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Chapter 17 Analyzing Starlight
17.7 Questions and Exercises
1: What two factors determine how bright a star appears to be in the sky?
2: Explain why color is a measure of a star’s temperature.
3: What is the main reason that the spectra of all stars are not identical? Explain.
4: What elements are stars mostly made of? How do we know this?
5: What did Annie Cannon contribute to the understanding of stellar spectra?
6: Name five characteristics of a star that can be determined by measuring its spectrum. Explain how you would use a spectrum to determine these characteristics.
7: How do objects of spectral types L, T, and Y differ from those of the other spectral types?
8: Do stars that look brighter in the sky have larger or smaller magnitudes than fainter stars?
9: The star Antares has an apparent magnitude of 1.0, whereas the star Procyon has an apparent magnitude of 0.4. Which star appears brighter in the sky?
10: Based on their colors, which of the following stars is hottest? Which is coolest? Archenar (blue), Betelgeuse (red), Capella (yellow).
11: Order the seven basic spectral types from hottest to coldest.
12: What is the defining difference between a brown dwarf and a true star?
13: If the star Sirius emits 23 times more energy than the Sun, why does the Sun appear brighter in the sky?
14: How would two stars of equal luminosity—one blue and the other red—appear in an image taken through a filter that passes mainly blue light? How would their appearance change in an image taken through a filter that transmits mainly red light?
15: Table 17.2 lists the temperature ranges that correspond to the different spectral types. What part of the star do these temperatures refer to? Why?
16: Suppose you are given the task of measuring the colors of the brightest stars, listed in Appendix J, through three filters: the first transmits blue light, the second transmits yellow light, and the third transmits red light. If you observe the star Vega, it will appear equally bright through each of the three filters. Which stars will appear brighter through the blue filter than through the red filter? Which stars will appear brighter through the red filter? Which star is likely to have colors most nearly like those of Vega?
17: Star X has lines of ionized helium in its spectrum, and star Y has bands of titanium oxide. Which is hotter? Why? The spectrum of star Z shows lines of ionized helium and also molecular bands of titanium oxide. What is strange about this spectrum? Can you suggest an explanation?
18: The spectrum of the Sun has hundreds of strong lines of nonionized iron but only a few, very weak lines of helium. A star of spectral type B has very strong lines of helium but very weak iron lines. Do these differences mean that the Sun contains more iron and less helium than the B star? Explain.
19: What are the approximate spectral classes of stars with the following characteristics?
- Balmer lines of hydrogen are very strong; some lines of ionized metals are present.
- The strongest lines are those of ionized helium.
- Lines of ionized calcium are the strongest in the spectrum; hydrogen lines show only moderate strength; lines of neutral and metals are present.
- The strongest lines are those of neutral metals and bands of titanium oxide.
20: Look at the chemical elements in Appendix K. Can you identify any relationship between the abundance of an element and its atomic weight? Are there any obvious exceptions to this relationship?
21: Appendix I lists some of the nearest stars. Are most of these stars hotter or cooler than the Sun? Do any of them emit more energy than the Sun? If so, which ones?
22: Appendix J lists the stars that appear brightest in our sky. Are most of these hotter or cooler than the Sun? Can you suggest a reason for the difference between this answer and the answer to the previous question? (Hint: Look at the luminosities.) Is there any tendency for a correlation between temperature and luminosity? Are there exceptions to the correlation?
23: What star appears the brightest in the sky (other than the Sun)? The second brightest? What color is Betelgeuse? Use Appendix J to find the answers.
24: Suppose hominids one million years ago had left behind maps of the night sky. Would these maps represent accurately the sky that we see today? Why or why not?
25: Why can only a lower limit to the rate of stellar rotation be determined from line broadening rather than the actual rotation rate? (Refer to Figure 17.14.)
26: Why do you think astronomers have suggested three different spectral types (L, T, and Y) for the brown dwarfs instead of M? Why was one not enough?
27: Sam, a college student, just bought a new car. Sam’s friend Adam, a graduate student in astronomy, asks Sam for a ride. In the car, Adam remarks that the colors on the temperature control are wrong. Why did he say that?
28: Would a red star have a smaller or larger magnitude in a red filter than in a blue filter?
29: Two stars have proper motions of one arcsecond per year. Star A is 20 light-years from Earth, and Star B is 10 light-years away from Earth. Which one has the faster velocity in space?
30: Suppose there are three stars in space, each moving at 100 km/s. Star A is moving across (i.e., perpendicular to) our line of sight, Star B is moving directly away from Earth, and Star C is moving away from Earth, but at a 30° angle to the line of sight. From which star will you observe the greatest Doppler shift? From which star will you observe the smallest Doppler shift?
31: What would you say to a friend who made this statement, “The visible-light spectrum of the Sun shows weak hydrogen lines and strong calcium lines. The Sun must therefore contain more calcium than hydrogen.”?
Figuring for Yourself
32: In Appendix J, how much more luminous is the most luminous of the stars than the least luminous?
33: For Exercise 17.33 through Exercise 17.38, use the equations relating magnitude and apparent brightness given in the section on the magnitude scale in The Brightness of Stars and Example 17.1.
34: Verify that if two stars have a difference of five magnitudes, this corresponds to a factor of 100 in the ratio that 2.5 magnitudes corresponds to a factor of 10; and that 0.75 magnitudes corresponds to a factor of 2.
35: As seen from Earth, the Sun has an apparent magnitude of about −26.7. What is the apparent magnitude of the Sun as seen from Saturn, about 10 AU away? (Remember that one AU is the distance from Earth to the Sun and that the brightness decreases as the inverse square of the distance.) Would the Sun still be the brightest star in the sky?
36: An astronomer is investigating a faint star that has recently been discovered in very sensitive surveys of the sky. The star has a magnitude of 16. How much less bright is it than Antares, a star with magnitude roughly equal to 1?
37: The center of a faint but active galaxy has magnitude 26. How much less bright does it look than the very faintest star that our eyes can see, roughly magnitude 6?
38: You have enough information from this chapter to estimate the distance to Alpha Centauri, the second nearest star, which has an apparent magnitude of 0. Since it is a G2 star, like the Sun, assume it has the same luminosity as the Sun and the difference in magnitudes is a result only of the difference in distance. Estimate how far away Alpha Centauri is. Describe the necessary steps in words and then do the calculation. (As we will learn in the Celestial Distances chapter, this method—namely, assuming that stars with identical spectral types emit the same amount of energy—is actually used to estimate distances to stars.) If you assume the distance to the Sun is in AU, your answer will come out in AU.
39: Do the previous problem again, this time using the information that the Sun is 150,000,000 km away. You will get a very large number of km as your answer. To get a better feeling for how the distances compare, try calculating the time it takes light at a speed of 299,338 km/s to travel from the Sun to Earth and from Alpha Centauri to Earth. For Alpha Centauri, figure out how long the trip will take in years as well as in seconds.
40: Star A and Star B have different apparent brightnesses but identical luminosities. If Star A is 20 light-years away from Earth and Star B is 40 light-years away from Earth, which star appears brighter and by what factor?
41: Star A and Star B have different apparent brightnesses but identical luminosities. Star A is 10 light-years away from Earth and appears 36 times brighter than Star B. How far away is Star B?
42: The star Sirius A has an apparent magnitude of −1.5. Sirius A has a dim companion, Sirius B, which is 10,000 times less bright than Sirius A. What is the apparent magnitude of Sirius B? Can Sirius B be seen with the naked eye?
43: Our Sun, a type G star, has a surface temperature of 5800 K. We know, therefore, that it is cooler than a type O star and hotter than a type M star. Given what you learned about the temperature ranges of these types of stars, how many times hotter than our Sun is the hottest type O star? How many times cooler than our Sun is the coolest type M star? |
The relationship between flow rate and temperature difference
Extracts from this document...
Investigation of Flow rate and temperature difference Introduction My coursework is based on the principle of how an electric shower works. The common assumption is that in order to alter the temperature i.e. increase or decrease it, there is a heating implement in place which does this, but my investigation has proven this is not the variable. There is a heating implement but the heat is kept constant. The temperature of the water coming out of the shower depends on the flow rate of the water. This investigation also involves the concept of energy transfer. The conservation law of energy states that "energy can neither be created nor destroyed but can only be transferred from 1 body to another through a medium". In this case the energy is being transferred is heat. The transfer is heat energy from the hot water in the metal container by the metal container ( medium - free electrons in metal ) to the water flowing through the pipe. I am investigating the effect of a changing flow rate on the temperature difference in a pipe. It will be important for me to carry out my experiment precisely and get a reasonable set of results because of all the variables involved such as temperature, and for the flow resistance in the pipe - viscosity, density, friction with pipe, specific heat capacity e.t.c. Factors affecting my investigation Viscosity The resistance to flow in a liquid can be characterised in terms of the viscosity of the fluid if the flow is smooth. Viscosity is a measure of the resistance of a fluid to deform under shear stress. It is commonly perceived as "thickness", or resistance to pouring. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. ...read more.
I kept the volume of water, and the temperature of the water in it, constant. The tricky part of my setup was keeping the rubber tubing wrapped around the metal container. I tried various methods, cello tape first of all but the adhesive property of the cello tape was lost once it came in contact with the water and it did come in contact with water often, seeing as my experiment was based around water. Finally I had to improvise and I smeared glue all around the metal container and then set the tubing around it and held it in place with a bit of glue at this point the tubing was stuck around it and then I froze it, by placing it in the freezer and leaving it for approximately 26 hours. By this time the glue was hard and dry and the tube was properly in place. I had some water in the metal container kept at a constant temperature in this case 70 degrees. All I did was pass water through the top opening, this water was kept at a constant temperature. It then flowed around the pipe gathering energy from the metal container the water in the metal container was as previously stated at constant temperature. This water was heated by an old boiling ring which was placed in the middle of the metal container not in direct contact with the edges of the container. The same problem arose of finding a way to regulate the heat giving of by the heating implement i.e the boiling ring, I solved this by starting the experiment only when the temperature was at 70 degrees and then estimating the time it took for it to rise by a degree ( this was done roughly and mentally ). I counted roughly in my head the time taken for it to rise from 70-71 degrees. ...read more.
Logic says this is a good set of results because as the flow rate decreases it means that the water flows slower in and out of the tube meaning that there is an increased time taken between water being poured in and then out. This also means that the water is in the tube and in contact with the radiation of heat energy from the metal container for a longer time. There is an increased energy transfer, if the power were to be worked out it would be constant because there is an increase in the energy transferred AND also the time taken, both these would cancel out. The flow rate could have been improved in a variety of ways like o Using a tube with less friction o Pumping the water in faster, e.t.c From the results it can be said that there is an inverse relationship (as one quantity increase, the other decreases). The flow rate is inversely proportional to the temperature difference. This could be shown mathematically by the formula F = k T Key F - flow rate of water T - temperature difference k - constant of proportionality I am going to plot 2 sets of graphs, the first one showing the temperature difference on the y axis and flow rate on the x axis, this should give me a graph with a negative (decreasing) gradient and the second the temperature difference on the y axis and the inverse of the flow rate on the x axis and this should give me a graph with a straight line from origin. I will now try and find the value of k, this is going to be done by finding the gradient of the graph and then finding the inverse of that figure. Finding the gradient of a graph is done by applying the formula y1-y2 = k x1-x2 0.856 - 0.536 = 0.32 = 0.1793, 1 = 0.1793, k = 1 9.500 - 11.350 -1.85 k 0.1793 K = and this has the unit of cm3/s per degrees Celsius. ...read more.
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Science, Tech, Math › Math Calculate a Confidence Interval for a Mean When You Know Sigma Known Standard Deviation Share Flipboard Email Print Formula for confidence interval of a mean when population standard deviation is known. C.K.Taylor Math Statistics Inferential Statistics Statistics Tutorials Formulas Probability & Games Descriptive Statistics Applications Of Statistics Math Tutorials Geometry Arithmetic Pre Algebra & Algebra Exponential Decay Functions Worksheets By Grade Resources View More By Courtney Taylor Professor of Mathematics Ph.D., Mathematics, Purdue University M.S., Mathematics, Purdue University B.A., Mathematics, Physics, and Chemistry, Anderson University Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra." our editorial process Courtney Taylor Updated April 19, 2018 In inferential statistics, one of the major goals is to estimate an unknown population parameter. You start with a statistical sample, and from this, you can determine a range of values for the parameter. This range of values is called a confidence interval. Confidence Intervals Confidence intervals are all similar to one another in a few ways. First, many two-sided confidence intervals have the same form: Estimate ± Margin of Error Second, the steps for calculating confidence intervals are very similar, regardless of the type of confidence interval you are trying to find. The specific type of confidence interval that will be examined below is a two-sided confidence interval for a population mean when you know the population standard deviation. Also, assume that you are working with a population that is normally distributed. Confidence Interval for a Mean With a Known Sigma Below is a process to find the desired confidence interval. Although all of the steps are important, the first one is particularly so: Check conditions: Begin by ensuring that the conditions for your confidence interval have been met. Assume that you know the value of the population standard deviation, denoted by the Greek letter sigma σ. Also, assume a normal distribution.Calculate estimate: Estimate the population parameter—in this case, the population mean—by use of a statistic, which in this problem is the sample mean. This involves forming a simple random sample from the population. Sometimes, you can suppose that your sample is a simple random sample, even if it does not meet the strict definition.Critical value: Obtain the critical value z* that corresponds with your confidence level. These values are found by consulting a table of z-scores or by using the software. You can use a z-score table because you know the value of the population standard deviation, and you assume that the population is normally distributed. Common critical values are 1.645 for a 90-percent confidence level, 1.960 for a 95-percent confidence level, and 2.576 for a 99-percent confidence level.Margin of error: Calculate the margin of error z* σ /√n, where n is the size of the simple random sample that you formed.Conclude: Finish by putting together the estimate and margin of error. This can be expressed as either Estimate ± Margin of Error or as Estimate - Margin of Error to Estimate + Margin of Error. Be sure to clearly state the level of confidence that is attached to your confidence interval. Example To see how you can construct a confidence interval, work through an example. Suppose you know that the IQ scores of all incoming college freshman are normally distributed with standard deviation of 15. You have a simple random sample of 100 freshmen, and the mean IQ score for this sample is 120. Find a 90-percent confidence interval for the mean IQ score for the entire population of incoming college freshmen. Work through the steps that were outlined above: Check conditions: The conditions have been met since you have been told that the population standard deviation is 15 and that you are dealing with a normal distribution.Calculate estimate: You have been told that you have a simple random sample of size 100. The mean IQ for this sample is 120, so this is your estimate.Critical value: The critical value for confidence level of 90 percent is given by z* = 1.645.Margin of error: Use the margin of error formula and obtain an error of z* σ /√n = (1.645)(15) /√(100) = 2.467.Conclude: Conclude by putting everything together. A 90-percent confidence interval for the population’s mean IQ score is 120 ± 2.467. Alternatively, you could state this confidence interval as 117.5325 to 122.4675. Practical Considerations Confidence intervals of the above type are not very realistic. It is very rare to know the population standard deviation but not know the population mean. There are ways that this unrealistic assumption can be removed. While you have assumed a normal distribution, this assumption does not need to hold. Nice samples, which exhibit no strong skewness or have any outliers, along with a large enough sample size, allow you to invoke the central limit theorem. As a result, you are justified in using a table of z-scores, even for populations that are not normally distributed. |
This is what Theoni Pappas writes on page 116 of her Magic of Mathematics:
"Take the year you were born. To this add the year of an important event in your life. To this sum add the age you will be at the end of 1994. Finally, add to this sum the number of years ago that the important event took place."
Here Pappas is showing us some "Number Magic." Notice that 1994 is the year in which she first wrote this book. The following is the example that she gives:
1953 -- Year born
1980 -- Year Pappas (claimed to have) traveled into outer space. (Really, Pappas?)
41 -- Age as of 1994
14 -- Years elapsed since 1994 of important event
3988 -- Total
According to Pappas, the answer is always 3988. Of course, this isn't really "magic" at all -- notice that there are two pairs that add up to 1994. The year of her birth and her age add up to 1994, as do the important event and the number of years since that event. So it's not surprising that the numbers always add up to 1994 * 2 = 3988.
Let me try another example. This time, I'll change it to the current year, 2017, so my four numbers should add up to 2017 * 2 = 4034. The year of my birth is 1980 -- hey, that's the same year that Pappas (supposedly) traveled into space. And as for the year of my important event -- I just think I'll choose the year 1994. After all, it's the year I took my favorite math class, Geometry. Oh, and a lady with the initials TP wrote one of my favorite books in that year.
1980 -- Year born
1994 -- Year of important event
37 -- Age at the end of this year
23 -- Years elapsed since important event
4034 -- Total
Unlike Pappas, here's something that really did travel into space -- Cassini. There was a special Google Doodle marking the space probe's arrival at the rings of Saturn. Again, I'm a science teacher, so I like to point out important scientific events. In this case, researchers are excited over the possibility that there is life on at least one of the ringed planet's moons. Cassini will orbit Saturn 22 times to gather data from the Cronian moons.
This is what I wrote last year about today's lesson:
Chapter 14 of the U of Chicago text is on Trigonometry and Vectors. Here's the plan:
Today, April 26th -- Lesson 14-1: Special Right Triangles
Tomorrow, April 27th -- Lesson 14-2: Lengths in Right Triangles
Friday, April 28th -- Activity (includes Lesson 14-3: The Tangent Ratio)
Monday, May 1st -- Lesson 14-4: The Sine and Cosine Ratios
Tuesday, May 2nd -- Lesson 14-5: Vectors
Wednesday, May 3rd -- Lesson 14-6: Properties of Vectors
Thursday, May 4th -- Lesson 14-7: Adding Vectors Using Trigonometry
Friday, May 5th -- ActivityMonday, May 8th -- Review for Chapter 14 Test
Tuesday, May 9th -- Chapter 14 Test
So the plan for this chapter is straightforward. The one thing to note is how the day that Lesson 14-4 would have occurred, there is a planned activity day. I've noticed how many texts, including the U of Chicago, discuss the tangent ratio in a separate lesson from sine and cosine. I suppose that in many ways, sine and cosine are alike in a way that tangent isn't. The sine or cosine of any real number is between -1 and 1, while the tangent can be any real number. Therefore the graphs of sine and cosine resemble each other. The tangent ratio involves two legs, while the sine and cosine ratios involve one leg and the hypotenuse. Even the name "cosine" includes the word "sine," while the name "tangent" doesn't include "sine."
Yet I will end up covering sine, cosine, and tangent all on the same day. In the past, I've seen many teachers simply teach SOH-CAH-TOA all in the same lesson, and then when they come to me for tutoring, they look at each triangle in the homework to determine whether sine, cosine, or tangent is needed to solve the problem. But as it turns out, all of the questions require tangent because the student is actually reading the tangent lesson in the text! If the student is going through all of that, then we might as well have all three trig ratios in the same lesson.
And so this is exactly what I'll do. This will then free a day for an activity. My planned activity will actually be one that I found off of another teacher's website. (Actually, I'm still debating whether to do the activity on Friday or on Thursday, since this teacher presents this activity before teaching the students about sine, cosine, and tangent.)
[2017 Update: The activity day is now on Friday and the trig ratios are on Monday. Thus the other teacher's original intent has been restored.]
But that's for later this week -- how about today's lesson? Lesson 14-1 of the U of Chicago text is on Special Right Triangles -- that is, the 45-45-90 and 30-60-90 triangles. The text emphasizes how these triangles are related to the regular polygons. In particular, the 45-45-90 and 30-60-90 triangles are half of the square and the equilateral triangle, respectively. We can obtain these regular polygons, in true Common Core fashion, by reflecting each right triangle over one of its legs. The regular hexagon is also closely related to the 30-60-90 triangle.
The questions that I selected from the text refers to these regular polygons and using the triangles to measure lengths related to the regular polygons. I mentioned today how I like to watch baseball over summer break -- well, a baseball "diamond" (really a square) appears on the worksheet. Also, a honeycomb, with its hexagonal bee cells, also appears.
The review questions that I selected are also preview questions. Two of the questions involve similar right triangles in preparation for geometric means in Lesson 14-2, and the other one is about how to simplify radicals, so we can explain in Lesson 14-4 why the sine and cosine of 45 degrees are usually written as sqrt(2)/2. |
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LMAO the correct answer is F hahhaha tough luck
it says that for every question, anyways, a double bond in the entity containing N follows the rules of octet rule, but why is NO3- better?
do u have a picture of the periodic table
i can post it just a min...
im looking at one already nvm
NO2+ doesn't have its best form as a resonance structure: |dw:1444197726202:dw|
oh i see
there are 3 equivalent resonant structures for NO3+
I see what your talking about for tsecond pic but the first one is a bit.... confusing
NO2+ doesnt have a resonant structure
no matter how u move those 4 bond lines around its the same bonds connected to the same atoms
i thought that since NO2- you can add another electron because ionic charge is negative so there would only be one bonding e opn each O which can make one double bond..
its NO2+ not -
Can you explain that more, I don't quite understand what you mean. Both oxygens are double bonded to the nitrogen atom in the center.
What do you mean by 'add another electron' I think this is what I don't understand the most.
like for lewis strucutres, when drawing them, if you have a lets say a negative polyatomic ion you add electrons depeing on the number of the ionic charge... unless that's wrong
Also @dan815 I wouldn't say that NO2+ doesn't have other resonance structures, I just drew two of them out right here: |dw:1444198406616:dw| It's just that the contributions of the wave function there is much smaller... (Don't worry about this part @dareintheren )
oh i see so the prob of that state is much lower?
would the prob of that go up with more kinetic energy in your substance
@dareintheren Can you show me how you would draw the Lewis structure of NO2+ and I'll help you figure it out. @dan815 Yes to the first and yes to the second. >:D
@dan815 At low temperatures the ground state is occupied most so think of \(O=N^+=O\) as being one continuous "box" where the electrons can travel freely between O,N,and O and higher energies correspond to 'nodes' in the box which are lower probability since they're higher energy (although we can tunnel to them so they're not zero probability). Now when you heat stuff up (increase average kinetic energy) the electrons don't move to higher energy states exactly. The relative distribution actually becomes more even. So at infinite temperatures you will actually have all states equally populated, which is actually a pretty useful approximation for high temperatures with finite states (not all solutions to the Schrodinger equation have infinite solutions, for instance an electron has only two spin states, +1/2 and -1/2). --- Ok back to @dareintheren Let me see what you've drawn here give me a sec.
oh so really when they are talking about like these electrons jumping down from higher energy levels and releasing photos, thats actually when these electrons are going back to more stable resonant structures?
i wonder how this photon is really created, like what is this strange thing that must come out of no where
@dareintheren Are you in organic chemistry or general chemistry right now and are you a chemistry major? I want to make sure I give you the right answer to this question cause there are varying levels of difficulty and I want to make sure you can understand it cause I think I need to explain this to you better since I think you have most of the idea correct.
@dan815 Make a new question with that photon thing in it and I can describe a little but I don't wanna hijack this dude's question anymore lol.
I'm in general, and nope not chem major
Has your class spoken at all about sigma or pi bonds? (sorry it's been a while and I forget if this is considered advanced or not)
ummm no not yet, but i did read up on it, sigma is end to end and pi is side to side.... forgot what they were for .... overlap/ hyrbdizaton right?
Yeah, exactly! A sigma bond is just a single bond by itself, but when you draw a double bond (just two lines) that actually represents a single sigma bond and a single pi-bond. So I guess I'll lay out how I thought to do this, I looked at the periodic table and found the total number of electrons that should be in the valence shells of the atoms, 5 for Nitrogen 6 for Oxygen, and there are two so 12 in all This gives us a total of 5+12=17 electrons to place on our molecule. But electrons are negative and we see that the molecule is NO2+ so we need to remove an electron, so we are only placing 16 electrons instead of 17. The positive charge comes from the protons in the nuclei of the atoms. So now we can start placing our electrons, generally electrons pair up, so instead of placing electrons, we can place electron pairs. 16/2=8 so 8 is just a more manageable number. Now we know that N is in the center, so let's go ahead and draw it out and start placing: |dw:1444199679236:dw| Since a bond is just a pair of electrons shared and we have to connect them, I used up 2, and we have 6 pairs left to place. Since the molecule is symmetric, we can imagine splitting up the pairs three to each side and there are two ways we might try to do this: |dw:1444199775696:dw| However the molecule on the left has Nitrogen with an incomplete octet, no good! So we are stuck with the right structure. From here we can see that by looking at formal charges that the plus charge goes on Nitrogen. So this is how I would solve this problem I hope this helps or if you want clarification I know this gets a little wordy there in the middle but I think your way of doing it is not quite right so I figured I'd just walk through all the steps.
However the molecule on the left has Nitrogen with an incomplete octet, no good! So we are stuck with the right structure , do you mean nitrogen doesn't have a full octet?
oh yes it is nvm
Yeah all good, specifically you can tell because it has 4, only 2 from each of the bonds it has. :)
but doesn't N have a lone pair? do it only needs one more bonding pair doesn't it?
It would have a lone pair if we placed one on it, but we didn't, we couldn't actually, even if we had did it lop-sided like this:|dw:1444200975612:dw| See, now the oxygen on the left and the nitrogen have a complete octet, however the oxygen on the right has only 6 valence electrons. So this is no good! We have to push that lone electron pair on nitrogen into a bond to give oxygen a complete octet.
|dw:1444201104385:dw| Maybe this makes it clear, this was the only possible way we could have placed these 16 electrons (8 pairs of electrons) on the molecule so that all atoms have a complete octet. 2 pairs on the left oxygen, 4 pairs into the bonds, and 2 more pairs on the oxygen on the right for a total of 8 electron pairs. Now you're probably used to seeing Nitrogen with a lone pair, in this case it wasn't possible to do that and obey the octet rule. Because of that, Nitrogen has a + charge on it since one of the electrons is being shared with the Oxygen. Keep asking questions if this doesn't seem to answer your question or make sense to you, I have gotten so used to this stuff that I might be skipping over something that is obvious to me that wasn't always obvious to me that I'm taking for granted haha. |
Note on the quasi-proper direct image with value in a Banach analytic set.
We give a rather simple proof of the generalization of Kuhlmann’s quasi-proper direct image theorem to the case of a map with values in a Banach analytic set. The proof uses a generalization of the Remmert-Stein’s theorem to this context.
32 H 02 - 32 K 05 - 32 C 25.
Quasi-proper map. Quasi-proper direct image. Banach analytic set.
The aim of the present Note is to give a rather simple proof for the generalization of Kuhlmann’s quasi-proper direct image theorem to the case of a map with values in Banach analytic set. N. Kuhlmann (and D. Mathieu in the Banach case, see [K.64], [K.66] and [M.00]) proved this direct image result for a ‘‘semi-proper’’ holomorphic map; this is a weaker hypothesis than quasi-proper, so their result is better than the one presented here. But the quasi-proper case fits well with the situation we are mainly interested in the study of f-analytic family of cycles (see [B.08], [B.13] and [B.15]222Nevertheless we prove the semi-proper case of the direct image theorem with values in , the space of finite type dimensional cycles in the complex space in [B.15] th. 2.3.2.). Our simpler argument via the generalization of the Remmert-Stein’s theorem does not work for a semi-proper map (see the remark following proposition 2.0.2). In the appendix we give an easy proof of the Remmert’s direct image theorem in the proper finite case (with values in an open set of a Banach space) which is used in our proof to make this Note self-contained modulo the theorem III 7.4.1. in [B-M 1].
1 A simple proof of Kuhlmann’s quasi-proper
direct image theorem
In order to show the strategy of proof for the generalization of Kuhlmann’s theorem with values in an open set of a Banach space, we shall begin by a simple proof of the finite dimensional case using the “usual” Remmert-Stein’s theorem.
Let a quasi-proper holomorphic map between irreducible complex spaces. Then is a closed analytic subset of .
The first point is to prove that is closed in . Let be a sequence in and suppose that converges to a point . Let be a neighbourhood of in and a compact subset in such that for any each irreducible component of meets . Then we may assume that for large enough lies in and so that is chosen in . So, up to pass to a sub-sequence we may assume that the sequence converges to . Then the continuity of implies hat . So is closed.
Consider now the integer and define the set
We want to show that is a closed analytic subset in . To show that is closed consider a sequence in converging to a point in . Let be a neighbourhood of in and a compact subset in such that for any each irreducible component of meets . Then for each large enough in order that lies in , we may find a point such that and with . So, up to pass to a subsequence we may assume that the sequence converges to . We have ; consider now a scale around adapted to . Then the map is proper and finite if is an open neighbourhood of which is small enough. For each the image of contains . So it contains also and the fibre has dimension in . So is in and is closed.
To prove the analyticity of we argue as above and we remark that for any holomorphic function with zero set we have for when is in . This will give (infinitely many) holomorphic equations for in using a Taylor expansion in of the holomorphic function . This proves the analyticity of . Remark that has dimension where .
Now we shall prove the theorem by induction on the integer . Note that, for the result is already proved. So let assume that the result is proved for and we shall prove it for . If the closed analytic set introduced above is equal to , the result is clear because . If not is an open dense set in because is irreducible and also is also an open dense set in . Moreover the map induced by is again quasi-proper. The induction hypothesis gives that is a closed analytic subset in and it has dimension bigger or equal to . As has dimension we may apply Remmert-Stein theorem to obtain that the closure of in is again an analytic set. This conclude the proof as we have , because we know that is closed.
The semi-properness of would be enough to get the fact that is closed.
The restriction of a quasi-proper map to a closed saturated analytic subset in 333A subset of is saturated for if implies . is again quasi-proper. This is also true for a semi-proper map.
When is quasi-proper to have quasi-properness for restricted to closed analytic subset it is enough for to be an union of some irreducible components of for .
This property of is not enough, in general, in the case of a semi-proper map . This is precisely what happen for the subset introduced below or in the proposition 2.0.2.
2 The case with values in a Banach analytic set
Let be a holomorphic map from a reduced complex space to a Banach analytic set . We say that is quasi-proper when for each point there exists a neighbourhood of in and a compact set in such that, for each , each irreducible component of meets .
Note that quasi-proper implies semi-proper444Recall that a continuous map between Hausdorff topological spaces is semi-proper if for any there exists a neighbourhood of in and a compact set in such that for any the fibre meets . This is a topological notion contrary to the quasi-properness which asks that is a complex space and that the fibres of are (closed) analytic subsets in (but no complex structure in needed on ). as the definition above implies that . In particular this condition implies that is closed in . So our result is local on and it is enough to prove the Kulmann’s theorem with value in an open set of a Banach space.
Let be a holomorphic map from a reduced and irreducible complex space to a Banach analytic set . We assume that is quasi-proper. Let , and define
Then is a closed analytic subset in which is finite dimensional (so is a reduced complex space embedded in ; see the theorem III 7.4.1 de [B-M 1]).
This result is not true in general for a semi-proper holomorphic map:
Let be an infinite connected cover of a complex manifold of dimension . Let and let be a relatively compact open set containing such that admit a continuous section where is a relatively compact open set in . Let be a sequence of points in with limit and let be a discrete sequence in such that for each . Let be the blow-up of at each point and put . Then is semi-proper and the subset of where the fibre of has dimension is exactely the subset which is not closed in .
First step of the proof: is closed in .
Let be a sequence in which converges to a point in . Let be an open neighbourhood of and a compact subset of such that for any each irreducible component of meets . For we have , and if is a dimensional irreducible component of , the intersection is not empty, and we may choose some . Up to pass to a sub-sequence, we may also assume that the sequence converges to a point . Of course we shall have . Choose now a scale such that is in and such that . This is possible because we know that . Then, up to shrink around , we may assume that for any we have . This means that for the scale is adapted to and with .
Define on the open set the holomorphc map
where is the projection and where is the open set in defined by the condition .
Second step: is a closed map with finite fibres.
The finiteness of fibres is obvious because for the intersection is empty, so is a compact analytic subset in a polydisc, so a finite set.
To show the closeness of , choose a closed set in and a sequence in converging to a point , and let be a sequence in such that . As is compact, we may assume, up to pass to a sub-sequence, that the sequence converges to some . But the limit of is in by assumption and this implies that is in . As it cannot be in because is in , lies in and so is in . This proves the closeness of .
Now the Remmert’s theorem in the proper finite case, but with values in a Banach analytic set555So we have to use the proper case with finite fibres of the Remmert’ theorem with values in an open set of a complex Banach space to prove this proposition. To make this Note self-contained (modulo the theorem III 7.4.1. in [B-M 1]) we give a simple proof of this case in the appendix (section 4). See also [B-M 1] chapter III for the general case., applies and shows that the image of is a reduced complex space. In this case it is clear that the cardinal of the fibres is locally bounded, so is bounded up to shrink and .
So, up to pass to a subsequence, we may assume that is constant equal to . Then it is easy to see, again up to pass to a sub-sequence and to shrink , that the sequence converges to a multiform graph. This implies that has dimension , so is in .
We shall denote the image of .
We shall prove now that, assuming that we choose small enough around the given point in , the set
is a closed analytic subset in . Remark that is contained in but it may be smaller that because the fibre of a point may be of dimension via a component of which does not meet .
Up to shrink and we may assume that is a closed analytic subset of an open set in some Banach space, and that we have a holomorphic map
where is a Banach space, such that the associated holomorphic map
satisfies . Then it is clear that we have .
Now remark that is a closed analytic subset of which is finite dimensional. So this implies that is also finite dimensional and we have .
If we cover the compact set by finitely many scales as above, we obtain that is locally a finite union of such as above and then is a finite dimensional analytic set of dimension near the point ; and, as we already know that is closed, it is a closed finite dimensional analytic set of dimension in .
Let a quasi-proper holomorphic map between an irreducible complex space and a Banach analytic set . Then f(M) is a closed analytic subset in S which is locally finite dimensional .
We shall prove the theorem by induction on the integer defined in the previous proposition. Note that for a given , we have always as we assume irreducible.
The case is clear because in this case we have .
Assume the theorem proved when for some and we shall prove it for .
From the previous steps we know that is a closed analytic subset in and that it has finite dimension . Let us consider now the map
induced by . It is clearly quasi-proper and for this map we have . So, by the induction hypothesis, the image of is a closed analytic subset in which is irreducible of finite dimension
Now we want to apply the Remmert-Stein theorem to conclude. This is clear in the case where is a finite dimensional complex space because the dimensions satisfy the desired inequality, but we want now to treat the case where is a Banach analytic set. As the problem is local, we may replace by a open set in a Banach space, and we apply the generalization of Remmert-Stein theorem obtained in the next section to conclude.
3 The Remmert-Stein’ theorem in a Banach space
Let an open set in a Banach space and let be a closed analytic susbset of dimension . Let a closed irreducible analytic subset in of finite dimension . Then is a closed irreducible analytic subset in of finite dimension equal to .
Here is a first step of the proof of this theorem.
Let be a closed analytic subset in an open set in a Banach space . Assume that has finite pure dimension and that is countable at infinity. Let and . Assume also that a dimensional linear subspace containing is given. Then there exists a linear closed codimension subspace in containing and , such is transversal to at and that the set is discrete and countable.
We shall make an induction on . The case is trivial. So let assume that the case is proved and consider of pure dimension . Choose for each irreducible component of a point which is not equal to 666As an irreducible component is not equal to .. Now choose an hyperplane containing and and such that is not in for each , and that is transversal to at . Such a hyperplane exists thanks to Baire’s theorem. Then is purely -dimensional and contains . The induction hypothesis gives us a co-dimension subspace containing and , transversal at to , such that is discrete and countable. Now the co-dimension of is exactly equal to and is transversal to at . This completes the proof.
Proof of the theorem 3.0.1.
We follow the proof of the proposition II 4.8.3 case (i) given in [B-M 1] but with an ambiant infinite dimensional complex Banach space.
The theorem 3.0.1 follows from this result as in the finite dimensional case.
First, the following point is not completely obvious : the local compactness of .
Of course and are locally compact by assumption. So it is enough to show that this union is locally compact near a point . Remark first that we have
and is closed in . So if we consider small enough, we have and is a compact set. Now we want to show that is compact. So it is enough to show that any sequence in admits a converging sub-sequence to a point in . Let
If goes to for some sub-sequence, then there is a Cauchy sub-sequence converging to some . If for all large enough , then the sequence is contained in the subset
This subset of is compact, because it is closed and a sequence in it cannot approach neither (as , any point satisfies ) nor , and is locally compact (so any discrete sequence has to approach the boundary of ).
Assume that has dimension with . For a fixed smooth point, let be a sub-manifold through containing (we use here the theorem III 7.4.1 in [B-M 1]) and choose a co-dimension plane transversal to at meeting some point in and such that is discrete.
Up to shrink and taking a small ball with center in , we may assume that an open neighbourhood of in is isomorphic to the product . And from our construction we have which is discrete.
We can choose two small balls in with center contained in such that and then we may find an open neighbourhood of in in order that the projection on of the set is proper (this means closed with compact fibres) and that its restriction to has finite fibres.
The choice of two arbitrary small balls with the first condition is easy because we know that
is discrete : the distances to of the points in are in a discrete subset in and we can choose with arbitrary small such that avoids these values. But the subset is compact and then, as is empty, the subset is compact. Then there exists an open neighbourhood of in such that the projection of on is proper :
Let be a compact neighbourhood in of the compact set (remember that we proved that is locally compact). Then for small enough the distance of a point in to the closed set is bounded below by a positive number, and so there exists an open neighbourhood of in such that does not meet . This is enough to prove the claim thanks to the following simple remark, as because .
Any compact analytic subset in is finite: assume that is a connected component of such a compact analytic subset. As compactness implies that there are at most finitely many , it is enough to prove that has at most one point.
Assume that are two points in . Choose a continuous linear form on such that . As the map is holomorphic, it has to be constant (maximum principle and connectness). Conclusion : has at most one point.
The end of the proof of the theorem 3.0.1 is now analogous to the end of the proof in the finite dimensional setting (we are in the “easy” case where the dimension of is strictly bigger than the dimension of ; see the remark before the lemma 4.8.7 in [B-M 1] ch.III).
Along the same line, it is not difficult to prove the analog of the Remmert-Stein’s theorem with equality of dimensions, assuming there exists an open set in meeting each irreducible component of and in which is an analytic subset.
In our proof we use in a crucial way the metric in the ambient Banach space. If the extension of this proof to an open set in a Frechet space may be easy, the case of a non metrizable e.l.c.s.s. (for instance a dual of Frechet space ) does not seem clear. Note that the proper case is proved with values in an open set of any e.l.c.s.s. in [B-M 1] ch.III.
4 Appendix: the direct image theorem in the proper finite case
An e.l.c.s.s. is a locally convex topological vector space which is separated and sequentially complete. For the definition we shall use of a holomorphic map on a reduced complex space with values in a e.l.c.s.s. see [B-M 1] ch. III definition 7.1.1.
Note that the general proper case of this result is given in loc. cit. th. III 7.4.3 but the proof of the proper finite case is much more simple.
To have a simple and self-contained proof (modulo the theorem III 7.4.1. in [B-M 1]) of the quasi-proper direct image theorem discussed in this Note, we shall give a easy proof of the direct image theorem in the proper finite case with values in an e.l.c.s.s.(this result has been used in the Banach case in our proof).
This allows to use the proof of the previous section also in the proper case, thanks to the following lemma.
Let a holomorphic map from a reduced complex space to an open set of an e.l.c.s.s. Assume that is closed with compact fibres (so is proper). Then is quasi-proper.
If is not quasi-proper at there exists, for each open neighbourhood of and for each compact in a point and an irreducible component of which does not meet . Now, for a fixed we may choose the points to be distinct. Let be a relatively compact open set in containing the compact fibre and let . Then for each we can find a point which is in . So the point is in the closed set . Consider the subset in . It is a closed set in (and also in ) because if , the point is limit of an ultra-filter of points in , and so is equal to , because the intersection of the for any ultra-filter is reduced to . But as and , this is a contradiction. So the map is quasi-proper at any point .
Let be a holomorphic map of a complex reduced space to an open set in a e.l.c.s.s. Assume that is closed with finite fibres (so proper and finite). Then is a closed analytic subset which is locally of finite dimension in .
As is the locally finite union of the sets , where is the set of irreducible components of because is quasi-proper and finite thanks to the previous lemma, we may assume that is irreducible. Let . As is a closed set in , it is enough to show that is an finite dimensional analytic subset near a point in .
Let be the ambiant vector space of . Choose a finite co-dimensional closed affine subspace containing in the e.l.c.s.s. such that the point is isolated in which is maximal for this property.
Such a exists because it is easy to construct a sequence of closed co-dimension affine subspaces in containing such that the germ of at gives a strictly decreasing sequence of analytic germs as long as is strictly bigger than the finite set . Then at the last step we have the equality of germs in . If is not maximal with this property, replace by a maximal closed affine subspace containing and containing with this property.
Write where is the closed co-dimension vector subspace in directing and let and be the projections.
Then the germ of analytic map is finite by construction. So its image is an analytic germ. If it is not equal to we can find a line through in such that . Then the affine space contradicts the maximality of . So the germ is surjective and we can find an open polydisc with center in such that is a sheeted branch covering. So we have a holomorphic map classifying the fibres of . Composed with the holomorphic map induced by we obtain that is a multiform graph of contained in (see [B-M 1] chapter III section 7.2). As we have in a neighbourhood of , we conclude that is a finite dimensional analytic subset in in a neighbourhood of .
[B.08] Barlet, D. Reparamétrisation universelle de familles f-analytiques de cycles … Comment. Helv. 83 (2008), pp. 869-888.
[B.13] Barlet Daniel, Quasi-proper meromorphic equivalence relations, Math. Z. (2013), vol. 273, pp. 461-484.
[B.15] Barlet, D. Strongly quasi-proper maps and the f-flattening theorem, math-arXiv:1504.01579 (41 pages)
[B-M 1] Barlet, D. et Magnússon, J. Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés, 22. Société Mathématique de France, Paris (2014).
[K.64] Kuhlmann, N. Über holomorphe Abbildungen komplexer Raüme Arch. der Math. 15, (1964) pp. 81-90.
[K.66] Kuhlmann, N. Bemerkungen über holomorphe Abbildungen komplexer Raüme Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedäachtnisfeier K. Weierstrass, (1966) pp.495-522.
[M.00] Mathieu, D. Universal reparametrisation of a family of cycles : a new approach to meromorphic equivalence relations Ann. Inst. Fourier (Grenoble) vol. 50 fasc. 4, (2000) pp. 1155-1189. |
(and what's wrong with the typical meta-analysis)
I've been reading the literature on whether business ethics classes have any effect on student attitudes. The literature has several features that I've come to associate with non-effects in psychology. Another literature like this is the literature attempting to find objective behavioral correlates of the subjective reports of imagery experiences. The same holds for the literature on the relationship between religiosity and moral behavior and the literature on the psychological significance of colored dreaming (back when people thought they mostly dreamed in black and white).
However, the typical review article or quantitative meta-analysis in these fields does not conclude that there is no effect. Below, I'll discuss why.
The features are:
(1.) A majority of studies show the predicted positive effects, but a substantial minority of studies (maybe a third) show no statistically significant effect.
(2.) The studies showing positive vs. negative effects don't fit into a clearly interpretable pattern -- e.g., it's not like the studies looking for X result almost all show effects while those looking for Y result do not.
(3.) Researchers reporting positive effects often use multiple measures or populations and the effect is found only for some of those measures or populations (e.g., for women but not men, for high-IQ subjects but not for low-IQ subjects, by measure A but not by measure B) -- but again not in a way that appears to replicate across studies or to have been antecedently predicted.
(4.) Little of the research involves random assignment and confounding factors readily suggest themselves (e.g., maybe participants with a certain personality or set of interests are both more likely to have taken a business ethics class and less likely to cheat in a laboratory study of cheating, an association really better explained by those unmeasured differences in personality or interest rather than by the fact that business ethics instruction is highly effective in reducing cheating).
(5.) Much of the research is done in a way that seems almost to beg the participants to confirm the hypothesis (e.g., participants are asked to report their general imagery vividness and then they are given a visual imagery task that is a transparent attempt to confirm their claims of high or low imagery vividness; or a business ethics professor asks her students to rate the wrongness of various moral scenarios, then teaches a semester's worth of classes, then asks those same students to rate the wrongness of those same moral scenarios).
(6.) There is a positive hypothesis that researchers in the area are likely to find attractive, with no equally attractive negative hypothesis (e.g., that subjective reports of imagery correlate with objective measures of imagery; or that business ethics instruction of the sort the researcher favors leads students to adopt more ethical attitudes).
The really striking thing to me about these literatures is that despite what seems likely some pretty strong positive-effect biases (features 4-6), still researchers in these areas struggle to show a consistent pattern of statistical significance.
In my mind this is the picture of a non-effect.
The typical meta-analysis will report a real effect, I think, for two reasons, one mathematical and one sociological. Mathematically, if you combine one-third null-effect studies with two-thirds positive-effect studies, you'll typically find a statistically significant effect (even with the typical "file-drawer" corrections). And sociologically, these reviews are conducted by researchers in the field, often including their own work and the work of their friends and colleagues. And who wants to devalue the work in their own little academic niche? See, for example, this meta-analysis of the business ethics literature and this one of the imagery literature.
In a way, the mathematical conclusion of such meta-analyses is correct. There is a mathematically discoverable non-chance effect underneath the patterns of findings -- the combined effects of experimenter bias, participants' tendency to confirm hypotheses they suspect the researcher is looking for, and unmeasured confounding variables that often enough align positively for positively-biased researchers to unwittingly take advantage of them. But of course, that's not the sort of positive relationship that researchers in the field are attempting to show.
For fun (my version of fun!) I did a little mock-up Monte Carlo simulation. I ran 10,000 sample experiments predicting a randomly distributed Y from a randomly distributed X, with 60 participants in each control group and 60 in each treatment group, adding two types of distortion: First, two small uncontrolled confounds in random directions (average absolute value of correlation, r = .08), and second a similarly small random positive correlation to model some positive-effect bias. (Both X and Y were normally distributed with a mean of 0 and a standard deviation of 1. The confounding correlation coefficients were chosen by randomly selecting r from a normal distribution centered at 0 with a standard deviation of 0.1; for the one positive-bias correlation, I took the absolute value.)
Even with only these three weak confounds, not all positive, and fairly low statistical power, 23% of experiments had statistically significant results at a two-tailed p value of < .05 (excluding the 5% in which the control group correlation was significant by chance). If we assume that each researcher conducts four independent tests of the hypothesis, of which only the "best", i.e., most positive, correlation is pursued and emphasized as the "money" result in publication, then 65% of researchers will report a statistically significant positive result, the average "money" correlation will be r = .28 (approaching "medium" size), and no researcher will emphasize in publication a statistically significant negative result.
Yeah, that's about the look of it.
(Slightly revised 10:35 AM.)
Update August 8th:
The Monte Carlo analysis finds 4% with a significantly negative effect. My wife asks: Wouldn't researchers publish those effects too, and not just go for the strongest positive effect? I think not, in most cases. If the effect is due to a large negative uncontrolled confound, a positive-biased researcher, prompted by the weird result, might search for negative confounds that explain the result, then rerun the experiment in a different way that avoids those hypothetized confounds, writing off the first as a quirky finding due to bad method. If the negative effect is due to chance, the positive-biased author or journal referee, seeing (say) p = .03 in the unpredicted direction, might want to confirm the "weird result" by running a follow-up -- and being due to chance it will likely be unconfirmed and the paper unpublished. |
is never proved or established, but is possibly disproved, in the course of experimentation. Fundamentals of Working with Data Lesson 1 - An Overview of Statistics Lesson 2 - Summarizing Data Software - Describing Data with Minitab II. required Name required invalid Email Big Data Cloud Technology Service Excellence Learning Data Protection choose at least one Which most closely matches your title? - select - CxO Director Individual Manager Remember to set it up so that Type I error is more serious. \(H_0\) : Building is not safe \(H_a\) : Building is safe Decision Reality \(H_0\) is true \(H_0\) is this content
Orangejuice is guilty Here we put "the man is not guilty" in \(H_0\) since we consider false rejection of \(H_0\) a more serious error than failing to reject \(H_0\). To lower this risk, you must use a lower value for α. Example 2 Hypothesis: "Adding fluoride to toothpaste protects against cavities." Null hypothesis: "Adding fluoride to toothpaste has no effect on cavities." This null hypothesis is tested against experimental data with a An example of a null hypothesis is the statement "This diet has no effect on people's weight." Usually, an experimenter frames a null hypothesis with the intent of rejecting it: that
The statistical practice of hypothesis testing is widespread not only in statistics, but also throughout the natural and social sciences. When we don't have enough evidence to reject, though, we don't conclude the null. If the significance level for the hypothesis test is .05, then use confidence level 95% for the confidence interval.) Type II Error Not rejecting the null hypothesis when in fact the
Examples: If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, but men predisposed to heart disease have a mean Reply Kanwal says: April 12, 2015 at 7:31 am excellent description of the suject. So that in most cases failing to reject H0 normally implies maintaining status quo, and rejecting it means new investment, new policies, which generally means that type 1 error is nornally What Are Some Steps That Scientists Can Take In Designing An Experiment To Avoid False Negatives Retrieved 10 January 2011. ^ a b Neyman, J.; Pearson, E.S. (1967) . "On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference, Part I".
If the medications have the same effectiveness, the researcher may not consider this error too severe because the patients still benefit from the same level of effectiveness regardless of which medicine Type 1 Error Psychology Screening involves relatively cheap tests that are given to large populations, none of whom manifest any clinical indication of disease (e.g., Pap smears). A typeI occurs when detecting an effect (adding water to toothpaste protects against cavities) that is not present. read this post here The lowest rate in the world is in the Netherlands, 1%.
Privacy Legal Contact United States EMC World 2016 - Calendar Access Submit your email once to get access to all events. Type 1 Error Calculator Paranormal investigation The notion of a false positive is common in cases of paranormal or ghost phenomena seen in images and such, when there is another plausible explanation. We could decrease the value of alpha from 0.05 to 0.01, corresponding to a 99% level of confidence. Diego Kuonen (@DiegoKuonen), use "Fail to Reject" the null hypothesis instead of "Accepting" the null hypothesis. "Fail to Reject" or "Reject" the null hypothesis (H0) are the 2 decisions.
This kind of error is called a type I error, and is sometimes called an error of the first kind.Type I errors are equivalent to false positives. http://boards.straightdope.com/sdmb/showthread.php?t=648404 required Name required invalid Email Big Data Cloud Technology Service Excellence Learning Data Protection choose at least one Which most closely matches your title? - select - CxO Director Individual Manager Probability Of Type 1 Error Type 2 would be letting a guilty person go free. Probability Of Type 2 Error This is as good as it gets in an Internet forum! :-) living_in_hell View Public Profile Find all posts by living_in_hell #12 04-17-2012, 10:16 AM Pleonast Charter Member
The errors are given the quite pedestrian names of type I and type II errors. news Thank you very much. The hypotheses being tested are: The man is guilty The man is not guilty First, let's set up the null and alternative hypotheses. \(H_0\): Mr. The value of alpha, which is related to the level of significance that we selected has a direct bearing on type I errors. Type 3 Error
You set out to prove the alternate hypothesis and sit and watch the night sky for a few days, noticing that hey…it looks like all that stuff in the sky is Since it's convenient to call that rejection signal a "positive" result, it is similar to saying it's a false positive. Connection between Type I error and significance level: A significance level α corresponds to a certain value of the test statistic, say tα, represented by the orange line in the picture have a peek at these guys heavyarms553 View Public Profile Find all posts by heavyarms553 #10 04-15-2012, 01:49 PM mcgato Guest Join Date: Aug 2010 Somewhat related xkcd comic.
Type II error can be made if you do not reject the null hypothesis. Type 1 Error Example Problems ISBN0-643-09089-4. ^ Schlotzhauer, Sandra (2007). After being deeply immersed in the world of big data for over 20 years, he shows no signs of coming up for air.
Assume also that 90% of coins are genuine, hence 10% are counterfeit. You've committed an egregious Type II error, the penalty for which is banishment from the scientific community. *I used this simple statement as an example of Type I and Type II Difference Between a Statistic and a Parameter 3. Power Of A Test Statistics Help and Tutorials by Topic Inferential Statistics What Is the Difference Between Type I and Type II Errors?
This is consistent with the system of justice in the USA, in which a defendant is assumed innocent until proven guilty beyond a reasonable doubt; proving the defendant guilty beyond a So you WANT to have an alarm when the house is on fire...because you WANT to have evidence of correlation when correlation really exists. The probability of a type II error is denoted by the beta symbol β. http://explorersub.com/type-1/type-1-and-2-error-examples.php A common example is relying on cardiac stress tests to detect coronary atherosclerosis, even though cardiac stress tests are known to only detect limitations of coronary artery blood flow due to
Summary Type I and type II errors are highly depend upon the language or positioning of the null hypothesis. Type II error A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. Please try again. Failing to reject H0 means staying with the status quo; it is up to the test to prove that the current processes or hypotheses are not correct.
The analogous table would be: Truth Not Guilty Guilty Verdict Guilty Type I Error -- Innocent person goes to jail (and maybe guilty person goes free) Correct Decision Not Guilty Correct It might have been true ten years ago, but with the advent of the Smartphone -- we have Snopes.com and Google.com at our fingertips. It is also good practice to include confidence intervals corresponding to the hypothesis test. (For example, if a hypothesis test for the difference of two means is performed, also give a For P(D|B) we calculate the z-score (225-300)/30 = -2.5, the relevant tail area is .9938 for the heavier people; .9938 × .1 = .09938.
A low number of false negatives is an indicator of the efficiency of spam filtering. Last edited by njtt; 04-15-2012 at 11:14 AM.. If a test with a false negative rate of only 10%, is used to test a population with a true occurrence rate of 70%, many of the negatives detected by the A problem requiring Bayes rule or the technique referenced above, is what is the probability that someone with a cholesterol level over 225 is predisposed to heart disease, i.e., P(B|D)=?
Null Hypothesis Type I Error / False Positive Type II Error / False Negative Display Ad A is effective in driving conversions (H0 true, but rejected as false)Display Ad A is Plus I like your examples. Common mistake: Neglecting to think adequately about possible consequences of Type I and Type II errors (and deciding acceptable levels of Type I and II errors based on these consequences) before The consistent application by statisticians of Neyman and Pearson's convention of representing "the hypothesis to be tested" (or "the hypothesis to be nullified") with the expression H0 has led to circumstances
Medicine Further information: False positives and false negatives Medical screening In the practice of medicine, there is a significant difference between the applications of screening and testing. |
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hence vv = 2v, or v = of the wheel equal only to
3v, and v=V, or the velocity of the velocity of the water.
To determine the Form and Dimensions of Gunpowder Magazines.
In the practice of engineering, with respect to the erection of powder magazines, the exterior shape is usually made like the roof of a house, having two sloping sides, forming two inclined planes, to throw off the rain, and meeting in an angle or ridge at the top; while the interior represents a vault, more or less extended, as the occasion may require; and the shape, or transverse section, in the form of some arch, both for strength and commodious room, for placing the powder barrels. It has been usual to make this interior curve a semicircle. But, against this shape, for such a purpose, I must enter my decided protest; as it is an arch the farthest of any from being in equilibrium in itself, and the weakest of any, by being unavoidably much thinner in one part than in others. Besides, it is constantly found, that after the centering of semicircular arches is struck, and removed, they settle at the crown, and rise up at the flanks, even with a straight horizontal form at top, and still much more so in powder magazines with a sloping roof; which effects are exactly what might be expected from a contemplation of the true theory of arches. Now this shrinking of the arches must be attended with other additional bad effects, by breaking the texture of the cement, after it has been in some de gree dried, and also by opening the joints of the voussoirs at one end. Instead of the circular arch therefore, we shall in this place give an investigation, founded on the true principles of equilibrium, of the only just form of the interior, which is properly adapted to the usual sloped roof.
