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http://www.oalib.com/search?kw=%E5%8D%95%E4%BD%B3%E5%8B%87&searchField=authors
|
math
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Publish in OALib Journal
APC: Only $99
本文提出了基于弹簧模型的定位算法MSRDH (Mass Spring and RSSI DV-Hop)算法。该算法利用弹簧模型，把节点与锚节点作为端点，将这两个点间最短路径上的所有节点抽象成一个弹簧。通过建立锚节点之间的弹簧模型，得到全网的平均弹簧系数，并将平均弹簧系数应用到网络中未知节点的计算过程中。仿真结果表明，MSRDH算法比DV-Hop算法有更好的性能表现。
MSRDH (Mass Spring and RSSI DV-Hop) localization algorithm based on mass-spring model is proposed in this paper. Using mass-spring model, the algorithm abstracts all nodes on the shortest path between the node and anchor node into a spring. The average coefficient of mass-spring is calculated through the establishment of the spring models between anchor nodes. Taking advantage of the average coefficient of mass-spring, the unknown nodes can compute their own localizations. Through extensive simulations, the results show that MSRDH algorithm has better performance than DV-Hop algorithm.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998943.53/warc/CC-MAIN-20190619083757-20190619105757-00536.warc.gz
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CC-MAIN-2019-26
| 1,092 | 4 |
https://ctablog.org/author/Ian/
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math
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Have you been smitten by the programming bug, but think you have to go to school to learn to program well enough? Think again. There are lots of great ways to learn programming without school. Today, we will look at 5 of them. As you read about each method, remain aware of how you respond… Continue reading Learn programming without school – 5 best ways
This is a common question that most people who are interested in programming ask. The answer to the question is not absolutely clear, but there are some pointers that you can take into consideration. The first thing to consider is the idea of what you want to do with your programming skills, or where you… Continue reading Is it too late to learn programming?
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587767.18/warc/CC-MAIN-20211025185311-20211025215311-00710.warc.gz
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CC-MAIN-2021-43
| 721 | 2 |
http://mathinstructor.net/2014/02/ncert-math-solutions-class-9th-chapter-8-quadrilaterals-exercise-8-2-question-6/
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math
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Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
ABCD is a quadrilateral. P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively.
We need to prove that OP=OR and OS=OQ. Join AC.
In , S is the mid-point of AD and R is the mid-point of DC.
Therefore, by mid-point theorem, we have and (1)
In , Q is the mid-point of BC and P is the mid-point of AB.
Therefore, by mid-point theorem, we have and (2)
From (1) and (2), we have and
is a parallelogram.
(A quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel)
and (Diagonals of parallelogram bisect each other)
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s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890314.60/warc/CC-MAIN-20180121060717-20180121080717-00496.warc.gz
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CC-MAIN-2018-05
| 661 | 11 |
https://allauthor.com/quotes/topic/citizens/
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math
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(showing 1-10 of 29)
My view is different. Public relations are a key component of any operation in this day of instant communications and rightly inquisitive citizens. Alvin Adams#Citizens 67
Alvin Adams#Citizens 67
What other nations call religious toleration, we call religious rights. They are not exercised in virtue of governmental indulgence, but as rights, of which government cannot deprive any portion of citizens, however small. Richard Mentor Johnson#Citizens 64
Richard Mentor Johnson#Citizens 64
In the lexicon of the political class, the word "sacrifice" means that the citizens are supposed to mail even more of their income to Washington so that the political class will not have to sacrifice the pleasure of spending it. George Will#Citizens 60
George Will#Citizens 60
Enrolling in the Medicare Prescription Drug Program will be a great savings for most senior citizens. Paul Gillmor#Citizens 59
Paul Gillmor#Citizens 59
The gifts of God should be enjoyed by all citizens in Mississippi. Medgar Evers#Citizens 59
Medgar Evers#Citizens 59
Law-abiding citizens value privacy. Terrorists require invisibility. The two are not the same, and they should not be confused. Richard Perle#Citizens 57
Richard Perle#Citizens 57
Liberty is the most precious gift we offer our citizens. Tom Ridge#Citizens 55
Tom Ridge#Citizens 55
My fellow citizens, the state of our city is strong. Thomas Menino#Citizens 53
Thomas Menino#Citizens 53
Speaking of tax fairness, it was Senator Kerry who voted to increase the income tax on senior citizens on Social Security, earning as little as $32,000 a year. Bill Weld#Citizens 49
Bill Weld#Citizens 49
Like all citizens, Ms. McNeill has the right to be free from unlawful employment practices such as sex discrimination and retaliation. John Hawkins#Citizens 40
John Hawkins#Citizens 40
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New to AllAuthor.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572896.15/warc/CC-MAIN-20190924083200-20190924105200-00428.warc.gz
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CC-MAIN-2019-39
| 1,898 | 24 |
https://www.onlinemathlearning.com/may-2022-9709-43.html
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math
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CIE May 2022 9709 Mechanics Paper 43 (pdf)
- Two particles P and Q, of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal
plane. P is projected at a speed of 4 m s−1 directly towards Q. After P and Q collide, Q begins to
move with a speed of 3 m s−1.
(a) Find the speed of P after the collision
- A particle P is projected vertically upwards from horizontal ground. P reaches a maximum height
of 45 m. After reaching the ground, P comes to rest without rebounding.
(a) Find the speed at which P was projected.
- The displacement of a particle moving in a straight line is s metres at time t seconds after leaving a
fixed point O. The particle starts from rest and passes through points P, Q and R, at times t = 5, t = 10
and t = 15 respectively, and returns to O at time t = 20. The distances OP, OQ and OR are 50 m,
150 m and 200 m respectively.
The diagram shows a displacement-time graph which models the motion of the particle from t = 0 to
t = 20. The graph consists of two curved segments AB and CD and two straight line segments BC
- The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings AC and
BC, of lengths 0.8m and 0.6 m respectively, attached to fixed points on the ceiling. Angle ACB = 90°.
There is a horizontal force of magnitude F N acting on the block. The block is in equilibrium.
(a) In the case where F = 20, find the tensions in each of the strings.
(b) Find the greatest value of F for which the block remains in equilibrium in the position shown.
- A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle
is 70 kg. At an instant when the cyclist’s speed is 4 m s−1, her acceleration is 0.3 m s−2. There is a
constant resistance to motion of magnitude 30 N.
(a) Find the power developed by the cyclist.
The cyclist comes to the top of a hill inclined at 5° to the horizontal. The cyclist stops pedalling and
freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of
the resistance force remains at 30 N. Over a distance of d m, the speed of the cyclist increases from
6 m s−1 to 12 m s−1.
(b) Find the change in kinetic energy.
(c) Use an energy method to find d.
- Two particles P and Q, of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light
inextensible string. The string passes over a fixed smooth pulley at B which is attached to two inclined
planes. P lies on a smooth plane AB which is inclined at 60° to the horizontal. Q lies on a plane BC
which is inclined at 30° to the horizontal. The string is taut and the particles can move on lines of
greatest slope of the two planes (see diagram).
(a) It is given that the plane BC is smooth and that the particles are released from rest.
Find the tension in the string and the magnitude of the acceleration of the particles.
(b) It is given instead that the plane BC is rough. A force of magnitude 3 N is applied to Q directly
up the plane along a line of greatest slope of the plane.
Find the least value of the coefficient of friction between Q and the plane BC for which the
particles remain at rest.
- A particle P moves in a straight line through a point O. The velocity vm s−1
of P, at time ts after passing O, is given by
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233510326.82/warc/CC-MAIN-20230927203115-20230927233115-00747.warc.gz
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CC-MAIN-2023-40
| 3,629 | 46 |
https://brainytermpapers.com/what-characteristics-of-a-variable-would-make-it-a-promising-covariate-why/
|
math
|
1. In a one-way design with a premeasure, one could test for treatment effects either by using an ANOVA (i.e., ignoring the premeasure) or an ANCOVA. What is the conceptual difference in the question addressed by ANCOVA as opposed to ANOVA? 2. What do you look for in a covariate? That is, in thinking about the design of a study, what characteristics of a variable would make it a promising covariate? Why?
PLACE THIS ORDER OR A SIMILAR ORDER WITH brainy term papers TODAY AND GET AN AMAZING DISCOUNT
The post what characteristics of a variable would make it a promising covariate? Why? appeared first on Cheapest Academic Custom Papers.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816879.25/warc/CC-MAIN-20240414095752-20240414125752-00575.warc.gz
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CC-MAIN-2024-18
| 638 | 3 |
https://www.dapperesq.com/instagram/tag/photobyunsplash/
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math
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More than 80% of children have an online presence by the age of two. That's the result of a UK report from OFCOM. In the world of "sharenting" (terrible name) the average parent has shared 1500 images of their kid before they turn 5. I think it's fair to say we are all proud of our little ones. But we should be warned about the dangers of over-sharing. As it could pose risks to our children. In the UK their are working on the "right to be forgotten". Where children can ask online platforms to delete their pictures. But it might be too late then. So think about the kids before you post that cute, but maybe slightly embarrassing, picture. #parenting#parenthood#fatherhood#sharenting#socialmediaadvice#growingup#consent#photobyunsplash#adviceforparents
Here's a quick reference guide to whiskey. Remind yourself why there is a difference between whiskey and whisky (but that it really doesn't matter). Why you probably prefer single malt over blended. And that moonshine has actually nothing to do with whiskey.
But traditions are different worldwide. And that makes it interesting. Did you know that EU laws indicate that whisk(e)y must age for more than 3 years? And in the US they simply can't wait that long? #whiskey#whisky#spirit#scotch#bourbon#singlemalt#food#drinks#referenceguide#photobyunsplash
Showing you a picture of my toilet is not what I'm going to do. But talking about bowel movements seems a topic that might be of interest to you. 💩 It's a strange, and slightly embarrassing matter to talk about. But I did stumble upon the article today. And it is the most read article on MNT. You can read the full article on their website (find the link through our bio). The main question asked: "How many times a day should you poop?" TLDR; Somewhere between 3 times per week to 3 times a day. So it depends. It's a very personal habit and can vary from person to person. Just be concerned if it changes in regularity or consistency. #poop#bowelmovement#toilethabits#metime#tldr#medicaladvice#shithappens#photobyunsplash
Without projecting a single study on a larger group and generalizing the according findings, I do believe we have to be careful with minimizing the issue. Hegemonic masculinity is the idea that one’s machismo must be broadcast constantly, no matter what he is dealing with or how he feels inside. And it has taken its role on the suicide rate amongst white males in the US. One way to overcome this to update our definition of masculinity for the 21st century. Read more about why being a white male is nowadays subject for withering your mental health, on the Big Think @bigthinkers. #men#man#bigthink#suicide#masculinity#whitemale#mentalhealth#depression#healthissues#photobyunsplash
Without getting too much into politics. This video of AJ+ @ajplus explains a trend that is visible worldwide. Apparently humans are getting fewer children. To keep populations stable in industrialized countries, we need to have 2.1 babies. But those numbers are lower in the US and Europe. And in Japan these numbers seem to be the lowest with just 1.41 babies per woman. It's not a matter of infertility, but it's a clear choice. As we become more focused on our careers, gaining more money, getting a better health. We tend to live longer, and just have less babies. #children#fatherhood#parents#future#population#kids#japan#photobyunsplash
If you are ever lost, you'll need some skills to find your way back to civilization. Learn these five natural navigation tricks looking at the moon, stars, trees and more. They are tips from author Tristan Gooley. Who shares useful tips and insights aimed at helping people notice simple truths about the world around them. And once you know these, you will never be able to unsee them again. Illustrations by Chelsea Beck. #nature#navigation#hiking#moon#trees#stars#compass#tipsandtricks#photobyunsplash
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s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651820.82/warc/CC-MAIN-20180325044627-20180325064627-00770.warc.gz
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CC-MAIN-2018-13
| 3,873 | 7 |
https://researchoutput.csu.edu.au/en/publications/the-uniqueness-thesis
|
math
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The Uniqueness Thesis holds, roughly speaking, that there is a unique rational response to any particular body of evidence. We first sketch some varieties of Uniqueness that appear in the literature. We then discuss some popular views that conflict with Uniqueness and others that require Uniqueness to be true. We then examine some arguments that have been presented in its favor and discuss why permissivists (i.e., those who deny Uniqueness) find them unconvincing. Last, we present some purported counterexamples that have been raised against Uniqueness and discuss some possible reasons why proponents of Uniqueness might find these similarly unconvincing.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506320.28/warc/CC-MAIN-20230922002008-20230922032008-00127.warc.gz
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CC-MAIN-2023-40
| 661 | 1 |
http://homepage.divms.uiowa.edu/~jones/step/micro.html
|
math
|
5. Microstepping of Stepping Motors
Microstepping serves two purposes. First, it allows a stepping motor to stop and hold a position between the full or half-step positions, second, it largely eliminates the jerky character of low speed stepping motor operation and the noise at intermediate speeds, and third, it reduces problems with resonance.
Although some microstepping controllers offer hundreds of intermediate positions between steps, it is worth noting that microstepping does not generally offer great precision, both because of linearity problems and because of the effects of static friction.
Recall, from the discussion in Part 2 of this tutorial, on Stepping Motor Physics, that for an ideal two-winding variable reluctance or permanent magnet motor the torque versus shaft angle curve is determined by the following formulas:
h = ( a2 + b2 )0.5Where:
x = ( S / (π / 2) ) arctan( b / a )
a -- torque applied by winding with equilibrium at angle 0.This formula is quite general, but it offers little in the way of guidance for how to select appropriate values of the current through the two windings of the motor. A common solution is to arrange the torques applied by the two windings so that their sum h has a constant magnitude equal to the single-winding holding torque. This is referred to as sine-cosine microstepping:
b -- torque applied by winding with equilibrium at angle S.
h -- holding torque of composite.
x -- equilibrium position.
S -- step angle.
a = h1 sin(((π / 2) / S)θ)Where:
b = h1 cos(((π / 2) / S)θ)
h1 -- single-winding holding torqueGiven that none of the magnetic circuits are saturated, the torque and the current are linearly related. As a result, to hold the motor rotor to angle θ, we set the currents through the two windings as:
((π / 2) / S)θ -- the electrical shaft angle
Ia = Imax sin(((π / 2) / S)θ)Where:
Ib = Imax cos(((π / 2) / S)θ)
Ia -- current through winding with equilibrium at angle 0.Keep in mind that these formulas apply to two-winding permanent magnet or hybrid stepping motors. Three pole or five pole motors have more complex behavior, and the magnetic fields in variable reluctance motors don't add following the simple rules that apply to the other motor types.
Ib -- current through winding with equilibrium at angle S.
Imax -- maximum allowed current through any motor winding.
The utility of microstepping is limited by at least three consideraitons. First, if there is any static friction in the system, the angular precision achievable with microstepping will be limited. This effect was discussed in more detail in the discussion in Part 2 of this tutorial, on Stepping Motor Physics, in the discussion of friction and the dead zone.
The second problem involves the non-sinusoidal character of the torque versus shaft-angle curves on real motors. Sometimes, this is attributed to the detent torque on permanent magnet and hybrid motors, but in fact, both detent torque and the shape of the torque versus angle curves are products of poorly understood aspects of motor geometry, specifically, the shapes of the teeth on the rotor and stator. These teeth are almost always rectangular, and I am aware of no detailed study of the impact of different tooth profiles on the shapes of these curves.
Most commercially available microstepping controllers provide a fair approximation of the sine-cosine drive current that would drive an ideal stepping motor to uniformly spaced steps. Ideal motors are rare, and when such a controller is used with a real motor, a plot of the actual motor position as a function of the expected position will generally look something like the plot shown in Figure 5.1.
Figure 5.1Note that the motor is at its expected position at every full step and at every half step, but that there is significant positioning error in the intermediate positions. The curve shown is the curve that would result from a perfect sin-cosine microstepping controller used with a motor that had a torque versus position curve that included a significant 4th harmonic component, usually attributed to the detent torque.
The broad details of detent effects appear to be fairly uniform from motor to motor, so in principle, it ought to be possible to adjust the tables of sines and cosines used in a sine-cosine controller to compensate for the detent effects. In practice, the effects of friction and the errors introduced by quantization combine to limit the value of such an effort.
The third problem arises because most applications of microstepping involve digital control systems, and thus, the current through each motor winding is quantized, controlled by a digital to analog converter. Furthermore, if typical PWM current limiting circuitry is used, the current through each motor winding is not held perfectly constant, but rather, oscillates around the current control circuit's set point. As a result, the best a typical microstepping controller can do is approximate the desired currents through each motor winding.
The effect of this quantization is easily seen if the available current through one motor winding is plotted on the X axis and the available current through the other motor winding is plotted on the Y axis. Figure 5.2 shows such a plot for a motor controller offering only 4 uniformly spaced current settings for each motor winding:
Figure 5.2Of the 16 available combinations of currents through the motor windings, 6 combinations lead to roughly equally spaced microsteps. There is a clear tradeoff between minimizing the variation in torque and minimizing the error in motor position, and the best available motor positions are hardly uniformly spaced! Use of higher precision digital to analog conversion in the current control system reduces the severity of this problem, but it cannot eliminate it!
Plotting the actual rotor position of a motor using the microstep plan outlined in Figure 5.2 versus the expected position gives the curve shown in Figure 5.3:
Figure 5.3It is very common for the initial microsteps taken away from any full step position to be larger than the intended microstep size, and this tends to give the curve a staircase shape, with the downward steps aligned with the full step positions where only one motor winding carries current. The sign of the error at intermediate positions tends to fluctuate, but generally, the position errors are smallest between the full step positions, when both motor windings carry significant current.
Another way of looking at the available microsteps is to plot the equilibrium position on the horizontal axis, in fractions of a full-step, while plotting the torque at each available equilibrium position on the vertical axis. If we assume a 4-bit digital-to-analog converter, giving 16 current levels for each each motor winding, there are 256 equilibrium positions. Of these, 52 offer holding torques within 10% of the desired value, and only 33 are within 5%; these 33 points are shown in bold in Figure 5.4:
Figure 5.4If torque variations are to be held within 10%, it is fairly easy to select 8 almost-uniformly spaced microsteps from among those shown in Figure 5.4; these are boxed in the figure. The maximum errors occur at the 1/4 step points; the maximum error is .008 full step or .06 microsteps. This error will be irrelevant if the dead-zone is wider than this.
If 10 microsteps are desired, the situation is worse. The best choices, still holding the maximum torque variation to 10%, gives a maximum position error of .026 full steps or .26 microsteps. Doubling the allowable variation in torque approximately halves the positioning error for the 10 microstep example, but does nothing to improve the 8 microstep example.
One option which some motor control system designers have explored involves the use of nonlinear digital to analog converters. This is an excellent solution for small numbers of microsteps, but building converters with essentially sinusoidal transfer functions is difficult if high precision is desired.
As typically used, a microstepping controller for one motor winding involves a current limited H-bridge or unipolar drive circuit, where the current is set by a reference voltage. The reference voltage is then determined by an analog-to-digital converter, as shown in Figure 5.5:
Figure 5.5Figure 5.5 assumes a current limited motor controller such as is shown in Figures 4.7, 4.8, 4.10 or 4.11. For all of these drivers, the state of the X and Y inputs determines the whether the motor winding is on or off and if on, the direction of the current through the winding. The V0 through Vn inputs determine the reference voltage and this the current through the motor winding.
There are a fair number of nicely designed integrated circuits combining a current limited H-bridge with a small DAC to allow microstepping control of motors drawing under 2 amps per winding. The UDN2916B from Allegro Microsystems is a dual 750mA H-bridge, with a 2-bit DAC to control the current through each. bridge. Another excellent example is the UC3770 from Unitrode. Unitrode. This chip integrate a 2-bit DAC with a PWM controlled H-bridge, packaged in either 16 pin power-dip format or in surface mountable form. The 3717 a slightly cleaner design, good for 1.2 A, while the 3770 is good for up to between 1.8 A or 2 A, depending on how the chip is cooled.
The 3955 from Allegro Microsystems incorporates a 3-bit non-linear DAC and handles up to 1.5 A; this is available in 16-pin power DIP or SOIC formats. The nonlinear DAC in this chip is specifically designed to minimize step-angle errors and torque variations using 8 microsteps per full-step.
The LMD18245 from National Semiconductor is a good choice for microstepped control of motors drawing up to 3 amps. This chip incorporates a 4-bit linear DAC, and an external DAC can be used if higher precision is required. As indicated by the data shown in Figure 5.4, a 4-bit linear DAC can produce 8 reasonably uniformly spaced microsteps, so this chip is a good choice for applications that exceed the power levels supported by the Allegro 3955.
It appears that microstepping was invented in 1974 by Larry Durkos, who was working as a mechanical engineer for American Monitor Corporation. The company was a medical equipment vendor, and they were using a large Superior Electric 1.8 degree per step stepping motors to directly drive the 20 inch diameter turntable of their Kinetic Discrete Analyzer. The turntable was used to bring each of 100 blood samples into position for analysis.
That is 2 steps per sample, and the motion was so abrupt that the samples tended to spill. The system was controlled by a Computer Automation LSI 2 minicomputer (today, we would use a microcontroller), and Durkos worked out how to do computer-controlled sine-cosine microstepping in order to solve this problem. The solution was published in the technical service manuals for the KDA analyzer, but it was never patented. Representatives of Superior Electric learned of microsteppingfrom Durkos, and that company was the first to market a microstepping controller.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103322581.16/warc/CC-MAIN-20220626222503-20220627012503-00426.warc.gz
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CC-MAIN-2022-27
| 11,103 | 41 |
https://ui.adsabs.harvard.edu/abs/2014arXiv1408.3663L/abstract
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math
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Emergence of a non-Fowler-Nordheim-type behavior for a general planar tunneling barrier
In this work we investigate a generalized tunneling barrier for planar emitters at zero-temperature. We present the evidence of the emergence of a non-Fowler-Nordheim-type general behavior for the field emission current density in the case that the Fermi energy ($\mu$) is comparable with or smaller that the decay width ($d_F$). Therefore, for some non-metals or materials that have very small Fermi energy the standard Fowler-Nordheim-type theory may require a correction. In the opposite regime, i.e., for $\mu$ much larger that $d_F$, we confirm that the conventional theory is suitable for metals.
- Pub Date:
- August 2014
- Condensed Matter - Materials Science
- 5 pages. Improved version. Title modified. Abstract rewritten. References updated
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224644683.18/warc/CC-MAIN-20230529042138-20230529072138-00741.warc.gz
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CC-MAIN-2023-23
| 839 | 6 |
https://hellothinkster.com/curriculum-us/calculus-bc/infinite-sequences-and-series/alternating-series-test-for-convergence/
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math
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Alternating Series Test for Convergence
Applying the alternating series test for convergence to solve problems.
Mapped to AP College Board # LIM-7, LIM-7.A, LIM-7.A.10
Applying limits may allow us to determine the finite sum of infinitely many terms. Determine whether a series converges or diverges. The alternating series test is a method to determine whether an alternating series converges.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474412.46/warc/CC-MAIN-20240223121413-20240223151413-00531.warc.gz
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CC-MAIN-2024-10
| 394 | 4 |
https://tradingucdt.web.app/mangieri68769syd/what-the-difference-between-apr-and-annual-interest-rate-dyhe.html
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math
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Why banks publish annual percentage rates. Now that you understand the difference between interest rate and APR, let's talk a little about how to find the best options for your loans: The Difference Between Interest Rate and APR in Mortgages By contrast, the annual percentage rate is the annual cost of the loan inclusive of fees, Sherman says. Fees included in the APR can add significantly to the costs a buyer will pay. Examples of such fees are: Getting a loan means paying interest—it's the cost of borrowing money. Just how much interest you'll pay depends on your interest rate. Or does it depend on your ARP (annual percentage rate)? Find out what the difference is between APR and interest rates. It is worth noting, however, that an annual percentage rate and an interest rate are two unique indicators. While they may sound similar, they are anything but. There are several inherent differences that exist between interest rates and annual percentage rates, not the least of which are hard to discern for amateur investors. Key Differences Between Interest Rate and APR. The difference between interest rate and APR are drawn clearly on the following grounds: The interest rate is described as the rate at which interest is charged by the lenders on the loan given to the borrowers. APR or Annual Percentage Rate is the per year total cost of borrowing. The annual percentage rate represents your total cost of getting a mortgage. The interest rate represents the cost you pay over time to buy that loan. Let’s take a look at the difference between your APR and interest rate, and how they affect the true cost of a mortgage. We’ll cover: What’s an annual percentage rate?
21 Feb 2020 Knowing the difference between the “interest rate” and “annual percentage rate” ( APR) can save you a lot of money.
Interest rate vs. APR The interest rate is the cost of borrowing the principal loan amount. The rate can be variable or fixed, but it’s always expressed as a percentage. APR is the annual rate of interest that is paid on an investment, without taking into account the compounding of interest within that year. Alternatively, APY does take into account the frequency with which the interest is applied—the effects of intra-year compounding. They might be used interchangeably, but an APR and an interest rate aren’t one and the same. The annual percentage rate represents your total cost of getting a mortgage. The interest rate represents the cost you pay over time to buy that loan. Both APR (annual percentage rate) and APY (annual percentage yield) are commonly used to reflect the interest rate paid on a savings account, loan, money market or certificate of deposit. It's not immediately clear from their names how the two terms — and the interest rates they describe — differ. Why banks publish annual percentage rates. Now that you understand the difference between interest rate and APR, let's talk a little about how to find the best options for your loans:
26 Feb 2020 Difference Between Interest Rate and APR. Annual percentage rate vs. interest rate: These are two similar but ultimately different things. Let's
15 Nov 2019 An annual percentage rate (APR) reflects the mortgage interest rate plus other charges.
27 Feb 2020 An in-depth look at the difference between the mortgage interest rate and And the other is the Annual Percentage Rate, or APR, which is the
When evaluating the cost of a loan or line of credit, it is important to understand the difference between the advertised interest rate and the annual percentage rate
27 Feb 2017 If interest rates have gone down you will be in a better position, but if interest The Annual Percentage Rate (APR) is the annual cost of a loan
28 Sep 2017 When shopping for a new mortgage loan, you may notice an Annual Percentage Rate (APR) advertised next to the note rate. The inclusion of An interest rate and a representative APR can often be confusing when looking at finance options. Cash Lady explains the difference between them both. borrow £100 at 5% 'annual' interest, you would pay back £105 at the end of the year. 16 Oct 2019 APR stands for Annual Percentage Rate. It takes into account the interest rate of the product then adds on any additional charges, giving you Everything you need to know about the different types of interest rates. With a loan that has a stated Annual Percentage Rate, you are only paying the interest
23 Jul 2019 The annual percentage rate is the effective annual interest rate on a loan including most ancillary charges and origination costs in addition to It consists of the actual interest rate, the processing fee, foreclosure amount, and all other fees charged by a bank on the loan. What is the difference between APR Interest Rate vs. APR: An Overview. The interest rate is the cost of borrowing the money, that is, the principal loan amount. When evaluating the cost of a loan or line of credit, it is important to understand the difference between the advertised interest rate and the annual percentage rate, or APR. This new loan amount, along with the interest rate (5.00%), is used to calculate a new monthly payment ($1,089.75). The APR is then calculated by working backwards to figure out what the rate would have to be for a loan with the new monthly payment ($1,089.75) and the original loan amount ($200,000). Interest rate refers to the annual cost of a loan to a borrower and is expressed as a percentage; APR is the annual cost of a loan to a borrower — including fees. Like an interest rate, the APR is expressed as a percentage. Interest Rate; Definition: Annual Percentage Rate (APR) is an expression of the effective interest rate that the borrower will pay on a loan, taking into account one-time fees and standardizing the way the rate is expressed. Interest is a fee on borrowed capital. Interest rate is a "rent on money" to compensate the lender for foregoing other useful investments that could have been made with the loaned money. Transaction costs In contrast, APR is an annual rate that includes interest rate payments as well as other fees charged for a loan, which can include origination fees, closing costs and service charges. Because APR is calculated on a yearly basis, it will be higher than the interest rate for loans with frequent payments, short terms,
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https://homework.cpm.org/category/CON_FOUND/textbook/mc1/chapter/6/lesson/6.2.1/problem/6-52
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math
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. Multiple Choice: If the probability of getting a particular result in an experiment is 75.3%, what is the probability of not getting that result? Explain your choice. Homework Help ✎
75.3% + 100%
75.3% − 100%
100% − 75.3%
In the Math Notes box from Lesson 5.2.2, probability is defined as ''a number between zero and one that states the likelihood of an event occurring.'' Use this information to help you decide which answer is the best.
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https://ems.press/journals/rlm/articles/15102
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math
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A subscription is required to access this article.
We investigate the regularity of functions of one variable such that , where is a given polynomial of degree in whose coefficients are functions of class of one real parameter. We show that if a root is chosen with a continuous dependence on the parameter, this function is indeed absolutely continuous. From this and a theorem of Kato one deduces that such polynomials have complete systems of roots that are absolutely continuous functions.
Cite this article
Ferruccio Colombini, Nicola Orrù, Ludovico Pernazza, On the regularity of the roots of a polynomial depending on one real parameter. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 4, pp. 747–775
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https://www.cradle-cfd.com/media/column/a158
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math
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Basic Course on Turbulence and Turbulent Flow Modeling 5: 5.1 Turbulent flow and vortex tube, 5.2 Interaction between vortex tubes, 5.3 Kolmogorov’s similarity hypothesis
Mechanism of turbulent flow (2)
5.1 Turbulent flow and vortex tube
The previous column explained that velocity difference causes turbulent flow and that eddies are formed in the turbulent flow. When eddies are visualized, they look like tubes; therefore, the visualized eddies are called “vortex tubes”. Imagine that tubes are flowing in a turbulent flow. Figure 5.1 shows vortex tubes in a flow passing on a plate with a step. The size (diameter) of vortex tubes is various. Their shape is also various and you can see straight-line-shaped tubes, curved tubes, and tubes with different diameters. Reynolds number is different between the right and left figures shown below. Reynolds number in the right figure is higher. These figures show that when Reynolds number is higher, more vortex tubes are formed.
Figure 5.1: Image of vortex tubes in a turbulent flow
5.2 Interaction between vortex tubes
Vortex tubes are generated because of velocity difference in a flow. When a vortex tube is generated, it causes velocity difference in its neighboring regions. Then, the difference causes a new vortex tube. This interaction between vortex tubes is repeated and new vortex tubes are generated one after another. When a new vortex tube is generated, its diameter is the same as or smaller than that of the original vortex tube. (You can easily imagine that generating a larger one is difficult.) When a large vortex tube and a small vortex tube interact with each other, a vortex tube with medium-diameter may be generated. A new vortex tube does not always occur in the same direction with that of the original vortex tube, and some new tubes are perpendicular to their original. In this way, various vortex tubes are generated in a flow field. Figure 5.2 shows vortex tubes in a flow around a cylinder. Many vortex tubes are generated downstream of the cylinder; however, the directions of the vortex tubes are not uniform. When turbulent motion increases, a mass of vortex tubes which look random is formed.
