power factor is a measure of how effectively equipment converts electric current to useful power output, such as heat, light, or mechanical motion. The power factor for a RL circuit is the ratio of the actual power dissipation to apparent power and can be summarized as follows:
- The power factor ranges from 0 to 1 and is sometimes expressed as a percentage.
- A 0 percent PF indicates a purely reactive load, while 100 percent PF indicates a purely resistive load.
- For circuits containing both resistance and inductive reactance, the power factor is said to be lagging (current lags) in some value between 0 and 1.
- The greater the power factor, the more resistive the circuit; the lower power factor, the more reactive the circuit.
- Circuit power factor is an indication of the portion of volt-amperes that are actually true power; a high PF indicates a high percentage of the total power is true power.
For many practical applications, the power factor of a circuit is determined by metering total circuit voltage, current, and power, as illustrated in the circuit of Figure 8. The power factor can then be determined by dividing the reading of the wattmeter by the product of the voltmeter and ammeter readings as follows:
Figure 8 Determining RL circuit power factor.
The power factor is not an angular measure but a numerical ratio with a value between 0 and 1. As the phase angle between the source voltage and current increases, the power factor decreases, indicating an increasingly reactive circuit. Any of the following equations can be used to calculate the power factor of a series RL circuit:
RL Series Circuit Example 4
Problem: For the series RL circuit shown in Figure 9, determine:
- Inductive reactance (XL).
- Impedance (Z).
- Current (I).
- Voltage drop across the resistor (ER) and inductor (EL).
- The angle theta (θ) and power factor (PF) for the circuit.
- True power (W), reactive power (VARs), apparent power (VA).
Figure 9 RL Circuit for Example 4.
- Step 1. Make a table and record all known values.
Step 2. Calculate XL and enter the value in the table.
Step 3. Calculate Z and enter the value in the table.
Step 4. Calculate IT, IR, and IL and enter the values in the table.
Step 5. Calculate ER and EL and enter the values in the table.
Step 6. Calculate the angle θ and PF for the circuit and enter the values in the table.
Step 7. Calculate the W, VARs, and VA for the circuit and enter the values in the table.
A real inductor has resistance due to the wire. It is impossible to have a pure inductance because all coils, relays, or solenoids will have a certain amount of resistance, no matter how small, associated with the coils turns of wire being used. This being the case, we can consider our simple coil as being a resistance in series with a pure inductance.
- Define the term impedance as it applies to AC circuits.
- What symbol is use to represent impedance?
- A circuit consists of a resistance of 20 Ω and an inductive reactance of 40 Ω connected in series and supplied from a 240-volt, 60-Hz source. Determine:
- The circuit impedance.
- Amount of current flow.
- The phase angle theta (θ) of the circuit.
- For the series RL circuit vector (phasor) diagram shown in Figure 10, determine the value of the voltage drop across the inductor.
Figure 10 Vector for review question 4.
5. The known quantities in a given series RL circuit are as follows: Resistance equals 8 Ω, inductive reactance equals 39 Ω, current equals 3 A, and the applied voltage is 120 volts, 60 Hz. Determine the following unknown quantities:
- Voltage across the resistor.
- Voltage across the inductor.
- Angle by which the applied voltage leads the current.
6. A wattmeter connected to a 240-volt, 60-Hz series RL circuit indicates a reading of 691 watts. A clamp-on ammeter used to measure current flow indicates a current of 4.8 A. Determine the:
- True power.
- Apparent power.
- Reactive power.
- Circuit power factor.
7. For the series RL circuit shown in Figure 11, determine:
- Apparent power.
- True power.
- Reactive power.
- Circuit power factor.
Figure 11 RL Series Circuit for review question 7.
8. Complete a table for all given and unknown quantities for the series RL circuit shown in Figure 12.
Figure 12 Circuit for review question 8.
9. The frequency to an RL series circuit is decreased. What effect will this have on the phase angle between the applied voltage and current? Why?
Review Questions – Answers
- The total opposition offered to the current flow in the AC circuit.
- (a) 44.7 Ω (b) 5.4 A (c) 63.4°
- 352 V
- (a) 40 Ω (b) 24 V (c) 117 V (d) 78.4 °
- (a) 691 Watts (b) 1152 VA (c) 921.6 VAR (d) 60%
- (a) 18.1 kVA, (b) 17 k Watts, (c) 6,380 VAR, (d) 94%
|E||I||R /XL /Z||W/ VA /VARs||PF|
|R||155 V||3.1 A||50 Ω||480.5 W||0|
|L||155 V||3.1 A||50 Ω||480.5 VARs||90|
|Total||220 V||3.1 A||70.7 Ω||682 VA||45||71 %|
9. The inductive reactance (XL) deceases causing the phase angle between the applied voltage and current to decrease.
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CC-MAIN-2023-23
| 11,864 | 116 |
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