For this purpose, put a DK the thickness of the arch at the top, x = any absciss DP of the required arch ADCM, U KR the corresponding absciss of the given exterior line KI, and y = PC RI their equal ordinates. Then by the principles of arches, in my tracts on that subject, it is found that cr or to a + x
ÿ x − x ÿ
or Q X supposing a constant quantity, and where a is some certain quantity to be determined hereafter. But KR or u isty, if t be put to denote
the tangent of the given angle of elevation KIR, to radius 1,
and then the equation is w= a + x =
: but at D the value of w isa, and at D being parallel to KI; therefore the
= 0, the curve correct fluent is
the correct fluent of which gives ya x hyp. log. of w+ √(x2-22)
Now, to determine the value of a, we are to consider that when the vertical line cr is in the position AL or MN, then w= ct becomes = AL or MN the given quantity e suppose, and y AQ or aмb suppose, in which position the c+(c2-a2) Last equation becomes bax hyp. log. (^2-22); and hence it is found that the value of the constant quantity vo, is which being substituted for it, in
the above general value of y, that value becomes
from which equation the value of the ordinate PC may always be found, to every given value of the vertical cı.
But if, on the other hand, PC be given, to find CI, which will be the more convenient way, it may be found in the following manner: Put A = log. of a, and c = xlog. of G+ √(e2-a2) ; then the above equation gives cy + A= log. of we + √(x2-a); again, put = the number whose log. is cy + A; then n = w + √(w2 - a2); and hence w
Now, for an example in numbers, in a real case of this
nature, let the foregoing figure represent a transverse vertical section of a magazine arch balanced in all its parts, in which the span or width AM is 20 feet, the pitch or height na is 10 feet, thickness at the crown DK = 7 feet, and the angle of the ridge LKN 112° 37′, or the half of it LKD = 56° 18′, the complement of which, or the elevation KIR, is 33° 41', the tangent of which is, which will therefore be the value of t in the foregoing investigation. The values of the other letters will be as follows, viz, DK=α=7; aq=b=10; DQ=h=10; AL=c=103: 3; A=log. of 7=8450980;
log. of 2.562070408591; cy + A = ·0408591y + 8450980 log. of n. From the general equation then, viz,
a2 + n2
+, by assuming y successively
equal to 1, 2, 3, 4, &c, thence finding the corresponding values of cy + ▲ or *0408591y + ·8450980, and to these, as common logs. taking out the corresponding natural numbers, which will be the values of n; then the above theorem will give the several values of or ci, as they are here arranged in the annexed table, from which the figure of the curve is to be constructed, by thus finding so many points in it.
Otherwise. Instead of making n the number of the log. cy + A, if we put m = the natural number of the log. `w + √(w2 — n2) cy only; then m = and am—w=√(w2 — a2), or by squaring, &c, am2-2amw+w2 = w2 — a2, and hence xa; to which the numbers being applied, the very same conclusions result as in the foregoing calculation and table.
m2 + 1
To construct Powder Magazines with a Parabolical Arch.
It has been shown, in my tract on the Principles of Arches of Bridges, that a parabolic arch is an arch of equilibration, when its extrados, or form of its exterior covering, is the very same parabola as the lower or inside curve. Hence then a parabolic arch, both for the inside and outer form, will be
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are circumstances which it were welon with any precision. A koum a other bulubky a sal be said, have mit Huset 2., tater til Geichstat Luca, That ingenious philosopher Da stupa kisprin showed that by die ingos parce o quitpower tity of elastie aur disengage wird, wuel collect in the space only super by the powder before it was footy was found to be dar 100 lines stronger that the weight or He then heated elasticity of the common atmospuene air. the same parcel of air IT the degree of red hot iron, auf found it in that temperature to be about 4 times as strong as before; whence be inferred, that the first strength of the sho |
Giuseppe Peano was born on August 27, 1858 at a farm near the village of Spinetta in northeastern Italy.
He graduated in 1880 with 'High Honours' and began his teaching career. Peano was a University assistant between 1880 and 1882. First to Enrico D'Ovidio and then to Angelo Genocchi (the chair of Infinitesimal Calculus). In 1881 Peano published his first paper. In the course of his life, Peano had over two hundred papers and books published (most of them on mathematics).
By 1882, due to Genocchi's ill health, Peano was in charge of the Infinitesimal Calculus course where the students made fun of him because of an inability to pronounce the letter 'r'.
His first major work a text book on calculus, credited to Genocchi, was published in 1884. On the July 27, 1887 he married Carola Crosio. The following year Peano's father died. Peano also published his first book dealing with mathematical logic. This book first uses the symbols for Union and Intersection of sets as are now used today.
In 1889 Peano is appointed 'Professor first class' at the Royal Military Academy in Turin where he also teaches. Peano's famous space-filling curve appears in a publication of his as a counter-example in 1890. He used it to show that a continuous curve cannot always be enclosed in an arbitrarily small region. In the same year he is also appointed 'Extraordinary Professor of Infinitesimal Calculus' at Turin University.
The following year he becomes a member of 'The Academy of Science' in Turin. This year (1891) sees Peano begin his ambitious task of creating an 'Encyclopedia of mathematics'. Known as the 'Formulario Project' it was to be a collection of all known formulas and theorems of mathematical science using a standard notation invented by Peano.
In 1895 he is promoted to 'Ordinary Professor' at Turin University.
The First International Conference of Mathematics is held in Zürich in 1897 where Peano is a key participant, presenting a paper on mathematical logic. At this time Peano starts to become increasingly occupied with 'Formulario' to the detriment of his other work.
In 1898 he presents a note to the 'Academy' about binary numeration and its ability to be used to represent the sounds of languages. He also becomes so frustrated with publishing delays (due to his demand that formulas be printed on one line) that he purchases a printing press.
Paris is the venue for the Second International Conference of Mathematics in 1900. The conference is preceded by the First International Conference of Philosophy where Peano is a member of the Patronage Committee. He presents a paper which poses the question of correctly formed definitions in mathematics (ie How do you define a definition?). This becomes one of Peano's main philosophical interests for the rest of his life. Peano meets Bertrand Russell and gives him a copy of 'Formulario'. Russell is so struck by Peano's innovative logical symbols that he leaves the conference for home where he studies Peano's text. Peano's followers present papers (using Peano's teachings) at the mathematics conference but Peano does not. A resolution is raised on the formation of an International Auxiliary Language that will make the spread of new mathematical (and commercial) ideas easier. Peano is in full support.
By 1901 Peano is at the peak of his mathematical career. He has made advances in the areas of analysis, foundation and logic. Peano has made many contributions to the teaching of calculus. He also contributed to the fields of differential equations and vector analysis. Peano played a key role in the axiomatization of mathematics and was a leading pioneer in the development of mathematical logic. In recognition of this Peano is made a "Knight of the Order of Saints Maurizio and Lazzaro". Peano has by this stage become heavily involved with the 'Formulario' project and his teaching begins to suffer. In fact he becomes so determined to teach his new mathematical symbols that the calculus in his course is neglected. As a result he is dismissed from the Royal Military Academy but retains his post at Turin University.
1903 sees Peano announce his work on an international auxiliary language called "Latino sine flexione" (Latin without flexions). This now becomes an important project (along with finding contributants for 'Formulario').
In 1905 Peano is made a "Knight of the Crown of Italy" and is also elected a Corresponding Member of the "Accademia dei Lincei" in Rome, the highest honour for an Italian scientist (Galileo was also a member in his time).
1908 is the year in which the final (5th) edition of 'Formulario Project', titled "Formulario Mathematico", is published. It contains 4200 formulas and theorems, all completely stated and most of them proved. The book received little use and most of the content was dated by this time. The comments and examples were written in "Latino sine flexions" which put most mathematicians off using the book. However, it was (is) a significant piece of mathematical literature. Also in this year Peano takes over the Chair of Higher Analysis at Turin. This lasts for only two years.
In 1910, Peano's mother dies.
For the next twenty-four years, Peano divides (not equally however) his energies between developing and promoting his and other artificial languages (he became a revered member of the international auxiliary language movement), teaching, and working on texts aimed for secondary schooling (including a dictionary of mathematics). He used his membership of the Accademia dei Lincei to present papers written by friends and colleagues who were not members (the Accademia recorded and published all presented papers given in sessions).
"He [Peano] was a man I greatly admired from the moment I met him for the first time in 1900 at a Congress of Philosophy, which he dominated by the exactness of his mind."
Bertrand Russell 1932
Reference: "Peano: Life and Works of Giuseppe Peano" Hubert C. Kennedy
See also: Peano axioms |
1 Copyright 2008 by the Foundation of the American College of Healthcare Executives 6/11/07 Version 9-1 CHAPTER 9 Time Value Analysis Future and present values Lump sums Annuities Uneven cash flow streams Solving for I and N Investment returns Amortization
2 9-2 Time Value of Money Time value analysis is necessary because money has time value. A dollar in hand today is worth more than a dollar to be received in the future. Why? Because of time value, the values of future dollars must be adjusted before they can be compared to current dollars. Time value analysis constitutes the techniques that are used to account for the time value of money.
3 9-3 Time Lines I% CF 0 CF 1 CF 2 CF 3 Tick marks designate ends of periods. Time 0 is the starting point (the beginning of Period 1); Time 1 is the end of Period 1 (the beginning of Period 2); and so on.
4 9-4 What is the FV after 3 years of a $100 lump sum invested at 10%? -$ % FV =? Finding future values (moving to the right along the time line) is called compounding. For now, assume interest is paid annually.
5 9-5 After 1 year: FV 1 = PV + INT 1 = PV + (PV x I) = PV x (1 + I) = $100 x 1.10 = $ After 2 years: FV 2 = FV 1 + INT 2 = FV 1 + (FV 1 x I) = FV 1 x (1 + I) = PV x (1 + I) x (1 + I) = PV x (1 + I) 2 = $100 x (1.10) 2 = $
6 9-6 After 3 years: FV 3 = FV 2 + I 3 = PV x (1 + I) 3 = 100 x (1.10) 3 = $ In general, FV N = PV x (1 + I) N.
7 9-7 Three Primary Methods to Find FVs Solve the FV equation using a regular (nonfinancial) calculator. Use a financial calculator; that is, one with financial functions. Use a computer with a spreadsheet program such as Excel, Lotus 1-2-3, or Quattro Pro.
8 Nonfinancial Calculator Solution % -$100 $ $ $ $100 x 1.10 x 1.10 x 1.10 = $
9 9-9 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Notes: (1) Set your calculator on P/YR = 1, END. (2) For lump sums, the PMT key is not used. Either clear the calculator before you start or enter PMT = 0.
10 9-10 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ Pv Present value % Rate Interest rate 5 6 $ =100*(1.10)^3 (entered into Cell A6) 7 8 $ =A3*(1+A4)^A2 (entered into Cell A8) 9 10 $ =FV(A4,A2,,-A3) (entered into Cell A10)
11 9-11 What is the PV of $100 due in 3 years if I = 10%? % PV =? $100 Finding present values (moving to the left along the time line) is called discounting.
12 9-12 Solve FV N = PV x (1 + I ) N for PV PV = FV N / (1 + I ) N. PV = $100 / (1.10) 3 = $100(0.7513) = $75.13.
13 Time Line Solution % $75.13 $82.64 $90.91 $100 $ = $ Note that the calculated present value ($75.13), when invested at 10 percent for three years, will produce the starting future value ($100).
14 9-14 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Either PV or FV must be negative on most calculators. Here, PV = Put in $75.13 today, take out $100 after three years.
15 9-15 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ Fv Future value % Rate Interest rate 5 6 $ =A3/(1+A4)^A2 (entered into Cell A6) 7 8 $ =PV(A4,A2,,-A3) (entered into Cell A8) 9 10
16 9-16 Opportunity Cost Rate On the last illustration we needed to apply a discount rate. Where did it come from? The discount rate is the opportunity cost rate. It is the rate that could be earned on alternative investments of similar risk. It does not depend on the source of the investment funds. We will apply this concept over and over in this course.
17 9-17 Opportunity Cost Rate (Cont.) The opportunity cost rate is found (at least in theory) as follows: Assess the riskiness of the cash flow(s) to be discounted. Identify security investments that have the same risk. Why securities? Estimate the return expected on these similarrisk investments. When applied, the resulting PV provides a return equal to the opportunity cost rate. In most time value situations, benchmark opportunity cost rates are known.
18 9-18 Solving for I Assume that a bank offers an account that will pay $200 after five years on each $75 invested. What is the implied interest rate? INPUTS OUTPUT N I/YR PV PMT FV 21.7
19 9-19 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ (75.00) Pv Present value 4 $ Fv Future value % =RATE(A2,,A3,A4) (entered into Cell A8) 9 10
20 9-20 Solving for N Assume an investment earns 20 percent per year. How long will it take for the investment to double? INPUTS OUTPUT N I/YR PV PMT FV 3.8? What is the Rule of 72?
21 9-21 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 $ (1.00) Pv Present value 4 $ 2.00 Fv Future value =NPER(A2,,A3,A4) (entered into Cell A8) 9 10
22 9-22 Types of Annuities Three-Year Ordinary Annuity I% PMT Three-Year Annuity Due PMT PMT I% PMT PMT PMT
23 9-23 What is the FV of a three-year ordinary annuity of $100 invested at 10%? % $100 $100 $ FV = $331
24 9-24 Financial Calculator Solution INPUTS OUTPUT N I/YR PV PMT FV Here there are payments rather than a lump sum present value, so enter 0 for PV.
25 9-25 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ (100.00) Pmt Payment % Rate Interest rate $ =FV(A4,A2,A3) (entered into Cell A8) 9 10
26 9-26 What is the PV of the annuity? % $ $ = PV $100 $100 $100
28 9-28 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ (100.00) Pmt Payment % Rate Interest rate $ =PV(A4,A2,A3) (entered into Cell A8) 9 10
29 9-29 What is the FV and PV if the annuity were an annuity due? % $100 $100 $100??
30 9-30 Switch from End to Begin mode on a financial calculator. Repeat the annuity calculations. First find PVA 3 = $ INPUTS OUTPUT N I/YR PV PMT FV Then enter PV = 0 and press FV to find FV = $
31 9-31 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ (100.00) Pmt Payment % Rate Interest rate 5 6 $ =PV(A4,A2,A3,,1) (entered into Cell A6) 7 8 $ =PV(A4,A2,A3)*(1+A4) (entered into Cell A8) 9 10
32 9-32 Financial Spreadsheet Calculator Solution Solution A B C D Nper Number of periods 3 $ (100.00) Pmt Payment % Rate Interest rate 5 6 $ =FV(A4,A2,A3,,1) (entered into Cell A6) 7 8 $ =FV(A4,A2,A3)*(1+A4) (entered into Cell A8) 9 10
33 9-33 Perpetuities A perpetuity is an annuity that lasts forever. What is the present value of a perpetuity? PMT PV (Perpetuity) =. I? What is the future value of a perpetuity?
35 9-35 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 4 $ 100 Value 1 Year 1 CF Value 1 Year 2 CF Value 1 Year 3 CF 7 (50) Value 1 Year 4 CF $ =NPV(A2,A4:A7) (entered into Cell A10)
36 9-36 Discussion Items Assume the cash flows on the previous slide are the cash flows from an investment. How much would you be willing to pay for these flows? What would be the financial benefit to you if you could buy it for less, say, $500?
37 9-37 Investment Returns The financial performance of an investment is measured by its return. Time value analysis is used to calculate investment returns. Returns can be measured either in dollar terms or in rate of return terms. Assume that a hospital is evaluating a new MRI. The project s expected cash flows are given on the next slide.
38 9-38 MRI Investment Expected Cash Flows (in thousands of dollars) $1,500 $310 $400 $500 $750? Where do these numbers come from?
39 9-39 Simple Dollar Return $1, $310 $400 $ 460 = Simple dollar return $500 $750? Is this a good measure?
40 9-40 Discounted Cash Flow (DCF) Dollar Return 0 8% $1, $310 $400 $500 $ 78 = Net present value (NPV) $750? Where did the 8% come from?
41 9-41 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 $ (1,500) Year 0 CF Value 1 Year 1 CF Year 2 CF Year 3 CF Value 1 Year 4 CF $ 78 =NPV(A2,A4:A7)+A3 (entered into Cell A10)
42 9-42 DCF Dollar Return (Cont.) The key to the effectiveness of this measure is that the discounting process automatically recognizes the opportunity cost of capital. An NPV of zero means the project just earns its opportunity cost rate. A positive NPV indicates that the project has positive financial value after opportunity costs are considered.
43 9-43 Rate of (Percentage) Return 0 10% $1, $310 $400 $500 $ 0.00 = NPV, so rate of return = 10.0%. $750
44 9-44 Financial Spreadsheet Calculator Solution Solution A B C D % Rate Interest rate 3 $ (1,500) Values Year 0 CF Year 1 CF Year 2 CF Year 3 CF Values Year 4 CF % =IRR(A3:A7,A2) (entered into Cell A10)
45 9-45 Rate of Return (Cont.) In capital investment analyses, the rate of return often is called internal rate of return (IRR). In essence, it is the percentage return expected on the investment. To interpret the rate of return, it must be compared with the opportunity cost of capital, in this case 10% versus 8%.
46 9-46 Intrayear Compounding Thus far, all examples have assumed annual compounding. When compounding occurs intrayear, the following occurs: Interest is earned on interest during the year (more frequently). The future value of an investment is larger than under annual compounding. The present value of an investment is smaller than under annual compounding.
48 9-48 Effective Annual Rate (EAR) EAR is the annual rate that causes the PV to grow to the same FV as under intrayear compounding. What is the EAR for 10%, semiannual compounding? Consider the FV of $1 invested for one year. FV = $1 x (1.05) 2 = $ EAR = 10.25%, because this rate would produce the same ending amount ($1.1025) under annual compounding.
49 9-49 The EAR Formula I Stated EAR = M M = = (1.05) = = 10.25%. Or, use the EFF% key on a financial calculator.
50 9-50 EAR of 10% at Various Compounding EAR Annual = 10%. EAR Q = ( /4) = 10.38%. EAR M = ( /12) = 10.47%. EAR D(360) = ( /360) = 10.52%.
51 9-51 Using the EAR 0 1 5% 2 3 $ month periods $100 $100 Here, payments occur annually, but compounding occurs semiannually, so we can not use normal annuity valuation techniques.
52 First Method: Compound Each CF % 2 3 $ $100 $ $331.80
53 9-53 Second Method: Treat as an Annuity Find the EAR for the stated rate: 2 ) EAR = ( = 10.25%. 2 Then use standard annuity techniques: INPUTS OUTPUT N I/YR PV PMT FV
54 9-54 Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with three equal payments.
55 9-55 Step 1: Find the required payments % -$1,000 PMT PMT PMT INPUTS OUTPUT N I/YR PV PMT FV
56 9-56 Step 2: Find interest charge for Year 1. INT t = Beginning balance x I. INT 1 = $1,000 x 0.10 = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = $ $100 = $
57 Step 4: Find ending balance at end of Year End bal = Beg balance - Repayment = $1,000 - $ = $ Repeat these steps for Years 2 and 3 to complete the amortization table.
58 9-58 BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $ TOTAL $1, $ $1,000 Note that annual interest declines over time while the principal payment increases.
59 $ Interest Principal Payments Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.
60 9-60 Conclusion This concludes our discussion of Chapter 9 (Time Value Analysis). Although not all concepts were discussed in class, you are responsible for all of the material in the text.? Do you have any questions?
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Solutions to Problems: Chapter 5 P5-1. Using a time line LG 1; Basic a, b, and c d. Financial managers rely more on present value than future value because they typically make decisions before the start
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Chapter 4 Discounted Cash Flow Valuation 4B-1 Appendix 4B Using Financial Calculators This appendix is intended to help you use your Hewlett-Packard or Texas Instruments BA II Plus financial calculator
Chapter 02 How to Calculate Present Values Multiple Choice Questions 1. The present value of $100 expected in two years from today at a discount rate of 6% is: A. $116.64 B. $108.00 C. $100.00 D. $89.00
TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate
FIN 301 Homework Solution Ch4 Chapter 4: Time Value of Money 1. a. 10,000/(1.10) 10 = 3,855.43 b. 10,000/(1.10) 20 = 1,486.44 c. 10,000/(1.05) 10 = 6,139.13 d. 10,000/(1.05) 20 = 3,768.89 2. a. $100 (1.10)
Chapter Time Value of Money Future Value Present Value Annuities Effective Annual Rate Uneven Cash Flows Growing Annuities Loan Amortization Summary and Conclusions Basic TVM Concepts Interest rate: abbreviated
Chapter 4 The Time Value of Money 4-2 Topics Covered Future Values and Compound Interest Present Values Multiple Cash Flows Perpetuities and Annuities Inflation and Time Value Effective Annual Interest
Kuwait University College of Business Administration Department of Finance and Financial Institutions Using )Casio FC-200V( for Fundamentals of Financial Management (220) Prepared by: Dalia A. Marafi Version
To begin, look at the face of the calculator. Almost every key on the BAII PLUS has two functions: each key's primary function is noted on the key itself, while each key's secondary function is noted in
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Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount
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The Time Value of Money Time Value Terminology 0 1 2 3 4 PV FV Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the current value of one or more future
Lease Analysis Tools 2009 ELFA Lease Accountants Conference Presenter: Bill Bosco, Pres. email@example.com Leasing 101 914-522-3233 Overview Math of Finance Theory Glossary of terms Common calculations
Chapters 5 and 6 Calculators Time Value of Money and Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills Be able
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126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,
To begin, look at the face of the calculator. Most keys on the 17BII have two functions: a key's primary function is noted in white on the key itself, while the key's secondary function is noted in gold
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REVIEW MATERIALS FOR REAL ESTATE ANALYSIS 1997, Roy T. Black REAE 5311, Fall 2005 University of Texas at Arlington J. Andrew Hansz, Ph.D., CFA CONTENTS ITEM ANNUAL COMPOUND INTEREST TABLES AT 10% MATERIALS
Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation
Time Value of Money Problems 1. What will a deposit of $4,500 at 10% compounded semiannually be worth if left in the bank for six years? a. $8,020.22 b. $7,959.55 c. $8,081.55 d. $8,181.55 2. What will
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Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.
Compounding Assumptions Financial Calculations on the Texas Instruments BAII Plus This is a first draft, and may contain errors. Feedback is appreciated The TI BAII Plus has built-in preset assumptions
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file:///f /Courses/2010-11/CGA/FA2/06course/m05intro.htm Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present
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McGraw-Hill/Irwin Copyright 2014 by the McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills Be able to compute: The future value of an investment made today The present value of cash
CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money
QUANTITATIVE METHODS THE TIME VALUE OF MONEY Reading 5 http://proschool.imsindia.com/ 1 Learning Objective Statements (LOS) a. Interest Rates as Required rate of return, Discount Rate and Opportunity Cost |
Fill in each gap with a suitable preposition.
Christine’s cat was missing and she started looking around the house (1) _____ it. She opened her closet and looked (2) _____but the cat was not there. She went to the bedroom and crawled (3) _____the floor to look (4) _____ the bed. She searched the kitchen and
bathroom but found nothing. Since her cat could not be seen anywhere inside the house, Christine decided to go (5) _____ to give it a try. Just as she was (6) _____ to open the door,
the doorbell rang. It was her neighbour, Mrs. Chan. Mrs. Chan pointed (7) _____ the top of a nearby tree. “Look, Christine! Your cat is (8) ____ the tree!” Mrs. Chan began to describe how she found the cat. (9) _____her, she was cleaning the windows when she heard a noise coming (10) ____the tree (11) ____ her house. She looked (12) ____and
saw the cat (13) ____ one of the branches high (14) _____the ground. (15) ____ ignoring the cat, Mrs. Chan decided to tell Christine. (16) _____ Mrs. Chan’s help, Christine was able to rescue it
(30,000+ Idioms With Examples)
(Advanced Spoken English Course)
Peter: The bus is supposed to come (17) ____ 8:20am and it’s already 8:30am. Jane: I hope it comes soon. I don’t want to be late. What time do you have to be (18)_____class?
Peter: My first class is (19)_____9:00 am, but my professor is never (20)_____time for class.
Jane: How lucky! My professor gives us penalty for being late. I usually arrive just (21)______ time, just before my professor starts taking marks off students who are late.
Peter: I understand it is always very hard to wake up (22)_____ the morning, especially (23)_____ Mondays.
Jane: Yes, I know. I tend to study late (24) _____ night and find it very hard to get up the next day. It gets worse (25)_____ the winter.
Peter: Oh! The bus is finally here. Let’s get (26) _____!
(50,000+ English Literature MCQs)
(Advanced Grammar MCQs)
• Prof. Charles Kao is one of the most prominent scientists in the world. All his students have great respect (27)____ him.
• The belief (28) ____ aliens and UFOs is absurd!
• Jessica is angry (29) ____ her boyfriend for not showing up for the movie.
• He apologized (30) ____ her (31)____ being late by treating her dinner.
• My parents are worried (32) ____ the family’s financial problems.
(20,000+ Linguistics MCQs)
(American Literature MCQs)
• Do not ask me to fix your laptop. I am not familiar (33) ____computers.
• This book belongs (34) ____ my sister.
• If you give (35) ____ now, you will never finish the project.
• Please take some time to think (36) ____ what you want to do in the future.
• Kate and Frank are in the library studying (37) ____ their test next week.
• The scientists have found the best solution (38) ____ the problem.
• The police decided to investigate the cause (39) ____ his death.
(Indian English Literature MCQs)
• (40)……..our study, there is a significant relationship between school shootings and playing violent video games.
• (41)……..our great effort, we failed to finish the project before the deadline.
• (42)………obtaining the government assistance, we were able to use an up-to-date sampling frame from which to select our sample.
• (43)………the economic downturns of many countries around the world, China’s economy has recorded phenomenal growth for at least five consecutive years.
• (44)……. great care and attention to details, our key projects proceeded successfully.
• (45)……..yourself, nobody thanked me.
• It is raining (46)……..the whole country.
• My car does 10 miles (47)……..the litre.
• He contributed $100,000 (48)……. the new building.
• I am (49)……..orders from the President.
• Please buy it if it’s(50)……$50.
• This problem is(51)……..any we have met before.
• We should choose peace (52)……..war.
• We flew from Paris to Bangkok (53)……. Dubai.
• You’ll forget her(54)………time.
• There is a modem(55)……..the computer.
• He lives(56)…….10 miles of his work.
• Anthony is(57)……..Rachel in the race.
• Nobody objected (58)…….you.
• (59)……..Anthony, he will arrive later.
• I have another car(60)……..this one.
• We must work strictly (61)…….the law.
• (62)…….your behaviour, I think you were wrong.
• (63)……you, I would have been on time.
• You open a door (64)…….its handle.
• My car is(65)…….your car.
• We will play tennis tomorrow,(66)……the rain.
• The cancellation was (67)…….the rain.
• He reacts bravely(68)……..danger.
• Can I pay pounds (69)…….dollars?
• I don’t like it, (70)……. what you say.
• I am writing to you (71)……..my mother.
• Is there a doctor(72)……..this plane?
• We have every drink(73)……..whisky.
• We didn’t go swimming,(74)……. the cold weather.
• She sat(75)……..the horse.
• I am(76)……..your project. I will support it.
• It was(77)………his dignity to do that.
• I can meet you(78)……….1pm and 2pm.
• Can you see someone in the distance, (79)………that house?
(Preposition Quiz Part 01)
(Preposition Quiz Part 02)
(Preposition Quiz Part 03)
(Preposition Quiz Part 04)
- According to
- In front of
- Instead of
- Thanks to
- According to
- In spite of
- As a result of
- In contrast to
- Ahead of
- Apart from
- As for
- Aside from
- As per
- As to
- But for
- By means of
- Close to
- Depending on
- Due to
- In face of
- In lieu of
- Irrespective of
- On behalf of
- On board
- Other than
- Owing to
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Name: Instructor: Course: Date: Deferential equations – second order Necessity is the mother of invention; it is the same with mathematics. A great discovery solves a great problem. Thinking mathematically could change and develop your life in many fields, including logical reasoning and problem-solving skills. Mathematics is vital in daily life, several types of employment, technology, medicine, economy, sciences environment and development in public decision-making. The earliest uses of mathematics included economic exchange, land allocation, drawing, and the recording of time, but now it is affecting the entire world. Study of mathematics satisfies a broad variety of interests and capabilities (Blanchard et al, 55). It forms the imagination, teaches clearly and logically.
Mathematics is challenging because it deals with complex ideas and difficult questions. Calculus is one of the most pertinent sections of the mathematics. Calculus deals with quantities, which are near other quantities with limits, so it is less static and more dynamic.
Studying calculus course is a window of opportunity to other courses related to Mathematics. Learners will have a better understanding as they advance their studies. Today calculus has multiple uses including navigation in outer space, predicting population sizes, estimating how fast coffee prices rise, and expecting the weather (Argawal & Donal, 22). Differential equation involves equations with not known function. The functions could belong to one or more variables relating the function’s value and derivatives. Differential equations are mainly used in engineering, economics and physics among other fields. Differential equations are applicable in many fields of science especially in situations of determining the relationship of quantities.
These quantities are continuously varying, and their levels of changing are postulated or known. In mechanical engineering, the movement of a body is defined by position and velocity. The formulas for differential equations differ depending on the form of equation. Studying differential equations is a broad field in pure mathematics as well as applied.
Every discipline using differential equations focuses on one or more types of these equations. Differential equations are significant in creating biological, physical or technical processes. Some of these processes include making designs of structures and buildings, determining motion of celestial bodies and interaction of neurons among others. When using these equations in real life situations, one may not get a specific solution. In such circumstances, the answer is estimated using numerical formulae. For example, Bessel’s differential equation is: This is a simplified form of a second order differential equation (Morris & Orley, 67). A mass on a spring will decelerate because of friction, and the magnitude is proportional to velocity. The differential equation will be in this form: C is the coefficient, which will be representing friction.
We could say m= 1 and 0 < c -2a and k= a2 + b2 Second order This is a category with several equations of different aspects. Some of the equations include linear and nonlinear, reduction of order, Euler Cauchy, linear independence and Wronskian among others (Argawal & Donal, 78). The explicit form of second order equation is: Second order linear equation could also have this form: The solution can be solved in the following way: Therefore, the solution for this equation is y = c1y1 + c2 y2 In this equation, P, Q, R, and G are direct forms of continuous functions. Such equations are mostly used in the study of spring motion. For example, a mass is fixed on a spring, and it exerts reasonable force, which is proportional to either extension or compression of the spring. All other forces are assumed to be constant.
The equation will use Newton’s second law (Blanchard et al, 88). M represents the mass on the spring and k stands for the measurement of the spring’s stiffness. Make m = k and use Euler’s theorem to get a solution. According to Euler’s theorem, the solution will take this format: This means there is a need to find out the unknown constants, which are A & B. We need to specify the state of a given time.
So t = 0, extension will be x = 1 and the particle will not be moving so it will be dx/dt =0. This will give the following equation: A =1 And B = 0 hence x (t) = cos t. Ordinary homogenous Ordinary homogenous equation has this general form (Argawal, Ravi & Dona, 61): The above equation can be solved in a closed format. This can happen by changing some variables. These variables are u = y/x and the equation will change to this form: It is common to have homogenous equations with constant coefficients. Such an equation is expressed as The general solution for this equation will be A homogenous equation can also take characteristic form: If the equation takes such a form, it is solved in the following criterion (Morris & Orley,75): If the equation is homogenous, linear and in the second order category, it will take this form: In this equation, a, b and c are constant.
This equation has an auxiliary equation, which is Solutions for the equations of this category are determined by the solution of the auxiliary equation. It begins with finding the roots of the auxiliary equation and they are given by the formulae of the quadratic equation (Morris & Orley, 107). The roots may be in various forms, which will determine the general solution. The following are examples of possible roots. Ordinary-Nonhomogenous Equations in this category take this form if it is free motion (Blanchard et al, 112). They are mostly used in calculating motions of different aspects.
They could be free motions or forced motions. Free motions are caused by gravity or the spring. If other external forces are involved, then it will be forced motion. Forced motion has this formula Solving a Nonhomogenous equation is combining both homogenous and Nonhomogenous general solution.
After they are combined, they form this solution The final solution format will have such an equation: Some equations will have coefficients, which are undetermined. For example, a student may be required to find the general solution for this equation: The first step is solving the characteristic equation (Argawal & Donal, 122). After getting these values, they should be substituted to the original equation (Morris & Orley, 132). Another example is finding the general solution of The characteristic equation is solved and the substitution (Morris & Orley, 111). The resulting equation will be: The final general solution will be as follows: Differential equations exist in different forms of categories. It is crucial to learn them since they are used in many fields. For one to understand integration, he or she will need to learn differential equations since they are dependent. They seem to have a long method of solving them, but practice enables learners to improve efficiency.
Learners who would like to advance Mathematics studies will need knowledge in differential equations. These equations are vital to professionals who deal with processes requiring the use of these equations (Blanchard et al, 131). Works Cited Agarwal, Ravi P, and Donal O’Regan.
Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems. New York: Springer, 2009. Print. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Pacific Grove, CA: Brooks/Cole Pub.
Co, 2010. Print. Morris, Max, and Orley E. Brown.
Differential Equations. Englewood Cliffs, N.J: Prentice-Hall, 2009. Print. |
Figure 1 shows a transmission system that
transmits power from an electric motor to an
industrial machine by means of V-belt and gear
transmission. The gears have a straight-toothed
tooth profile. The electric motor is delivered with
a V-belt mounted on the shaft journal. The
following data are given for the electric motor:
Nominal power, P = 60 kW, speed n1 = 625 rpm.
Due to the tension wheel, it can be assumed that
the pull in the tight part (T1) of the belt on the
pulley is twice as large as the pull in the slack part
(T2) and that both act parallel to the z-axis (see
Fig. 1 (b)) . The intermediate shaft is made of steel
(E335) and carries a pulley and a gear. It is
supported by a single-row ball bearing and a
cylindrical roller bearing. The shaft of the
industrial machine and the intermediate shaft lie
in the same vertical plane.
a) What is the effective torque of the industrial
machine if the belt drive is assumed to have an
efficiency of 90%?
b) Assume module m = 5 and that there is
clearance engagement between the gears, and
determine the base circle diameter (db), number
of teeth (z1) and tooth thickness (s) on gears 1.
c) A simplified calculation model of the
intermediate shaft together with mounted pulley,
bearings and gears are shown in FIG. 1 (c) above.
Fig. 1 (c) Simplified calculation model for the
Based on the nominal power of 60 kW and the
dimensions given in the figure:
i) Draw the torque diagram for loading the
intermediate shaft in both x-y and x-z
planes. How big is the largest bending
moment? The weight of the shaft, gears and
pulley can be neglected.
ii) At a point on the intermediate shaft, the
voltages acting on a voltage element have
been calculated as shown next to it. Use
Mohr's circle and find the magnitude and
direction of the main voltages for this
voltage element. Enter the voltages on a
sketch of a main voltage element.
d) An examination shows that the machine runs at
almost full load for 75% of the operating time and
otherwise with half load. What will be the nominal
service life of bearing A, in number of hours, if you
choose bearing type 6310 which has the following
bearing capacities: C = 65.0 kN, C0 = 38.0 kN?
e) FIG. 1 (d) shows a curvature surface diagram
for the intermediate shaft (see also Fig. 1 (a))
calculated on a plane where the shaft is most
loaded. The diagram is based on a shaft with a
constant diameter (Ø50). Use the linker
(curvature surface) method and calculate the max.
the displacement C at end C of the shaft. Tip: the
weight of the shaft, gears and pulley is neglected.
Fig. 1 (d) Deformation image and curvature
f) The pulley is rigidly connected (mounted) to the
intermediate shaft by a single-shrink connection.
As shown in Fig. 1 (a), the connection is
dimensioned with fit tolerance Ø45H6 / p5. The
following data are given for the shrink connection:
- Diametral press nozzle in mounted condition:
min = 10 μm, max = 37 μm. - Influence
coefficients are calculated and specified as
follows: Shaft: a = 5.0 * 10-5 Belt pulley (hub):
n = 6.5 * 10-5. How large must the width of the
contact surface between the pulley and the
intermediate shaft be in order to transmit a torque
of 1.5 kNm?
A design draft for a manual water pump is shown
in FIG. 2 below. In order to be able to optimize the
water flow, it is recommended that the piston has
a vertical speed of vs = 50 mm / s. The pumping
force F is assumed to be perpendicular to the
crank arm BD at all times.
a) Determine the number of instantaneous poles
for the mechanism, and state the location of these
at the moment shown in the figure on a sketch of
the mechanism to scale.
b) Use graphical method and find the angular
velocity 2 of the crank arm BD.
c) The column CE is made of aluminum alloy (E =
70 GPa, Rp02 = 160 MPa) with an outer diameter
of 20 mm and a wall thickness of 4 mm. i) Make
the necessary assumptions and assess whether
the column is in the Euler area. ii) How great is the
safety against buckling of the column with respect
to Euler voltage when the angle θ = 00?
Fig. 2 Design draft of manual water pump
Figure 3 shows the calculation model for a wheel
suspension in a trolley. The axle is bolted to the
car body at A and is supported against the car
body by a helical spring at B. The axle cannot
rotate, but it can swing in the vertical plane, the
plane shown in the figure. The wheel, which is in
contact with the road surface at all times, is
mounted on the axle at C.
By a random check, an imbalance has been found
in the wheel which can be stated as if the wheel's
mass of m = 25 kg is placed eccentrically in
relation to its axis of rotation at a distance e = 2
mm. Due to the fact that the mass of the car body
is so large in relation to the mass of the wheel, one
can disregard vibrations in the car body.
a) Neglect the mass of the axle and critically
calculate the angular velocity of the wheel in the
axle suspension. Tip: By considering the
relationship between an arbitrary force F
(centrifugal force) acting in C and the deformation
at the same point, one can derive the resulting
spring constant indicated by:
b) How large is the deformation amplitude of the
helical spring at B when the wheel turns at an
angular velocity of = 120 rad / s. Also determine
the alternating force that is transmitted to the car
body via the coil spring due to. the unbalanced
mass of the wheel.
We now consider that the following data are
known about the helical spring:
Material: cold drawn steel wire with 2 mm
Number of active turns: n = 10
Spring length in unloaded condition: L0 = 56 mm
Spring index: C = 10. c) To avoid buckling in the
spring, its compression should not exceed 25 mm.
i) How large is then the largest shear stress in
ii) Also determine the safety against tight turns in |
In philippines how much per cubic meter of gravel and sand Products As a leading global manufacturer of crushing grinding and mining equipments we offer advanced reasonable solutions for any size reduction requirements including In philippines how much per cubic meter of gravel and sand quarry aggregate and different kinds of mineralsConsulting Service
How heavy is Portland cement Calculate how many cubic meters ( m3 ) of Portland cement are in 1 kilogram ( 1 kg kilo ) Specific unit weight of Portland cement amount properties converter for conversion factor exchange from 1 kilogram kg kilo equals = 0 00066 cubic meters m3 exactly for the masonry material type To convert Portland cement measuring units can be useful when building.Custom Concrete (Northern) installed a new paste plant at the Goldcorp Inc mine in 2009 This particular plant has a 2 25 cubic meter twin shaft mixer At this particular installation we are using 100 tailings The cement content varies from 3 to 12 depending on the mine requirements The tailings are screened to produce a 1 inch minus product.
How much sand cement and gravel do i need to cover an area of 20m2 at 2 deep Measurements and Units 1 meter = approx 39 37 inches 1 square meter = 39 37 ^2 multiply that by 20 to find area in square inches How many kilograms makes 1 cubic meter of gravel Conversions The conversion is 1 cubic meter = 2406 53 kg So in this case 2 406.Both of these concrete calculators make an allowance for the fact that material losses volume after being mixed to make concrete Calculator are provided for a general mix and a paving mix the different ratio of materials are General mix 1 5 cement all in ballast or 1 2 3 cement sharp sand gravel and.
Cement quantity for concrete mixing KnowledgePoint With a 1 3 6 mix for 100 cubic meter of concrete you would need 420 x 50kg bags of cement 56 cubic meters of fine aggregate (sand and gravel passing through a 5mm sieve) and 79 cubic metres of coarse aggregate (gravel.1 elf of sand in the philippines how many cubic meter Products As a leading global manufacturer of crushing grinding and mining equipments we offer advanced reasonable solutions for any size reduction requirements including 1 elf of sand in the philippines how many cubic meter quarry aggregate and different kinds of minerals.
Concrete mix ratio The strength of concrete mixture depends on the ratio in which these four ingredients are mixed Concrete mix ratio of 1 3 3 On mixing 1 part cement 3 parts sand with 3 parts aggregate produces concrete with a compressive strength of 3000 psi.Sep 08 2014 a cubic metre of concrete weighs circa 2 4 tonnes 5300lbs depending on what you are using it for depends on what strength you make it a good mix for foundations is 9 1 a really strong mix is 6 1 problem is people do not understand the structure of concrete so a mix @9 1 is 6parts gravel 3 sharp sand 1 cement @6 1 4 gravel 2 sand 1 cement where people dont.
Feb 15 2021 Concrete is a solid block available in the 16 regular dye colors 1 Obtaining 1 1 Breaking 1 2 Post generation 2 Usage 2 1 Note Blocks 3 Sounds 4 Data values 4 1 ID 4 2 Block data 4 3 Block states 5 History 6 Trivia 7 Issues 8 Gallery 9 References 10 External links Concrete requires a.Get the best deals on Industrial Cement Mixers Shop with Afterpay on eligible items CEMENT MIXER CONCRETE MIXER 2 2 CUBIC FT 450WATT AU $399 00 AU $383 00 postage 63L Kartrite Portable Cement Concrete Mixer Electric Construction Sand Gravel AU $209 00 AU $999 00 postage Electric Portable Concrete Cement Mortar Plaster Mixer.
Feb 15 2021 Concrete is a solid block available in the 16 regular dye colors 1 Obtaining 1 1 Breaking 1 2 Post generation 2 Usage 2 1 Note Blocks 3 Sounds 4 Data values 4 1 ID 4 2 Block data 4 3 Block states 5 History 6 Trivia 7 Issues 8 Gallery 9 References 10 External links Concrete requires a pickaxe to be mined When mined without a pickaxe it drops nothing Concrete powder becoming solid concrete.Total weight of concrete ingredients = 50+115+209+27 5 = 401 5 say 400 kg Density of concrete = 2400 kg cum So 1 bag of cement produces = 400 2400 = 0 167 cum No of bags required for 01 cum of concrete = 1 0 167 = 5 98 bags ~ 6 bags From above if the concrete mix is 1 2 4 to get a cubic meter of concrete we require 1 Cement = 6 bags.
In the past 20 years we devote to producing mining equipments sand making machines and industrial grinding mills Cost of 1 cubic meter of M30 concrete in India.For estimating purposes you can make about 1 cubic yard of concrete with 5 1 2 94 pound bags of cement 17 cubic feet of sand and 18 cubic feet of gravel (It takes about forty 80 pound bags of prepackaged materials (like Quikrete) to make 1 cubic yard of concrete ).
Jul 21 2019 Handy guide on factors to multiply per cubic on different concrete class mixtures Just multiply each factor to the total volume (in cubic meter) and you'll get the needed amount of cement (40Kg bag) sand (cu m ) and gravel (cu m ) Class C (2 000 psi) Cement 5 634 Bags Sand 0 4794 cu m Gravel 0 898 cu m Class B (2 500 psi).How heavy is Portland cement Calculate how many cubic meters ( m3 ) of Portland cement are in 1 ton (short) ( 1 sh tn ) Specific unit weight of Portland cement amount properties converter for conversion factor exchange from 1 ton (short) sh tn equals = 0 60 cubic meters m3 exactly for the masonry material type To convert Portland cement measuring units can be useful when building with.
Trust but verify How much cement you poured the better to check Please note that the cost of sand and gravel is specified in the program for 1 ton Vendors also announced a price per cubic meter of sand or gravel or gravel Proportion of sand depends on its origin such as river sand is heavier than a career 1 cubic meter of sand weighs.Aug 11 2012 Concrete mixture and proportionsNote Commercially available Portland cement bags come in 40kg and 50kg quantitiesMixture Class Proportion Cement in Bag Sand Gravel Cement 40 kg 50 kg Cubic Meter Cubic Meter Sand Gravel Class AA 1 1 3 12 9 5 0 5 1 0 Class A 1 2 4 9 0 7 0 0 5 1 0 Class B 1 2 5 7 5 6 0 0 5 1 0 Class C 1 3 6 6 0 5 0 0 5 0.
Depending on the structural concrete strength required and condition in which the concrete will be used we advise the customers on selection of type class of cement to be used maximum amount of cement per cubic meter of concrete as well as maximum water cement ratio the use of admixture to modify the concrete workability etc.Waste rock was often processed into gravel or cement and used for road and railroad construction (AECB) has only promulgated rough guidelines and it decides together with the mine and mill operators on the necessity of measures to be taken involving a total of 14 36 million cubic meters of tailings To prevent seepage of.
FULLER'S RULE states mixing of a total of 42 volumes of cement sand and gravel will produce a 27 volumes of concrete The unit for this one is in cubic foot Conversion of 1 0 meter = 3 281 ft hence (42 27cu ft) x (3 281ft m)^3=54 94 or rounded off to a good number of 55 That is how the constant of 55 comes from Hope it helps Tumugon I.Haledon Quarry Asphalt 400 402 Central Ave Haledon NJ 07508 Sparta Quarry 217 Limecrest Rd Lafayette NJ 07848 Franklin Quarry 280 Cork Hill Rd Franklin NJ 07416.
The cement mixed with fine aggregate produces cement mortar for masonry or with sand and gravel produces concrete Ans 1440 kg m3 is density of cement Density of cement in kg m3 density of cement measured in Kg m3 (kilograms cubic meter) density is the ratio of mass to volume.Really over estimated there 1 cubic metre of concrete contains approx 3 400 kg of cement and about 2ton of sand gravel mix 2 metres could be done easy enough even with a small mixer and 2 people in 3 hrs Did a 1 5 metres in Feb on an afternoon with a small mixer on my own Mixed poured and finished in just over 4 hours.
1 Part AfriSam High Strength Cement + 2 parts coarse sand and 2 parts stone Quantities per m 3 of Concrete To make a mix for 1 cubic metre of Ultra High Strength Concrete you will need 7 7 bags of AfriSam High Strength Cement + 0 63 cubic metres of coarse sand + 0 63 cubic metres of stone We suggest adding an extra 10 for wastage.Concrete Calculator Keohane Readymix concrete is supplied in cubic metres from a minimum of 1 cubic metre upwards in 0 5 cubic metre quantities Once you have your measurements use our Concrete Calculator below to work out how much you need Select the basic shape of your space – rectangle circle or triangle then insert your.
Crushed aggregate one inch crushed concrete recycled asphalt 3 8 inch pea gravel 2 inch sewer filter rock and concrete and asphalt aggregate blends all weigh in at 1 07 tons Pit run gravel follows at 1 25 tons per cubic meter regardless of whether it is 2 inch or 4 inch.Gravel Cost Per Square Foot Gravel costs $0 50 to $2 50 per square foot on average when covering an area 12 deep (a cubic foot) depending on the type Base gravel crusher run slate shale and crushed concrete are the cheapest at $0 50 to $1 50 per square foot while colored and decorative gravel run $3 per square foot Return to Top Gravel Delivery Cost.
How Much Does Concrete Cost Per Cubic Metre Concrete cost per cubic metre sits at average $250 to $300 m3 for the material only whereas it can start at low $160 m3 and reach even $550 m3 plus GST You also need to count in $60 to $80 per hour for concreters’ labour rates when calculating concreting costs on a per cubic metre basis.Feb 26 2018 An 80 pound bag yields approximately 0 60 cubic feet of concrete 60 pound bags yield 0 45 cubic feet and 40 pound bags just 0 30 cubic feet The actual yield is approximate because the amount of water added to the mix may vary 60 to 80 pounds is a lot of weight to lift carry and handle. |
Table of Contents
- 0 is a Rational Number
- Understanding Rational Numbers
- The Characteristics of Zero
- Evidence Supporting Zero as a Rational Number
- Zero as a Fraction
- Zero as a Terminating Decimal
- Zero in the Number Line
- Common Misconceptions
- Zero Divided by Zero
- Zero as an Irrational Number
- Q: Is zero a whole number?
- Q: Can zero be a denominator?
- Q: Is zero an even or odd number?
- Q: Can zero be a prime number?
- Q: Is zero a rational number?
When it comes to numbers, there are various classifications that help us understand their properties and relationships. One such classification is the distinction between rational and irrational numbers. While most people are familiar with rational numbers, there is often confusion surrounding the inclusion of zero in this category. In this article, we will explore the concept of rational numbers, delve into the characteristics of zero, and provide evidence to support the claim that zero is indeed a rational number.
Understanding Rational Numbers
Before we can establish whether zero is a rational number, it is essential to have a clear understanding of what rational numbers are. Rational numbers are those that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, any number that can be written in the form p/q, where p and q are integers and q is not equal to zero, is considered a rational number.
For example, the numbers 1/2, -3/4, and 5/1 are all rational numbers. These numbers can be expressed as fractions, and their decimal representations either terminate or repeat indefinitely. It is this property of terminating or repeating decimals that distinguishes rational numbers from irrational numbers.
The Characteristics of Zero
Zero, denoted by the symbol 0, is a unique number with distinct characteristics. It is the additive identity, meaning that when added to any number, it does not change the value of that number. For example, 5 + 0 = 5 and -3 + 0 = -3. Additionally, zero is the only number that is neither positive nor negative.
Zero also plays a crucial role in arithmetic operations. When multiplied by any number, the result is always zero. For instance, 0 × 7 = 0 and 0 × (-2) = 0. However, when zero is used as the divisor in a division operation, it leads to undefined results. This is because division by zero violates the fundamental principles of mathematics and leads to contradictions.
Evidence Supporting Zero as a Rational Number
Now that we have established the characteristics of zero, let us examine the evidence that supports its classification as a rational number.
Zero as a Fraction
One of the most compelling arguments for zero being a rational number is its representation as a fraction. Zero can be expressed as the fraction 0/1, where the numerator is zero and the denominator is any non-zero integer. This satisfies the definition of a rational number, as it is the quotient of two integers with a non-zero denominator.
Zero as a Terminating Decimal
Another piece of evidence supporting zero as a rational number is its decimal representation. When zero is expressed as a decimal, it terminates after the decimal point. In other words, there are no repeating digits or an infinite sequence of decimals. For example, 0.0, 0.00, and 0.000 are all representations of zero as a decimal. This aligns with the characteristic of rational numbers, which have decimal representations that either terminate or repeat.
Zero in the Number Line
Visualizing zero on the number line further reinforces its classification as a rational number. The number line represents all real numbers, including both rational and irrational numbers. Zero falls precisely in the middle of the number line, equidistant from the positive and negative numbers. This positioning indicates that zero can be expressed as a fraction with a positive or negative numerator and a non-zero denominator, satisfying the definition of a rational number.
Despite the evidence supporting zero as a rational number, there are some common misconceptions that lead to confusion. Let’s address a few of these misconceptions:
Zero Divided by Zero
One misconception is that zero divided by zero is equal to one. However, division by zero is undefined in mathematics, and any attempt to assign a value to it leads to contradictions. Therefore, zero divided by zero is not equal to one or any other number.
Zero as an Irrational Number
Another misconception is that zero is an irrational number. Irrational numbers are those that cannot be expressed as fractions and have decimal representations that neither terminate nor repeat. However, zero does not fit this definition, as it can be expressed as a fraction and has a terminating decimal representation.
Q: Is zero a whole number?
A: Yes, zero is considered a whole number. Whole numbers include all the natural numbers (positive integers) and zero.
Q: Can zero be a denominator?
A: No, zero cannot be a denominator. Division by zero is undefined in mathematics and leads to contradictions.
Q: Is zero an even or odd number?
A: Zero is an even number. It is divisible by 2 without leaving a remainder.
Q: Can zero be a prime number?
A: No, zero cannot be a prime number. Prime numbers are defined as positive integers greater than one that have no positive divisors other than one and themselves.
Q: Is zero a rational number?
A: Yes, zero is a rational number. It can be expressed as the fraction 0/1, where the numerator is zero and the denominator is any non-zero integer.
In conclusion, zero is indeed a rational number. It satisfies the definition of a rational number as it can be expressed as the quotient of two integers, with a non-zero denominator. Zero’s representation as a fraction, its termination as a decimal, and its position on the number line all provide evidence supporting its classification as a rational number. While misconceptions may exist, understanding the characteristics and properties of zero helps clarify its status as a rational number. |
Types of Samples
Although there are different methods that might be used to create a sample, they generally can be grouped into one of two categories: probability samples or non-probability samples.
The idea behind this type is random selection. More specifically, each sample from the population of interest has a known probability of selection under a given sampling scheme. There are four categories of probability samples described below.
Sampling The most widely known type of a random sample is the simple random sample. This is characterized by the fact that the probability of selection is the same for every case in the population. Simple random sampling is a method of selecting n units from a population of size N such that every possible sample of size n has equal chance of being drawn. Imagine you want to carry out a survey of 100 voters in a small town with a population of 1,000 eligible voters. We could write the names of all voters on a piece of paper, put all pieces of paper into a box and draw 100 tickets at random. These 100 form our sample where every name in the box had the same probability of being chosen.
Sampling In this form of sampling, the population is first divided into two or more mutually exclusive segments based on some categories of variables of interest in the research. It is designed to organize the population into homogenous subsets before sampling, then drawing a random sample within each subset. With stratified random sampling the population of N units is divided into subpopulations of units respectively. These subpopulations, called strata, are non-overlapping and together they comprise the whole of the population.
When these have been determined, a sample is drawn from each, with a separate draw for each of the different strata. The primary benefit of this method is to ensure that cases from smaller strata of the population are included in sufficient numbers to allow comparison. For example, you may be interested in how job satisfaction varies by ethnicity among a group of employees at a firm. To explore this issue, we need to create a sample of the employees of the firm. However, the employee population at this particular firm is predominantly Trinidadians, as the following chart illustrates:
If we were to take a simple random sample of employees, we may end up with very small numbers of Jamaicans, Barbadians and Puerto Ricans. That would not be good for research, since we might end up with too few cases for comparison in one or more of the smaller groups. Instead of taking a simple random sample from the population, a stratified sampling method can be used to ensure that appropriate numbers of elements are drawn from each ethnic group in proportion to the percentage of the population as a whole. For example, if we want a sample of 1000 employees - we would stratify the sample by ethnicity (group of Trinidadians employees, group of Jamaican employees, group of Barbadian employees and group of Puerto Rican employees), then randomly draw out 750 employees from the Trinidadian group, 90 from the Barbadian group, 100 from the Jamaican group and 60 from the Puerto Rican group. This yields a sample that is proportionately representative of the firm as a whole.
This method of sampling is at first glance very different from simple random sampling. In practice, it is a variant of simple random sampling that involves some listing of elements - every nth element of list is then drawn for inclusion in the sample. Say you have a list of 10,000 people and you want a sample of 1,000.
Creating such a sample includes three steps:
- Divide number of cases in the population by the desired sample size. In this example, dividing 10,000 by 1,000 gives a value of 10.
- Select a random number between one and the value attained in Step 1. In this example, we choose a number between 1 and 10 - say we pick 7.
- Starting with case number chosen in Step 2, take every tenth record (7, 17, 27, etc.).
More generally, suppose that the N units in the population are ranked 1 to N in some order (e.g., alphabetic). To select a sample of n units, we take a unit at random, from the 1st k units and take every k-th unit thereafter.
The advantages of systematic sampling method over simple random sampling include:
- It is easier to draw a sample and often easier to execute without mistakes. This is a particular advantage when the drawing is done in the field.
- Intuitively, you might think that systematic sampling might be more precise than simple random sampling. In effect it stratifies the population into n strata, consisting of the 1st k units, the 2nd k units, and so on. Thus, we might expect the systematic sample to be as precise as a stratified random sample with one unit per stratum. The difference is that with the systematic one the units occur at the same relative position in the stratum whereas with the stratified, the position in the stratum is determined separately by randomization within each stratum.