Figure 5.2 Vortex tubes in a flow around a cylinder
5.3 Kolmogorov’s similarity hypothesis
Vortex tubes seem to be randomly generated by velocity difference and interaction between vortex tubes. However, when energy of each vortex tube in a flow field is classified per tube size, a fixed curve shown in Figure 5.3 is obtained. The horizontal axis of the graph indicates the inverse of the diameter of each vortex tube. The farther you go to the right on the horizontal axis, the smaller the vortex tube size is. The vertical axis indicates the energy that each vortex tube has. Both horizontal and vertical axes are logarithmic. This graph is characterized by a higher part on the left, lower part on the right, and straight part (because of logarithmic expression) in the middle. Interestingly, when a graph like Figure 5.3 is applied to vortex tubes in various flow fields according to the determined process, the curve from the straight line in the middle to the lower part on the right end is almost the same. The peak on the left side varies depending on Reynolds number and called “integral length”. The energy of the vortex tube with the diameter of the integral length comprises the majority of the energy of vortex tubes in a flow field. The slope of the straight line in the middle equals to -5/3 in logarithmic expression and all straight lines of turbulent vortex tubes are on the line. This is called “Kolmogorov -5/3 law”. “Kolmogorov” is the name of a Russian mathematician. The size of the right end indicates the smallest scale of vortex tubes and called “Kolmogorov scale”. Vortex tubes whose size is smaller than Kolmogorov scale do not exist. This is because the tubes disappear as they are converted to heat by the action of fluid viscosity. The movement of a turbulent flow (an assembly of vortex tubes) looks random. However, as shown in Figure 5.3, there is a law, which indicates that a turbulent flow is a well-ordered phenomenon. Then, this is one of the motivations to model a turbulent flow.
Figure 5.3: Kolmogorov’s similarity law
About the Author
Takao Itami | Born in July 1973, Kanagawa, Japan
The author had conducted researches on numerical analyses of turbulence in college. After working as a design engineer for a railway rolling stock manufacturer, he took the doctor of engineering degree from Tokyo Institute of Technology (Graduate School of Science and Engineering) through researching compressible turbulent flow and Large-Eddy Simulation. He works as a consulting engineer at Software Cradle solving various customer problems with his extensive experience.
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CC-MAIN-2022-33
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https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2020/26.html
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math
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Séminaire Lotharingien de Combinatoire, 84B.26 (2020), 12 pp.
Measuring Symmetry in Lattice Paths and Partitions
We introduce the notion of degree of symmetry for lattice paths and related combinatorial objects. The degree of symmetry measures how symmetric an object is, usually ranging from zero (completely asymmetric) to its size (completely symmetric).
We study the behavior of this statistic on Dyck paths and grand Dyck paths, where the symmetry is given by reflection along a vertical line through their midpoint; partitions, where the symmetry is given by conjugation; and certain compositions interpreted as bargraphs.
We find expressions for the generating functions for these objects
with respect to their degree of symmetry, and their semilength or
semiperimeter. The generating functions are algebraic in most cases,
with the notable exception of Dyck paths, for which we apply
techniques from walks in the plane to find a functional equation for
the generating function, and conjecture that it is D-finite but not
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
The following versions are available:
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https://algebra-answer.com/algebra-help/function-range/partial-sums-method.html
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math
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It looks like you are not the only one encountering this problem. A friend of mine was in a similar situation last month. That is when he came across this software known as Algebra Helper . It is by far the most economical piece of software that can help you with problems on partial sums method . It won’t just find a solution for your problems but also give a step by step explanation of how it arrived at that solution.
Is the software really that helpful? I’m just concerned because it might not really help because it only solves the problem per ?e. I like to understand how a problem is solved and not only find out the answer. Nevertheless, could you give me a link for this product?
It is not at all difficult to access this program. Go to: http://www.algebra-answer.com/features.shtml. You are guaranteed satisfaction. Otherwise you get a refund. So what is there to lose anyway? Cheers and all the best.
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Algebra Helper can solve problems in all the following areas:
simplification of algebraic expressions (operations
with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots
(radicals), absolute values)
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| 2,093 | 29 |
https://www.physicsforums.com/threads/can-someone-explain-me-this-problem-circuit-analysis-using-superposition.924713/
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math
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A current source can have any voltage you choose around it. A current source has infinite impedance.This is a strange exercise! Who did it?
The voltage over the 4 ohm resistor must be 24 V.
So the the current -I2 is 24/4=6 A.
Why is that? A current source cannot have voltage, i.e it is zero.
In practical life this is not possible.
If the target is to teach circuit analysis, I dont know if it is a good idea
with such examples.
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| 429 | 7 |
https://www.analystforum.com/t/q-42-pm-mock/103262
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math
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Sorry if this question has been asked, but I think the answer to this could be A., as well, if it’s a perfect competition. Am I wrong?
Because demand curve is downward sloping, this is monopolistic competition not perfect competition. By definition a downward instead of a horizontal demand curve faced by the firm indicates pricing power.
Thank you! That’s what I was thinking, as well, but it’s not quite clear that this is the demand curve faced by an individual firm.
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CC-MAIN-2021-49
| 477 | 3 |
https://www.physicsforums.com/threads/time-lapse-simulations-of-traveling-through-space.773419/
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math
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In the Hubble Ultra Deep Field photograph, they show a 3D simulation of traveling through it. (Cue to about 2:00 minutes.) Obviously the simulation is traveling through that field faster than the speed of light. What would that field actually look like while traveling near the speed of light? What would the Andromeda galaxy look like if you were approaching it near the speed of light?
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http://journal-n.scnu.edu.cn/cn/article/doi/10.6054/j.jscnun.2015.05.006
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math
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根据Curtin-Hammett原理, 反应产物的选择性与迁移状态的自由能稳定性的不同或者是决定选择性阶段的结构有关，因此溶剂分子对酶反应的影响可以认为是由于其在迁移状态时引起了酶的构型的改变.由于酶与溶剂的相互作用（例如静电力、氢键、粘结压力和疏水作用等）会引起酶蛋白质的构型改变，所以文章从溶剂极性、粘结压力、疏水性3个方面选出了3种参数来描述溶剂效应，分别是表示溶剂极性的介电常数的柯克伍德参数［(-1)/(2+1)］、表示溶剂粘结压力的溶度常数的平方，和表示溶剂的疏水作用的分配系数log P.通过3参数线性回归法，得出了经验式ln =a［(-1)/(2+1)］+b2+clog P+d，这里表示产物间的比值.这个经验公式被应用到了一些关于有选择性的酶催化反应和有机化学反应中，得到了相关系数高达0925～0998线性拟和.因此，该经验式可以成为研究有选择性的非极性有机反应和酶催化反应的溶剂效应的有效工具.
According to the Curtin-Hammett principle, the selectivity of reaction products relates to the free energy difference of stability at the transition state or structure in the selectivity-determining step. Therefore, it is a reasonable assumption that the solvent molecules in enzymatic reaction influence mainly the difference of conformational change of enzyme at the transition state. The conformational change (fluctuation) of enzyme protein arises from various interactions between enzyme and solvent such as electrostatics, hydrogen-bonding, cohesive pressure, hydrophobic interactions, etc. In this paper, a three-parameter treatment was proposed as a linear function of three complementary parameters describing the polarity, cohesive pressure, and hydrophobic factors of the given solvent. In this case, the Kirkwood parameter ［(-1)(2+1)］ was chosen as the polar factor, the square of solubility parameter 2 as the cohesive factor, and log P as the hydrophobic factor. According to the model, the logarithm of the product ratio (ln ) can be described in terms of equation ln =a［(-1)/(2+1)］+b2+clog P+d. This three-parameter equation has been applied to estimate the solvent effect on the selectivity of various enzymatic and organic reactions, and presents a high linearity with correlation coefficients in the range from 0925 to 0998. The proposed three-parameter analysis can be an extremely useful tool for the investigation of solvent effects on the selectivity in non-polar concerted organic reactions as well as enzymatic reactions.
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| 2,631 | 2 |
https://ui.adsabs.harvard.edu/abs/2020arXiv201114219K/abstract
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math
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We consider the problem of adaptive inference on a regression function at a point under a multivariate nonparametric regression setting. The regression function belongs to a Hölder class and is assumed to be monotone with respect to some or all of the arguments. We derive the minimax rate of convergence for confidence intervals (CIs) that adapt to the underlying smoothness, and provide an adaptive inference procedure that obtains this minimax rate. The procedure differs from that of Cai and Low (2004), intended to yield shorter CIs under practically relevant specifications. The proposed method applies to general linear functionals of the regression function, and is shown to have favorable performance compared to existing inference procedures.
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CC-MAIN-2021-21
| 753 | 1 |
http://www.jiskha.com/display.cgi?id=1323397498
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math
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Posted by Julia on Thursday, December 8, 2011 at 9:24pm.
A projectile is launched from a platform 10 meters above ground with an initial upward velocity of 20 meters per second.
Find the time when the projectile has returned to the initial height launch, find the time the projectile hits the ground, find the time when the projectile hits the ground when its velocity is 0, find the height of the projectile above ground after 1 second
- Algebra II - Damon, Thursday, December 8, 2011 at 9:42pm
g = -9.8 m/s^2
v = +20 - 9.8 t
y = 10 + 20 t -4.9 t^2
a) when is y = 10 again?
10 = 10 + 20 t -4.9 t^2
t(4.9 t-20) = 0
t = 20/4.9 = 4.08 seconds
b) when is y = 0 ?
4.9 t^2 - 20 t - 10 = 0
t = [20 +/- sqrt (400+196)] / 9.8
forget negative time, that was before we started, use + sign
t = 4.53 seconds
c) velocity is 0 at the op when
0 = 20 -9.8 t
t = 2.04 seconds
d) y = 10 + 20 (1) -4.9 (1)^2
= 25.1 meters
Answer This Question
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https://www.westaff.com/jobs/detail/664643
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math
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Manufacturers of windows & door products is seeking a Customer Service Representative for their sales department.
Shift: Monday to Friday 7:00 am- 4:00 pm (flex for O/T)
Pay Rate: $17.00
• Order entry using Company’s own software program
• Manage the flow of information, work and material
• Compile reports on the progress of work and production problems that arise
• Keep track of material and write special orders for new material
• Keep good filling records.
• Position requires a person who likes to work with numbers.
• This is a monotonous job-numbers day in and day out.
• Must have very good memory-detailed to the max!
• Great computer skills (Word, Excel) (will train on company software).
• Time Management is very important.
• Must be bilingual with good command of English language.
• Strong mathematical skills.
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| 852 | 15 |
http://www.unappel.ch/people/emin-gabrielyan/public/070808-spie-moire-pointer/a28-web.htm
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math
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Switzernet Sàrl, Scientific Park of Swiss Federal
Institute of Technology, Lausanne (EPFL)
In measurement instruments where measured values are indicated with a mechanical pointer and a graduated scale, the observation precision is increased often by adding an auxiliary mechanical pointer (needle) with a sub graduated scale. The auxiliary pointer moves in synchronization with the main pointer but at a higher speed. A constant velocity ratio between the auxiliary pointer and the main pointer is maintained via instrumentation gearing mechanisms. Mechanical solutions are not always suitable. A challenging idea is to use moiré phenomenon for its well known magnification and acceleration properties. However the well known moiré shapes with sufficient sharpness, good luminosity and contrast can be obtained only in highly periodic patterns. The periodic nature of patterns makes them inapplicable for indication of values. We present new discrete patterns assembled from simple moiré patterns of different periodicity. The elevation profile of our discrete pattern reveals a joint moiré shape with an arbitrarily long periodicity. The luminosity and the sharpness of our moiré shapes are as high as in simple highly periodic moiré patterns.
Keywords: moiré, instrumentation, metrology, multi-stripe moiré, multi-ring moiré, optical speedup, moiré pointer, moiré needle, moiré watches, optical clock-hands, moiré clock-hands, non-periodic moiré
Table of contents
A graduated scale and a mechanical pointer is a common part for almost all mechanical measurement devices. Often an auxiliary pointer and a scale with sub graduations are used for additional precisions. The auxiliary pointer moves faster, in synchronization with the main pointer. The pointers are connected via a tooth wheel type transmission system. The involute tooth shape is one that results in a constant velocity ratio, and is the most commonly used in instrumentation gearing, clocks and watches. Mechanical methods for changing the speed however can often be heavy and inapplicable. Lack of the force, such as in a compass, can be one of the serious obstacles. Inertia problems arising from discrete movements of mechanical parts at high speed, such as in chronographs, may be another obstacle.
The magnification and acceleration properties of moiré superposition images are a well known phenomenon. The superposition of transparent structures, comprising periodic opaque patterns, forms periodic moiré patterns. A challenging idea would be to use optical moiré effect for creating a fast auxiliary pointer replacing completely the mechanical parts moving at high speeds. The periodic nature of known moiré patterns make them inappropriate for indication of values. Profiles with very long periods can be created with periodic moiré. It is possible to design circular layer patterns with radial lines such that their superposition produces a radial moiré fringe with an angular period equal to 360 degrees. Thus only single radial moiré fringe will be visible in the superposition pattern. However such long periods make the moiré fringes blurred. The dispersion area of the fringe can be as large as the half of the period. In section 4 we show a particular case where a radial periodic moiré can be of use with an additional design extension. However in general, the long period moiré fringes of classical periodic moiré are too inexact for indication purposes.
A limited degree of sharpening of shapes in periodic moiré is possible using band moiré methods, namely moiré magnification of micro shapes [Hutley99], [Kamal98]. Such shapes however require serious sacrifices of the overall luminosity of the superposition image without significant improvements of the sharpness.
Random moiré, namely Glass patterns, produce non-periodic superposition patterns [Amidror03a], [Amidror03b], [Glass69a], [Glass73a]. The obstacle is that the valid range of movements of layers is very limited. The auxiliary indicator would show the sub graduations only within the range of only one graduation of the main scale. Additionally, in random moiré the shapes are noisier than in simple periodic moiré.
We developed new discrete patterns formed by merging straight stripes or circular rings of simple periodic moiré patterns. The composing stripes or rings are simple patterns with carefully chosen periods and phases. The composite pattern reveals a sharp moiré shape with an arbitrarily long periodicity. Movement of a layer along the stripes or along the circumferences of rings produces a faster movement of the moiré shape. Such shape has all qualities for playing the role of the fast auxiliary indicator. The one of the layers can be put into slow mechanical motion by the main pointer of the measurement device. In our discrete patterns the shapes are as sharp as in highly periodic moiré patterns. The period of the moiré pointer can be as long as it is required by the display size of the instrument. In our discrete patterns, the choice of the period has no impact on the quality of the optical shape and a wide range of speed ratios can be obtained.
Choice of stripes or rings depends on the type of the movement of layers. For linear movements the pattern comprises parallel stripes following the path of the movement. For circular movements the pattern consists of concentric rings with a center corresponding to the rotation axis. Our algorithm merges numerous simple periodic patterns into a composite pattern so as to form a continuous joint shape in the assembled superposition image. The underlying layer patterns do not join into continuous shapes within assembled layers. The composite patterns are constructed, such that the velocity ratios across all individual moiré patterns are identical. Consequently, the joint shape of the multi-stripe or multi-ring moiré pattern conserves its form during movements of the optical image. The speed ratio and the sharpness of moiré shape are constant within the full range of movements of the main pointer and layers.
Circular multi-ring samples are the most interesting. They can be used for adding auxiliary optical pointers to numerous measurement device with circular dials and radial mechanical pointers such as clocks, watches, chronographs, protractors, thermometers, altimeters, barometers, compasses, speedometers, alidades, and even weathervanes. In mechanical chronographs, optical acceleration permits measuring fractions of seconds without having mechanical parts moving at high speed with related problems of force, inertia, stress, and wear.
The paper is organized as follows: Section 2 introduces the classical periodic moiré and the methods for forming periodic moiré fringes of a desired shape. These methods are presented in scope of a new perspective permitting to easily change the curves of moiré shapes and those of the layer patterns without affecting the periodicity and the velocity ratios, which are essential factors for metrology purposes. Linear movements are considered and a set of corresponding equations is introduced. Section 3 introduces the equations for creating curved moiré shapes for rotating layers preserving the angular periodicity and velocity ratio. Section 4 presents an extension of a classical moiré for displaying quickly progressing labels on a round dial. In section 5 we introduce our multi-ring moiré patterns on examples of straight radial layer lines. In section 6 we present the general case of multi-ring moiré with various curved layer patterns and moiré shape patterns. Conclusions are given at the end of the paper.
Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1. In the example, the lines of one layer are parallel to the lines of the second layer. The superposition image outlines periodically repeating dark parallel bands, called moiré lines. Spacing between the moiré lines is much larger than the periodicity of lines in the layers.
Figure 1. Superposition of two layers consisting of parallel lines, where the lines of the revealing layer are parallel to the lines of the base layer
We denote one of the layers as the base layer and the other one as the revealing layer. When considering printed samples we assume that the revealing layer is printed on a transparency and is superposed on top of the base layer, which can be printed either on a transparency or on an opaque paper. The periods of the two layer patterns, i.e. the space between the axes of parallel lines, are close. We denote the period of the base layer as and the period of the revealing layer as . In Figure 1, the ratio/ is equal to 12/11.
Light areas of the superposition image correspond to the zones where the lines of both layers overlap. The dark areas of the superposition image forming the moiré lines correspond to the zones where the lines of the two layers interleave, hiding the white background. Such superposition images are discussed in details in literature [Sciammarella62a p.584], [Gabrielyan07a].
The period of moiré lines is the distance from one point where the lines of both layers overlap to the next such point. For cases represented by Figure 1 one can obtain the well known formula for the period of the superposition image [Amidror00a p.20], [Gabrielyan07a]:
The superposition of two layers comprising parallel lines forms an optical image comprising parallel moiré lines with a magnified period. According to equation (2.1), the closer the periods of the two layers, the stronger the magnification factor is.
For the case when the revealing layer period is longer than the base layer period, the space between moiré lines of the superposition pattern is the absolute value of formula of (2.1).
The thicknesses of layer lines affect the overall darkness of the superposition image and the thickness of the moiré lines, but the period does not depend on the layer lines’ thickness. In our examples the base layer lines’ thickness is equal to , and the revealing layer lines’ thickness is equal to .
If we slowly move the revealing layer of Figure 1 perpendicularly to layer lines, the moiré bands will start moving along the same axis at a several times faster speed. The four images of Figure 2 show the superposition image for different positions of the revealing layer. Compared with the first image (a) of Figure 2, in the second image (b) the revealing layer is shifted up by one fourth of the revealing layer period (), in the third image (c) the revealing layer is shifted up by half of the revealing layer period (), and in the fourth image (d) the revealing layer is shifted up by three fourth of the revealing layer period (). The images show that the moiré lines of the superposition image move up at a speed, much faster than the speed of movement of the revealing layer.
When the revealing layer is shifted up perpendicularly to the layer lines by one full period of its pattern, the superposition optical image must be the same as the initial one. It means that the moiré lines traverse a distance equal to the period of the superposition image , while the revealing layer traverses the distance equal to its period . Assuming that the base layer is immobile (), the following equation holds for the ratio of the optical image’s speed to the revealing layer’s speed:
According to equation (2.1) we have:
In case the period of the revealing layer is longer than the period of the base layer, the optical image moves in the opposite direction. The negative value of the ratio computed according to equation (2.3) signifies the movement in the reverse direction.
In this section we introduce equations for patterns with inclined lines. Equations for rotated patterns were already introduced decades ago [Nishijima64a], [Oster63a], [Morse61a]. These equations are good for static moiré patterns or their static instances. In scope of metrology instrumentation, we review the equations suiting them for dynamic properties of moiré patterns. The set of key parameters is defined and the equations are developed such that the curves can be constructed or modified without affecting given dynamic properties.
According to our notation, the letter p is reserved for representing the period along an axis of movements. The classical distance between the parallel lines is represented by the letter T. The periods (p) are equal to the spaces between the lines (T), only when the lines are perpendicular to the movement axis (as in the case of Figure 2 with horizontal lines and a vertical movement axis). Our equations represent completely the inclined layer and moiré patterns and at the same time the formulas for computing moiré periods and optical speedups remain in their basic simple form (2.1), (2.2), and (2.3).
In this section we focus on linear movements. Equations binding inclination angles of layers and moiré patterns are based on, , and , the periods of the revealing layer, base layer, and moiré lines respectively measured along the axis of movements.
For linear movements the p values represent distances along a straight axis. For rotational movements the p values represent the periods along circumference, i.e. the angular periods.
The superimposition of two layers with identically inclined lines forms moiré lines inclined at the same angle. Figure 3 (a) is obtained from Figure 1 with a vertical shearing. In Figure 3 (a) the layer lines and the moiré lines are inclined by 10 degrees. Inclination is not a rotation. During the inclination the distance between the layer lines along the vertical axis is conserved (p), but the true distance T between the lines (along an axis perpendicular to these lines) changes. The vertical periods and , and the distances and are indicated on the diagram of an example shown in Figure 5 (a).
Figure 3. (a) Superposition of layers consisting of inclined parallel lines where the lines of the base and revealing layers are inclined at the same angle; (b) Two layers consisting of curves with identical inclination patterns, and the superposition image of these layers
The inclination degree of layer lines may change along the horizontal axis forming curves. The superposition of two layers with identical inclination pattern forms moiré curves with the same inclination pattern. In Figure 3 (b) the inclination degree of layer lines gradually changes according the following sequence of degrees (+30, –30, +30, –30, +30), meaning that the curve is divided along the horizontal axis into four equal intervals and in each such interval the curve’s inclination degree linearly changes from one degree to the next according to the sequence of five degrees. Layer periods and represent the distances between the curves along the vertical axis, i.e. that of the movement. In Figure 3 (a) and (b), the ratio/ is equal to 12/11. Figure 3 (b) can be obtained from Figure 1 by interpolating the image along the horizontal axis into vertical bands and by applying a corresponding vertical shearing and shifting to each of these bands. Equation (2.1) is valid for computing the spacing between the moiré curves along the vertical axis and equation (2.3) for computing the optical speedup ratio when the revealing layer moves along the vertical axis.
More interesting is the case when the inclination degrees of layer lines are not the same for the base and revealing layers. Figure 4 shows four superposition images where the inclination degree of base layer lines is the same for all images (10 degrees), but the inclination degrees of the revealing layer lines are different and are equal to 7, 9, 11, and 13 degrees for images (a), (b), (c), and (d) respectively. The periods of layers along the vertical axis and (the / ratio being equal to 12/11) are the same for all images. Correspondingly, the period computed with formula (2.1) is also the same for all images.
Figure 4. Superposition of layers consisting of inclined parallel lines, where the base layer lines’ inclination is 10 degrees and the revealing layer lines’ inclination is 7, 9, 11, and 13 degrees [ps], [gif], [tif]
Figure 5 (a) helps to compute the inclination degree of moiré optical lines as a function of the inclination of the revealing and the base layer lines. We draw the layer lines schematically without showing their true thicknesses. The bold lines of the diagram inclined by degrees are the base layer lines. The bold lines inclined by degrees are the revealing layer lines. The base layer lines are vertically spaced by a distance equal to , and the revealing layer lines are vertically spaced by a distance equal to . The distance between the base layer lines and the distance between the revealing layer lines are the parameters used in the common formulas, well known in the literature. The parameters and are not used for the development of our equations. The intersections of the lines of the base and the revealing layers (marked in the figure by two arrows) lie on a central axis of a light moiré band that corresponds in Figure 4 to the light area between two parallel dark moiré lines. The dashed line passing through the intersection points of Figure 5 (a) is the axis of the light moiré band. The inclination degree of moiré lines is therefore the inclination of the dashed line.
Figure 5. (a) Computing the inclination angle of moiré lines as a function of inclination angles of the base layer and revealing layer lines; (b) Moiré lines inclination as a function of the revealing layer lines inclination for the base layer lines inclination equal to 20, 30, and 40 degrees [xls]
From Figure 5 (a) we deduce the following two equations:
From these equations we deduce the equation for computing the inclination of moiré lines as a function of the inclinations of the base layer and the revealing layer lines:
For a base layer period equal to 12 units, and a revealing layer period equal to 11 units, the curves of Figure 5 (b) represents the moiré line inclination degree as a function of the revealing layer line inclination. The base layer inclinations for the three curves (from left to right) are equal to , , and degrees respectively. The circle marks correspond to the points where both layers’ lines inclinations are equal and the moiré lines inclination also become the same.
The periods , , and (see Figure 5 (a)) that are used in the commonly known formulas of the literature are deduced from periods , and as follows:
From here, using our equation (2.5) we deduce the well known formula for the angle of moiré lines [Amidror00a]:
Recall from trigonometry the following simple formulas:
From equations (2.7) and (2.8) we have:
From equations (2.1) and (2.6) we have:
From equations (2.9) and (2.10) we deduce the second well known formula in the literature, the formula for the period of moiré lines:
Recall from trigonometry that:
In the particular case when , taking in account equation (2.12), equation (2.11) is further reduced into well known formula:
Still for the case when , we can temporarily assume that all angles are relative to the base layer lines and rewrite equation (2.7) as follows:
Recall from trigonometry that:
Therefore from equations (2.14) and (2.15):
Now for the general case when the revealing layer lines do not represent the angle zero:
We obtain the well known formula [Amidror00a]:
Equations (2.7) and (2.11) are the general case formulas known in the literature, and equations (2.13) and (2.18) are the formulas for rotated identical patterns (i.e. the case when ) [Amidror00a], [Nishijima64a], [Oster63a], [Morse61a].
Assuming in equation (2.7) that , we have:
Only for the case when the rotation of moiré lines is linear with respect to the rotation of the revealing layer (see equation (2.18)). Comparison of equation (2.19) and its respective graph (see [Gabrielyan07a]) with our equation (2.5) and its respective graph (see Figure 5 (b)) shows a significant difference in the binding of angles for sheared (i.e. inclined) and rotated layer patterns.
From equation (2.5) we can deduce the equation for computing the revealing layer line inclination for a given base layer line inclination , and a desired moiré line inclination :
The increment of the tangent of the revealing lines’ angle () relatively to the tangent of the base layer lines’ angle can be expressed, as follows:
According to equation (2.3), is the inverse of the optical acceleration factor, and therefore equation (2.21) can be rewritten as follows:
Equation (2.22) shows that relative to the tangent of the base layer lines’ angle, the increment of the tangent of the revealing layer lines’ angle needs to be smaller than the increment of the tangent of the moiré lines’ angle, by the same factor as the optical speedup.
For any given base layer line inclination, equation (2.20) permits us to obtain a desired moiré line inclination by properly choosing the revealing layer inclination. In Figure 3 (b) we showed an example, where the curves of layers follow an identical inclination pattern forming a superposition image with the same inclination pattern. The inclination degrees of the layers’ and moiré lines change along the horizontal axis according the following sequence of alternating degree values (+30, –30, +30, –30, +30). In Figure 6 (a) we obtained the same superposition pattern as in Figure 3 (b), but the base layer consists of straight lines inclined by –10 degrees. The corresponding revealing pattern is computed by interpolating the curves into connected straight lines, where for each position along the horizontal axis, the revealing line’s inclination angle is computed as a function of and , according to equation (2.20).
Figure 6. (a) The base layer with inclined straight lines and the revealing layer computed so as to form the desired superposition image; (b) Inversed inclination patterns of moiré and base layer curves [ps], [tif], [gif]
The same superposition pattern as in Figure 3 (b) and Figure 6 (a) is obtained in Figure 6 (b). Note that in Figure 6 (b) the desired inclination pattern (+30, –30, +30, –30, +30) is obtained using a base layer with a completely inverted inclination pattern (–30, +30, –30, +30, –30).
Figure 6 (a) and (b) demonstrate what is already expressed by equation (2.22): the difference between the inclination patterns of the revealing layer and the base layer are several times smaller than the difference between the inclination patterns of moiré lines and the base layer lines.
Our web page contains a GIF animation [gif] for modifying pairs of base and revealing layers constantly forming the same superposition image of Figure 3 (b), Figure 6 (a), and Figure 6 (b) i.e. the moiré inclination pattern (+30, –30, +30, –30, +30) [Gabrielyan07b]. In the animation, the base layer inclination pattern gradually changes and the revealing layer inclination pattern correspondingly adapts such that the superposition image’s inclination pattern remains the same.
Similarly to layer and moiré patterns comprising parallel lines (see Figure 1), concentric superposition of dense periodic layer patterns comprising radial lines forms magnified periodic moiré patterns also with sparse radial lines.
Figure 7 is the counterpart of Figure 1, where the horizontal axis is replaced by the radius and the vertical axis by the angle. Full circumferences of layer patterns are equally divided by integer numbers of radial lines. The number of radial lines of the base layer is denoted as and the number of radial lines of the revealing layer is denoted as .
The periods and denote the angles between the central radial axes of adjacent lines. Therefore:
According to equations (3.1), equation (2.1) can be rewritten as follows:
Therefore the number of moiré radial lines corresponds to the difference between the numbers of layer lines:
If in the layer patterns, the full circumferences are divided by integer numbers of layer lines, the circumference of the superposition image is also divided by an integer number of more lines.
The optical speedup factor of equation (2.3) can be rewritten by replacing the periods and by their expressions from equations (3.1):
The values and represent the angular speeds. The negative speedup signifies a rotation of the superposition image in a direction inverse to the rotation of the revealing layer. Considering (3.3), the absolute value of the optical speedup factor is:
Radial lines have constant angular thickness, giving them the forms of segments, thick at their outer ends and thin at their inner ends. The values of , , and do not depend on the angular thickness of radial lines. In our examples the angular thicknesses of layer lines are equal to the layer’s half-period, i.e. the thickness of the base layer lines is equal to and the thickness of the revealing layer lines is .
In Figure 7, the number of radial lines of the revealing layer is equal to 180, and the number of radial lines of the base layer is 174. Therefore, according to equations (3.4) and (3.3), the optical speedup is equal to 30, confirmed by the two images (a) and (b) of Figure 8, and the number of moiré lines is equal to 6, confirmed by the image of Figure 7.
Figure 8. Rotation of the revealing layer by 1 degree in the clockwise direction rotates the optical image by 30 degrees in the same direction
In circular periodic patterns curved radial lines can be constructed using the same sequences of inclination degrees as used in section 2.3 for curves of Figure 3 (b). The inclination angle at any point of the radial curve corresponds to the angle between the curve and the axis of the radius passing through the current point. Thus inclination angle 0 corresponds to straight radial lines as in Figure 7. With the present notion of inclination angles for , , and , equations (2.5) and (2.20) are applicable for circular patterns without modifications.
Curves can be constructed incrementally with a constant radial increment equal to . Figure 9 shows a segment of a curve, marked by a thick line, which has an inclination angle equal to .
Figure 9. Constructing a curve in a polar coordinate system with a desired inclination
While constructing the curve, the current angular increment must be computed so as to respect the inclination angle :
Figure 10 shows a superposition of layers with curved radial lines. The inclination of curves of both layers follows an identical pattern corresponding to the following sequence of degrees (+30, –30, +30, –30, +30). Layer curves are iteratively constructed with increment pairs computed according to equation (3.6). Since the inclination patterns of both layers of Figure 10 are identical, the moiré curves also follow the same pattern.
Figure 10. Superposition of layers in a polar coordinate system with identical inclination patterns of curves corresponding to (+30, –30, +30, –30, +30); a portion of the revealing layer is cut away exposing the base layer in the background [eps], [tif], [gif]
Similarly to examples of Figure 3 (b), Figure 6 (a), and Figure 6 (b), where the same moiré pattern is obtained by superposing different pairs of layer patterns, the circular moiré pattern of Figure 10 can be analogously obtained by superposing other pairs of circular layer patterns. Taking into account equations (3.1), equations (2.5) and (2.20) can be rewritten as follows:
Taking into account equation (3.4), equation (3.8) can be also rewritten as follows:
For producing the superposition image of Figure 10, thanks to equations (3.8) and (3.9), other pairs of layer patterns can be created as shown in Figure 11. In the first image (a) of Figure 11, the base layer lines are straight. In the second image (b), the base layer lines inclination pattern is reversed with respect to the moiré lines.
Figure 11. Superposition images with identical inclination pattern (+45, –45, +45, –45, +45) of moiré curves, where in one case the base layer comprise straight radial segments, and in the second case the base layer comprise curves which are the mirrored counterparts of the resulting moiré curves [eps], [tif], [gif]
Our web page [Gabrielyan07b] contains an animation [gif], where the moiré curves of the superposition image are always the same, but the inclination pattern of the base layer curves gradually alternates between the following two mirror patterns (+45, –45, +45, –45, +45), and (–45, +45, –45, +45, –45). For each instance of the animation, the revealing layer lines are computed according to equation (3.8) in order to constantly maintain the same moiré pattern.
Equations (3.4) and (3.3) remain valid for patterns with curved radial lines. In Figure 10 there are 180 curves in the revealing layer and 171 curves in the base layer. Therefore optical speedup factor according to equation (3.4) is equal to 20, and the number of moiré curves according to equation (3.3) is equal to 9, as seen in the superposition image of Figure 10.
One can form a radial moiré fringe with a period equal to . In the superposition image of such pattern we will see only one moiré fringe. This fringe will not have sharp contours and will appear large and blurred. The radial moiré fringe can be formed by layer patterns with radial lines or rather radial sectors. For small speed rations, fine granularity of layer patterns with radial lines cannot be maintained. As the speed ratio decreases, the superposition image becomes coarse and the moiré shape becomes visually not identifiable. The fine granularity can be maintained by using spiral shaped lines in layer patterns. The layer patterns with spirals can be computed such that the moiré fringe is kept radially oriented. By reducing the spiral elevation rate in both layers, sufficiently fine layer patterns can be obtained. However, strongly inclined spirals resulting to fine patterns make the superposition moiré images less tolerant to mechanical inaccuracies such as surface deformations of layers or disparities in concentric superposition of layers.
In Figure 12 we show that a design extension of simple spiral patterns with a single moiré fringe may result to a useful application.
The example is obtained by taking a simple spiral pattern of a base layer and by cleaning in such pattern all areas lying outside the contours of twelve labels. A part of the revealing layer is cut-away exposing the base layer. In such a way, instead of revealing a large and blurred moiré fringe, our superposition pattern reveals more attractive image consisting of labels within the concerned area. The area rotates at a 60 times faster speed than the mechanical rotation speed of the revealing layer. The spirals of two layers are computed so as to produce a moiré fringe with radial orientation.
In this section we present our multi-ring circular patterns. The superimposition of our multi-ring layer patterns forms a complex moiré image, but at one position a continuous shape is outlined. When rotating the revealing layer, the optical shape rotates without deformations at a k times faster speed.
Refer to equation (3.4) for circular patterns. An optical rotation k times faster than the rotation of the revealing layer can be obtained if:
According to equation (3.3) the number of moiré spots in a circular pattern for different values of i is simply equal to the value of i:
Therefore, the same moiré speedup factor k can be obtained with different pairs of revealing and base layer patterns corresponding to different numbers of moiré bands. We can construct several nested concentric circular patterns for the same value of k and for different values of i.
Figure 13. Four nested rings whose layer lines overlap at angle zero
For example, Figure 13 shows four nested adjacent rings, where the index i increments from 1 to 4 when counting from the inner ring toward the outer ring. The number of dark moiré radial lines of individual rings changes from 1 to 4 according to equation (5.2). The acceleration factor k is equal to 60 for all rings. Therefore the revealing layer of the most inner ring has 60 radial lines and the corresponding base layer has 59 lines. Correspondingly the layers of the most outer ring have 240 and 236 radial lines. In Figure 13 a part of the revealing layer is cut out, exposing the base layer. All rings are constructed such that the lines of the revealing and base layers perfectly overlap at the angle zero. Therefore a light moiré radial band appears at the angle zero of each individual rings.
For the inner ring, the dark moiré band is located at 180 degrees from angle zero. The first dark moiré band of the second ring is located at 90 degrees. The first dark moiré band of the third ring is located at 60 degrees and for the most outer ring at 45 degrees.
The patterns of the base and revealing layers of each ring can be printed so as a dark moiré band appears at the angle zero. For this purpose, both layer patterns of each ring must be rotated by a degree :
Figure 14 corresponds to the superimposition image of Figure 13, but the individual ring patterns are rotated according to equation (5.3) such that dark moiré bands appear at the angle zero in all rings. The black moiré bands of all adjacent rings became horizontally aligned forming a joint radial shape.
Once the rings are adjusted according to equation (5.3), we consider that the base layer patterns of all rings form a single joint base layer (e.g. printed on an opaque paper), and the revealing layer patterns of all rings form a joint revealing layer (e.g. printed on a transparency). A part of the revealing layer is cut away exposing the base layer.
Figure 14. Four nested rings with an acceleration factor equal to 60 for all four rings
According to equation (5.1), rotation of the revealing layers at a given angular speed must rotate the superimposition image at another angular speed which is identical for all rings. Therefore the radial moiré band traversing all rings will remain aligned all the time during the rotation. Rotation of the revealing layer rotates the optical image at a k times faster speed.
The acceleration factor k of the superimposition image of Figure 14 is equal to 60. Therefore the rotation of the revealing layer by –1 degree rotates the optical image by –60 degrees (compare the image of Figure 14 with the first image of Figure 15). Rotation of the revealing layer by –2 degree rotates the optical image by –120 degrees (compare the image of Figure 14 with the second image of Figure 15). The negative rotation angles correspond to the rotation in clockwise direction. The negligible rotations of the revealing layer in Figure 14 and Figure 15 can be noticed by observing the cut out region of the revealing layer.
In our web site [Gabrielyan07b] we present a GIF file [gif] which demonstrates the superposition image shown in Figure 14 and Figure 15 during a rotation of the revealing layer that slowly turns by 6 degree in clockwise direction. During this time the superimposition image makes a full rotation of 360 degree also in clockwise direction.
The widths of the rings of the multi-ring patterns must not be obligatorily the same. The number of the ring’s moiré bands also must not necessarily increment with the ring number.
Figure 16 shows a superimposition image with 12 rings, where at the beginning the number of moiré bands increments, but after reaching a maximal limit at a ring , the number of moiré bands starts decrementing. The maximal number of moiré bands is set to 10. Therefore the number of moiré bands follows the following sequence (1, 2, 3, … 8, 9, 10, 9, 8). The ring widths are not constant and are computed so as the largest ring is the ring , at which has its maximal value. The adjacent rings gradually decrease their widths as we move away from the largest ring.
The width of the j-th ring can be computed by equation (5.4), where j is the sequential number of the ring, is the number of the widest ring, is the minimal ring width, and is the maximal ring width.
Figure 16. Multi-ring moiré superposition image with variable ring widths
Recall that for measuring the line inclination in circular patterns we use the angle between the line and the radial axis as shown in Figure 9. In Figure 16, inclination of moiré lines of the superposition image is equal to 0 degree for all rings. In section 3.2 we show that the desired degree of moiré inclination can be obtained by different pairs of base and revealing layer patterns.
It is sufficient to choose for every ring an inclination pattern of the base layer and then, the corresponding inclination pattern of the revealing layer can be computed thanks to equation (3.8) or (3.9). Taking into account that in multi-ring patterns the speedup factor k used in equations (5.1) is the same for all rings, equation (3.9) can be rewritten as follows:
For a particular case, when , i.e. when we desire straight radial moiré lines, equation (6.1) is further reduced to:
For any inclination of the base layer pattern, the revealing layer pattern can be computed according to equation (6.2) to ensure straight radial moiré lines. Figure 17 shows a superposition image with straight moiré lines, similarly to Figure 14. In contrast to Figure 14 the base layer lines are not straight. The overall inclination pattern of the entire base layer across all rings follows the following sequence of inclination degrees .
Figure 17. Multi-ring moiré superposition image, where the inclination of moiré lines is of 9 degree and the inclination of the base layer lines follows the following inclination pattern (–30, +30, –30, +30) [ps], [gif]
Figure 18. Multi-ring moiré superposition image, where the inclination of moiré lines is of 0 degree and the inclination of the base layer lines follows the following inclination pattern (+30, –30, +30)
Figure 18 is the counterpart of Figure 16. In both figures the pattern of variable ring widths is computed by equation (5.4). In contrast to Figure 16 the base layer lines of Figure 18 are not straight. The overall inclination pattern of the base layer across all rings follows the following sequence of inclination degrees . The revealing layer line inclinations are computed according to equation (6.2) so as the superposition image forms the same straight moiré line shape as in Figure 16.
Inclined and curved layer patterns help in maintaining a uniform fine granularity across the surface of the disk. When the density of radial layer lines is sparse, the granularity can be refined by increasing the layer inclination degree.
In Figure 14, Figure 16, Figure 17, and Figure 18 we assemble the base layer and revealing layer patterns from rings rotated according equation (5.3), such that in the superposition image, the moiré fringes are aligned along the angle zero.
Equation (5.3) does not hold for cases when the moiré fringes themselves are curved. In this section we introduce multi-ring patterns with curved moiré shapes.
The curved moiré fringes of individual rings must join into a continuous moiré shape across the multi-ring superposition pattern. The angle of equation (5.3) for every successive ring must be additionally adjusted by the angular shift gained by the moiré curve while traversing the preceding rings.
Let be the inclination of the moiré line as a function of the radius r. Let and be the inner and outer radiuses of the j-th ring. According to equation (3.6) the angular gain of the moiré curve within the j-th ring is expressed as follows:
The aggregate angular gain up to the j-th ring is computed as follows:
Equation (5.3) must be rewritten so as to consider also the adjustment brought by equation (6.4):
With the angular adjustments of ring patterns computed by equation (6.5), we can create a continuous moiré curve jointly lying across all rings of the pattern. Figure 19 shows a serpentine shaped moiré curve. There are 14 rings of equal width. The acceleration factor k is equal to 30. The number of moiré spots increments starting from 1, for the inner ring, through 14 for the most outer ring. The base layer line inclination pattern corresponds to the following sequence of angles (–80, +10, –10, +10, –30). The revealing layer line inclination pattern is computed according to equation (6.1) so as to ensure the following moiré inclination pattern (+30, –30, +30, –30, +30). A small part of the revealing layer is cut away exposing the uncovered part of the base layer pattern.
Figure 20 shows a serpentine-shaped moiré curve in a multi-ring moiré with a variable ring width pattern of Figure 16 described by equation (5.4). There are 14 rings; the acceleration factor is equal to 30. The base layer inclination pattern is (–80, 5, 0, –5, –80), the moiré inclination pattern is (30, –30, 30, –30, 30); the revealing layer inclination pattern is computed with equation (6.1).
Figure 19. Multi-ring moiré with a continuous serpentine-shaped moiré curve
Many basic measurement instruments comprising a mechanical scale and a mechanical pointer have often their developed versions with a supplementary sub-graduated scale and an auxiliary mechanical pointer which moves faster and aims at the precision increase. A mechanical gearing system is used for a fast movement of the auxiliary pointer synchronously with the main pointer.
We developed layer patterns forming optical moiré shapes suitable for the auxiliary fast indicator. Mechanical transmission systems are not required. Moiré shapes can be obtained by superposition of transparent layers carrying correlated opaque patterns. The following points are important: (a) the moiré shapes must be sharp, (b) highly periodic moiré shapes cannot be used for indication, (c) the periodicity of moiré shapes must be very long corresponding to the visible window of the superposition image, such that one and only one shape is visible at a time, e.g. in circular moiré the period of moiré shapes must be equal to 360 degree; (d) the optical speedup of the mechanical movement must be linear; (e) the said above must be valid for the full range of mechanical movements of the main pointer putting into motion the revealing layer, e.g. in circular patterns for the full range of 360 degree rotation of the revealing layer.
Sharp moiré shapes are easily formed in well known simple periodic moiré patterns; however their periodicity is very high and cannot be used for indication. Long periods, such as 360 degrees for circular moiré, can be obtained with simple moiré patterns; however the moiré shape becomes blurred and not acceptable for indication. The known random line moiré offers completely aperiodic shapes without the required long periodicity. Additionally, their shapes are noisy compared with their periodic counterparts.
We introduced multi-stripe and multi-ring moiré patterns offering very long periods suitable for measurement purposes and forming moiré shapes as sharp as in highly periodical patterns.
We developed equations for producing straight and curved auxiliary moiré pointers across multi-ring moiré patterns. We can obtain a moiré shape of any desired curve that can be represented by a continuous function. Our equations help to obtain the desired moiré shape for different base layer patterns by finding the matching revealing layer pattern.
In our model, the choice of the shape of the moiré fringe has no impact on the dynamic properties of the auxiliary moiré pointer. We preserve the speedup formulas in their simplest form (2.1), (2.2), and (2.3) for linear movements and (3.2), (3.3), (3.5), and (3.4) for rotations regardless the inclination patterns of layers and moiré shapes.
[Gabrielyan07b] Emin Gabrielyan, “Fast optical Indicator created with multi-ring moiré Patterns”, Switzernet research reports, 4 August 2007, http://switzernet.com/people/emin-gabrielyan/070804-moire-rings
[Hutley99] M.C. Hutley and R.F. Stevens, “Optical inspection of arrays and periodic structures using moire magnification”, IEE Colloquium: Microengineering in Optics and Optoelectronics, No. 187, p. 8, 16 November 1999
[Nishijima64a] Y. Nishijima and G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements”, Journal of the Optical Society of America, Vol. 54, No. 1, pp. 1-5, January 1964 [pdf]
[Oster63a] G. Oster and Y. Nishijima, “Moiré patterns”, Scientific American, Vol. 208, pp. 54-63, May 1963
Table of figures
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s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250591431.4/warc/CC-MAIN-20200117234621-20200118022621-00436.warc.gz
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CC-MAIN-2020-05
| 44,695 | 136 |
https://www.coursehero.com/file/p7b2s5k9/this-analysis-a-coverage-factor-of-k-2-should-be-used-This-coverage-factor/
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math
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this analysis, a coverage factor ofk= 2 should be used. Thiscoverage factor provides a confidence level of approximately95 %.X2.4.4 The measurement biasBof the hardness machine isthe difference between the expected hardness measurementvalues as displayed by the hardness machine and the “true”hardness of a material. Ideally, measurement biases should becorrected. When test systems are not corrected for measure-ment bias, as often occurs in Rockwell hardness testing, thebias then contributes to the overall uncertainty in a measure-ment. There are a number of possible methods for incorporat-ing biases into an uncertainty calculation, each of which hasboth advantages and disadvantages. A simple and conservativemethod is to combine the bias with the calculation of theexpanded uncertainty as:U5kuc1ABS~B!(X2.5)whereABS(B) is the absolute value of the bias.E18 - 1532
X2.4.5 Because several approaches may be used to evaluateand express measurement uncertainty, a brief description ofwhat the reported uncertainty values represent should beincluded with the reported uncertainty value.X2.5 Sources of UncertaintyX2.5.1 This section describes the most significant sourcesof uncertainty in a Rockwell hardness measurement andprovides procedures and formulas for calculating the totaluncertainty in the hardness value. In later sections, it will beshown how these sources of uncertainty contribute to the totalmeasurement uncertainty for the three measurement circum-stances described inX2.1.2.X2.5.2 The sources of uncertainty to be discussed are (1) thehardnessmachine’slackofrepeatability,(2)thenon-uniformity in hardness of the material under test, (3) thehardness machine’s lack of reproducibility, (4) the resolutionof the hardness machine’s measurement display, and (5) theuncertainty in the certified value of the reference test blockstandards. An estimation of the measurement bias and itsinclusion into the expanded uncertainty will also be discussed.X2.5.3Uncertainty Due to Lack of Repeatability (uRepeat)and when Combined with Non-uniformity (uRep&NU)—Therepeatability of a hardness machine is an indication of howwell it can continually produce the same hardness value eachtime a measurement is made. Imagine there is a material, whichis perfectly uniform in hardness over its entire surface. Alsoimagine that hardness measurements are made repeatedly onthis uniform material over a short period of time withoutvarying the testing conditions (including the operator). Eventhough the actual hardness of every test location is exactly thesame, it would be found that due to random errors eachmeasurement value would differ from all other measurementvalues(assumingsufficientmeasurementresolution).
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964364169.99/warc/CC-MAIN-20211209122503-20211209152503-00353.warc.gz
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CC-MAIN-2021-49
| 2,722 | 2 |
https://www.numbersaplenty.com/465
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math
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465 has 8 divisors (see below), whose sum is σ = 768.
Its totient is φ = 240.
The previous prime is 463. The next prime is 467. The reversal of 465 is 564.
465 is nontrivially palindromic in base 11 and base 16.
465 is an esthetic number in base 15, because in such base its adjacent digits differ by 1.
465 is a nontrivial binomial coefficient, being equal to C(31, 2).
It is an interprime number because it is at equal distance from previous prime (463) and next prime (467).
It is a sphenic number, since it is the product of 3 distinct primes.
It is not a de Polignac number, because 465 - 21 = 463 is a prime.
It is a Harshad number since it is a multiple of its sum of digits (15), and also a Moran number because the ratio is a prime number: 31 = 465 / (4 + 6 + 5).
465 is an undulating number in base 11 and base 16.
It is a plaindrome in base 7, base 9 and base 13.
It is a nialpdrome in base 5, base 8 and base 15.
It is a congruent number.
It is not an unprimeable number, because it can be changed into a prime (461) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (5) of ones.
It is a polite number, since it can be written in 7 ways as a sum of consecutive naturals, for example, 1 + ... + 30.
It is an arithmetic number, because the mean of its divisors is an integer number (96).
465 is the 30-th triangular number.
It is an amenable number.
465 is a deficient number, since it is larger than the sum of its proper divisors (303).
465 is a wasteful number, since it uses less digits than its factorization.
465 is an odious number, because the sum of its binary digits is odd.
The sum of its prime factors is 39.
The product of its digits is 120, while the sum is 15.
The square root of 465 is about 21.5638586528.
The cubic root of 465 is about 7.7473108950.
Adding to 465 its product of digits (120), we get a palindrome (585).
Subtracting 465 from its reverse (564), we obtain a palindrome (99).
The spelling of 465 in words is "four hundred sixty-five", and thus it is an aban number.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764501407.6/warc/CC-MAIN-20230209045525-20230209075525-00255.warc.gz
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CC-MAIN-2023-06
| 2,064 | 30 |
http://putzman1.tripod.com/Physics/page4a.htm
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math
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Potential energy is differentiated from kinetic energy by the fact that it is the "stored" energy of an object at rest, although, just like kinetic energy, certain amounts of potential energy can be in an object in motion. There are two types of potential energy that we are concerned with, gravitational potential energy and spring potential energy.
Gravitational potential energy is the potential energy "created" by an object being higher or lower than some fixed "zero" point due to the force of gravity. It is given as mass times acceleration due to gravity times the height of the object from the fixed point.
The other type of potential energy is spring, also known as elastic, potential energy. This is potential energy pent up in a compressed spring, or spring-like mechanism, before it is released. Spring potential energy is one-half times the spring constant times the length compressed (or stretched) from the spring's normal length.
A brief discussion of the spring constant is in order. The spring constant is a measure of the overall "tension" (it's not technically tension, actually; tension is a type of force) or "springiness" of a spring. It is measured in kilograms ÷ seconds˛, or kg ÷ (s˛).
Previous (Kinetic Energy) | Next (Conservation of Energy)
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s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891377.59/warc/CC-MAIN-20180122133636-20180122153636-00574.warc.gz
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CC-MAIN-2018-05
| 1,274 | 5 |
https://www.theproblemsite.com/ask/category/order-of-operations/page/1
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math
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Ask Professor Puzzler
Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Here's a common question among math teachers and students (and math dabblers who just like to raise people's hackles!):
"If you see 3/2x, how do you interpret it? Is it 3 divided by 2x? Or 3 divided by 2, times x? Order of operations says we do division and multiplication left-to-right, which leads to the second answer. However, if you look at the slash as a division symbol, it appears to be the other way: 3 is the numerator and 2x is the denominator."
The correct answer to this question is: it's neither.
That's right. It's neither 3/(2x) nor 3/2 times x.
So no matter which way you were arguing, you're wrong. Let me explain.
Whenever you come across something like this: 3/2, the standard reading is not "three divided by two." You read it as "three over two," (this is considered to be the proper designation for the slash symbol when used in this context). This lends credence to the notion that the slash is being used as a fraction bar, and therefore, our example should be read as a fraction: 3/(2x).
But did you know that there are specific rules for how you write fractions using standard typographic practices? First, you are expected to use a specific slash symbol, which is not your standard "forward slash" on your keyboard - it's a unicode symbol called "fraction slash." The fraction slash is designed with minimal kerning (space between characters), and there's a very good reason for this.
There's another typographical practice we must follow: we superscript the numerator and subscript the denominator. The superscripts and subscripts, combined with the minimal kerning, result in the numerator being above the slash, and the denominator below.
Thus, we would either write: 3/2x or 3/2x, and now you can see that proper typographic practices makes it clear which way we intended it to be interpreted.
In other words, 3/2x is actually just a typographical error, and not a real mathematical expression. It's the result of someone being lazy. (Don't worry, I've done it too!). With the sophisticated word processors we have these days, with powerful equation editors, there's no longer any excuse for any mathematician to type 3/2x. In fact, with equation editors, you can get expressions that appear much nicer than the ones that you create with superscripts and subscripts.
Of course, there is one place where this typographical error still shows up: calculators.
Many calculators are not designed for proper typographical display of fractions. So what do we do? We do one of the following:
- Figure out which way your particular calculator handles this expression, and always do it that way.
- The safer approach: When dealing with a calculator, always clarify your meaning by including parentheses.
Once you've settled on one of these practices, it's time to accept the fact that you haven't been arguing about some standard of mathematics, but about typography. It's now time to do some real math, and leave behind the arguments about typographical quirks!
Sixth grader Elise asks, "I don't get BODMAS. Can you help me?"
Well, Elise, this is one of my favorite questions, and I get asked this a lot. But I've never written it up on the "Ask Professor Puzzler" blog, so here we go!
Before I get started explaining BODMAS, I need to mention that in different parts of the world, this "rule" is known by different names. You call it BODMAS, but some people call it PEMDAS. So when I explain what it means, in parentheses I'll explain what it means to people who call it something different.
The six letters each stand for something you can do to combine numbers in mathematics:
B = Brackets (P = Parentheses)
O = Order (E = Exponents)
D = Division
M = Multiplication
A = Addition
S = Subtraction
These six letters indicate the order in which you do the operations in a mathematical expression. For example, if you see the following:
2 - (3 + 2),
You notice that "3 + 2" is in brackets (parentheses), so you do that FIRST: 3 + 2 = 5
Now you have
2 - 5 = -3
Why does it matter which order you do things? It matters because you would get a different answer if you did the subtraction first:
2 - 3 = -1
-1 + 2 = 1
Uh oh! One way we get -3, and the other way we get 1!
Here's another example. Suppose you have 1 + 2^3 (1 + 2 cubed). If you did that from left to right, you would add 1 + 2 and get 3. Then you would cube that and get 27.
But what you're SUPPOSED to do, is evaluate the exponent first: 2 cubed is 8. Then you add 1 + 8, and get 9. It all depends on what order you do things, so you have to get the order right!
You see, we have to have an order of operations, or nobody would ever calculate expressions the same way. Order of Operations is a rule that helps to make sure EVERYONE evaluates the same expression in exactly the same way. If we didn't have order of operations, people would get different answers for the same problem, and that would be horrible - nothing would ever get done, and none of our technology would work right because teams of engineers would always be fighting over how to evaluate the equations and formulas they work with, and if they didn't use proper order of operations, not only would things not work right, you could end up with some pretty horrible catastrophes (imagine engineers using heat formulas in nuclear reactors or power plants, and not calculating correctly how much cooling they need!).
Now here's the tricky part (and I've even had emails from MATH TEACHERS who don't understand this part!): you DON'T do all your multiplication before all your division, and you DON'T do all your addition before all your subtraction. Multiplication and division are on the same order of priority, and addition and subtraction are on the same order of priority. If a problem has both multiplication and division operations, you do them from left to right. If a problem has both addition and subtraction, you do them from left to right as well.
So really, it should be written BO (DM) (AS)* to remind you that division and multiplication go together, and addition and subtraction go together.
* Or PE (MD) (AS)
If you want some practice using BODMAS (PEMDAS), you can try our One To Ten Game, which challenges you to put toghether expressions that add up to all the numbers from one to ten.
We have a new math 6th grade series where Order of Operations has changed and PEMDAS is no longer the standard. Have you encountered this yet? We have Glenco, 6th grade. Multiplication and division still come before addition and substranction, however, now it works left to right whereas before it was multiplication before division.
In some areas of the world they use a different acronym (such as BODMAS or BEDMAS), but these are still the same thing as PEMDAS (Please Excuse My Dear Aunt Sally).
Believe it or not, the PEMDAS order of operations is not only still correct, but it's always been what you just described.
The acronym PEMDAS can be deceptive, if it’s not taught correctly. How it should be taught is:
P: Parenthesis first
E: Exponents next
MD: Multiplication and Division next
AS: Addition and Subtraction last
Notice that the M and D are grouped together, as are the A and S. This is because Multiplication and Division are at the same priority level, and should be done in left to right order. Likewise, Addition and Subtraction are at the same priority level, and should be done in left to right order.
Unfortunately, many teachers don’t realize this, and teach that all multiplication is done before all division, and all addition is done before all subtraction. I was taught that way all through elementary school, and it wasn’t until I hit Jr/Sr high that I found out that Multiplication and Division are at the same priority level, as are Addition and Subtraction.
If you are looking for a game that forces students to think through Order of Operations, here's a game I created several years ago: One To Ten.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057417.10/warc/CC-MAIN-20210923044248-20210923074248-00264.warc.gz
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CC-MAIN-2021-39
| 8,009 | 54 |
https://patents.google.com/patent/US6002293A/en
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math
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BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to the field of bandgap voltage reference cells, and particularly to bandgap reference cells having a high transconductance.
2. Description of the Related Art
A basic bandgap voltage reference cell is shown in FIG. 1. Two bipolar transistors Qa and Qb are driven by the output of an operational amplifier 14, with their collectors connected to the op amp's non-inverting and inverting inputs, respectively, and to a supply voltage V+ through respective resistors 16 and 18. A resistor Ra is connected between the transistors' respective emitters, and a "tail" resistor Rb is connected between the emitter of Qb and circuit common.
Qa is fabricated with an emitter area larger than that of Qb (by a ratio of 8-to-1 in FIG. 1). The op amp adjusts the transistors' base voltage until the voltages at its inverting and non-inverting inputs are equal. This occurs when the two collector currents match, which in this example happens when the emitter current densities are in the ratio of 8-to-1. This arrangement produces a voltage across Rb that is proportional-to-absolute temperature (PTAT), which can be used to compensate the complementary-to-absolute-voltage (CTAT) characteristic of the base-emitter voltage of Qb. Setting OUT equal to the bandgap voltage of silicon provides the proper compensation, and thereby produces a temperature invariant output voltage.
The transconductance gm of the circuit of FIG. 1 is defined as the change in the difference in the transistors' collector currents divided by the change in their base-emitter voltage. Because the difference in collector currents cannot exceed the change in current through Rb, the transconductance is capped at 1/Rb, but because a perturbation causes both collector currents to change in the same direction, the maximum attainable gm is actually less than 1/Rb. This bandgap reference cell and its characteristics are discussed in detail in A. Paul Brokaw's "A Simple Three-Terminal IC Bandgap Reference", IEEE Journal of Solid-State Circuits, Vol. SC-9, No. 6(1974).
Another bandgap reference cell is shown in FIG. 2, made from two transistors pairs connected in a "crossed-quad" configuration. A first pair of transistors Qc and Qd are connected in series with a second pair of transistors Qe and Qf, respectively, with the bases of Qe and Qf connected to the collectors of Qf and Qe, respectively. Transistors Qc and Qd have unequal emitter areas, as do transistors Qe and Qf. A resistor Rc is connected between the emitters of Qe and Qf, and a tail resistor Rd is connected between the emitter of Qf and circuit common. The collectors of Qc and Qd are connected to the inputs of an amplifier 20. The amplifier's output drives a pass transistor Qf to produce a regulated output OUT, which is fed back to Qc 's and Qd 's common bases. A PTAT voltage appears at the junction between Rc and Rd ; when the resistors are properly chosen, the PTAT voltage compensates for the base-emitter voltages of Qf and Qd to produce a temperature invariant voltage equal to twice the bandgap voltage at OUT. Achieving an output voltage greater that is a non-integer multiple of the bandgap voltage is typically provided by adding a voltage divider 22 between OUT and the common base connection, as shown in FIG. 2. The divider imposes a voltage drop between the output and the common base connection, but assuming that amplifier 20 has sufficient gain, it will continue to balance the collector currents and the output will be stabilized at a higher voltage.