In some instances the sampling unit consists of a group or cluster of smaller units that we call elements or subunits (these are the units of analysis for your study). There are two main reasons for the widespread application of cluster sampling. Although the first intention may be to use the elements as sampling units, it is found in many surveys that no reliable list of elements in the population is available and that it would be prohibitively expensive to construct such a list. In many countries there are no complete and updated lists of the people, the houses or the farms in any large geographical region.
Even when a list of individual houses is available, economic considerations may point to the choice of a larger cluster unit. For a given size of sample, a small unit usually gives more precise results than a large unit. For example a SRS of 600 houses covers a town more evenly than 20 city blocks containing an average of 30 houses apiece. But greater field costs are incurred in locating 600 houses and in traveling between them than in covering 20 city blocks. When cost is balanced against precision, the larger unit may prove superior.
Social research is often conducted in situations where a researcher cannot select the kinds of probability samples used in large-scale social surveys. For example, say you wanted to study homelessness - there is no list of homeless individuals nor are you likely to create such a list. However, you need to get some kind of a sample of respondents in order to conduct your research. To gather such a sample, you would likely use some form of non- probability sampling.
To reiterate, the primary difference between probability methods of sampling and non-probability methods is that in the latter you do not know the likelihood that any element of a population will be selected for study.
There are two primary types of non-probability sampling methods:
Availability sampling is a method of choosing subjects who are available or easy to find. The primary advantage of the method is that it is very easy to conduct, relative to other methods. A researcher can merely stand out on his/her favorite street corner and distribute surveys. One place this method is popular is in university courses. For example, all students taking introductory sociology courses would have been given a survey and compelled to fill it out.
The primary problem with availability sampling is that you can never be certain what population the participants in the study represent. The population is unknown, the method for selecting cases is haphazard, and the cases studied probably do not represent any population you could come up with.
Quota sampling is designed to overcome the most obvious flaw of availability sampling. Rather than taking just anyone, you set quotas to ensure that the sample you obtain represents certain characteristics in proportion to their prevalence in the population. Note that for this method, you have to know something about the characteristics of the population ahead of time. Say you want to make sure you have a sample proportional to the population in terms of gender - you have to know what percentage of the population is male and female, then collect sample until yours matches.
The primary problem with this form of sampling is that even when we know that a quota sample is representative of the particular characteristics for which quotas have been set, we have no way of knowing if sample is representative in terms of any other characteristics. If we set quotas for gender and age, we are likely to attain a sample with good representativeness on age and gender, but one that may not be very representative in terms of income, education or other factors. |
To determine the number of significant figures in a number use the following 3 rules: Non-zero digits are always significant. Any zeros between two significant digits are significant. A final zero or trailing zeros in the decimal portion ONLY are significant.
What are the 4 rules for counting significant figures?
- Annotation category:
- RULES FOR SIGNIFICANT FIGURES.
- All non-zero numbers ARE significant.
- Zeros between two non-zero digits ARE significant.
- Leading zeros are NOT significant.
- Trailing zeros to the right of the decimal ARE significant.
How many significant figures does 0.001 have?
The number of significant figures in 0.001 is 1, while in 0.100 it is 3 .
How many significant figures are there in 432?
432 has 3 significant figures.
How many significant figures does 0.0040 have?
The number 0.0040 has two significant figures. The first three zeros mark the decimal place and are not significant. The last zero indicates accurate measurement to the third decimal place with some uncertainty at the fourth decimal place, so the 4 and last 0 are considered significant.
How many significant figures does 30.00 have?
Correct option b 4Explanation :In a number with decimal zeros on the right of the last non-zero digit are significant. Therefore 30.00 has four significant figures.
How many significant figures does 0.0086 have?
And the same kind of story applies to the next number 0.0086; it has two significant figures, only the 8 and the 6 are significant this 0.00 business here is just to put the 8 and the 6 in their proper place values.
How many significant figures does 1500.00 have?
Thus, in 1,500, the two trailing zeros are not significant because the number is written without a decimal point; the number has two significant figures. However, in 1,500.00, all six digits are significant because the number has a decimal point.
How many significant figures does 9010.0 have?
4) 9010.0 grams: This has five significant figures (the final zeros are significant because there is a decimal shown) and is precise to the nearest 0.1 gram.
How many significant figures does 0.250 have?
Answer: The number of sig figs. In the number 0.25, it has 2 sig figs and 0.250 has 3 sig figs.
What is the significant figure of 501?
501 has 3 significant figures.
How many significant figures does 10 have?
The number “10.” is said to have two significant digits, or significant figures, the 1 and the 0. The number 1.0 also has two significant digits. So does the number 130, but 10 and 100 only have one “sig fig” as written. Zeros that only hold places are not considered to be significant.
How many significant figures are present in the number 10450?
Answer and Explanation: Thus, there are four significant figures in the given number.
How many significant figures does 37.0 have?
Zeros will count when they are trapped by nonzero digits like in 4509 (4 sig figs) or when the zeros follow both the decimal and the nonzero digits like in 37.0 or . 370 (both with 3 sig figs). The number 370 only has 2 sig figs because the zero follows the nonzeros but it does not follow the decimal.
How many significant figures does the number 0.03 have?
3. All zeros at the left of the number are NOT significant. Example: 0.00032 has 2 sig. figs, 0.03 has 1 sig.
How many significant figures are in the number 300?
Yes, 300. has 3 sig figs because of the decimal which makes the zeros significant.
What are the 5 Rules for significant figures?
- Rule 1. All non-zero digits are significant.
- Rule 2. Zeros between non-zero digits are significant.
- Rule 3. Leading zeros are never significant.
- Rule 4. In a number with a decimal point, trailing zeros, those to the right of the last non-zero digit, are significant.
- Rule 5.
- Certain Digit.
- Uncertain Digits.
How many significant figures does 0.00120 have?
Hence, 0.00120 have 3 significant digits.
How many significant figures does 600 have?
Without a decimal point, the previous number would be 500, the following number would be 700. 100, 200, 300, 400, 500, 600, etc. In this way, there is only 1 significant figure.
How many significant figures are in the measurement 0.020 km?
Explanation: 0.020 has two significant figures. The 2 is significant because all non-zero numbers are signficant. The second 0 is significant because all zeros at the end of a decimal are significant.
How many significant figures are in the number 2000?
The rules for significant figures state that trailing zeroes that do not follow a decimal place are not significant as they are simply placeholders. Thus the only significant figure in 2000 is 2.
How many significant figures are in the measurement 463.090 m?
Answer and Explanation: All non-zeros and zeros between the two significant figures or captive zeros are significant. Trailing zeros to the right of the decimal point are also significant. Given measurement is 463.090 m. Therefore, the significant figure in the measurement is 6.
How many significant figures are there in 5000?
When written 5000. there are four significant digits while 5000 could contain one, two, three, or four significant digits. The nomber of significant figures used implies a certain maximum error range. Three , four, and five significant figures imply maximum errors of 1%, 0.1%, and 0.01% respectively.
How many significant figures does 20.00 have?
For example, 20.00 contains four significant digits. All the zeros that are on the right of the last non-zero digit, after the decimal point, are significant.
How many significant figures are there in the number 1008500?
In this case, you have 1 , 8 , and 5 as non-zero, and thus significant, digits. |
- Who is the father of mathematics in the world?
- Which country has the best mathematician?
- What country is first in math?
- Who is the mother of math?
- Who is the best scientist in the world?
- Why is Pi 22 divided 7?
- Who are the top 10 mathematicians?
- Who invented math?
- Who is a math genius?
- Is 0 a real number?
- What are the four branches of mathematics?
- Who invented calculus?
- What was used before zero?
- Who invented 0 in India?
- Who invented Number 1?
- Who is the greatest mathematician in the world?
- Who invented 0?
Who is the father of mathematics in the world?
ArchimedesBiography of Archimedes Archimedes (287 BC–212 BC) is known as Father of Mathematics.
He was born in the seaport city of Syracuse on the greek island of Sicily; his father was an astronomer.
He was fortunate enough to be born into a family who encouraged him to get an education..
Which country has the best mathematician?
18 Jan 7 Countries That Have Smart Mathematics Students#1: SINGAPORE. According to an international benchmarking study, Singapore ranked as the #1 country to have students performing their best in Mathematics and Science. … #2: AUSTRALIA. … #3: RUSSIA. … #4: IRAN. … #5: JAPAN. … #6: CHINA. … #7: INDIA.
What country is first in math?
Singapore is the smartest country in the world, followed by Hong Kong, South Korea, Taiwan, Japan, Finland, Estonia, Switzerland, Netherlands and Canada rounding out the top 10.
Who is the mother of math?
1. HYPATIA. Hypatia (c. 355–415) was the first woman known to have taught mathematics.
Who is the best scientist in the world?
1- Albert Einstein (1879-1955) Arguably the most influential scientist the world has ever seen. Einstein has a reputation for the greatest originality of thought. His theories of relativity enhance our understanding of the universe.
Why is Pi 22 divided 7?
But as you can see, 22/7 is not exactly right. In fact π is not equal to the ratio of any two numbers, which makes it an irrational number. A really good approximation, better than 1 part in 10 million, is: 355/113 = 3.1415929…
Who are the top 10 mathematicians?
Top 10 Greatest Mathematicians8 Isaac Newton and Wilhelm Leibniz.7 Leonardo Pisano Blgollo.6 Alan Turing.5 René Descartes.4 Euclid.3 G. F. Bernhard Riemann.2 Carl Friedrich Gauss.1 Leonhard Euler.More items…•
Who invented math?
Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof.
Who is a math genius?
Srinivasa Ramanujan , a Mathematical Genius. Srinivasa Ramanujan, the brilliant twentieth century Indian mathematician, has been compared with all-time greats like Euler, Gauss and Jacobi, for his natural mathematical genius. It may be impossible to define who a mathematical genius is, or, genius for that matter.
Is 0 a real number?
Real numbers consist of zero (0), the positive and negative integers (-3, -1, 2, 4), and all the fractional and decimal values in between (0.4, 3.1415927, 1/2). Real numbers are divided into rational and irrational numbers.
What are the four branches of mathematics?
The main branches of pure mathematics are:Algebra.Geometry.Trigonometry.Calculus.Statistics and Probability.
Who invented calculus?
Isaac NewtonCalculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century.
What was used before zero?
Western civilization didn’t adopt arithmetic with zeros until about 700 years later, based on the work of the thirteenth-century Italian mathematician Leonardo Fibonacci. Thanks to zero, we can punch any number into our calculators using just ten keys.
Who invented 0 in India?
AryabhataWhat is widely found in textbooks in India is that a mathematician and astronomer, Aryabhata, in the 5th century used zero as a placeholder and in algorithms for finding square roots and cube roots in his Sanskrit treatises.
Who invented Number 1?
Hindu-Arabic numerals, set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the decimal number system. They originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi, about the 12th century.
Who is the greatest mathematician in the world?
Euler and Newton are considered as the best mathematicians. Gauss, Weierstrass and Riemann are considered as the best theorist. Archimedes is often considered as the greatest mathematical genius who ever lived.
Who invented 0?
MayansThe first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. |
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Transcript of Chapter 09
- 1. Estimation and Confidence Intervals Chapter 9
- Define a point estimate.
- Define level of confidence.
- Construct a confidence interval for the population mean when the population standard deviation is known.
- Construct a confidence interval for a population mean when the population standard deviation is unknown.
- Construct a confidence interval for a population proportion.
- Determine the sample size for attribute and variable sampling.
3. Point and Interval Estimates
- Apoint estimateis the statistic (or estimate), computed from sample information, which is used to estimate the population parameter.
- Aconfidence interval estimateis a range of values constructed from sample data so that a specific probability can be assigned that the interval will include the true value of the population parameter. The specified probability is called the level of confidence.
4. Factors Affecting Confidence Interval Estimates
- The factors that determine the width of a confidence interval are:
- 1.Thesample size ,n.
- 2.Thevariability in the population , usually estimated bysif unknown .
- 3.The desiredlevel of confidence (the probability) .
5. Interval Estimates - Interpretation
- For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated.Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population
6. Characteristics of the t-distribution
- 1. It is, like thezdistribution, acontinuous distribution .
- 2. It is, like thezdistribution,bell-shapedandsymmetrical .
- 3. There isnot one t distribution , but rathera family of t distributions . Alltdistributions have a mean of 0, but their standard deviations differ according to thesample size,n. (These approach 1 as n increases)
- 4. Thet distribution is more spread outandflatter at the center than the standard normal distributionAs the sample size increases, however, thetdistribution approaches the standard normal distribution,
7. Comparing the z and t Distributions whennis small 8. Confidence Interval Estimates for the Mean
- UseZ -distribution
- If the population standard deviation is known or the sample is greater than 30.
- Uset -distribution
- If the population standard deviation is unknown and the sample is less than 30 and normally distributed.
9. When to Use thezortDistribution for Confidence Interval Computation 10. Confidence Interval for the Mean Example using the t-distribution
- A tire manufacturer wishes to investigate the tread life of its tires. Asample of 10tires driven 50,000 miles revealed asample mean of 0.32 inchof tread remaining with astandard deviation of 0.09inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?
11. Students t-distribution Table 12.
- The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent.
Confidence Interval Estimates for the Mean Using Minitab 13. Confidence Interval Estimates for the Mean By Formula 14. Confidence Interval Estimates for the Mean Using Minitab 15. Confidence Interval Estimates for the Mean Using Excel 16. Using the Normal Distribution to Approximate the Binomial Distribution
- To develop a confidence interval for a proportion, we need to meet the following assumptions.
- 1. The binomial conditions, discussed in Chapter 6, have been met. Briefly, these conditions are:
- a. The sample data is the result of counts.
- b. There are only two possible outcomes.
- c. The probability of a success remains the same from one trial to the next.
- d. The trials are independent. This means the outcome on one trial does not affect the outcome on another.
- 2. The valuesnandn (1- ) should both be greater than or equal to 5. This condition allows us to invoke the central limit theorem and employ the standard normal distribution, that is,z , to complete a confidence interval.
17. Confidence Interval for a Population Proportion
- The confidence interval for a population proportion is estimated by:
18. Confidence Interval for a Population Proportion -Example
- The union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-fourths of the union membership must approve any merger. A random sample of2,000current BBA members reveals1,600plan to vote for the merger proposal. What is the estimate of the population proportion?
- Develop a95 percentconfidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why?
19. Finite-Population Correction Factor
- A population that has a fixed upper bound is said to be finite.
- For a finite population, where the total number of objects isNand the size of the sample isn , the following adjustment is made to the standard errors of the sample means and the proportion:
- However, if n / N< .05 , the finite-population correction factor may be ignored.
20. Effects on FPC when n/N Changes Observe that FPC approaches 1 whenn/Nbecomes smaller 21. Confidence Interval Formulas for Estimating Means and Proportions with Finite Population Correction C.I. for the Mean ( ) C.I. for the Proportion ( ) C.I. for the Mean ( ) 22. CI For Mean with FPC - Example
- There are 250 families in Scandia, Pennsylvania. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75.
- Develop a 90 percent confidence interval for the population mean.
- Interpret the confidence interval.
- Given in Problem:
- N 250
- n 40
- s - $75
- Since n/N = 40/250 = 0.16, the finite population correction factor must be used.
- The population standard deviation is not known therefore use the t-distribution (may use the z-dist since n>30)
- Use the formula below to compute the confidence interval:
23. CI For Mean with FPC - Example 24. Selecting a Sample Size
- There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population.They are:
- The degree of confidence selected.
- The maximum allowable error.
- The variation in the population.
25. Sample Size Determination for a Variable
- To find the sample size for a variable:
- A student in public administration wants to determine the mean amount members of city councils in large cities earn per month as remuneration for being a council member. The error in estimating the mean is to be less than $100 with a 95 percent level of confidence. The student found a report by the Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size?
- Given in the problem:
- E , the maximum allowable error, is $100
- The value ofzfor a 95 percent level of confidence is 1.96,
- The estimate of the standard deviation is $1,000.
Sample Size Determination for a Variable-Example 27.
- A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence.Based on similar studies the standard deviation is estimated to be$20.00 . How large a sample is required?
Sample Size Determination for a Variable- Another Example 28. Sample Size for Proportions
- The formula for determining the sample size in the case of a proportion is:
29. Another Example
- The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet.If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact?Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.
30. Another Example
- A study needs to estimate the proportion of cities that have private refuse collectors. The investigator wants the margin of error to be within .10 of the population proportion, the desired level of confidence is 90 percent, and no estimate is available for the population proportion. What is the required sample size?
31. End of Chapter 9 |
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This primitive concept, familiar from undergraduate physics and mathematics, applies equally in general relativity. Find and save ideas about Vector calculus on Pinterest. The LATEX and Python les Introduction to a surface integral of a vector field; Scalar surface integral examples; Vector surface integral examples; Integration Synopsis The integrals of multivariable calculus; Length, area, and volume factors; The fundamental theorems of vector calculus Gradient theorem for line integrals An introduction to conservative vector fields Shed the societal and cultural narratives holding you back and let free step-by-step Stewart Calculus textbook solutions reorient your old paradigms.
Read more about popularity Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. For the newcomer to general relativity we warmly recom-mend Schutz Vector Calculus Vector calculus is the fundamental language of mathematical physics. For these reasons, we are providing these notes. So I decided to retake the course, and the book we're using this time is Vector Calculus by Miroslav Lovric. Buy Vector Calculus on Amazon.
Introduction to vector calculus.
Vector Calculus 20E, Fall Calculus with Vector Functions — In this section here we discuss how to do basic calculus, i. The best introductory textbook on multivariable calculus for the rank beginner that I know is Vector Calculus by Peter Baxandall and Hans Liebeck. It has two major branches, differential calculus and integral calculus. This sheaf carries a vector field whose direction is normal to the surface, and whose magnitude is the inverse of the thickness.
I can easily get A's on it even the first time. Re-studied two more times. To prove that these formulas work for arbitrarily large integers k, we can use a method called mathematical induction. Scalar Quantities. It has been used for the past few years here at Georgia Tech. We begin with vectors.
MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. You may feel embarrassed to nd out that you have already forgotten a number of things that you learned di erential calculus. This tutorial is a guide for serious students who want to dig deeply into the subject. Some knowledge of vector calculus is a prerequisite for the videos, but no Vector-Valued Functions3. Free delivery on qualified orders. Students, teachers, and professionals turn to Dover for low-priced works on advanced and elementary calculus, calculus of variations, fractional calculus, technical calculus, vector calculus, and more.
Ex: Velocity, Acceleration. Vector calculus is the fundamental language of mathematical physics. Book Review. In multivariable calculus, we progress from working with numbers on a line to points in space. If you do not have an Adobe Acrobat Reader, you may down-load a copy, free of charge, from Adobe. The prerequisites are the standard courses in single-variable calculus a. Vector Calculus Marsden; Anthony Tromba and a great selection of similar New, Used and Collectible Books available now at great prices.
Nonetheless, social scientists also stand to gain a great deal from a good knowledge of vector calculus. It covers the derivative, the integral, and a variety of applications. A great text perhaps the best for an undergraduate introduction into analytical mechanics would be John Taylor's Classical Mechanics.
Tromba and Jerrold E. I will be taking 23A in the fall quarter and was wondering whether anyone had a PDF of the textbook. Concepts of rank, basis, linear transformations, and vector spaces are closely related. Vector Calculus. At CCNY, this is how calculus and vector calculus works. Many calculus books will have a section on vectors in the second half, but students would not like to start reading there. Our treatment is closer to that Wald and closer still to Misner, Thorne and Wheeler Includes number of downloads, views, average rating and age.
Test Bank for Precalculus Mathematics for Calculus 6th Edition
Vector Analysis All the multi-variables calculus books which I have a lot barely touch these topics. This book covers calculus in two and three variables. Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world. When drawing a vector in 3-space, where you position the vector is unimportant; the vector's essential properties are just its magnitude and its direction.
Vector Calculus: Understanding the Dot Product. A with i. A vector has both magnitude and direction whereas a scalar has only magnitude.
Chapter 8 deals with eigenvalue problems. The classical theorems of vector calculus are amply illustrated with figures, worked examples, and physical applications. The author takes time to build and prove each theorem the way it ought to be done. Matrix calculus From too much study, and from extreme passion, cometh madnesse. For example T x,y,z can be used to represent the temperature at the point x,y,z. For calc 3, which is mutli-variable calculus, basic vector calculus everything until vector fields and beyond are not consider as basic vectors are taught.
I have tried to be somewhat rigorous about proving This is a very important topic in Calculus III since a good portion of Calculus III is done in three or higher dimensional space. The vector points in the direction of flow see Fig. Calculus books do not clearly indicate which topics are actually difficult. Independence of Path 75 Being a senior Maths student I have given my opinion about the best five books on Vector Analysis or Vector Calculus here in this video.
Vector Calculus, 2 nd ed.
ISBN 13: 9780840068071
The material is also contained in a variety of other mathematics books, but then we would not want to force students to acquire another book. We will also be taking a look at a couple of new coordinate systems for 3-D space. Ask the provider about this item. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. Best Books on Vector Calculus list of books on Vector Calculus names of the books on Vector Calculus together with the brief introduction about the books Read More. Our interactive player makes it easy to find solutions to Vector Calculus, Books A La Carte Edition 4th Edition problems you're working on - just go to the chapter for your book.
While vector calculus can be generalized to dimensions , this chapter will specifically focus on 3 dimensions vector calculus Download vector calculus or read online here in PDF or EPUB. We start with vectors Now in its fifth edition, Vector Calculus helps students gain an intuitive and solid understanding of this important subject. Now in its fifth edition, Vector Calculus helps students gain an intuitive and solid understanding of this important subject. This has got to be the most insipid exposition of multivariable calculus I've ever had the misfortune of having to read.
Only 3 left in stock - order soon. The Fundamental Theorem for Line Integrals 74 Very minimal writing or notations in margins not affecting the text. I stumbled across this terrific and very underrated book while searching for a modern treatment of functions of several variables that could be used by bright undergraduates without the use of manifolds or differential forms.
Download with Google Download with Facebook or download with email. Book: Vector Calculus Corral Thumbnail: The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. There have been many offers from people around the world to translate my books into other languages. The book we used then was Vector Calculus by Marsden and Tromba: a truly awful textbook.
It's not a big, heavy book, so that is a plus. Compare e. Fourier sums and integrals, as well as basic ordinary di erential equation theory, receive a quick review, but The mathematical study of change, calculus has two major branches: differential calculus and integral calculus. The response will be emailed to you. At each molecule, we could draw an arrow representing the speed and direction of motion.
Math 20E. Most of the same websites that sell books to students are also interested in buying books. We provide a list of quotes below. If you decide to sell your book to one of the sites, they will provide you free shipping labels. After you print the label, simply drop the book in the mail with the shipping label and you'll have that sweet sweet cash or Amazon store credit if you sold through Amazon Trade-in before you know it! We have sent an email on your behalf to the book's seller. If the book is still available they will be in touch with you shortly. Thank you for using SlugBooks, and please come back at the end of the term to list your book for other students.
This book may be a custom edition only available through your bookstore. Save money by searching for another book above! About The Book This best selling author team explains concepts simply and clearly, without glossing over difficult points. Read more are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking.
Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors' attention to detail and clarity, the same as found in James Stewart's market-leading Calculus text, is what makes this text the market leader.
Online Buyback Sites. Rational Functions. Chapter 3 Review.
Chapter 3 Test. Exponential Functions. The Natural Exponential Function. Logarithmic Functions. Laws of Logarithms. Exponential and Logarithmic Equations. Modeling with Exponential and Logarithmic Functions. Chapter 4 Review. Chapter 4 Test. Cumulative Review Test: Chapters 2, 3, and 4. The Unit Circle. Trigonometric Functions of Real Numbers.
Trigonometric Graphs. More Trigonometric Graphs. Inverse Trigonometric Functions and Their Graphs.
Modeling Harmonic Motion. Chapter 5 Review. Chapter 5 Test. Angle Measure. Trigonometry of Right Triangles. Trigonometric Functions of Angles. Inverse Trigonometric Functions and Triangles. The Law of Sines. The Law of Cosines. Chapter 6 Review. Chapter 6 Test. Trigonometric Identities. Addition and Subtraction Formulas.
Basic Trigonometric Equations. More Trigonometric Equations. Chapter 7 Review. Chapter 7 Test. Cumulative Review Test: Chapters 5, 6, and 7. Polar Coordinates. Graphs of Polar Equations. Plane Curves and Parametric Equations. Chapter 8 Review. Chapter 8 Test. |
See my Do Now in my Strategy folder that explains my beginning of class routines.
Often, I create do nows that have problems that connect to the task that students will be working on that day. Today I want students to review writing expressions and creating and solving equations from the previous lessons.
Problem one can be difficult for some students. If students struggle to write an expression, I ask them what they know and what they are trying to model. I want students to understand that twice the amount of Elizabeth’s crayons would be 2e, and three more than that would be 2e + 3 or 3+2e.
For problem 2 and 3 I ask a student to share and explain their thinking. I call on other students to share if they agree and disagree and why. Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.
I ask students, “What is the difference between an algebraic expression and an algebraic equation?” I want students to recognize that an expression shows the relationship between a variable and numbers. An algebraic equation sets two expressions equal to each other, like the balance we used in the previous lesson. Solving an equation is the process of figuring which values make the equation true. I also review that if a variable is right next to a number, the two are being multiplied together. I remind students they can also use the dot to show multiplication, but that we should avoid using the x symbol because it can be confused as a variable.
I tell students that they must read through the scenarios and decide whether equation a or b fits that scenario. Students participate in a Think Write Pair Share. I call on students to share out their thinking. I ask students, “How do you know that equation works? Why can’t the other equation work?” Even though both scenarios include the same values, I want them to recognize the difference between equation a and b.
Then I ask students to work with their partner to figure out the value of x for equation a and b. What value must x be in order to make the equation true? Students are engaging with MP2: Reason abstractly and quantitatively. Some students may guess a value and check the answer and then readjust as needed. Other students may use the inverse operation to figure out the value of x. I ask two students who used these two strategies to show and explain their work. I do not explicitly teach students to use the inverse operations step by step. In the past when I have taught this way, students have memorized the procedure oftentimes without having a deep understanding of what they were doing and why it worked.
We work on Part A together. I have a volunteer read the situation out loud. I ask, “What is going on?”, “What do we know?” and “What does the variable represent?”. If students struggle to see that the expression would be 2w, I substitute 11 in for w. What if the plant grew for 11 weeks, how tall would it be? Sometimes it’s easier for students to replace the variable with a number to understand what operation they would use. I ask students how to set up the equation and how to solve for w. I require students to use substitution to prove that their value for x works. I ask students, “Are there any other values that would also work for x?”. I want students to realize that for this particular equation, x can only be 8.
Students move into their groups. As students work, I walk around and monitor student progress. Students are engaging in MP2: Reason abstractly and quantitatively and MP4: Model with mathematics.
If students are struggling, I may ask them the following questions:
If students are correctly working through the examples, they can move onto the challenge questions.
During group work time, I walked around to monitor student progress. A couple groups struggled with Part E. One group misread the problem and created the expression “m + 10”. I asked them to reread the situation and explain their expression. I asked them, “If I had some money and then gave them 10 dollars, would I have more or less money?” They were able to recognize that the operation had to be subtraction. I asked them whether the expression should be “m -10” or “10 – m”. They explained that it would be “m – 10” since Lisa started with m dollars, not 10 dollars.
A few groups struggled with part H. At first a few groups wrote “3n” as their expression. I asked them, “How many lines does the poem have?” and “What would 3n mean in this situation?” I wanted students to recognize that 3n would represent each student reading the entire poem. Students realized that they needed to use division.
I ask students to turn to Part G. I ask students to share their thinking about the expression, equation, and the number of points Kevin Garnett made. I ask other students to share if they agree or disagree with the other students’ thinking and why. I want students to understand that if Paul Pierce made 26 points, then Kevin Garnett must have made 12 points since 12 + 14 = 26. As a challenge question I ask, “Garnett and Pierce made a total of 50 points. How many points did each player make?” Students participate in a Think Pair Share. I want students to be able to extend the pattern and see that Pierce still has to make 14 more points than Garnett. Some students may guess and check, while others may work backwards. If students are still working on it, I tell them they can continue thinking about it after they complete their ticket to go.
After we talked about Part G, I asked, “What if Paul Pierce still made 14 more points than Kevin Garnett, but together they scored a total of 50 points?” I had students talk about it with their partner for a minute. I started a table on the whiteboard and recorded student guesses. With each guess, I asked other students to use the information to find the other player’s points and the total to see if it matched the problem. It was interesting to see student guesses. They were able to keep the pattern of Paul Pierce’s points being 14 more, but they struggled to get the total to 50 points. Eventually a few students said that if Kevin Garnett scored 18 points, Paul Pierce would score 32 points and 18 + 32 = 50. Students were excited to take on the challenge and figure out a complex problem.
I collected the tickets to go to see what students understood about translating algebraic expressions and equations as well as what gaps in understanding they had. I corrected the tickets to go and grouped them in the following way:
These students struggled to model a situation with an expression. Often they were able to recognize that the answer to problem 3c was 7 hours, but their expression and equation did not match. This particular student created the equation “12 + h = 84” and then added 12 seven times to get 84. She did not see the disconnect between her answer of 7 hours and her own equation. Another common struggle was that students could create an expression, but did not know how to make it into an equation.
These students were able to create an expression to model most situations, but they usually made one mistake either writing their equation or answer 1c. Some students just copied their expression without creating an equation. Other students created an equation including the answer 7, rather than “h”. Other students made a mistake answering problem 1c. This particular student made a mistake dividing and said that 9 x 12 = 81. It is clear that she understands the algebra concepts in this lesson, but needs extra practice on her multiplication and division facts.
These students were able to correctly model situations with expressions and solve equations. They also understand the difference between an expression and an equation.
For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work. Most students were in the proficient and approaching mastery category. For the few students who were novices, they will be in a group together during the next lesson. I will work with their group to address the issues I’ve noticed from this ticket to go. I will also pass back the tickets to go to students so they can see and correct their work. |
Knockers made the journey to Farningham last Sunday. This was a particularly long journey for Calver, who had shown impressive commitment to make it from Wolverhampton. In a surprising turn of events, Knockers fielded 11 players despite two drop-outs on Sunday morning - John Mitchell and Anton deserve particular thanks for making up the numbers at very late notice. Finn lost the toss, however, the Farningham skipper chose to bowl first - a decision possibly influenced by Scott informing him that we only had one batsman.
Charlie and Scott opened the innings for Knockers. It quickly looked like bowling first was a good decision as both opening bowlers were swinging the ball considerably. Scott was unlucky to get a good ball, which he edged behind. Ziggy was next to the crease. While he managed to deal with the swing, he fell quickly following an unfortunate run out looking for two. JT was next in and after briefly looking solid he too edged behind. Knockers were struggling as this wicket led to a score of 16 for 3. However, Charlie had been looking increasingly comfortable, while he watched his partners fall at the other end. His strong cutting belied the fact that this was his first game back after a long break. One particularly nice drive led to Snellers wondering whether Charlie could be recruited to play Saturday cricket as well. Calver was the next man in and continued his fine form. He built a very good partnership with Charlie as both batsmen transferred the pressure back onto the bowlers. Charlie reached his 50 first and was quickly followed by Calver. Knockers looked set for a big total, having reached 118 off 18 overs at drinks.
This partnership continued after drinks, although Ziggy (possibly looking for revenge for the earlier run out) gave Charlie out stumped. This decision was overturned by Scott who pointed out that the keeper did not have the ball, when he removed the bails. Charlie fell shortly afterwards caught at square leg. This brought Snellers to the crease. He had spent the afternoon trying to talk himself down the order as he 'didn't feel like it', however the threat of being made to bowl was eventually enough to convince him to bat. Once Snellers started to bat he looked in very good form picking up two sixes in his first few balls. Calver also scored freely as they pushed Knockers past 200 in the 27th over. Calver was out LBW shortly after this, bringing John to the crease. John looked comfortable but fell to a good catch as he selflessly looked to push the rate. Kuts was next in. After playing his trademark sweep to his first two balls, he strayed from the sweep and was caught. Matt then formed a good partnership with Snellers as they both scored freely and put Knockers into a very strong position. Snellers, having reached his 50, fell and Finn joined Matt for the last over, which they took for 14.
Despite being 16 for 3, Knockers reached 294 for 8 in 36 overs (the club's fourth highest total).
Knockers were eager to get into the field, knowing that they had a good total on the board and a strong bowling line up. Anton and Matt opened the bowling. Both bowled with good control and restricted the Farningham batsmen effectively. Matt picked up the first wicket as he bowled one of the openers. Kuts took over from Anton down the hill as Matt continued to induce play-and-misses from the other end. Kuts quickly found his rhythm and bowled very nicely. The RRR climbed to over 10 by the time Finn decided to replace Matt and Kuts with the deception of Scott and John. John made a break-through as he tempted the remaining opener out of his crease for Calver to take off the bail. Scott almost picked up a wicket: Farningham's no.3 having already made 50 of their 80 runs hit the ball powerfully to Kuts, who unfortunately dropped a hard chance, however, the batsman fell over while playing the shot and (sadly) had to retire hurt with a sprained ankle.
As Farningham were keeping up with the run rate, Finn turned to Ziggy and Charlie to get some control back. Ziggy quickly delivered picking up three quick wickets of dangerous-looking batsmen. He was well supported in the field as Finn held a catch at deep mid-wicket and Snellers held two at long-on. Snellers' second catch was a particularly impressive diving catch, which even he was impressed with. Ziggy did not enjoy his wickets as much as the rest of the team as he took to celebrating with his head in his hands leading Scott to wonder whether he was exchanging brain cells for wickets. Charlie bowled well up the hill and was unlucky not to get an edge. As Ziggy was struggling, the opening bowlers returned to finish the game. Both bowled well and Matt picked up another wicket as finally a flashing stroke resulted in an edge which was held by Calver.
This was another good win for Knockers to extend our unbeaten run.
|Batsman|| How Out|| 4s|| 6s|| Runs|
|Scott Landers|| caught|| 0|| 0|| 0|
|Charlie Adam|| caught|| 0|| 0|| 73|
|David Bowen|| run out|| 0|| 0|| 2|
|James Thompson|| caught|| 0|| 0|| 3|
|Richard Calver|| lbw|| 0|| 0|| 81|
|Paul Seldon|| caught|| 0|| 0|| 70|
|John Mitchell|| caught|| 0|| 0|| 5|
|Christian Kutner|| caught|| 0|| 0|| 6|
|Matt Watts|| not out|| 0|| 0|| 23|
|Finn Kinsler O'Sullivan|| not out|| 0|| 0|| 7|
|Anton Kinsler O'Sullivan|| dnb|| 0|| 0|| 0|
|Bowler|| Overs|| Mdns|| Runs|| Wkts|
|Anton Kinsler O'Sullivan|| 6.0|| 3|| 17||0|
|Matt Watts|| 8.0|| 3|| 24||2|
|Christian Kutner|| 5.0|| 1|| 25||0|
|John Mitchell|| 5.0|| 0|| 58||1|
|Scott Landers|| 4.0|| 0|| 50||0|
|Charlie Adam|| 4.0|| 0|| 29||0|
|David Bowen|| 4.0|| 0|| 29||3| |
So whats he do now. Now I am gonna turn to the publication you've seen. The publication I assigned. It really consists of two parts. Number one. Number two, it is dialectic in the classic philosophic sense. He's taking a series of hypotheses that he thinks every Cartesian will accept and showing the Descartes claims in his rules of impact are false.
Okay, so he's going to show in effect Descartes is inconsistent with his own positions because taking, working with Cartesian hypotheses you can show that the very claims Descartes makes are false. So what are the hypotheses. Number one, anybody once moved continues to move if nothing prevents it at the same constant speed along a straight line.
Okay, that's of course our principal of inertia, etc. Hypotheses two. Whatever the cause of the rebound of hard bodies for mutual contact when they collide with one another we pause it, and when two equal bodies with equal speed collide directly with one another from opposite directions, each rebounds with the same speed with which it approached.
That's a Cartesian principle, I think it's rule number one. If they approach one another this way, and they're hard bodies and they're equal in size, they bounce off the same way. Hypothesis three. When two bodies collide with one another, even if both together are further subject to another uniform motion, they will move each other with respect to a body that is carried by the same common motion, no differently than if this motion extraneous to all were absent.
That's the principle of relativity. Okay, so in effect saying, the same thing happens whether you refer to one frame or another. Now what he does is use number 1 and number 2 to take other problems, change the frame of reference so that it meets the conditions of one and two, solves the problem right off of those, transfers back to the original problem.
So, we have to equal bodies approaching one another at different velocities. Adjust one in the boat, so that, relative to the boat, they're moving in exactly the same speed. Invoke hypothesis two to get the solution relative to the boat. Now transform back relative to the ground. Now I say this generally, we've got two frames of reference, they're moving with respect to one another.
We've got a prototypical solution for one frame of reference. Convert any other problem into the prototypical frame problem for that frame of reference, by simply adjusting the philosophy of the other frame of reference and that's how we get solutions. And the first two propositions reject two principles of Descartes.
I can't remember which ones they actually are at the moment, so I have to look at notes. Proposition one, it says, "Exchange at rest." So what happens, according to Descartes, on exchange of rest is, as I recall, if this ball is at rest and this one strikes it it bounces back perfectly and the one at rest doesn't change, if they're equal size.
The contest is won by the one at rest. To the contrary, they come back at opposite speeds. And then the other rule that's false, exchange at unequal speeds, that's rule three. At any rate, the point that's made, propositions one and two show that Descartes' rules six and three are wrong.
And he gets that only out of the relativity principle and two Cartesian principles. Now what you can't guarantee is Cartesians will accept the relativity principle. That's the only fly in the ointment in this part at all, but he's now gonna do some more. Hypothesis Four. If a larger body meets a smaller one at rest, it will give it some of its motion, and hence lose something of its own.
Descartes said, if a smaller, if a larger body meets a smaller body they go off together. Oh no, I'm sorry, I have, I'm misleading you. That hypothesis is Descartes. He says that very thing. The larger body meets a smaller. Proposition three. A body, however large, is removed by impact, is moved by impact by a body, however small, and moving at any speed.
Descartes had expressly denied that. If one body's larger than another, the smaller one can't make it move. Now you could see how he's gonna solve that. He's gonna take this case, change the frame of reference and then change it back. And sure enough, when you do that, the smaller body's causing the larger one to move.
Hypothesis five. When two bodies meet each other, if after impulse, one of them happens to conserve all the motion that it had, then likewise nothing will be taken from or added to the motion of the other. That's a Cartesian principle because the total motion has to stay the same.
Consequence. Whenever two bodies collide with one another, the speed of separation is the same with respect to each other as that of approach. That violates almost every one of the Cartesian rules. Okay. Two more consequences before we get fancy. This of course is the one that you already saw.
If two bodies each collide again at the speed at which they rebounded. You didn't see this was the next one. Which they rebounded from impulse, after the second impulse each will acquire the same speed at which it was moved toward the first collision. That is if you could somehow or other collide and then collide a second time you restore the original position.
And then proposition six is the one. When two bodies collide with one another, the same quantity of motion in both taken together does not always remain after impulse what it was before, but can be either increased or decreased denying Descartes conservation of motion. He does not, in the paper you read, state the principle of conservation of momentum.
That's only in the 1669 paper. But he does state some of the, the kinetic energy principle, the masses times velocity squared. Now, he can't solve the general problem with just Cartesian principles. So he's got to add something more to solve the general problem. What he adds is his version of Torricelli's principle, which we're gonna hear again and again as the night goes on.
For in mechanics, it is a most certain axiom that the common center of gravity of bodies cannot be raised by a motion that arises from their weight. So if we get a motion from a falling, and you go back up, it can't go higher. Okay. And with that he can do a reductio proof for the following case.
If two bodies, the speeds of which correspond inversely to the magnitudes. So the larger body is moving slower, the smaller body is moving faster, in proportion to their magnitudes. Collide with each other from opposite directions, each will rebound at the same speed at which it approached. And he proves that by proving if they don't do that, you will violate this version of Torricelli's principle.
Get a reductio on each part leaving you with, this is the case. Now you can see how he can do any problem. Because, given any two balls approaching one another at any speed of any magnitude. He just adjusts the speed so that the conditions of proposition eight are satisfied.
They're approaching at a speed inversely proportional to their masses. Mass is not his word, that's Newton's word, later. Inversely proportional to their bulks, you solve it in that frame of reference. You take the original problem, transform it into a frame of reference that meets proposition eight, solve the problem there, take it back into the other frame of reference, you got the general solution, and that's how you got that formula I showed before.
So what have we done here? We've taken a Cartesian problem: Impact. Use Cartesian assumptions plus one Galilean assumption if you'll let Torricelli's principle be Galilean and we've given a solution to, a solution to impact for hard bodies, generally of all kinds, hit on impact. And I say it's Galilean, because what you actually do with the Torricelli Principal is you let objects fall from height, that's the way they're going to acquire their velocity.
So if they have acquired their velocity and they're now in impact and they don't rebound in exactly the right way, they won't have enough velocity to get back up. Fair enough? Now, I'll make the point I really wanna make. We've got a Galilean solution to a Cartesian problem.
And to cap it all off, we've got a Galilean way of testing it. And what he does, of course, in the paper you read, he didn't just do one body. If 100 bodies in double proportion are given in order. That's consecutive squares. And the motion begins from the greatest.
One finds by tearing out the calculation according to the precept of the rule set forth in proposition nine. But abbreviated in the compendium that the speed of the smallest body is to the speed at which the greatest is moved approximately as 14 billion, 760,000 to 1. A hundred bodies in a row going from the largest to the smallest in geometric progression gets you a 14 million amplification of the effect of the largest one on the smallest one.
That's a nice striking thing to test. And of course, that's the point of this kids' toy, but of course we need to do it with different sized spheres and to test it that way. Okay, but again, perfect Galilean form of testing, a really stunning result. Now, one is not gonna do it with 100, but two or three would be very compelling.
And by the way, that's how they've started testing this, with ballistic pendiums at the royal society, etc. We'll talk about that later. The royal of society results are disappointing. They had a lot of trouble setting this experiment up. Huygens says he's done the experiment and Christopher Wren said he had done the experiment at Oxford and I believe it's possible to do.
I just don't think the royal society carried it out as well as it might. Last thing about collision. Then we go on to another topic. This phrase at the bottom here, "The result is not alien to reason and agrees above all with experiments.". I want you to hear that phrase 'cause it's gonna come up repeatedly in all of Huygens' work.
Not alien to reason. As we sit back and adopt reasonable principles consistent with the world around us as we normally observe it, we develop mathematics, we get striking results out of the theory. Once we have those, those are testable, when it agrees with those two you're covered. Fair enough?
That's the style. It's not really different from Galileo in style. I wanna go a little bit longer. That's all I'm gonna do about percussion. This is the nearest we get to a general solution till Newton's Principia. And Newton's Principia then gives us the general solution that we use today where we no longer require hard bodies.
And, you'll see it first week of next semester. I hope you're impressed. Now, Wren. I repeat, Christopher Wren got the same general solution. He didn't draw the consequences. Okay. In particular didn't draw the consequence. Decartes' conservation of motion is wrong, but there is a replacement for it, conservation of linear momentum in any one direction. |
Math 103 Summer, 2003
work in progress!
SUBJECT TO REVISION!
Homework Assignments due:
(* = interesting but optional)
General Reading Assignments (Revised 7-6-2003)
||7/7 Pythagorean Activities
5-8, 11, 12, *13
||7/9 1.2: 1-3,
Web Surfing Activity.
|7/10Work on a portfolio entry.
4.1: 7, 8, 9
||7/15 4.1: 14
(based on 7)
4.1: 10, 24
5.1: 6 (g,h,i)
Design Symmetry Assignment
(now due 7/17)
|7/17 Tesselation Thursday!
Assignment on symmetry.
Start on Lineland paper-due
||7/21 4.2: 9,10
Models of the platonic solids from templates.
Cave Metaphor- online.
Finish 8.1: 6,9
Begin Plato essay due 7-28
|7/24 Dual Tessellations.
||7/ 28 Plato essay
||7/29 See assignment on
Make two tori: one from two annuli, one from a single "rectangle."
assignment on the torus and maps and surfaces
See assignment on Zeno.
|7/31 Symmetry Day.
See assignment on Projection
13.4 : 2-6
||8/4 10.1: 4,5
|8/5 Portfolios and projects due
by 5 pm.
|8/6.See assignment on coordinates
Make a Desargues' Configuration.
See Final paper assignment
||Chapter and pages for Reading
||Comments, Web Sites to Visit, and other things
|7-7 to 7-9
K & M
|Introduction, Preface, and Part I.
1.1 Measurement pp 1-8
1.1 pp 8-12
1.2 Polygons pp13-16
is available on the web. Perception
Over 30 proofs of
the Pythagorean theorem!
Applets that visualize proofs of the Pythagorean Theorem
Japanese Site with Tangram Puzzles
Here's a web page with many
annotated Tangram references
references related to scissors congruence- dissections.
|7-9 to 7-14
K & M
4.1 Reg. and Semi Regular Tesselations pp 85 - 91
4.1 Dual Tilings pp91- 93
|A wealth of materials can be found by going to this Tesselation
This might be a good time to visit Rug
patterns and Mathematics exhibit plus...
Thursday is Tessellation
Day: Wear to class clothing that has a tiling pattern on
|K & M
K & M
|6.1 Flatlands pp 180 -184
5.1 Kaleidoscopes (1 mirror)127-130
5.1 (2 mirrors)130 - 134
5.1 (3 or 4 mirrors) 134 -135
5.2 Point symmetry 138- 146
5.3 Frieze Patterns 147-155
* 5.4 Wallpaper Patterns
*5.5 Islamic Lattice Pattern
|You might want to visit the Kali:
Symmetry group page now .
|4.2 Irregular Tilings pp94-107
*4.3 Penrose Tilings
7.1 Pyramids, Prisms, and Anti Prisms pp 208 -215
7.2 The Platonic solids pp 216-221
7.3 Archimedean Solids pp 224-228
of the cave. (On line.)
|You might want to look at Penrose tilings by downloading Winlab
You can look at polyhedra by downloading Wingeom
Platonic solids is an interesting site with Java viewers for
interactive manipulation created by Peter Alfeld of Univ. of Utah.
|7-23 to 7-24
||7.4 Polyhedral Transformations
8.1 Symmetries of Polyhedra
||6.2 The Fourth Dimension
10.2 Optical illusions
11.2 Map Projections
Paradoxes and the infinite.
The Fourth dimension
Networks and Euler's formula
Euler's formula, the torus.
Visualization of 4d hypercube (Java applet).
|7-31 to 8-4
|pp 1-3, Sections 3,6, 11, 13
13.3 More on Surfaces
|More on Euler's applications.
Symmetry Day: Bring to class an example of a
natural or synthetic physical object that has a non- trivial group of symmetries
together with your description of those symmetries. You may bring either
the physical object itself or a sketch of the object.
Durer and perspective drawing
Surfaces in topology
The Moebius strip, The
Klein bottle, orientability, and dimension.
Constructing surfaces in general
The Infinite (Zeno's
Paradoxes and the infinite.)
|8-5 to 8-7
|Sections 11, 13
|Projection and Ideal elements.
Color problems .
||Continuation of Projective geometry
Conics, Euclidean and Non-Euclidean Geometry
List for Portfolio Entries
The following list contains suggestions for finding resources as
well as the names of resources that may be used for one or more portfolio
entries. Before reading an article in one of these resources thoroughly
it is advisable to scan it quickly to see that it contains something of
interest to yourself. Your portfolio entry can report on the content of
your reading, illustrate it by examples, and/or follow up on it with your
own response and creativity.
The content of the portfolio
entry should relate specifically and directly to some visual mathematics.
Personal observations , philosophical musings, and aesthetical judgments
are not adequate connections to something visual by themselves to qualify
as mathematical content.
These articles may also be useful in developing a deper level of understanding
on a topic which will suppport your term project. I
will add to this list as the term progresses.
Several chapters from the course text will not be covered in class but
can be used for portfolio entries. An entry based on our text should report
on a selection of the included exercises along with the content of the
Use my collection of Visual
Mathematics web sites for surfing visual mathematics and geometry.
Use articles from old Scientific American magazines (located outside my
office at Library 48)
(Older issues) Martin Gardiner's articles are usually short and clear enough
to provide material for one or even two even entries.
(More recent issues) Ian Stewart 's articles are similar and about as playful
as the Gardner pieces.
Some issues have had articles on special topics that are relevant
to our interests. These are usually longer and require a little more effort
to digest - though well worth the effort.
"Topology" by Tucker and Bailey, 1950, pp 8-24.
A number of liberal arts / mathematics textbooks contain chapters that
would be suitable for reporting.
Mathematics: the Man-made Universe by Sherman Stein.
Excursions into Mathematics by Beck, Bleicher, and Crowe.
What is Mathematics? by Courant and Robbins.
The World of Mathematics by Newman.
The library has a collection of films and videos that are relavant to our
For All Practical Purposes (COMAP)
Some of the history of mathematics videos from the Open University Series
There are several non-text mathematics books and collections of essays.
Two of these are - K. Devlin's Mathematics: The Science of Patterns
and T. Banchoff's Beyond the Third Dimension. Martin Gardiner has
many books full of puzzles and recreations many of which are relevant.
The Problems of Mathematics by Ian Stewart.
The Mathematical Experience by Philip Davis and Reuben Hersh
Special Assignment Problems and Projects
(These may be assigned - watch for due dates on assignment schedule)
1. We are still trying to describe the cube to a Flatlander,
this time using the transformation of the framework of the cube onto the
plane by central projections.
Show the image of the cube on the plane under the following projection
A. One square of the cube is in Flatland and the center of projection
is above the cube directly over the center of the cube.
B. Only one edge of the cube is in Flatland and the center of
projection is above the cube directly over the center of the edge in Flatland.
C. One square of the cube is in Flatland and the center of projection
is higher than the cube and not directly over any part of the cube.
2. Parallels on the torus and the sphere. Let's call an arc on a sphere
(or the torus) a sline segment
if it arises from a cross section of the sphere
(or the torus) by a plane that passes through the center of the sphere
( or the torus).
A. Draw a figure showing some sline segments on a sphere with the planes
through its center and some sline segments on a torus with the planes through
B. Is the following statement true for any sphere? for any torus?
"Any two sline segments on a sphere (or a torus) can be extended so
that they will intersect."
3. Find two drawings, paintings, prints, or photographs that have noticable
perspective in the composition. Make a sketch or photocopy of the works
and locate at least one "infinite" point on the horizon (ideal) line on
each of your figures. Find at least three lines in each of your figures
that meet at the infinite point.
Due Thursday 8-7.
During the sessions we have covered many topics in class and through the
readings. Choose two topics we have studied for examples in writing a paper
(1-3 pages) discussing one of the following statements:
A. The study of visual mathematics in two
dimensions has much in common but also some noticable differences with
its study in three dimensions.
B. The amazing thing about mathematics is
how it is able to turn even the simplest things into abstractions and can
make the subtlest of concepts clear through a figure.
Due Wednesday 8-6:
Coordinates and conics.
Coordinate geometry is a tool used in intermediate algebra courses
to investigate the conic curves. Recall the basic idea is that a point
with coordinates (x,y) will lie on a curve in the coordinate plane if and
only if the numbers x and y make an equation determining the curve true.
For example, a circle with center (0,0) and radius 5 is determined by the
X 2 + Y 2 = 25. We can check that the point
with coordinates (3,4) is on the circle by verifying that 3 2
+ 4 2 = 25.
1. Each of the following equations determines a conic curve.
Plot 10 points for each equation on a standard rectangular coordinate graph.
Connect these points with straight line segments to give a polygon that
will approximate the curve.
a. 4X 2 + Y 2 = 25
[an ellipse] b. X 2
- Y 2 = 9 [an hyperbola] c.
X 2 - Y = 4
2. Draw three separate projective planes including a system of
coordinates with the horizon line and lines for X=1, 2, 3, 4, and 5, and
Y=1, 2, 3, 4, and 5 as well as the X and Y axes. For each of the previous
equations, plot 6 points on a projective coordinate plane that correspond
to 6 of the 10 points plotted previously on the standard plane. Connect
these points with straight line segments in the projective plane.
1. Suppose three lines l , m, and k form a triangle.
[Draw a large figure to illustrate this situation.] Draw ten points on
line l perspective with 10 points on line m with center O.
Use these ten points on line m to draw 10 points on line k
in perspective with center O'. Draw the lines connecting the corresponding
points on line l and k. Describe the figure that these lines
2. Draw a figure showing a tessellation of the projective plane on one
side of the horizon line by parallelograms.
3. Draw a figure illustrating a black and white chess board in perspective.
Central Projection. [see Figure 7 in A&S.]
On a line mark 11 points that are separated one from the next by one
inch. At the middle point draw a circle of radius one inch as in Figure
7. Find the points on the circle that correspond to the points on
the line as in Figure 7. Describe the relation of a point on the
circle to the corresponding point on the line with regard to the point
O where the circle and the line touch.
July 30. Look up "Zeno's
Paradoxes" in the Encyclopedia (Britannica). Draw a figure that
illustrates the paradox of Achilles and the Tortoise. Describe a common
context today to which Zeno's argument about Achilles and the Tortoise
could be applied. Using your situation, discuss where the accumulation
of small and infinitely divisible intervals is incorrectly compared with
the accumulation of equally sized intervals.
Tuesday, July 29 1. The fourth dimension can be used
to visualize and keep track of many things involvimg four distinct qualities
that can be measured in some fashion.
A. For example, a 13 card bridge hand can be thought of as a point
in four dimensions where the coordinates represent the number of cards
of each suit present in the hand. In this context the point with coordinates
( 2, 4, 6, 1) might represent a hand with 2 clubs, 4 diamonds, 6 hearts
and 1 spade.
Using this convention discuss briefly the following representations
of bridge hands: (0, 0, 0, 13), (0, 0, 6, 7), (3, 3, 3, 4).
Suppose a bridge hand is represented by the point with coordinates
(x, y, z, w).
Explain why x + y + z + w = 13.
B. Describe another context where four dimensions can be used in representing
some features of the context.
2. Hypercubes in Higher Dimensions.
The 16 vertices of the 4-dimensional hypercube can be described by
the collection of ordered quadruples (a,b,c,d) where the numbers
a, b, c, and d are either 0 or 1.
Write a similar description of the 8 vertices of the 3-dimensional
Write a description of the vertices of the 5-dimensional hypercube.
How many vertices does the 5-dimensional hypercube have? How many vertices
does the 6-dimensional hypercube have? How many vertices does the
10-dimensional hypercube have? What can you say about the vertices for
the hypercube of dimension N?
Wednesday, July 30. Surfaces.
A. Describe 5 physical objects that have surfaces that are topologically
equivalent to a (one hole) torus. Bring one example to class on Tuesday.
B. Describe 5 physical objects that have surfaces that are topologically
equivalent to a torus with two or more holes. Bring one example to class
Bring in 3 different world maps. Describe
how each map deals with lines of longitude, latitude, and the poles.
Thursday, July 31st, is Symmetry Day: Bring to class
an example of a natural or synthetic physical object that has a non-trivial
group of symmetries together with your written description of those symmetries.
[You may bring either the physical object itself or a sketch of the object.]
Wednesday, July 30. Casting Torus Shadows on Flatland.
The sphere is still trying to explain some of the features of the torus
to a Flatlander. This time the sphere has decided to show the Flatlander
different shadows that are cast by the projection of the torus onto Flatland.
A. Draw three different shadows that the torus could cast.
B. Do you think it is possible to make a torus that would cast a shadow
on Flatland that completely covers a circle and its interior? If so, describe
some of the features of such a torus. If not, give some reasons for your
belief. In other words, is it possible that a Flatlander might mistake
a torus for a sphere based on the shadow it casts?