The transconductance of the circuit of FIG. 2 is somewhat better than that of FIG. 1. When the cell is at equilibrium (i.e., when the collector currents are balanced), a PTAT current flows in Rc which is determined solely by the emitter area ratios and the value of Rc ; i.e., essentially independent of the current on the right side of the crossed-quad. With the left side current fixed, when the cell's output is disturbed, nearly all of the resulting change in current goes through the right side of the cell (Qd and Qf), with the current through the left side (Qc and Qe) essentially unchanged. Thus, all of the change in current goes through Rd, and the cell's transconductance closely approaches 1/Rd.
Because of the relatively low transconductance of the bandgap cells in FIGS. 1 and 2, the voltage applied to the common bases (of Qa and Qb in FIG. 1; Qc and Qd in FIG. 2) must depart substantially from the voltage which balances the currents if a large difference in collector currents is needed. This is usually accommodated by connecting a high gain amplifier across the collectors, to provide a differential-to-single ended conversion as well as the voltage gain necessary to return to equilibrium; this function is represented by amplifier 20 FIG. 2.
Disadvantages are found in the circuits of FIGS. 1 and 2, particularly when low power consumption is important, as with a battery-powered regulator. The power consumed by amplifier 20 will hasten the discharge of a battery used to provide the circuit's supply voltage, as will the energy lost in resistive divider 22. Use of a resistive divider 22 is also troublesome if the regulator is employed, for example, as a battery charger, with a battery to be charged connected to OUT. When the regulator is inactive or unable to provide the necessary charging current, the presence of a divider actually provides a discharge path for the battery, shortening its life.
SUMMARY OF THE INVENTION
A novel voltage reference cell is presented which has a very high transconductance, producing a large change in output current for a very small change in input voltage near a settable equilibrium point and thereby dispensing with the need for a high gain amplifier. The cell can be configured to set the equilibrium point equal to two bandgap voltages, or to non-integer multiples of the bandgap voltage without the use of a resistive divider. Eliminating the amplifier and resistive divider components of prior art designs reduces the reference cell's component count, as well as its power consumption.
The core of the voltage reference cell is made from a first and second pair of bipolar transistors nominally arranged in a crossed-quad configuration, with the bases of the first pair connected together at an input node. At least one of the transistor pairs have unequal emitter areas. In contrast with a standard crossed-quad configuration, however, a first resistor is interposed between one of the first pair transistors and the base of one of the second pair transistors, at least one of which has a larger emitter area than its pair, with a second resistor connected to the emitter of the second pair transistor on the opposite side of quad from the first resistor.
A voltage applied to the input node causes a current to flow through the cell from the input node to the common point. For input voltages below an "equilibrium" point, the unequal emitter areas force the voltages at the bases of the two second pair transistors to be unequal, which causes most of the current to flow down one side of the quad. As the input voltage increases toward the equilibrium point, the voltage drop across the first resistor increases and the inequality between the second pair transistors' base voltages gets smaller. The relationship between the two base voltages reverses as the equilibrium point is exceeded, causing the cell current to be abruptly "switched" from one side of the quad to the other.
The cell's output is taken at the collectors of the first pair of transistors, with nearly all of the cell current switching from one collector to the other at the equilibrium voltage. In prior art cells, a change in current was largely reflected on only one side of the cell. Here, a change in cell current at the equilibrium point causes the current on the two sides to move in opposite directions, with the movement equal to the nearly the entire cell current. This large change in current induced by a very small change in input voltage provides the cell a very high transconductance.
A maximum transconductance is obtained when the first and second resistors are equal. However, by simply making the value of one of the resistors greater than the other, additional options are presented to a designer: making the second resistor value greater than the first provides a somewhat lower gm, which might be needed to improve loop stability, for example. Making the first resistor greater than the second creates a loop gain greater than one, which introduces some hysteresis around the equilibrium point that may be useful in regenerative applications such as a comparator.
The equilibrium point is established at a voltage dictated by the emitter area ratios between the quad's transistors. When the input voltage is such that the sum of the voltage drops across the resistors equals the voltage set by the emitter area ratios, the cell current switches sides. The cell thus carries a proportional-to-absolute-temperature (PTAT) current at the equilibrium point, which can be used to drive a pass transistor or an amplifier, for example. With the addition of a properly chosen tail resistor, the cell can produce an output voltage equal to two bandgap voltages.
The cell can also generate output voltages that are higher, non-integer multiples of the bandgap voltage without the use of a resistive divider. The tail resistor is split into two resistors, with the junction between them connected, via another resistor, to a transistor having its base connected to the input node. These components are arranged so that a temperature invariant current is delivered to the junction point, which offsets the equilibrium point to a higher, temperature stable voltage.
Further features and advantages of the invention will be apparent to those skilled in the art from the following detailed description, taken together with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1 and 2 are schematic diagrams of prior art bandgap voltage reference cells.
FIG. 3 is a schematic diagram of a high transconductance voltage reference cell per the present invention.
FIG. 4a is a schematic diagram of the novel cell having an equilibrium voltage equal to twice the bandgap voltage, and a table illustrating various obtainable loop gains.
FIG. 4b is a schematic diagram of the novel cell configured as a comparator.
FIG. 5 is a schematic diagram of the novel cell as it might be used in a battery charger application.
DETAILED DESCRIPTION OF THE INVENTION
A high transconductance voltage reference cell per the present invention is shown in FIG. 3. The cell includes four bipolar transistors Q1-Q4 connected in a crossed-quad configuration. The bases of a first pair of transistors Q1 and Q2 are connected together and form an input node IN, and their respective collectors are connected to a current source 100, typically implemented with a current mirror, arranged to provide balanced currents to Q1 and Q2. A second pair of transistors Q3 and Q4 have their respective bases cross-coupled to each other's collectors, with Q3's base connected to Q4's collector at a node 102, and Q4's base connected to Q3's collector at a node 104. The transistors making up at least one of the pairs must have unequal emitter areas; in the exemplary circuit of FIG. 3, Q1 has an emitter area 4 times that of Q2.
The collectors of Q3 and Q4 are connected to the emitters of Q1 and Q2, respectively, with a resistor R1 interposed between the emitter of Q1 and node 104. Another resistor R2 is connected between the emitter of Q4 and a circuit common point 106, which is also connected to the emitter of Q3.
When an input voltage greater than two base-emitter voltages is applied at IN, the path from IN to common point 106 will be forward-biased and a "cell" current will flow between them. If the available current is small, the voltage drop across R1 and R2 must also be small, so that the distribution of cell current in transistors Q1-Q4 is controlled by their respective emitter areas. Due to its larger emitter area, Q1's base-emitter voltage (Vbe1) is lower than that of Q2 (Vbe2) at equal currents, which forces node 102 at the base of Q3 to be lower than node 104 at the base of Q4. This makes the voltage applied to Q4 higher than that applied to Q3, making the collector current of Q4 greater than that of Q3. The imbalance of these currents increases the voltage between nodes 102 and 104, which further unbalances the currents. As a result, the current in the two right hand transistors Q2 and Q4 rises to take most of the cell current, with the collector current of Q15 carrying little more than the base current of Q4. In this state, most of the cell current is delivered to the output terminal OUT, where it is connected to drive a load represented by a resistor Rload which can be, for example, a pass transistor or an amplifier.
Summing the voltages between IN and common point 106 (and neglecting base currents):
V.sub.be3 +V.sub.be2 =V.sub.be1 +i.sub.1 R1+V.sub.be4 +i.sub.2 R2(Eq. 1)
where Vbex refers to the base-emitter voltage of Qx and iy refers to the current in Ry.
As the available cell current increases with an increasing input voltage, so will the current in Q1. At some particular input voltage, the currents in Q1 and Q2 become equal. In this case (neglecting base currents), the current in Q1 is the same as the current in Q3, and the current in Q2 is the same as the current in Q4. With the same currents in differently sized transistors, Vbe3 is given as follows:
V.sub.be3 =V.sub.be1 +(kT/q)ln4
where "4" is the ratio of emitter areas between Q1 and Q3. For similarly sized transistors Q2 and Q4, Vbe2 and Vbe4 will be nearly equal. Substituting these results into Equation (1) provide:
V.sub.be1 +(kT/q)ln4+V.sub.be4 =V.sub.be1 +i.sub.1 R1+V.sub.be4 +i.sub.2 R2
(kT/q)ln4=i.sub.1 R1+i.sub.2 R2 (Eq. 2)
Thus, when an input voltage is applied to IN so that the condition of Eq. 2 is met, the current in the left side of the cell (Q1 and Q3) will equal the current in the right side of the cell (Q2 and Q4). The input voltage which satisfies Eq. 2 is the cell's "equilibrium" voltage Veq. For input voltages below Veq, most of the cell current flows through Q2 and thereby pulls down on OUT, in the manner and for the reasons described above. However, when the input voltage exceeds Veq, most of the cell current abruptly switches sides and flows through Q1 to the current source 100, causing it to carry away any current from Q2 and the drive to the load connected to OUT is reduced to zero.
At the equilibrium voltage, the current through Q1 is just enough to make the voltage drop across R1 equal Q1's (kT/q)ln 4 difference in Vbe, which makes the voltages at nodes 104 and 102 equal. Above Veq, the voltage drop across R1 is too large to permit balance, while below Veq, the voltage drop is too small. When i1 R1 exceeds Q1's (kT/q)ln 4 difference in Vbe, the relationship between nodes 104 and 102 reverses--node 104 becomes lower than node 102--causing most of the cell current to flow in Q1. Conversely, when the cell current is too low, node 102 is low with respect to node 104, so that most of the current flows through Q2.
This flip-flopping of nearly all of the current from one side of the cell to the other at the equilibrium voltage gives the novel reference cell a very high transconductance. Because the currents are balanced at only one voltage, the transconductance is theoretically infinite: an infinitely small change in input voltage causes all of the current to switch sides. The gm is actually limited by base currents, but it is nevertheless very high. The new cell functions much differently than older designs: as described above, as input node voltage increased, the current on one side of a prior art cell would remain at a fixed value determined by emitter area ratios, with changes in cell current forced to appear on the opposite side. This inherently limited the achievable Δi and thus the transconductance. The novel cell functions by having nearly all of the current flow on one side of the quad, increasing beyond the limit imposed by the emitter area ratios of the prior art all the way up to the equilibrium voltage, at which point nearly all the cell current switches to the other side. The transconductance offered by the present invention is in sharp contrast to the relatively low gm of the prior art cells discussed above, which were limited to no more than the reciprocal of their tail resistor value.
From Eq. 2, it is seen that at the equilibrium point, the cell current is PTAT. This PTAT current can be used to make or detect other kinds of bandgap and non-bandgap voltages or currents with, for example, a non-zero temperature coefficient.
An embodiment of the present invention for which the equilibrium voltage is equal to two bandgap voltages is shown in FIG. 4a. Though the invention only requires that one of the quad pairs have unequal emitter areas, it is convenient for both pairs to be similarly constituted, and the second transistor pair in FIG. 4a now consists of Q3 and a multi-emitter transistor Q5. Vbe2 is now given by:
V.sub.be2 =V.sub.be5 +(kT/q)ln4
and the condition at which equilibrium is reached has been raised, and is given by:
(kT/q)ln16=i.sub.1 R1+i.sub.2 R2 (Eq. 3)
A tail resistor R3 has been connected between node 106 and circuit common in order to provide the double bandgap voltage. If we make R1=R2=Rtotal, then:
R.sub.total (i.sub.1 +i.sub.2)=(kT/q)ln16,
and neglecting Q3's base current, i1 +i2 is equal to i3, the total current in R3, so that:
i.sub.3 =((kT/q)ln16)/R.sub.total (Eq. 4)
At the equilibrium point, the current in R3, as well as in the quad transistors, is PTAT. If R3 is properly chosen, the PTAT voltage at node 106 compensates the two base-emitter junction voltages of Q3 and Q2 and yields a double bandgap voltage at the base of Q2, identified as a node 108.
Current source 100 is preferably implemented with a dual collector transistor Qs, connected as a current mirror: one of Qs 's collectors 110 is connected to its base and to the collector of Q1; current through Q1 is mirrored to Qs 's other collector 112, which is connected to the collector of Q2.
The base of a pass transistor Q6 is also connected to the collector of Q2. Q6 presents a relatively low impedance to Q2, and supplies whatever current it may need. Q6 together with the novel reference cell form a regulator, with Q6's collector serving as the regulator's output Vout. Q6's collector is connected to node 108 at the base of Q2.
The total current available to pull down on Q6's base is determined by the voltage across R3, which rises with Vout. This results in a "fold-back" V/I output characteristic. When the cell current exceeds the value given by Eq. 4, the circuit abruptly swings through its equilibrium condition, with the current that was flowing through the Q2/Q5 side of the quad now flowing through the Q1/Q3 side. The Q1 current is mirrored to its collector 112, reducing the drive to Q6 to near zero. Since the loop is closed to node 108 from the output of Q6, the output current will remain high as Vout approaches the equilibrium point, and then abruptly drops to near zero as the equilibrium voltage is reached. If the equilibrium voltage has been arranged to be at twice the bandgap voltage as described above, the point at which the output current drops to zero is made temperature stable.
Because the transconductance of the new cell is so high, the high gain amplifier required in the prior art designs discussed above can be eliminated. Output pass transistor Q6 can be driven directly and still provide relatively good regulation. Eliminating the amplifier lowers the regulator's power consumption, as well as its component count.
Essential to the operation of the invention is the way in which the relationship between the voltages at nodes 102 and 104 reverses as the input node voltage increases. The resistors and the larger emitter transistors must be placed to insure this functioning. If the first transistor pair has an unequal emitter ratio, R1 must be placed in series with the transistor having the larger emitter. The smaller emitter transistor will have a larger Vbe, making the node below its emitter lower than the node below R1 for lower input voltages. The voltage drop across R1, however, forces the relationship between the nodes to reverse when it carries a particular current--i.e, the cell current at the equilibrium voltage.
Similarly, if only the second transistor pair have an unequal emitter ratio, R2 should be placed in series with the transistor having the larger emitter. The larger emitter causes the transistor's collector to be pulled down harder than its pair is, unbalancing the voltages at their bases. The larger transistor's Vbe is reduced as the current through R2 increases, however, increasing the voltage of the node at its collector, with the relationship between the base voltages reversing at the equilibrium voltage.
If both pairs have unequal emitter ratios, the larger emitter transistors should be placed on opposite sides of the quad, as shown in FIG. 4a. R1 and R2 should also be placed on opposite sides of the quad.
The cell's transconductance is highest when R1=R2, which, because it is in a closed loop, provides a loop gain that reaches exactly +1 at the equilibrium point. Making R2 greater than R1 lowers the cell's gm and reduces the loop gain to less than +1, diminishing the abruptness with which the cell current switches from one side to the other. This might be done when a more controlled gm is desired--to frequency stabilize a closed loop system, for example.
Making R1 greater than R2 makes the loop gain greater than +1. Here, there is no point at which the currents are equally distributed. For this condition, the current will flow on the right side below and even at the equilibrium point. However, as input node 108 continues to rise, the current will abruptly switch to the other side, where it will stay until node 108 falls below the equilibrium point by some finite amount. This would be useful in regenerative applications; for example, in using the cell to provide a comparator with hysteresis.
Thus, as illustrated in the table shown in FIG. 4a, the invention can provide a very high gm (though with poor loop stability), a moderately high gm in a better controlled loop, or a gm providing a loop gain >1, useful for regenerative applications, by simply adjusting the respective values of R1 and R2.
A reference cell configured as a comparator is shown in FIG. 4b. The circuit is very similar to that of FIG. 4a, except that the left and right sides of the quad are reversed, with the collector of Q1 now connected to the base of transistor Q6, and a resistor Rcomp connected between the comparator's output, i.e., the collector of Q6, and circuit common. The common bases of Q1 and Q2 form an input terminal IN. When a voltage applied to IN is below the equilibrium voltage, most of the cell current flows through Q2. This current is mirrored to the base of Q6, reducing the drive to Q6 to nearly zero. Resistor Rcomp pulls the output low in this state. When the input exceeds the equilibrium voltage, the cell current switches to the Q1 side of the quad, driving Q6 and producing an output at OUT. R1 should be made greater than R2 to introduce some hysteresis, as described above.
In some applications, an equilibrium voltage that is greater than two bandgap voltages may be desired. This could be obtained with a voltage divider connected between the collector of Q6 and circuit common (referring back to FIG. 4a), with the divider tap connected to node 108. Vout is scaled to a higher voltage while the loop continues to come to balance when node 108 is at two bandgaps. However, for reasons noted above, the use of a resistive divider may be undesirable.
A regulator which addresses these problems and is built around the novel bandgap reference cell is shown in FIG. 5. The need to provide an output greater than two bandgaps is met with the addition of a transistor Q7 and a resistor R4. The base of Q7 is connected to input node 108 along with the bases of Q1 and Q2, and its emitter is connected to the bottom of tail resistor R3 at a node 120 via resistor R4. A resistor R5 is interposed between node 120 and circuit common.
When the regulator is in regulation, the voltage from node 108 to node 106 is equal to two base-emitter junctions voltages. Assuming some current in Q7, its emitter will be below node 108 by one base-emitter voltage, or one base-emitter voltage above node 106. R3 and R5 are selected such that, at equilibrium, the PTAT voltage across R3+R5 compensates two base-emitter voltages, so that approximately half of the PTAT voltage compensates a single base-emitter voltage. R3 and R5 are selected so that approximately half the PTAT voltage is at node 120; this compensates Q7 and makes the voltage from the emitter of Q7 to node 120 temperature invariant. Resistor R4 spans this voltage, so that its current is also temperature invariant.
R4's temperature invariant current (at equilibrium) flows in R5, adding to the voltage already present and compensating the quad. Since this additional voltage is constant, it simply offsets the equilibrium point to a higher, temperature stable voltage at node 108. This higher voltage can be adjusted by adjusting R4.
Alternative arrangements for establishing a higher equilibrium voltage are possible. For example, R4 could be connected to node 106 instead of node 120, causing a complementary-to-absolute-temperature (CTAT) voltage to be added to the output. The resulting temperature coefficient could be compensated by adding some resistance in the R3, R5 path to increase the PTAT voltage component, and the values of R4 and R3+R5 could be adjusted together to set the equilibrium voltage at a value higher than two bandgap voltages. Connecting R4 to node 120 is preferred, however, to reduce the interaction between R4 and R3+R5 and thereby facilitate trimming.
The regulator shown in FIG. 5 is advantageously used as a battery charger, to charge a battery 130 connected to Vout. The circuit shown charges the battery at a relatively high rate if its voltage is below full charge, without exceeding some maximum value when the battery is at a very low voltage. The battery charger is itself powered by a battery with a voltage Vbatt. An inverter is made from transistors Q8 and Q9 and is driven by a signal Vmon which monitors the value of Vbatt with respect to Vout ; Vmon is high when Vbatt is sufficiently greater than Vout. The output of the inverter controls a transistor Q10 connected between Vout and node 108. In normal operation, Vbatt exceeds Vout and Vmon is high. The inverter turns on Q10, connecting Vout to node 108. However, if Vbatt becomes discharged, or is removed from the circuit, Vmon goes low, turning off Q10 and disconnecting the load battery 130 from node 108. This prevents inadvertent discharge of the load battery 130.
As the node 108 voltage rises, the current that results in R3 and R5 flows mostly through Q2 and Q5 to the base of Q6. A maximum charging current is established by controlling the values of R3 and R5. Voltage Vout rises as the battery 130 approaches a fully charged condition; when Vout reaches the equilibrium voltage, the cell current switches from the right side to the left side, and the charging current to the battery is reduced to a low "maintenance" level.
The load battery 130 presents a low impedance when near full charge, so that loop stability is unlikely to be a problem. Thus, for this battery charger application, R1 and R2 are preferably made equal to provide the highest possible transconductance. If a higher impedance load were being driven, a lower transconductance may be preferable, which is easily achieved by making R1 smaller than R2.
Though the novel high transconductance reference cell has been described and shown as made from npn bipolar transistors, it is obvious that it can be similarly constructed of pnp transistors (with a corresponding inversion of supply voltage polarity and current flow direction), with no difference in the invention's function or performance advantages.
While particular embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Accordingly, it is intended that the invention be limited only in terms of the appended claims.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039626288.96/warc/CC-MAIN-20210423011010-20210423041010-00178.warc.gz
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CC-MAIN-2021-17
| 28,636 | 72 |
https://ctse5040.wordpress.com/2016/09/13/calculators-from-four-function-to-graphed-function/
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math
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For decades, one question has prevailed among math teachers and researchers: Do calculators help or hinder the teaching of mathematics? In a technology research brief, the National Council for Teachers of Mathematics (NCTM) synthesizes four decades of research regarding this question. Their conclusion? Almost unanimously, research, for decades, has shown that the use of calculators in math classrooms raises student achievement. Because of the overwhelming evidence that calculators improve student achievement, the question we should be asking is how teachers can make calculator use in the classroom more effective.
An important element required for the effective use of calculators in the classroom is connecting calculator usage with mathematical reasoning. Calculators should be more than a four-function task servant. Instead, teachers can use guided lessons in order to step students through mathematical processes, employing tools such as graphing capabilities to illustrate concepts.
For example, a graphing calculator such as the TI-84 Plus can be used to illustrate the effects of manipulating constants of an equation in slope intercept form. A teacher might have a student graph the following three equations:
- Y= (1/4)x
Using a graphing calculator, a student can visualize how changing the coefficients of x affect the slope (steepness) of a line. As an extension, a teacher might prompt students to graph equations with negative coefficients (negative slope). Similarly, students can see how adding/subtracting constants at the end of the equation affects the y-intercept. Allowing students to individually investigate slope and y-intercept enables them to personalize the teaching and discover the math principle themselves.
This activity is similar to other methods that can be used to incorporate calculators effectively in the classroom. In calculus, graphing calculators can be used to show the relationship between a function and its derivative. In order to connect this to relational understanding of this concept, a teacher could first teach how to take the derivative, explaining that the derivative is a “slope-generating function” which gives the instantaneous rate of change for any given value of x. Then, a graphing calculator can be used to graph a function and its derivative, as shown below.
The situations described above exemplify effective use of calculators in math classrooms. As NCTM has showed us, calculators do enhance student achievement. But the new question remains: How will you commit to making calculators effective in your classroom?
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CC-MAIN-2019-09
| 2,589 | 7 |
http://emp.byui.edu/BrownD/SPSS/Instr_Graphs/scatterplots_spss.htm
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math
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Creating Scatterplots using SPSS
Start SPSS if you haven’t already and turn in your text to pages 88 and 89. Read these two pages carefully. We will construct scatterplots for these data, starting with a plot of sales vs. items sold (that is, we’re going to try to reproduce Figure 2.1). Open the data file ta02_001.por, look over the data and make sure you know what it all means. Go the “Graphs” menu and select “Scatter/Dot…” A dialog box appears. We are going to make just one scatterplot (for starters), and we have no grouping variables, so this is a very simple situation. So we select “Simple Scatter” and click “Define.”
A new dialog box appears. To say, “sales versus items sold” means “sales” goes on the vertical axis (or y-axis) and “items” goes on the horizontal axis (or x-axis). So put those two variables into their respective boxes in the dialog box. If you want to add a title, you can do that by clicking “Title…” and filling in the boxes appropriately. Once you’re done, click “Continue” and “OK.” That’s all there is to it! You should see this in your Output Viewer (except your title is probably different than the one shown here.)
Now you may interpret to your heart’s content.
If you want to make more than one scatterplot at once, you can. Let’s use the same data set for this. In particular, let’s see how the variables “sales,” “check,” and “card” relate to each other. Go to Graphs->Scatter/Dot…,” select “Matrix Scatter” and click “Define.” Put the variables [sales], [chck], and [card] in the “Matrix Variables:” box. Add a title, if you like, and click “OK.” You should get something very like this:
First, note that we have three different scatterplots here. You may think there are six, but look very carefully, and you’ll see that the one in the row labeled “chck” (check) and in the column “sales” is the mirror image of the one in the row labeled “sales” and the column labeled “chck.” In fact, the three plots in the upper right portion of the matrix are mirror images of the ones in the lower left part.
Now, the labels on the axes tell you which variables are plotted in each plot. For example, in the lower left-hand corner, we have a plot of “card” versus “sales.” Looks pretty linear to me, though we may have an outlier (the point farthest right.) This isn’t too surprising, as an increase in credit card sales should be associated with an increase in sales.
Next to this plot, on the right of it, we have a plot of “card” versus “chck.” There seems to be no recognizable pattern in this plot, and there could be three or four outliers.
In the next row up (the middle row), and in the left-most column, we have a plot of “chck” versus "sales.” Again, there seems to be a positive linear association between these two variables. It seems fairly strong, at that. The uppermost point (which is also the point farthest right) could be an outlier.
These are the only three plots we need to interpret, since the other three are mirror images of these three.
Examples of scatterplots and their interpretation:
Pearson’s (linear) correlation coefficient
Simple linear correlation
Simple linear regression
David E. Brown
232 Ricks Building 208-496-1839 voice
Rexburg, ID 83460 208-496-2005 fax
Please do not call me at home.
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CC-MAIN-2017-51
| 3,393 | 18 |
https://www.hindawi.com/journals/jdd/2011/376548/fig6/
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math
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Figure 6: Fractions of drug molecules in inner and outer leaflets of donor and acceptor liposomes. The quantities , , , and are plotted according to (26) for and . The broken lines show the biexponential behaviors of the sums and . The time is plotted in units of the inverse rate constant . Note also and are the effective rate constants for the decay.
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CC-MAIN-2017-09
| 353 | 1 |
https://www.reddit.com/r/statistics/
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math
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use the following search parameters to narrow your results:
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Weekly /r/Statistics Discussion - What problems, research, or projects have you been working on? - April 08, 2020 (self.statistics)
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Weekly /r/Statistics Discussion - What problems, research, or projects have you been working on? - May 20, 2020 (self.statistics)
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Question[Q] Multilevel GLM Modelling questions: when 3-level and binomial. (self.statistics)
submitted 3 hours ago * by KantIsCool
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submitted 5 hours ago by T0bbi
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REDDIT and the ALIEN Logo are registered trademarks of reddit inc.
π Rendered by PID 20652 on r2-app-0dfc54d2f0a47e8ea at 2020-05-25 19:08:07.399986+00:00 running 197782e country code: US.
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| 5,492 | 79 |
https://www.thecollegepapers.com/mario-decided-invest-620-tax-refund-rather-spending-found-bank-pay-4-interest-compounded-quarterly-mario-deposits-entire-620-refund-not-deposit/
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math
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Posted: March 27th, 2017
Mario decided to invest his $620 tax refund rather than spending it. He found a bank that would pay him 4% interest, compounded quarterly. Mario deposits the entire 0 refund and does not deposit or withdraw any other amount.
a. write an equation that models the growth of the investment.
b. how many years will it take for the initial investment to double?
Place an order in 3 easy steps. Takes less than 5 mins.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224652235.2/warc/CC-MAIN-20230606045924-20230606075924-00690.warc.gz
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CC-MAIN-2023-23
| 437 | 5 |
https://www.letssolveques.com/wp-corporation-produces-products-x-y-and-z-from-a-single-raw-material-input-in-a-joint-production-process-budgeted/
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math
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WP Corporation produces products X, Y, and Z from a single raw material input in a joint production process. Budgeted data for the next month is as follows:
Product X Product Y Product Z
Units produced 1,500 2,000 3,000
Per unit sales value at split-off $19.00 $21.00 $24.00
Added processing costs per unit $7.00 $7.50 $7.00
Per unit sales value if processed further $29.00 $29.00 $30.00
The cost of the joint raw material input is $149,000. Which of the products should be processed beyond the split-off point?
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363376.49/warc/CC-MAIN-20211207105847-20211207135847-00338.warc.gz
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CC-MAIN-2021-49
| 511 | 7 |
https://www.jiskha.com/display.cgi?id=1219975824
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math
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posted by Soojung .
When 2.50 mol of Mg3N2 are allowed to react according to the following equation, how many moles of H2O also react?
Mg3N2 + 6H2O ---> 3Mg(OH)2 + 2NH3
2.5 mol Mg3N2 x (6 mols H2O/1 mol Mg3N2) = mols H2O. Note that the numbers in the factor (that part inside the parentheses) actually are the coefficients in the balanced equation AND that the unit we don't want to keep (mols Mg3N2) cancel and the unit we want at the end (mols H2O) remain unaffected and is the unit for the answer.
15 mol H20
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CC-MAIN-2017-47
| 511 | 5 |
https://phantran.net/sensitivity-and-scenario-analysis/
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math
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Uncertainty means that more things can happen than will happen. Therefore, whenever managers are given a cash-flow forecast, they try to determine what else may happen and the implications of these possible surprise events. This is called sensitivity analysis.
Put yourself in the well-heeled shoes of the financial manager of the Otobai Company in Osaka. You are considering the introduction of a high-performance electric scooter for city use. Your staff members have prepared the cash-flow forecasts shown in Table 10.1. Since NPV is positive at the 20% opportunity cost of capital, it appears to be worth going ahead, but before you decide, you want to delve into these forecasts and identify the key variables that determine whether the project succeeds or fails.
The project requires an initial investment of ¥15 billion in plant and machinery, which will have negligible further value when the project comes to an end. As sales build up in the early and middle years of the project, the company will need to make increasing investments in net working capital, which is recovered in later years. After year 6, the company expects sales to tail off as other companies enter the market, and the company will probably need to reduce the price of the scooter. The cost of goods sold is forecast to be 50% of sales; in addition, there will be fixed costs each year that are unrelated to the level of sales. Taxes at a 40% rate are computed after deducting straight-line depreciation.
These seem to be the important things you need to know, but look out for unidentified variables that could affect these estimates. Perhaps there could be patent problems, or perhaps you will need to invest in service stations that will recharge the scooter batteries. The greatest dangers often lie in these unknown unknowns, or “unk-unks,” as scientists call them.
Having found no unk-unks (no doubt you will find them later), you conduct a sensitivity analysis with respect to the required investment in plant and working capital and the forecast unit sales, price, and costs. To do this, the marketing and production staffs are asked to give optimistic and pessimistic estimates for each of the underlying variables. These are set out in the second and third columns of Table 10.2. For example, it is possible that sales of scooters could be 25% below forecast, or you may be obliged to cut the price by 15%. The fourth and fifth columns of the table shows what happens to the project’s net present value if the variables are set one at a time to their optimistic and pessimistic values. Your project appears to be by no means a sure thing. The most dangerous variables are cost of goods sold and unit sales. If the cost of goods sold is 70% of sales (and all other variables are as expected), then the project has an NPV of – ¥10.7 billion. If unit sales each year turn out to be 25% less than you forecast (and all other variables are as expected), then the project has an NPV of – ¥5.9 billion.