A. Draw the three regular tilings and one semiregular tiling of
B. Use a red pencil to mark the center of each of the polygons.
Join any two centers of polygons that share a common side. This should
give new tilings of the plane in red.
C. Describe the new tilings you obtain in part B. These are called
the duals of the original tilings.
D. For each of the 8 tilings (from parts A and B) make a list of
E. Compare the symmetries of each tiling with the symmetries of
its dual tiling. Explain any connections you notice between these symmetries
Plato and Shadows: The Greek philosopher Plato describes a situation
where a person lives in a cave and can only perceive what happens outside
the cave by observing the shadows that are cast on the walls of the cave
from the outside.
Write a brief essay discussing a situation in the contemporary world
where indirect experiences are used to make observations. How are the observations
made? How are they connected to the actual situation? Do you think the
inferences made from the observations are always accurate? [3 or 4 paragraphs
should be adequate.]
7-21 Lineland Paper: Imagine you are a Flatlander talking
to a Linelander.
Write an explanation of symmetry to a Linelander from the point of
view of a Flatlander. Discuss and illustrate the kinds of symmetry that
are possible in Lineland. Which Flatland symmetries (if any) would you
associate with Lineland symmetries? Explain the association briefly.
Here are some terms you might use in your discussion: Reflection
Rotation Translation Orientation
7-16&17-03 Find (or create) three graphic designs
(in advertisements, logos, or icons) that have (i) reflection symmetry
only, (ii) rotational symmetry only, and (iii) reflection and rotational
7-16&17-03Classifications by symmetry:
It is often useful to classify visual objects by their
symmetries. For example, the letter "T" as it appears on this page has
only a reflection symmetry determined by a vertical line, whereas the letter
"I" has two reflection symmetries and one rotational symmetry of order
2 (a half turn).
Group the following letters together in different classes determined
by the number and types of symmetries they have as printed on this page.
[It is up to you to determine the appropriate classes.]
A B C D E F G H
I J K L M N O P Q R S T U V W X Y Z |
In this paper we discuss the conservation of angular momentum of light in single scattering of circularly polarized light from a spherical, non-absorbing particle. We show that the angular momentum carried by the incident wave is distributed in the scattered waves between terms related to polarization or spin and to orbital angular momentum, respectively. We also show that, in all scattering directions, a constant ratio exists between the flux density of the total angular momentum and the intensity.
©2006 Optical Society of America
In elastic scattering from a non absorbing spherical particle, two parameters of the electromagnetic field are conserved: the energy and the angular momentum component along the propagation direction. Linear momentum as a whole is also conserved, of course, but some of it is transferred to the particle leading to radiation pressure. The conservation of energy (which is a scalar quantity) leads to a normalization condition for the integrated energy flux density, which is further used in defining the scattering cross section[1, 2]. Similarly, the conservation of angular momentum should be related to the angular momentum flux density. The continuity conditions for the angular momentum density can be described by three equations (one for each component of the vector) or one equation for a tensor.
We are interested in calculating the angular moment flux density of the electromagnetic field which results from scattering of a circularly polarized wave from a non-absorbing spherical particle. It was demonstrated that scattering of circularly polarized wave does not exert torque on the particle and that transfer of angular momentum from the field to the particles is mediated only by absorption. Therefore, the angular momentum of the field is preserved and it should be interesting to know how the flux density of the angular momentum is distributed between the spin or polarization term which will be henceforth designated by s and the orbital angular momentum (OAM), which will be denoted by l. The total angular momentum flux density, j, is given by the sum of these two terms. The problem of torques applied to particles was treated extensively in the context of particle manipulation or “optical tweezers” . In this paper we will limit ourself to cases in which no torque is applied and we will discuss the angular momentum carried by the scattered electromagnetic wave.
2. Scattering effects on spin angular momentum
Let us consider an incident plane wave which is monochromatic (angular frequency ω), left circularly polarized, has an amplitude E 0 and propagates in the direction ẑ (the ‘hat’ denotes a unit vector). This is the simplest example of a paraxial wave which has the general form
(where F (x,y,z) is a slowly varying spatial envelope). For such a wave one can employ expressions for angular momentum flux density of a paraxial wave[6, 7, 8] and write the z component of the angular momentum flux density can be written as
The first term relates to the transverse distribution of the field and is the OAM term. The second term reflects the angular momentum carried by circular polarization is zero for linear polarization and. For a class of paraxial beams for which the transverse field distribution can be written as
and recognizing that we find that
In Eq. (4)σ is the wave helicity: +1 for left circular polarization, -1 for right circular polarization and 0 for any linear polarization. The beams described by Eq. (3) are called vortex beams and they exhibit an axial singularity since the phase is indeterminate on the beam axis. Due to destructive interference, this phase singularity leads to an intensity null. The integer factor m is called the topological charge of the singularity and it is an invariant of the propagation. This follows from the fact that m relates directly to the orbital angular momentum and, under the paraxial approximation, each term of the angular momentum is conserved independently.
The flux density of the angular momentum of the incident plane wave which is carried by the spin term is equal to I 0/ω where I 0 = ε 0 /2 (ε 0 being the permittivity of vacuum and c the velocity of light). This corresponds to a quantum description in which each photon in the incident field carries a spin angular momentum of +ħ. The rate at which the angular momentum is removed from the incident field can be derived from the the scattered power and is given by σsc I 0/ω ẑ. This is the source term for the angular momentum of the scattered field and it should be recovered by integrating over all directions the angular momentum flux density of the scattered field.
From general scattering theory it is known that the scattered electric field in the far zone can be written as
where the matrix S⃡ is the scattering matrix in the circular polarization basis, the L and R subscripts designate the left and right circular polarizations. In Eq(5), k is the wave number and r is the distance from the center of the scattering particle which is both the origin of the coordinates frame and the reference point for the angular momentum calculations. The scattering matrix in the circular base can be related to the more usual amplitude scattering matrix given in terms of the parallel and perpendicular electric field components with respect to the scattering plane. For a spherical particle, this scattering matrix is diagonal with elements S 2 (θ), S 1 (θ) and the relation is
The notation used in this paper is illustrated in Figure 1.
It is important to remember that scattered field components are given in terms of a coordinate frame which is rotated such that the local scattered field is given in terms of component is a plane which is perpendicular to the scattering direction. However, in the following it is necessary to express all the scattered waves in the same reference frame such that spatial derivatives can be made consistently. The most convenient system of coordinates is the one associated with the incident wave and, in this case, it is found that the rotation introduces a phase term exp(iϕ) where ϕ is the azimuth angle. We note here that in the case of an incident wave which is right circularly polarized, the phase factor is exp(-iϕ).
The scattered field can now be written as
and expanding further the left and right circular unit vectors in terms of the locally transverse unit vectors
one finally obtains
where the notation
Sθ (θ) = [SLL (θ)+SLR (θ)]/√2 and Sϕ (θ) = i[SLL (θ)-SLR (θ)]/ √2 has been used.
Having found E, we can now proceed to calculate the angular momentum flux density through a radially oriented infinitesimal area in the radiation zone. When the wave is approximated locally as a plane wave the spin term can be found using Eq. (2) :
This result is physically reasonable: the spin is simply the difference of intensities of the two radially outgoing orthogonal circular polarization components divided by the angular frequency of radiation. The common phase term exp(iϕ) does not play a roll in this calculation.
Due to axial symmetry, all the components of s average to zero when an integration over the angles is performed except for the z component which is
For a Rayleigh scatterer, one can immediately find that the expression in Eq. (11) reduces to
In order to calculate the total scattered flux, the spin flux density must be integrated over a sphere of radius r to obtain
The bar superscript in Eq. (13) indicates integration over a sphere of arbitrary radius, in the far field. Notably, one can see that only half of the angular momentum flux removed from the incident wave is contained in the spin term. Of course, this conclusion is similar to the results obtained for the case of a field radiated by a rotating dipole.
In order to evaluate the more complex case of a Mie scatterer let us rewrite Eq. (11) as
where IL (θ),IR (θ) are the scattered intensities with left and right hand circular polarization respectively and V(θ) is the fourth Stokes parameter. In our case the incident wave is characterized by a Stokes vector of the form [1,0,0,1] while the scattered wave Stoke vector is [F 11 (θ) ,F 21 (θ) ,F 34 (θ), F 44 (θ)] where F⃡ (θ) is the angle dependant 4×4 scattering matrix which is block diagonal for spherical particles. Since we obtained that for a left hand circular incident wave V (θ) = F 44 (θ) , we can calculate how much of the angular momentum flux density is contained in the spin term, normalized by the scattered angular momentum:
It is worth mentioning that this expression is similar to the asymmetry factor defined by MacKintosh and John (designated there as A) but note that their definition lacks the cos(θ) in the numerator and therefore it reflects the overall helicity flip rather then the z component of the spin.
Numerical evaluations of Eq. (15) based on the Mie theory suggest that a good approximation of this ratio is
where g = 〈cos(θ)〉 is the so-called scattering asymmetry parameter. For highly forward scattering which is helicity preserving, this ratio is close to 1 meaning that the total angular momentum is concentrated in the polarization term. Some examples are presented in Fig. 2.
The calculations were made for several relative indices of refraction: 1.09 which is comparable to the case of silica spheres in water, 1.18 which represents polystyrene spheres in water and a hypothetical higher contrast material with a relative index of refraction of 1.25. One can observe that the agreement with the dependence suggested in Eq. (16) is excellent for g below 0.1 and above 0.75. A higher order polynomial which will be better in the intermediate range is of course possible. It is also interesting to examine the mean helicity (the Mackintosh_John parameter) of the scattered field which is illustrated in Fig 3. One can note that up to g = 0.7 there is a good linear dependence with a slope of about 1.3.
3. Scattering effects on orbital angular momentum
In order to evaluate this expression we employ the spherical coordinates form of the gradient operator and recall that the unit vectors have to be differentiated as well; for example and Also note that the common phase term, exp(iϕ), is important in this case and should be considered in the differentiation. Accounting only for the terms that contribute to the z component, we obtain after some algebra that
This result, together with the one expressed in Eq. (11) indicate that sz (θ) and lz (θ) sum up to an expression which is proportional to the scattered intensity. For each θ, the total angular momentum flux density is simply the intensity divided by the angular frequency of the radiation. Integrating over a spherical surface in the far field leads indeed to a manifestation of the conservation of angular momentum flux:
Moreover, we note again that at each θ, the ratio jz (θ)/I (θ) is constant. In the quantum description one may say that a photon scattered in any direction carries the same angular momentum as an incident photon, but, in different directions it is distributed differently between the spin and OAM terms. It is now evident that in the forward and backward directions the angular momentum is carried only by the polarization. In other words the helicity is fully preserved in the forward direction and it is fully reversed in the backward direction. For some angles, which for a Rayleigh scatterer is only 90°, on the other hand, the scattered light is linearly polarized and therefore does not carry any spin angular momentum. At these angles the angular momentum is carried only by OAM term.
Figure 4 illustrates the three dimensional distribution of the spin term (relative to the scattered intensity) plotted as a function of scattering direction for several values of the size parameter x = 2πa/λ, where a is the radius of the particle and λ is the wavelength. The orbital angular momentum distribution is the complementary one (as it will be given by one minus the spin). It is interesting to note that for small size parameters the changes in the normalized distribution are small. For large particles on the other hand, in which Mie resonances are dominant, one can observe that there are directions in which the spin term is negative- meaning that the normalized orbital term is larger then 1. The transition to such behavior happens for particles with a size parameter of about π (i.e. a particle diameter which is approximately equal to the wavelength).
In order to illustrate the distinction between spin and orbital angular momenta one may use the following “gedanken” experiment. Let us consider a small dielectric sphere which is slightly absorbing, and is placed in the far field of a Rayleigh scatterer. In the exact forward direction the particle will rotate about itself due to the absorption of circularly polarized light. At of 90° with respect to the direction of incidence, on the other hand, the test particle will “orbit” the scattering particle due to the phase gradient in the scattered field. We would like to emphasize that this effect is not due to absorption and that, of course, the test particle will experience a radial force due to radiation pressure.
Finally, we note that the orbital angular momentum of scattered light can also be interpreted as the result of a slight direction dependant shift of the apparent origin of the scattered waves, a shift which introduces an “impact parameter” of the order of λ/2π. The origin of this shift can be identified in the classical electromagnetic theory where it is known that the Poynting vector of radiation from a rotating dipole (equivalent to a Rayleigh scatterer illuminated by a circularly polarized light) spirals in the near field. As a result the far field radiation seems to be emerging not from the center of the dipole but from a shifted position as illustrated qualitatively in Figure 5. In this figure the incident light direction is into the page and the scattered wave is observed in a transverse plane.
The angle α is approximated by λ/2πr and the angular momentum at the point r can then be calculated to be krα = 1. This means of course that the orbital angular momentum carries all the angular momentum. In our gedanken experiment, the slight tilt of the k-vector induces a rotating motion of the test particle placed at r. This is the mechanism which couples angular momentum from the electromagnetic field to the medium.
In this paper we elucidated the mechanism by which the scattering from a spherical non-absorbing particle is transferring angular momentum carried by an electromagnetic field from the spin term to the orbital angular momentum term. The relative part of the angular momentum contained in the spin term can be calculated exactly and it ranges from 0.5 for a Rayleigh scatterer to 1 in the case of a highly forward scatterer. In any scattering direction the ratio between the total angular momentum flux density carried and the intensity, is constant. Moreover, this ratio has the same value as the corresponding one evaluated for the incident wave. The orbital angular momentum complements the spin angular momentum and in some directions may carry all the angular momentum. The orbital angular momentum carried by the scattered field mediates the transfer of angular momentum to the medium.
References and links
1 . H.C. van de Hulst , Light Scattering by Small Particles ( Dover publications, New York , 1981 ).
2 . R. G. Newton , Scattering Theory of Waves and Particles , second edition ( Dover publications, New York , 2002 ).
3 . S.M Barnett , “ Optical Angular-momentum flux ,” J. Opt. B: Quantum Semiclass. Opt. 4 S7 – S16 ( 2002 ). [CrossRef]
4 . P.L. Marston and J.H. Crichton , “ Radiation torque on a sphere caused by a circularly polarized electromagnetic wave ,” Phys. Rev. A 30 2508 – 2516 ( 1984 ). [CrossRef]
5 . H. He , M. E. J. Friese , N. R. Heckenberg , and H. Rubinsztein-Dunlop , “ Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity ,” Phys. Rev. Lett. 75 , 826 – 829 ( 1995 ). [CrossRef] [PubMed]
6 . J.D. Jackson , Classical Electrodynamics , ( Wiely, New York , 1975 ).
7 . L. Allen , S. M. Barnett , and M. J. Padgett , Optical Angular Momentum ( Institute of Physics Pub., Bristol , 2003 ). [CrossRef]
8 . S. J. van Enk and G. Nienhuis , “ Eigenfunction description of laser beams and orbital angular momentum of light ,” Opt. Comm. 94 , 147 – 158 ( 1992 ). [CrossRef]
9 . G Moe and W Happer , “ Conservation of Angular-Momentum for Light Propagating in a Transparent Anisotropic Medium ,” J. Phys. B-Ato. Mol. Opt. Phys. 10 , 1191 – 1208 ( 1977 ). [CrossRef]
10 . F. C. MacKintosh and S. John , “ Diffusing-wave spectroscopy and multiple scattering of light in correlated random media ,” Phys. Rev. B 40 , 2383 – 2406 ( 1989 ). [CrossRef]
11 . W. Gouch , “ The angualr momentum of radiation ,” Eur. J. Phys. 7 , 81 – 87 ( 1986 ). |
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Some of the worksheets for this concept are Graphing rational Asymptotes and holes graphing rational functions Kuta graphing rational functions Graphing rational functions work 2 Graphing simple rational functions Graphing rational functions Haat chapter 3 review supplement name graphing rational Rational. This quiz is about rational function graphs and their specific components.
Solving A Rational Function And Graphing It Rational Function Graphing Algebra
Youll be asked to recognize what certain parts of a graph represent and to calculate.
Graphs of rational functions worksheet. Graphing Translations of Simple Rational Functions To graph a rational function of the form y a x h k follow these steps. A x0 B x0 x3 C x3 D x3 E x0 x3 2. Here is a graph of the curve along with.
That is if pxandqx are polynomials then px qx is a rational function. In each of the graphs below only half of the graph is given. You will need to download these for the Activity.
2Reduce the rational function to lowest terms if possible. For what values of x does the graph of fx have a vertical asymptote. First we have to find hole if any.
Worksheet analyze each function and predict the location of any vertical asymptotes horizontal asymptotes holes points of discontinuity x and y intercepts domain and range. Y 2x 1 x – 1 Solution. Each Group is Given a Rational Function and must.
Create your own worksheets like this one with infinite algebra 2. You will need to have the students count off by fours and assign them each a student number. None horz asym 0 x intercep ts.
2 4 3 4 x x x y 2 4 5 4 x x x y 2 4 4 x x x y A B C Complete the assignment on pp134 to 136. 3×5 x1 1 x 2x 3 1 2x 3 The last example is both a polynomial and a rational function. Rational Functions Group Activity Directions.
Then sketch the graph. _x-13_y -12 0 2. Ad Download over 20000 K-8 worksheets covering math reading social studies and more.
207 Graphs of Rational Functions 1. To find hole of the rational function we have to see whether there is any common factor found at both numerator and denominator. Rational functions A rational function is a fraction of polynomials.
1 f x 4 x 3 x y 8 6 4 2 2 4 6 8 8 6 4 2 2 4 6 8 2 f x x2 7x 12 2×2 2x 12 x y 8 6 4 2 2 4 6 8 8 6 4 2 2 4 6 8 3 f x 1 x 4 x y. Worksheet 26A Rational functions MATH 1410 SOLUTIONS For each of the rational functions given below do the following. Graphing Rational Functions Practice Identify the holes vertical asymptotes x-intercepts horizontal asymptote and domain of each.
Below we have provided the Rational Functions and Their Graphs Group Activity. About This Quiz Worksheet. X and y Intercepts 1 point.
Now let us take an example and explore the graphs of rational functions. Worksheet by kuta software llc 8 answers to graphing rational functions 1 x y 8 6 4 22468 8 6 4 2 2 4 6 8discontinuities. Step 3 Draw the two branches of the hyperbola so that they pass through the plotted points and approach the.
Discover learning games guided lessons and other interactive activities for children. Worksheet by Kuta Software LLC Kuta Software – Infinite Precalculus Graphs of Rational Functions Name_____ Date_____ Period____-1-For each function identify the points of discontinuity holes intercepts horizontal asymptote domain limit behavior at. This can sometimes save time in graphing rational functions.
Domain and Range 1 point. 1Find the domain of the rational function. Graphing Rational Functions Date_____ Period____ Identify the points of discontinuity holes vertical asymptotes x-intercepts and horizontal asymptote of each.
Write the equation for each graphed rational function. 1 to 4 first before trying the questions below. If a function is even or odd then half of the function can be.
1 print and separate the 8 cards. Horizontal Asymptotes 1. The numerator is pxandthedenominator is qx.
Worksheet 35Rational Functions Show all work. All answers must be given as simplified exact answers. Rational Functions MATH 1330 Precalculus 229 Recall from Section 12 that an even function is symmetric with respect to the y-axis and an odd function is symmetric with respect to the origin.
Step 1 Draw the asymptotes x h and y k. Graph the rational function given below. Graphing rational functions worksheet 1 horizontal asymptotes answers.
Graphing Rational Expressions – Displaying top 8 worksheets found for this concept. Match the equation of each rational function with the most appropriate graph. No Calculators are permitted unless specified otherwise.
Each card contains an equation and graph. Step 2 Plot points to the left and to the right of the vertical asymptote. How to sketch the graph of a rational function.
Ad Download over 20000 K-8 worksheets covering math reading social studies and more. Discover learning games guided lessons and other interactive activities for children. 3Find the x- and y-intercepts of the graph of the rational function.
Graphs of Rational Functions Mathematics Start Practising In this worksheet we will practice graphing rational functions whose denominators are linear determining the types of their asymptotes and describing their end behaviors. Let 2 2 3 x fx xx.
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In physics, work is the energy that is transferred to a body when it is moved along a path by a force. When the force is conservative (non-dissipative) the work is independent of the path. Work done on a body is accomplished not only by a displacement of the body as a whole from one place to another but also, for example, by compressing a gas, by rotating a shaft, and even by directing small magnetic particles within a body along an external magnetic field.
When Newton founded classical mechanics in his 1687 magnum opus Principia, he defined, of course, implicitly energy and work. However, it took more than a century before these concepts were recognized amidst the classical mechanics equations and seen as useful and worthy of independent consideration. A quantity close to kinetic energy was noticed quite early; Leibniz coined the name vis viva (live force) in 1695 for the quantity mv2 that—except for the factor ½— is what we now call kinetic energy. The fact that Leibniz used the name "vis", which usually was restricted to mean "force", shows that neither work (a form of energy, a scalar) nor force (a vector) were fully grasped concepts at that time.
The modern statement: "work is force times path" is due to French physicist Gustave-Gaspard Coriolis, who gave the correct formula for change in kinetic energy associated with work, although he spoke of force vive (including the modern factor ½), instead of kinetic energy (the latter term was coined by William Thomson and Peter Guthrie Tait in their 1867 book Treatise on Natural Philosophy). Coriolis' terminology was taken over by Jean-Victor Poncelet who did much to propagate the terms "work" and "path", especially among engineers.
Hence, since the early 19th century, the terms "work" and "energy" have well-defined, quantitative, meanings in physics, and both can be measured with a real number as outcome. The physics definitions are inspired by, but differ somewhat from, their more qualitative meanings in daily life. For instance, in physics work can be negative.
As stated, work is a form of energy and consequently it has the same dimension as energy [force times length, or mass times (length/time)2]. In SI units the unit is joule; in cgs units it is erg; in US units it is foot-pound.
Definition of mechanical work
The mechanical work W is defined as force times path length: When a constant force F acts on a body along a straight path and the body is moved over a length |s| along this path, then
When the force is directed along the path (F and s parallel), α = 0, cosα = 1, and we have simply W = |F| |s|. The positive work W is converted into an increase of kinetic energy T,
- W = ΔT ≡ T1 − T0 > 0.
In the next section this relation will be proven and shown to be a consequence of Newton's second law: F = m a.
When a force is perpendicular to the path, α = 90°, cosα = 0, the force performs no work. Examples of such a situation are the centrifugal force on a mass in uniform circular motion and the gravitational force acting on a satellite in a circular orbit. If a body is in uniform (i.e., has constant speed) straight motion and the only force acting on it is perpendicular to its path, then the body will persist in its uniform straight motion (Newton's first law).
When a force F is anti-parallel to the path (α = 180°), it performs negative work W < 0, since cosα = −1, and |F| and |s| are both greater than zero. The negative work done by the force is converted into a decrease of the kinetic energy T,
- W = ΔT < 0
(ΔT < 0 because T decreases, i.e., T1 < T0).
For example, think of an old-fashioned type of cannon that shoots a cannon ball of mass m straight up. The gun powder explosion gives the ball initial kinetic energy T0 = ½ m v02. The gravitational attraction of the earth performs negative work (force downward, motion of the cannon ball upward), until at the highest point the speed v1 and the kinetic energy T1 = ½ m v12 of the ball are zero. The amount of work performed is W = ΔT = 0−T0 = −T0. At the highest point the motion of the cannon ball reverts direction, it starts falling to earth, and from there on the gravitational attraction performs positive work (direction of motion and force are parallel). The kinetic energy increases again until it achieves its original value T0 = ½ m v02 at the point where the cannon ball arrives again at the cannon. The total work done by the gravitation is zero, the work done on the cannon ball going up cancels the work done on the ball going down. (In this example we ignore friction by the air).
When the the force is conservative (non-dissipative), the work is independent of path, and when furthermore the path is a closed curve, the total work is zero (one way the work is positive and the other way the work is equally large in absolute value, but negative).
As the gravitational field is conservative, we just saw an example of a mass (a cannon ball) making a closed path in a conservative force field. The potential energy U of a mass m close to the surface of the earth is equal to U = mgh, where g is the gravitational acceleration and h is the height. When the cannon is positioned at height h = 0, the cannon ball starts with potential energy zero, U0 = 0. The work W done by the gravitational field on the cannon ball going upward has two effects: it decreases its kinetic energy—as just discussed—and it increases its potential energy, U1 > U0, so that
- W = −ΔU ≡ −(U1 − U0) = U0 − U1 < 0.
At the highest point, h1, the potential energy is maximum U1 = mgh1 (equal to minus the work performed) and the kinetic energy is zero (this is the point where the motion of the cannon ball changes direction, for a small amount of time its speed is zero). The work done by a conservative force converts a decrease in kinetic energy into an increase of potential energy and conversely:
- W = ΔT = −ΔU, so that ΔT + ΔU = 0.
When the cannon ball dropping down is at height h = 0 again, its kinetic energy is maximum (equal to the energy imparted to it by the exploding gun powder) and its potential energy is zero again.
Example of mechanical work
Work to lift a mass m in the gravitational field of the earth. Close to the surface of the earth, the attractive force is constant and equal to the gravitational acceleration g times m. The work W to lift the mass to a height h is,
When h is positive, and the mass is at rest before and after the lifting (no change in kinetic energy), the work is completely converted into potential energy: U(h) = mgh.
This textbook example seems to contradict energy conservation. Before its lifting, the mass is at rest on the ground with mechanical energy T + U = 0. After its lifting, the mass has still kinetic energy zero, but non-zero potential energy. In other words, its energy has changed during the lifting.
The solution to this paradox is that the agent that does the lifting (a human being, an electric motor as used in elevators, exploding gunpowder, etc.) is left out of the equation. The lifting agent loses at a minimum the energy mgh. (The agent may lose more than mgh because its efficiency may be less than unity; energy may be lost in friction, by incomplete transfer of energy, etc.). The energy of the mass plus the energy of the agent (plus possibly dissipated energy) is conserved.
Secondly, the example seems to contradict the relation W = ΔT (work gives change in kinetic energy), because evidently there is no change in kinetic energy, and yet work has been performed on the mass.
The solution to this apparent contradiction is that the gravitational force Fg must be considered simultaneously with the force Fl exerted by the lifting agent. The two forces work in opposite direction, and apparently |Fl| > |Fg|, otherwise the mass wouldn't go up. However, a net force will accelerate the mass and give it kinetic energy. The solution to this problem is somewhat ingenious. At first the net force is assumed to be small and positive (directed upward) and a small amount of work ΔW is performed, resulting in a small increase in kinetic energy ΔT = ΔW. Once the mass has small speed, the lifting force is decreased so that the net force vanishes, Fl = −Fg. Then the mass pursues its motion upward in accordance with Newton's first law (no net force, no change in velocity). When the mass has arrived close to its endpoint h, the lifting force Fl is decreased to a value just below the gravitational force Fg, so that now the net force performs the work −ΔW that exactly decreases the kinetic energy to zero again. In a very good approximation Fl = −Fg over the whole lifting process and the total work of the net force is
When the force is not constant along the path, or the path is not straight, it is possible to compute the work by infinitesimal calculus. One chooses N + 1 consecutive points on the path
This divides the path in N pieces Δsi, which are small enough to assume that the force is constant, F(ri), on the i-th piece. By their definition as the differences of two vectors, the pieces are straight. The (approximate) total work is obtained by summing the work done along the individual small pieces,
Obviously, when the total path is straight and the force is constant over the path (as in the figure in the previous section), two points give an exact result, i.e., N = 1 suffices.
To improve the approximation one chooses the sampling points ri closer and closer, which makes the pieces Δsi smaller and smaller, so that their lengths go to zero. The limit is the path integral
where we wrote the upper limit (endpoint of the curve) now as r1 (when the number of intervals N goes to infinity, writing the upper limit as rN becomes meaningless).
To show that the work is converted into an increase in kinetic energy, we write for one mass m
where we used F = m a (Newton's second law) and that the acceleration a is the second derivative of r with respect to time. Integrate
where we used that Δs is tangent to the path, i.e.,
Note that from the usual definition of the time derivative of a vector function used here, follows dr ≡ ds and that accordingly the limits in the path integral are r0 ≡ r(t0) and r1 ≡ r(t1). The following important result is obtained that connects the work and change in kinetic energy over the path,
To show that work by a conservative force is converted into decrease in potential energy, we use the relation between force and potential, valid for conservative fields
and by the chain rule
In total, the conservation of mechanical energy follows
The following remarks are based on a paper by Grattan-Guinness.
In 1753 Daniel Bernoulli wrote about "live forces which the man produces during his work". He was close to the statement (proved earlier in this article) that work gives (a change of) kinetic energy (live force).
During the industrial revolution attention of many engineers and scientists became focused on machines (steam engines, waterwheels, windmills, etc.) that were constructed to deliver mechanical work. In that context classical mechanics was seen with fresh eyes and it inspired Lazare Carnot to derive in 1779 a formula for the change in live forces that comes close to the modern equation. It contains the angle between the velocity (ds/dt) and the acceleration (F/m) and integrates over time, hence Carnot's formula is almost (in modern notation):
Next developments came from French physicists and engineers educated at the École Polytechnique, an engineering school which was founded in revolutionary France (1794). Claude-Louis Navier, recognizing energy conservation, derived
where (ξ, η, ζ) is the acceleration of particle with mass m. This is already the modern formula, but not yet the modern terminology. Navier published this formula in 1819, in a new edition of Architecture Hydraulique, an old work (c. 1738) written by Forest de Bélidor. The old work was revised and extended by Navier, (who, parenthetically, referred in it to the 1779 work of Lazare Carnot). As mentioned in the introduction, this work culminated in the formula of the Polytechnician Coriolis: work is path times force.
- G. Coriolis, Du Calcul de l'Effet des Machines, Paris (1829). Also the second edition of 1844 (see Google books) uses the term force vive and not énergie cinétique.
- I. Grattan-Guinness, Work for the Workers: Advances in Engineering Mechanics and Instruction in France, 1800-1830, Annals of Science, vol. 41, (1984), pp. 1–33
- V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1978) |
Presentation on theme: "Chapter 7 Properties of Matter"— Presentation transcript:
1 Chapter 7 Properties of Matter Physics Beyond 2000Chapter 7Properties of Matter
2 States of matter Solid state Liquid state Gas state (will be studied in Chapter 8.)
3 Points of viewMacroscopic: Discuss the relation among physics quantities.Microscopic: All matters consist of particles. The motions of these particles are studied. Statistics are used to study the properties.
4 Solids F = k.e where k is the force constant of the spring Extension and compression (deformation) of solid objects– elasticity .Hooke’s law for springs– The deformation e of a spring is proportional to the force F acting on it, provided the deformation is small.F = k.e where k is the force constant of the spring
5 Hooke’s law for springs Force FDeformation eextensioncompressionF = k.ee1e2F1F2Natural lengthExtensionCompression
6 Hooke’s law for springs Force FDeformation eextensioncompressionF = k.eThe slope of the graphrepresents the stiffnessof the spring.A spring with large slopeis stiff.A spring with small slopeis soft.
7 Energy stored in a spring It is elastic potential energy.It is equal to the work done W by the external force F to extend (or compress) the spring by a deformation e.
8 Energy stored in a spring It is also given by the area under the F-e graph.extensionFeForce
9 Example 1 Find the extension of a spring from energy changes. Find the extension of the spring by using Hooke’s law.
12 More than springsAll solid objects follow Hooke’s law provided the deformation is not too large.The extension depends onthe nature of the materialthe stretching forcethe cross-sectional area of the samplethe original length
13 stress Stress is the force F on unit cross-sectional area A. Unit: PaStress is a measure of the cause of a deformation.Note that A is the cross-sectional area of the wirebefore any stress is applied.
14 strain Strain is the extension e per unit length. If is the natural length of the wire,theneFStrain expresses the effect of the strain on the wire.
15 Example 3Find the stress and the strain of a wire.strainstress
16 Young modulus EYoung modulus E is the ratio of the tensile stress σ applied to a body to the tensile strain ε produced.Unit: Pa
17 Young modulus EThe value of E is dependent on the material.
18 Young modulus E and force constant k F = k.eSo k depends on E (the material), A (thethickness) and (the length).
20 Experiment to find Young modulus Suspend two long thin wire as shown.The reference wire cancompensate for thetemperature effect.The vernier scale is tomeasure the extensionof the sample wire.samplewirereferencewirelevelmetervernierscaleweight
22 Experiment to find Young modulus Adjust the weight so that vernier scale to read zero.Measure the diameter of the sample wire andcalculate its cross-section area A.samplewirereferencewirevernierscaleweight
26 Experiment to find Young modulus referencewiresampleweightvernierscaleMeasure the length of the sample wire.
27 Experiment to find Young modulus Add weight W to the sample wire and measure its extension e .The force on the wire isF = W = mg.samplewirereferencewirevernierscaleF = W = mg where m is the added mass.weight
38 The stress-strain curve OALBCDA: proportional limit Between OA, the stressis proportional to the strain.Point A is the limit of thisproportionality.permanentstrain
39 The stress-strain curve OALBCDL: elastic limitBetween AL, thestrain can be back tozero when the stress isremoved.i.e. the wireis still elastic.Usually the elasticlimit coincides withthe proportional limit.permanentstrain
40 The stress-strain curve B: yield pointBetween LB, the wirehas a permanentdeformation whenthe stress is removed.i.e. the wire is plastic.At point B, there isa sudden increase ofstrain a small increasein stress.stress σstrain εOALBCDpermanentstrain
41 The stress-strain curve OALBCDC: breaking stressThis is the maximumstress.Beyond this point,the wire extendsand narrows quickly,causing a constrictionof the cross-sectionalarea.permanentstrain
42 The stress-strain curve OALBCDD: breaking pointThe wire breaksat this point.This is the maximumstrain of the wire.permanentstrain
44 Energy stored in the extended wire The area under the stress-strain graph =stresswhere Fe is the elastic potentialenergy andA is the volume of thewire.σεstrain
45 Properties of materials StiffnessStrengthDuctilityToughness
46 Stiffness It indicates how the material opposes to deformation. Young modulus is a measure of the stiffness of a material.A material is stiff if its Young modulus is large.A material is soft if its Young modulus is small.
47 StrengthIt indicates how large the stress the material can stand before breaking.The breaking stress is a measure of the strength of the material.A material is strong if it needs a large stress to break it.A material is weak if a small stress can break it.
48 DuctilityIt indicates how the material can become a wire or a thin sheet.A ductile material enters its plastic stage with a small stress.ε
49 Toughness A tough material is one which does not crack readily. The opposite is a brittle material.A brittle material breaks over a very short time without plastic deformation.
52 Graphical representation for various materials stress σstrain εglassmetalrubber
53 Elastic deformation and plastic deformation In elastic deformation,the object will be back to its original shape when the stress is removed.In plastic deformation, there is a permanent strain when the stress is removed.
55 FatigueMetal fatigue is a cumulative effect causing a metal to fracture after repeated applications of stress, none of which exceeds the breaking stress.
56 CreepCreep is a gradual elongation of a metal under a constant stress which is well below its yield point.
57 Plastic deformation of glass Glass does not have any plastic deformation.When the applied stress is too large, the glass has brittle fracture.
58 Plastic deformation of rubber Deformation of rubber would produce internal energy.The area in the loop represents the internal energy produced per unit volume.stress σstrain εloadingunloadingHysteresis loop
59 Model of a solid Microscopic point of view A solid is made up of a large number of identical hard spheres (molecules).The molecules are attracted to each other by a large force.The molecules are packed closely in an orderly way.There are also repulsion to stop the molecules penetrating into each other.
60 Structure of solidCrystalline solid: The molecules have regular arrangement. e.g. metal.Amorphous solid: The molecules are packed disorderly together. e.g. glass.
61 Elastic and plastic deformation of metal Metal has a structure of layers.Layers can slide over each other under an external force.layerlayer
62 Elastic and plastic deformation of metal When the force is small, the layer displaces slightly.Force
63 Elastic and plastic deformation of metal When the force is removed, the layer moves back to its initial position.The metal is elastic.
64 Elastic and plastic deformation of metal When the force is large, the layer moves a large displacement.Force
65 Elastic and plastic deformation of metal When the force is removed, the layer settles down at a new position.The metal has a plastic deformation.New structureInitial structure
66 Intermolecular forces The forces are basically electrostatic in nature.The attractive force results from the electrons of one molecule and the protons of an adjacent molecule.The attractive force increases as their separation decreases.
67 Intermolecular forces The forces are basically electrostatic in nature.When the molecules are too close, their outer electrons repel each other. This repulsive force prevents the molecules from penetrating each other.
68 Intermolecular forces The forces are basically electrostatic in nature.Normally the molecules in a solid have a balance of the attractive and repulsive forces.At the equilibrium position, the net intermolecular force on the molecule is zero.
69 Intermolecular separation r It is the separation between the centres of two adjacent molecules.ro is the equilibriumdistance.r = roThe force on each moleculeis zero.ro
70 Intermolecular separation r It is the separation between the centres of two adjacent molecules.rorr > roThe force on the moleculeis attractive.
71 Intermolecular separation r It is the separation between the centres of two adjacent molecules.rorr < roThe force on the moleculeis repulsive.
72 Intermolecular forces ro is the equilibrium separationrepulsiverroattractiveThe dark line is theresultant curve.
73 Intermolecular separation Suppose that a solid consists of N molecules with average separation r.The volume of the solid is V.What is the relation among these quantities?
74 Intermolecular separation Example 6.Mass = density × volumeThe separation of molecules in solid and liquid is of order m.
75 Intermolecular potential energy ro is the equilibrium separationIntermolecularforcerroThe potential energyis zero for large separation.Potential energy-εThe potential energy is a minimumat the equilibrium separation.
76 Intermolecular potential energy ro is the equilibrium separationIntermolecularforcerroWhen they move towardseach other from far away, thepotential energy decreasesbecause there is attractive force.The work done by externalforce is negative.Potential energy-εThe potential energy is a minimumat the equilibrium separation.
77 Intermolecular potential energy ro is the equilibrium separationWhen they are further towardseach other after the equilibriumposition, the potential energy increasesbecause there is repulsive force.The work done by external force ispositive.IntermolecularforcerorPotential energy-εThe potential energy is a minimumat the equilibrium separation.
78 Force and Potential Energy U = potential energyF = external forceand
79 Variation of molecules If the displacement of two neighbouring molecules is small, the portion of force-separation is a straight line with negative slope.FattractiveIntermolecularforcerepulsiverrorro
80 Variation of molecules The intermolecular force isF = -k. Δrwhere k is the force constant between moleculesand Δr is the displacementfrom the equilibrium position.FrroSo the molecule is insimple harmonic motion.
81 Variation of molecules So the molecule is insimple harmonic motion.withω2 =where m is the massof each molecule.Frro
82 Variation of molecules However this is only a highly simplified model.Each molecule is under more than one force from neighbouring molecules.
83 The three phases of matter Solid, liquid and gas states.In solid and liquid states, the average separation between molecules is close to ro.IntermolecularforcerroPotential energy-ε
84 The three phases of matter Solid, liquid and gas states.In gas state, the average separation between molecules is much longer than ro.IntermolecularforcerroPotential energy-ε
85 Elastic interaction of molecules All the interactions between molecules in any state are elastic. i.e. no energy loss on collision between molecules.
86 Solids When energy is supplied to a solid, the molecules vibrate with greater amplitude until melting occurs.IntermolecularforcerroPotential energy-ε
87 Solids On melting, the energy is used to break the lattice structure. IntermolecularforcerroPotential energy-ε
88 Liquids Molecules of liquid move underneath the surface of liquid. When energy is supplied to a liquid, the molecules gain kinetic energy and move faster. The temperature increases.
89 LiquidsAt the temperature of vaporization (boiling point), energy supplied is used to do work against the intermolecular attraction.The molecules gain potential energy. The state changes.The temperature does not change.
90 Gases Molecules are moving at very high speed in random direction. -ε IntermolecularforcerroPotential energy-ε
91 Gases The average separation between molecules is much longer than ro IntermolecularforcerroPotential energy-ε
92 Gases The intermolecular force is so small that it is insignificant. rroPotential energy-ε
93 Example 7 There are 6.02 1023 molecules for one mole of substance. The is the Avogadro’s number.
94 Example 8The separation between molecules depend on the volume.
95 Thermal expansion In a solid, molecules are vibrating about their equilibrium position.Potential energyrro-ε
96 Thermal expansion Suppose a molecule is vibrating between positions A and B about the equilibriumposition.Potential energyrro-εAB
97 Thermal expansionNote that the maximum displacement from the equilibriumposition is not the same on each side because the energy curveis not symmetrical about the equilibrium position.Potential energyrro-εABA’B’C’
98 Thermal expansion The potential energy of the molecule varies along the curve A’C’B’ while the molecule isoscillating along AB.Potential energyrro-εABA’B’C’
99 Thermal expansion The centre of oscillation M is mid-way from the positions A and B. So point Mis slightly away from the equilibrium position.Potential energyrro-εABA’B’C’M
100 Thermal expansionWhen a solid is heated up, it gains more potential energy and the points A’ and B’ move up the energy curve. The amplitude of oscillation is also larger.Potential energyMABrroA’B’-εC’
101 Thermal expansion The molecule is vibrating with larger amplitude between new positions AB.Potential energyMABrroA’B’-εC’
102 Thermal expansion The centre of oscillation M , which is the mid-point of AB, is further away from the equilibrium position.Potential energyMABrroA’B’-εC’
103 Thermal expansionAs a result, the average separation between moleculesincreases by heating. The solid expands on heating.Potential energyMABrroA’B’-εC’
104 Absolute zero temperature At absolute zero, the molecule does not vibrate. Theseparation between molecules is ro. The potentialenergy of the molecule is a minimum.Potential energyrro-εC’
105 Young Modulus in microscopic point of view Consider a wire made up of layers of closely packed molecules.When there is not any stress, the separation between two neighbouring layer is ro.ro is also the diameter of each molecule.rowire
106 Young Modulus in microscopic point of vies The cross-sectional area of the wire iswhere N is the number ofmolecules in each layerarea of one molecule=roroA
107 Young Modulus in microscopic point of vies When there is not an external force F, the separation between two neighbouring layer increases by r.Fro+ r
108 Young Modulus in microscopic point of vies The strain isFro+ r
109 Young Modulus in microscopic point of vies Since the restoring force between two molecules in the neighbouring layer is directly proportional to N and r, we have F = N.k.r where k is the force constant between two molecules.Fro+ r
110 Young Modulus in microscopic point of vies and F = N.k.rFro+ r
111 Young Modulus in microscopic point of vies Thus, the Young modulus isFro+ r
112 Example 9Find the force constant k between the molecules.
113 Density Definition: It is the mass of a substance per unit volume. where m is the massand V is the volumeUnit: kg m-3
114 Measure the density of liquid Use hydrometerupthrustweight
115 PressureDefinition: The pressure on a point is the force per unit area on a very small area around the point.orUnit: N m-2 or Pa.
116 Pressure in liquidPressure at a point inside a liquid acts equally in all directions.The pressure increases with depth.
117 Find the pressure inside a liquid = density of the liquidh = depth of the point Xsurface ofliquidXh
118 Find the pressure inside a liquid Consider a small horizontal area A around point X.Xhsurface ofliquidA
119 Find the pressure inside a liquid The force from the liquid on this area is the weight W of the liquid cylinder above this areaXhsurface ofliquidAW
120 Find the pressure inside a liquid W = ?W = hAgXhsurface ofliquidAW
121 Find the pressure inside a liquid W = hAg and P =Xhsurface ofliquidAW
122 Find the pressure inside a liquid As there is also atmospheric pressure Po on the liquidsurface, the total pressure at X isPoXhAPsurface ofliquid
124 Force on a block in liquid Consider a cylinder of area A and height L ina liquid of density .Lh1h2P1P2A
125 Force on a block in liquid The pressure on its top area is P1 = h1g + PoThe pressure on its bottom area is P2 = h2g + PoLh1h2P1P2A
126 Force on a block in liquid The pressure difference P = P2 – P1 = Lgwith upward direction.Lh1h2P1P2A
127 Force on a block in liquid So there is an upward net force F = P.A= Vg where V is the volume of the cylinder.Lh1h2AF
128 Force on a block in liquid This is the upthrust on the cylinder.Upthrust = Vgh1h2FV
129 Force on a block in liquid Upthrust = VgNote that it is also equal to the weight of theliquid with volume V.h1h2FV
130 Force on a block in liquid The conclusion: If a solid is immersed in a liquid, the upthrust on the solid is equal to the weight of liquid that the solid displaces.h1h2FV
131 Force on a block in liquid The conclusion is correct for a solid in liquid and gas (fluid).h1h2FV
132 Archimedes’ Principle When an object is wholly or partially immersed in a fluid, the upthrust on the object is equal to the weight of the fluid displaced.upthrustupthrust
133 Measuring upthrust W spring-balance The reading of the spring-balance is W,which is the weightof the object.objectbeakerliquidThe reading of thecompressionbalance is B, whichis the weight of liquidand beaker.compressionbalanceB
134 Measuring upthrust spring-balance Carefully immerse half the volume of the object in liquid.objectbeakerliquidWhat would happen tothe reading of thespring-balance and thatof the compressionbalance?compressionbalance
135 Measuring upthrust spring-balance The reading of the spring- balance decreases.Why?objectbeakerliquidThe difference in the readingsof the spring-balance givesthe upthrust on the object.compressionbalance
136 Measuring upthrust spring-balance The reading of the compression balanceincreases.Why?objectbeakerliquidThe difference in the readingsof the compression balancegives the upthrust on theobject.compressionbalance
137 Measuring upthrust spring-balance Carefully immerse object the whole object in liquid.objectbeakerliquidWhat would happen tothe reading of thespring-balance and thatof the compression balance?compressionbalance
138 Measuring upthrust spring-balance Carefully place the object on the bottom of the beaker.objectbeakerliquidWhat would happen tothe reading of thespring-balance and thatof the compressionbalance?compressionbalance
139 Law of floatationA floating object displaces its own weight of the fluid in which it floats.weight of the object= upthrust= weight of fluid displacedweightupthrust
140 float or sink? ‘ = density of the object = density of the fluid If ‘ > , then the object sinks in the fluid.If ‘ < , then the object floats in the fluid.density is smaller than density is larger than
141 Manometer liquid of density connect to the fluid Same level X Y A manometer can measure the pressure difference of fluid.Note that the pressure on the same level in the liquid must be the same.liquid ofdensity connectto the fluidSame levelXY
143 Manometer The pressures at points A and B are equal. Po = atmospheric Po+P= fluid pressureh = differencein heightABliquid ofdensity
144 Manometer The pressure at A = Po+P The pressure at B = Po + hg liquid ofdensity Po = atmosphericpressurePo+P= fluid pressureh = differencein heightBA
145 Manometer The pressure difference of the fluid P = hg liquid ofdensity Po = atmosphericpressurePo+P= fluid pressureh = differencein heightBA
146 Liquid in a pipeConsider a pipe of non-uniform cross-sectional area with movable piston at each end.The fluid is in static equilibrium.Same levelhx = hYYXstatic fluid
147 Liquid in a pipeThe manometers show that the pressures at points X and Y are equal.Same levelhx = hYYXstatic fluid
148 Liquid in a pipeThe pressures at points M and N on the pistons are also equal.Same levelhx = hYMNstatic fluid
149 Liquid in a pipeThere must be equal external pressures on the pistons to keep it in equilibrium.PM = PNSame levelhx = hYMNPMPNstatic fluid
150 Liquid in a pipeAs F = P.A , the external forces are different on the two ends.FM > FNSame levelhx = hYMNFMFNstatic fluid
151 Liquid in a pipe Same level hx = hY M N FM FN static fluid Note that the net force on the liquid is still zero to keep it in equilibrium.There are forces towards the left from the inclined surface.Same levelhx = hYMNFMFNstatic fluid
152 Fluid Dynamics Fluid includes liquid and gas which can flow. In this section, we are going to study the force and motion of a fluid.Beurnoulli’s equation is the conclusion of this section.
153 Turbulent flow Turbulent flow: the fluid flows in irregular paths. We will not study this kind of flow.
154 Streamlined flowStreamlined flow (laminar flow) : the fluid moves in layers without fluctuation or turbulence so that successive particles passing the same point with the same velocity.
155 Streamlined flowWe draw streamlines to represent the motion of the fluid particles.
156 Equation of continuity Suppose that the fluid is incompressible. That is its volume does not change. Though the shape (cross-sectional area A) may change.
157 Equation of continuity At the left end, after time t, the volume passing is A1.v1. tAt the right end, after the same time t, the volume passing is A2.v2. t
158 Equation of continuity As the volumes are equal for an incompressible fluid,A1.v1. t = A2.v2. t A1.v1 = A2.v2
160 Pressure difference and work done x2A2Suppose that an incompressible fluid flows from position 1 to position 2 in a tube.Position 2 is higher than position 1.There is a pressure difference P at the two ends.pPosition 2x1h2A1P+PPosition 1h1
161 Pressure difference and work done x2A2Work done by the external forces is(P+P).A1.x1 - P.A2.x2PPosition 2x1h2A1P+PPosition 1h1
162 Pressure difference and work done x2A2Work done by the external forces is(P+P).A1.x1 - P.A2.x2A1x1=A2x2=V= volume of fluid that movesPPosition 2x1h2A1P+PPosition 1h1
163 Pressure difference and work done x2A2Work done by the external forces is(P+P).A1.x1 - P.A2.x2= P .VWith V =Work done = Pwhere m is the mass of the fluid and ρis the density of the fluidpPosition 2x1h2A1P+PPosition 1h1
164 Bernoulli’s principle x2A2v2In time t, the fluid moves x1 at position 1 and x2 at position 2.x1 = v1.t andx2 = v2.tP2Position 2x1h2A1P1v1Position 1h1
165 Bernoulli’s principle x2A2v2In time t, the fluid moves x1 at position 1 and x2 at position 2.x1 = v1.t andx2 = v2.tWork done by external pressure =(P1-P2)P2Position 2x1h2A1P1v1Position 1h1
166 Bernoulli’s principle x2A2v2Work done by external pressure =(P1-P2)Increase in kinetic energy =P2Position 2x1h2A1P1v1Position 1h1
167 Bernoulli’s principle x2A2v2Increase in kinetic energy =Increase in gravitatioanl potential energy =mgh2 – mgh1P2Position 2x1h2A1P1v1Position 1h1
168 Bernoulli’s principle x2A2v2P2The left hand side is the workdone by external pressure. Itis also the energy suppliedto the fluid.The right hand side isthe increase in energyof the fluid.Position 2x1h2A1P1v1Position 1h1
186 Ball floating in air force due to spinning thrust from the air blower weight of the ballairWhat is the direction of spinning of the ball?
187 Ball floating in air thrust from the force due to pressure air blower differenceairweight of the ballairIt is spinning in clockwise direction.
188 Ball floating in airForce due topressuredifference
189 Air blown out through a funnel What would happen to the light ball?
190 Air blown out through a funnel Force due topressure differenceForce due topressure differenceweightIt is sucked to the top of the funnel.
191 Yacht sailing A yacht can sail against the wind. Note that the sail is curved.
192 Yacht sailing A yacht can sail against the wind. The pressure difference produces a net force F.A component of F pushes the yacht forward.
193 Yacht sailingThe yacht must follow a zig-zag path in order to sail against the wind.windpath
194 JetsWhen a stream of fluid is ejected rapidly out of a jet, air close to the stream would be dragged along and moves at higher speed.This results in a low pressure near the stream.low pressureairfluid
195 Jets: Bunsen burner The pressure near the jet is low. Air outside is pulled into the bunsen burner through the air hole.airgas |
We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow.
We study a backward stochastic differential equation (BSDE) whose terminal condition is an integrable function of a local martingale and generator has bounded growth in z. When the local martingale is a strict local martingale, the BSDE admits at least two different solutions. Other than a solution whose.A backwards martingale is a stochastic process that satisfies the martingale property reversed in time, in a certain sense. In some ways, backward martingales are simpler than their forward counterparts, and in particular, satisfy a convergence theorem similar to the convergence theorem for ordinary martingales. The importance of backward martingales stems from their numerous applications. In.As application, we give a solution to reflected backward stochastic differential equations when the barrier is an optional positive process neither right-continuous nor left-continuous. Previous article in issue; Next article in issue; MSC. 60G20. 60H05. 60H15. Keywords. Optional Snell envelopes. Split stopping times. Reflected backward stochastic differential equations. Mertens decomposition.
Martingales Last updated by Serik Sagitov: May 23, 2013 Abstract This Stochastic Processes course is based on the book Probabilities and Random Processes by Geo rey Grimmett and David Stirzaker. Chapters 7.7-7.8, and 12. 1 De nitions and examples Example 1 Martingale: a betting strategy. Let X n be the gain of a gambler doubling the bet after each loss. The game stops after the rst win. X 0.
This is illustrated by a simple shift between two martingale measures such that the price process is uniformly integrable under one of them, but not under the other. To prepare for a more systematic study of dynamics on the space of martingale measures, we consider different versions of the optimal transport problem for measures on path space.
Lp Solutions of Re ected Backward Stochastic Di erential Equations with Jumps Song Yao Abstract Given p 2(1;2), we study Lp solutions of a re ected backward stochastic di erential.
Abstract. We prove that a continuous -supermartingale with uniformly continuous coeffcient on finite or infinite horizon, is a -supersolution of the corresponding backward stochastic differential equation.It is a new nonlinear Doob-Meyer decomposition theorem for the -supermartingale with continuous trajectory. 1. Introduction. In 1990, Pardoux-Peng () proposed the following nonlinear.
Martingale Theory with Applications 34. Unit aims. To stimulate through theory and examples, an interest and appreciation of the power of this elegant method in probability theory. And to lay foundations for further studies in probability theory. Unit description. The theory of martingales is of fundamental importance to probability theory, statistics, and mathematical finance. This unit is a.
Theorem 233 Let X n F n n 0 1 2 be a uniformly integrable submartingale Then X. Theorem 233 let x n f n n 0 1 2 be a uniformly School DeAnza College; Course Title MATH 10; Type. Notes. Uploaded By eelompanda. Pages 44 This preview shows page 36 - 38 out of 44 pages.
It is easiest to think of this in the nite setting, when the function X: !R takes only nitely many values. Then, as you might already suspect from (1.2), to check if Xis measurable its.
Pr Backward submartingale convergence Suppose that X n n N is a. Pr backward submartingale convergence suppose that x School Peking Uni. Course Title MATHEMATIC 2012080032; Type. Essay. Uploaded By victory1832. Pages 12 This preview shows page 7 - 10 out of 12 pages.
Uniformly integrable family of functions, continuation of proof of an explicit formula for generator of an Ito diffusion, Dynkin's formula and its applications. Mon Feb 29th More examples for applications of Dynkin's formula (such as calculation of expected value of the first exist time of Brownian motion from a ball), the characteristic operator of an Ito diffusion and its coincidence with.
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On g pEvaluations with L Domains under Jump Filtration Song Yao Abstract Given p2(1;2), the unique Lp solutions of backward stochastic di erential equations with jumps (BSDEJs) al.
I.3. Uniformly integrable martingales 16 I.4. Stochastic integration 18 I.5. Selected results from stochastic calculus 26 I.6. The Skorokhod space 29 I.7. Martingale representation of square-integrable martingales 46 Chapter II. Backward Stochastic Di erential Equations with Jumps 51 II.1. Introduction 51 II.2. A simple counterexample and its.
Constrained Quadratic Risk Minimization via Forward and Backward Stochastic Di erential Equations Yusong Li and Harry Zhengy Abstract In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical nance. We model the asset dynamics with random market coe cients and portfolio strategies with convex constraints. Following the convex duality.
Backward stochastic differential equation, Infinite horizon, Reflected barriers, Stochastic optimal control, Stochastic differential game. I. Introduction. Nonlinear backward stochastic daerential equations (BSDE's in short) have been independently introduced by Pardoux and Peng (18) and DdEe and Epstein (7). It has already been discovered by Peng (20) that, coupled with a forward SDE, such. |
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find them all?
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after
every cut you can rearrange the pieces before cutting straight
through, can you do it in fewer?
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
Every day at noon a boat leaves Le Havre for New York while another
boat leaves New York for Le Havre. The ocean crossing takes seven
days. How many boats will each boat cross during their journey?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can you find a way of representing these arrangements of balls?
What is the shape of wrapping paper that you would need to completely wrap this model?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
How many moves does it take to swap over some red and blue frogs? Do you have a method?
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
A huge wheel is rolling past your window. What do you see?
When dice land edge-up, we usually roll again. But what if we
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Exchange the positions of the two sets of counters in the least possible number of moves
Can you fit the tangram pieces into the outline of this plaque design?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
This article looks at levels of geometric thinking and the types of
activities required to develop this thinking.
Can you fit the tangram pieces into the outlines of the chairs?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Can you fit the tangram pieces into the outline of the rocket?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
Can you fit the tangram pieces into the outline of this sports car?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
Can you fit the tangram pieces into the outlines of these people?
Can you fit the tangram pieces into the outline of these convex shapes?
Can you fit the tangram pieces into the outline of this telephone?
Can you fit the tangram pieces into the outlines of these clocks? |
2 Crystal Structure Objectives Relationships between structures-bonding-properties of engineering materials.Arrangements in crystalline solidsGive examples of each: Lattice, Crystal Structure, Unit Cell and Coordination NumbersDescribe hard-sphere packing and identify cell symmetry.Define directions and planes (Miller indices) for crystalsOutlineStructure of the Atom and Atomic BondingElectronic Structure of the AtomLattice, Unit Cells, Basis, and Crystal StructuresPoints, Directions, and Planes in the Unit CellCrystal Structures of Ionic MaterialsCovalent Structures
3 Crystal Structure Crystal Structure Lattice- A collection of points that divide space into smaller equally sized segments.Unit cell - A subdivision of the lattice that still retains the overall characteristics of the entire lattice.Atomic radius - The apparent radius of an atom, typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number).Packing factor - The fraction of space in a unit cell occupied by atoms.Types of Crystal StructureBody centered cubic (BCC)Face centered cubic (FCC)Hexagonal close packed (HCP)
4 Crystal StructureA number of metals are shown below with their room temperature crystal structure indicated. There are substances without crystalline structure at room temperature; for example, glass and silicone. All metals and alloys are crystalline solids, and most metals assume one of three different lattice, or crystalline, structures as they form: body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close-packed (HCP).Aluminum (FCC)Chromium (BCC)Copper (FCC)Iron (alpha) (FCC)Iron (gamma) (BCC)Iron (delta) (BCC)Lead (FCC)Nickel (FCC)Silver (FCC)Titanium (HCP)Tungsten (BCC)Zinc (HCP)
5 Number of Lattice Points in Cubic Crystal Systems Crystal StructureNumber of Lattice Points in Cubic Crystal SystemsIn the SC unit cell: point / unit cell = (8 corners)1/8 = 1In BCC unit cells: point / unit cell = (8 corners)1/8 + (1 center)(1) = 2In FCC unit cells: point / unit cell = (8 corners)1/8 + (6 faces)(1/2) = 4Relationship between Atomic Radius and Lattice ParametersIn SC, BCC, and FCC structures when one atom is located at each lattice point.
6 Crystal Structure Packing Factor In a FCC cell, there are four lattice points per cell; if there is one atom per lattice point, there are also four atoms per cell. The volume of one atom is 4πr3/3 and the volume of the unit cell is a0 3
7 Crystal Structure Density Density of BCC iron, which has a lattice parameter of nm.Atoms/cell = 2, a0 = nm = 10-8 cmAtomic mass = g/molVolume of unit cell = = (2.866 10-8 cm)3 = cm3/cellAvogadro’s number NA = 6.02 1023 atoms/mol
8 Crystal Structure Points, Directions, and Planes in the Unit Cell GeometryUnit Cell: The basic structural unit of a crystal structure. A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition.RRRRaAtomic configuration in Face-Centered-CubicAtomic configuration in Simple Cubic
10 Crystal Structure Unit Cells Types A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition.Primitive (P) unit cells contain only a single lattice point.Internal (I) unit cell contains an atom in the body center.Face (F) unit cell contains atoms in the all faces of the planes composing the cell.Centered (C) unit cell contains atoms centered on the sides of the unit cell.Face-CenteredPrimitiveBody-CenteredEnd-CenteredCrystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monclinic, triclinic, trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice.
11 Crystal Structure Counting Number of Atoms Per Unit Cell Counting Atoms in 3D CellsAtoms in different positions are shared by differing numbers of unit cells.• Vertex atom shared by 8 cells => 1/8 atom per cell.• Edge atom shared by 4 cells => 1/4 atom per cell.• Face atom shared by 2 cells => 1/2 atom per cell.• Body unique to 1 cell => 1 atom per cell.Simple Cubic8 atoms but shared by 8 unit cells. So,8 atoms/8 cells = 1 atom/unit cellHow many atoms/cell forBody-Centered Cubic?And, Face-Centered Cubic?
12 Crystal Structure Atomic Packing Fraction for FCC Face-Centered-Cubic ArrangementAPF = vol. of atomic spheres in unit celltotal unit cell vol.No. of atoms per unit cell =Volume of one atom=Volume of cubic cell =“R” related to “a” by4/cell4R3/3a3= 0.74
13 Crystal Structure APF for BCC Again, For BCC BCC: a = b = c = a and angles a = b =g= 90°.2 atoms in the cubic cell: (0, 0, 0) and (1/2, 1/2, 1/2).
14 Crystal Structure A B C FCC Stacking Highlighting the stacking the faces
15 Crystal Structure FCC Stacking ABCABC.... repeat along <111> direction gives Cubic Close-Packing (CCP)Face-Centered-Cubic (FCC) is the most efficient packing of hard-spheres of any lattice.Unit cell showing the full symmetry of the FCC arrangement : a = b =c, angles all 90°4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0)
16 Crystal Structure A B A HCP Stacking Highlighting the stacking the cellLayer ALayer BLayer A
17 Crystal Structure HCP Stacking ABABAB.... repeat along <111> direction gives Hexagonal Close-Packing (HCP)Unit cell showing the full symmetry of the HCP arrangement is hexagonalHexagonal: a = b, c = 1.633a and angles a = b = 90°, g = 120°2 atoms in the smallest cell: (0, 0, 0) and (2/3, 1/3, 1/2).
18 Crystal Structure Crystallographic Points, Directions, and Planes. To define a point within a unit cell….Express the coordinates uvw as fractions of unit cell vectors a, b, and c(so that the axes x, y, and z do not have to be orthogonal).pt. coord.pt.x (a) y (b) z (c)0 0 0origin1 0 01 1 11/2 0 1/2
19 Crystal Structure Crystallographic Points, Directions, and Planes. Crystallographic directions and coordinates.Direction B1. Two points are 1, 1, 1 and 0, 0, 02. 1, 1, 1, -0, 0, 0 = 1, 1, 13. No fractions to clear or integers to reduce4. Direction A1. Two points are 1, 0, 0, and 0, 0, 02. 1, 0, 0, -0, 0, 0 = 1, 0, 04. Direction C1. Two points are 0, 0, 1 and 1/2, 1, 02. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 13. 2(-1/2, -1, 1) = -1, -2, 2
20 Crystal Structure Crystallographic Points, Directions, and Planes. Procedure:Any line (or vector direction) is specified by 2 points.The first point is, typically, at the origin (000).Determine length of vector projection in each of 3 axes in units (or fractions) of a, b, and c.X (a), Y(b), Z(c)Multiply or divide by a common factor to reduce the lengths to the smallest integer values, u v w.Enclose in square brackets: [u v w]: direction.abcDIRECTIONS will help define PLANES (Miller Indices or plane normal).5. Designate negative numbers by a barPronounced “bar 1”, “bar 1”, “zero” direction.6. “Family” of directions is designated as <110>.
21 Crystal Structure Crystallographic Points, Directions, and Planes. Figure 2.9 Crystallographic planes and interceptsPlane B1. The plane never intercepts the z axis, so x = 1, y = 2, and z =2.1/x = 1, 1/y =1/2, 1/z = 03. Clear fractions:1/x = 2, 1/y = 1, 1/z = 04. (210)Plane A1. x = 1, y = 1, z = 12.1/x = 1, 1/y = 1,1 /z = 13. No fractions to clear4. (111)Plane C1. We must move the origin, since the plane passes through 0, 0, 0. Let’s move the origin one lattice parameter in the y-direction. Then, x = ∞ , y = -1, and z = ∞2.1/x = 0, 1/y = 1, 1/z = 03. No fractions to clear.4 (o1-o)
22 Crystal Structure t r s Miller Indices for HCP Planes 4-index notation is more important for planes in HCP, in order to distinguish similar planes rotated by 120o.tAs soon as you see , you will know that it is HCP, and not cubic!Find Miller Indices for HCP:Find the intercepts, r and s, of the plane with any two of the basal plane axes (a1, a2, or a3), as well as the intercept, t, with the c axes.Get reciprocals 1/r, 1/s, and 1/t.Convert reciprocals to smallest integers in same ratios.Get h, k, i , l via relation i = - (h+k), where h is associated with a1, k with a3, i with a2, and l with c.Enclose 4-indices in parenthesis: (h k i l) .rs
23 Crystal Structure Miller Indices for HCP Planes What is the Miller Index of the pink plane?The plane’s intercept a1, a3 and c at r=1, s=1 and t= , respectively.The reciprocals are 1/r = 1, 1/s = 1, and 1/t = 0.They are already smallest integers.We can write (h k i l) = (1 ? 1 0).Using i = - (h+k) relation, k=–2.Miller Index is
24 Crystal Structure Linear Density in FCC LD =Number of atoms centered on a direction vectorLength of the direction vectorExample: Calculate the linear density of an FCC crystal along [1 1 0].ASKa. How many spheres along blue line?b. What is length of blue line?ANSWER2 atoms along [1 1 0] in the cube.Length = 4RXZ = 1i + 1j + 0k =
25 Planar Packing Density in FCC Crystal StructurePlanar Packing Density in FCCPD =Area of atoms centered on a given planeArea of the planeExample: Calculate the PPD on (1 1 0) plane of an FCC crystal.Find area filled by atoms in plane: 2R2Find Area of Plane: 8√2 R2Hence,Always independent of R!
27 Crystal Structures of Ionic Materials Factors need to be considered in order to understand crystal structures of ionically bonded solids are:Ionic RadiiElectrical NeutralityConnection between Anion PolyhedraVisualization of Crystal Structures Using ComputersThe cesium chloride structure, a SC unit cell with two ions (Cs+ and CI-) per lattice point. (b) The sodium chloride structure, a FCC unit cell with two ions (Na+ + CI-) per lattice point
28 Crystal Structures of Ionic Materials The zinc blende unit cell, (b) plan view.Fluorite unit cell, (b) plan view.
29 Crystal Structures of Ionic Materials The perovskite unit cell showing the A and B site cations and oxygen ions occupying the face-center positions of the unit cell.
30 Crystal Structure Covalent Structures Tetrahedron and (b) the diamond cubic (DC) unit cell. This open structure is produced because of the requirements of covalent bondingThe silicon-oxygen tetrahedron and the resultant β-cristobalite form of silica
31 Crystal Structure Covalent Structures Tetrahedron and (b) the diamond cubic (DC) unit cell. This open structure is produced because of the requirements of covalent bonding |
Estimating Firm-Anticipated DEFENSE ACQUISITION COSTS WITH A VALUE-MAXIMIZING FRAMEWORK.
If we assume that the cost of research and development of a contract is predetermined and only known by the contractor, fluctuations during the contract are determined simply by the initial bid. If the initial bid is lower than actual cost, there will be cost overruns. Alternatively, if we assume that contractors have a great amount of control over their costs of development, they should fine-tune their cost performance throughout the project to maximize their value. As we develop our theory, only one of these assumptions must be true.
For cost-plus contracts, the government pays all cost overruns and the contractor is virtually unaffected. The only way that a contractor might be immediately impacted is if there is specific language inserted into the contract detailing a risk-sharing scheme, or if the government decides to cut the number of end-product units ordered in response to overruns. While risk-sharing contracts are difficult to empirically evaluate due to a dearth of available data, changes in quantity ordered are well documented for any Major Defense Acquisition Program (MDAP). This reduction in quantity ordered will hurt the firm as most contracts have a research phase, resulting in a predetermined, negotiated rate for the cost of each manufactured unit. Production revenue can often be a significant portion of a government contractor's cash flow in an MDAP.
Whether we assume that the contractor can adjust its costs or that it bids with a fixed final cost in mind, the market will respond to new projects and fluctuations in value in the same manner. It should also not matter if the diminished contract value is due to an incentive scheme or changes in quantity ordered. This article will estimate the effect that new contracts and later amendments have on a company's value. Once the effects are estimated, government contractors can then use them to determine optimal incentives for cost control and cost predictability.
We developed the model to take into account the market's reaction to the initial award amount, changes in development costs, and changes in the value of quantities ordered after development. Intuition tells us that the stock price will go up with a new contract award and will go down as less quantity is ordered. Our theory states that as development costs go up, the government will cut quantity ordered. Given this relationship, we expect that as development costs increase, firm value will decrease. We will compile evidence that development cost increases in a predominantly "cost-plus" environment will only affect firm value if the government changes quantity ordered in response.
After we estimate our model's coefficients, we will develop a framework for designing optimal contract incentives. Since firms will not purposely lower their own value, we can find the ratio of quantity cuts to development cost overruns that forces the firm to either bid more closely to their estimations of cost or control costs below a certain desired level. We can understand this ratio, [E.sup.1], as an elasticity where we measure the percentage change in quantity ordered for a percentage change in development costs.
Cost overruns in defense acquisition have been a problem for as long as the military has looked to industry to deliver innovative products. Most of the relevant literature views contracting for major, innovative defense projects through an insurance framework with moral hazard.
Cummins (1977) observes that prior to the 1960s, the government's primary focus was on limiting contractor profits. To this end, cost-plus contracts, where the government bore all of the risk of cost overruns, were the primary contract types. As awareness of the risk-sharing problem became better understood, the government shifted to greater use of contracts with incentives built in to share the risk between both parties. This shift of risk to the firm controlled cost overruns, but required a greater fixed fee to firms.
The basic framework of the risk-sharing contract is [pi] = [alpha] - (1-y) C, where n is firm profit, a is a fixed fee, y is the fraction of cost overruns borne by the government, and C is the cost overruns. To ascertain a pareto-optimal contract, Cummins built upon a profit maximizing framework with contingent contracts and positive externalities gained by the firm from contract completion. In this article, we will use a value-maximizing framework. The effective difference is that a firm will likely need some market-designated positive profit to undertake a project in our framework, whereas a firm would accept a contract with zero accounting profit in Cummins' model.
Cummins showed that if we take the government's objective to be lowering total cost vice cost overruns and firm profit, risk-sharing contracts can be ineffective. Hiller and Tollison (1978) provide empirical evidence to support this theory. As we increase the risk sharing in a contract, the firm requires a greater fixed fee. Any savings from increased firm attention to cost may be offset by the increased fixed fee. We could see lower cost overruns, but the same--or even greater-total cost. It is therefore much more important to focus on controlling total cost vice cost overruns. Our model will allow the government to calculate expected total cost prior to the awarding of a contract.
Weitzman (1980) takes the analysis of risk-sharing contracts in another direction. His research presents the risk-sharing contract equation and a utility maximization framework to determine an efficient sharing ratio between the principal (government) and the agent (contractor). Weitzman's chief result is simple: if the sharing rate is high, the firm will bid higher and in turn make bids a more reliable indicator of final cost. We will obtain a similar result, but our "sharing rate" will be between development costs and quantity cuts, vice a contracted agreement on risk sharing. Additionally, we will show our sharing rate to impact total cost, not just cost overruns.
Goel (1995) builds upon Weitzman by including an auction framework. His research develops a model where the principal designates a sharing rate, y, and then allows various agents to bid on the contract. Under the model, the principal is able to coax agents to bid closer to their expectations of cost by increasing the amount of risk borne by the agent. Some differences between Goel and our article are a lack of bidding framework in this article, but the addition of a firm value perspective along with empirical investigation. Because the bidding framework does not affect the underlying result shared between Goel and Weitzman--that a higher sharing rate results in a bid that is more reflective of cost estimations--this article's omission of a bidding framework should not impact its results.
Intuition tells us that markets will generally respond positively to any mention of additional profit flowing to a publicly traded firm. Conversely, any news that suggests the possibility of a loss of profit should result in a negative reaction. Our model defines how markets reward and punish defense contractors for different kinds of news. We will then test our model to see if the market encompasses new information in the manner we predict.
Rational markets should reward firms for securing a new contract. The firm should enjoy a larger increase in value if the contract is larger. Our initial model will look just at time period zero, T0, to determine how much of an effect the award has on stock price. Equation 1 shows this where P0 is the percentage change in stock price and IAt is the initially awarded contract amount divided by the value of the firm.
[P.sub.0] = [[beta].sub.1] * I[A.sub.t] (1)
Next, we need an equation that explains the change in price for all further time periods. Firm value fluctuation during the life of the contract should primarily be affected by changes in the value of the contract. Defense contracts adjust through two main avenues: changes in quantity ordered and changes in development costs. Many MDAPs are of the cost-plus variety, meaning that an overrun in cost will be borne by the government, not the firm. Assuming our model's firm is undertaking a cost-plus contract, its value should only be affected by a change in quantity ordered. Equation 2 shows this where [Q.sub.t] is the percentage change in quantity ordered.
[P.sub.t>0] = [[beta].sub.2] * [Q.sub.t] (2)
We can account for all time periods with Equation 3. In [T.sub.0] of Equation 3, I[A.sub.t] will be equal to the initial amount of the contract whereas [Q.sub.t] will be zero. In all other time periods, IA will be zero whereas [Q.sub.t] will likely be nonzero numbers as long as the contract is still active.
[P.sub.t] = [[beta].sub.1] * I[A.sub.t] + [[beta].sub.2] * [Q.sub.1] (3)
In our model, the government can decide [Q.sub.t] at any given time. If it requires less of the final product, [Q.sub.t] will be negative. If it requires more, [Q.sub.t] will be positive. We suspect that the primary reason that the government will change [Q.sub.t] is a change in [D.sub.t], or the percentage change in development costs. As development costs go up, the government must save costs elsewhere and cut [Q.sub.t]. Equation 4 gives us the relationship between [Q.sub.t] and [D.sub.t], where [E.sub.1] is the amount of [Q.sub.t] that the government will change for a given amount of [D.sub.t]. (Note that [E.sub.1] can be thought of as the elasticity of changes in quantity with respect to development cost fluctuations.) We will use this equation to empirically estimate [E.sub.1] in the Results section of this article.
[Q.sub.t] = [E.sub.1] * [D.sub.t] (4)
So that we can express our change in stock price equation only in terms controllable or estimable by the firm, we substitute Equation 4 into Equation 3 to get Equation 5.
[P.sub.t] = [[beta].sub.1] * I[A.sub.t] + [[beta].sub.2] * [E.sub.t] * [D.sub.t] (5)
The point at which our firm will either not bid on a project given known costs (or will avoid reaching if costs are controllable) is the point where the net change in value of the firm across the project is zero. Condition 1 gives us this mathematically, where i is the number of time periods.
Condition 1: (1 + [P.sub.0]) * [[PI].sup.i-1.sub.t=1] (1 + [P.sub.t]) = 1
If we assume a constant change in development costs over the life of the project, we can sum Equation 5 across all time periods. This allows us to calculate the total value change across the entire project to determine the optimal government reaction to a development cost increase. We use notation P to denote the net percentage change in firm value for the duration of the project. If we take D to be the total development cost growth for the duration of the project and assume constant cost growth, we get each annual cost growth to be D . The variable "i" is the number of years required for project completion.
P = [1 + [[beta].sub.1] * IA] * [1 + [[beta].sub.2] * [[E.sub.1] * [D.sup.1/i]].sup.i] - 1 (6)
By solving Equation 6 for [E.sub.1] where P = 0 (from condition 1), we can estimate the appropriate amount that quantity should be reduced if we know the government's opinion of the acceptable development cost growth of the program.
[E.sub.1] = [D([-.sup.1/i])] * [-1 + [(1/([[beta].sub.1] * IA + 1)).sup.1/i]]/[[beta].sub.2] (7)
If the government sets [E.sub.1] equal to its result from Equation 7, the firm would lose net value from the project if it has greater cost growth than D. Because the firm will avoid losing net value, it will avoid having development cost growth greater than D. It can avoid such cost growth either by implementing cost controls where possible, or by bidding closer to the actual cost if it has a reasonable initial estimate.
Our predicted coefficient for all variables in all equations is displayed in Table 1. We will test Equations 3, 4, 5, and 6 empirically to estimate all relevant coefficients. We will then use those coefficients to develop our methodology for incentive creation and cost estimation from Equation 7.
Table 1 displays all of our empirically evaluated variables and our expected coefficients for each variable in each equation. The only variable that we have not previously discussed is SP. It represents the change in the Standard & Poor's 500 Index (S&P 500) for the given time period. This variable will control for exogenous market variation.
We collected data from Selected Acquisition Reports (SARs) on 20 MDAPs that are published at least annually, sometimes quarterly. The 20 MDAPs were awarded to seven separate contractors between 2000 and 2014. The SARs give top-level summaries of changes in costs categorized by Support, Quantity, Engineering, Estimation, Economic, and Schedule. We categorize Engineering, Estimation, and Schedule cost changes as developmental costs. Quantity changes will help us determine our Q variable, whereas Support and Economic changes will be their own category that we suspect will have little impact on the model.
Hough (1992) discusses the difficulties associated with using SARs to study cost overruns. The most significant problem is that when incentive contracts are used, the costs borne by the contractor are not identified in the SAR. The SAR only identifies costs to the government, though it is intuitive that as the development cost grows in the SAR, a firm engaged in a risk-sharing contract will bear a corresponding cost. It is therefore difficult to dissect the effect that the SAR-measured development cost overruns have on firm value and the effects of the presumed collinearity that the SAR measurement has with firm-borne costs. Since our data do not consist of completely cost-plus contracts, we sanitize our results from this effect by testing for this collinearity in three ways.
Because our data will not exclusively consist of cost-plus contracts, we need to test whether or not [D.sub.t] is impacting the stock price directly or indirectly, as hypothesized. To do this, we can compare the value of [[beta].sub.2] * [E.sub.1], as estimated from Equations 3 and 4, and compare it to our empirical estimation from Equation 5. If the values are close to each other, then our data conform to our theoretical model. If there is a significant difference, then a change in [D.sub.t] is having an effect on firm value independent of its impact on [Q.sub.t].
The second test will be to simply regress both [Q.sub.t] and [D.sub.t] against [P.sub.t]. This will allow us to control for either variable and determine which one is primarily driving [P.sub.t].
For a third way to test whether [D.sub.t] is directly or indirectly changing firm value, we will look to Equation 6. We break [D.sub.t] into [Dq.sub.t] and [Dn.sub.t], where [Dq.sub.t] is [D.sub.t] for all cases where [Q.sub.t] is nonzero and [Dn.sub.t] is [D.sub.t] for all cases where [Q.sub.t] is zero. This way we break up the variable into two categories: one where [D.sub.t] plausibly could have impacted Q and one where it could have no impact.
[P.sub.t] = [[beta].sub.1] * I[A.sub.t] + [[beta].sub.2] * [Dq.sub.t] + [[beta].sub.3] * [Dn.sub.t] (8)
We combined data from the last SAR for each year (usually from December) with the last stock quote of the corresponding company for the same year as reported by Google Finance. While it is unusual to rely on annual stock data for an event study, it is not possible in this instance to use daily, weekly, or monthly data. This is because the information included in a SAR flows to the market at varying times, while the SAR simply summarizes it. It is therefore problematic to decide when to designate the event when using more frequent market return data. This hurts the accuracy of this study, but does not negate its findings. For instance, Holthausen and Leftwich (1986) successfully used 300 days to define their event horizon in another event study. We additionally seek to mitigate this problem by including a larger dataset than is typical for this topic.
The predominant firm awarded each MDAP contract was used as the corresponding company. All cost growths are noted in real dollars with the specified program's start year as the baseline. Hough notes that simply using the real dollar values from the SAR is inadequate since initial cost estimates are based on certain inflation assumptions. To account for this, we discount the cost growth associated with economic factors.
To calculate production cost growth, we normalize to the contemporarily approved quantity, vice normalizing to the baseline-approved quantity. The selected calculation method is as reported in SARs.
In several instances, a contract was ongoing for several years and then was transformed into a new contract. Because the SAR data are very high-level and provide no amplifying information to ascertain estimated changes in quantity and development costs, we treated the change as the final year of the project. This slightly reduces the number of observations, but should not impair the validity of our findings.
Our data include 321 data points at the project level and, when aggregated into firm-level data, 113 data points. We perform our analysis with project-level data.
The following MDAPs appeared in our data:
* AESA--Active Electronically Scanned Array (Radar)
* SM-6--Standard Missile-6 (Rocket Intercept Missile [RIM]-174 Standard Extended Range Active Missile)
* SDBII--Small Diameter Bomb II
* MH-60--Military Helicopter-60 (Seahawk)
* AEHF--Advanced Extremely High Frequency (Satellite)
* ACS--Aegis Combat System (Integrated Command and Control/ Weapons Control System)
* JSF--Joint Strike Fighter (Fighter, Strike, and Ground Attack Aircraft)
* Land Warrior--Integrated Soldier System (Weapon, Helmet, Computer, Digital and Voice Communications, Positional and Navigation System, Protective Clothing, Individual Equipment)
* EFV--Expeditionary Fighting Vehicle
* Stryker--Interim Armored Vehicle
* T-AKE--Auxiliary Cargo (K) and Ammunition (E) Ship, Military Ship Classification (MSC) Manned
* Bradley Upgrade--Infantry Fighting Vehicle
* Comanche--RAH-66 Helicopter
* F/A-18E/F--Ai craft Variants (Based on McDonnell Douglas F/A18 Hornet)
* EA-18G--Boeing Growler (Electronic Attack Aircraft)
* CH-47F--Boeing Chinook (Twin-Engine, Tandem Rotor Heavy-Lift Helicopter)
* P-8A--Boeing Poseidon (Navy Maritime, Patrol, Reconnaissance Aircraft)
* FBCB2--Force XXI Battle Command Brigade and Below (Communications Platform to Track Friendly/Hostile Forces on Battlefield)
* Global Hawk--Unmanned Aircraft System
Summary statistics for our project-level data are summarized in Table 2.
The key variables in our project-level data are summarized in Table 3. Strongly correlated variables are in bold and include Q with D, Q with Dq, and D with Dq. Only Q and D are both used in the same regression, and we should expect them to be negatively correlated as observed.
Validity Test of Data
Since our theory relies on the premise that development cost changes impact firm value, primarily through their impact on order quantities, we must test our data to see that they conform to this assumption. We do this through three regressions, two of which will be discussed in this section, while the third will be discussed in the Results section. For the first test, we will split [D.sub.t] into two variables: one that has value when [Q.sub.t] moves in the same time period, [Dq.sub.t], and one that has value when [Q.sub.t] does not move in the same time period, [Dn.sub.t], as in Equation 8. Table 4 defines and shows the results of this test regression.
[Dq.sub.t] is significant at the 95 percent level in our first model and significant at the 90 percent level in the second. The more interesting result is that [Dn.sub.t] is completely insignificant in both models, suggesting that a change in development costs without a change in quantity ordered has no impact on firm value. In other words, without a corresponding change in quantity ordered, variations in development costs are not associated with a change in firm value.
Our second test for data validity will be to regress both [D.sub.t] and [Q.sub.t] against [P.sub.t] to ascertain whether [D.sub.t] becomes insignificant when controlled for [Q.sub.t]. We will use firm-level data for this regression, as project-level data result in insignificance for both variables. Table 5 shows the results of this regression.
As we can see, [D.sub.t], is completely insignificant when controlled for [Q.sub.t]. This indicates that our data are valid. We will discuss our final test in the Results section as we need to complete our primary analysis prior to completing the discussion on our final test. The test will also indicate that our data conform to our theoretical assumptions.
We tested all regressions herein with OLS (Ordinary Least Squares) and random effects and fixed effects models, with the firm as the panel variable and observed minimal variation from the basic regression. To simplify the reading of these results, we have only included the OLS version of each regression.
Our first OLS model (Table 2) looks at Equation 3, with the addition of the SP variable (percent change in S&P 500 Index) and a constant in one of the models. Data are at the project level, with each year being a data point. Table 6 shows the results of this regression.
We can see that only in the model with a constant is our primary variable of interest, [Q.sub.t], significant at the 95 percent level, though the point estimate is within 20 percent in either model. Its coefficient is positive as predicted, meaning that as the government cuts quantity ordered, the firm's value decreases as well. We can also see that an increase in the S&P 500 and the initial contract award are correlated to an increase in firm value. With our next model, we try to tease out the relationship between a change in development costs and a change in quantity costs. Equation 4 from the model section of this article was [Q.sub.t] = [E.sub.1] * [D.sub.t]. Our model in Table 7 is simply [D.sub.t] regressed against [Q.sub.t] to find [E.sub.1]. With the assumption that the government adjusts [Q.sub.t] as a reaction to [D.sub.t], we understand [E.sub.1] to be the ratio of [Q.sub.t] changes for a given [D.sub.t]. In other words, the elasticity of [Q.sub.t] with respect to [D.sub.t].
Accepting the no constant model as our primary model because the constant is not significant at even the 90 percent level, we can understand the coefficient of [D.sub.t] to mean that for every percent that an MDAP's development costs increase from initial estimates, the government will order about 0.17 percent less quantity. Our 95 percent confidence interval for [E.sub.1] lies between -0.1838 and -0.1562.
We also tested the Pearson correlation between the two variables and found it to be 0.8072, indicating a significant inverse relationship between them.
We can also analyze the relationship between Q and D by viewing them on a scatterplot. As we can see in Figure 1, which contains project-level data, the correlation appears negative.
Shifting our initial model from Table 6 to have [D.sub.t] replace [Q.sub.t], we would expect that the coefficient for [D.sub.t] would simply be [Q.sub.t] coefficient multiplied by [E.sub.1] whereas all other coefficients remain the same. The OLS model for Equation 5 is depicted in Table 8, and the far right column is our theory's prediction of the model.
Whereas the no constant model has a higher [R.sup.2], our constant model has all variables significant at the 95 percent level. For this reason, and because our model from Table 2 used the constant model as its primary one, we will use the constant model. In the far right, we can see our theoretical predictions for the OLS model with a constant and that all values are close to actual estimations (well within the 95 percent confidence interval). Our empirically estimated coefficient for [D.sub.t] is -0.125 with a 95 percent confidence interval of -0.232 to -0.018. This places our theoretical prediction well within the limits of our actual estimations and lends significant credence to the validity of both our data and theory.
All regressions including a constant were tested for heteroscedasticity using the Breusch-Pagan/Cook-Weisberg test. We were not able to reject the null hypothesis of homoscedasticity for any regression. We also observed the residual plot for every regression and found no clear evidence of any specification errors. Figure 2 shows the residual plot for equation 5. The plot is not uniformly distributed, but there is no clearly identifiable pattern suggesting omitted variable bias or another specification error. Various specifications all yield qualitatively similar plots.
Our theory, that even in a cost-plus contract, defense contractors' firm value fluctuates indirectly with development cost changes through the government's quantity cutting response, presents a clear framework for building effective incentives to mitigate potential cost overruns. Our data show that as development costs rise from initial estimates, the quantity ordered by the government decreases. Because our theory and model show that a decrease in quantity ordered leads to a lowered firm value, development cost overruns that lead to less quantity ordered should have a similar effect. The only difference in a dollar of cost overrun and a dollar cut from the final order is the ratio between the two. If the government cuts 25 cents of the final order for every dollar of development cost overruns, the harm to the firm from development cost overruns will not be as strong as a 1:1 ratio. The government is the decider of this elasticity and can therefore determine how great the disincentive is for a firm to allow development costs to climb. It is hard to imagine the government determining at the outset of a contract how much quantity they will cut based on development costs, but by establishing a reputational [E.sub.1], the government can effectively achieve the same objective. [E.sub.1] will simply not be as flexible from contract to contract.
Our models all support the theory as we would expect. Changes in quantity ordered are positively correlated and changes in development costs are negatively correlated with changes in firm value. More convincingly, the magnitudes of these correlations are roughly the same as the ratio that the government chooses. Firm value increases about 0.51 percent per percentage increase in end product purchases. Firm value decreases about 0.12 percent for every percentage increase in development costs. If our theory proved exactly correct, given our estimated ratio of quantity cost changes to development cost changes (~ -.17), firm value should decrease approximately 0.9 percent for every percentage increase in development costs. This value is only 25 percent away from our point estimate and is well within reasonable confidence intervals. Further, when we look at changes in development costs that occurred concurrently to changes in quantity ordered versus those that did not, our theory is further supported. Development costs with no quantity changes have no effect on firm value while those with quantity changes associated do have an effect.
If we take our theory and solve for the ratio of changes in quantity ordered to changes in development costs [E.sub.1] = [D([-.sup.1/i])] * [-1 + (1/[([[beta].sub.1] * IA + 1)).sup.1/i]]/[[beta].sub.2], the government can determine the optimal ratio to incentivize firms as desired. If the government wants the end cost to be below a certain amount and the firm can control costs, it must create incentives such that the firm will lose value on the project if it exceeds that amount. For a $70 billion firm bidding $20 billion on a project that will last 5 years, we might create a graphic as in Figure 3.
If the government does not want total cost to exceed $40 billion, it should set the ratio of quantity ordered cut for a development cost overrun at approximately -0.19. Given our estimated values and the firm's aversion to losing value, the firm will allow development costs to only grow an acceptable amount.
We create Figure 1 by calculating [E.sub.1] for all reasonable values of [D.sub.1]. We can then calculate total cost for each value of [E.sub.1] by adding the development cost effects and quantity-ordered effects to the initial bid. The government could also use Figure 3 to better predict final costs of research and development. All it needs to know is [E.sub.1], and it can then ascertain a firm's incentives to control costs.
If the government alternatively believes that the firm knows its costs, but has no control over them, it can seek to incentivize a realistic bid. The harsher the incentives (lower [E.sub.1]), the more closely the bid will reflect firm expectation of cost. If we assume a $70 billion firm that has known costs of $20 billion for development for a 5-year project, we can build Figure 4.
As we can see, the harsher the incentive, the closer the firm's minimum bid gets to its actual estimate. We can rearrange our chart to give us the desired percentage of the total estimate that the government can reasonably expect all bids to at least reach.
We create Figure 4 by calculating [E.sub.1] for every reasonable [D.sub.1], given the firm's expected development costs. We then infer the bid from [D.sub.1].
This framework has several important implications for policy makers. If the government seeks to control costs or at least obtain an accurate estimation of the firm's expectations of cost, it can use Equation 7 and our estimated coefficients to design an optimal contract. While this article looks at the ratio of quantity cuts to development cost overruns, we could easily calculate the profit lost from the quantity cuts and determine a more straightforward cost-sharing ratio with the same coefficients. A dollar of profit lost from quantity cuts should not impact the firm differently than a dollar of profit lost from a cost-sharing scheme.
The government might be encouraged to cancel projects at a lower threshold than is current policy. It might also be encouraged to establish firmer top-line budgets for projects, from development to production. The firm would then understand that as more funds were used in development, a smaller share of the funds would go toward production. When a higher deterrent to cost overruns is established through a demonstrated willingness to cut quantity ordered, we should see more reasonable bidding, less cost overruns, and lower total cost of future projects.
With an [E.sub.1] established by the government, the firm should bid in a predictable manner, given its own expectations of the final cost of development. The government can then use the bid it receives to estimate the firm's true cost expectations. For instance, if we know that a firm should bid 50 percent of its expected cost for an established [E.sub.1], then the government should budget twice what is called for in the contract bid. This insight could allow the government to more accurately forecast expenses and improve contract stability. This stability could lead to lower costs to the taxpayer.
Cummins, J. M. (1977). Incentive contracting for national defense: A problem of optimal risk sharing. The Bell Journal of Economics, 8(1), 168-185.
Goel, R. K. (1995). Choosing the sharing rate for incentive contracts. The American Economist, 39(2), 68-72.
Hiller, J. R., & Tollison, R. D. Incentive versus cost-plus contracts in defense procurement. The Journal of Industrial Economics, 26(3), 239-248.
Holthausen, R., & Leftwich, R. (1986). The effect of bond rating changes on common stock prices. Journal of Financial Economics, 17(1), 57-89.
Hough, P. G. (1992). Pitfalls in calculating cost growth from Selected Acquisition Reports (Report No. N-3136-AF). Santa Monica, CA: RAND.
Weitzman, M. L. (1980). Efficient incentive contracts. The Quarterly Journal of Economics, 94(4), 719-730.
LTJG Sean Lavelle, USN, is a naval flight officer serving with VP-26 on the P-8A Poseidon in Jacksonville, Florida. He holds a BS in Economics from the U.S. Naval Academy and a Master's in Finance from Johns Hopkins University.
(E-mail address:: email@example.com)
Caption: FIGURE 1. PROJECT-LEVEL Q VS. D SCATTERPLOT
Caption: FIGURE 2. EQUATION 5 RESIDUAL PLOT
Caption: FIGURE 3. DEVELOPMENT COSTS VS. STRENGTH OF FIRM PENALTIES
Caption: FIGURE 4. BID ACCURACY VS. STRENGTH OF FIRM PENALTIES
TABLE 1. VARIABLE DESCRIPTION TABLE 1: VARIABLE DESCRIPTIONS Variable Symbol Description IA [[beta].sub.1] The Initially Awarded Contract Amount Divided by the Market Cap of the Firm. Only nonzero during time period of initial award. Q [[beta].sub.2] Change in Cost of Quantity Divided by Initial Contract Amount D Eq 4: Change in Cost of [[beta].sub.3] Development Divided by Eq 5: [E.sub. 1] * [[beta].sub.3] Initial Contract Amount Dq [[beta].sub.3] Change in Cost of Development Divided by Initial Contract Amount if Q Changed in Same Time Period Dn [[beta].sub.4] Change in Cost of Development Divided by Initial Contract Amount if Q did not Change in Same Time Period SP [[beta].sub.5] Percentage Change in S&P 500 Variable Expected Coefficient IA Equation 3: Positive Equation 5: Positive Equation 6: Positive Q Equation 3: Positive D Equation 4: Negative Equation 5: Negative Dq Equation 6: Negative Dn Equation 6: Insignificant SP Positive TABLE 2. SUMMARY STATISTICS Project-Level P SP IA Data Mean 0.144073 0.043694 0.013118 Standard Error 0.013847 0.010968 0.004444 Median 0.145455 0.105877 0 Mode 0.535274 -0.10139 0 Standard Deviation 0.240243 0.189972 0.079493 Sample Variance 0.057717 0.036089 0.006319 Kurtosis 0.543315 0.128455 148.0833 Skewness 0.011312 -0.81439 10.99353 Range 1.314126 0.722675 1.175556 Minimum -0.59603 -0.40967 0 Maximum 0.718095 0.313007 1.175556 Project-Level D Q Data Mean 0.017552 -0.00057 Standard Error 0.011839 0.002507 Median 0 0 Mode 0 0 Standard Deviation 0.211774 0.044844 Sample Variance 0.044848 0.002011 Kurtosis 264.9717 161.3609 Skewness 15.69839 -10.383 Range 3.846937 0.912608 Minimum -0.22689 -0.67047 Maximum 3.62005 0.242139 TABLE 3. PROJECT-LEVEL DATA CORRELATIONS Variables IA Q D Dq Dn SP IA 1 Q 0.0015 1 D -0.0147 -0.8073 1 Dq -0.0153 -0.8113 0.9953 1 Dn 0.0057 -0.0005 0.0996 0.0031 1 SP -0.1124 0.038 0.0527 0.056 -0.0232 1 TABLE 4. CATEGORIZED [D.sub.t] REGRESSION [P.sub.t] = [[beta].sub.1] * [IA.sub.t] + With No [[beta].sub.3] * [Dq.sub.t] + [[beta].sub.4] Constant Constant * [Dn.sub.t] +[[beta].sub.5] * [SP.sub.t] + Constant [SP.sub.t] .6584133 .7914403 (.000) (.000) [IA.sub.t] .4269925 .675944 (.004) (.000) [Dq.sub.t] -.1238065 -.0868289 (.024) (.158) [Dn.sub.t] -.26261614 -.3981645 (.641) (.530) Constant .1432938 N/A (.000) [R.sup.2] .2825 .3647 Observations 300 300 Note: Parentheses contain P-Values. TABLE 5. REGRESSION WITH [Q.sub.t] AS CONTROL VARIABLE [P.sub.t] = [[beta].sub.1] * [IA.sub.t] + With No [[beta].sub.2] * [Q.sub.t] + [[beta].sub.3] * Constant Constant [D.sub.t] + [[beta].sub.5] & [SP.sub.t] + Constant [SP.sub.t] .0006096 .0006593 (.000) (.000) [IA.sub.t] .2742662 .4170445 (.010) (.000) [D.sub.t] -.0101116 .0363141 (.918) (.275) [Q.sub.t] .4029882 .4397591 (.003) (.002) Constant .072123 N/A (.000) [R.sup.2] .4487 .4975 Observations 112 112 Note: Parentheses contain P-Values. TABLE 6. EQUATION 3 REGRESSION [P.sub.t] = [[beta].sub.1] * [IA.sub.t] + With No [[beta].sub.2] * [Q.sub.t] + [[beta].sub.5] * Constant Constant [SP.sub.t] + Constant [SP.sub.t] .6468396 .7822163 (.000) (.000) [IA.sub.t] .428198 .6746299 (.004) (.000) [Q.sub.t] .4995067 .4489935 (.052) (.121) Constant .1106027 N/A (.000) [R.sup.2] .2786 .3280 Observations 300 300 Note: Parentheses contain P-Values. TABLE 7. EQUATION 4 REGRESSION [Q.sub.t] = [E.sub.1] & With Constant No Constant [D.sub.t] + Constant [D.sub.t] -.1709295 -.1699817 (.000) (.000) Constant .0024306 N/A (.103) [R.sup.2] .6516 .6487 Observations 320 320 Note: Parentheses contain P-Values. TABLE 8. EQUATION 5 REGRESSION [P.sub.t] = [[beta].sub.1] * With No Predicted [IA.sub.t] + [[beta].sub.2] * Constant Constant [E.sub.1] * [D.sub.1] + [[beta].sub.5] * [SP.sub.t] + Constant [SP.sub.t] .6589882 .7930062 .6468396 (.000) (.000) [IA.sub.t] .4268835 .676214 .428198 (.004) (.000) [D.sub.t] -.1251678 -.0898093 -0.08538 (.022) (.142) Constant .1121303 .1430959 (.000) [R.sup.2] .2823 .3275 N/A Observations 300 300 300 Note: Parentheses contain P-Values.
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|Publication:||Defense A R Journal|
|Date:||Jul 1, 2017|
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LIGO announced that they have detected gravitational waves from a black hole merger. If verified it will be the ultimate confirmation of Einstein’s theory of general relativity. But what are gravitational waves, and why is their detection such a big deal?
Water is perhaps the most common example of a wave. Drop a stone into a calm pond and you can see ripples expand over the surface of the water. This occurs because water is a fluid. When the stone is dropped into the water, it pushes the water around it out of the way. The water closest to the stone is pushed into the surrounding water, causing it to bunch up a bit. As the bunched up water tries to go back to its original state, it pushes into water further out. Thus a ripple moves through the water.
This is the basic process of any wave. A disturbance in a material affects the region around it causing the disturbance to move through the material. Thus we have ripples in water, sound waves in air, and even seismic waves from earthquakes. Since it takes time for the disturbance to move through the material, waves move through the material at a finite speed.
For a long time it was thought that waves could only move through a physical material, like sound through air. When it was shown in the 1600s that light travels at a finite speed, it sparked much debate over whether light was made of “particles” as Newton suggested, or whether it was a wave traveling through some material. In the early 1800s experiments showing light’s wave behavior seemed to answer the question in favor of waves. We now know light’s behavior is more subtle than that, but in the 1800s it seemed clear that light was definitely a wave. So there must be a material through which light propagates.
The most popular candidate was known as the luminiferous aether. The difficulty with the aether was that it would have to be completely invisible and insubstantial to physical objects, so there was no way to detect it directly. The only evidence that the aether existed came from the fact that light travels in waves. Odd as the aether was, it clearly had to exist since waves always move through some material.
Then in the late 1800s it was found that the speed of light was always the same regardless of ones motion through the aether. This is a deeply un-wavelike behavior. If light really moved through the aether at a particular speed, the Earth’s motion through the aether should make the speed of light appear faster or slower at different times of the year. An unchanging speed of light meant our assumption about the aether must be wrong.
In the early 1900s it was shown that light waves could be explained through special relativity. Instead of moving through a material, the energy fields of electricity and magnetism could be disturbed like a fluid. Light waves are thus waves of electromagnetic energy moving through space. Because physical objects are made of charged particles that have electromagnetic fields, this meant they could never travel through space faster than the speed of light. Relativity showed us that waves didn’t need a physical material to travel through. Waves could move through fields of energy.
But if physical objects couldn’t move faster than light, what about gravity? In Newton’s model of gravity, masses exert forces on each other instantly. As a result, energy could pass from one mass to another with infinite speed. It seemed odd that light energy could only travel at the speed of light while gravitational energy could travel instantly. This puzzle led Einstein to develop a general theory of relativity.
The basic idea of special relativity is that no frame of reference can be favored over any other. This allows the speed of light to be the same in all frames of reference, but it does so by making the behavior of space and time relative to the observer rather than an absolute background. Even the concept of “now” is relative. In general relativity the central idea is the principle of equivalence. Since all bodies fall at the same rate regardless of their mass, a body floating freely in space must be equivalent to a body falling freely. As a result, gravity is not a force between masses, but rather an effect of spacetime curvature. Under Eisntein’s model space and time become flexible and relative, and take on a fluid-like behavior. Masses moving through space should create disturbances in spacetime, just as running your hand through water creates ripples. If general relativity is correct, then there must be gravitational waves.
Over the years various tests of general relativity have confirmed the theory works, and so it has widely assumed that gravitational waves exist. But observing gravitational waves directly has been notoriously difficult. Even the strongest of gravitational waves would be extraordinarily weak, and since they are a warping of spacetime itself, effects such as the finite speed of gravity are impossible to measure. The best evidence we’ve had so far has been indirect evidence. In the 1970s observations of a pulsar orbiting another star found that it slowly spirals closer to its companion. According to relativity this is because gravitational waves radiate away from the binary system, causing it to lose energy and spiral closer together.
But without a direct observation of gravitational waves, there is the chance that general relativity could be wrong. We could, for example, come up with a model that gives us all the effects of Einstein’s theory without gravitational waves. It wouldn’t be as elegant as general relativity, but it would work. Gravitational waves are an absolute necessity for general relativity, and if they don’t exist the model is wrong. So even though we expect gravitational waves to exist, without proof there would always be a small bit of doubt about relativity.
That’s why we’ve been looking for gravitational waves for so long, and that’s why the result is so important.
This post originally appeared on Forbes. |
Create a number using only the digits 4,4,3,3,2,2,1 and 1.
So I can only be eight digits.
You have to make sure the ones are separated by one digit, the twos are separated by two digits the threes are separated with three digits and the fours are separated by four digits.
At a dinner party, many of the guests exchange greetings by shaking hands with each other while they wait for the host to finish cooking.
After all this handshaking, the host, who didn't take part in or see any of the handshaking, gets everybody's attention and says: "I know for a fact that at least two people at this party shook the same number of other people's hands."
How could the host know this? Note that nobody shakes his or her own hand.
Assume there are N people at the party.
Note that the least number of people that someone could shake hands with is 0, and the most someone could shake hands with is N-1 (which would mean that they shook hands with every other person).
Now, if everyone at the party really were to have shaken hands with a different number of people, then that means somone must have shaken hands with 0 people, someone must have shaken hands with 1 person, and so on, all the way up to someone who must have shaken hands with N-1 people. This is the only possible scenario, since there are N people at the party and N different numbers of possible people to shake hands with (all the numbers between 0 and N-1 inclusive).
But this situation isn't possible, because there can't be both a person who shook hands with 0 people (call him Person 0) and a person who shook hands with N-1 people (call him Person N-1). This is because Person 0 shook hands with nobody (and thus didn't shake hands with Person N-1), but Person N-1 shook hands with everybody (and thus did shake hands with Person 0). This is clearly a contradiction, and thus two of the people at the party must have shaken hands with the same number of people.
Pretend there were only 2 guests at the party. Then try 3, and 4, and so on. This should help you think about the problem.
Search: Pigeonhole principle
Four people need to cross a rickety bridge at night. Unfortunately, they have only one torch and the bridge is too dangerous to cross without one. The bridge is only strong enough to support two people at a time. Not all people take the same time to cross the bridge. Times for each person: 1 min, 2 mins, 7 mins and 10 mins. What is the shortest time needed for all four of them to cross the bridge?
It is 17 mins.
1 and 2 go first, then 1 comes back. Then 7 and 10 go and 2 comes back. Then 1 and 2 go again, it makes a total of 17 minutes.
A bad king has a cellar of 1000 bottles of delightful and very expensive wine. A neighboring queen plots to kill the bad king and sends a servant to poison the wine.
Fortunately (or say unfortunately) the bad king's guards catch the servant after he has only poisoned one bottle.
Alas, the guards don't know which bottle but know that the poison is so strong that even if diluted 100,000 times it would still kill the king. Furthermore, it takes one month to have an effect.
The bad king decides he will get some of the prisoners in his vast dungeons to drink the wine. Being a clever bad king he knows he needs to murder no more than 10 prisoners – believing he can fob off such a low death rate – and will still be able to drink the rest of the wine (999 bottles) at his anniversary party in 5 weeks time.
Explain what is in mind of the king, how will he be able to do so?
Think in terms of binary numbers. (now don’t read the solution, give a try).
Number the bottles 1 to 1000 and write the number in binary format.
bottle 1 = 0000000001 (10 digit binary)
bottle 2 = 0000000010
bottle 500 = 0111110100
bottle 1000 = 1111101000
Now take 10 prisoners and number them 1 to 10, now let prisoner 1 take a sip from every bottle that has a 1 in its least significant bit. Let prisoner 10 take a sip from every bottle with a 1 in its most significant bit. etc.
prisoner = 10 9 8 7 6 5 4 3 2 1
bottle 924 = 1 1 1 0 0 1 1 1 0 0
For instance, bottle no. 924 would be sipped by 10,9,8,5,4 and 3. That way if bottle no. 924 was the poisoned one, only those prisoners would die.
After four weeks, line the prisoners up in their bit order and read each living prisoner as a 0 bit and each dead prisoner as a 1 bit. The number that you get is the bottle of wine that was poisoned.
1000 is less than 1024 (2^10). If there were 1024 or more bottles of wine it would take more than 10 prisoners.
A swan sits at the center of a perfectly circular lake. At an edge of the lake stands a ravenous monster waiting to devour the swan. The monster can not enter the water, but it will run around the circumference of the lake to try to catch the swan as soon as it reaches the shore. The monster moves at 4 times the speed of the swan, and it will always move in the direction along the shore that brings it closer to the swan the quickest. Both the swan and the the monster can change directions in an instant.
The swan knows that if it can reach the lake's shore without the monster right on top of it, it can instantly escape into the surrounding forest.
How can the swan succesfully escape?
Assume the radius of the lake is R feet. So the circumference of the lake is (2*pi*R). If the swan swims R/4 feet, (or, put another way, 0.25R feet) straight away from the center of the lake, and then begins swimming in a circle around the center, then it will be able to swim around this circle in the exact same amount of time as the monster will be able to run around the lake's shore (since this inner circle's circumference is 2*pi*(R/4), which is exactly 4 times shorter than the shore's circumference).
From this point, the swan can move a millimeter inward toward the lake's center, and begin swimming around the center in a circle from this distance. It is now going around a very slightly smaller circle than it was a moment ago, and thus will be able to swim around this circle FASTER than the monster can run around the shore.
The swan can keep swimming around this way, pulling further away each second, until finally it is on the opposite side of its inner circle from where the monster is on the shore. At this point, the swan aims directly toward the closest shore and begins swimming that way. At this point, the swan has to swim [0.75R feet + 1 millimeter] to get to shore. Meanwhile, the monster will have to run R*pi feet (half the circumference of the lake) to get to where the swan is headed.
The monster runs four times as fast as the swan, but you can see that it has more than four times as far to run:
[0.75R feet + 1 millimeter] * 4 < R*pi
[This math could actually be incorrect if R were very very small, but in that case we could just say the swan swam inward even less than a millimeter, and make the math work out correctly.]
Because the swan has less than a fourth of the distance to travel as the monster, it will reach the shore before the monster reaches where it is and successfully escape. |
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The net force on the elevator is 3200x4 Newtons. This comprises the sum of a downward force due to gravity of 3200xg, and the upward force equal to the tension in the cables, so: 3200x4 = T-3200xg, so: T = 3200(4 + g). Opps.. lbs not kg, so the elevator mass is ~3200*.4536 ~=1451.5 kg and ft/s not m/s, so ~4 ft/s 4*0.3048 ~=1.22 m/s so we have: Historically, the main technical problem has been considered the ability of the cable to hold up, with tension, the weight of itself below any given point. The greatest tension on a space elevator cable is at the point of geostationary orbit, 35,786 km (22,236 mi) above the Earth's equator. Problem 14.4: Pressure and buoyant force of block suspended in water Please wait for the animation to completely load. The blue liquid is an oil with ρ = 850 kg/m 3 (position is given in centimeters and the dimension of the oil containers into the screen is 20 cm) . Energy Physics Problems - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. Soal fisika tentang hukum kekekalan energi
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physics questions and answers. Problem 4, Elevator Problem, From Lecture (15 Points) 1002 Kg. The Cabin Is Pulled Upward By A Cable, W The Tension In The Cable Is 5187.54 N, What Is The Mass Of The Cabin Accelerates Upward, The Tension ...
Physics 1 Advanced Practice Force, Tension and Friction Problems 1. A 1.45 kg squirrel accelerates down a playground slide set at an angle of 38.0 degrees. Its rate of acceleration is a constant 2.63 m/s2. a. Write a net force equation representing all of the forces acting on the slope of the slide. b. Solve for the coefficient of kinetic ... A 550N physics student stands on a bathroom scale in an elevator. As the elevator starts moving the scale reads 450N. Draw a free body diagram of the problem and find the magnitude and direction of the acceleration of the elevator.
Mar 08, 2015 · The “Elevator Problem” is a physics phenomenon observed in an everyday experience that students can relate to quite easily. It is presented to our IP3 (K9 students) right after the introduction of Newton’s 2nd Law, with the students having a good understanding of the forces of weight and normal contact as well as what makes a resultant force.
Sep 27, 2014 · It is #55 in chapter 3 problems. #55 …A spider is resting after starting to spin its web. The gravitational force on the spider makes it exert a downward force of .150 N on the junction of three strands of silk. The junction is supported by different tension forces in the two stands above it so that the resultant force on the junction is zero.