Trendy consultants sometimes use a tornado diagram such as Figure 10.1 to illustrate the results of a sensitivity analysis. The bars at the summit of the tornado show the range of NPV outcome due to uncertainty about the level of sales. At the base of the tornado you can see the more modest effect of uncertainty about investment in working capital and the level of fixed costs.1
1. Value of Information
The world is uncertain, and accurate cash-flow forecasts are unattainable. So, if a project has a positive NPV based on your best forecasts, shouldn’t you go ahead with it regardless of the fact that there may be later disappointments? Why spend time and effort focusing on the things that could go wrong?
Sensitivity analysis is not a substitute for the NPV rule, but if you know the danger points, you may be able to modify the project or resolve some of the uncertainty before your company undertakes the investment. For example, suppose that the pessimistic value for the cost of goods sold partly reflects the production department’s worry that a particular machine will not work as designed and that the operation will need to be performed by other methods. The chance of this happening is only 1 in 10. But, if it does occur, the extra cost would reduce the NPV of your project by ¥2.5 billion, putting the NPV underwater at +2.02 – 2.50 = – ¥0.48 billion. Suppose that a ¥100 million pretest of the machine would resolve the uncertainty and allow you to clear up the problem. It clearly pays to invest ¥100 million to avoid a 10% probability of a ¥2.5 billion fall in NPV. You are ahead by -0.1 + .10 X 2.5 = ¥0.15 billion. On the other hand, the value of additional information about working capital is small. Because the project is only marginally unprofitable, even under pessimistic assumptions about working capital, you are unlikely to be in trouble if you have misestimated that variable.
2. Limits to Sensitivity Analysis
Sensitivity analysis boils down to expressing cash flows in terms of key project variables and then calculating the consequences of misestimating those variables. It forces the manager to identify the crucial determinants of the project’s success and indicates where additional information would be most useful or where design changes may be needed.
One drawback to sensitivity analysis is that it always gives somewhat ambiguous results. For example, what exactly does optimistic or pessimistic mean? The marketing department may be interpreting the terms in a different way from the production department. Ten years from now, after hundreds of projects, hindsight may show that the marketing department’s pessimistic limit was exceeded twice as often as that of the production department, but what you may discover 10 years hence is no help now. Of course, you could specify that when you use the terms “pessimistic” and “optimistic,” you mean that there is only a 10% chance that the actual value will prove to be worse than the pessimistic figure or better than the optimistic one. However, it is far from easy to extract a forecaster’s notion of the true probabilities of possible outcomes.
Another problem with sensitivity analysis is that the underlying variables are likely to be interrelated. For example, if inflation pushes prices to the upper end of your range, it is quite probable that costs will also be inflated. And if sales are unexpectedly high, you may need to invest more in working capital. Sometimes the analyst can get around these problems by
defining underlying variables so that they are roughly independent. For example, it made more sense for Otobai to look at cost of goods sold as a proportion of sales rather than as a dollar value. But you cannot push one-at-a-time sensitivity analysis too far. It is impossible to obtain expected, optimistic, and pessimistic values for total project cash flows from the information in Table 10.2.
Sensitivity analysis boils down to expressing cash flows in terms of key project variables and then calculating the consequences of misestimating the variables. It forces the manager to identify the underlying variables, indicates where additional information would be most useful, and helps to expose inappropriate forecasts.
3. Scenario Analysis
If the variables are interrelated, it may help to consider some alternative plausible scenarios. For example, perhaps the company economist is worried about the possibility of a sharp rise in world oil prices. The direct effect of this would be to encourage the use of electrically powered transportation. The popularity of hybrid cars after a recent oil price increases leads you to estimate that an immediate 20% rise in the price of oil would enable you to increase unit sales by 10% a year. On the other hand, the economist also believes that higher oil prices would stimulate inflation, which would affect selling prices, costs, and working capital. Table 10.3 shows that this scenario of higher oil prices and higher inflation would on balance help your new venture. Its NPV would increase to ¥6.9 billion.
Managers often find such scenario analysis helpful. It allows them to look at different, but consistent, combinations of variables. Forecasters generally prefer to give an estimate of revenues or costs under a particular scenario than to give some absolute optimistic or pessimistic value.
5 thoughts on “Sensitivity and Scenario Analysis”
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https://www.nextbigfuture.com/2006/06/costs-in-synthetic-genetics-and.html
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math
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A new sequencer produced by 454 Life Sciences Corporation can sequence the genome (3.165 billion base pairs) for $2.2M. Making a hypothetical biological Intel 8088 (3500 transistor. A DNA transistor should take up 450 base pairs. 450 base pairs * 3500 transistors = 1.575M base pairs. 1.575M bp * $1.23 per bp = a total cost of ~$1.94M. the parts in the registry of biological parts tend to be 900 base pairs long (as of the end of 2005).
A computer simulation of the ribosome undertaken at Los Alamos National Lab involving 2.64M atoms was done in 2005. This type of simulation is a very important step towards understanding the ribosome, and then re-engineering it.
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| 667 | 2 |
http://casascevisa.com/epub/calculus-and-ordinary-differential-equations-modular-mathematics-series
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math
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By David Pearson
Professor Pearson's booklet begins with an advent to the realm and a proof of the main frequent features. It then strikes on via differentiation, particular capabilities, derivatives, integrals and onto complete differential equations.
As with different books within the sequence the emphasis is on utilizing labored examples and tutorial-based challenge fixing to realize the boldness of scholars.
Read Online or Download Calculus and Ordinary Differential Equations (Modular Mathematics Series) PDF
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Exhibits essentially how the research of concrete regulate platforms has encouraged the advance of the mathematical instruments wanted for fixing such difficulties. The Aizerman and Brockett difficulties are mentioned and an advent to the speculation of discrete keep watch over structures is given.
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It's the major objective of this e-book to boost at an available, average point an $L_2$ conception for elliptic differential operators of moment order on bounded soft domain names in Euclidean n-space, together with a priori estimates for boundary-value difficulties by way of (fractional) Sobolev areas on domain names and on their limitations, including a similar spectral idea.
In anglo-american literature there exist various books, dedicated to the appliance of the Laplace transformation in technical domain names equivalent to electrotechnics, mechanics and so forth. mainly, they deal with difficulties which, in mathematical language, are ruled through ordi nary and partial differential equations, in numerous bodily dressed types.
Additional resources for Calculus and Ordinary Differential Equations (Modular Mathematics Series)
5 Composition of two functions - it depends on the order! .. Getting Functions Together .. • .. • ... 6 Composition of two functions 27 • t; f2. two' can be carried out in two different ways: 'square' and then 'add two' gives you the functionf(x) == x 2 + 2; 'add two' and then 'square' gives youf(x) == (x + 2)2. It will be useful to have a notation which distinguishes between these two different ways of taking the composition of a pair of functions. Given functions 11 and h, the composition that we get from letting 12 operate first, then 11, will be written as 11 0 f2- If 11 acts first, followed by 12, this will be written as/2 0 It.
Functions need be ... just functions. The ideas, techniques and applications of calculus which form the subject matter of this book will of necessity focus primarily on that core of functions which have tolerably smooth behaviour. Calculus may itself be fairly characterized, from a modern standpoint, as an attempt to bring a little order and regularity to the world of functions. Though content to remain for the most part within this well-ordered framework, we should also seek from time to time to glimpse the boundaries of our world, and to make ourselves aware of the wider, infinite universe beyond.
3 1. A point P moves along the x-axis during the time interval from t == 0 to t == 2, the x coordinate of the point being given as a function of t by x(t) == 2t - t2 (0 ~ t ~ 2). Describe in words the motion of P from t == 0 to t == 2. e. x coordinate) of P, as a function of t; (b) the distance travelled by P, up to time t; (c) the velocity of P, as a function of t; and (d) the speed of P, as a function of t. Express the speed as a function of distance travelled, for values of t in the interval 0 ~ t S 1, and hence or otherwise sketch a graph of speed against distance travelled.
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http://www.thermistor.com/calculators
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math
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Please Choose A Calculator
Calculators for Temperature and Thermistor Applications
Beta Value Calculator - The approximate relationship between the resistance and temperature for a NTC thermistor.
R – T (Resistance vs. Temperature Tables) – A table showing the standard resistance at each temperature point.
Steinhart-Hart Calculator - The Steinhart–Hart equation is a model of the resistance of a semiconductor at different temperatures.
- T is the temperature (in Kelvin)
- R is the resistance at T (in ohms)
- A, B, and C are the Steinhart-Hart coefficients which vary depending on the type and model of thermistor and the temperature range of interest. (The most general form of the applied equation contains a (ln(R))2 term, but this is frequently neglected because it is typically much smaller than the other coefficients, and is therefore not shown above.)
Basic Thermistor Formulas
To find degree tolerance – Add standard part tolerance to maximum deviation and divide by the NTC value.
To find percent tolerance – Multiply degree tolerance by NTC value.
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https://blog.waikato.ac.nz/physicsstop/2009/09/09/complex-numbers/
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math
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Sometimes the divide between physics and mathematics is a thin one. Particularly between theoretical physics (which is what I do most of) and mathematics. The difference is that physicists have to keep one foot loosely planted in reality. It’s true that sometimes it just the tip of a little toe that’s behind the reality-line, but look hard enough and you’ll find realism in all that a physicist does.
One example is a physicist’s love of complex numbers. What’s a complex number? Well, think of your square numbers: One times one is one, two times two is four, three times three is nine, etc. Now turn the problem around – for example, what number when squared is 36? Answer, six. But let’s make it awkward. What number squared is minus one? Well, it’s not one (one squared is one), and neither is it minus one (because minus one squared is also one). Put minus one in a calculator and hit the square-root button and you’ll get (if your calculator is like mine) an error. Why? There is no real number that when squared equals minus one.
But this needn’t stop the mathematician. if there isn’t a real solution, he just makes one up, and calls it an ‘imaginary’ number. He calls it by the letter ‘i’. So i squared equals minus one. Beyond reality, yes, but maths is not limited by reality. The mathematician now proceeds to construct a whole algebra (complex algebra) around this concept. For example, you can add, multiply, square, squareroot, take the exponential of etc complex numbers.
Now this last one is why a physicist gets interested. It turns out that the exponential of an imaginary number is related to the cosine and sine functions. And the cosine and sine come almost everywhere in physics. But problems using exponentials are much easier than using sine and cosine. So the physicist turns his problem from a real-world problem to one that is in the ‘imaginary’ (or ‘complex’) world, does his complex algebra on it, then transforms it back into the real world, to get the result. It’s a neater way of doing things. Note that reality remains, because at the end of the journey he’s back in the real world. (It’s only a toe that needs to be behind the line, the rest of the body can well and truly be beyond it.)
If that has confused you, it’s rather like journeying from Auckland to Wellington. You can travel overland, which will take you a long time and you’ll have to cross rivers and go halfway up mountains on the way, but you’ll get there. Or you can go to the airport, leave the ground, travel there quickly, and land again (don’t forget the landing bit – or you’ll be a mathematician). You end up in the reality of Wellington by either route; one just happens to be quicker and simpler, and the fact that it’s off the ground doesn’t matter.
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| 2,821 | 5 |
https://www.courseeagle.com/questions-and-answers/let-f-x--x3--
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math
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Let f (x) = x3 - x. In Examples 3 and 7 in Section 2.7, we showed that f'(x) = 3x2 - 1 and f"(x) = 6x. Use these facts to find the following. (a). The intervals on which f is increasing or decreasing. (b). The intervals on which f is concave upward or downward. (c). The inflection point of f.
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http://melaniemasonauthor.blogspot.com/2012/05/dreams.html
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math
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I am always amazed at how vivid dreams can be. Lately I have been dreaming about doing math, Algebra to be specific. I know this seems random, but I have been helping my friend with her Algebra homework and apparently it stuck in my brain. For the last few nights I have been dreaming about doing Algebra problems.
Then every once in a while I dream random things and sometimes I think that those things will make great stories and sometimes I think that those random things will be good things to happen in my stories and then sometimes they are just random things that I wonder where in heaven's name my brain came up with that.
Here's to dreams :)
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| 650 | 3 |
https://www.librarything.com/author/halmospaulr-1
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math
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Naive Set Theory 237 copies
Finite-Dimensional Vector Spaces 153 copies
Measure Theory 118 copies
I Want to Be a Mathematician: An Automathography 79 copies, 1 review
Linear Algebra Problem Book 32 copies, 1 review
Lectures on Boolean Algebras 26 copies
Algebraic logic 26 copies
Lectures on Ergodic Theory 19 copies
I Have a Photographic Memory 6 copies, 1 review
Mathematics as a creative art 2 copies
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http://www.pokemonelite2000.com/forum/showpost.php?p=1362689&postcount=11
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math
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I'm quite the sensitive person... No, I'm not gay. (Not that there's anything wrong with them...)
Thanks, I'm glad you liked mine! =]
I'm very smart, but thick at the same time... If you know what I mean...
So I got confused in the 2nd person thingy... It's really hard to write like that, and well done! But some of us find it confusing. =[
OMG, I actually pointed out a fault! (Only a slight one. XD)
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https://perpustakaan.jakarta.go.id/book/detail?cn=INLIS000000000836172
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math
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"It's the same thing my life is now, ma'am," Desi thrust the notebook into the middle of the desk. Mrs. Amanah, who is also a math teacher, smiled bitterly at the sight of a straight-line equation with variables self-defined by Desi, x1: education, x2: intelligence. What caught his eye was the constant a: sacrifice.
"Education requires sacrifice, ma'am. Sacrifice is a fixed value, constant, unchanging."
That said, based on research in the middle of nowhere, generally the idealism of young people who have just graduated from college lasts for at most 4 months. After that they would become complainers, grumblers, and abusers like many others, then be sadly dragged down by the torrent of the great river of routines and bureaucratic pleasantries and then submit to a bad system.
In the reality of such a life, how far has Desi dared to maintain his idealism of being a math teacher in a remote school?
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https://ez.analog.com/thread/45599-what-is-the-input-impedance-of-ad8513
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math
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i go through the data sheet i dint get the input impedance of AD8513 .. anyone can please suggest me to how to get it?
AD8513 is a JFET input operational amplifier with a pA range of bias current. Since it has a very low input bias current, its input impedance range will be in GΩ to TΩ range. If you look into other precision JFET operational amplifier product like ADA4610 and ADA4000, their input impedance is in TΩ range too.
If you want to solve it mathematically, you can also try to use the following formula. Common mode input resistance is the ratio of the change in input common mode voltage to the change in input bias current. For the differential input resistance, it is the ratio of the change in input differential voltage to the change in input offset current.
I hope this helps!
thank you JGunao...
Retrieving data ...
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http://www.platinumgmat.com/
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math
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GMAT Prep From Platinum GMAT
GMAT Prep MaterialsProblem Solving · Data Sufficiency · Sentence Correction · Reading Comprehension · Critical Reasoning Number Properties · Combinatorics · AWA Essay Template · GMAT Idioms Average Scores & GPAs · Rankings · School Profiles
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A test maker is to design a probability test from a list of 21 questions. The 21 questions are classified into three categories: hard, intermediate and easy. If there are 7 questions in each category and the test maker is to select two questions from each category, how many different combinations of questions can the test maker put on the test?
Correct Answer: A
The order of questions is not important and this is a combinations problem.
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http://amcm.pcz.pl/?id=view_html&volume=15&issue=1&article=1
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math
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Certain inequalities connected with the golden ratio and the Fibonacci numbers
Journal of Applied Mathematics and Computational Mechanics
CERTAIN INEQUALITIES CONNECTED WITH THE GOLDEN RATIO AND THE FIBONACCI NUMBERS
Marcin Adam, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń[*], Roman Wituła
Institute of Mathematics, Silesian
University of Technology
[email protected], [email protected], [email protected], [email protected]
Abstract. In the present paper we give some condensation type inequalities connected with Fibonacci numbers. Certain analytic type inequalities related to the golden ratio are also presented. All results are new and seem to be an original and attractive subject also for future research.
Keywords: Fibonacci numbers, golden ratio, Perrin constant
The notion of Fibonacci numbers and the golden ratio may be found in many branches of mathematics, including number theory, geometry, algebra, matrix theory, numerical methods, classical analysis, dynamical systems, and even spectral analysis or music (see monographs [1-3], and selective papers [4-7]). Despite such a large spread occurrence of Fibonacci numbers and the golden ratio in mathematics, still some areas of mathematics tend to be poorly represented by these objects. In our opinion, a good example of such a niche (considered also in this paper) is the area of analytic inequalities. We believe this paper opens up a new stage of discoveries, and the inequalities presented here will be classical ones in the considered area of mathematics.
The main results of the paper are presented in two sections. In the first one we investigated the condensation type inequalities associated with the Fibonacci numbers. In the second one we discuss several analytic type inequalities related to the golden ratio and Perrin constant.
2. Condensation type inequalities connected with Fibonacci numbers
We begin with the following inequality based on basic properties of the Fibonacci numbers. Let us recall that the Fibonacci numbers are defined by the following linear recurrence relation
with . As a result of the definition we get .
Theorem 1. Let and be a finite sequence of real numbers such that two inequalities are satisfied:
Then there is an index such that
Proof. We prove this by contradiction. Let us suppose that for each index we have
that can be easily shown by induction. Indeed, from (3) we have
for the initial step and
for the inductive one.
From (4), on account of (1) we obtain
Next, let us denote where Then the left-hand side of the previous inequality is equal to the following:
Now from (2) it follows directly that for all Indeed, and since , there is So finally we obtain
which contradicts to (4). This completes the proof.
As a direct conclusion of this result we obtain the following generalization:
Corollary 2. Let and be a finite sequence of real numbers satisfying condition 1 of the previous theorem. If, additionally,
for some then there is an index such that
Remark 3. For the inequality (2) see also Chern and Cui paper .
3. Inequalities connected with the golden ratio
Let denote the golden ratio, i.e. and let .
Theorem 4. The following golden ratio type inequalities hold:
for ; the equality sign is attained for only. The function is increasing on interval and decreasing on each of the intervals and , where (see Fig. 1). We note that
2. The function
is increasing on each of the intervals and , and decreasing on interval , where . We note that and
3. The function
is decreasing on and increasing on .
Moreover, we have .
4. Let us set
for , and
for . Then the function is increasing on each of intervals and , and decreasing on interval . On the other hand, the function is increasing on each of the intervals and , and decreasing on Furthermore we obtain
Moreover, if then (see Figs. 2-4)
and the minimum of this function is attained in , we have
Fig. 1. Plot of the function
Fig. 2. Plot of the function
Fig. 3. Plot of the function
Fig. 4. Plot of the function in the interval (the domain of this one is equal to
Proof. We consider the following functions:
Computing derivatives of these functions we obtain
It is easy to check that the function is decreasing on and is increasing on , so we have for which is equivalent to the inequality for (the equality sign is attained here only for ). By numerical calculations, we proved that the function is increasing on interval and decreasing on intervals and , where .
Similarly as , also is decreasing on and is increasing on , so we obtain
We have . By numerical calculations we proved that the function is increasing on intervals and , and decreasing on interval . The function is decreasing on ) and increasing on . Hence, function has a local minimum at the point which is equal to .
Corollary 5. By item 1, the following inequality holds
In equivalent form, we obtain
Proof. We have
which implies (5) for since and .
Corollary 6. By item 2, the following inequality holds
Corollary 7. By item 3, the following inequality holds
More precisely, the function
is decreasing on interval and increasing on interval .
Remark 8. Closely connected to the golden ratio is the so-called Perrin constant defined to be the only real zero of the so-called Perrin polynomial (see [9-11])
In relation to inequalities
from point 4 of Theorem 4, we are interested in the equivalent of these inequalities for the Perrin constant , i.e. the inequalities of the type
Since we have
so we are interested when the following system of equations hold:
which implies that is a real solution of the following equation
Finally, by numerical calculations we get precisely two triplets of real numbers being the solution of system (6):
For these solutions we can deduce the following inequalities:
A) the first collection of five inequalities for the first triple (a, b, c) of solutions (see Fig. 5)
for and where the equality sign is taken only for .
is increasing on , . Moreover, is decreasing on two intervals and , and
B) for the second triple (a, b, c) of solutions similar inequalities can be obtained, however, due to the volume of the paper, they will be omitted.
Fig. 5. Plot of the functions and for the first triple - on the left, and for the second triple - on the right
In the paper certain inequalities connected with the golden ratio and the Fibonacci numbers are discussed. We were able to accomplish the intended overall goal of the paper, even with some excess (see in particular the results of point 4 of Theorem 4). Generalization from Remark 8 connected with the Perrin’s polynomial and constant is quite natural and in fact well-defined, but did not completely fulfill our expectations of aesthetic nature. We believe that the research subject matter indicated in this paper is still open and can encourage (especially Fibomaniacs) for active reflection.
Dunlop R., The Golden Ratio and Fibonacci Numbers, World Scientific, Singapore 2006.
Vajda S., Fibonacci and Lucas Numbers, and the Golden Section Theory and Applications, Dover Publications Inc., New York 2008.
Hoggatt V.E., Fibonacci and Lucas Numbers, The Fibonacci Association, Santa Clara 1979.
Mongoven C., Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae 2013, 41, 175-192.
Wituła R., Słota D., Hetmaniok E., Bridges between different known integer sequences, Annales Mathematicae et Informaticae 2013, 41, 255-263.
Słanina P., Generalizations of Fibonacci polynomials and free linear groups, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2015.1031073.
Herz-Fischler R., A “very pleasant” theorem, College Mathematics Journal 1993, 24, 4, 318-324.
Chern S., Cui A., Fibonacci numbers close to a power of 2, The Fibonacci Quarterly 2014, 52, 4, 344-348.
Hetmaniok E., Wituła R., Lorenc P., Pleszczyński M., On an improvement of the numerical application for Cardano’s formulae in Mathematica software (in review).
Wituła R., Lorenc P., Różański M., Szweda M., Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Śląskiej, Seria Matematyka Stosowana 2014, 4, 17-34.
Dubickas A., Hare K.G., Jankauskas J., There are no two nonreal conjugates of a Pisot number with the same imaginary part, arXiv:1410.1600v1 [math.NT].
[*] Currently, the fourth author, Barbara Smoleń, is a student of mathematics and this paper is a part of her Bachelor's thesis written under the supervision of Prof. Roman Wituła.
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https://www.jiskha.com/display.cgi?id=1240181571
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math
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posted by kim .
how would you determine a formula for cot(2x) using only cot(x)? Thenhow would you determine a formula for sec(2x) using only sec(x)?
cot 2x = 1/tan 2x
= [1 - 1/cot^2(x))/(2/cotx)
= [(cot^2(x) - 1)/cot^2(x)]/[2/cotx]
= (cot^2(x) - 1)/(2cotx)
I don't know if that is the simplest way, I just sort of followed by nose.
you can try sec 2x by noting
sec 2x = 1/cos 2c
then using cos2x = 2cos^2(x) - 1
follow the above procedure.
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| 440 | 11 |
https://brainly.com/question/68327
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math
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At the start of "The Necklace," why does Mathilde's husband go through "great trouble" to get an invitation to the ball?
A. He knows that attending the ball will lead to his getting a promotion at work.
B. He has bought Mathilde a new dress and wants her to wear it for some special event.
C. He thinks attending such an elegant affair will make Mathilde happy.
D. He wants to prove to Mathilde that he is very good at his job.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282202.61/warc/CC-MAIN-20170116095122-00338-ip-10-171-10-70.ec2.internal.warc.gz
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CC-MAIN-2017-04
| 427 | 5 |
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3891&option_lang=eng
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math
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This article is cited in 1 scientific paper (total in 1 paper)
Statistical theory of viscoelastic properties of fluids
F. M. Kuni
Mori's method of projection operators is used to derive equations for the mass density, momentum density, and momentum-current density. By means of Bogolyubov's condition of
correlation weakening, averaged equations (in the linear approximation in the amplitude deviations from equilibrium) of causal-retarded nature are obtained. Unlike the previously known
equations, space and time dispersiou are taken into account in these equations completely.
Symmetry relations are established for the transport coefficients. It is shown that if space
and time dispersion are ignored, the equations go over into the usual Maxwell theologic
equations for the stress-tensor deviator and the relaxation pressure. Rigorous microscopic
expressions are obtained for the times of shear relaxation and pressure relaxation;
these differ from the ones found previously by nonrigorous application of the Chapman–Enskog procedure for the elimination of the time derivatives. Rigorous microscopic expressions
are also obtained for the shear and bulk moduli of viscous fluids.
PDF file (2125 kB)
Theoretical and Mathematical Physics, 1974, 21:2, 1105–1115
F. M. Kuni, “Statistical theory of viscoelastic properties of fluids”, TMF, 21:2 (1974), 233–246; Theoret. and Math. Phys., 21:2 (1974), 1105–1115
Citation in format AMSBIB
\paper Statistical theory of viscoelastic properties of fluids
\jour Theoret. and Math. Phys.
Citing articles on Google Scholar:
Related articles on Google Scholar:
This publication is cited in the following articles:
Yu. V. Gurikov, “Generalized hydrodynamics of a van der Waals liquid”, Theoret. and Math. Phys., 28:2 (1976), 764–772
|Number of views:|
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| 1,808 | 23 |
http://www.lotrgfic.com/viewstory.php?sid=3238&index=1
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math
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Five moments in Barliman’s life when Gandalf was there.
Notes: Written for the 2015 b2mem, for starbrow’s prompt “I'd love to see a 5 times (or 5 + 1) story about Gandalf stopping off in The Prancing Pony for a drink on his way to/from the Shire, and what Barliman thinks of him, and how that changes over time.”
Since I chose the drabble format, I could not adhere too closely to the prompt. I hope it’s still an interesting enough reading.
Also for lotr_community March Challenge: b2mem prompt.
Table of Contents
Categories: The Lord of the Rings
Characters: Bree-lander: Barliman Butterbur, Other: Gandalf/Mithrandir/Ólorin
Times: 3-Third Age: late
Warnings: 2. mild violence
Series: 91. 2015 March Challenge: Back to Middle-earth Month
Chapters: 1 | Word count: 520 | Read Count: 1022
Completed: Yes | Updated: 05/05/15 | Published: 05/05/15
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| 896 | 12 |
https://www.warriorforum.com/off-topic-forum/1237880-why-you-here-warrior-forum.html
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math
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I'm curious, what is the reason that you are going through threads today?
Did you come here just to browse anything in hope to find interesting threads?
Did you have a particular topic in mind that you wanted to explore further?
Did you come to post a new thread about your current problem and get advice?
Maybe you're here just to chat or give advice?
I myself haven't visited the Warrior forum in a long time, but wanted to come here to absorb the language that you all use when you describe your problems, opinions, statements and so on.
I'm looking to put myself in your shoes so I can keep the content that I create as simple and valuable as possible for the situation that the viewer is in, if that makes sense.
I'd appreciate any comments: Why are you here on Warrior forum?
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| 781 | 8 |
http://mathforum.org/kb/message.jspa?messageID=8911403
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math
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MC, I think a lot of people on this site are wondering what I am: When do you intend to get a text or other info source and actually LEARN what people have discovered about infinite sets? There are many here (especially Angela) who know more about this than I ever will. But Cantor's basic idea was this: He matched up infinite sets using a 1-1 correspondence (For every x in the first set, there's exactly one y in the second set, and vice versa.). Now you'll either put in the time and effort to learn Cantor's system, or you won't. Along the way, you may notice this about math philosophy: The more math you know, the better you can philosophize about it. Incidentally, the people here have probably been a great deal more patient with you than anyone ever was with them. I'll have no further comment on this thread. Ben
> Date: Thu, 2 May 2013 21:27:22 -0400 > From: [email protected] > To: [email protected] > Subject: Re: RE: How does infinitesimal exist? > > Um, sorry if I came across as a jerk, I didn't mean to. Angela, I am trying to point out that talking about cardinality is a way of talking about size for infinite sets. But that the whole idea of size isn't even rational without number or distance or some other metric since as soon as that is introduced, it makes infinite division an immediate contradiction. Thus talking about cardinality in terms of size, and then talking about differently sized cardinalities, doesn't make much sense to me. Once you divorce size from the equation, it is gone. > > Also, if looking at the real number line divided into one unit segments, couldn't one ignore the irrationals and see this also as an infinity of infinities of rationals (since each one unit is infinitely divided into rationals), and thus argue that the cardinality of the reals is the same as the cardinality of the rationals, which is aleph-naught? Am I way off here? > > Sorry, this is just super interesting to me, and yes I know that the generally accepted view is that the cardinality of the reals is greater than the cardinality of the rationals. I would agree with that if the rationals were somehow limited to not divide out infinitely, but flying in the face of that is the fact that between any two rationals, I can actually find an infinity of other rationals. So theoretically the rationals are not limited from infinite division, despite this somehow breaking the definition of a rational number. Its all very weird. Thanks for any insight.
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| 2,487 | 2 |
http://www.math.mcgill.ca/darmon/qvnts/12-13/wan.html
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math
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Daqing Wan (UC Irvine).
Given a polynomial f(x) over a finite field Fq of q elements,
we are interested in estimating the number of distinct values taken by
f(x) as x runs over the finite field Fq.
This problem has a long history,
and has received increasing attentions in recent years. In this lecture,
we try to give a comprehensive introduction to this topic and explain
the various points of view (analytic, algebraic/group theoretic,
geometric and algorithmic).
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CC-MAIN-2018-51
| 466 | 9 |
https://www.wilsonhillacademy.com/academics/algebra/
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math
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The wonder of mathematics is most truly observed and appreciated by students when they are led to transfer their concrete mathematical knowledge to abstract algebraic generalizations. A primary goal in Algebra I at WHA is to facilitate that vital transfer. Students cannot help but be inspired by the beauty of algebra as they discover how it provides a basic language to describe so many aspects of the created world.
Students will study mathematical symbols and the rules for manipulating those symbols. Learning algebra helps to develop critical thinking skills using problem solving, logic, patterns, and reasoning. Topics covered include algebraic expressions, including polynomials and rational expressions, solving linear and quadratic equations, inequalities and systems of equations, and radicals and exponents. Emphasis is placed on problem solving and graphing.
Algebra I is a required course for virtually every postsecondary school program. WHA recognizes that this course is critically important as a gatekeeper to higher-level mathematics and seeks to maximize the experience for every student enrolled. It forms the basis for advanced studies in many fields, including mathematics, science, engineering, medicine, and economics. While algebra is used directly in many professions, especially those in science and math, the problem-solving strategies learned and practiced in Algebra I at WHA will enhance all critical thinking.
Prerequisites: Pre-Algebra is recommended; this course may not be developmentally appropriate for students who have not yet reached the 8th grade.
Gaining a deeper insight and developing a greater appreciation for mathematical models is the goal of this college preparatory course. Students will develop and practice higher level abstract thinking and reasoning skills that will help with data interpretation, proportions, measurements and equations.
Algebra 2 builds on the foundation of Algebra 1. Topics covered include real and complex number systems, solving equations and inequalities, and functions including linear, quadratic, exponential, logarithmic, and rational. Analytic geometry, sequences and series, statistics and probability, and matrices and determinants are also covered.
Most colleges and universities still require Algebra 2 (college algebra) for admission; it is particularly important for those considering studying science, technology or engineering in college. Research shows that students who successfully complete Algebra 2 are more likely to graduate from college.
Prerequisites: Algebra 1; Geometry (previously or concurrently).
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| 2,602 | 8 |
https://www.shaalaa.com/question-bank-solutions/statements-i-some-players-are-singers-ii-all-singetsate-tall-syllogism_102315
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math
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The question contains two statements and two conclusions numbered I and II.
You have to take the two given statements to be true even if they seem to be at variance from commonly known facts and decide which of the given conclusion(s) logically follow(s) from the two given statements.
Answer (1) If only conclusion I follows.
Answer (2) If only conclusion II follows.
Answer (3) If neither I nor II follows.
Answer (4) 1 f both I and II follows.
I. Some players are singers.
II. All singers are tall.
I. Some players are tall.
II. All players are tall.
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| 553 | 10 |
https://hubpages.com/family/answer/32206/do-you-think-the-balloon-parents-should-have-gone-to-jail-at-all-and-if-not-what-should-be-done
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math
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They wasted resources and generated a hoax that had many people praying and hoping their child was okay. Jail I think is not the answer but rather restitution in a number of ways. Not just $$$. So what other punishments should have been considered.
sort by best latest
You can help the HubPages community highlight top quality content by ranking this answer up or down.
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CC-MAIN-2017-34
| 369 | 3 |
http://www.quizquestionmaster.com/free-quizzes/Driving_Theory
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math
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Driving Theory Quiz
Driving Theory is something that luckily most of us did not have to do if we are over the age of about 30.
However these days anyone who wants to learn to drive in the UK must first pass a theory test that has both a multiple choice question and a hazard perception element to it.
So whether you have been driving for 50 minutes or 50 years, what do you know about the theory of driving? Let's find out!
Note if you are studying currently for the test or know someone who is, you can see all the official questions at Driving Theory Test Questions.
2) How many hazard perception clips are there in the theory test?
3) How many marks is the case study element of the theory test worth?
4) How much does it cost to take a theory test?
5) How much does a car practical test cost to take during the week?
6) What does MOT stand for?
7) Which organisation runs the theory and practical driving tests?
8) What is the stopping distance if you are travelling at 30mph?
Once you've finished tackling the Driving Theory Quiz, then it's time to check your answers!
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s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815934.81/warc/CC-MAIN-20180224191934-20180224211934-00003.warc.gz
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| 1,092 | 14 |
http://www.studymode.com/essays/5-Common-Sampling-Techniques-Used-In-1389013.html
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math
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discuss any five (5) common sampling techniques used in business research. Support you answer with relevant examples.
Simple random sampling:
The simple random sampling is one of the most widely-used random sampling method. The term “random” here does not mean a haphazard selection as many people think. The “random” in this method means each member of the population has equal opportunities being chosen be subject and no one in the identified population who could not be selected in this method. For example, the teacher wants to choose 5 people in QTB class to stand up and introduce themselves. In order to perform random sampling, each member has to have a specific number as an ID, and those number are put in a random sample list or termed sampling frame. In the example, sampling frame would be the class list. The mechanical and primitive method would be the lottery method. Each number is placed in a bowl or a container and mixed thoroughly. After that, the researcher picks numbers tags from the container without any awareness of what numbers that are. All the individuals bearing the numbers picks are the subjects for the study. Thank to advance technology; another way to perform this sampling would be using computer or calculator to do a random selection from the population. There are two types: sampling without replacement and with replacement in simple random sampling. In the first example about choosing students to introduce themselves, the student who have talk about themselves could not be chosen one more time. So the teachers need to remove their number out of the sampling frame. When we look at another example like lottery, the numbers are picked, and then they are put back to the container. Those numbers, which are put back, may be selected more than once. Simple Random sampling has its advantages and disadvantages. On one hand, the best thing about this random sampling is that it is easy to perform. Moreover it is also considered as an unbiased random selection since every member is given equal chances of being selected. On the other hand, there is the most obvious limitation of simple random sampling method is its sampling frame required. The sampling frame must be complete and up to date, which is not usually available for large population.
Stratified random sampling:
Stratified random sampling, which is a variant of probability sampling technique, is used when population may have different value for the responses of interest. The researcher wants to highlight particular subgroups in the whole population. In this case, unlike the simple random sampling, we divide the population into groups that are called strata, than randomly selects the final subjects from the different strata. Each individual or unit in a stratum has same opportunity to be chosen. In order to give equal chance to each unit, the researcher must apply the simple random sampling within the different strata and more important is that the strata must be non-overlapping. Having overlapping means that some units will have higher chance to be chosen as subject. For example, to choose students introducing themselves at BA1, the teacher would first organize the class into groups like Asian, European, American and so on. After dividing students into groups, the teacher chooses randomly students from each group. By doing that, the teacher certainly does not miss any continent, which could happen when the teacher just used the simple random sampling. There are two types of stratified random sampling: Proportionate and disproportionate stratified random sampling. In proportionate stratified random sampling, each stratum has the same sampling fraction. Take dividing groups to introduce themselves as an example, the teacher chooses a sampling fraction of a half, this means a half of students each group are selected to introduce themselves. In disproportionate stratified random sampling, there are different...
Please join StudyMode to read the full document
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| 4,001 | 6 |
https://courses.thoughtleader.school/mmc/mini-mental-model-encyclopedia/9-4-teams-and-problem-solving
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math
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"The team can only get stuck at a solution that's a local optima for everyone on the team. That means the team has to be better than the people in it. So what we want, right, you want people with different local optima. You want people to get stuck in different places. Well how do we get it? We don't. We've already looked at this twice, right? We looked at it first perspective perspectives. So if you coat it this way and I coat it this way, then we're going to get stuck in different places. We also want people with different heuristics. If I look in this direction and this direction, and you look in this direction and this direction, and we add us together, we look in all four of those directions. So what we want, is we want diverse perspectives, and we want diverse heuristics. And that diversity will give us different local optima, and those different local optima will mean that we take the intersections, and we end up with better points. That's sort of the big idea."
Scott Page Model Thinking MOOC Course
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| 1,021 | 2 |
http://www.androidblip.com/android-apps/dextorlab.speakingcalculator.html
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math
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About Voice Calculator
Voice Calculator for android is very convinient to use Voice Calculator which calculates the numbers by voice command, and shows correct answers. it is very simple to use while you are doing any kind of paper calculation and you have to use calculator at the same time as well then go for Voice Calculator. this is very quick and easy to use for everyone. You can perform faster calculation then typing or pressing keys.
Simple speak calculator simulator is also called talking calculator as calculator for kids because this speech calculator, audio calculator makes learning of kids through basic calculator, basic calculations.
This Speak Calculator easy to use.
Latest calculator where you don't have to use any keys to calculate. You can perform simple as well as complicated calculation by just speaking. It's very easy to use and very quick to calculate. You can perform faster calculation then typing or pressing keys.
You can carry out long calculations too.
It's very easy to use and very quick to calculate.
Never Miss An Update!
Follow Android Blip on Facebook / Google+
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| 1,104 | 9 |
http://oxfordindex.oup.com/view/10.1093/oi/authority.20110803100053505
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math
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A formulation of gauge theories in which space and time are taken to be discrete, rather than continuous. At the end of calculations in lattice gauge theories it is necessary to take the continuum limit. Lattice gauge theory is used to make calculations for gauge theories with strong coupling, such as quantum chromodynamics, in which many of the important features of the theory cannot be obtained by perturbation theory. Lattice gauge theory is particularly suitable for numerical and computational calculations. Techniques from statistical mechanics can be used in lattice gauge theory. Difficulties arise with putting fermions on a lattice, although various remedies have been used to overcome these difficulties.
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| 718 | 1 |
http://www.ambrosini.us/wordpress/2009/01/knightian-uncertainty/
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math
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Uncertainty, rather than just plain risk, leads to over-insurance:
A simple example can reinforce this point. Suppose two investors, A and B, engage in a swap, and there are only two states of nature, X and Y. In state X, agent B pays one dollar to agent A, and the opposite happens in state Y. Thus, only one dollar is needed to honour the contract. To guarantee their obligations, each of A and B put up some capital. Since only one dollar is needed to honour the contract, an efficient arrangement will call for A and B jointly to put up no more than one dollar. However, if our agents are Knightian, they will each be concerned with the scenario that their counterparty defaults on them and does not pay the dollar. That is, in the Knightian situation the swap trade can happen only if each of them has a unit of capital. The trade consumes two rather than the one unit of capital that is effectively needed.
This is a classic negative externality and so Caballero suggest government intervention when so-called Knightian uncertainty — the fear of unknown unknowns — rears its head (e.g. now).
Just because I have a single tracked mind… I’m wondering if the fiscal stimulus will do anything to calm fears of the unknown unknowns.
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| 1,241 | 4 |
http://owtermpaperyfkb.gnomes-inc.info/coursework-mathematics-t.html
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math
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Coursework mathematics t
1: a minimum grade of “c” is required by the department in each course counted toward the math/stat requirement for the bs in applied mathematics. Mathematics coursework stpm 2017 a essay on nature vs nurture debate : november 13, 2017college tip: rising seniors, start working on your apply texas essays. Methodology mathematics t coursework sem 2 2016 how to do the introduction for math t coursework 2017 what should be the content inside introduction. The operation of the jury session 2016 t stpm mathematics coursework versus the verbal, left brain, the authors lifespan, the authors in b zimmerman & d. Leighton ’81, a professor in the department of mathematics and a member of csail, will be awarded at the marconi society’s annual awards dinner in bologna. (for malaysian schools by henry tan) mathematics t stpm home numbers and sets polynomials sequence and series. Coursework mathematics t the master of financial mathematics is designed to meet the strongly increasing demand for skilled quantitative finance professionals to provide expert support to a wide range of industries including banking, insurance, utility companies, investment companies and companies affected by international exchange. Stpm maths-t term 1 assignment 2012 - scribd stpm maths t assignment (semester 1) 2012 document stpm mathematics (t) assignment b documentstpm 954 math t coursework 2013 [sem 2.
Study notes, guides and examination papers for all malaysian form 6 mathematics m students. The old further mathematics t paper is the malaysian examination council has decided that further mathematics will not be i want coursework. Mathematics t coursework stpm 2015 introduction, term 3 2015 term 3 mathematics stpm mathematics t coursework 2012 introduction, mathematics t coursework 2015 semester 1 universo online2013 common app versions dissertation chapter 1 structure practice ap english rhetorical analysis essaymathematics t coursework stpm 2015. Information about math video courses, education courses, and idaho engineering outreach video format for the mat mathematics program. Essay about multiculturalism is the aim mathematics t coursework four are identified the atmospheric mechanisms whereby greenhouse gases cause a scientist or philosopher does, by posing wide range of forms which clearly makes them appealing to a well - being and expressing an explanation in a world where long - term study mobility.
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A course in mathematics on a topic outside the current undergraduate offerings. Mathematics course descriptions and sample syllabi updated tue dec 29 20:21:48 pst 2015 course descriptions and sample syllabi developmental math courses keep in mind that taking approved high school. The access to these courses demonstrated by the fact that dozens of federal programs have made teaching and learning in science, technology, engineering, and math.
Coursework mathematics t
Free math courses are available online and don't require students to register or pay tuition, but most of these courses also don't lead to college credit. On this page you can read or download coursework mathematics t 2016 term 3 in pdf format.
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- Stpm mathematics t, mathematics s and further mathematics syllabuses of maths t and maths s are the same stpm mathematics t.
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- In this coursework i am going to investigate the relationship between t-numbers and t-totals i will find the relationship between t-number and t-totals by looking at the pattern of the sequence, which goes up by 5 each times when you add the t-total with 5.
Manual math (t) 2016 (1) - download as pdf file (pdf) the maximum marks for this paper is 180restricted mathematics (t) coursework stpm 2016. Coursework mathematics t 2015 sem 2, graphic thesis essay on why following instructions is important project mathematics t coursework stpm 2015 sem 1 accounting darius lynch from dallas was looking for. Mathematics coursework stpm 2017 sem 2, stpm 2017 term 2 mathematics (t) coursework on kk lee mathematics. If a program requires mathematics courses beyond first year, students should not take mat133y1 preparatory course, mat138h1 - introduction to proofs. But when the time they integrate cloud computing informatics phd engineering - information science is mathematics gcse t totals coursework is are content to adult students exactly like mike russo, the post exam shows that there is a requirement.
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| 5,902 | 13 |
http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=6413
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math
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The Opening Bell Curve
Posted By: Daniel Murphy
Date: Monday, 4 June 2007, at 3:36 p.m.
Has anyone already done any work with the value of winning/losing the opening roll in a money game session or at any particular match score?
The 30 possible opening roll equities (cubeless equities, Tom Keith's bkgm.com gnubg 0-ply cubeless rollouts) are:
-0.1670 -0.1234 -0.1035 -0.0638 -0.0545 -0.0155 -0.0125 -0.0102 -0.0088 -0.0072 -0.0066 -0.0056 -0.0053 -0.0032 -0.0024 0.0024 0.0032 0.0053 0.0056 0.0066 0.0072 0.0088 0.0102 0.0125 0.0155 0.0545 0.0638 0.1035 0.1234 0.1670 Average equity attained from the opening roll = 0.0000 Average equity attained from winning the opening roll = 0.0393 Median equity attained from winning the opening roll = 0.0102
In the gammonlife.com forum there is a post from a Jakob Garal, Dr.Eng. His own site is www.fairbg.com and he thinks that the vagaries of winning the opening roll are so unfair that we need an entirely new system -- his system -- for scoring match play.
His suggestions are going nowhere, certainly, but in the category "if it ain't broke, don't fix it" I'm wondering if anyone's given any thought to the probability of fairness of the distribution of the opening roll.
Messages In This Thread
BGonline.org Forums is maintained by Stick with WebBBS 5.12.
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http://www.urbandictionary.com/author.php?author=Big+Daddy+P-Stop
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math
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A dance move where one puts a large frown on his face, crosses his eyes and moves around making crab pincers with his hands. It is essential to get up into peoples person space while preforming the Crab Hands.
" Yo dude, what are going to do for the dance tonight "
" I dunno...fuck it, i'll just Crab Hands."
" That's whatsup"
#crab#hands#dance#big pimpin#fuck bitches get money
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| 379 | 5 |
http://brendatonn.com/details.php?ebook=11060
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math
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An Infinitely Large Napkin
by Evan Chen
Publisher: MIT 2016
Number of pages: 615
The Napkin project is a personal exposition project aimed at making higher math accessible to high school students. Topics: Basic Algebra and Topology; Linear Algebra and Multivariable Calculus; Groups and Rings; Complex Analysis; Quantum Algorithms; Algebraic Topology; Category Theory; Differential Geometry; etc.
Home page url
Download or read it online for free here:
by John Hutchinson - Australian National University
The goal is to introduce you to contemporary mainstream 20th and 21st century mathematics. If you are doing this course you will have a strong interest in mathematics, and probably be in the top 5% or so of students academically.
by MacGregor Campbell - Annenberg Foundation
Mathematics Illuminated is a text for adult learners and high school teachers. It explores major themes of mathematics, from humankind's earliest study of prime numbers, to the cutting-edge mathematics used to reveal the shape of the universe.
by William J. Meese - arXiv.org
My goal with the book is to provide some kind of bridge for mathematics between the high-school-level and college-level for physics students. My focus is to help modify your thinking of how math is used, rather than just pummel you with algorithms...
This book is about the topic of mathematical analysis, particularly in the field of engineering. This will build on topics covered in Probability, Algebra, Linear Algebra, Calculus, Ordinary Differential Equations, and others.
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| 1,533 | 14 |
https://www.physicsforums.com/threads/how-to-cause-cavitation.604527/
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math
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So I'm designing an impeller and rather than how to prevent cavitation, I'd like to know how to cause cavitation. I've read up on fluid shear stress and Couette flow, but this all appears to be based on relative velocity dV, of parallel discs, separated by a small distance dX, and fluid of viscosty μ. Shear Stress, τ = μ dV/dx. I also read that cavitation occus when pressure exceeds vapour pressure for the given fluid a the given temperature (3.2 kN/m2 for water @ 25 oC). I'm happy to accept all this. What I don't think I understand is how the geometry of the impeller affects cavitation. Videos of a cavitating marine propeller show the cavitation bubbles all the way along the length of the leading edge, right from the hub to the tip, but not on the hub (despite the hub having the same relative velocity as the leading edge, close to the hub). This suggests to me that geometry does play a role in caviaton, not just relative velocity. I'd exect an impeller with a very square leading edge to cavitate at a lower relative veloity than a very sharp leading edge. Is this a reasonable thought? In water at 25 oC at 1500rpm, my 80mm impeller seems to produce a shear stress of 22.4 Pa Nowhere near the 3'200 Pa required for cavitation. Further calculations seem to indicate that to get an 80mm impeller to cavitate would require it to rotate at 215'000 rpm! Surely cavitation is dependant on more than just diameter? I have Solidworks Simulation and plan on verifying results with this, but I'd like to have some idea if what Solidworks tells me is correct. Thanks for ANY help anyone can provide!
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| 1,607 | 1 |
https://www.coursehero.com/file/6409965/ASE-365-HW-3/
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math
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Homework 3 1. Problem 3.1 in the textbook. Note that the “natural frequency” ω n is without springs k 1 and k 2 and the “resonance frequency” ω r is with k 1 and k 2 , which are just added to the system for testing. Use FBDs to analyze the system in both configurations to get EOMs and natural/resonance frequencies. 2. Problem 3.4 in the textbook. The EOM for this problem, without the pressure excitation, is derived in Example 2.2 using energy considerations. Derive an EOM that takes p(t) into account by considering the forces that act on the fluid in the direction of the axis of the tube.
This is the end of the preview.
access the rest of the document.
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| 670 | 3 |
http://mathhelpforum.com/pre-calculus/95212-domain-range.html
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math
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find the domain of
find the range of
Follow Math Help Forum on Facebook and Google+
for the first one you should have which means .
so the domain is
The OP already posted this question in this thread: http://www.mathhelpforum.com/math-he...symptotes.html .
for the second one,you have and so the range is
(still need confirmation)
View Tag Cloud
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| 345 | 9 |
https://www.hackmath.net/en/word-math-problems/time?tag_id=154
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math
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Time + expression of a variable from the formula - practice problems
Number of problems found: 128
- Two companies
The two companies made a work together in 100 days. The first company would do it in 150 days, how many days would the work take the second company?
- Radio radius
Two friends have shortwave radios with a range of 13 km. The first of them travels by train at a speed of 48 km per hour along a straight section of track, from which the second of the friends is 5 km away. How long will radio friends be allowed for both
- The speed
The speed (v) of a moving object is inversely proportional to the time (t) traveled is written as?
- Last wagon
A 200 m long train passes through the 700 m long tunnel so that a time of 1 minute elapses from the entry of the locomotive into the tunnel to the exit of the last wagon from the tunnel. Find the speed of the train.
- 10 workers
10 workers will perform the assigned work in 8 days. How long will it take to get the job done if 6 workers work for six days and then another six join them?
- At what
At what speed did the motorist drive when he reacted by noticing the obstacle by 0.8 s; the magnitude of the opposite acceleration during braking was 6.5 m/s ^ 2 and the car ran to a stop track 35 m?
If someone gives u discount on "X" at 5% for 60 days. Then the discounted amount is 80000. find the actual amount? Discounted Amount=80000 rate=5% Period=60 days Actual amount=?
- The satellite
The satellite orbiting the Earth at an altitude of 800 km has a speed of 7.46 km/s. For how long would it have to move from the start to the orbit to reach this speed if it evenly accelerated its motion in a straight line? What is the acceleration of sate
- Cost structure
You are currently trying to decide between two cost structures for your business: one that has a greater proportion of short-term fixed costs and another that is more heavily weighted to variable costs. Estimated revenue and cost data for each alternative
- Maya is
Maya is 14 years old. Her brother Jorge is 3 years more than half her age. Which of the following is the correct expression for Jorge's age?
- Expression money per time
You started this year with $196 saved and you continue to save an additional $19 per month. Write an algebraic expression to represent the total amount of money saved after m months.
- The temperature 13
The temperature in Toronto at noon during a winter day measured 4°C. The temperature started dropping 2° every hour. Which inequality can be used to find the number of hours, x, after which the temperature will measure below -3°C?
- The temperature 12
The temperature of a liquid was 25°C before it was warmed at a rate of 10°C per minute for 5 minutes. It was then cooled at 3.5°C per minute for 6 minutes. What was the temperature of the liquid after cooling?
- Hour salary + fix
Devin recently hired a contractor to do some necessary work. The contractor gave a quote of $395 for parts plus $62 an hour for labor. Let x represent the number of labor hours worked. Write an algebraic expression to represent the total cost for the rep
The road from the cottage to the shop 6 km away leads either along a straight road that the bicycle can drive at a speed of 18 km/h or by "shortcut". It measures only 3.6 km. But the road from the cottage is all uphill - at a speed of 8 km/h, you can go h
- The car
The car weight 1280 kg, increased its speed from 7.3 m/s to 63 km/h on a track of 37.2 m. What force did the car engine have to exert?
- Škoda cars
At 8:30 am, Škoda 120 started from Prievidza at 60 km/h in the direction of Bratislava. At 9:00 started from Prievidza at the same direction Škoda Rapid at 90 km/h. Where and when will Škoda Rapid catch up Skoda 120?
- Tractor vs car
Behind the tractor, which was traveling at an average speed of 20 km/h, a car started in two hours, which overtook it in 1 hour. At what speed was the car moving?
- The stadion
John can ran around a circular track in 20 seconds and Eddie in 30 seconds. Two seconds after Eddie starts, John starts from the same place in opposite direction. When will they meet?
Charlie is saving money to buy a pair of headphones for $225. He has $37 so far, and he can save $15 per week. In how many weeks will he have enough money to buy the headphones?
Do you want to convert time units like minutes to seconds? Time - practice problems. Expression of a variable from the formula - practice problems.
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http://www.ingentaconnect.com/content/klu/matg/2000/00000032/00000002/00220653
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math
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From Celsius to Chaos to Cyclones: Using Temperature-Conversion Equations to Introduce Advanced Mathematical Concepts in Earth Science Courses
Author: Knox, J.A.
Source: Mathematical Geology, Volume 32, Number 2, February 2000 , pp. 203-215(13)
Abstract:The Fahrenheit-to-Celsius temperature-conversion equation is a basic component of many introductory earth science courses. Despite its simplicity, it presents a challenge to students and instructors alike because residents of the United States are unfamiliar with the Celsius scale. By solving for the point at which these two temperature scales are equal, it is possible to use the equations for temperature conversion as a springboard to more advanced topics. It is demonstrated that temperature-conversion equations and chaotic equations can be solved using identical numerical and graphical techniques. As a result, the fundamental concepts of chaos theory and numerical methods can be introduced to students in the context of the simplest equations in the earth sciences. These solution methods are applied to the quantitative theory of the extratropical cyclone as an example of the utility and broad scope of this educational approach.
Document Type: Regular Paper
Affiliations: Department of Geography and Meteorology, Valparaiso University, Mueller Hall, Valparaiso, Indiana 46383; [email protected]
Publication date: February 1, 2000
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https://www.hackmath.net/en/word-math-problems/area?tag_id=118
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math
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Area + The Law of Cosines - practice problems
Number of problems found: 4
- Calculate triangle
In the triangle ABC, calculate the sizes of all heights, angles, perimeters and its area, if given a-40cm, b-57cm, c-59cm
- Quadrilateral oblique prism
What is the volume of a quadrilateral oblique prism with base edges of length a = 1m, b = 1.1m, c = 1.2m, d = 0.7m, if a side edge of length h = 3.9m has a deviation from the base of 20° 35' and the edges a, b form an angle of 50.5°.
- Circular railway
The railway connects in a circular arc the points A, B, and C, whose distances are | AB | = 30 km, AC = 95 km, BC | = 70 km. How long will the track from A to C?
The sides of the parallelogram are 8 cm and 6 cm long, and the diagonals' angle is 60°. What is its area?
Cosine rule uses trigonometric SAS triangle calculator. Area - practice problems. The Law of Cosines - practice problems.
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| 892 | 10 |
http://www.wyzant.com/Williamsburg_New_York_NY_Math_tutors.aspx?sort=27&pagesize=5&pagenum=2
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math
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...I have also worked at least two days a week at a second high school in Queens. Before coming to this country, I worked for twenty years as a middle school math
I have a B.A. in Math
ematics. I recently re-took Intermediate Algebra and Trigonometry, Precalculus,...
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| 267 | 3 |
http://archive.org/stream/MenOfMathematics/TXT/00000126.txt
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math
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it must have been ripe port, to judge by what Sylvester got out
of the decanter.
Cayley and Sylvester came together again professionally
when Cayley accepted an invitation to lecture at Johns Hopkins
for half a year in 1881-82. He chose Abelian functions, in which
he was researching at the time, as his topic, and the sixty-seven-
year-old Sylvester faithfully attended every lecture of Ms
famous friend. Sylvester had still several prolific years ahead of
Mm, Cayley not quite so many.
We shall now briefly describe three of Cayley's outstanding
contributions to mathematics in addition to his work on the
theory of algebraic invariants. It has already been mentioned
that he invented the theory of matrices, the geometry of space
of n dimensions, and that one of his ideas in geometry threw a
new light (hi Klein's hands) on non-Euclidean geometry. We
shall begin with the last because it is the hardest.
Desargues, Pascal, Poncelet, and others had created projec-
tive geometry (see chapters 5,13) in which the object is to dis-
cover those properties of figures which are invariant under
projection. Measurements - sizes of angles, lengths of lines -
and theorems which depend upon measurement, as for example
the Pythagorean proposition that the square on the longest side
of a right angle is equal to the sum of the squares on the other
two sides, are not projective but metrical, and are not handled
by ordinary projective geometry. It was one of Cayley's greatest
achievements in geometry to transcend the barrier which,
before he leapt it, had separated projective from metrical pro-
perties of figures. From his higher point of view metrical geo-
metry also became projective, and the great power and flexi-
bility of projective methods were shown to be applicable, by
the introduction of 'imaginary' elements (for instance points
whose co-ordinates involve V — 1) to metrical properties.
Anyone who has done any analytic geometry will recall that
two circles intersect in four points, two of which are always
*imaginary'. (There are cases of apparent exception, for
example concentric circles, but this is close enough for our
purpose.) The fundamental notions in metrical geometry are the
distance between two points and the angle between two lines.
M.H.—VOL. n. K 433 *
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https://mathematics.jhu.edu/about/archive/page/4/
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math
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Speaker: Daniel Zaharopol Executive Director, Art of Problem Solving Foundation; Director, Bridge to Enter Advanced Mathematics Date/Location: 3-4 pm, October 19, 2017 in Maryland 201 Title: Advanced Mathematical Pathways for Underserved Students Abstract: Those of us in science and math careers had many experiences to help us get here. From our own independent projects, to a teacher who […]
News & Announcements Archive
Richard Melrose (MIT) will give Fall 2017 Kempf Lectures.
Xiaojun Huang will give Spring 2017 Monroe H. Martin Lectures.
HUB article: Two professors in the Johns Hopkins Mathematics Department are among 40 mathematicians across the country chosen for 2017-2018 fellowships by the Simons Foundation.
The Baltimore-Washington Metro Area Differential Geometry Seminar is a new joint seminar between the math departments of University of Maryland College Park, Howard University, and Johns Hopkins University. We meet twice a year, alternating between locations, for a day of talks on various topics in geometry. The next meeting will be held at JHU on April 22, 2017.
The Johns Hopkins-University of Maryland Algebra and Number Theory Day is a day-long meeting held once each semester, featuring speakers on topics in and around the areas of algebra, number theory, algebraic geometry, representation theory, and algebraic topology.
Congratulations to Professor David Savitt, who was named to the 2017 class of the Fellows of the American Mathematical Society, “for contributions to number theory and service to the mathematical community”.
Luis Caffarelli will give Spring 2017 Kempf Lectures.
JAMI 2017: Local zeta functions and the arithmetic of moduli spaces: A conference in memory of Jun-Ichi Igusa. March 22-26, 2017.
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CC-MAIN-2019-43
| 1,751 | 10 |
https://phoneia.com/en/education/sum-of-fractions-with-equal-numerator/
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math
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Perhaps it may be best for it, before moving forward with an explanation of the correct way in which a sum of fractions of equal numerator should be resolved, is to review some definitions, which will allow us to understand this operation and its methods of resolution within its indicated context.
In this sense, it may also be prudent to define this theoretical revision to two specific notions: the first, the definition of fractions itself, since this will allow to take into bear in mind the nature of the mathematical expression on the basis of which it takes place this operation.
On the other hand, it will also be useful to throw lights on the definition of sum of fractions, since it is the mathematical process that serves as the framework for this case in which the fractions involved have equal numerators, and different denominators. Here are each of these concepts:
Therefore, it will begin by saying that mathematics points to fractions as one of the two expressions with which fractional numbers count. Thus, the different authors point out that fractions can be described as the approach of a division between integers, where each of the elements that make up it, are understood as follows:
- Numerator: Set by the element or number that occupies the top of the expression. This element accomplishes the task of indicating the entire part that the expression is given to the rendering task.
- Denominator: The denominator will be the number or element that occupies the bottom of the fraction. According to what the different mathematical sources indicate, the denominator will be the element of the fraction that gives an entire account, based on which the fraction is established.
Sum of fractions
Similarly, it will be entirely helpful to review the definition offered by the Mathematics on the Sum of Fractions, which is described as an operation in which it is intended to determine a total value based on the combination or additions of the values of each of the fractions that participate in the operation, and which they exercise as the additions of the operation.
Likewise, mathematics indicates that it is the homogeneity or heterogeneity of the fractions involved in the addition operation, which determines the appropriate method of resolution of this operation.
Sum of fractions of equal numerator
With these operations in mind, it is perhaps much easier to approach an explanation of the Sum of fractions of equal numerator, and that as well as the different situations with respect to the shape and values of the elements of the fractions involved, will determine how the operation should be resolved.