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Elevator problem finding acceleration.wmv. Physics: Newtons 2nd Law Elevator Problem (With images) These pictures of this page are about:Physics Elevator Problems. Does All Mass Have Weight? « electrogravityphysics.com. Velocity and Acceleration.Tension Formula is made use of to find the tension force acting on any object. It is useful for problems. Tension is a force so it is expressed in Newtons (N). Tension Solved Examples . Underneath are questions based on tension which may be useful for you. Problem 1: A 8 Kg mass is dangling at the end of a string. If the acceleration of the mass is Nov 16, 2019 · The weight of the liquid ($\pi r^2 h \rho g$) is balanced by the upward force due to surface tension ($2\pi r \sigma \cos\theta$). This formula can also be derived using pressure balance. The capillary rise experiment is used to measure the surface tension of a liquid. Solved Problems on Capillary Rise Problem from IIT JEE 2014 Find (a) tension T 1 , (b) tension T 2 (c) tension T 3 (d) angel pheta. 2. The system in the picture is in equilibrium. A concrete block of mass 225 kg hanfs from the end of the uniform strut whose mass is 45.0 kg. Find (a) the tension T in the cable and the (b) horizontal and (c) vertical force components on the strut from the hinge. 3. teaches physics through the physics problems. Menu Skip to content. An object of mass m is hanging by a string from the ceiling of an elevator. The elevator is moving upward, but slowing down.The elevator accelerates downward at a constant rate of 1.2 m/s 2 2. A crane accelerates a 175 kg load is upward. The tension in the cable is 2000 N. Find the magnitude and direction of the elevator’s acceleration.What is the solution to the following system mc015 1 jpg 8 40 32 10 2 3 20 4 5 40 8 10As this practice physics problems and answers for tension, it ends happening brute one of the favored book practice physics problems and answers for tension collections that we have. This is why you remain in the best website to see the amazing ebook to have. Tension Force Physics Problems Two Cables With Hanging Mass Static Equilibrium. Elevator Physics Problem Normal Force On A Scale Apparent Weight. Box With Pulley.Three box pulley problem with slopes and friction using the "black box" method. Three boxes (3.75 kg, 5.50 kg and 12.0 kg) are tied together by two ropes and hang from a pulley as shown. The coefficient of friction between the ground and the boxes is 0.250. What is the acceleration of each box, and the tension in ropes A and B? Physics Problems (video solutions) ... What are the hinge forces and the tension in the cable? ... Various questions about the motion of an elevator speeding up ... The "Elevator Problem" is a classic problem in physics. The situation is this: "You are standing on a bathroom scale in an elevator. You are holding an apple. (Yes, people are staring at you...) You weigh 500 Newtons, so your mass is about 50 kg." This assignment is a step-by-step analysis of the elevator problem. Jan 6, 2016 - This is a classic 2nd law problem to calculate the apparent weight in an elevator accelerating upward. More information Physics: Newtons 2nd Law Elevator Problem Review Problems for Introductory Physics 1 May 20,2019 Robert G.Brown,Instructor Duke University PhysicsDepartment Durham, NC27708-0305 [email protected]
This Physics Girl video highlights seven different surface tension experiments that you can easily try at home or in the classroom. Have any of these around? A plate, a glass, a penny, an index card, a paperclip, an eye dropper, a cup of coffee, dish soap, or some food coloring… and if you’re teaming up with an adult, get a pan and stove, too. Apr 24, 2013 · The Newton is the unit used for Force and is equivalent to the force required to give a mass of 1 kg an acceleration of 1 ms^-2. We are going to calculate the tension force in newtons that exists in each supporting cable. We can begin this problem by imagining the weight of the box as a force exerted downward and coinciding with the z-axis. Physics 4A Chapter 5 HW Problems - Free download as Word Doc (.doc), PDF File (.pdf), Text File (.txt) or read online for free. A. The tension is greater than 9800 N. B. The tension is less than 9800 N. C. The tension equals 9800 N. D. It cannot be determined from the information given. Correct Answer: A. Explanation: The forces on the elevator are the tension upward and the weight downward, so the net force on the elevator is the difference between the two. In physics, tension is the force exerted by a rope, string, cable, or similar object on one or more Let's say in our example problem that our object is no longer accelerating upwards but instead is "I was not capable of solving problems based on tension. But after going through this page, I am now...Apr 05, 2013 · The weight of the elevator is a force downwards due to gravity. Friction always acts in the opposite direction of motion, so friction is also acting downwards. In total, the force acting downwards on the elevator is 12000 N + 5000 N = 17000 N. The tension in the cable is pulling the elevator up, so the total force upwards is equal to 20000 N. Baptist courier jobsProblems Of Physics Assessment Ap Physics 1 2016 Free by physicsassessment.blogspot.com. Incline Plane With Friction And Tension Physics Challenge Problem by youtube.com. Pulley Problems by real-world-physics-problems.com. Tension String Forces Problems With Solutions by problemsphysics.com The physics of the circulation controlled airfoil is complex and poorly understood, particularly with regards to jet stall, which is the eventual breakdown of lift augmentation by the jet at some sufficiently high blowing rate. This section of The Physics Hypertextbook is a gathering place for problems where the forces are balanced in all directions. practice problem 1. A 100 N sign is to be suspended by two cables. Determine the tension in each cable for the given angles…Welcome in Collection of Solved Problems in Physics. This collection of Solved Problems in Physics is developed by Department of Physics Education, Faculty of Mathematics and Physics, Charles University in Prague since 2006. The Collection contains tasks at various level in mechanics, electromagnetism, thermodynamics and optics. What happens when you jump in a moving elevator? Do you weigh more when you're going up and less when you're going down? Carol Hedden explores the relationship between gravity, weight, and relative motion, using a moving elevator to explain the fascinating physics. Review Problems for Introductory Physics 1 May 20,2019 Robert G.Brown,Instructor Duke University PhysicsDepartment Durham, NC27708-0305 [email protected] 026009593 and tax idProblems with solutions and explanations on tension and forces using free body diagram and Newton's laws. Several problems with solutions and detailed explanations on systems with strings, pulleys and inclined planes are presented.I just noticed a weird problem with the elevators on my S.E.5a, which is probably one of the reasons I've been crashing so much. Is this a problem with my joystick, or a known fault with the game, or something I can fix?The Elevator Ride Interactive is a simulation depicting the forces acting upon an elevator rider while ascending and descending. The emphasis on the Interactive is on communicating the sensations of weightlessness and weightiness experienced by a rider. Physics Assignment Help, Determine tension in pulley problems, Three equal weights A, B, C of mass 2 kg each are hanging on a string passing over a fixed frictionless pulley as shown. Tension in the string connecting weights B and C is approximately. Answers: Acceleration of the system = a = (4 For m 2 we have the tension up (-), and the weight down (-). m 2 g = (2.7)(9.8 N/kg) = 26.46 N . Our equation is: 26.46 - T = 2.7a Where T is the tension in the string, and a is the acceleration. (Tension is up, which is the negative direction) China Physics Elevator Problem manufacturers - Select 2020 high quality Physics Elevator Problem products in best price from certified Chinese Sourcing Guide for Physics Elevator Problem: Transportation is not just about moving an object from point A to point B, it's a process of value...Unit 8: Worksheet 6: MORE Additional Practice Problems. p. 1, 10/4/2009 Practice of a 2D tension problem. A force board (or force table) is a common physics lab apparatus that has three (or more) strings or cables attached to a center ring which is in equilibrium. The strings or cables exert forces upon the center ring in three different ... Forces in Physics, tutorials and Problems with Solutions Free tutorials on forces with questions and problems with detailed solutions and examples. The concepts of forces, friction forces, action and reaction forces, free body diagrams, tension of string, inclined planes, etc. are discussed and through examples, questions with solutions and ... Physique Physics Concepts Physical Science Elevator Classroom Ideas Bar Chart Law Engineering Design. Shows you how to calculate the acceleration and tension in the string of a two block system on an inclined plane with friction.Elevator Problems. Elevator problems are actually easy. There are only two forces acting on the person in the elevator, gravity and the normal force. (In the problem the normal force may be referred to as “the force between the person and the floor”, “the force on the scale” -if the person is standing on a scale – or “the apparent weight”.) The force of gravity (or weight) of the person is constant (mg). As this practice physics problems and answers for tension, it ends happening brute one of the favored book practice physics problems and answers for tension collections that we have. This is why you remain in the best website to see the amazing ebook to have. Physics elevator problem tension
Problem 14.4: Pressure and buoyant force of block suspended in water Please wait for the animation to completely load. The blue liquid is an oil with ρ = 850 kg/m 3 (position is given in centimeters and the dimension of the oil containers into the screen is 20 cm) . Mar 04, 2011 · elevator cabs A and B are connected by a short cable and can be pulled upward or lowered by the cable above cab A. Cab A has mass 1650 kg; cab B has mass 1390 kg. A 13 kg box of catnip lies on the floor of cab A. The tension in the cable connecting the cabs is 1.93 ×104 N. What is the magnitude of the normal force on the box from the floor? Physics Teacher, 15, 2, 99-100, Feb 77 Discusses the physics involved in a falling elevator and whether an occupant of a falling elevator will be unharmed if he jumps up as the elevator crashes. (MLH)
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The specific heat capacity of the sample was calculated by equation 1 from the dsc data obtained (a, b, and c in figure 1) figure 1 dsc measurement of specific heat capacity. What is the difference between specific heat capacity, heat capacity, and molar heat capacity - duration: 12:29 the organic chemistry tutor 12,226 views. Specific heat of a substance is the heat capacity per unit mass thus, heat capacity = mass x specific heat thus, heat capacity = mass x specific heat the specific heat is essentially a measure of how thermally insensitive a substance is to the addition of. Phy pack 10 – specific heat capacity of solid 2 produced by the physics group university of cape coast as part of a delphe funded project title: determination of the specific heat capacity of a solid by the method of mixtures.
So of course mah/g is a measure of stored charge and is some times called specific capacity, and c/g is clearly specific charge these quantities are measures of how many ions and electrons can be. A material's specific heat capacity is a measurement of how much energy it takes to change the temperature of a sample by some fixed amount, divided by the mass of the sample. To measure the specific heat capacity by the method of mixtures physics homework help and to measure the specific heat capacity by the method of mixtures physics assignments help available 24/7, as well as assignments experts and tutors also available online for hire for physics projects. The profile of the estimated values of the msw specific gravity according to the results of cptu testing by the method [1, 2] is shown in fig 9 types of soil behavior i c : 2 -organic soils, 3.
Episode 607-2: the specific heat capacity of water and aluminium (word, 37 kb) it is useful to compare electrical methods of measuring the specific heat capacity of a solid and liquid including the continuous flow calorimeter for a liquid. Thermal diffusivity, specific heat capacity and thermal conductivity of glassy carbon and molybdenum were measured in the temperature range from 300 to 1100 k laser flash method simultaneous measurement solid materials specific heat capacity thermal conductivity thermal diffusivity. Specific heat capacity of a solid specific heat capacity of a substance is defined as the amount of the heat energy required to raise the temperature of a unit mass of a substance through 1 degree. Specific heat of a solid substance can be determined by the “method of mixture” using the concept of the “law of heat exchange” ie heat lost by hot body = heat gained by cold body the method of mixture based on the fact that when a hot substance is mixed with a cold substance, the hot body loses heat and the cold body absorbs heat.
Student 1 of 2 practical 11 specific heat capacity of a solid purpose the aim of this experiment is to measure the specific heat capacity of a solid using an electrical method. 1 the specific thermal capacity can be determined from the relationship: mass x specific thermal capacity x temperature / time = current (amps) x pd (volts) the ratio temperature rise / time can be obtained from the slope of a graph of temperature plotted against time. Specific gravity of soil by pycnometer method equipment for specific gravity test of soil the major measuring equipment in this test is pycnometer this is a glass jar of 1 litre capacity that is fitted at its top by a conical cap made of brass it has a screw type cover as shown in figure-1. The specific heat of a material is related to heat capacity, except that specific heat doesn’t depend on an object’s mass, though it still depends on the type of material specific.
Specific heat capacity of a solid by an electrical method the method is very similar to that for a liquid except that there is no container the solid under test is a lagged cylinder with holes drilled for the thermometer and the heater element. The software calculates the specific heat capacity with respect to the specific heat capacity of the sapphire standard slide 10: sapphire method the slide summarizes the results obtained using the sapphire method. Measurement of specific heat capacities there are several simple methods for measuring the specific heat capacities of both solids and liquids, such as the method of mixtures, but we will consider here only electrical methods. To determine the specific heat capacity of a given metallic solid using the method of mixtures 2 2 to obtain the values of specific heat capacities of different. By this method, the volume of a solid sample is determined by comparing the weight of the sample in air to the weight of the sample immersed in a liquid of known density the volume of the sample is equal to the difference in the two weights divided by the density of the liquid.
Specific heat capacity, or simply specific heat, is the amount of heat required to change the temperature of a substance as water requires more time to. I have an assignment to find the specific heat capacity of water we did an experiment in class where we used an electric kettle with power output of 1850w-2200w to heat up 1,400g of water (we actually used 1,400 ml of tap water but we were told to assume that the tap water has the same density as. Abstract: a substantial number of upper-level science students and practicing physical science teachers demonstrate confusion about thermal equilibrium, heat transfer, heat capacity, and specific heat capacity the traditional method of instruction, which involves. The specific thermal capacity of aluminium is 900 j/kg °c the specific thermal capacity of water is 4200 j/kg °c 2 it takes more energy to raise the same temperature of water by each °c than it does to raise the temperature of the same mass of aluminium.
To determine the specific heat capacity of a given metallic solid using the method of mixtures 2 to obtain the values of specific heat capacities of different materials from other groups and compare them. The specific gravity of whole mix is the average specific gravity of all solid particles present in that mix for most of the soils, specific gravity lies in the range of 265 to 280 it has no units. In this experiment, the specific heat of a solid was determined by the method of mixtures the latent heat of fusion and heat of vaporization of water were determined specific heat specific heat is the amount or quantity of heat per unit of mass of a substance required to. |
“I wouldn’t know a spacetime continuum or a warp core breach if they got into bed with me.” –Patrick Stewart
It’s the end of the week once again, and so it’s time for another Ask Ethan segment! There have been scores of good questions to choose from that were submitted this month alone (and you can submit yours here), but this week’s comes from our reader garbulky, who asks:
Why does gravity decrease the further away you are from the object? I’ve read that it does decrease with distance squared but not why it does this.
This question seems so simple, and yet the answer — to the limits of our understanding — is nothing short of profound.
Physics, and science in general, doesn’t normally address the question of why when it comes to natural phenomena; it normally sticks to how. You give me an overarching theory, such as a set of laws, and physical objects with specific properties, such as a set of particles, and science tells you how those objects behave according to the predictions of that theory. Gravity is no exception.
For centuries, Newtonian gravitation was the most successful theory describing forces on the largest scales, saying that every object in the Universe that has a mass exerts an attractive force on every other object in the Universe with a mass, and that the magnitude of that force is proportional to the mass of both objects and inversely proportional to the distance between them. That’s what Newton’s law of universal gravitation says, and what it tells us is — in principle — how any system of particles will behave under the influence of gravity.
Can we say something intelligent about why gravity works this way, though? Let’s think about our own neighborhood for a minute.
The Sun, the largest mass in our Solar System, is orbited in circles and ellipses by practically every known object, from planets to asteroids and (most) comets. There’s something special about circles and ellipses that we don’t normally think about as special: they’re stable, closed orbits, meaning that these objects return to the same point they started at after what we call a year.
That alone, mathematically, tells you something incredibly interesting. You see, all forces are vectors, meaning they have magnitudes and directions. In the case of our Solar System, the direction of the force on each object is (to an excellent approximation) towards the center of the Sun. Want something to go around the Sun in a closed orbit? Guess what.
You only have two options! One is to have a force that obeys an inverse-square law (like gravity does), and the other is to have a force that increases linearly with distance (like a spring does), and there’s a theorem that proves those are the only two possibilities!
So it could have gotten stronger or weaker as the distance increased, but only in one particular way, or we wouldn’t have stable, closed orbits.
And since those are the types of orbits required to have stable, moderate temperatures necessary for life, we sure did luck out that these are the laws governing our Universe!
Now there are some forces where the force increases as your distance from the object increases: the strong force is a great example! And there’s even an example of a type of force that has no direction and is constant everywhere: that’s what dark energy is, permeating all of space equally!
The thing is, though, saying that gravity is an inverse-distance-squared force is an incomplete story. In fact, the very fact that we have an orbit in our Solar System that very clearly isn’t closed is how we wound up replacing Newtonian gravity with our modern theory of gravity: General Relativity!
We’ve come up with a whole host of predictions that have been borne out by experiment and observation, including the gravitational bending of light, the different orbital mechanics of systems with large masses and small distances, gravitational redshift, and many, many others.
But the greatest advance that’s related to this question of the strength of gravity is the knowledge that all orbiting bodies do not technically obey an inverse-squared force law.
All orbits under General Relativity come from forces that behave ever so slightly stronger than inverse-square laws, and this means that they will eventually decay over long enough timescales. The innermost planets will have their orbits decay first, followed by progressively outer worlds, because the distance is larger. Eventually, in the absence of all other phenomena, everything would spiral into the gravitational source at the center of all orbital systems.
For an object like Earth that orbited an imaginary, infinitely-long-lived Sun, it would take something like 10150 years for the orbit to decay, but it means that a true stable, closed orbit is a phantasm, something that doesn’t really exist in this Universe!
At least, in a Universe governed by General Relativity, which is the best law of nature we have to describe gravity. In the weak-field limit (an approximation) — when masses are small and distances are large — this can be shown to reduce to Newtonian gravity, which is where the inverse-square-law-with-distance comes from!
But why do we have General Relativity as the theory that governs gravitation in this Universe, with the particular details that it has? I can’t say for certain; no one can.
Which means I have to resort to the standard cop-out answer: the force of gravity is this way because the laws of nature cause it to be. We can imagine a Universe where those laws are different, but this is the one we’ve got, and we don’t fully understand why the laws are this way any deeper than that. We can observe phenomena, infer the laws, test them in new and spectacular ways, and maybe someday we’ll understand why the laws are this way. In the meantime, this is the best answer we’ve got! |
2 edition of first course in modern mathematics. found in the catalog.
first course in modern mathematics.
Anderson, Marie B. SC.
|Series||Heinemann"s modern mathematics series|
|LC Classifications||QA106 .A48|
|The Physical Object|
|LC Control Number||67076816|
Additional Pure Mathematics: A Modern Course by Chow, W.K., So, P.F. and a great selection of related books, art and collectibles available now at Holt Rinehart and Winston; Author division. Holt Rinehart and Winston is currently considered a "single author." If one or more works are by a distinct, homonymous authors, go ahead and split the author. Includes. Holt Rinehart and Winston is composed of 27 names. You can examine and separate out names. Combine with.
Description. This book provides a complete course for first-year engineering mathematics. Whichever field of engineering you are studying, you will be most likely to require knowledge of the mathematics presented in this textbook. Created by Modern States, a non-profit education alliance dedicated to college access for all, in partnership with edX from Harvard and MIT. "Freshman Year for Free Takes Aim at Student Debt" Choose one or many free courses that interest you from great Universities. Explore Courses. Courses prepare you for widely accepted Advanced Placement and.
The books introduced “Modern Mathematics” topics such as Set theory, Matrices, Vectors and Mappings alongside the more traditional topics. An innovative feature of the course is the series of Computer topics which appeared from book 4 onwards. The intention was that these would form an introduction to computer studies. This book is intended to be used by children ages 5 to 6. Other age groups will also benefit from the book. Anyone can use this book globally, although the curriculum may differ slightly from one region to the other. This is so because the core content of Mathematics is the same around the world.
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Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web. I tried to select only the works in book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out.
A Course in Modern Geometries In geometry, I went the opposite way: studying Cederberg's book first before moving to the more advanced one.
I like her clear presentation and especially the part on matrix representations of groups of transformations. This book would be a Cited by: A First Course in Linear Algebra is an introductory textbook aimed at college-level sophomores and juniors. Typically students will have taken calculus, but it is not a prerequisite.
The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear /5(11). Basic modern mathematics: First-[second] course [Robert E Eicholz] on *FREE* shipping on qualifying offers.
A First Course in Complex Analysis With Applications. by Dennis Zill and Patrick Shanahan. Review: This book gives students an accessible introduction to the world of complex analysis and how its methods are used.
A First Course in Complex Analysis is reader-friendly to the newcomer and therefore is ideal for use by both undergrads as well as. This text, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course in linear algebra for science and engineering students who have an understanding of basic algebra.
All major topics of linear algebra are available in detail, as well as proofs of important theorems/5(7). A First Course in Applied Mathematics is an ideal book for mathematics, computer science, and engineering courses at the upper-undergraduate level.
The book also serves as a valuable reference for practitioners working with mathematical modeling, computational methods, and the applications of mathematics in their everyday work. Pure Mathematics book. Read 11 reviews from the world's largest community for readers. This well-established two-book course is designed for class teachi /5.
Great deals on Mathematics Books Get cozy and expand your home library with a large online selection of books at Fast & Free shipping on many items. OCLC Number: Notes: Previous edition: Includes index. Description: vi, pages: illustrations ; 23 cm.
Series Title: Heinemann modern mathematics. Starting with symbolizing sentences and sentential connectives, this work proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables.
Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Additional Physical Format: Online version: Eicholz, Robert E.
Basic modern mathematics. Palo Alto, Calif., Addison-Wesley Pub. (OCoLC) A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by E.
Whittaker and G. Watson, first published by Cambridge University Press in Author: E. Whittaker and G. Watson. “The author aims to present a few topics that he considers important, striking an excellent balance between the need to be concise and to include all of the fundamental methods. this is a valuable book for both mathematics students and applies researchers, and fills a gap in the literature by providing a rigorous and modern introduction to the basic theory of statistics.” (Marco Bee.
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales.
Long a standard textbook for graduate use in both Britain and America, this classic of modern mathematics remains a lucid, if advanced introduction to higher mathematics as used in advanced chemistry and physics courses. ( views) Mathematical Formula Handbook by Wu-ting Tsai - National Taiwan University, I think a good first book is 'A First Course in Mathematical Analysis' by David Alexandar Brannan and can suggest it as well as several that have already been mentioned on this page, but this one has the advantage that it was a byproduct of the Open University and is thus totally designed for self-study.
The History of Mathematics: A Brief Course, Second Edition. Author(s): Written by one of the world's leading experts on the history of mathematics, the book details the key historical developments in the field, providing an understanding and appreciation of how mathematics influences today's science, art, music, literature, and society.
Ordinary differential equations of first order. Differential Equations with YouTube Examples. A First Course in Ordinary Differential Equations. Advanced Topics In Introductory Probability.
Introductory Finite Volume Methods for PDEs. Fibonacci Numbers and the Golden Ratio. Discrete Dynamical Systems. Mathematics - Free of Worries at the.
MTH Mathematics and the Modern World with Algebra Review This is a general education course required by many non-technical degree programs, mostly those in the liberal or performing arts. Material covered includes everything in MTH with the addition of a review of elementary algebra skills.
Undergraduates seriously interested in mathematics are encouraged to elect an upper-level mathematics seminar. This is normally done during the junior year or the first semester of the senior year. The experience gained from active participation in a seminar conducted by a research mathematician is particularly valuable for a student planning.Solution Manual A First Course in Mathematical Modeling 5th 5E Frank R.
Giordano; William P. Fox; Year: ISBN ISBN Math Textbook Precalculus Trigonometry Elementary Teacher Mathematics The Book Manual Ebooks Modeling.This book is the most elegant book I read in mathematics. It's more fun than reading any novel or story. The amazing Ian Stewart talent in conveying the importance of modern mathematics is just unique.
The book is more like a story to me, you can read it all along without interruption/5. |
Have you ever been given a problem to solve in chemistry class that left you feeling overwhelmed and clueless? If so, then this guide is for you. We’ll provide an overview of the concepts behind stoichiometry and discuss how a calculator can help make the process of solving stoichiometry problems easier. By taking the time to understand the basics of stoichiometry, you’ll set yourself up for success in your chemistry classes!
Stoichiometry is the study of the quantitative relationship between the reactants and products in a chemical reaction. In other words, it’s the study of how much of each reactant is needed to produce a given amount of product. In order to understand stoichiometry, you need to be familiar with the terms “molarity” and “moles.”
Molarity is a measure of the concentration of a solution. It’s expressed as moles per liter (mol/L). For example, if you have a 0.5 molar (0.5 M) solution of hydrochloric acid, that means that there are 0.5 moles of hydrochloric acid per liter of solution.
Moles is a unit of measurement that corresponds to the number of atoms or molecules in a sample. For example, one mole of oxygen gas contains 6.02 x 10^23 oxygen atoms. That might seem like a lot, but it’s actually a very small number when you compare it to the number of atoms in even a small sample of material.
Now that you know what molarity and moles are, let’s talk about how to use them to solve stoichiometry problems. To do this, we’ll need to use a calculator.
Definition of Stoichiometry and Its Applications
In order to understand what stoichiometry is, one must first know what the word “stoichiometry” means. Stoichiometry comes from two Greek words: “stoicheion” meaning element, and “metron” meaning measure.
In other words, stoichiometry is the study of measuring quantities of elements in a chemical reaction. The mole is the SI unit for amount of substance. It allows us to relate the masses of reactants and products involved in a chemical reaction to the number of moles of each. This is extremely important because it allows us to predict how much product will be formed from a given reactant quantity, or how much reactant is needed to produce a given quantity of product.
Now that we know what stoichiometry is, let’s talk about some of its applications. One example where stoichiometry calculations are used frequently is in baking recipes. When baking, it is important to use the correct ratios of ingredients in order to produce the desired outcome; too much or too little of any one ingredient can result in an undesirable end product.
Another common application for stoichiometry occurs when making detergents and cleaning solutions. The specific amounts of each component must be carefully measured in order for the solution to work correctly and achieve its intended purpose. Lastly, pharmaceutical companies heavily rely on accurate stoichiometric calculations when creating new drugs and medications; even a small error could have potentially dangerous consequences.
How To Use a Calculator for Stoichiometry Problems
To use a calculator for stoichiometry problems, start by inputting the molar masses of the reactants and products. Next, input the moles of each reactant. Once you have your moles, you can calculate the mole ratio between the two substances. To do this, divide the moles of one substance by the other. This will give you the mole ratio between them. Finally, use this mole ratio to determine how many moles of product will be produced from the given reactants.
Working Through Examples of Stoichiometry Calculations
When it comes to chemistry, stoichiometry is the study of the quantitative relationships between reactants and products in chemical reactions. In order to understand and solve problems related to stoichiometry, it is important to be able to work through examples using a calculator.
To start, let’s consider the following reaction: 2H2 + O2 –> 2H2O
In this reaction, we know that for every two molecules of hydrogen gas (H2), we will need one molecule of oxygen gas (O2) in order to produce two molecules of water (H2O). We can use this information to calculate the amount of each reactant needed in order to produce a specific amount of product.
For example, let’s say we want to know how many moles of hydrogen gas we would need in order to produce 10 moles of water. We can use the following equation to calculate this:
10 moles H2O x (1 mole H2 / 2 moles H2O) = 5 moles H2
This equation tells us that in order to produce 10 moles of water, we would need 5 moles of hydrogen gas. We can then use this same equation to calculate the amount of oxygen gas needed by rearranging it as follows:
10 moles H2O x (1 mole O2 / 2 moles H2O) = 5 moles O2
Tips for Solving Stoichiometric Problems Easily
When it comes to solving stoichiometric problems, there are a few things that you can do to make the process easier. First, it is important to have a clear understanding of the concepts behind stoichiometry. This will allow you to better understand the problem and how to solve it.
Second, make sure that you have a good calculator that can help you with the calculations. There are many different types of calculators available, so make sure that you choose one that will be easy for you to use. Finally, practice solving stoichiometric problems before you take your test or exam. This will help you get used to the process and ensure that you are able to correctly solve the problem.
Stoichiometry can be a complex concept to understand, but with the right information and practice it can become much easier. We hope this comprehensive guide has provided you with all of the tools necessary to help you better understand stoichiometry and use a calculator to solve problems related to it. With enough practice, you will soon find yourself solving these kinds of problems in no time! Good luck! |
This advanced undergraduate-level text was recommended for teacher education by The American Mathematical Monthly and praised as a "most readable book." An ideal introduction to groups and Galois theory, it provides students with an appreciation of abstraction and arbitrary postulational syste... read more
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SINC APPROXIMATION OF ALGEBRAICALLY DECAYING FUNCTIONS††footnotetext: Key words. sinc methods, sinc interpolation, algebraically decaying functions, Lambert-W function, polynomial order of convergence, approximation on real-line.
D. O. SYTNYK
Abstract. An extension of sinc interpolation on to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of error estimates. First covers a wider class of functions with the algebraic order of decay on . The second type of error estimates governs the case when the order of function’s decay can be estimated everywhere in the horizontal strip of complex plane around . The numerical examples are provided.
We begin by introducing some necessary notation. Let
By in the paper we denote the class of functions analytic in the horizontal strip
and such, that the quantity
is bounded. Next, for some given and integer we define a sinc interpolation polynomial as
The following classical result characterize the accuracy of interpolation of by for the case, when is exponentially decaying.
Theorem (Stenger [6, p. 137])Assume that the function is bounded by
with some > 0. Then the error of term sinc interpolation of by , satisfies the following estimate
Here is some constant dependent on and independent on . In this paper we extend the results of the above theorem to a class of algebraically decaying functions on . All theoretical considerations are given in sections 1,2. Section 3 is devoted to numerical examples and discussion.
1. Interpolation of functions with algebraic decay on real line
In this section we study the convergence of sinc interpolation for the class of algebraically decaying functions. Specifically, we consider the situation when the function satisfies
provided that in (0.3) is chosen as
Here denotes a positive branch of the Lambert-W function, and is the constant independent of :
Proof. For any fixed the error of sinc interpolation can be represented as follows [6, equation (3.1.29)]
Bound of the first term on the right-hand side of this formula was obtained in Theorem 3.1.3 from . For this term satisfies
where is some constant to be determined later. For the second term we get
The above sequence of inequalities is justified as long as satisfy (1.1) with some . For such , truncation error (1.6) decays algebraically as . In order to balance it with exponentially decaying discretization error (1.5) one needs to solve for the equation
Let and assume that is some fixed parameter. Then, equation (1.7) takes the form
which has a unique solution
Now, let us come back to the determination of . The smallest suitable for (1.5) can be defined as follows
Its not hard to see that the maximum is attained at . Therefore, the value of :
is clearly greater than one, for any , . To get (1.4) we apply the identity to the above formula for and rearrange the result accordingly
The presence of in estimate (1.2) makes it harder to perceive the asymptotic behavior of the interpolation error intuitively. To fix that we recall a well-established result on the asymptotic properties of , valid for any :
whence it is clear that the error of sinc interpolation provided by Theorem 1.1 is asymptotically equal to as . To analyze the error for small we note that, in the view of (1.7), is bounded by the exponent with a strictly decreasing negative argument. Consequently, for any , , the error lies within the interval and decreases as .
One might conclude from the foregoing analysis that a simple asymptotic formula can be used to redefine (1.3) in terms of logarithms, which are computationally more favorable than the Lambert-W function. To explore this possibility we set
and study the corresponding error terms of the approximation. Discretization error (1.5) is positive and monotonically decreasing in for any , since is monotonic. The principal part of truncation error (1.6) has one global maximum at :
To guarantee a monotonous decrease of the truncation error for all we must require , which yields The aforementioned formula for is thereby reduced to
For such , the error of sinc interpolation will be bounded by (1.2) with
With an additional a priori knowledge about we should be able to improve the convergence properties of described by Theorem 1.1. The following improvement of (1.2) offers a more realistic balance of discretization and truncation errors, presuming that both and are known.
2. Interpolation of functions with algebraic decay in the strip
Corollary 1.3 is difficult to apply as it is, because the evaluation of requires computation of the contour integral over . In order to make this result more applicable we note, that if , for some , then uniformly with respect to , for all [2, Proposition 6]. Hence, for any there exist a nonempty subspace of , such that its elements satisfy
with some .
Proof. For small values of we proceed as follows
Here and is real and imaginary part of correspondingly. To evaluate the last limit we employ Corollary 1 from . It offers a convergent expansion of the Hurwitz-Lerch zeta function , when its second parameter is an integer number
The expression on the right of (2.5) is bounded and uniformly convergent to the left-hand side for any , , such that . Therefore
which leads us to the bound
For large , the integral from (2.4) can be estimated as follows
3. Examples and discussion
In this section we consider several examples of the developed approximation method. As a measure of experimental error we use a discrete norm
defined on a uniform grid . With such choice of the specified discrete norm ought to capture the contribution from both the descretization and truncation parts of the error. To experimentally check the convergence of we repeat the approximation procedure on a sequence of grids determined by
where is integer. Then, the largest possible value of such that remains analytic in , is equal to . For the purpose of the illustration we set , , then , , . The behaviour of an error for the values of , calculated by three different formulas (1.3), (1.10), (2.3), is depicted in Fig. 1.
Predictably, the maximum of for calculated by (1.10) (see central plot from Fig. 1) is superior to the error with calculated by (1.3) (left plot from Fig. 1 ). The value of calculated by (2.3) is close to the one obtained from (1.10), that is why the error function (see plot on the right from Fig. 1) is close to obtained with help of (1.10). One can see a discernible spike in the error function from central plot of Fig. 1 at . The values of on the left of corresponds to the discretization error, whilst the values on the right of corresponds to the truncation error. The magnitude of those errors almost match. This highlight the fact that the chosen is quite close to the theoretically optimal value.
In this example we set as
and choose formula (1.3) for the evaluation of . The function is meromorphic and bounded in for any smaller than the imaginary part of zeros of . The zeros of the polynomial part of this expression lie closer to the real line than any zero of , so . Therefore it is safe to set . For given we can also explicitly find the parameters of algebraic decay bound (1.1): , .
Note, that for a more general function the corresponding can be calculated numerically from a sequence of its values.
For explicitly given the possible values of can be calculated numerically as well, for example using
Analytic routine from Maple .
The graphs of the approximated function and the error of its interpolation by are given in Fig. 2.
The data from in Table 1 demonstrates that the approximation method presented by Theorem 1.1 converges to . The magnitude of the observed approximation errors are consistent with the estimate provided by (1.2). Moreover the estimated value of from (1.2) remains bounded by for all . All this prove the effectiveness of the developed method.
- 1. Bateman H. Higher transcendental functions. — 1983 edition. — Tata-mcgrawhill Book Company Ltd., Bombay, 1953. — Vol. 1.
- 2. Butzer P. L., Schmeisser G., Stens R. L. Basic Relations Valid for the Bernstein Space and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from // Journal of Fourier Analysis and Applications. — 2013. — Apr. — Vol. 19, no. 2. — P. 333–375.
- 3. Ferreira E., Kohara A., Sesma J. New properties of the lerch’s transcendent // Journal of Number Theory. — 2017. — Vol. 172. — P. 21 – 31. — Access mode: http://www.sciencedirect.com/science/article/pii/S0022314X16302219.
- 4. Maplesoft maple, Maplesoft, a division of Waterloo Maple Inc. — Waterloo, Ontario, 2016. — Access mode: https://www.maplesoft.com/products/Maple/.
- 5. On the LambertW function / R.M. Corless, G.H. Gonnet, D.E.G. Hare et al. // Advances in Computational Mathematics. — 1996. — Vol. 5, no. 1. — P. 329–359. — Access mode: http://dx.doi.org/10.1007/BF02124750.
- 6. Stenger F. Numerical methods based on Sinc and analytic functions. — Berlin, Germany / Heidelberg, Germany / London, UK / etc. : Springer-Verlag, 1993. — Vol. 20 of Springer Series in Computational Mathematics. — P. xv + 565. — ISBN: 0-387-94008-1 (New York), 3-540-94008-1 (Berlin).
Institute of mathematics, NAS of Ukraine, 3, Tereschenkivska St., 01004, Kyiv, Ukraine
E-mail address: , |
Characterization of Lipschitz Spaces via Commutators of Maximal Function on the -Adic Vector Space
In this paper, we give characterization of a -adic version of Lipschitz spaces in terms of the boundedness of commutators of maximal function in the context of the -adic version of Lebesgue spaces and Morrey spaces, where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a Lipschitz norm involving the John-Nirenberg-type inequality for homogeneous Lipschitz functions, which is new in the -adic field context.
1. Introduction and Statement of Main Results
For a prime number , let be the field of -adic numbers. It is defined as the completion of the field of rational numbers with respect to the non-Archimedean -adic norm . This norm is defined as follows: . If any nonzero rational number is represented as , where and are integers which are not divisible by , and is an integer, then . It is not difficult to show that the norm satisfies the following properties:
It follows from the second property that when , then . From the standard -adic analysis , we see that any nonzero -adic number can be uniquely represented in the canonical serieswhere are integers, , . Series (2) converges in the -adic norm because .
The space consists of points , where , . The -adic norm on is for . Denote by , the ball with center at and radius , and by the sphere with center at and radius , . It is clear that and .
Since is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Harr measure on (up to positive constant multiple) which is translation invariant. We normalize the measure so thatwhere denotes the Harr measure of a measurable subset of . From this integral theory, it is easy to obtain that and for any .
In what follows, we say that a (real-valued) measurable function defined on is in , , if it satisfies
Here the integral in equation (4) is defined as follows:
For a function , we defined the Hardy-Littlewood maximal function of on by the following equation:
The maximal commutator of with a locally integrable function is defined by the following equation:
The first part of this paper is to study the boundedness of when the symbol belongs to a Lipschitz space (see in Section 2). Some characterizations of the Lipschitz space via such commutator are given. Our first result can be stated as follows:
Theorem 1. Let be a locally integrable function and , then the following statements are equivalent.(1);(2) is bounded from to for all with and ;(3) is bounded from to for some with and ;(4) satisfies the weak-type estimates, namely, there exists a positive constant such that for any ,(5) is bounded from to .
We remark that the boundedness of the commutators of maximal function were pretty much unknown on the -adic vector space. However, in Euclidean case, the mapping properties of the maximal commutator have been studied intensively by many authors, see [11–21].
It is well-known that the Morrey space introduced by Morrey in 1938, is connected to certain problem in elliptic PDE . Later the Morrey spaces were found to have many important applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients and the potential analysis, see [23–27] and so on. Next, we introduce the -adic version of Morrey space on -adic vector space.
For and . The -adic version of Morrey space is defined by the following equation:where
It is well known that if , then and .
Theorem 2. Let be a locally integrable function and . Assume that , and . Then, if and only if is bounded from to .
Theorem 3. Let be a locally integrable function and . Assume that , , , and . Then, if and only if is bounded from to .
On the other hand, the classical commutator of the -adic version of maximal function with a locally integrable function can be defined by the following equation:
In the Euclidean setting, the commutator of maximal function as in (11) has attracted much more attention. For examples, Milman and Schonbek established a commutator result by real interpolation techniques. As an application, they obtained the -boundedness of the commutator of maximal function when and . This operator can be used in studying the product of a function in and a function in BMO (see for instance). Bastero et al. studied the necessary and sufficient conditions for the boundedness of on spaces when . Zhang and Wu extended their results to commutators of the fractional maximal function. The results in [30, 31] were extended to variable Lebesgue spaces in [32, 33]. Recently, Zhang studied the commutator when belongs to Lipschitz spaces. Some necessary and sufficient conditions for the boundedness of on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in . For more information about the characterization of the commutator of maximal operator, see also [19, 20] and the references therein.
We would like to remark that operators and essentially differ from each other. For example, is positive and sublinear, but is neither positive nor sublinear. Motivated by the papers mentioned above, in this paper, the second part of this paper aims to study the mapping properties of the commutator when belongs to some Lipschitz spaces. More precisely, we will give some new necessary and sufficient conditions for the boundedness of on -adic vector spaces, by which some new characterizations for certain subclasses of Lipschitz spaces are obtained.
Theorem 4. Let be a locally integrable function and . Assume that and . Then, the following statements are equivalent.(1) and ;(2) is bounded from to ;(3)There exists a constant such that
Where denote the maximal operator with respect to a -adic ball which is defined by the following equation:
Here, the supremum is taken over all the -adic with for a fixed -adic ball .
Theorem 5. Let be a locally integrable function, and . Then, there exist a positive such that for any ,
Theorem 6. Let be a locally integrable function and . Assume that , , and . Then, the following statements are equivalent(1) and ;(2) is bounded from to .
Theorem 7. Let be a locally integrable function and . Assume that , , , and . Then, the following statements are equivalent.(1) and ;(2) is bounded from to .
Remark 1. Theorem 6 is essentially the Adams-type result and Theorem 7 is the Spanne-type result. Also, Theorem 7 can be immediately proved via Theorem 6.
The rest of the present paper is organized as follows: in Section 2, we gave some definitions and lemmas. The Proof of Theorems 1–3 are presented in Section 3. In Section 4, we gave the proof of Theorems 4–7. By we mean that with some positive constant independent of appropriate quantities. The positive constants varies from one occurrence to another. For a real number , , is the conjugate number of , that is, .
2. Preliminary Definitions and Lemmas
To prove our main results, we need the following definitions and lemmas.
Definition 1. Assume that . The -adic version of homogeneous Lipschitz spaces (see ) is the space of all measurable functions on such thatFor , the -adic version of Lipschitz spaces is the space of all measurable functions on such thatwhere denotes the average of over , i.e., . When , we use as .
We shall give the -adic version of John-Nirenberg inequality for homogeneous Lipschitz functions in the following lemma.
Lemma 1. Let , then there exist some depending on and such thatfor any and for any -adic , where denotes the family of all the -adic balls which is defined as follows:
Kim in observed several interesting properties on the family .
Lemma 2. The family has the following properties:(a)If , then either or .(b) if and only if .
Proof of Lemma 1. Let be a fixed -adic ball and be some positive real number which will be determined later. Applying the -adic version of the Calderón-Zygmund decomposition (see Corollary 3.4 in , or see also Theorem 5.16 in ) of for relative to to obtain a pair-wise disjoint family of -adic balls (by Lemma 2) which satisfies the following equation:andCombining and equation (20) gives the following equation:We denote . For any , we haveBy using equation (19), we can infer that for any Thus, equations (23) and (24) now implies.For any , setObviously, is a decreasing function and . Thus, we have the following equation:Hence, for any , we conclude thatTaking , then is also a fixed positive number and for any ,Using induction argument for any , we obtain the following equation:Thus, for , we have the following equation:Notice that this inequality is also true for , due to . Therefore, for any , we conclude thatThis finishes the proof of the lemma.
Kim in gave the property of as in the Euclidean case.
Lemma 3. For , the distribution function of on defined by the following equation:
If for , then we have the following equation:
We also need the -adic version of Lebesgue differential theorem, which is due to Kim .
Lemma 4. If , then we have that for a.e.
The following lemma is about some properties of Lipschitz space .
Lemma 5. If is given, then we have the following properties:(a)For , then there exists a constant such that .(b)The norm is equivalent to the norm on Lipschitz space.(c)For any with , where is the constant given in Lemma 1,
Proof. .(a)For , using Lemmas 1 and 3 we give the following equation: where . Hence, we conclude that .(b)For , by Hölder’s inequality it is obvious that for any . Thus, we have that . Hence, by (a) we can obtain the equivalence of those two norms.(c)We first observe thatFor any . Then, it follows from Lemma 1 and the -adic version of changing the order of integration thatfor any , provided that . Therefore, we have the following equation:This completes the proof of the lemma.
To prove the theorems, we need the following key lemma. In Euclidean case, the Lipschitz space coincides with some Morrey-Compana to space and can be characterized by mean oscillation which is due to DeVore and Sharpley , Janson et al. and Harboure et al. (see also Paluszyński ).
Lemma 6. For and , then the homogeneous Lipschitz space coincides with the space .
Proof. Using the property (b) of Lemma 5, we only need to prove that for some constant ,Now, we give the proof of the right side of inequality (42). Assume that . Given and , set and . Then we have the following equation:We estimate only the first term of the right side, since the others follow similar lines. Letting for and , then by Lemma 4, we show thatThus, we obtain .
On the other hand, for and , we have the following equation:Since and (b) of Lemma 2 implies that . Hence, by using above inequality, we can infer thatfor any . Thus, we have . This concludes the proof of the lemma.
In , Taibleson introduced and studied the Riesz potentials on local fields. Let us recall the definition of Riesz potentials on -adic vector space as follows:where with .
Let us recall the Hardy-Littlewood-Sobolev inequality for the Riesz potentials on -adic vector space. More details, the interested reader may refer to the book .
Lemma 7. Let be a complex number with and such that . Then, the following statements are held.(a)If , then is bounded from to .(b)If , then is bounded from to .Let be a complex number with and be locally integrable function on , the -adic version of fractional maximal function of is given by the following equation:Using the definition of the -adic version of Riesz potential in equation (47), we give the following inequality:So inequality (49) gives the following equation:Using inequality (50) and Lemma 7, we give the boundedness of -adic version of fractional maximal function on the -adic vector space.
Lemma 8. Let be a complex number with and such that . Then, the following statement are held.(a)If , then is bounded from to .(b)If , then is bounded from to .We now state the boundedness of on Morrey space which will be useful in proving our results. This lemma is similar to the results of ones from [39–41]. In Euclidean setting, Adams and Spanne studied the corresponding results, and see also .
Lemma 9. Let , and .(1)If , then is bounded from to .(2)If and , then is bounded from to .
Proof. (1) Let and . Then, we split the following equation:First, we estimate . Using the definition of , we obtain the following equation:On the other hand, by Hölder’s inequality, we have that Combining equations (52) and (53) gives the following equation:Takingthen equation (54) becomes as follows: |
In the very first, you simply plug numbers into the equation. It is an excellent notion to test to be certain that every constant was properly named. To fix that issue, we do a before and following equation. There are four main constant acceleration equations you'll want to fix all problems in this way.
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The Foolproof Acceleration Formula Physics Strategy
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After all the very best thing about physics is the fact that it can be utilized to address real world issues. It's a great notion to be as systematic as possible when it has to do with analyzing the circumstance. Additionally, a lot of content on Science Trends is written by the main author of the peer-reviewed research in which they're writing.
Ok, I Think I Understand Acceleration Formula Physics, Now Tell Me About Acceleration Formula Physics!
This calculator will figure out the unknown values and offer the derived equations which were used to come across the solution. Combine the initial two equations together in a way that will eliminate time for a variable. Physical quantities that are completely specified by just giving out there magnitude are called scalars. The exact same equation used to find out the worth of g on Earth' surface can likewise be employed to find out the acceleration of gravity on the surface of different planets.
The Downside Risk of Acceleration Formula Physics
This may appear to be lots of trouble, but nevertheless, it can be quite helpful if your constants change values often. It's complicated to memorize each and every arrangement of the 2 equations and I recommend that you practice creating new combinations from the original equations. This equation appears to be logical enough. These equations are called kinematic equations.
The Characteristics of Acceleration Formula Physics
At some point, the front tires will start to slip before the rear tires, that is the definition of understeer. The ideal rear holds a lot more weight than the left rear. The same is true for the rear tires. Let's look at speed first.
In everyday English, the term acceleration is often used to refer to a state of raising speed. It is the rate at which an object moves from one place to another. It is different in that is relative to whatever circumstance it is applied to.
Yet acceleration doesn't have anything to do with going fast. It's a great notion to be as systematic as possible when it has to do with analyzing the circumstance. Additionally, a lot of content on Science Trends is written by the main author of the peer-reviewed research in which they're writing.
I have to pause here briefly to impose one particular limitation. We've got a little set of quite powerful formulae that can be combined to get results despite unknown variables. Information about one of the parameters can be utilized to determine unknown information concerning the other parameters. A simulation is frequently the simplest approach to calculate such outcomes. |
45 Questions (1 – 45)
90 Questions (46 – 135)
45 Questions (136 – 180)
Q. 1 Which of the following statements is “not” true for the halogens?
A. All form monobasic oxyacids
B. Chlorine has the highest electron gain enthalpy
C. All but fluorine show positive oxidation states
D. All are oxidizing agents
Q. 2 The correct order of atomic radii in group 13 elements are:
A. B < Al < In < Ga < Tl
B. B < Ga < Al < In < Tl
C. B < Ga < Al < Tl < In
D. B < Al < Ga < In < Tl
Q. 3 In the structure of CIF₃ , the number of lone pairs of electrons on the central atom “Cl” is:
Q. 4 The correct order of the N-compounds in its decreasing order of oxidation states is:
A. HNO₃, NO, N₂, NH₄Cl
B. NH₄Cl, N₂, NO, HNO₃
C. HNO₃, NH₄Cl, NO, N₂
D. HNO₃, NO, NH₄Cl, N₂
Q. 5 Which one of the following elements is unable to form MF₆⁻³ ion:
Q. 6 Considering Ellingham diagram, which of the following metals can be used to reduce alumina?
Q. 7 The compound A on treatment with Na gives B, and with PCl₅ gives C. B and C react together to give diethyl ether. A, B and C are in the order
A. C₂H₅OH, C₂H6, C₂H₅Cl
B. C₂H₅OH, C₂H₅ONa, C₂H₅Cl
C. C₂H₅Cl, C₂H6, C₂H₅OH
D. C₂H₅OH, C₂H₅Cl, C₂H₅ONa
Q. 8 Hydrocarbon (A) reacts with bromine by substitution to form an alkyl bromide which by Wurtz reaction is converted to gaseous hydrocarbon containing less than four carbon atoms (A) is :
A. CH = CH
C. CH₃ – CH₃
D. CH₂ = CH₂
Q. 9 The compound C₇H₈ undergoes the following reactions given in the figure. The product ‘C’ is:
Q. 10 Which oxide of nitrogen is not a common pollutant introduced into the atmosphere both due to natural and human activity?
Q. 11 Following solutions were prepared by mixing different volumes of NaOH and HCl of different concentrations:
a. 60mL M/10 HCl + 40mL M/10 NaOH
b. 55mL M/10 HCl + 45mL M/10 NaOH
c. 75mL M/5 HCl + 25mL M/5 NaOH
d. 100mL M/10 HCl + 100mL M/10 NaOH
pH of which one of them will be equal to 1?
Q. 12 On which of the following properties does the coagulating power of an ion depend?
A. The magnitude of the charge on the ion alone
B. The sign of charge on the ion alone
C. Both magnitude and sign of the charge on the ion
D. Size of the ion alone
Q. 13 The solubility of BaSO₄ in water is 2.42 x 10⁻³ gL⁻¹ at 298K.The value of its solubility product (Ksp) will be (Given molar mass of BaSO₄=233g/mol⁻¹)
A. 1.08 x 10⁻¹⁰ mol²/L⁻²
B. 1.08 x 10⁻⁸ mol²/L⁻²
C. 1.08 x 10⁻¹⁴ mol²/L⁻²
D. 1.08 x 10⁻¹² mol²/L⁻²
Q. 14 Given van der Waals constant for NH₃, H₂, O₂ and CO₂ are respectively 4.17, 0.244, 1.36 and 3.59, which one of the following gases is most easily liquefied?
Q. 15 Match the metal ions given in column I with the spin magnetic moments of the ions given in column II (Given in figure (1)) and assign the correct code:
A. a(iv), b(v), c(ii), d(i)
B. a(iii), b(v), c(i), d(ii)
C. a(iv), b(i), c(ii), d(iii)
D. a(i), b(ii), c(iii), d(iv)
Q. 16 Iron carbonyl, Fe(CO)₅ is:
Q. 17 The geometry and magnetic behaviour of the complex [Ni(CO)₄] are:
A. square planar geometry and diamagnetic
B. tetrahedral geometry and paramagnetic
C. square planar geometry and paramagnetic
D. tetrahedral geometry and diamagnetic
Q. 18 Which one of the following ions exhibits d-d transition and paramagnetism as well?
Q. 19 The type of isomerism shown by the complex [CoCl₂(en)₂] is:
A. geometrical isomerism
B. linkage isomerism
C. ionisation isomerism
D. coordination isomerism
Q. 20 Identify the major products P, Q and R (Among options (1), (2), (3), (4)) in the following sequence of reaction given in the figure:
Q. 21 Which of the following compounds can form a zwitterion?
C. Benzoic acid
Q. 22 Which of the following molecules represent the order of hybridisation sp², sp², sp, sp from left to right atoms?
A. HC ≡ C – C ≡ CH
B. CH₃ – CH = CH – CH₃
C. CH₂ = CH – CH = CH₂
D. CH₂ = CH – C = CH
Q. 23 Which of the following carbocations is expected to be most stable (Among options (1), (2), (3), (4)):
Q. 24 Which of the following is correct with respect to -I effect of the substituents?(R = alkyl)
A. -NH₂ < -OR < -F
B. -NR₂ > -OR > -F
C. -NH₂ > -OR > -F
D. -NR₂ < -OR < -F
Q. 25 Magnesium reacts with an element (X) to form an ionic compound. If the ground state electronic configuration of (X) is 1s² 2s² 2p³, the simplest formula for this compound is:
Q. 26 Iron exhibits bcc structure at room temperature. Above 900°C, it transforms to fcc structure. The ratio of density of iron at room temperature to that at 900°C (assuming molar mass and atomic radii of iron remains constant with temperature) is:
Q. 27 Which of the following statement is wrong (Among (1), (2), (3), (4))?
Q. 28 Consider the following species:
CN⁺, CN⁻, NO and CN
Which one of these will have the highest bond order?
Q. 29 In the reaction given in the figure, the electrolyte involved is:
A. dichloromethyl cation (CHCl₂)
B. dichlorocarbene (:CCl₂)
C. dichloromethyl anion (CHCl₂)
D. formyl cation (CHO)
Q. 30 Carboxylic acids have higher boiling points than aldehydes, ketones and even alcohols of comparable molecular mass. It is due to their:
A. formation of intramolecular H-bonding
B. formation of intermolecular H-bonding
C. more extensive association of carboxylic acid via van der Waals force of attraction
D. formation of carboxylate ion
Q. 31 Compound A, C₈H₁₀O is found to react with NaOI (produced by reacting Y with NaOH) and yields a yellow precipitate with characteristic smell.
A and Y are respectively (Among (1), (2), (3), (4)):
Q. 32 The correct difference between first and second order reactions is that:
A. The rate of first order reaction does not depend on the reactant concentrations; the rate of a second order reaction does depend on reactant concentrations.
B. The rate of first order reaction does depend on the reactant concentrations; the rate of a second order reaction does not depend on the reactant concentrations.
C. a first order reaction can be catalyzed; a second order reaction cannot be catalyzed.
D. the half time of a first order reaction does not depend on the [A]0; the half life of a second order reaction does not depend on the [A]0
Q. 33 Among the CaH₂, BeH₂, BaH₂, the order of the ionic character is:
A. BeH₂ < CaH₂ < BaH₂
B. BaH₂ < BeH₂ < CaH₂
C. BeH₂ < BaH₂ < CaH₂
D. CaH₂ < BeH₂ < BaH₂
Q. 34 Consider the change in oxidation state of Bromine corresponding to different emf values as shown in the diagram.
Then the species undergoing disproportionation is:
Q. 35 In which case is the number of molecules of water maximum?
A. 18 mL of water
B. 10⁻³ mol of water
C. 0.00224L of water vapours at 1 atm and 273 K
D. 0.18 g of water
Q. 36 Regarding cross-linked or network polymers, which of the following statements is incorrect?
A. They contains covalent bonds between various linear polymer chains.
B. They contains strong covalent bonds in their polymer chains
C. Examples are bakelite and melamine
D. They are formed from bi and tri functional monomers.
Q. 37 Nitration of aniline in strong acidic medium also gives m-nitroaniline because:
A. In spite of substitutions nitro group always goes to only m-position
B. In acidic (strong) medium aniline is present as anilinium ion
C. In absence of substituents nitro groups always goes to m-position
D. In electrophilic substitution reactions amino group is meta directive
Q. 38 Which of the following oxides is most acidic in nature?
Q. 39 The difference between amylose and amylose and amylopectin is:
A. Amylopectin have 1 → 4 α-linkage and 1 → 6 α-linkage
B. Amylose is made up of glucose and galactose
C. Amylopectin have 1 → 4 α-linkage and 1→ 6 β-linkage
D. Amylose have 1 → 4 α-linkage and 1 → 6 β-linkage
Q. 40 A mixture of 2:3 g formic acid and 4.5 g oxalic acid is treated with conc. H₂SO₄. The evolved gaseous mixture is passed through KOH pallets. Weight (in g) of the remaining product at STP will be:
Q. 41 For redox reaction given in the figure, what is the correct coefficients of the reactants for the balanced equation are:
Q. 42 The correction factor ‘a’ to the ideal gas equation corresponds to:
A. density of the gas molecules
B. forces of attraction between the gas molecules
C. electric field present between the gas molecules
D. volume of the gas molecules
Q. 43 Which one of the following conditions will favour maximum formation of the product in the reaction:
A₂(g) + B₂(G) ⇔ X₂(g) ΔrH = – X kJ?