As for the specific case of sum of fractions of equal numerator, the following method should then be followed:
- Once the transaction has been raised, the elements of each of the fractions inherent in the sum should be reviewed.
- When it has already been determined that in fact each of the fractions participating in the operations has equal numerator, regardless of the value of the denominators, then the sum is expressed as a single fraction.
- In this way, the values of the denominators are then summed, while the value of the numerator common to all is maintained. The total obtained is interpreted as the final result of the operation.
Examples of Sum of fractions of equal numbered
However, it may be necessary to give a specific example, which allows you to see in a practical way how such a sum is performed, that is, the fraction sum operation where they have equal numerator, as shown below:
Add the following fractions:
September 21, 2019
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| 3,588 | 19 |
http://www.tutorcircle.com/consecutive-exterior-angles-sgHki.html
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math
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Two straight lines, which exist in a plane, are intersected by a straight line, which is called as a transversal line. This transversal line creates total 8 angles on these two lines and each line has 4 angle like a traversal line XY, which intersect two lines ‘AB’ and ‘DE’ and creates 8 angles.
Angles on straight line
AB: $\angle$i, $\angle$j, $\angle$k, $\angle$l
Angles on Straight Line
DE: $\angle$m, $\angle$n, $\angle$o, $\angle$p
When two angles from opposite groups opposite in same side of traversal, then both angle pair are known as an Consecutive Exterior Angles.
i and $\angle$p, $\angle$j and $\angle$o are Consecutive Interior Angles
We use following steps for evaluation of consecutive exterior angles:
First we calculate first consecutive exterior angle from any of two consecutive interior angles.
Now it’s easy to find out other consecutive exterior angle from given transversal line by using following rule.
Consecutive Exterior Angle 2 = 180 - Consecutive Exterior Angle 1
Suppose we have to find consecutive exterior angle of a transversal line, whose first consecutive exterior angle is equal to 60 degree. The other consecutive exterior angle is,
Consecutive Exterior Angle 2
= 180 - Consecutive Exterior Angle 1
= 180 - 60
So, the other consecutive exterior angle of given transversal line is equal to ‘120 degree’.
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CC-MAIN-2018-30
| 1,356 | 16 |
https://www.arxiv-vanity.com/papers/1304.0533/
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math
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Energy–momentum tensor from the Yang–Mills gradient flow
The product of gauge fields generated by the Yang–Mills gradient flow for positive flow times does not exhibit the coincidence-point singularity and a local product is thus independent of the regularization. Such a local product can furthermore be expanded by renormalized local operators at zero flow time with finite coefficients that are governed by renormalization group equations. Using these facts, we derive a formula that relates the small flow-time behavior of certain gauge-invariant local products and the correctly-normalized conserved energy–momentum tensor in the Yang–Mills theory. Our formula provides a possible method to compute the correlation functions of a well-defined energy–momentum tensor by using lattice regularization and Monte Carlo simulation.
B01, B31, B32, B38
Although lattice regularization provides a very powerful non-perturbative formulation of field theories, it is unfortunately incompatible with fundamental global symmetries quite often. The most well-known example is chiral symmetry Nielsen:1980rz ; Nielsen:1981xu ; supersymmetry is another infamous example Dondi:1976tx , as is, needless to say, translational invariance. When a regularization is not invariant under a symmetry, it is not straightforward to construct the corresponding Noether current that is conserved and generates the symmetry transformation through Ward–Takahashi (WT) relations. This makes the measurement of physical quantities related to the Noether current in a solid basis very difficult. To solve this problem, one can imagine at least three possible approaches.
The first approach is an ideal one: One finds a lattice formulation that realizes (a lattice-modified form of) the desired symmetry. If such a formulation comes to hand, the corresponding Noether current can easily be obtained by the standard Noether method. The best successful example of this sort is the lattice chiral symmetry Kaplan:1992bt ; Neuberger:1997fp ; Hasenfratz:1997ft ; Hasenfratz:1998ri ; Neuberger:1998wv ; Hasenfratz:1998jp ; Luscher:1998pqa ; Niedermayer:1998bi , which can be defined with a lattice Dirac operator that satisfies the Ginsparg–Wilson relation Ginsparg:1981bj . Although this is certainly an ideal approach, it appears that such an ideal formulation does not always come to hand, especially for spacetime symmetries (see, e.g., Ref. Kato:2008sp for a no-go theorem for supersymmetry).
The second approach is to construct the Noether current by tuning coefficients in the linear combination of operators that can mix with the Noether current under lattice symmetries.111Here, we assume that fine tuning of bare parameters to the target (symmetric) theory is done. For example, for the energy–momentum tensor—the Noether current associated with the translational invariance and rotational and conformal symmetries Callan:1970ze ; Coleman:1970je —one can construct a conserved lattice energy–momentum tensor by adjusting coefficients in the linear combination of dimension operators Caracciolo:1988hc ; Caracciolo:1989pt 222A somewhat different approach on the basis of the supersymmetry has been given in Refs. Suzuki:2013gi ; Suzuki:2012wx .; the overall normalization of the energy--momentum tensor has to be fixed in some other way.333It might be possible to employ “current algebra” for this, as for the axial current Bochicchio:1985xa . Although this method is in principle sufficient when the energy–momentum tensor is in “isolation”, i.e., when the energy–momentum tensor is separated from other composite operators, as in the on-shell matrix elements, it is not obvious a priori whether one can control the ambiguity of possible higher-dimensional operators that may contribute when the energy–momentum tensor coincides with other composite operators in position space. This implies that it is not obvious whether the energy–momentum tensor constructed in the above method generates correctly-normalized translations (and rotational and conformal transformations) on operators through WT relations. (If the energy–momentum tensor generates correctly-normalized translations, it is ensured Fujikawa:1980rc (see also Sect. 7.3 of Ref. Fujikawa:2004cx ) that the trace or conformal anomaly Crewther:1972kn ; Chanowitz:1972vd is proportional to the renormalization group functions Adler:1976zt ; Nielsen:1977sy ; Collins:1976yq .)
The third possible approach is to utilize some ultraviolet (UV) finite quantity. Since such a quantity must be independent of the regularization adopted (in the limit in which the regulator is removed), there emerges a possibility that one can relate the lattice regularization and some other regularization that preserves the desired symmetry. This methodology can be found e.g. in Ref. Luscher:2004fu (see also Ref. Luscher:2010ik ), where an ultraviolet finite representation of the topological susceptibility is derived. Although the derivation of the representation itself relies on a lattice regularization that preserves the chiral symmetry Kaplan:1992bt ; Neuberger:1997fp ; Hasenfratz:1997ft ; Hasenfratz:1998ri ; Neuberger:1998wv ; Hasenfratz:1998jp ; Luscher:1998pqa ; Niedermayer:1998bi , one can use any regularization (e.g., the Wilson fermion Wilson:1975hf ) to compute the representation because it must be independent of the regularization.
In the present paper, we consider the above third approach for the energy–momentum tensor, by taking the pure Yang–Mills theory as an example. For this, we utilize the so-called Yang–Mills gradient flow (or the Wilson flow in the context of lattice gauge theory) whose usefulness in lattice gauge theory has recently been revealed Luscher:2010iy ; Luscher:2010we ; Luscher:2011bx ; Borsanyi:2012zs ; Borsanyi:2012zr ; Fodor:2012td ; Fodor:2012qh ; Fritzsch:2013je ; Luscher:2013cpa . A salient feature of the Yang–Mills gradient flow is its robust UV finiteness Luscher:2011bx . More precisely, any product of gauge fields generated by the gradient flow for a positive flow time is UV finite under standard renormalization. Such a product, moreover, does not exhibit any singularities even if some positions of gauge fields coincide. The basic mechanism for this UV finiteness is that the flow equation is a type of the diffusion equation and the evolution operator in the momentum space acts as an UV regulator for . This property of the gradient flow implies that the definition of a local product of gauge fields for positive flow times is independent of the regularization. In our present context, there is a hope of relating quantities obtained by the lattice regularization and the dimensional regularization with which the translational invariance is manifest.
As noted in Ref. Luscher:2011bx , on the other hand, a local product of gauge fields for a positive flow time can be expanded by renormalized local operators of the original gauge theory with finite coefficients. Those coefficients satisfy certain renormalization group equations that, combined with the dimensional analysis, provide information on the coefficients as a function of the flow time. Because of the asymptotic freedom, one can then use the perturbation theory to find the asymptotic behavior of the coefficients for small flow times.
By using the above properties of the gradient flow, one can obtain a formula that relates the small flow-time behavior of certain gauge-invariant local products and the energy–momentum tensor defined by the dimensional regularization. Since the former can be computed by using the Wilson flow with lattice regularization Luscher:2010iy ; Luscher:2010we ; Luscher:2011bx ; Borsanyi:2012zs ; Borsanyi:2012zr ; Fodor:2012td ; Fodor:2012qh ; Fritzsch:2013je ; Luscher:2013cpa and the latter is conserved and generates correctly-normalized translations on composite operators, our formula provides a possible method to compute the correlation functions of a correctly-normalized conserved energy–momentum tensor by using Monte Carlo simulation.
In the present paper, we follow the notational convention of Ref. Luscher:2011bx unless otherwise stated.
2 Yang–Mills theory and the energy–momentum tensor
2.1 The energy–momentum tensor with dimensional regularization
In the present paper, we consider the Yang–Mills theory defined in a dimensional Euclidean space. The action is given by
from the Yang–Mills field strength
and then the mass dimension of the bare gauge coupling is .
Assuming that the theory is regularized by the dimensional regularization (for a very nice exposition, see Ref. Collins:1984xc ), one can define the energy–momentum tensor for the system (2.1) simply by (see, e.g., Ref. Freedman:1974gs )
up to terms attributed to the gauge fixing and the Faddeev–Popov ghost fields, which are irrelevant in correlation functions of gauge-invariant operators. Note that the mass dimension of the energy–momentum tensor is .
The advantage of dimensional regularization is its translational invariance. Because of this property, the energy–momentum tensor naively constructed from bare quantities, Eq. (2.4), is conserved and generates correctly-normalized translations through a WT relation,
where it is understood that the derivative on the right-hand side is acting all positions in a gauge-invariant operator . Used in combination with dimensional counting and gauge invariance, this WT relation implies that the energy–momentum tensor is finite Joglekar:1975jm ; Nielsen:1977sy and thus, in the minimal subtraction (MS) scheme,444Here, we define the renormalized operator by subtracting its vacuum expectation value. In the perturbation theory using dimensional regularization, this subtraction is automatic.
The finiteness of the energy–momentum tensor (2.4) provides further useful information on the renormalization of dimension gauge-invariant operators. The gauge coupling renormalisation with dimensional regularization is defined by
where is the renormalization scale and is the renormalization factor. In the MS scheme,
From the rotational invariance that the dimensional regularization keeps, we see that the operator-renormalization possesses the following structures:555Here again, we define renormalized operators by subtracting their vacuum expectation values.
Since the left-hand side is finite for , in the MS scheme in which only pole terms are subtracted, we infer (by considering the cases, and ) that
2.2 Implications of the trace anomaly
By Eq. (2.6), this relation is equivalent to
i.e., the contraction with the metric and the minimal subtraction, the subtraction of poles, do not commute; this is a peculiar but legitimate property of the dimensional regularization Collins:1984xc .
In the MS scheme in which only pole terms are subtracted, this implies
and Eq. (2.14) then shows
3 Yang–Mills gradient flow and the small flow-time expansion
The Yang–Mills gradient flow defines a dimensional gauge potential along a fictitious time , according to the flow equation
where the dimensional field strength and the covariant derivative are defined by
respectively. The initial condition for the flow is given by the dimensional gauge potential in the previous section:
In Eq. (3.1), the last term is introduced to suppress the evolution of the field along the direction of gauge degrees of freedom. Although this term breaks the gauge symmetry, it does not affect the evolution of any gauge-invariant operators Luscher:2010iy . Note that the mass dimension of the flow time is .
Now, from the field strength extended to the dimension (3.2), we define a dimensional analogue of the energy–momentum tensor by
Although this is similar in form to the original energy–momentum tensor (2.4), it is not obvious a priori how this dimensional object and Eq. (2.4) are related (or not). To find the relationship between them is the principal task of the present paper. We also use the density operator studied in Ref. Luscher:2010iy :
Now, as shown in Ref. Luscher:2011bx , for , any correlation function of is UV finite after standard renormalization in the dimensional Yang–Mills theory. This property holds even for any local products of such as Eqs. (3.5) and (3.6). Also, for small flow times, a local product of can be regarded as a local field in the dimensional sense because the flow equation (3.1) is basically the diffusion equation along the time and the diffusion length in is . These properties allow us to express, as explained in Sect. 8 of Ref. Luscher:2011bx , and as an asymptotic series of dimensional renormalized local operators with finite coefficients. Considering the gauge invariance and the index structure, for , we can write
where abbreviated terms are the contributions of operators with a mass dimension higher than or equal to . For Eq. (3.6), we similarly have
We note that, when the renormalized gauge coupling is fixed, (3.5) is traceless for ,
because (3.6) is finite Luscher:2010iy and does not produce a singularity (this explains why there is no number expectation value term in Eq. (3.7)). Thus, considering the trace part of Eq. (3.7), we see that the coefficients and are not independent and are related by, for ,
because of the trace anomaly (2.15).
This expression relates the energy–momentum tensor (2.6) and the short flow-time behavior of gauge-invariant local products defined by the gradient flow. Thus, once the coefficients are known, one can extract the energy–momentum tensor from the behavior of the combination on the right-hand side.
4 Renormalization group equation and the asymptotic formula
4.1 Renormalization group equation for the coefficients
We now operate
on both sides of Eq. (3.7). Since the left-hand side of Eq. (3.7), i.e., Eq. (3.5), is entirely expressed by bare quantities through the flow equation (3.1) and the initial condition (3.4), the action of (4.1) on the left-hand side identically vanishes. On the right-hand side, this vanishing must hold in each power of . Thus we infer that
Similarly, for Eq. (3.8), we have
By the standard argument and from the fact that dimensionless quantities can depend on the renormalization scale only through the dimensionless combination , the above renormalization group equations imply that
where the dependence on the renormalized gauge coupling and on the renormalization scale has been explicitly written. In these expressions, the running coupling is defined by
and we introduce a variable
In the one-loop order, the running couping (4.15) is given by
where is the parameter in the one-loop level,
and the integral appearing in Eqs. (4.12) is
In the small flow-time limit , and the running coupling (4.17) becomes very small thanks to the asymptotic freedom. Thus, the right-hand sides of Eqs. (4.11)–(4.14) allow us to compute the small flow-time behavior of the coefficients by using the perturbation theory.
4.2 Lowest-order approximation and the asymptotic formula
simply because our energy–momentum tensor (2.4) is proportional to . If we apply the right-hand side of Eq. (4.11) to this expression by substituting Eq. (4.17), however, it depends on while the left-hand side of Eq. (4.11) does not. This shows that should depend on and through a particular combination as (for )
This is the relation that we were seeking: One can obtain the correctly-normalized conserved energy–momentum tensor from the small flow-time behavior of gauge-invariant products given by the Yang–Mills gradient flow. It is interesting to note that the leading behavior is completely independent of the detailed definition of the gradient flow; the structure and coefficients follow solely from the finiteness of the local products and the renormalizability of the Yang–Mills theory. The sub-leading corrections in the asymptotic form, i.e., the coefficients and , depend on the detailed definition of the gradient flow; in the Appendix, we compute the constants and and we have
Finally, a possible method to determine the factor in Eq. (4.29), i.e., the flow time in the unit of the one-loop parameter (4.18), for small flow times is to use the expectation value of the density operator, Eq. (3.6). For this quantity, by applying Eqs. (4.13) and (4.17) to the result of the one-loop calculation, Eqs. (2.28) and (2.29) of Ref. Luscher:2010iy (specialized to the pure Yang–Mills theory), we have the asymptotic form,
In the present paper, we have derived a formula that relates the short flow-time behavior of some gauge-invariant local products generated by the Yang–Mills gradient flow and the correctly-normalized conserved energy–momentum tensor in the Yang–Mills theory. Our main result is Eq. (4.29). The right-hand side of Eq. (4.29) can be computed by the Wilson flow in lattice gauge theory with appropriate discretizations of operators, Eqs. (3.5) and (3.6) (see, e.g., Refs. Luscher:2010iy ; Borsanyi:2012zs ). Here, the continuum limit must be taken first and then the limit is taken afterwards; otherwise our basic reasoning does not hold.
Although the formula (4.29) should be mathematically correct, the practical usefulness of Eq. (4.29) is a separate issue and has to be carefully examined numerically.888We hope to return to this problem in the near future. Since the lattice spacing must be sufficiently smaller than the square-root of the flow time for our reasoning to work, the reliable application of Eq. (4.29) will require rather small lattice spacings. One also worries about contamination by higher-dimensional operators (i.e., the terms in Eqs. (3.7) and (3.8)) and the finite-size effect which we have not taken into account in the present paper. If our strategy turns to be practically feasible, it provides a completely new method to compute correlation functions containing a well-defined energy–momentum tensor. It is clear that the present approach to the energy–momentum tensor on the lattice is not limited to the pure Yang–Mills theory although the treatment might be slightly more complicated with the presence of other fields. The application will then include the determination of the shear and bulk viscosities (see, e.g., Refs. Meyer:2007ic ; Meyer:2007dy ), the measurement of thermodynamical quantities (see Ref. Giusti:2012yj and references cited therein), the mass and the decay constant of the pseudo Nambu–Goldstone boson associated with the (approximate) dilatation invariance (see Ref. Appelquist:2010gy and references cited therein), and so on.
It is also clear that our basic idea, that operators defined with lattice regularization and in the continuum theory can be related through the gradient flow is not limited to the energy–momentum tensor. For example, it might be possible to construct an ideal chiral current or an ideal supercurrent on the lattice, from the small flow-time limit of local products. It would be interesting to pursue this idea.
The possibility that the Yang–Mills gradient flow (or the Wilson flow) can be useful for defining the energy–momentum tensor in lattice gauge theory was originally suggested to me by Etsuko Itou. I would like to thank her for enlightening discussions. I would also like to thank Martin Lüscher for a clarifying remark on the precise meaning of Eq. (3.8). This work is supported in part by a Grant-in-Aid for Scientific Research 23540330.
Appendix A One-loop calculation of coefficient functions
For calculational convenience, we define the coefficient functions and by
Equation (3.5) then becomes (for ),
Comparison with Eq. (3.7) then shows
To find the coefficient functions and in Eq. (A.1), we consider the correlation function
For , there are flow-line Feynman diagrams (Figs. 1–17) that contribute to this correlation function. In the figures, gauge potentials at the flow time , , are represented by small filled squares; the open circle denotes the flow-time vertex and the full circle is the conventional vertex in the Yang-Mills theory. We refer the reader to Ref. Luscher:2011bx for the details of the Feynman rules for flow-line diagrams.
To read off the coefficient functions and in Eq. (A.1) from the correlation function (A.8), we consider the vertex functions, i.e., amputated diagrams in which the external propagators of the original Yang–Mills theory are truncated. Therefore, Figs. 11, 14 and 17, which provide only the conventional wave function renormalization, should be omitted in the computation of and .999More precisely, these diagrams are different from conventional Feynman diagrams in that the propagators carry an additional factor (in the Feynman gauge), where is the external momentum. This factor is, however, irrelevant in the present computation of the coefficients of operators with the lowest number of derivatives. On the other hand, the flow-line propagators Luscher:2011bx , the arrowed straight lines in the diagrams, should not be truncated because these are not propagators in the quantum field theory but instead represent time evolution along the flow time.
The tree-level contribution to the vertex function is
and, here and in what follows, the alternating-sign symbol implies
This tree-level result was used in obtaining Eq. (4.20).
The vacuum expectation value in the lowest order is
Now, as an example of the computation of one-loop flow-line Feynman diagrams, we briefly illustrate the computation of Fig 14. A straightforward application of the Feynman rules in Ref. Luscher:2011bx in the “Feynman gauge” in which the gauge parameters are taken as , yields the expression,
To find the coefficients and in Eq. (A.1), we write this vertex function as
and find the coefficients of
respectively, in . For this, we first exponentiate the denominators in Eq. (A.13) by using
We then simply expand the integrand with respect to the external momenta and to . The flow-time evolution factor in the integrand makes the integral (A.13) UV finite for any dimension . On the other hand, there always exists a complex domain of such that the integral is infrared finite; this provides the analytic continuation of the integral such that
In Table 1, we summarize the contribution of each diagram computed in the above method in the unit of
In the last line of the table, “ factors” implies the contributions of the one-loop operator renormalization factors, (2.13) and (2.22), through the tree-level diagram, Eq. (A.9) (recall Eq. (2.10)). We see that those operator renormalization factors precisely cancel the residues of and make the coefficients and finite; this is precisely what we expect from the general argument. From the results in the table, we then have
Finally, comparison with the formulas (A.4), (A.7), (4.21) and (4.23) shows the results quoted in Sect. 4, Eqs. (4.30) and (4.31). Note that the coefficients of in the explicit one-loop calculation (Eqs. (A.23) and (A.24)) are in agreement with those by the general argument on the basis of the renormalization group equations and the trace anomaly (Eqs. (4.21) and (4.23)). This agreement provides a consistency check for our one-loop calculation and supports the correctness of our reasoning.
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| 25,515 | 112 |
https://morpherhelmet.co.uk/how-to-calculate-betting-odds/
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math
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The betting odds are very vital for the new players to learn as these allow them to understand how likely an event is to occur, and how likely your chance of winning is.
The probability of the occurrence of the event can be converted and presented in the forms of these betting odds. The three odds almost present the same thing with very little difference.
Calculation Method of Various Betting Odds
Calculation Of Fractional Odds
- These odds are written with a slash or hyphen in between the numbers like 7/1 and are read as seven-to-one.
- If the player stakes a total of $10 at the odds of 7/1, then he will receive a total of $70 stake added with $10 profit, which means a total of $80.
- In this odd, the total return can be calculated as: [Stake × Numerator/Denominator] + Stake. Here, the numerator/denominator is the fractional odds.
- Let, 5/1 can be calculated as 1/(5+1)= 0.16, which means that there is a 16% chance of winning.
Calculation of Decimal Odds
- These odds represent the total amount the player wins for every $1 that is wagered and the number instead of showing the profits reflects the total payout. The amount of stake is already added to the decimal which makes it easy to calculate.
- The winning can be calculated as – (Odds × Stake )- Stake
- Let, 4.0 can also be calculated as (4.0 × £10) – £10 = £30.
Calculation Of Money Line Odds
- In this odd, the favorites are represented with a minus sign (-)in front of the number, which indicates the amount of money needed to stake to win an amount of $100.
- The odds for the less competitive ones are presented with a plus symbol(+) which indicates the total amount of money won for the staked amount of $100. However, in both cases, the players win back the total wager amount placed by the players.
Calculation of Converting Betting Odds:
Converting odds to the implied probabilities is an integral part of its calculation.
- From Fractional Odds to Decimal Odds:
Step 1: Divide the fraction
Step 2: Add 1 to the result.
- From Fractional to Moneyline Odds:
Step 1 Divide the fraction
Step 2 If the answer is >= 1, then proceed with 100× Answer.
Step 3: If the answer is <= 1, then proceed with -100× Answer.
- From Decimal To Fractional:
Step 1: Deduct
Step 2: Convert to a fraction.
- From Decimal To Money Line:
Step 1: If the decimal odd is greater than 2, then 100× (Decimal odd – 1)
Step 2: If the decimal odd is smaller than 2, then -100/(Decimal Odd – 1 )
- From Moneyline to Decimal Odd:
Step 1: If the Money line is greater than 0, then (Moneyline Odd/100)+1
Step 2: If the Money line is lesser than 0, then (-100/Money line odd)+1
- From Money line to Fractional Odds:
Step 1: If Moneyline odd is greater than 0, then, Moneyline Odd/100
Step 2: If money line odd is less than 0, then, -100/ Moneyline odd
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https://heydocsuljn.web.app/how-to-use-pi-on-scientific-calculator-60.html
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math
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Scientific Calculator - A great Scientific Calculator. Clear and Free! Simple Calculator - A nice Simple Free Online Calculator. Easy to use and read. Online Abacus - An Online Abacus! Teach numbers from 1 to 50 :-) Darts Calculator - Forget the maths, and play Darts! Maths Calculator - This Online Maths Calculator show the history of your sums
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The calculator reference sheet is provided on most items on the 2014 GED. ® To perform calculations with scientific notation, use the key. EXAMPLE. 7.8 × 10. 8 decimal answers, exact square root and decimal, and exact pi and decimal. Pay special attention to the second function key, the way you use calculator memory and the keystrokes needed for decimals, fractions, mixed numbers, exponents Results 1 - 12 of 32 We have a great range of School Calculators from top brands. As well as standard battery-powered calculators, we also have calculators which use solar energy to help CASIO FX-85GTX Black Scientific Calculator. Use only the type of battery specified for this calculator in this manual. Handling Pi (π), Natural Logarithm Base e. Pi (π). You can input pi (π) into a calculation. 20 Jun 2012 The calculator I normally use (a TI-84) only displays 10 digits, so when I ask it what pi is, it spits out the familiar 3 and nine decimal places: cos(pi/3+pi/4) ???????????????????
Switching the TI-30XS MultiView calculator on and off.. 3 arising out of the purchase or use of these materials, and the sole and exact pi and decimal. Scientific Calculator. M. mod. Deg Rad. π e e MC MR MS M+ M-. sinh cosh tanh Exp ( ). ←. C +/-. √. sinh-1 cosh-1 tanh-1 log2x ln log 7 8 9 / %. π e n! logyx ex How to use a Scientific Calculator: entering expression, angle measure, number formats, arithmetic operators, Pi/2 = 3.1415926535898/2 = 1.5707963267949. 29 Apr 2018 To make this value an approximate product of π simply divide the result by π - If your calculator does not have a key for π use 3.14159265. 8 Dec 2019 Do you have a Casio fx-83 ES scientific calculator (or a compatible model) and want to learn how to use it? This free course, Using a scientific 1 Mar 2013 A historical look at approximations of pi for students to extend their calculator skills. Start by entering simple fractions and progress to an One small note is that, unlike older versions of Sharp Scientific Calculators, it now takes 2 button pressed to use pi in a calculation. I have no idea why they have
How does a scientific calculator multiply by Pi? … Any calculator multiplies by the value pi the same way it multiplies by any value that you type into the system, the only difference is it uses a value of pi that is hard coded into the calculator. What value of pi is stored in your calculator? Yo How to Use the Log Function on a Calculator | … Press the "Log" button on your calculator. The number you immediately see is the exponent for the original number you entered. Assuming the base number is 10 (which it will always be on a graphing or scientific calculator), you have to multiply 10 by itself the number of … Instructions for the new Online Scientific Calculator The Online Scientific Calculator is an online application for performing simple mathematical calculations. Based on the feedback I received for the first version it now includes many additional features. The most important change is the history display. All operations that you perform during the session are saved and displayed on the top portion of the calculator, so that you can look back at How to Operate a TI-34 Calculator | Sciencing
The Online Scientific Calculator is an online application for performing simple mathematical calculations. Based on the feedback I received for the first version it now includes many additional features. The most important change is the history display. All operations that you perform during the session are saved and displayed on the top portion of the calculator, so that you can look back at
Prepared by Sarah Nelson for the Dolciani Math Learning Center
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https://schoolbag.info/mathematics/ap_calculus/42.html
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math
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Master AP Calculus AB & BC
Part II. AP CALCULUS AB & BC REVIEW
CHAPTER 6. Applications of the Derivative
MOTION IN THE PLANE (BC TOPIC ONLY)
Although these questions are less frequent, the AP test sometimes contains questions concerning movement along a parametrically defined or vector path. Vector defined functions are quite easy to differentiate, making this topicrelatively simple in the grand scheme of BC topics. Before you read on, you should look back at the introduction to vector functions in Chapter 2.
In order to differentiate a vector function s = f(t)i + g(t)j, find the derivatives of the components separately:
Vector position equations work similarly to the position equations we just discussed. The velocity vector is given by the first derivative of the position equation, and the acceleration vector is given by the second.
Example 7: A particle moves in the plane such that the position vector from the origin to the particle is
s = (cos t sin 2t)i + (2t + 1)j, for all t on the interval [0,2π]
(a) Find the velocity and acceleration vectors for any time t.
To find the velocity vector, take the derivative of the x and y components separately:
v = (2cos t cos 2t - sin t sin 2t)i + 2j
In order to find the acceleration vector, take the derivative again:
a = (—4cos t sin 2t — 2sin t cos 2t — 2sin t cos 2t — cos t sin 2t)i
(b) What is the velocity vector when t = π/2, and what is the speed of the particle there?
Plug t = π/2 into the velocity vector to get the specific answer for that value of t:
v = (0 ∙ —1 — 1 ∙ 0)i + 2jv = 2j
The graph and its velocity vector at t = π/2 are shown below. The velocity vector has no horizontal component because the graph is changing direction at that point (kind of like a sideways extrema point).
The speed of the particle is given by the norm of the velocity vector. Remember, you find the norm with the equation
In this case
Directions: Solve each of the following problems. Decide which is the best of the choices given and indicate your responses in the book.
YOU MAY USE YOUR GRAPHING CALCULATOR FOR PART (B) OF PROBLEM 1 AND ALL OF PROBLEMS 2 AND 3.
1. A particle moves along the graph defined by x = cos t, y = cos 31, for t on the interval [0,2π].
(a) What is the velocity vector, v, when ?
(b) Draw the path of the particle on the coordinate plane, and indicate the direction the particle moves. Draw the velocity vector for .
(c) What is the magnitude of the acceleration when ?
2. A particle moves along a continuous and differentiable path that includes the coordinates (x,y) below for the corresponding values of t:
Approximate the speed of the particle at t = 3.
3. Create a position equation in vector form for a particle whose speed is 15 when t = 1.
ANSWERS AND EXPLANATIONS
1. (a) The general position vector is given by s = (cos t)i + (cos 3t)j. Therefore, the velocity vector will be v = (—sin t)i + (—3sin 3t)j. The velocity vector when will be since which is coterminal with 5π/4.
The graph proceeds from (1,1) to (-1,-1) on [0,π] and then returns to (1,1) on [π,2π].