A. low temperature and high pressure
B. high temperature and low temperature
C. high temperature and high pressure
D. low temperature and low pressure
Q. 44 The bond dissociation energies of X₂, Y₂ and XY of in the ratio of 1 : 0.5 : 1. ΔH for the formation of XY is -200 KJ/mol⁻¹.The bond dissociation energy of X₂ will be:
A. 200KJ mol⁻¹
B. 400 KJ mol⁻¹
C. 800 KJ mol⁻¹
D. 100 KJ mol⁻¹
Q. 45 When initial concentration of the reactant is doubled, the half life period of a zero order reaction.
A. is halved
B. remains unchanged
C. is tripled
D. is doubled
Q. 46 Which of the following is an occupational respiratory disorder?
Q. 47 Calcium is important in skeletal muscle contraction because it:
A. binds to troponin to remove the masking of active sites on actin for myosin
B. prevents the formation of bonds between the myosin cross bridges and the actin filament
C. detaches the myosin head from the actin filament
D. activates the myosin ATPase by binding to it
Q. 48 Which of the following gastric cells indirectly help in erythropoiesis?
A. chief cells
B. parietal cells
C. goblet cells
D. mucous cells
Q. 49 Match the items given in column I with those in column II (Given in the figure) and select the correct option:
A. a(iii), b(ii), c(i)
B. a(ii), b(iii), c(i)
C. a(i), b(iii), c(ii)
D. a(i), b(ii), c(iii)
Q. 50 Which of the following hormones can play a significant role in osteoporosis?
A. Aldosterone and Prolactin
B. Parathyroid hormone and Prolactin
C. Estrogen and Parathyroid hormone
D. Progesterone and Aldosterone
Q. 51 Which of the following is an amino acid derived hormone?
Q. 52 Which of the following structures or regions is incorrectly paired with its function?
A. Medulla oblongata: It controls respiration and cardiovascular reflexes
B. Corpus callosum: band of fibres connecting the left and right cerebral hemispheres
C. Hypothalamus: production of releasing hormones and regulation of temperature, hunger and thirst
D. Limbic system: consists of fibre tracts that interconnect different regions of brain; controls movements.
Q. 53 The transparent lens in the human eye is held in its place by
A. ligaments attached to ciliary body
B. smooth muscles attached to the ciliary body
C. smooth muscles attached to the iris
D. ligaments attached to the iris
Q. 54 Among the following set of examples for divergent evolution, select the incorrect option.
A. forelimbs of man, bat and cheetah
B. eye of octopus, bat and man
C. brain of bat, man and cheetah
D. heart of bat, man and cheetah
Q. 55 In which disease does mosquito transmitted pathogen cause chronic inflammation of lymphatic vessels?
C. Ringworm disease
Q. 56 Which of the following is not an autoimmune disease?
C. Alzheimer’s disease
D. Rheumatoid arthritis
Q. 57 Conversion of milk to curd improves its nutritional value by increasing the amount of:
A. Vitamin D
B. Vitamin E
C. Vitamin B12
D. Vitamin A
Q. 58 Which of the following characters represent ‘inheritance of blood groups’ in human:
c. Multiple allele
d. Incomplete dominance
e. Polygenic inheritance
A. b, c and e
B. a, c and e
C. b, d and e
D. a, b and c
Q. 59 The similarity of bone structure in the forelimbs of many vertebrates is an example of:
B. adaptive radiation
C. convergent evolution
Q. 60 Which of the following animals does not undergo metamorphosis?
Q. 61 Which one of these animals is not a homeotherm?
Q. 62 Which of the following features is used to identify a male cockroach from a female cockroach?
A. Presence of a boat shaped sternum on the 9th abdominal segment
B. Presence of anal cerci
C. Forewings with darker tegmina
D. Presence of caudal styles
Q. 63 Which of the following organisms are known as chief producers in the oceans?
Q. 64 Ciliates differ from all other protozoans in
A. using flagella for locomotion
B. having two types of nuclei
C. using pseudopodia for capturing prey
D. having a contractile vacuole for removing excess water
Q. 65 Identify the vertebrate group of animals characterized by crop and gizzard in its digestive system
Q. 66 The amnion of mammalian embryo is derived from:
A. ectoderm and mesoderm
B. ectoderm and endoderm
C. mesoderm and trophoblast
D. endoderm and mesoderm
Q. 67 Hormones secreted by the placenta to maintain pregnancy are:
A. hCG, hPL , progestogens, prolactin
B. hCG, progestogens, estrogens, glucocorticoids
C. hCG, hPL, progestogens, estrogens
D. hCG, hPL, estrogens, relaxin, oxytocin
Q. 68 The contraceptive ‘SAHELI’
A. blocks estrogen receptors in the uterus, preventing eggs from getting implanted
B. is a post coital contraceptive
C. is an IUD
D. increases the concentration of estrogen and prevents ovulation in females.
Q. 69 The difference between spermiogenesis and spermiation is:
A. In spermatogenesis spermatids are formed, while in spermiation spermatozoa are formed
B. In spermatogenesis spermatozoa are formed, while in spermiation spermatozoa are released from sertoli cells into the cavity of seminiferous tubules
C. In spermatogenesis spermatozoa from sertoli cells are released into the cavity of seminiferous tubules, while in spermiation spermatozoa are formed
D. In spermatogenesis spermatozoa are formed while in spermiation spermatids are formed
Q. 70 In a growing population of a country:
A. pre-reproductive individuals are more than the reproductive individuals
B. pre-reproductive individuals are less than the reproductive individuals
C. reproductive and pre-reproductive individuals are equal in number
D. reproductive individuals are less than the post reproductive individuals
Q. 71 Match the items given in the column I with those in column II (Given in the figure) and select the correct option given below:
A. a(ii), b(i), c(iii), d(iv)
B. a(i), b(ii), c(iv), d(iii)
C. a(iii), b(iv), c(i), d(ii)
D. a(i), b(iii), c(iv), d(ii)
Q. 72 Which part of poppy plant is used to obtain the drug ‘smack’ ?
Q. 73 Which of the following population interactions widely used in medical sciences for the production of antibiotics?
Q. 74 All of the following are included in “ex-situ conservation” except
A. wildlife safari parks
B. seed banks
C. botanical gardens
D. sacred groves
Q. 75 Match the items given in column I with those column II (Given in the figure) and select the correct option given below:
A. a(iii), b(ii), c(iv), d(i)
B. a(iv), b(i), c(ii), d(iii)
C. a(ii), b(iii), c(i), d(iv)
D. a(i), b(ii), c(iii), d(iv)
Q. 76 Match the items given in column I with those in column II and select the correct option given below:
A. a(iv), b(v), c(ii), d(iii)
B. a(v), b(iv), c(i), d(iii)
C. a(v), b(iv), c(i), d(ii)
D. a(iv), b(i), c(ii), d(iii)
Q. 77 A woman has an X-linked condition on one of her X chromosomes. This chromosome can be inherited by:
A. only daughters
B. both sons and daughters
C. only grandchildren
D. only sons
Q. 78 AGGTATCGCAT is a sequence from the coding strand of a gene. What will be the corresponding sequence of the transcribed mRNA?
Q. 79 Match the items given in column I with those in column II (Given in the figure) and select the correct option given below:
A. a(iii), b(ii), c(i)
B. a(iii), b(i), c(ii)
C. a(ii), b(iii), c(i)
D. a(i), b(iii), c(ii)
Q. 80 According to Hugo de Vries, the mechanism of evolution is:
A. multiple step mutations
B. minor mutations
C. phenotypic variations
Q. 81 All of the following are part of an operon except
A. an operator
B. a promoter
C. an enhancer
D. structural genes
Q. 82 Which of the following events does not occur in rough endoplasmic reticulum?
A. protein folding
B. phospholipid synthesis
C. cleavage of signal peptide
D. protein glycosylation
Q. 83 Which of these statements is incorrect?
A. enzymes of TCA cycle are present in mitochondrial matrix
B. oxidative phosphorylation takes place in outer mitochondrial membrane
C. glycolysis operates as long as it is supplied with NAD that can pick up hydrogen atoms
D. glycolysis occurs in cytosol
Q. 84 Select the incorrect match:
A. lampbrush chromosome – diplotene bivalents
B. polytene chromosome – oocytes of amphibians
C. submetacentric chromosome – L-shaped chromosomes
D. allosomes – sex chromosomes
Q. 85 Which of the following terms describe human dentition?
A. thecodont, diphyodont, homodont
B. pleurodont, diphyodont, heterodont
C. pleurodont, monophyodont, homodont
D. thecodont, diphyodont, heterodont
Q. 86 Nissil bodies are mainly composed of
A. proteins and lipids
B. free ribosomes and RER
C. nucleic acids and SER
D. DNA and RNA
Q. 87 Many ribosomes may associate with a single mRNA to form multiple copies of a polypeptide simultaneously. Such strings of ribosomes are termed as:
D. Polyhedral bodies
Q. 88 Match the items given in the column I with those in column II (Given in figure) and select the correct option given below:
A. a(iii), b(i), c(ii)
B. a(ii), b(i), c(iii)
C. a(i), b(ii), c(iii)
D. a(i), b(iii), c(ii)
Q. 89 Match the item in the column I with the column II (Given in the figure) and select the correct option given below:
A. a(iii), b(ii), c(i), d(iv)
B. a(iv), b(iii), c(ii), d(i)
C. a(i), b(iv), c(ii), d(iii)
D. a(iii), b(i), c(iv0, d(ii)
Q. 90 Which of the following options correctly represents the lung conditions in asthma and emphysema , respectively.
A. inflammation of bronchioles, decreased respiratory surface
B. decreased respiratory surface, inflammation of bronchioles
C. increased respiratory surface, inflammation of bronchioles
D. increased number of bronchioles, increased respiratory surface
Q. 91 The stage during which separation of the paired homologous chromosomes begins is
Q. 92 Which of the following is true for nucleolus?
A. Larger nucleoli are present in dividing cells.
B. It is a site for active ribosomal RNA synthesis.
C. It takes part in spindle formation.
D. It is a membrane-bound structure.
Q. 93 Stomatal movement is not affected by
B. CO₂ concentration
C. O₂ concentration
Q. 94 Which among the following is not a prokaryote ?
Q. 95 Which of the following is not a product of light reaction of photosynthesis ?
Q. 96 Stomata in grass leaf are
A. Dumb-bell shaped
B. Barrel shaped
D. Kidney shaped
Q. 97 The Golgi complex participates in
A. Fatty acid breakdown
B. Activation of amino acid
C. Respiration in bacteria
D. Formation of secretory vesicles
Q. 98 The two functional groups characteristic of sugars are
A. hydroxyl and methyl
B. carbonyl and hydroxyl
C. carbonyl and phosphate
D. carbonyl and methyl
Q. 99 A ‘new’ variety of rice was patented by a foreign company, though such varieties have been present in India for a long time. This is related to
C. Lerma Rojo
D. Sharbati Sonora
Q. 100 Select the correct match:
A. Ribozyme – Nucleic acid
B. G. Mendel – Transformation
C. T.H. Morgan – Transduction
D. F₂ x Recessive parent – Dihybrid cross
Q. 101 Which of the following is commonly used as a vector for introducing a DNA fragment in human lymphocytes ?
B. pBR 322
C. λ phage
D. Ti plasmid
Q. 102 In India, the organisation responsible for assessing the safety of introducing genetically modified organisms for public use is
A. Indian Council of Medical Research (ICMR)
B. Genetic Engineering Appraisal Committee (GEAC)
C. Research Committee on Genetic Manipulation (RCGM)
D. Council for Scientific and Industrial Research (CSIR)
Q. 103 The correct order of steps in Polymerase Chain Reaction (PCR) is
A. Extension, Denaturation, Annealing
B. Denaturation, Annealing, Extension
C. Denaturation, Extension, Annealing
D. Annealing, Extension, Denaturation
Q. 104 Use of bioresources by multinational companies and organisations without authorisation from the concerned country and its people is called
Q. 105 Winged pollen grains are present in
Q. 106 After karyogamy followed by meiosis, spores are produced exogenously in
Q. 107 Which one is wrongly matched ?
A. Uniflagellate gametes – Polysiphonia
B. Unicellular organism – Chlorella
C. Gemma cups – Marchantia
D. Biflagellate zoospores – Brown algae
Q. 108 Match the items given in Column I with those in Column II (Given in figure) and select the correct option given below:
A. a → i, b → iv, c → iii, d → ii
B. a → iii, b → iv, c → i, d → ii
C. a → ii, b → iv, c → iii, d → i
D. a → iii, b → ii, c → i, d → iv
Q. 109 What is the role of NAD⁺ in cellular respiration ?
A. It functions as an enzyme.
B. It is the final electron acceptor for anaerobic respiration.
C. It is a nucleotide source for ATP synthesis.
D. It functions as an electron carrier.
Q. 110 Oxygen is not produced during photosynthesis by
A. Green sulphur bacteria
Q. 111 Double fertilization is
A. Fusion of two male gametes of a pollen tube with two different eggs
B. Syngamy and triple fusion
C. Fusion of two male gametes with one egg
D. Fusion of one male gamete with two polar nuclei
Q. 112 In which of the following forms is iron absorbed by plants ?
B. Both ferric and ferrous
C. Free element
Q. 113 Which of the following elements is responsible for maintaining turgor in cells?
Q. 114 Which one of the following plants shows a very close relationship with a species of moth, where none of the two can complete its life cycle without the other ?
Q. 115 Pollen grains can be stored for several years in liquid nitrogen having a temperature of
Q. 116 Niche can be defined as:
A. All the biological factors in the organism’s environment
B. The functional role played by the organism where it lives
C. The range of temperature that the organism needs to live
D. The physical space where an organism lives
Q. 117 Which of the following is a secondary pollutant ?
Q. 118 World Ozone Day is celebrated on
A. 5th June
B. 22nd April
C. 16th September
D. 21st April
Q. 119 In stratosphere, which of the following elements acts as a catalyst in degradation of ozone and release of molecular oxygen ?
Q. 120 What type of ecological pyramid would be obtained with the following data ?
Secondary consumer : 120 g
Primary consumer: 60 g
Primary producer: 10 g
A. Inverted pyramid of biomass
B. Upright pyramid of biomass
C. Upright pyramid of numbers
D. Pyramid of energy
Q. 121 Natality refers to
A. Death rate
B. Number of individuals entering a habitat
C. Number of individuals leaving a habitat
D. Birth rate
Q. 122 Which of the following has proved helpful in preserving pollen as fossils ?
C. Oil Content
D. Cellulosic intine
Q. 123 Which of the following pairs is wrongly matched ?
A. Starch synthesis in pea : Multiple alleles
B. T.H. Morgan : Linkage
C. XO type sex determination : Grasshopper
D. ABO blood grouping : Co-dominance
Q. 124 Select the correct match:
A. Alec Jeffreys – Streptococcus pneumoniae
B. Francois Jacob and Jacques Monod – Lac operon
C. Matthew Meselson and F. Stahl – Pisum sativum
D. Alfred Hershey and Martha Chase – TMV
Q. 125 Which of the following flowers only once in its lifetime ?
A. Bamboo species
Q. 126 Select the correct statement:
A. Franklin Stahl coined the term “linkage”.
B. Transduction was discovered by S. Altman.
C. Spliceosomes take part in translation.
D. Punnett square was developed by a British scientist.
Q. 127 Offsets are produced by
A. Meiotic divisions
D. Mitotic divisions
Q. 128 The experimental proof for semiconservative replication of DNA was first shown in a
Q. 129 Select the wrong statement:
A. Cell wall is present in members of Fungi and Plantae.
B. Mitochondria are the powerhouse of the cell in all kingdoms except Monera.
C. Pseudopodia are locomotory and feeding structures in Sporozoans.
D. Mushrooms belong to Basidiomycetes.
Q. 130 Casparian strips occur in
Q. 131 Which of the following statements is correct ?
A. Ovules are not enclosed by ovary wall in gymnosperms.
B. Stems are usually unbranched in both Cycas and Cedrus.
C. Horsetails are gymnosperms.
D. Selaginella is heterosporous, while Salvinia is homosporous
Q. 132 Pneumatophores occur in
B. Submerged hydrophytes
C. Carnivorous plants
D. Free-floating hydrophytes
Q. 133 Sweet potato is a modified
C. Tap root
D. Adventitious root
Q. 134 Secondary xylem and phloem in dicot stem are produced by
A. Apical meristems
B. Axillary meristems
D. Vascular cambium
Q. 135 Plants having little or no secondary growth are
D. Deciduous angiosperms
Q. 136 The power radiated by a black body is P and it radiates maximum energy at wavelength, λ0. If the temperature of the black body is now changed so that it radiates maximum energy at wavelength 3/4λ0 , the power radiated by it becomes nP.The value of n is
Q. 137 Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by ΔI on applying a force F, how much force is needed to stretch the second wire by the same amount ?
A. 9 F
C. 4 F
D. 6 F
Q. 138 A sample of 0.1 g of water at 100°C and normal pressure (1.013×10⁵ Nm⁻²) requires 54 cal of heat energy to convert to steam at 100°C. If the volume of the steam produced is 167.1 cc, the change in internal energy of the sample, is
A. 104.3 J
B. 84.5 J
C. 42.2 J
D. 208.7 J
Q. 139 A small sphere of radius ‘r’ falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to
Q. 140 An electron falls from rest through a vertical distance h in a uniform and vertically upward directed electric field E. The direction of electric field is now reversed, keeping its magnitude the same. A proton is allowed to fall from rest in it through the same vertical distance h. The time of fall of the electron, in comparison to the time of fall of the proton is
C. 10 times greater
D. 5 times greater
Q. 141 A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is 20 m/s² at a distance of 5 m from the mean position. The time period of oscillation is
A. 2 π s
B. 1 s
C. 2 s
D. π s
Q. 142 The electrostatic force between the metal plate of an isolated parallel plate capacitor C having charge Q and area A, is
A. independent of the distance between the plates.
B. inversely proportional to the distance between the plates.
C. proportional to the square root of the distance between the plates.
D. linearly proportional to the distance between the plates.
Q. 143 A tuning fork is used to produce resonance in glass tube. The length of the air column in the tube can be adjusted by a variable piston. At room temperature of 27°C two successive resonances are produced at 20 cm and 73 cm of column length. If the frequency of the tuning fork is 320 Hz, the velocity of sound in air at 27°C
A. 330 m/s
B. 300 m/s
C. 350 m/s
D. 339 m/s
Q. 144 The ratio of kinetic energy to the total energy of an electron in a Bohr orbit of the hydrogen atom. is
A. 1 : 1
B. 1 : -2
C. 2 : -1
D. 1 : -1
Q. 145 When the light of frequency 2v0 (where v0 is threshold frequency), is incident on a metal plate, the maximum velocity of electrons emitted is v1. When the frequency of the incident radiation is increased to 5v0, the maximum velocity of electrons emitted from the same plate is v2. The ratio of v1 to v2 is
A. 1 : 2
B. 2 : 1
C. 4 : 1
D. 1 : 4
Q. 146 For a radioactive material, half-life is 10 minutes. If initially there are 600 number of nuclei, the time taken (in minutes) for the disintegration of 450 nuclei is
Q. 147 An electron of mass m with an initial velocity V⃗ =V₀î(V₀ > 0) enters an electric field E⃗ = −E₀î (E₀ = constant > 0) at t=0. If λ₀ is its de-Broglie wavelength initially, then its de-Broglie wavelength at time t is
A. λ₀/(1 + (eE₀/mV₀)t)
D. λ₀(1 + (eE₀/mV₀)t)
Q. 148 An inductor 20 mH, a capacitor 100 μF and a resistor 50 Ω are connected in series across a source of emf, V = 10 sin 314 t. The power loss in the circuit is
A. 0.79 W
B. 1.13 W
C. 2.74 W
D. 0.43 W
Q. 149 A metallic rod of mass per unit length 0.5 kg m⁻¹ is lying horizontally on a smooth inclined plane which makes an angle of 30° with the horizontal. The rod is not allowed to slide down by flowing a current through it when a magnetic field of induction 0.25 T is acting on it in the vertical direction. The current flowing in the rod to keep it stationary is
A. 7.14 A
B. 11.32 A
C. 14.76 A
D. 5.98 A
Q. 150 A thin diamagnetic rod is placed vertically between the poles of an electromagnet. When the current in the electromagnet is switched on, then the diamagnetic rod is pushed up, out of the horizontal magnetic field. Hence the rod gains gravitational potential energy. The work required to do this comes from
A. the current source
B. the induced electric field due to the changing magnetic field
C. the lattice structure of the material of the rod
D. the magnetic field
Q. 151 Current sensitivity of a moving coil galvanometer is 5 div/mA and its voltage sensitivity (angular deflection per unit voltage applied) is 20 div/V. The resistance of the galvanometer is
A. 40 Ω
B. 500 Ω
C. 250 Ω
D. 25 Ω
Q. 152 A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy (Kt) as well as rotational kinetic energy (Kr) simultaneously. The ratio Kt : (Kt + Kr) for the sphere is
A. 7 : 10
B. 2 : 5
C. 10 : 7
D. 5 : 7
Q. 153 The kinetic energies of a planet in an elliptical orbit about the Sun, at positions A, B and C are KA, KB and KC respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S as shown in the figure. Then
A. KA < KB < KC
B. KB > KA > KC
C. KB < KA < KC
D. KA > KB > KC
Q. 154 If the mass of the Sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following is not correct ?
A. Raindrops will fall faster
B. g’ on the Earth will not change
C. Time period of a simple pendulum on the Earth would decrease.
D. Walking on the ground would become more difficult.
Q. 155 A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere ?
A. Angular velocity
B. Angular momentum
C. Rotational kinetic energy
D. Moment of inertia
Q. 156 Unpolarised light is incident from air on a plane surface of a material of refractive index ′μ′. At a particular angle of incidence ′i′, it is found that the reflected and refracted rays are perpendicular to each other. Which of the following options is correct for this situation ?
A. Reflected light is polarised with its electric vector parallel to the plane of incidence
B. i = tan⁻¹(1/μ)
C. i = sin⁻¹(1/μ)
D. Reflected light is polarised with its electric vector perpendicular to the plane of incidence
Q. 157 In Young’s double slit experiment the separation d between the slits is 2 mm, the wavelength λ of the light used is 5896 Å and distance D between the screen and slits is 100 cm. It is found that the angular width of the fringes is 0.20°. To increase the fringe angular width to 0.21° (with same λ and D) the separation between the slits needs to be changed to
A. 1.8 mm
B. 1.7 mm
D. 1.9 mm
Q. 158 An astronomical refracting telescope will have large angular magnification and high angular resolution, when it has an objective lens of
A. small focal length and large diameter
B. small focal length and small diameter
C. large focal length and large diameter
D. large focal length and small diameter
Q. 159 A carbon resistor of (47±4.7) kΩ is to be marked with rings of different colours for its identification. The colour code sequence will be
A. Violet – Yellow – Orange – Silver
B. Green – Orange- Violet- Gold
C. Yellow – Green – Violet – Gold
D. Yellow – Violet – Orange – Silver
Q. 160 A set of ‘n’ equal resistors, of value R each, are connected in series to a battery of emf E and internal resistance ‘R’ The current drawn is I. Now, the ‘n’ resistors are connected in parallel to the same battery. Then the current drawn from battery becomes 10 I. The value of ‘n’ is
Q. 161 A battery consists of a variable number ‘n’ of identical cells (having internal resistance ‘r’ each) which are connected in series. The terminals of the battery are short-circuited and the current I is measured. Which of the graphs shows (among (1), (2), (3), (4)) the correct relationship between I and n ?
Q. 162 A body initially at rest and sliding along a frictionless track from a height h (as shown in the figure) just completes a vertical circle of diameter AB = D. The height h is equal to
A. 3/2 D
B. 5/4 D
C. 7/5 D
Q. 163 Three objects, A : (a solid sphere), B : (a thin circular disk) and C : (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed ω to about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation
A. WC > WB > WA
B. WA > WC > WB
C. WB > WA > WC
D. WA > WB > WC
Q. 164 Which one of the following statements is incorrect ?
A. Rolling friction is smaller than sliding friction
B. Coefficient of sliding friction has dimensions of length
C. Frictional force opposes the relative motion
D. Limiting value of static friction is directly proportional to normal reactions
Q. 165 A moving block having mass m, collides with another stationary block having mass 4m. The lighter block comes to rest after collision. When the initial velocity of the lighter block is v, then the value of coefficient of restitution (e) will be
Q. 166 An em wave is propagating in a medium with a velocity V⃗ = Vî. The instantaneous oscillating electric field of this em wave is along +y axis. Then the direction of oscillating magnetic field of the em wave will be along :-
A. –z direction
B. –x direction
C. –y direction
D. +z direction
Q. 167 The refractive index of the material of a prism is √2 and the angle of the prism is 30°. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is :-
Q. 168 The magnetic potential energy stored in a certain inductor is 25 mJ, when the current in the inductor is 60 mA. This inductor is of inductance
A. 0.138 H
B. 13.89 H
C. 1.389 H
D. 138.88 H
Q. 169 An object is placed at a distance of 40 cm from a concave mirror of focal length 15 cm. If the object is displaced through a distance of 20 cm towards the mirror, the displacement of the image will be:-
A. 30 cm away the mirror
B. 36 cm towards the mirror
C. 30 cm towards the mirror
D. 36 cm away the mirror
Q. 170 In the circuit shown in the figure, the input voltage Vi is 20 V, VBE = 0 and VCE = 0. The values of IB, IC and β are given by :-
A. IB = 40 μA, IC = 10 mA, β = 250
B. IB = 40 μA, IC = 5 mA, β = 125
C. IB = 20 μA, IC = 5 mA, β = 250
D. IB = 25 μA, IC = 5 mA, β = 200
Q. 171 In a p-n junction diode , change in temperature due to heating
A. affects only reverse resistance
B. affects the overall V-I characteristics of p-n junction
C. does not affect resistance of p-n junction
D. affects only forward resistance
Q. 172 In the combination of the following gates output Y can be written in terms of input A and B as :
Q. 173 A toy car with charge q moves on a frictionless horizontal plane surface under the influence of a uniform electric field E⃗ . Due to the force q E⃗ , its velocity increases from 0 to 6 m/s in one second duration. At that instant the direction of the field is reversed. The car continues to move for two more seconds under the influence of this field. The average velocity and the average speed of the toy car between 0 to 3 seconds are respectively
A. 2 m/s , 4 m/s
B. 1.5 m/s , 3 m/s
C. 1 m/s , 3.5 m/s
D. 1 m/s , 3 m/s
Q. 174 A block of mass m is placed on a smooth inclined wedge ABC of inclination θ as shown in the figure. The wedge is given an acceleration ‘a’ towards right .The relation between a and θ for the block to remain stationary on the wedge is
A. a= g/cosec θ
B. a= g tan θ
C. a= g cos θ
D. a= g/sin θ
Q. 175 A student measured the diameter of the small steel ball using the screw gauge of least count 0.001 cm .The main scale reading is 5 mm and zero of circular scale division coincides with 25 divisions above the reference level. If screw gauge has a zero error of -0.004 cm , the correct diameter of the ball is :
A. 0.521 cm
B. 0.529 cm
C. 0.053 cm
D. 0.525 cm
Q. 176 The moment of the force F = 4î + 5ĵ – 6k̂ at (2,0,-3) about the point (2, -2,-2) is given by
A. -8î – 4ĵ – 7k̂
B. -7î – 4ĵ – 8k̂
C. -7î – 8ĵ – 4k̂
D. -4î – ĵ – 8k̂
Q. 177 The volume (V) of a monatomic gas varies with its temperature (T), as shown in the graph. The ratio of work done by the gas, to the heat absorbed by it, when it undergoes a change from state A to state B, is
Q. 178 The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20 cm , the length of the open organ pipe is :
A. 13.2 cm
B. 16 cm
C. 12.5 cm
D. 8 cm
Q. 179 The efficiency of an ideal heat engine working between the freezing point and boiling point of water, is:
A. 26.8 %
Q. 180 At what temperature will the rms speed of oxygen molecules become just sufficient from escaping from Earth’s atmosphere?
Given: Mass of oxygen molecule (m) = 2.76 x 10⁻²⁶ kg
Boltzmann’s constant kB = 1.38 x 10⁻²³ J K⁻¹)
A. 2.508 x 10⁴ K
B. 1.254 x 10⁴ K
C. 5.016 x 10⁴ K
D. 8.360 x 10⁴ K |
Karoly Albert November 4, 2019 Number
But, teaching the addition facts doesn’t have to be like this. In this article, you’ll learn everything you need to know to teach your child the addition facts—without killing your kid’s love of math or wanting to poke your eyeballs out in the process.
These graph will produce a data set, of which the student will have to make a five number summary. You may select the amount of data, the range of numbers to use, as well as the number of problems. This worksheet is useful for teaching how to make box and whisker plots.
This list of activities to help students think about goal setting comes from the Supporting Transition and Education through Planning and Partnerships Program, a program designed to help students with learning disabilities succeed in college. It’s not just helpful for students with learning disabilities though—the activities can be helpful for anyone who wants to improve their goal setting skills.
These Function Table Worksheets are great for giving students practice in computing the outputs for different linear equations. You may select between four different types of equations. These Function Table Worksheets will generate 12 function table problems per worksheet. These Function Table Worksheets are appropriate for 4th Grade, 5th Grade, 6th Grade, and 7th Grade.
This Properties Worksheet is a great handout for reinforcing the different properties of Mathematics. This handout include the Associative Property, Commutative Property, Distributive Property, Identity Property, Additive Inverse Property, Multiplicative Inverse Property, Addition Property of Zero, Multiplication Property of Zero, Property of Equality, Reflexive Property, Symmetric Property, and Transitive Property. This Properties Worksheet is ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students.
We can’t all be master mathematicians or famous doctors. Who would create the music we listen to while we relax or exercise? Who would paint the artworks we love to admire? Who would write the books we love to read? Each and every child has their own unique talents and abilities. It is so important to encourage these, instead of pushing some ancient notion on your child.
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REFERENCE LINKING PLATFORM OF KOREA S&T JOURNALS
> Journal Vol & Issue
Research in Mathematical Education
Journal Basic Information
Journal DOI :
Korea Society of Mathematical Education
Editor in Chief :
Volume & Issues
Volume 11, Issue 4 - Dec 2007
Volume 11, Issue 3 - Sep 2007
Volume 11, Issue 2 - Jun 2007
Volume 11, Issue 1 - Mar 2007
Selecting the target year
A Comparison of Chinese Secondary School Mathematics In- and Pre-service teachers' beliefs about Mathematics, Mathematics Teaching and Learning
Jin, Meiyue ; Guo, Yanmin ; Dai, Feng ; Jia, Ping ;
Research in Mathematical Education, volume 11, issue 4, 2007, Pages 221~234
A comparison of mathematics teachers' personal beliefs between in- and pre-service teachers for Chinese secondary schools (grades 7-12) about mathematics theories, teaching and learning has been studied. In-service teachers' beliefs are close to constructivist's aspect and pre-service teachers' beliefs are close to absolutist's views. Based on the results, we give some suggestions to both teacher education and in-service teachers' training.
Study on the Ability Level Test of Mathematics Application
Dan, Qi ; Song, Naiqing ; Zhang, Nan ;
Research in Mathematical Education, volume 11, issue 4, 2007, Pages 235~246
The background and the existing problems in teaching of mathematics application were analyzed. Based on the mathematics knowledge which from the simple application, complicate application, and synthetically application to the mathematics modeling, the ability level test questions of mathematics application was worded out to help the teaching and learning in mathematics application.
The Determination of Elementary School Students' Successes in Choosing an Operation and the Strategies They Used While Solving Real-World Problems
Soylu, Yasin ;
Research in Mathematical Education, volume 11, issue 4, 2007, Pages 247~263
Problem solving takes place not only in mathematics classes but also in real-world. For this reason, a problem and the structure of problem solving, and the enhancing of success in problem solving is a subject which has been studied by any educators. In this direction, the aim of this study is that the strategy used by students in Turkey when solving oral problems and their achievements of choosing operations when solving oral problems has been researched. In the research, the students have been asked three types of questions made up groups of 5. In the first category, S-problems (standard problems not requiring to determine any strategy but can be easily solved with only the applications of arithmetical operations), in the second category, AS-SA problems (problems that can be solved with the key word of additive operation despite to its being a subtractive operation, and containing the key word of subtractive operation despite to its being an additive operation), and in the third category P-problems (problematic problem) take place. It is seen that students did not have so much difficulty in S-problems, mistakes were made in determining operations for problem solving because of memorizing certain essential concepts, and the succession rate of students is very low in P-problems. The reasons of these mistakes as a summary are given below:
Because of memorizing some certain key concepts about operations mistakes have been done in choosing operations.
Not giving place to problems which has no solution and with incomplete information in mathematics.
Thinking of students that every problem has a solution since they don't encounter every type of problems in mathematics classes and course books.
Exploration and Practice in Training Research Mathematics Teachers in the Middle Schools Located in the Countryside of Western China
Ding, Fengchao ;
Research in Mathematical Education, volume 11, issue 4, 2007, Pages 265~274
The middle school mathematics teachers of the countryside in western China have little chance to further their education and strengthen their research abilities. In this program, we should offer them chance to learn new educational ideas and teaching methods. There will be demonstration lectures and video teaching of course standards followed by group discussion and personalized instructions. These activities prepare the teachers to give their own demonstration lectures. This again is followed by additional personalized instructions. The teachers come to the program with very limited knowledge of these techniques but will leave with the ability to write educational scientific research papers.
Characteristics of Problem on the Area of Probability and Statistics for the Korean College Scholastic Aptitude Test
Lee, Kang-Sup ; Kim, Jong-Gyu ; Hwang, Dong-Jou ;
Research in Mathematical Education, volume 11, issue 4, 2007, Pages 275~283
In this study, we gave 132 high school students fifteen probabilities and nine statistics problems of the Korean College Scholastic Aptitude Test and then analyzed their answer using the classical test theory and the item response theory. Using the classical test theory (the Testian 1.0) we get the item reliability (
), and using the item response theory (the Bayesian 1.0) we get the item difficulty (
) and discrimination (
). From results, we find out what and why students could not understand well. |
Irish Geometry Conference 2019 - Titles and Abstracts
Speaker: Wilhelm Klingenberg
Title: On a conjecture of Toponogov on complete convex surfaces
In 1995, Victor Andreevich Toponogov authored the following conjecture: “Every smooth strictly convex and complete classical surface of the type of a plane has an umbilic point, possibly at infinity“. He proceeded to prove this in case the gradients of mean and Gauss curvatures on the surface under consideration are bounded. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle, of the general case. It proceeds indirectly by showing that a counterexample gives rise to a Riemann Hilbert boundary problem for holomorphic discs with negative index. However, Mean Curvature Flow serves to prove existence of a holomorphic disc in a geometrization of the space of oriented lines of R^3.
Speaker: Vladimir Dotsenko
Title: Homotopy type of the moduli space of stable rational curves
I shall show that the rational cohomology of the moduli space of stable rational curves is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way. Time permitting, I shall talk about a few classes of spaces for which similar results are available, and a few other conjectural classes of spaces like that.
Speaker: Graham Ellis
Title: The cuspidal cohomology of an arithmetic group
I'll start with the definition of the cohomology of a group, and go on to explain what is meant by the cuspidal cohomology of an arithmetic group such as SL(n,O) with O the ring of integers of an algebraic number field. I'll propose a method, based on simple homotopy collapses, for calculating this cohomology on a computer.
Speaker: Madeeha Khalid
Title: Examples of Moduli of vector bundles on K3 surfaces
We define K3 surfaces and describe some examples of algebraic K3 surfaces. Mukai showed in his seminal works (1980’s) that, under certain conditions, moduli spaces of vector bundles on K3’s are also isomorphic to K3’s. We consider modifications of a classical example, first noted by Mukai, and produce concrete examples of moduli spaces that have slightly different features. Joint work with Colin Ingalls.
Speaker: Eduardo Mota Sanchez
Title: Constant Mean Curvature Surfaces and Heun's Differential Equations.
The generalised Weierstrass representation for surfaces with constant mean curvature allows to describe any conformal constant mean curvature immersion in R3, H3 or S3 with four ingredients: a Riemann surface, a base point, a holomorphic loop Lie algebra valued 1-form and the initial condition for a 2x2 linear system of ODEs. Associating to this linear system a second order differential equation from the class of Heun's Differential Equations, we can prescribe certain kind of singularities in the constructing method that appear in the resulting surface. Regular singularities produce asymptotically Delaunay ends in the surface and irregular singularities produce irregular ends.
We discuss global issues such as period problems and asymptotic behavior involved in the construction of this kind of surfaces. Finally we show how to construct new parametric families of constant mean curvature surfaces in R3 with genus zero that possess at least one irregular end using these methods.
Speaker: Arne Rueffer
Title: Bridgeland stability conditions for the category of holomorphic triples
Stability conditions on triangulated categories have been introduced by Bridgeland in 2005. They generalise stability concepts on the abelian category Coh(C) of coherent sheaves on a smooth projective curve C to a triangulated category. A particular example of a triangulated category in which the stability space is completly understood, is the bounded derived category of Coh(C). Drawing on this knowledge we describe the stability space of the bounded derived category of the abelian category of holomorphic triples. This category has objects phi:E1→E2, where E1, E2 are coherent sheaves and phi is a morphism between them.
Speaker: Stephen Buckley
Title: Quasihyperbolic geodesics are hyperbolic quasigeodesics
The hyperbolic metric, known also as the Poincaré metric for simply connected domains, is an important tool in complex analysis and complex geometric function theory. The quasihyperbolic metric is a key tool in quasiconformal analysis, and has many other applications. Both of these metrics are defined in the setting of hyperbolic plane domains, but they are not in general bilipschitz equivalent. Nevertheless. we prove that a geodesic curve for either one of them is in a certain precise sense not far from being a geodesic curve for the other, regardless of the domain. More precisely, we prove that, as curves, hyperbolic and quasihyperbolic quasigeodesics are quantitatively the same (with no quantitative dependence on the domain). We also show that a domain is Gromov hyperbolic with respect to one of these metrics if and only if it is Gromov hyperbolic with respect to the other.
Speaker: Mark Walsh
Title: The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary
The problem of whether or not a given smooth closed manifold admits a Riemannian metric with positive scalar curvature (psc-metric) has recieved considerable attention over the years. In particular, in the case of simply connected manifolds (of dimension not equal 4) this problem is completely understood. More recently, a good deal of progress has occurred in understanding the topology of the space of psc-metrics for a given manifold as well its various moduli spaces. Less is understood about the related problem for manifolds with boundary, where metrics are required to satisfy certain boundary constraints. In this talk I will provide some background to this problem before presenting some new results. |
EFFECT OF SUSTAINABLE INFRASTRUCTURAL DEVELOPMENT ON ECONOMIC DEVELOPMENT OF NIGERIA
In this research work on the effect of sustainable infrastructural development on economic development of Nigeria. The researcher examined the effect of sustainable infrastructure on the gross domestic product (GDP)of Nigeria. The impact of infrastructural development on the Gross Fixed Capital Formation (GFCF) of Nigeria. Explore the impact that infrastructural development has on Nigeria’s economic growth. Data for the study was sourced through CBN Annual report and journal articles related to the subjects matter. The data collected was analyzed using SPSS. The results of the study shows that The results as presented in the coeficiente revealed that calculated t-statistics (t = -2.723) for parameter GDP is greater than tabulated t-statistics at 0.05 level of significance. The regression equation also revealed that GDP accounted for -0.881 unit for every increase in infrastructure expenses. The coefficient of determinant (R2) 0.921 indicating that 92% of variation in GDP increase is caused by variation infrastructure expenses. The relationship between GDP and infrastructure expenses is high, positive and statistically significant at 0.05 level (r=0.960, p<0.05). The overall regression model is statistically significant in terms of its overall goodness of fit (f = 12.22, p < 0.05). As a result of this the study accepts the alternative hypothesis meaning that Sustainable infrastructure affects the gross domestic product (GDP) of Nigeria. It was also observed that infrastructural development has great impact on the Gross Fixed Capital Formation (GFCF) of Nigeria. Based on the findings the researcher recommends that to attain significant accelerated development over the next 10 -15 years, Nigeria will have to expand its infrastructure development funding in tangible capacities by 24% of GDP over 10 years or 18% of GDP over 15 years to catch-up with most Asian countries. This of course is based on the assumptions that Asian countries will maintain a modest growth rate of 6%/annum with spending on infrastructure remaining in the average 6% range.
TABLE OF CONTENT
Title Page- - - - - - - - - i
Approval Page - -- - - - - - - ii
Dedication - - - - - - - iii
Acknowledgment - - - - - - - iv
Abstract - - - - - - - - - v
1.1 Background of the Study- - - - - 1
1.2 Statement of the Problem- - - - - 4
1.3 Objective of the Study - - - - - - 4
1.4 Research Questions- - - - - - - 5
1.5 Statement of Hypotheses - -- - - - 6
1.6 Significance of the Study- - - - - - 7
1.7 Scope and Limitations of the Study - - 8
1.8 Operational Definition of Terms - - - 9
2.1 Conceptual Framework - - - - - 11
2.2 Theoretical Framework - - - - - 14
2.3 Empirical Framework - - - - - - 18
2.3.1 The Importance of Infrastructure in Economic
Growth and Sustainable Development - - - 18
2.3.2 Inadequate Infrastructure and Economic Growth
and Sustainable Development- - - - - 24
2.3.3 Nigeria’s Current Infrastructure Base - - 33
2.3.4 Infrastructure Procurement Strategies - - 36
2.3.5 Improving Critical Infrastructure for Sustainable
Development in Nigeria - - - - - - 44
3.1 Research Design - - - - - - -49
3.2 Sources of Data Collection- - - - - -51
3.3 Area of the study - - - - - - -51
3.4 Population of the Study- - - - - - -52
3.5 Sample Size Determination - - - - - -52
3.6 Research Instrumentation - - - - - -52
3.7 Validity of the instrument - - - - -53
3.8 Reliability of the instrument - - - - -54
3.9 Model specification - - - - - - -54
3.10 Description of variables - - - - - -55
3.11 Method of Data Analysis - - - - - - -55
4.2 Data Analysis - - - - - - - -57
4.3 Testing of Hypothesis - - - - - -61
4.4 Discussion of Findings - - - - - -65
5.1 Summary of Findings - - - -- - - 67
5.2 Conclusion - - - - - - - 68
5.3 Recommendations - - - - - - 67
References - - - - - - - 73
1.1 Background of the study
The attainment of sustainable economic growth remains a paramount objective of every country. A primary source required for achieving this objective is through increased domestic productivity. However, for this to occur, such country must be able to create sufficient domestic physical capital to stimulate such desired economic growth. In other words, fixed capital formation is a major contributor, catalyst and determinant of a country’s economic growth.
Gross Fixed Capital Formation (GFCF) according to the World Bank (2014) refers to fixed assets accumulation such as land improvements, equipment, machinery construction of roads and railways, building of schools etcetera, required for augmenting a country’s economic productivity. This definition reiterates and captures the predictions of Romer (2008) and Lucas (2007) Growth Models which stipulates that increased growth rates can be achieved by increasing capital accumulation. Also, the building of schools leads to improved educational enrolment rate which will enhance the quality of human capital. The improvement of human capital in this regards will ensure innovation, invention and enhancement of productivity in the economy. Likewise, the investment in machinery and equipment will also increase the efficiency of labour productivity. Furthermore, Bakare (2011) explained capital formation as the “proportion of present income saved and invested in order to augment future output and income". This definition buttresses the importance of savings as an integral element needed for creating (GFCF) and enhancing economic growth. Therefore, it can be concluded that a country with low domestic marginal propensity to save is likely to have poor capital formation which potentially impedes economic growth and vice versa. This is because, such country will have an insufficient pool of loanable funds for domestic investment into physical capital. More importantly, the availability of quality physical capital attracts Foreign Direct Investment (FDI) inflow, which is an integral macro-economic variable necessary for increasing a country’s economic prosperity. In a broader perspective, capital formation in the financial economics lingual refers to savings drives, developing of capital and secondary markets and privatizing financial institutions (Ray, 2013). Ray, (2013) opined that GFCF results in increased production in the long run which eventually causes share prices to rise, thus increasing profitability which in the end has a positive spillover effect on a country’s economic growth. Based on the discussion so far, an intuitive conclusion that a key precondition for ensuring and enhancing sustainable economic growth is through increased fixed capital formation. This study is geared towards investigating the effect of sustainable infrastructural development on economic growth in Nigeria.
1.2 Statement of the Problem
In recent years, Nigeria has experienced increased infrastructural transformation in terms of building of more schools, road, telecommunication facilities and etcetera. However, there are only a few studies found to have investigated the impact that these infrastructural development has on Nigeria’s economic growth. Thus, the aim of this study is geared towards contributing to the existing studies by investigating the contribution and impact that infrastructural development has on Nigeria’s economic growth.
1.3 Objectives of the study
The aim of this research work is to examine the effect of sustainable infrastructural development on economic development of Nigeria. The specific objectives of this research work include the following:
1. To examine the effect of sustainable infrastructure on the gross domestic product (GDP) of Nigeria.
2. To evaluate the impact of infrastructural development on the Gross Fixed Capital Formation (GFCF) of Nigeria.
3. To explore the impact that infrastructural development has on Nigeria’s economic growth.
4. To investigate whether there is causal relationship existing between infrastructural development and economic growth in Nigeria. |
The question of home field advantage deserves some mention early on. There are three ways to treat home field advantage: as a score adjustment, as a schedule strength adjustment, or ignoring it altogether. The last option is unacceptable, given that teams clearly perform better at home than they do on the road. In my many years of ranking various sports and leagues, I have solved for the home field factor and always found it to be significantly positive. (If home field advantage didn't measurably help teams play better, I would find negative values as often as positive.)
Of the remaining options, adjusting game scores is also troublesome. If the home field factor is 3.5 points, does this mean a 3-point win become a loss, while a 4-point win remains a win? What if the team with the 3-point win scored a touchdown with PAT in the final 30 seconds? Had they known (and cared) that I would consider a 3-point home win a loss, wouldn't they have gone for a 2-point conversion instead? In short, a win needs to remain a win because that's what the teams are worried about.
This leaves the final option of treating home field as an adjustment to the quality of one's opponent. This intuitively makes sense; playing the #25 team on the road may be like playing the #10 team at home. Likewise, a team that played an excess of road games indeed faced tougher competition than they would have with a balanced schedule.
This may seem like an obvious and trivial point, but it is something that the most accepted computer ratings do not agree on.
In hockey and soccer, tere are many possessions, with either a score (1 point) or not (0 points). In other sports, more than one point value per possession is possible. (Note: in baseball, one possession is one inning, not one at-bat.) There are two ways of addressing this complication. One is to consider all possible combinations of scores that would create S points; to do so is quite difficult but possible. A simpler solution is to divide S1 and S2 by a "typical" number of points per score. Tests of the full-blown probabilities indicate that the "typical" number of points is not the average, but rather the average weighted by the number of points scored. For example, if a football teams scores field goals (3 points) and touchdowns (7 points) with equal frequency, the scaling factor is 5.8. Using actual frequencies of scoring types and factoring in safeties, missed PATs, and two-point conversions, the value is 6.2. In basketball, I use a value of 2.15; in baseball 2.65. (To take the factorial of a non-integer, replace it with the gamma function.)
Another note involves the treatment of overtime games. My research has found that overtime results are 50-50 propositions. In other words, the better team wins only 50% of overtime games. Perhaps it is 50.5%, but the deviation from 50% is too small to be accurately measured. Thus G(sa,sb) is set to zero for an overtime game, regardless of the final score.
Implicit in the binomial statistics is the assumption that the odds of scoring on any possession is constant over the course of the game. This is not true, as players and coaches will adjust their tactics based on how the game is progressing. This is modeled in two different ways. The first way reflects the changes in coaching strategy -- the winning team will try to protect the lead by playing conservatively, while the losing team will try any measure to get back into the game. Effectively, this means that both teams' scoring odds are lowered by the winning team's strategy changes, and both are raised by the losing team's strategy changes. In football and basketball, the changes made by the losing team tend to outweigh those made by the winning team; this raises F since a team in either of those sports can effectively prolong the game. The defensive changes tend to be more important in the other sports, lowering F values.
A second element of adjustments for a lead is the tendency of players to play up or down somewhat to the level of their opponent. This is seen empirically by the fact that it is much easier to predict a basketball game's margin of victory than it is to predict the total score. In other words, a team with a 15-point lead will tend to let up a little bit, which prevents the lead from getting much bigger. Unfortunately, this lowers the leading team's X value and raises the opposing team's X value. For the purposes of this section, we can model this also by changes (reductions) to F. In most sports the change is quite small (around 0-30%); in basketball it is nearly a factor of two.
Note that this correction assumes that all teams use similar measures to avoid running up the score. This is true for the most part, but points out the reason why margin of victory should not be used in postseason selections such as the BCS or NCAA basketball tournament. If teams know that margin of victory is important, they will try to run up the score, thus destroying the validity of a margin-of-victory rating.
A discussion of the need for and calculation of priors is given in the constructing a ranking page. As described there, priors are used to constrain the overall set of team rankings to be within a typical range.
An alternate use of priors is to enhance the numerical stability early in the season, when insufficient data exists to draw significant statistical conclusions. If one has a guess of how good the team actually is, then the team's rating should equal that guess before games have been played and move away from that as data is collected. I treat this prior data the same as games, in that a team that has not yet played some minimum number of games has its schedule padded with ties against a team of its guessed strength. Thus, if the minimum is 6 games (the value used for my football ratings) and a team with a preseason rating of +0.9 has played 4 games, then 2 games are added that are treated as ties against a team of strength 0.9. This adjustment is made in the three main ratings (standard, simple, and predictive) and for the improved RPI. An analogous adjustment is made in the college pseudo-poll. Teams with priors in use have a "P" shown at the end of their rating lines.
Schedule strengths have been calculated several ways, but what is the best way of doing it? Consider two cases. Team 1 is an outstanding team that plays most of its games against mediocre teams (i.e. teams ranked near the middle of the set). Team 2 is also an outstanding team, but it plays half of its games against other outstanding teams and half against horrible teams. According to an RPI ranking, the two teams would have the same (or nearly identical) schedule strengths, since the RPI uses the straight average of the opponents' winning averages. However, it is clear that Team 2 challenged itself much more. Given average luck, team 2 probably beat all of the horrible teams but only half of the excellent teams, thus winning 75% of its games. Team 1, on the other hand, probably won 90% of the games against mediocre teams.
Thus the key factor is the team's most likely winning percentage against its schedule. To calculate this, use the principles described above. Given the team's ranking, its opponents' rankings, and the home field advantage, the odds of winning is:
P(win) = integral(x=-inf,dr) exp(-0.5*(x^2)) / sqrt(2*pi),
One must then make this calculation for each game the team has played, giving the average number of wins. Dividing by the number of games gives the team's average winning percentage against its schedule. Setting P(win) to this value and solving for dr by inverting the integral calculation thus gives the "dr" value of a "typical" opponent, where "typical" means an opponent that, if played in every single game at a neutral site, would give the same average winning percentage as the set of opponents a team actually played. I define this as the schedule strength.
Conference ratings are calculated in a similar manner. A conference's rating equals the rating of a team that would be expected to go 0.500 against the teams in the conference with all games played at a neutral site. Again, this calculation is most sensitive to teams in the middle of the conference, which are the games that can go either way.
Because the schedule strength depends on a team's own strength, two teams playing identical schedules will not have the same schedule strength rating. While this is intentional (what matters is how the schedule affects the team playing that schedule), there may be times in which it is important to have all schedules rated on the same scale. To do this, a second schedule measure is provided, expected losses (ELOSS). For pro sports, ELOSS gives the number of losses an average team would have if playing the team's schedule. For college sports, ELOSS gives the number of losses an average ranked (top 25 except for hockey, which is top 15) team would have against the schedule.
A common complaint about computer rankings is that they appear to completely overlook head-to-head game results, and will happily put one team immediately ahead of a team it lost to. As I have shown in a study, this is indeed the statistically correct treatment. Given three games between three teams, each going 1-1, it is more likely that there were two minor upsets and one favorite winning than that there was one significant upset and two favorites winning.
That said, if one wishes to accurately rank a pair of teams, the head-to-head result should be given greater importance. The reason has nothing to do with perceived importance of such games, but rather the fact that teams match up better against some teams than they do against others. So back to the three team example, where team A beat team B, B beat team C, and C beat A. If matchup effects do not exist, team A is more likely than not worse than team B and better than team C. However, accounting for matchups, team A is probably better than team B, in the sense that they would probably win the rematch.
In my rankings, I account for this factor only in the probable superiority (standard) ranking, where it can be done fairly easily because the ranking is based on team-by-team comparisons.
Note: if you use any of the facts, equations, or mathematical principles on this page, you must give me credit. |
On the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan equation.
We give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan master equation and the related problem of attractivity. We also provide a rigorous mathematical proof of the uniqueness of the steady-state configuration for a large class of open quantum systems with finite dimensional Hilbert space that is based on a particular assumption about the structure of the Lindblad operators. Our results apply to systems such as dissipative lattices of interacting bosons with truncated Fock space and dissipative spin systems.
During the last few decades systems of cold atoms, molecules and trapped ions, as well as photons, have been extensively used in so many fields of research that they seem to represent the key to understand and investigate the most fundamental laws of quantum mechanics Intro3 ; haroche ; Intro1 ; Intro2 ; carusotto . Indeed, thanks to the impressive progress which has been made especially in optics, it is possible to use these systems to engineer microscopic Hamiltonians for the purpose of quantum simulation Intro3b ; hartmann ; noh : by using lasers one can achieve those critical regimes at which the system dynamics becomes completely ruled by principles of quantum mechanics. However, keeping systems in the proper working conditions
can be quite a hard task: the coupling between the microscopic degrees of freedom and the surroundings naturally leads to losses. If these losses are negligible, the quantum system can be approximate as an ideal and closed system which follows the standard quantum theory. If it is not possible to neglect the leakage induced by the environment one needs to change point of view and use the formalism of open quantum systems.
The first examples of a rigorous treatment of the open quantum system dynamics were given during the 70s by Davies while considering a harmonic oscillator daviesoscillator and a N-level atom daviesnlevel coupled to a heat bath and by Pulè pule in the context of a single spin coupled to an infinite bath of harmonic oscillators. During the same years, Kossakowski and Ingarden kossakowski ; ingarden , while attempting to provide a mathematical framework suitable for the description of the irreversible dynamics proper of the open quantum systems, introduced the concept of quantum dynamical semigroup: while in the case of closed systems it is possible to move both “forward” and “backwards” in time by using unitary operators which are elements of a one-parameter group, in the case of quantum open systems time-translations form a semigroup i.e. elements do not have an inverse.