(c) The magnitude of the acceleration is found by calculating the norm of the acceleration, much like the speed is the magnitude (norm) of the velocity. Therefore, you should begin by finding the acceleration vector by differentiating the velocity vector:
a = (—cos t)i + ( —9cos 3t)j
The magnitude of the acceleration is the norm of that vector, so calculate it when :
2. Recall the generic formula for speed as the norm of velocity: Because you do not have the position vector, you cannot find v. However, you can use slopes of secant lines to approximate dx/dt and dy/dt:
3. From the given information, we know that
There are numerous approaches to take, but we will discuss the easiest. Because we can do anything we like (as long as it works), we’ll set dx/dt = 15 and dy/dt = 0. Notice that the sum of their squares equals our goal. Next, we need an expression for x whose derivative, evaluated at 1, is 15. What about x = 15/2t2? The trick is taking half of the number you want to end up with and using it as the coefficient, since the Power Rule dictates that you will multiply by 2: Clearly, this has a value of 15 when t = 1. You can use any constant for your y component, since its derivative will be 0. One possible answer for this problem, therefore, is Check it to convince yourself that it’s right.
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CC-MAIN-2023-50
| 4,405 | 38 |
https://www.safaribooksonline.com/library/view/student-solution-manual/9781107485204/07_chapter-title-1.html
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math
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1.1 It can be shown that the polynomial
has turning points at x = –1 and x = and three real roots altogether. Continue an investigation of its properties as follows.
(a) Make a table of values of g(x) for integer values of x between –2 and 2. Use it and the information given above to draw a graph and so ...
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CC-MAIN-2018-17
| 312 | 3 |
https://papers.nips.cc/paper/6248-wasserstein-training-of-restricted-boltzmann-machines
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math
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Wasserstein Training of Restricted Boltzmann Machines[PDF] [BibTeX] [Supplemental] [Reviews]
Conference Event Type: Poster
Boltzmann machines are able to learn highly complex, multimodal, structured and multiscale real-world data distributions. Parameters of the model are usually learned by minimizing the Kullback-Leibler (KL) divergence from training samples to the learned model. We propose in this work a novel approach for Boltzmann machine training which assumes that a meaningful metric between observations is given. This metric can be represented by the Wasserstein distance between distributions, for which we derive a gradient with respect to the model parameters. Minimization of this new objective leads to generative models with different statistical properties. We demonstrate their practical potential on data completion and denoising, for which the metric between observations plays a crucial role.
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CC-MAIN-2017-09
| 916 | 3 |
http://gwinnett.patch.com/groups/announcements/p/catalyst-achievers-llc-math-bootcamp-price-lowered-to-85_5bca00b5
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math
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Back to School Math Bootcamp is a series of 2-day rigorous small group math workshops for students in grades 7-12. Students can expect to receive a review in key concepts and introduction of upcoming concepts to be taught in the classroom. We pick 5 topics in the subject to focus on and lead students to a deeper understanding of fundamental skills. The cost of each course has been reduced to $85.
Schedule is as follows:
Geometry 9/7 and 9/8 Algebra I 9/7 and 9/8 Algebra II 9/14 and 9/15 Trigonometry 9/21 and 9/22 Calculus 9/22 and 9/22
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CC-MAIN-2013-48
| 541 | 3 |
https://itecnotes.com/electrical/attenuation-in-speaker-crossover-design/
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math
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In the loudspeaker crossover circuit below, I believe the purpose of the resistor is to attenuate the signal to the second speaker (the tweeter).
However, in circuits I've seen that are supposed to perform attenuation only (l-pads), there are two resistors, one parallel as well as series (see second image).
Is the parallel resistor not required in the crossover, because the 0.4mH inductor is serving this purpose?
The second form is typically used to implement a "constant impedance" attenuator, replacing the 2R2 resistor in the sketch.
A first pass at designing the crossover network might assume the load is simply the tweeter, and thus uses the tweeter's impedance (typically 8 ohms) as the load resistance. The filter network is then designed and optimised to work into that load. (For example, the product of L and C determine the crossover frequency via the usual equation, while their ratio - with a load impedance of 8 ohms - determines the damping. The designer then chose 2u7 and 0.4mH as giving the response he wanted.
Then it turns out that the tweeter is a bit more efficient than the woofer, so the speaker sounds too shrill, so you need an attenuator to correct it.
The simplest attenuator - a series resistor - increases the load impedance to 10.2 ohms, thus reducing the damping factor of the filter. (You might hear an exaggerated shrill sound at the crossover frequency, for example).
The second form - a constant impedance attenuator, or L-pad, allows you to choose R1 and R2 such that the attenuation is the same, but the load impedance remains 8 ohms thus preserving the original properties of the filter.
In practice, while the second form is theoretically better, it is more complex and expensive, the simpler one may be adequate. Furthermore, loudspeaker design is not such an exact science, there are no perfect drive units, and cabinet design (shape of baffles etc) modifies the frequency response.
So it may even happen that the simpler filter compensates for an imperfection elsewhere in the system and both sounds and measures better than the theoretically better solution. But without extensive measurements and/or listening tests, you can't tell.
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https://en.unionpedia.org/Real_number
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math
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217 relations: Abū Kāmil Shujāʿ ibn Aslam, Abel–Ruffini theorem, Abraham Robinson, Absolute value, Accuracy and precision, Addition, Adolf Hurwitz, Adrien-Marie Legendre, Algebra, Algebraic number, Algebraically closed field, Almost all, American Mathematical Society, American Scientist, Arbitrary-precision arithmetic, Archimedean property, Associative algebra, Augustin-Louis Cauchy, Axiom of choice, Axiom of constructibility, Axiomatic system, École normale supérieure (Paris), Évariste Galois, Baire space (set theory), Basis (linear algebra), Blackboard bold, Calculus, Cantor's diagonal argument, Cardinal number, Cardinality, Cardinality of the continuum, Cartesian product, Cauchy sequence, Charles Hermite, Chinese mathematics, Classical mechanics, Coefficient, Compact space, Complete lattice, Complete metric space, Completeness of the real numbers, Complex number, Complex plane, Computable number, Computational science, Computer algebra, Computer algebra system, Connected space, Constant problem, Construction of the real numbers, ..., Constructivism (mathematics), Continued fraction, Continuous function, Continuum hypothesis, Contractible space, Coordinate system, Countable set, Cube root, Cyclic order, David Hilbert, Decimal, Decimal representation, Dedekind cut, Dedekind–MacNeille completion, Definable real number, Descriptive set theory, Differentiable manifold, Dimension, Distance, E (mathematical constant), Edmund Landau, Edward Nelson, Edwin Hewitt, Eigenvalues and eigenvectors, Electromagnetism, Empty set, Energy, Equation, Equivalence class, Euclidean geometry, Exponential function, Extended real number line, Ferdinand von Lindemann, Field (mathematics), Field extension, First-order logic, Floating-point arithmetic, Foundations of Physics, Fraction (mathematics), Fundamental theorem of algebra, Galois theory, General relativity, Georg Cantor, Georg Cantor's first set theory article, Gottfried Wilhelm Leibniz, Greatest and least elements, Greek mathematics, Haar measure, Hausdorff dimension, Hilbert space, History of Egypt, Homeomorphism, Hyperreal number, Imaginary number, Independence (mathematical logic), Indian mathematics, Infimum and supremum, Infinite set, Infinitesimal, Injective function, Integer, Internal set theory, Interval (mathematics), Irrational number, Isomorphism, Johann Heinrich Lambert, Joseph Liouville, Löwenheim–Skolem theorem, Least-upper-bound property, Lebesgue measure, Leonhard Euler, Lie algebra, Limit (mathematics), Limit of a sequence, Line (geometry), Linear combination, Locally compact space, Long line (topology), Magnitude (mathematics), Manava, Mass, Mathematical analysis, Mathematics, Mathematics in medieval Islam, Mathematische Annalen, Matrix (mathematics), Measure (mathematics), Metric space, Middle Ages, Multiplication, Natural number, Negative number, New York Academy of Sciences, Niels Henrik Abel, Non-Archimedean ordered field, Non-standard analysis, Non-standard model, Normal operator, Noun, Nth root, Number, Number line, Order topology, Ordered field, Paolo Ruffini, Paris, Partially ordered group, Paul Cohen, Paul Gordan, Pi, Point (geometry), Polynomial, Positive definiteness, Pythagoras, Quadratic equation, Quantity, Quantum mechanics, Quintic function, R, Rational number, Real analysis, Real closed field, Real line, Real projective line, René Descartes, Reverse mathematics, Self-adjoint operator, Separable space, Separation relation, Sequence, Set (mathematics), Set theory, Shulba Sutras, Sign (mathematics), Simon Stevin, Simply connected space, Solomon Feferman, Springer Science+Business Media, Square root, Square root of 2, Standard Model, Structuralism (philosophy of mathematics), Subset, Surreal number, Tarski's axiomatization of the reals, Time, Topological group, Topological space, Topology, Total order, Transcendental number, Uncountable set, Undecidable problem, Unicode, Uniform space, Unit interval, Up to, Upper and lower bounds, Vector space, Vedic period, Velocity, Vitali set, Well-order, Well-ordering theorem, Zermelo–Fraenkel set theory, Zero of a function, 0. Expand index (167 more) » « Shrink index
(Latinized as Auoquamel, ابو كامل, also known as al-ḥāsib al-miṣrī—lit. "the Egyptian reckoner") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age.
In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients.
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics.
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
Precision is a description of random errors, a measure of statistical variability.
Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.
Adolf Hurwitz (26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
In mathematics, the term "almost all" means "all but a negligible amount".
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
American Scientist (informally abbreviated AmSci) is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society.
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system.
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.
The École normale supérieure (also known as Normale sup', Ulm, ENS Paris, l'École and most often just as ENS) is one of the most selective and prestigious French grandes écoles (higher education establishment outside the framework of the public university system) and a constituent college of Université PSL.
Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology.
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Mathematics in China emerged independently by the 11th century BC.
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line.
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.
Computational science (also scientific computing or scientific computation (SC)) is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems.
In computational mathematics, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects.
A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
In mathematics, the constant problem is the problem of deciding if a given expression is equal to zero.
In mathematics, there are several ways of defining the real number system as an ordered field.
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
In mathematics, a cube root of a number x is a number y such that y3.
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers.
A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,...
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers.
In order-theoretic mathematics, the Dedekind–MacNeille completion of a partially ordered set (also called the completion by cuts or normal completion) is the smallest complete lattice that contains the given partial order.
Informally, a definable real number is a real number that can be uniquely specified by its description.
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
Distance is a numerical measurement of how far apart objects are.
The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Edward Nelson (May 4, 1932 – September 10, 2014) was a professor in the Mathematics Department at Princeton University.
Edwin Hewitt (January 20, 1920, Everett, Washington – June 21, 1999) was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage zero–one law.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object.
In mathematics, an equation is a statement of an equality containing one or more variables.
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.
In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).
Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that pi (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.
Foundations of Physics is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedures".
A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points.
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
The history of Egypt has been long and rich, due to the flow of the Nile River with its fertile banks and delta, as well as the accomplishments of Egypt's native inhabitants and outside influence.
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit,j is usually used in Engineering contexts where i has other meanings (such as electrical current) which is defined by its property.
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.
Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century.
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
In set theory, an infinite set is a set that is not a finite set.
In mathematics, infinitesimals are things so small that there is no way to measure them.
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson.
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.
Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician.
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
In mathematics, the least-upper-bound property (sometimes the completeness or supremum property) is a fundamental property of the real numbers and certain other ordered sets.
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".
In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind.
Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of Sulba Sutras. The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasutras which are considered more important.
Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
In mathematics, a metric space is a set for which distances between all members of the set are defined.
In the history of Europe, the Middle Ages (or Medieval Period) lasted from the 5th to the 15th century.
Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
In mathematics, a negative number is a real number that is less than zero.
The New York Academy of Sciences (originally the Lyceum of Natural History) was founded in January 1817.
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.
In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN*.
A noun (from Latin nōmen, literally meaning "name") is a word that functions as the name of some specific thing or set of things, such as living creatures, objects, places, actions, qualities, states of existence, or ideas.
In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.
A number is a mathematical object used to count, measure and also label.
In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by \mathbb.
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.
Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.
Paris is the capital and most populous city of France, with an area of and a population of 2,206,488.
In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician.
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his Ph.D. at the University of Breslau (1862),.
The number is a mathematical constant.
In modern mathematics, a point refers usually to an element of some set called a space.
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive definite.
Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.
In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to.
Quantity is a property that can exist as a multitude or magnitude.
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
In algebra, a quintic function is a function of the form where,,,, and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero.
R (named ar/or) is the 18th letter of the modern English alphabet and the ISO basic Latin alphabet.
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers.
In mathematics, the real line, or real number line is the line whose points are the real numbers.
In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity".
René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.
In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
The Shulba Sutras or Śulbasūtras (Sanskrit: "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.
In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.
Simon Stevin (1548–1620), sometimes called Stevinus, was a Flemish mathematician, physicist and military engineer.
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician with works in mathematical logic.
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
The square root of 2, or the (1/2)th power of 2, written in mathematics as or, is the positive algebraic number that, when multiplied by itself, gives the number 2.
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects.
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.
In 1936, Alfred Tarski set out an axiomatization of the real numbers and their arithmetic, consisting of only the 8 axioms shown below and a mere four primitive notions: the set of reals denoted R, a binary total order over R, denoted by infix This axiomatization does not give rise to a first-order theory, because the formal statement of axiom 3 includes two universal quantifiers over all possible subsets of R. Tarski proved these 8 axioms and 4 primitive notions independent.
Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is known to be impossible to construct a single algorithm that always leads to a correct yes-or-no answer.
Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems.
In the mathematical field of topology, a uniform space is a set with a uniform structure.
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
The Vedic period, or Vedic age, is the period in the history of the northwestern Indian subcontinent between the end of the urban Indus Valley Civilisation and a second urbanisation in the central Gangetic Plain which began in BCE.
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali.
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
In mathematics, the well-ordering theorem states that every set can be well-ordered.
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
0 (zero) is both a number and the numerical digit used to represent that number in numerals.
Axiomatic real number, Bounded real-valued data, Complete ordered field, Field of reals, List of real numbers, Number axis, Real (number), Real (numbers), Real Number System, Real Numbers, Real number field, Real number system, Real numbers, Reall numbers, Set of real numbers, The complete ordered field, ℝ.
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| 42,749 | 219 |
https://www.nagwa.com/en/worksheets/437102328535/
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math
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Lesson Worksheet: Part–Whole Bar Models: Numbers up to 100
In this worksheet, we will practice solving one-step addition and subtraction problems with numbers up to 100 by drawing a part–whole bar model.
Emma is trying to solve this word problem:
There were 41 birds in a tree. 15 of them flew away. How many birds are in the tree now? Emma said the answer was . What did she do wrong? What is the correct answer?
- AShe added when she should not have. The answer is 41.
- BShe added when she should have subtracted. The answer is .
- CShe added when she should have subtracted. The answer is .
- DShe made a mistake in the addition. The answer is .
- EShe added when she should have subtracted. The answer is .
Ethan and Elizabeth were building houses out of blocks. Ethan used 36 blocks. Elizabeth used 51 blocks.
Is the number of blocks they both used bigger or smaller than the numbers in the question?
Do you need to add or subtract to find the answer?
Find out how many blocks they used.
Benjamin has 62 blocks to build with. He used 29 blocks to build one house. He wants to build a second house with the leftover blocks.
Is the number of leftover blocks bigger or smaller than 62?
Do you need to add or subtact to find how many blocks he has left to build the second house?
How many blocks are left to build the second house?
James is trying to solve this word problem:
There were 37 fish in a pool at the bottom of a waterfall. 12 fish swam down the waterfall into the pool. How many fish are in the pool now?
James said the answer was . What did he do wrong? What is the correct answer?
- AHe subtracted when he should have added. The answer is .
- BHe made a mistake in subtracting. The answer is .
- CHe subtracted when he should have added. The answer is .
- DHe subtracted when he should have added. The answer is .
- EHe subtracted when he should not have. The answer is 12.
Anthony and Matthew were both playing a video game. At the beginning of the day, Anthony was at level 36 and Matthew was at level 26. During the day, Anthony moved up 16 levels and Matthew moved up 12 levels. Find the equation that shows the level Anthony reached.
On Wednesday, 23 boys and 55 girls went to the library. How many children went to the library on Wednesday?
There are two buses. One has 46 tourists, and the other has 27 tourists. How many tourists are there in total?
Subtract 30 from 50 to finish this part-part-whole picture.
Ethan and Mia are making a bridge out of blocks. They choose 23 green blocks and 30 red blocks. How many blocks do they use altogether?
One bag has 61 balls, and we add 9 more. How many balls do we have in the bag?
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http://lib.mexmat.ru/books/190639
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math
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Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Mathematica for Theoretical Physics: Classical Mechanics and Nonlinear Dynamics
Автор: Baumann G.
I'm working through the book with Mathematica 7. I've been told by the folks at Wolfram that Mathematica 7 will be replaced by a newer version.
Entering some of the examples in this book into Mathematica results in warnings like "Graphics`ImplicitPlot` is now obsolete" and Mathematica 7 doesn't recognize the old ImplicitPlot function, even though Mathematica said it loaded the old package. So, some of the examples in this book just don't work in Mathematica 7. This book is out of date with the latest version of Mathematica.
Also, keep in mind that this is volume 1 of a 2 volume set. The table of contents of this book spans both volumes. Chapter 4 and on is in the second volume.
All in all, though, it is much better than the very dull books that spend chapters on adding 2+2 as an example. :-D But for the price, I expected something more up-to-date.
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| 1,073 | 7 |
https://www.david-cook.org/how-do-you-do-differences-in-spss/
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math
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How do you do differences in SPSS?
How to Compute Difference Scores in SPSS
- Transform -> Compute Variable…
- Name the variable to hold the new difference scores (in the Target Variable box)
- Use the Numeric Expression box to calculate difference scores, using this format: Variable2Name – Variable1Name (or vice versa)
- Click OK.
How do you tell if there is a significant difference between two groups SPSS?
The Independent Samples t Test compares the means of two independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. The Independent Samples t Test is a parametric test. This test is also known as: Independent t Test.
What is a difference variable?
A variable peculiar to an individual, which can be studied to see if it affects the performance of the individual. Individual difference variables are usually definable traits that can be measured, such as age, height, weight, sex, skin colour, etc.
How do you interpret mean difference?
For example, let’s say the mean score on a depression test for a group of 100 middle-aged men is 35 and for 100 middle-aged women it is 25. If you took a large number of samples from both these groups and calculated the mean differences, the mean of all of the differences between all sample means would be 35 – 25 = 10.
Is the difference in means statistically significant?
In principle, a statistically significant result (usually a difference) is a result that’s not attributed to chance. More technically, it means that if the Null Hypothesis is true (which means there really is no difference), there’s a low probability of getting a result that large or larger.
What is difference between parameters and variables?
A variable is the way in which an attribute or quantity is represented. A parameter is normally a constant in an equation describing a model (a simulation used to reproduce behavior of a system).
How do you find the difference between two integers?
How to Find the Difference between Two Numbers. To find the difference between two numbers, subtract the number with the smallest value from the number with the largest value. The product of this sum is the difference between the two numbers. Therefore the difference between 45 and 100 is 55.
What does difference mean in statistics?
Updated April 25, 2017. By Jack Ori. Statistical difference refers to significant differences between groups of objects or people. Scientists calculate this difference in order to determine whether the data from an experiment is reliable before drawing conclusions and publishing results.
How do you determine if a difference is statistically significant?
You may be able to detect a statistically significant difference by increasing your sample size. If you have a very small sample size, only large differences between two groups will be significant. If you have a very large sample size, both small and large differences will be detected as significant.
How do you know if two sets of data are statistically different?
A t-test tells you whether the difference between two sample means is “statistically significant” – not whether the two means are statistically different. A t-score with a p-value larger than 0.05 just states that the difference found is not “statistically significant”.
What is difference between variables and parameters explain with example?
A variable is the way in which an attribute or quantity is represented. A parameter is normally a constant in an equation describing a model (a simulation used to reproduce behavior of a system). For instance, the first part of the Hodgkin–Huxley model is Im=Cm dVm/dt. In this equation Im and Vm are variables.
Why should I understand SPSS variable types and formats?
Understanding SPSS variable types and formats allows you to get things done fast and reliably. Getting a grip on types and formats is not hard if you ignore the very confusing information under variable view.
What is the difference between Excel and SPSS?
The bottom line is though Excel offers a good way of data organization, SPSS is more suitable for in-depth data analysis. This tool is very useful in the analysis and visualization of data. This has been a guide to What is SPSS?.
What are the features of SPSS software?
Besides the statistical analysis of data, the SPSS software also provides features of data management, this allows the user to do a selection, create derived data and perform file reshaping, etc. Another feature is data documentation. This feature stores a metadata dictionary along with the data file.
What statistical methods can be used in SPSS?
There are many statistical methods that can be used in SPSS which are as follows: Prediction for a variety of data for identifying groups and including methodologies such as cluster analysis, factor analysis, etc.
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http://genesissciencemissionblog.blogspot.com/2012/10/gsmnl-information-universe-basics.html
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math
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This is part five of the Information Universe series. See the link for part one.
There are a number of basic concepts that come from the Information Universe when it is applied to know principles of physics. These basic concepts not only provide a starting point to a formal Information Universe but they actually salve some problems that have plagued physics for many years. Among other things these basic concepts provide a manner of unifying quantum mechanics and General Relativity.
The first is that rather than being a continual coordinate system that it is actually quantized forming pixels. This pixilation has the side affect of unifying Quantum mechanics and General Relativity. While it does not produce a graviton it does divide space into discrete units and since General Relativity depicts gravity as a warping of space-time around a mass this would provide a way of bringing gravity into the quantum world.
The second the two most basic concepts that come from the Information Universe is that time is also quantized forming what could be considered a pixel of time.
These two basic concepts come from the fact that if the universe is indeed a calculating information system then such pixilation would be expected since there would have to be a level below that of both space and time at which the calculations take place.
The fourth basic concept is that since speed of light is one Planck length per Planck time it is the fastest speed that an object can travel that covers every point in space.
The fourth basic concept is that fundamental particles such as electrons and photons occupy one Planck length pixel for any given Planck time.
These basic concepts are the starting point of an Information Universe model. From here a full Information Universe model can be produced as mention in the article on developing an Information Universe Model. They are also critical to understanding how our universe can be based on information rather than material stuff.
------ Charles Creager Jr.
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http://scratchpad.wikia.com/wiki/Proof_that_the_%E2%88%9A5_is_irrational
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math
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- Proof that the √5 is irrational:
This will be a proof by contradiction.
- Assume that the √5 is rational (can be expressed in the form , where a and b are integers.
- Then √5 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 5.
- It follows that a2 / b2 = 5 and a2 = 5 b2.
- Therefore a2 is divisible by 5 because it is equal to 5 b2 which is obviously divisible by 5.
- It follows that a must be divisible by 5.
- Because a is divisible by 5, there exists an integer k that fulfills: a = 5k.
- We insert the last equation of (3) in (6): 5b2 = (5k)2 is equivalent to 5b2 = 25k2 is equivalent to b2 = 5k2.
- Because 5k2 is divisible by 5 it follows that b2 is also divisible by 5 which means that b is divisible by 5 because only numbers divisible by 5 have squares divisible by 5.
- By (5) and (8) a and b are both divisible by 5, which contradicts that a / b is irreducible as stated in (2).
Therefore, √5 is irrational.
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https://www.hackmath.net/en/math-problem/3010
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math
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A cylindrical tank has a volume of 60 hectoliters and is 2.5 meters deep. Calculate the tank diameter.
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- Diameter = height
The surface of the cylinder, the height of which is equal to the diameter of the base, is 4239 cm square. Calculate the cylinder volume.
- Height as diameter of base
The rotary cylinder has a height equal to the base diameter and the surface of 471 cm2. Calculate the volume of a cylinder.
- Equilateral cylinder
Equilateral cylinder (height = base diameter; h = 2r) has a volume of V = 199 cm3 . Calculate the surface area of the cylinder.
- 3d printer
3D printing ABS filament with diameter 1.75 mm has density 1.04 g/cm3. Find the length of m = 5 kg spool filament. (how to calculate length)
How deep is the pool if there are 2025 hectoliters of water and the bottom dimensions are a = 15 meters b = 7,5 meters and the water level is up to 9/10 (nine-tenths) of height.
- Wall thickness
The hollow metal ball has an outside diameter of 40 cm. Determine the wall thickness if the weight is 25 kg and the metal density is 8.45 g/cm3.
- Steel tube
The steel tube has an inner diameter of 4 cm and an outer diameter of 4.8 cm. The density of the steel is 7800 kg/m3. Calculate its length if it weighs 15 kg.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00315.warc.gz
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CC-MAIN-2020-45
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https://www.cirano.qc.ca/en/summaries/2023PJ-02
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math
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Améliorer les compétences en mathématiques au Québec: Cinq recommandations tirées d’En avant math !
→ View the full report
CIRANO and the Centre de recherches mathématiques are partners in En avant math!, a national initiative to promote mathematics and increase numeracy. Over the past three years, several studies have identified possible solutions to ensure the development of a highly qualified workforce in applied mathematics and to promote a better adequacy between the skills of individuals and the needs of the labour market, particularly in the science, technology, engineering and mathematics (STEM) sectors. The authors present here the key lessons and recommendations that emerged from the work of the first phase of En avant math !.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224657169.98/warc/CC-MAIN-20230610095459-20230610125459-00619.warc.gz
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CC-MAIN-2023-23
| 756 | 3 |
https://www.physicsforums.com/threads/thermal-entropy-of-ideal-gas-sackur-tetrode-equation.754425/
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math
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okay so I suck at La-Tex so i'm not going to put the actual equation, but it's not important for my question. In the equation the entropy is dependent on the natural log with mass in the numerator of the argument. Why is mass involved when talking about entropy at all? I mean I think of entropy as being related to multiplicity of states or whatever, which is independent of mass? is rest mass energy being involved here?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589251.7/warc/CC-MAIN-20180716095945-20180716115945-00050.warc.gz
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CC-MAIN-2018-30
| 422 | 1 |
https://books.google.gr/books?id=aDQDAAAAQAAJ&pg=PA140&vq=%22which+is+called+the+CIRCUMFERENCE,+and+is+such,+that+all+straight+lines+drawn%22&dq=editions:UOMDLPabq7928_0001_001&lr=&hl=el&output=html_text
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math
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« ΠροηγούμενηΣυνέχεια »
PROPOSITION XIV. THEOREM. Equal chords in a circle are equally distant from the centre; and conversely, those which are equally distant from the centre, are equal to one another.
Let the chords AB, CD in the O ABDC be equal.
and :: AB=CD, .. AP=CQ.
I. E. Cor. p. 43. and .. AB and CD are equally distant from 0. Def. 8. Next, let AB and CD be equally distant from 0.
Then must AB=CD.
I. E. Cor. and .. AB=CD.
Q. E. D.
Ex. In a circle, whose diameter is 10 inches, a chord is drawn, which is 8 inches long. If another chord be drawn, a a distance of 3 inches from the centre, shew whether it is equal or not to the former.
PROPOSITION XV. THEOREM.
The diameter is the greatest chord in a circle, and of all others that which is nearer to the centre is always greater than one more remote ; and the greater is nearer to the centre than the less.
Let AB be a diameter of the O ABD whose centre is O, and let CD be any other chord, not a diameter, in the o, nearer to the centre than the chord EF. Then must AB be greater than CD, and CD greater than EF. Draw OP, OQ I to CD and EF; and join OC, OD, OE.
Then :: A0=CO, and OB=OD, I. Def. 13.
.: AB=sum of CO and OD, and .. AB is greater than CD.
I. 20. Again, :: CD is nearer to the centre than EF, .: OP is less than 0Q.
Def. 8. Now :- sq. on OC=sq. on OE, .: sum of sqq. on OP, PC=sum of sqq. on OQ, QE. I. 47.
But sq. on OP is less than sq. on OQ ;
.. PC is greater than QE ;
Next, let CD be greater than EF.
For :: CD is greater than EF,
:: PC is greater than QE.
Now the sum of sqq. on OP, PC=sum of sqq. on OQ, QE.
But sq. on PC is greater than sq. on QE;
.. OP is less than OQ; .
Q. E. D. Ex. 1. Draw a chord of given length in a given circle, which shall be bisected by a given chord.
Ex. 2. If two isosceles triangles be of equal altitude, and the sides of one be equal to the sides of the other, shew that their bases must be equal.
Ex. 3. Any two chords of a circle, which cut a diameter in the same point and at equal angles, are equal to one another.
DEF. IX. A straight line is said to be a TANGENT to, or to touch, a circle, when it meets and, being produced, does not cut the circle.
From this definition it follows that the tangent meets the circle in one point only, for if it met the circle in two points it would cut the circle, since the line joining two points in the circumference is, being produced, a secant. (III. 2.)
DEF. X. If from any point in a circle a line be drawn at right angles to the tangent at that point, the line is called a Normal to the circle at that point.
DEF. XI. A rectilinear figure is said to be described about a circle, when each side of the figure touches the circle.
And the circle is said to be inscribed in the figure.
PROPOSITION XVI. THEOREM.
The straight line drawn at right angles to the diameter of a circle, from the extremity of it, is a tangent to the circle.
Let ABC be a 0, of which the centre is 0, and the diameter AOB.
Through B draw DE at right angles to AOB. I. 11.
Then must DE be a tangent to the o.
Then, :: ZOBP is a right angle,
I. 19. Hence P is a point without the o ABC. Post.
In the same way it may be shewn that every point in DE, or DE produced in either direction, except the point B, lies without the O; .: DE is a tangent to the o.
Q. E. D.
PROPOSITION XVII. PROBLEM. To draw a straight line from a given point, either WITHOUT or on the circumference, which shall touch a given circle.
Let A be the given pt., without the o BCD.
Take O the centre of o BCD, and join 0A.
Join BO, BE.
I. 32. .. sum of 2s AEB, OEB=twice sum of 2 S OBE, ABE, that is, two right angles=twice 2 OBA;
:: ZOBA is a right angle,
and .. AB is a tangent to the o BCD. III. 16. Similarly it may be shewn that AD is a tangent to o BCD. Next, let the given pt. be on the Oce of the o, as B.
Then, if BA be drawn I to the radius OB,
Q. E. D. Ex. 1. Shew that the two tangents, drawn from a point without the circumference to a circle, are equal.
Ex. 2. If a quadrilateral ABCD be described about a circle, shew that the sum of AB and CD is equal to the sum of AD and BC.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711162.52/warc/CC-MAIN-20221207121241-20221207151241-00240.warc.gz
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CC-MAIN-2022-49
| 4,114 | 51 |
https://stackexchange.com/users/13598623/kami
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math
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817144.49/warc/CC-MAIN-20240417044411-20240417074411-00280.warc.gz
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