In 1976, Lindblad Lindblad and Gorini, Kossakowski and Sudarshan Gorini derived the general form of the generator of completely positive quantum dynamical semigroups and introduced the master equation nowadays known as the LGKS master equation which prescribes the time-evolution of an open quantum system weakly coupled to a Markovian environment. Between 1976 and 1978 a series of theoretical papers concerning the uniqueness of the solution of the LGKS master equation appeared in literature. In spohn1 and spohn2 Spohn gave a set of sufficient conditions for uniqueness based respectively on the properties of the decay-rate matrix and on the properties of the Lindblad operators entering in the generator of the dynamical semigroup. In frigerio1 and frigerio2 Frigerio considered under which sufficient conditions a dynamical semigroup possessing a faithful normal stationary state admits a unique equilibrium state. Frigerio in frigerio1 also derived, for this particular class of problems, the equivalence between irreducibility and the uniqueness of the equilibrium state. Evans in evans provided a necessary and sufficient condition for irreducibility based on the study of von Neumann algebras.
After three decades, the impressive technological progress made in optics has led to a renewed and growing interest for the physics of open quantum systems. As a consequence, several new interesting results concerning the structure of Hilbert spaces and the characterisation of the time-evolution of open quantum systems baum1 ; baum2 ; ticozzi1 ; ticozzi2 ; schirmer ; baum3 , the structure of the state space i.e. the convex set of density matrices byrd , a set of sufficient conditions for the uniqueness of the steady-state solution of the LGKS equation schirmer as well as the role of symmetries and conserved quantities albert have been derived.
The main purpose of this paper is to review the most relevant theoretical papers concerning the uniqueness of the steady-state solution of the LGKS and whether a dynamical semigroup is relaxing i.e. the steady-state solution is attractive for all the initial conditions, and to provide a mathematically rigorous proof of the uniqueness of steady-state solutions in systems where some of the dissipation channels are induced by Lindblad operators with a particular structure. Our results find application in the field of open quantum systems with bosons and spins.
The paper is organised as follows. In Section II we briefly review the main ideas of the formalism of dynamical semigroups, introducing the LGKS equation and its dual expression. In Section III we review the most relevant theoretical results concerning the issues discussed above: the uniqueness of the steady-state configuration and the conditions under which a dynamical semigroup is relaxing. Section IV represents the core of this work. Here we show and compare our work to the results already available in literature. In Section V we draw our conclusions.
Ii Theory of dynamical semigroups
In this Section, we proceed in the same way as Lindblad in Lindblad who gave a description of the theory of dynamical semigroups starting from the formalism introduced by Kossakowski in kossakowski for the description of non-hamiltonian systems.
According to the standard formalism used in quantum mechanics, to every quantum system it is possible to associate a separable Hilbert space and admissible states of our quantum system are represented by a self-adjoint positive semidefinite linear operator of unit trace i.e. density operators. The set of all the density operators, denoted by , is a convex set: given any couple of states , , also , with is a element of (in literature this set is also denoted as , the set of trace class operators on , see e.g. spohnrev ). Observables correspond to self-adjoint operator on and they are elements of , the set of bounded operators on . The mean value of an observable at a state , that is , can be obtained evaluating , where denotes the trace operation.
Let be the Hamiltonian of our physical system. The time-evolution of the system is given by the family of two-parameters operators acting on the set . has the form
where denotes the time ordering operator, and
is the Liouville operator (also called Liouvillian) of the system () and denotes the commutator between the operators and . The relation (1) is usually written as the von Neumann equation of motion for the system state
with starting condition , being the identity map. Given the system state at the time , the system state at is given by .
In general, is a semi-group under the following composition law
A physical system for which its dynamical semi-group can be extended to a group , with the introduction of the inverse operator , is called Hamiltonian system, since is completely determined by the Hamiltonian . In this case the Hamiltonian is the generator of the group. This is the case of ordinary (closed) quantum systems. A physical system for which the dynamical semi-group cannot be extended to a group is called non-Hamiltonian. In this case we denote the semigroup by . This is the case of open quantum systems, where one has a total system consisting of two parts, but is interested in characterising only the dynamics of one of the two subsystems. Let us call these two parts and : is the so called environment or reservoir, while denotes the subsystem of interest, the open part of the total system which is coupled to . While the total system can be considered as a Hamiltonian system, suffers a time evolution which is non-Hamiltonian. This means that, if we consider a state of the total system , the reduced density matrix , which describes the state of the subsystem as a part of the larger system , does not evolve in time according to the von Neumann equation. If one considers temporarily homogeneous semigroups (Markov approximation), that is semigroups in which all the elements () are functions only of so that the composition law becomes
the time-evolution of the reduced density matrix can be cast in the following form
where is the generator of the dynamical semigroup . The expression of has been independently derived in the 70s by Lindblad Lindblad and Gorini, Kossakowski and Sudarshan Gorini . In general, the rhs of Eq.(6) has the following expression
where is the Hamiltonian of the subsystem , are positive decay rates and are operators acting on which are usually called Lindblad operators, is a set of indices and denotes the adjoint of which in finite dimension is the conjugate transposed of . The decay rates and the explicit form of the Lindblad operators depend both on the particular global system under study. People usually refers to Eq.(7) as the Lindblad or LGKS master equation.
For completeness we also report the dual expression of Eq.(7) which reads
Iii Uniqueness of the steady-state configuration relaxing dynamical semigroups
After the derivation of the explicit expression of the generator of quantum dynamical semigroups, several theoretical papers concerning the uniqueness of the steady-state configuration i.e. the fixed point of the equation , and sufficient conditions under which a dynamical semigroup is relaxing have been published: a semigroup is called relaxing if there exists a state such that for every state
It is worth noting that while “relaxingness” implies the convergence to a unique fixed-point configuration, the converse is not always true: given a steady-state configuration it is not true a priori that any initial configuration will approach to it as time goes to infinity. This is true if and only if the fixed-point is unique.
In spohn1 Spohn derived a simple sufficient condition for a completely positive dynamical semigroup to be relaxing. However, as pointed out by the author, this result applies only to physical systems where every “coupling degree of freedom” is effectively coupled to a reservoir. In spohn2 Spohn gave another sufficient condition for a dynamical semigroup to be relaxing, based this time on the self-adjointness of the linear span of the Lindblad operators, that is (also denoted by and nowadays usually called ), and on the triviality of the commutant of these operators, that is , where the commutant is the set of all the operators commuting simultaneously with all the and indicates all the operators multiple of the identity . While the self-adjointness requirement can be easily checked by an explicit analysis of the set , proving the triviality of the commutant can be a complex task. In frigerio1 Frigerio provided a sufficient condition for the equivalence between the uniqueness of the stationary state and the triviality of the commutant . However, as pointed out by Spohn in spohnrev , in frigerio1 one needs to assume by hypothesis that the semigroup has a maximum rank invariant state i.e. a faithful invariant state, therefore a priori this condition restricts considerably the applicability domain of this theoretical result. In frigerio2 the author generalised the result given by Spohn in spohn2 , showing that if the dynamical semigroup has at least one stationary state and if is a self-adjoint set with , then is faithful and we have
for all and for all .
Evans evans provided a necessary and sufficient condition for the irreducibility of a dynamical semigroup i.e. the uniqueness of the steady-state configuration: a dynamical semigroup is irreducible if and only if the commutant . This result may look quite similar to those listed above, however there are few aspects which make it more applicable: it does not assume any particular structure for the Lindblad operators or their linear span (for example self-adjointness as assumed in spohn2 ; frigerio2 ), it provides a necessary and sufficient condition and in addition it does not assume any particular form of the steady-state solution, for example faithfulness.
More recently a new series of theoretical papers have been published and it is worth comparing them to the classical work mentioned above. In 2008 and 2009, Baumgartner, Narnhofer and Thirring baum1 and Baumgartner and Narnhofer baum2 analysed the properties of quantum dynamical semigroups in a general fashion. In particular, they pointed out that there is a relation between the decomposability into invariant subspaces of the Hilbert space associated to an open quantum system i.e. the conservation of a given set of projectors and the presence of dynamical symmetries (this issue has been studied in details in albert ). According to their results, the conservation of a projector (with ) is possible if and only if it commutes both with the Hamiltonian and all the Lindblad dissipators . In other words, a projector is conserved if and only if it is an element of the von Neumann algebra (thanks to the hermiticity of , ). If in this set there are projectors with orthogonal support, then there exist more than a single fixed point for the quantum dynamical semigroup. When the only projector conserved by the time-evolution is the identity (trivial projection operation), this result is coherent with what has been derived by Evans in evans . This analysis clarifies the relation between the irreducibility criterion given by Evans and the uniqueness of the steady-state configuration.
For our purposes, the most relevant contribution given by Baumgartner, Narnhofer and Thirring is the characterisation of the geometry of paths in the space given in baum1 . These paths can be classified by considering the eigenmatrices and the corresponding eigenvalues of the generator in Eq.(7). In particular, the fact that there are no eigenvalues on the imaginary axis, as proved in Theorem 17 baum1 , is an extremely relevant result. Indeed, it guarantees that in the case of a unique fixed point the semigroup is relaxing.
Iv Theoretical results
In the following sections we provide a set of sufficient conditions for the irreducibility of dynamical semigroups governing the time-evolution of single and composite quantum systems.
Our sufficient conditions are based on the assumption that among all the Lindblad operators entering in the LGKS some have a particular structure. As we show in the following, this structure is exactly the one characterising for example ladder operators in spin systems and annihilation and creation operators in boson systems.
We stress that our results guarantee the uniqueness of the steady-state solution and, as a consequence of what has been derived in baum1 , its attractivity i.e. any starting condition approaches to the same fixed point as .
iv.1 Single quantum system
Let us first consider the case in which we have a single quantum system described by a Hilbert space having dimension . Let us suppose to have a LGKS master equation in which at least one among the Lindblad operators , let us call it , has the following structure
where is the element in the -th row and -th column of the matrix representing the operator and being the element of the canonical basis for the square matrices, whose elements are all 0 except for the element in the -th row and -th column whose value is 1. As we use this expression in the followings, we also report the structure of the operator :
Our proof is based on the following observation:
since the set , then the commutant black .
Therefore, if one shows that , that is the only operators which commute with both and are multiples of the identity, then also and as a consequence, thanks to the result provided by Evans, the dynamical semigroup is irreducible and the steady-state solution is unique.
Let us consider a generic operator . It can be decomposed in terms of the set in the following way:
we obtain that
The operator belongs to the commutant if and only if the two matrices in Eq.(16) and in Eq.(17) are both equal to , that is the matrix with all entries equal to zero. Since the matrices are all linearly independent, this happens if and only if the coefficient multiplying each matrix is 0.
Since in Eq.(16) the first summation does not depend on and by hypothesis all the , we have that (all the elements in the first column of the matrix except are zero). In the same way, by considering Eq.(17), since the summation in the second term does not depend on , all the coefficients for in the first term must be equal to zero. This means that (all the elements in the first row of the matrix except are zero).
Let us now consider in Eq.(16) the coefficients for . In this case, for we obtain
while for , since , we have that . In the same way, by considering in Eq.(17), we obtain again that from and since we obtain . This means that after the steps described above the only entries of the matrix in the first two columns and rows which could be non-zero are the diagonal entries and . In addition, . If we now consider the terms for in Eq.(16) and the terms for in Eq.(17), we obtain that and that and .
In this way at the step ( , from Eq.(16) by setting one obtains
while from Eq.(17) by setting one ontains
where and denote respectively the -th column and the -th row of the matrix .
Thanks to the constraint obtained by the previous steps, these two equations reduce to
In this way, after a finite number of steps, we obtain that the matrix is diagonal and proportional to . For the arbitrariness in the choice of , we conclude that any matrix which commutes simultaneously with both and must be of the form , with . Therefore, we have that
In the case of spin systems, where one has usually that and , since any operator that commutes with both and also commutes with , by Schur’s Lemma . This happens because the is a -dimensional irreducible representation of the Lie algebra of the group georgi .
From Proposition 1, it follows that Evans result applies to any finite-dimensional quantum open system governed by a LGKS equation where al least one of the Lindblad operators is in the form (11).
iv.2 Composite quantum systems
Let us now consider the case in which we have a quantum system described by a total Hilbert space that is the tensor product of Hilbert spaces with finite dimension , that is , where .
Let us consider the following set of operators acting on the total Hlbert space
In order to show the validity of Eq. (27) we proceed by induction. In Proposition 1 we have shown that the thesis is true for . Let assume the thesis true for and let us consider a matrix in the set of the bounded operators for the total Hilbert space . The matrix can be decomposed in the following way
where is the matrix of elements with .
Let us now divide the operators into two sets: a first set given by and a second set . All the operators in the set are in the form
where and are the Lindblad operators for -composite problem, while those in the set are in form
By using the following property of the Kronecker product
being ,, and matrices for which the matrix product makes sense, we obtain the following expressions for commutators between and the elements of
If we now set the two commutators in Eq.(32) to be equal to , since the are linearly independent, this happens if and only if all the commutators and are equal to . Since we have assumed the thesis to be true for , this happens if and only if all the blocks are in the form , with .
Therefore, we have that
Let us now consider the commutator with the elements of . We have
where we used the explicit form of the operators and given respectively in Eq. (11) and Eq. (12), the property (31) and the commutation rule in Eq. (15). Compare now Eq.(34) to Eq. (16), and compare Eq.(35) to Eq. (17). These two set of expressions have the same structure, so it is easy to see that if we require Eq.(34) and Eq.(35) to be equal to the matrix , we get a set of equations similar to those obtained in the proof of Proposition 1, with the only exception that now these equations give a set of constraint for the matrices , that is on the . Therefore, after a finite number of steps, we get that the matrix is proportional to . For the arbitrariness in the choice of , we conclude that a matrix belongs to the commutant if and only if it is proportional to . Since we have shown in Proposition 1 that the thesis is true for and here above we have shown that its validity for implies also the validity of the thesis for , by induction principle we have shown that for any :
We are now in the position to state the following theorems.
Given an open quantum system described by a Hilbert space with finite dimension , whose dynamics is governed by a dynamical semigroup with generator given in Eq.(7), if at least one of the Lindblad operators is in the form given in Eq.(11), then the dynamical semigroup is irreducible i.e. the steady-state solution is unique.
It follows from Observation 1, Proposition 1 and the results provided by Evans in evans . ∎
Given a composite open quantum system described by a Hilbert space with and with , whose time-evolution is governed by a dynamical semigroup with generator given in Eq.(7), if at least among the Lindblad operators are in the form given in Eqs.(24)-(26), with given by Eq. (11), then the dynamical semigroup is irreducible i.e. the steady-state solution is unique.
It follows from Observation 1, Proposition 2 and the results provided by Evans in evans . ∎
The results shown in this section give a sufficient condition for the irreducibility of the dynamical semigroup which governs the time-evolution of a wide variety of problems in low energy physics. It is easy to see that an operator in the form (11) describes for example the annihilation operator of a system of bosons with a truncated Fock space with maximum occupation number or the ladder operator for a spin with . Indeed, by using the two functions
with and and by using the standard representation for the annihilation operator and the ladder operator ,
one finds respectively for boson an spin systems that
Is it possible to generalise our results to the case where the Hilbert spaces associated to the single subsystems have different dimension , and morever to the case where every subsystem describes a different physical systems? The answer is yes in both cases and it follows from the fact that in the proof of Proposition 2 we did not use that the subsystems had the same dimension or that the operators entering in Eqs.(24)-(26) were the same operator. Therefore, we have the following theorem:
Let us consider a composite open quantum system made up of subsystems, described by a Hilbert space , where all the Hilbert spaces have finite dimension i.e. , . Let us suppose the time evolution of this composite open quantum system to be governed by a dynamical semigroup whose generator is in the form given in Eq. (7). If at least among the Lindblad operators entering in the generator have the following structure
where the all the operators are in the following form
then the dynamical semigroup is irreducible i.e. the steady-state configuration is unique.
Our studies have been motivated by the examples shown in schirmer to prove the applicability of Condition 3 (page 4). However, while they formulated their results starting from the decomposition of the Hilbert space, our results are formulated in terms of operator algebras and thanks to the assumption on the structure of the and operators apply in principle to a larger class of problems.
It is interesting to observe that in what we have shown the Hamiltonian operator does not play a relavant role. However, this is not the case if one relaxes the hypothesis of Theorem 2 and Theorem 3, for example by removing one or more of the relevant operators. In this case, the Hamiltonian plays a fundamental role. Indeed, it has been shown for example in prosen by Prosen that for a one-dimensional chain of spin-1/2 particles with dissipators only at the boundaries, the result provided by Evans in evans can be exploited to prove the uniqueness of the steady-state solution.
V Summary and Conclusions
We have briefly reviewed the most relevant theoretical results developed during the last forty years concerning the problem of the uniqueness of the steady-state solution of the LGKS master equation and the problem of the determination of conditions under which a dynamical semigroup is relaxing. In addition we have provided a sufficient condition, based on the assumption of a particular structure of the Lindblad operators and the results derived by Evans in evans , which guarantees the irreducibility of the quantum dynamical semigroup under analysis i.e. the uniqueness of the steady-state solution. As pointed out above, our results are closely related to some of those derived by Schirmer and Wang in schirmer . However, while they derived their results working in Schödinger picture, considering the Hilbert space and the set of invariant states under the action of the Lindblad operators, here we have considered the problem from an another perspective, that is in terms of operator algebras. As shown in the previous section, the structure assumed for the Lindblad operators is compatible with those of annihilation operators in boson systems with truncated Fock space and ladder operators in spin systems. However, our results provide a sufficient condition for irreducibility which at least in principle applies to a wider variety of quantum open systems.
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posted by Kaitlyn .
Posted earlier but I wrote it incorrectly.
How fast would a rocket have to be going in order to completely leave the solar system? (Assume that the Sun is the only mass in the solar system and that the rocket is beginning its journey from Earth's orbit?)
With the earth being the only planet in the solar system, the final velocity of the rocket, perpendicular to the earth/sun radial, would need to be
Vesc= sqrt2(4.68772x10^21)/93,000,000(5280)) = 138,177fps = 94,212mph.
Since the earth's velocity around the sun is 66,618mph, the rocket's orbital velocity of ~25,000 and the velocity of 912mph acquired from the earth's rotational speed at KSC Launch site, the rocket need only provide a final burnout velocity of 26,682 mph.
You might find the following of some interest.
How long is a round trip to mars?
It depends on the configuration of the planets at launch, the size of the rocket, and the weight of the payload. It could be as long as 970 days or as quick as 240 days. The following will explain this in more detail.
The time for a space probe, launched from Earth, to reach the planet Mars, or any planet for that matter, is a function of the location of the planets relative to one another at the time of launch, the final velocity of departure from earth orbit, and the velocity direction, at burnout of the rocket stage. The exact phasing and distances of the planets from each other, dictates the required magnitude and direction of the velocity. There are fundamentally two extremes to examining the time required to travel to a planet in a direct planet to planet flight. One requires a minimal expenditure of rocket energy but results in the longest trip time. The other requires a huge expenditure of rocket energy but results in a shorter trip time, relatively speaking.
The minimum energy, one way trip time, is ~259 days (assuming the average radii of the Earth's and Mar's orbits) while the fast track approach could get you there as fast as 70 days or less, depending on the final burnout velocity of the rocket stage. It is also possible to launch at a non-optimum phasing of the planets and take longer than the 259 days, such as the recent Mars Global Surveyor, which took ~10 months to arrive at Mars in September 1997. The Pathfinder spacecraft, traveled on a fast track trajectory, reaching Mars in ~7 months, on July 4, 1997. The following will hopefully shed some additional light on the subject for you.
The minimum energy approach for a probe to reach any planet on its own is by means of the Hohmann Transfer Orbit. By minimum energy, I mean the lowest possible final velocity of the probe as it departs its earth orbit. The Hohmann Transfer Orbit (HTO) is an elliptical orbit that is tangent to both of the orbits of the planets between which the transfer is to be made. In other words, a probe placed into a heliocentric orbit about the Sun would leave the influence of the earth with a velocity vector tangent to the earth's orbit and arrive at the destination planet's orbit with a velocity vector tangent to its orbit. One of the focii of this elliptical orbit is the Sun and the orbit is tangent to both the Earth's orbit and the target planet's orbit.
Lets explore what it takes to send a space probe to Mars. Lets assume a launch vehicle has already placed our probe and its auxiliary rocket stage in a circular, 250 mile high, low Earth orbit (LEO) with the required orbital velocity of 25,155 feet per second, fps.
By definition, a probe being launched on a journey to another planet, must be given sufficient velocity to escape the gravitational pull of Earth. A probe that is given the exact minimum escape velocity, will depart the Earth on a parabolic trajectory, and just barely escape the gravitational field. This means that its velocity will be approaching zero as its distance from the center of the Earth approaches infinity. If however, we give our probe more than minimal escape velocity, it will end up on a hyperbolic trajectory, and with some finite residual velocity as it approaches infinity, or the edge of the Earth’s sphere of influence. This residual velocity that the probe retains is called the "hyperbolic excess velocity." When added to the velocity of the Earth in its orbit about the Sun, the result is the heliocentric velocity required to place the probe on the correct Hohmann transfer trajectory to rendezvous with Mars
It is worth noting at this point that it is somewhat naive to talk about a space probe reaching infinity and escaping a gravitational field completely. It is somewhat realistic, however, to say that once a probe has reached a great distance (on the order of 500,000 to 1,000,000 miles) from Earth, it has, for all intensive purposes,
escaped. At these distances, it has already slowed down to very near its hyperbolic excess velocity. It has therefore become convenient to define an imaginary sphere surrounding every gravitational body as the body's "sphere of influence", SOI. When a space probe passes through this SOI, it is said to have truly escaped. Over
the years, it has become difficult to get any two people to agree on exactly where the SOI should be located but, nevertheless, the fictitious boundary is widely used in preliminary lunar and interplanetary trajectory studies.
As stated earlier, a probe that departs from a 250 mile high LEO with the minimal escape velocity of 36,605 feet per second will end up at the edge of the Earth's SOI with near zero velocity and remain there forever. We must then give our probe a final Earth departure velocity in excess of minimal escape velocity. Lets back into this required velocity in the following manner.
The final heliocentric velocity required by our probe for a Hohmann transfer orbit to Mars is ~107,350 fps, relative to the Sun. Since the probe, at the beginning of its journey, picks up and retains the velocity of the Earth about the Sun (97,700 fps), we can subtract this from our required heliocentric velocity to determine our required hyperbolic excess velocity. Thus, 107,350 - 97,700 = 9,650 fps, the hyperbolic excess velocity required by our probe at the edge of the Earth's SOI. To achieve this hyperbolic excess velocity, on the hyperbolic escape trajectory from our 250 mile high parking orbit, requires a final rocket burnout velocity of 36,889 fps, relative to the Earth. The velocity vector must nominally be perpendicular to the radius from the Sun, through the center of the Earth and parallel to the Earth's direction around the Sun. Since our circular parking orbit velocity is already 25,155 fps, we need therefore only add a velocity increment (delta V) of 11,734 fps with our auxiliary rocket stage burn.
Our launch vehicle has already delivered us to our 250 mile LEO. The auxiliary stage now fires, imparting its impulsive velocity change of 11,734 fps to push us out of the circular orbit onto our required hyperbolic orbit from which it escapes the Earth altogether. The journey would take ~259 days to travel the 180 degrees around the Sun and arrive in the vicinity of Mars. Using the more exact perihelion and aphelion distances for Mars, ~128,416,000 miles and ~154,868,000 miles, and ~93,000,000 miles for the Earth's distance from the sun, the Hohmann Orbit Transfer trip times would be ~238 days and ~281 days respectively. Obviously, the phasing of the planets is important for this type of transfer and the launch must occur at a carefully predetermined time such that both Mars, and the probe, will reach the heliocentric rendezvous vicinity at the same time.
The disadvantage of this type of trajectory is that when the spacecraft has reached Mars, the earth is too far past the position from which a reverse Hohmann Transfer trajectory could return the spacecraft back to earth. So the spacecraft would either stay in this elliptical trajectory until such time as the earth might meet up with it in the future or it could be placed into a permanent orbit around Mars. Obviously such a mission would be unmanned.
If it was desirable to place a manned spacecraft into a Martian orbit after a Hohmann Transfer trip to Mars, and eventually return it to Earth, they would have to remain there for 450 days until the planets were in the exact reverse configuration that would allow a return Hohmann Transfer trajectory back to Earth. Needless to say, such a mission would require a huge amount of life support equipment and supplies and last a total of 970 days, or 2 years, 7 months and 24 days.
If bigger rockets were available to deliver the spacecraft out of earth orbit, it could reach Mars in shorter times during what is called opposition class missions. The planets are considered in opposition when the Sun, Earth, and Mars are all on a straight line and the Sun and Mars are on opposite sides of the Earth. One version of such a mission could take a total of 240 days. The outbound trip time would take 120 days with the spacecraft reaching Mars at opposition, passing Mars by and returning to Earth in the reverse of the trajectory it took to get to Mars in the first place. This would take a bigger delivery rocket than that required to send the spacecraft on a Hohmann Transfer trajectory. Another version, also taking a total of 240 days, is one where the spacecraft goes into Martian orbit for 30 days and then returns to Earth. The outbound trip in this case would require 105 days and the spacecraft would reach Mars 15 days before opposition, go into orbit around Mars, explore for 30 days until 15 days after opposition, and then depart Mars on a 105 day return trip. This mission, though taking the same 240 days to complete, would require a still larger upper stage delivery rocket as the trip time is further reduced to 105 days. Clearly, there are other, more expensive, choices of departure velocities for even shorter trip times. The choice is primarily a function of how much money one is willing to spend on a launch vehicle and the launch site hardware. |
About This Chapter
Logical Problem Solving in Mathematics - Chapter Summary
If you are in need of a resource that can help you better understand logical problem solving in mathematics, you've reached the right place. Our lessons provide in-depth analyses of this subject by closely examining connective reasoning, the three-way principle, conditional statements and more. By the end of this chapter, you will be able to do the following:
- Describe logic and critical thinking in mathematics
- Discuss conditional statements in math
- Differentiate between inductive, deductive and connective reasoning
- Explain Polya's Four-Step Problem-Solving Process
- List and describe the mathematical principles for problem solving
- Define the three-way principle of mathematics
- Use estimation to solve math problems
- Solve problems using mathematical models
The lessons in this chapter are available as short videos that average about 8 minutes and feature clickable timelines you can use to skip to key topics. They are also accessible as full transcripts you can read online or print to review offline. With each lesson is a short quiz designed to check your knowledge of its contents. To get a comprehensive review of this chapter, be sure to take our practice exam.
1. Critical Thinking and Logic in Mathematics
Logic has its own unique language and way of defining what is true and false. Watch this video lesson to learn how you can critically think in the language of logic while working with math.
2. Conditional Statements in Math
Sometimes, what is true in the mathematical world of logic is false in the real world. Check out this lesson to learn how to identify conditional statements and how you can differentiate between what is logically true and what is true in reality.
3. Reasoning in Mathematics: Inductive and Deductive Reasoning
Many people think that deductive and inductive reasoning are the same thing. It is assumed these words are synonymous. They are not. This lesson reveals the reality of these two types of reasoning.
4. Reasoning in Mathematics: Connective Reasoning
Connective reasoning is reasoning that has an operation, or a way to connect two phrases. The five main logic connectives will be reviewed in this lesson.
5. Polya's Four-Step Problem-Solving Process
Problem solving can be a problem. Any problem is solved easier with an action plan. Polya's 4-Step Problem-Solving Process is discussed in this lesson to help students develop an action plan for addressing problems.
6. Mathematical Principles for Problem Solving
Solving problems is not just a simple, straightforward process. There are a few principles that can help you as you approach any problem solving scenarios. This lesson covers those principles with examples.
7. The Three-Way Principle of Mathematics
What methods are there to solve and understand mathematical problems? This lesson will review three methods to understand mathematical problems (verbal, graphical, and by example). Each will be illustrated with examples.
8. Solving Mathematical Problems Using Estimation
Estimating is a method of calculating a result that is close to, but not exactly, the correct answer to a math problem. Why would you ever need to do this? This lesson reviews estimating and answers the question as to why you would do it.
9. Using Mathematical Models to Solve Problems
Mathematical modeling simply refers to the creation of mathematical formulas to represent a real world problem in mathematical terms. This lesson reviews the creation and pitfalls of mathematical models.
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Other chapters within the WEST-B Reading, Writing & Mathematics: Practice & Study Guide course
- Finding the Main Idea of Reading Passages
- Understanding Reading Selections
- Evaluating Written Arguments
- Point of View, Tone & Meaning
- Figurative Language & Expressions
- Audience & Argument in Written Communication
- Organizing an Essay
- Essay Revision Strategies
- Parts of Speech & Writing Conventions
- Grammar & Sentence Structure
- Working with Fractions & Mixed Numbers
- Decimals, Percents & Operations
- Measurement Concepts & Application
- Basic Geometry Overview
- Properties & Applications of Triangles in Geometry
- Measuring Closed Figures & 3D Shapes
- Probability, Statistics & Evaluating Data
- Sequences & Algebraic Expressions
- Linear Equation Basics
- Properties of Equality
- WEST-B Reading, Writing & Mathematics Flashcards |
Food and Drug Administration
VERSION NO.: 1.4
- 4.2.1 Accuracy, Precision, and Uncertainty
- 4.2.2 Error and Deviation; Mean and Standard Deviation
- 4.2.3 Random and Determinate Error
- 4.2.4 The Normal Distribution
- 4.2.5 Confidence Intervals
- 4.2.6 Populations and Samples: Student's t Distribution
- 4.2.7 References
Statistical procedures used to describe measurements of samples in the ORA laboratory allow regulatory decisions to be made in as unbiased manner as possible. The following are numerically descriptive measures commonly used in ORA laboratories.
The accuracy of a measurement describes the difference between the measured value and the true value. Accuracy is said to be high or low depending on whether the measured value is near to, or distant from, the true value. Precision is concerned with the differences in results for a set of measurements, regardless of the accuracy. Applied to an analytical method as used in an ORA laboratory, a highly precise method is one in which repeated application of the method on a sample will give results which agree closely with one another. Precision is related to uncertainty: a series of measurements with high precision will have low uncertainty and vice versa. Terms such as accuracy, precision, and uncertainty are not mathematically defined quantities but are useful concepts in understanding the statistical treatment of data. Exact mathematical expressions of accuracy and precision (error and deviation), will be defined in the next section.
As an example of these terms, consider shooting arrows at a target, where the "bull's eye" is considered the true value. An archer with high precision (low uncertainty) but low accuracy will produce a tightly clustered pattern outside the bull's eye; if low precision (high uncertainty) and low accuracy, the pattern will be random rather than clustered, with the bull's eye being hit only by chance. The best situation is high accuracy and high precision: in this case a tight cluster is found in the bull's eye area. This example illustrates another important concept: accuracy and precision depend on both the bow and arrow, and the archer. Applied to a laboratory procedure, this means that the reliability of results depends on both the apparatus/instruments used and the analyst. It is extremely important to have a well trained analyst who understands the method, applies it with care (for example by careful weighing and dilution), and uses a calibrated instrument (demonstrated to be operating reliably). Without all of these components in place, it is difficult to obtain the reliable results needed for regulatory analysis.
The concepts of accuracy and precision can be put on a mathematical basis by defining equivalent terms: error and deviation. This will allow the understanding of somewhat more complicated statistical formulations used commonly in the ORA laboratory.
If a set of N replicate measurements x1, x2, x3,…,xn , were made (examples: weighing a vial N times, determining HPLC peak area of N injections from a single solution, measuring the height of a can N times, …), then:
Ei = xi - μ
The definition of error often has little immediate practical application, since in many cases μ, the true value, may not be known. However, the process of calibration against a known value (such as a chemical or physical standard) will help to minimize error by giving us a known value with which to compare an unknown.
The deviation, a measure of precision, is calculated without reference to the true value, but instead is related to the mean of a set of measurements. The mean is defined by:
Note: this is the arithmetic mean of a set of observations. There are other types of mean which can be calculated, such as the geometric mean (see the section on "Application of Statistics to Microbiology" below), which may be more accurate in special situations.
Then, the deviation, di, for each measurement is defined by:
di = xi - X
Using the example of the archer shooting arrows at a target, the deviation for each arrow's position is the distance from the arrow's position to the calculated mean of all of the arrow's positions.
Finally, the expression of deviation most useful in many ORA laboratory applications is s, the standard deviation:
where s = standard deviation, and other terms are as previously defined.
The standard deviation is then a measure of precision of a set of measurements, but has no relationship to the accuracy. The standard deviation may also be expressed in relative terms, as the relative standard deviation, or RSD:
Whereas the standard deviation has the same units as the measurement, the RSD is dimensionless, and expressed as a percentage of the mean.
Standard deviation as defined above is the correct choice when we have a sample drawn from a larger population. This is almost always the case in the ORA laboratory: the sample which has been collected is assumed to be "representative" of the larger population (for example, a batch of tablets, lot of canned goods, field of wheat) from which it has been taken. As it is taken through analytical steps in the laboratory (by subsampling, compositing, diluting, etc.) the representative characteristic of the sample is maintained.
If the entire population is known for measurement, the standard deviation s is redefined as σ, the population standard deviation. The formula for σ differs from that of s in that (N-1) in the denominator is replaced by N. The testing of an entire population would be a rare circumstance in the ORA laboratory, but may be useful in a research project.
Statistical parameters such as mean and standard deviation are easily calculated today using calculators and spreadsheet formulas. Although this is convenient, the analyst should not forget how these parameters are derived.
Recall the definition of error in section 4.2.2 above. Errors in measurement are often divided into two classes: determinate error and non-determinate error. The latter is also termed random error. Both types of error can arise from either the analyst or the instruments and apparatus used, and both need to be minimized to obtain the best measurement, that with the smallest error.
Determinate error is error that remains fixed throughout a set of replicate measurements. Determinate error can often be corrected if it is recognized. Examples include correcting titration results against a blank, improving a chromatographic procedure so that a co-eluting peak is separated from the peak of interest, or calibrating a balance against a NIST-traceable standard. In fact, the purpose of most instrument calibrations is to reduce or eliminate determinate error. Using the example of the archer shooting arrows at a target, calibration of the sights of the bow would decrease the error, leading to hitting the bull's eye.
Random error is error that varies from one measurement to another in an unpredictable way in a set of measurements. Examples might include variations in diluting to the mark during volumetric procedures, fluctuations in an LC detector baseline over time, or placing an object to be weighed at different positions on the balance pan. Random errors are often a matter of analytical technique, and the experienced analyst, who takes care in critical analytical operations, will usually obtain more accurate results.
In the introduction to this chapter, it was briefly mentioned that statistics is derived from the mathematical theory of probability. This relationship can be seen when we consider probability distribution functions, of which the normal distribution function is an important example. The normal distribution curve (or function) is of great value in aiding understanding of measurement statistics, and to interpret results of measurements. Although a detailed explanation is outside the scope of this chapter, a brief explanation will be beneficial. The normal distribution curve describes how the results of a set of measurements are distributed as to frequency; assuming only random errors are made. It describes the probability of obtaining a measurement within a specified range of values. It is assumed here that the values measured (i.e. variables) may vary continuously rather than take on discrete values (the Poisson distribution, applicable to radioactive decay is an example of a discrete probability distribution function; see discussion under "Statistics Applied to Radioactivity"). The normal distribution should be at least somewhat familiar to most analysts as the "bell curve" or Gaussian curve. The curve can be defined with just two statistical parameters that have been discussed: the true value of the measured quantity, μ, and the true standard deviation, σ. It is of the form:
An example of two normal curves with the same true value, μ, but two different values of σ is shown below (this was calculated using an Excel ® spreadsheet, using the formula above and an array of x values):
Some properties of the normal distribution curve that are evident by inspection of the graph and mathematical function above go far in explaining the properties of measurements in the laboratory:
- In the absence of determinate errors, the measurement with the most probable value will be the true value, μ.
- Errors (i.e. x-μ), as defined previously, are distributed symmetrically on either side of the true value, μ; errors greater than the mean are equally as likely as errors below the mean.
- Large errors are less likely to occur than small errors.
- The curve never reaches the y-axis but approaches it asymptotically: there is a finite probability of a measurement having any value.
- The probability of a measurement being the true value increases as the standard deviation decreases.
The confidence interval of a measurement or set of measurements is the range of values that the measurement may take with a stated level of uncertainty. Although confidence intervals may be defined for any probability distribution function, the normal distribution function illustrates the concept well.
Approximately 68% of the area under the normal distribution curve is included within ±1 standard deviation of the mean. This implies that, for a series of replicate measurements, 68% will fall within ±1 standard deviation of the true mean. Likewise, 95% of the area under the normal distribution curve is found within about ± 2σ (to be precise, 1.96 σ), and approximately 99.7% of the area of the curve is included within a range of the mean ±3σ. A 95% confidence interval for a series of measurements, therefore, is that which includes the mean ± 2σ. An example of the application of confidence limits is in the preparation of control charts, discussed in Section 7.6 below.
In the above discussion, we are using the true standard deviation, σ (i.e. the population standard deviation). In most real life situations, we do not know the true value of σ. In the ORA laboratory, we are generally working with a small sample which is assumed to be representative of the population of interest (for example, a batch of tablets, a tanker of milk). In this case, we can only calculate the sample standard deviation, s, from a series of measurements. In this case, s is an estimate of σ, and confidence limits need to be expanded by a factor, t, to account for this additional uncertainty. The distribution of t is called the Student's t Distribution. Further discussion is beyond the scope of this chapter, but tables of t values, which depend on both the confidence limit desired and the number of measurements made, are widely published.
The following are general references on statistics and treatment of data that may be useful for the ORA Laboratory:
- Dowdy, S., Wearden, S. (1991). Statistics for research (2nd ed.). New York: John Wiley & Sons.
- Garfield, F.M. (1991). Quality assurance principles for analytical laboratories. Gaithersburg, MD: Association of Official Analytical Chemists.
- Taylor, J. K. (1985). Handbook for SRM users (NBS Special Publication 260-100). Gaithersburg, MD: National Institute for Standards and Technology. |
EI Machine to Profile Golf Shafts
Dave Tutelman -- November 14, 2008
|The engineering description of a flex profile is the
variation of EI
along the shaft. EI is a structural term, the abbreviation of "E times
This is different from the way clubfitters think of shaft profiles.
When you use a frequency meter or a NeuFinder to measure a profile,
each measurement is a weighted average of stiffness over a
section of the shaft from the tip to some some point X, and you vary X
to make a
graph of the profile. EI measures the local stiffness at X, not an average from X to the tip.
- E stands for
of elasticity", a measure of the stiffness of the material. In
particular, it says how much force it takes to stretch the material a
given amount. Steel would have a very high E, and rubber a very low E.
(Do not confuse elasticity with strength. Strength is the force it
takes to damage the material, not to stretch it.)
- I stands for
moment of inertia", a measure of stiffness of the cross sectional geometry of the
shaft. Thicker walls or a bigger diameter make for a higher I.
How it works
support a beam (say, a golf shaft) at two points and apply a
force halfway between those points, you will deflect (bend) the beam.
It is pretty easy to calculate how much the
beam flexes. The well-known formula for deflection at the middle, where the force is applied,
So the way to measure the EI along a shaft is to support part
of it between two points and do one of two things:
- y is the deflection
- F is the force
- L is the length between the supports
you compute EI by solving the equation above:
- Either apply a force in the middle between the
measure the deflection, or
- Deflect the middle a known amount, and measure the
force required to do it.
What is an EI profile? The
stiffness of a golf shaft varies along its length; the
EI can easily be 3-4
times higher at the butt than at the
tip. An EI profile is a graph of the EI over the length of the shaft.
The way you measure it is to go through the steps above (apply force,
measure deflection, and compute EI) for a series of points along the
length of the shaft, then make a graph of those measurements.
formula works well if the stiffness doesn't vary much over the length
shaft between the
supports. But the EI may change quickly enough that it can vary
significantly even in the short distance between the supports. If EI
varies too much between the supports, the formula loses accuracy. In
the diagram, the close-together (green) set of supports does not see
too much change in EI, while the farther-apart (blue) pair of supports
sees a substantial change. If we consider the measured EI as the EI
halfway between the supports, the green is obviously more accurate.
So the trick is to find a
combination of a force and a distance between supports so that:
Obviously these criteria are in conflict. The art of designing an EI
machine is finding a good tradeoff of distance and force.
- You are able to deflect the shaft enough to measure
deflection and force with some precision (implies a big force
or a big distance).
- The distance is small enough so that the EI is fairly
constant between the supports (implies a small distance).
- You don't damage the
shaft (implies a small force).
Description of my EI machine
Here are a couple of pictures of a machine I made to measure the EI
profile of golf shafts.
The first picture is an overview of the machine. What you see is an
orange shaft resting on two supports 11" apart. The shaft is pre-loaded
with a small weight hanging on it in the middle of the 11"
span. There is also a 15-pound weight hooked to a storage loop on the machine;
this picture the weight is not loading the shaft, but just waiting to be used.
second picture is a
closeup of the business end of the measurement. It shows:
- The hook of the pre-load weight assembly draped over
shaft, in the middle of the 11" span.
- The 15-pound weight now hooked to the bottom of the
weight. The shaft is now being bent by 15 more pounds than it was
- A dial indicator measuring exactly how much the shaft
deflected by the additional 15 pounds. The probe of the dial indicator
rests on a flat spot machined into the top of the hook.
- A ruler to indicate the position on the shaft
reading. The measured distance is that from the weight (at the middle
of the span) to the tip of the shaft.
pictures were taken, I have replaced the dial indicator with a digital
- Better precision. The dial
reads to .001", and
the digital reads to .0005". Precision of this magnitude is indeed an
issue, because 15 pounds only flexes the butt of the shaft .015" to
.020" with an
- No arithmetic. The digital
a zero/tare button.
The precision issue
As noted back in the first section, the trick of designing a good EI
meter is to balance the need for:
don't have a neat solution to the problem. This machine represents a
tradeoff, with all parameters in a very workable range but certainly
not ideal. A distance of 11" allows an EI variation of over 50% in many
shafts. A force of 15 pounds is not really close to the damage point.
It was chosen because a machine that applied force with removable
weights was easy to build and use; a much larger force would have made
machine difficult to use. But, even if there were a different mechanism
to apply the force, I would not want to use a force many times larger,
because damage to the shaft would be a definite possibility.
- A small distance between the supports (to
keep EI constant over the span).
- A large distance between the supports (for
- A small force (to avoid damaging the shaft).
- A large force (for measurement precision).
look at what sort of precision we can expect from the machine. Here is
a sample EI profile measured using the EI machine. In this graph EI is
in units of pound inches squared. (No, not pounds per square inch; it's
multiplication, not division.)
The points on the graph are the result of solving the original equation. The solution
|15 * 113
We measured y at 5" intervals, and
computed EI using the formula. Simple! But perhaps
not very precise.
The values of y that we measured ranged from .0183 (near the butt we
get very small deflections, because the butt is stiff) to .0562 (near
the tip). But bear in mind that the smallest distance we can measure is
only .0005, and that is with the digital indicator; the dial indicator
has a resolution of .001.
That means that our measurements cannot be more precise than the
resolution, and that is .0005. So the precision of the measurements
One percent is a pretty good resolution. Three percent is probably good
enough for profiling, but not good enough for shaft matching nor
quality control of shafts. And stiffer shafts will show even smaller
deflections, meaning that the precision can be as coarse as 5% or 6%.
Again, it will demonstrate the general shape of a profile, but
you would not want to use the measurement for anything else.
I have several ways in mind that the precision can be increased. But
the existing instrument gives profile shapes, and that is all I intend
to use it for. My frequency meter and NF-4 are quite sufficient to do
matching, and are more convenient for profiling. I am exploring a
computer algorithm from M. Brillouette ("On Measuring the Flexural
Rigidity Distribution of Golf Shafts", Science and Golf IV, 2002) to
convert cantilever measurements (like frequency or NF-4 measurements)
to EI profiles. If I can mechanize it with an Excel spreadsheet, and if
my measurements prove its value, then I can use my NF-4 to yield EI profiles as well. If so, I will probably abandon my EI
machine as not being worth the lab space.
Precision vs Accuracy
In another article, I
point out the difference between precision and accuracy. The EI machine
is an extreme example, because there is actually a tradeoff between
precision and accuracy. Most of the simple things you can do to increase accuracy will decrease precision.
We have seen that there is an inherent inaccuracy due to the 11"
spacing between the supports. The EI value can change by as much as 50%
over that distance. So the measurement is not really the EI at the
center of measurement, but rather a weighted average IE over the 11"
span. You can increase the accuracy by decreasing the distance between
Let's do that and see what happens. Let's look at the measurement at
31" from the tip, where we measured the deflection at .0183", which
computed to an EI of 22700 pound inches squared. We'll cut the span in
half, from 11" to 5½". According to the equation, the new deflection
This is eight times smaller than it was with a span of 11". (Not
surprising. Deflection depends on the cube of length. We divided the
length by 2, and 2 cubed is 8.) So what happened to the precision? It
|15 * 5.53
48 * 22700
| = .0022"
This is really bad precision! The result of improving
accuracy by a factor of 2 was that precision went to hell in a
EI machine shows a shaft's flex profile as the stiffness at the point
where the measurement is taken. That may or may not be the best way for
custom clubmakers to look at a profile, but it definitely is the way
shaft design engineers see profiles.
A simple EI machine is
easy to build and easy to understand. It is much harder to build one
that combines precision and accuracy. Precision demands larger
deflections, while accuracy demands a smaller span between the supports
-- which are in conflict with each other..
like to thank Don Johnson of BTM Clubs, who shared photos and details
of the EI machine that he built. Knowing what he did and what gave him
problems saved me a lot of time and resulted in a simpler design from
the outset. Thanks, Don.
This article is based on the "description" section, which was originally published in April 2006.
Last modified - Nov 15, 2008 |
Sal factors -4t^2-12t-9 as -1(2t+3)^2. Created by Sal Khan and Monterey Institute for Technology and Education.
Want to join the conversation?
- why can -4t^2-12t-9 only be solved by (a+b)(a+b)=a^2+2ab+b^2 and not by x^2+(a+b)x+ab?Thank you!(8 votes)
- because x^2+(a+b)x+ab has a constant 1 as leading coefficient.
while -4t^2-12t-9 has a -4 as leading coefficient.
we will need something like this to solve -4t^2-12t-9 :
(ax + b) (ax + b) = a^2x^2 + 2abx + b^2(5 votes)
- What is the exact meaning of Binomial?(5 votes)
- Why is this type of a quadratic expression called a perfect square, when only the term and the last term are perfect squares?(3 votes)
- It's called a perfect square trinomial because it is created from squaring a binomial.
(a+b)^2 = a^2+2ab+b^2(3 votes)
- Why did he took the negative one out at the very beginning?(2 votes)
- It is much easier to factor trinomials when the leading coefficient is a positive number. It is also easier to see that the trinomial is a perfect square if the leading coefficient is a positive number. If you look at -4t^2 and -9, you aren't going to recognize them as perfect squares because a perfect square will always be positive. This is why Sal would have factored out the -1.
Hope this helps.(5 votes)
- The answer could also be (-2t-3)(2t+3) ? But its not a perfect square...?(4 votes)
- It is the same thing as (-1)(2t+3)^2 because all you did in that factorization is multiply one of the binomials by -1.(2 votes)
- At time2:50in the video, he says that we know the 3 is positive because if it were negative, we would get -12t. Since the 12t is positive, we know that the 3 is positive. But since we square rooted 2t (the number the multiply 3 with to get 12t), 2t could also be positive or negative. So, wouldn't it also be true if both 3 and 2t were negative? -3 * -2t = 12t?(2 votes)
- Does it also work without factoring out the negative one? i got (2t-3)^2(1 vote)
- Without removing the negative 1, the trinomial is not a perfect square. So, yes, you need to remove the negative one.
If you multiply your factors, you will find that they actually don't create: -4t^2 - 12t -9.
(2t-3)^2 = 4t^2 -6t -6t +9 = 4t^2 -12t +9
Your results have sign errors on the 1st and last term.
Hope this helps.(3 votes)
- If the "perfect square" method doesn't work, then does it mean that the trinomial is infactorable?(2 votes)
- No it doesn't. You could always use long division method and the Quadratic Formula. http://www.regentsprep.org/regents/math/algtrig/ate3/QuadLe21.gif(1 vote)
- If the 2ab term is positive, can't both a and b be negative, as well?(2 votes)
- Yes, they can! Because if both a and b are negative and are multiplied together, then the negatives cancel out. Also in the a^2 and b^2 part, if they were both negative, then when they were squared the negatives would again cancel out. Hope this helps! (-:(1 vote)
- I know that "perfect square type polynomials" follow (Ax+B)^2 form. But do they also follow the (Ax-B)^2 form?(2 votes)
We need to factor negative 4t squared minus 12t minus 9. And a good place to start is to say, well, are there any common factors for all of these terms? When you look at them, well these first two are divisible by 4, these last 2 are divisible by 3, but not all of them are divisible any one number. Will, but you could factor out a negative 1, but even if you factor out a negative 1-- so you say this is the same thing as negative 1, times positive 4t squared plus 12t plus 9-- you still end up with a non-one coefficient out here and on the second degree term, on the t squared term. So you might want to immediately start grouping this. And if you did factor it by grouping, it would work, you would get the right answer. But there is something about this equation that might pop out at you that might make it a little bit simpler to solve. And to understand that, let's take a little bit of a break here on the right hand side, and just think about what happens if you take a plus b times a plus b, if you just have a binomial squared. Well you have a times a, which is a squared. Then you have a times that b, which is plus ab. Then you have b times a, which is the same thing is ab. And then you have b times b, or you have b squared. And so if you add these middle two terms, right here, you're left with a squared plus 2ab plus b squared. This is the square of a binomial. Now, does this right here, does 4t squared plus 12t plus 9 fit this pattern? Well the 4t squared is a squared. So this right here is a squared. If that is a squared right there, then what does a have to be? If this is a squared, then a would be equal to the square root of this. It would be 2t. And if this is b squared, let me do that in a different color. If this right here is b squared, if the 9 is b squared, right there, then that means that b is equal to 3. It's equal to the positive square root of the 9. Now, this number, right here-- and actually it doesn't have to just be equal to 3, it might have been negative 3 as well. It could be plus or minus 3. But this number here, is it 2 times ab? Right? That's the middle term that we care about. Is it 2 times ab? Well if we multiply 2t times 3, we get 6t. And then if we multiply that times 2, you get 12t. This right here, 12t, is equal to 2 times 2t times 3. It is 2 times ab. And if this was a negative 3, we would look to see if this was a negative 12, but this does work for positive 3. So this it does fit the pattern of a perfect square. This is a square of a binomial. So if you wanted to factor this-- the stuff on the inside, you still have that negative 1 out there, the 4t squared plus 12t plus 9-- you could immediately say, well that's going to be a plus b times a plus b. Or 2t plus 3 times 2t plus 3, or you could just say, it's 2t plus 3 squared. It fits this pattern. And, of course, you can't forget about this negative 1 out here. You could have also solved it by grouping, but this might be a quicker thing to recognize. This is a number squared. That's another number squared. If you take each of those numbers that you're squaring, take their product and multiply it by 2, you have that right there. So this is a perfect square. |
If three positive integers a, b, and c are in the relation ab = c, it is said that a and b are divisors or factors of c, or that a divides c (written a|c), and b divides c. The number c is said to be a multiple of a and a multiple of b.
The number 1 is called the unit, and it is clear that 1 is a divisor of every positive integer. If c can be expressed as a product ab in which a and b are positive integers each greater than 1, then c is called composite. A positive integer neither 1 nor composite is called a prime number. Thus, 2, 3, 5, 7, 11, 13, 17, 19, … are prime numbers. The ancient Greek mathematician Euclid proved in his Elements (c. 300 bc) that there are infinitely many prime numbers.
The fundamental theorem of arithmetic was proved by Gauss in his Disquisitiones Arithmeticae. It states that every composite number can be expressed as a product of prime numbers and that, save for the order in which the factors are written, this representation is unique. Gauss’s theorem follows rather directly from another theorem of Euclid to the effect that if a prime divides a product, then it also divides one of the factors in the product; for this reason the fundamental theorem is sometimes credited to Euclid.
For every finite set a1, a2, …, ak of positive integers, there exists a largest integer that divides each of these numbers, called their greatest common divisor (GCD). If the GCD = 1, the numbers are said to be relatively prime. There also exists a smallest positive integer that is a multiple of each of the numbers, called their least common multiple (LCM).
A systematic method for obtaining the GCD and LCM starts by factoring each ai (where i = 1, 2, …, k) into a product of primes p1, p2, …, ph, with the number of times that each distinct prime occurs indicated by qi; thus,
Then the GCD is obtained by multiplying together each prime that occurs in every ai as many times as it occurs the fewest (smallest power) among all of the ai. The LCM is obtained by multiplying together each prime that occurs in any of the ai as many times as it occurs the most (largest power) among all of the ai. An example is easily constructed. Given a1 = 3,000 = 23 × 31 × 53 and a2 = 2,646 = 21 × 33 × 72, the GCD = 21 × 31 = 6 and the LCM = 23 × 33 × 53 × 72 = 1,323,000. When only two numbers are involved, the product of the GCD and the LCM equals the product of the original numbers. (See the table for useful divisibility tests.)
If a and b are two positive integers, with a > b, two whole numbers q and r exist such that a = qb + r, with r less than b. The number q is called the partial quotient (the quotient if r = 0), and r is called the remainder. Using a process known as the Euclidean algorithm, which works because the GCD of a and b is equal to the GCD of b and r, the GCD can be obtained without first factoring the numbers a and b into prime factors. The Euclidean algorithm begins by determining the values of q and r, after which b and r assume the role of a and b and the process repeats until finally the remainder is zero; the last positive remainder is the GCD of the original two numbers. For example, starting with 544 and 119:
- 1. 544 = 4 × 119 + 68;
- 2. 119 = 1 × 68 + 51;
- 3. 68 = 1 × 51 + 17;
- 4. 51 = 3 × 17.
Thus, the GCD of 544 and 119 is 17.
From a less abstract point of view, the notion of division, or of fraction, may also be considered to arise as follows: if the duration of a given process is required to be known to an accuracy of better than one hour, the number of minutes may be specified; or, if the hour is to be retained as the fundamental unit, each minute may be represented by 1/60 or by
In general, the fractional unit 1/d is defined by the property d × 1/d = 1. The number n × 1/d is written n/d and is called a common fraction. It may be considered as the quotient of n divided by d. The number d is called the denominator (it determines the fractional unit or denomination), and n is called the numerator (it enumerates the number of fractional units that are taken). The numerator and denominator together are called the terms of the fraction. A positive fraction n/d is said to be proper if n < d; otherwise it is improper.
The numerator and denominator of a fraction are not unique, since for every positive integer k, the numerator and denominator of a fraction can each simultaneously be multiplied by the integer k without altering the fractional value. Every fraction can be written as the quotient of two relatively prime integers, however. In this form it is said to be in lowest terms.
The integers and fractions constitute what are called the rational numbers. The five fundamental laws stated earlier with regard to the positive integers can be generalized to apply to all rational numbers.
Adding and subtracting fractions
From the definition of fraction it follows that the sum (or difference) of two fractions having the same denominator is another fraction with this denominator, the numerator of which is the sum (or difference) of the numerators of the given fractions. Two fractions having different denominators may be added or subtracted by first reducing them to fractions with the same denominator. Thus, to add a/b and c/d, the LCM of b and d, often called the least common denominator of the fractions, must be determined. It follows that there exist numbers k and l such that kb = ld, and both fractions can be written with this common denominator, so that the sum or difference of the fractions is obtained by the simple operation of adding or subtracting the new numerators and placing the value over the new denominator. |
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- August 30, 2000 at 12:00 am #2167
With so many people getting piercings in different locations on their bodies, I was wondering if this practice still (or ever has) denoted sexual orientation. If piercing does classify sexual orientation, how so? In other words, left ear, nipple, etc. vs. right ear, nipple, etc. And what about facial, tongue and genitalia piercings? Thank you for your response – it’s all rather confusing.
User Detail :Name : Karen R., Gender : F, Sexual Orientation : Straight, Race : White/Caucasian, Religion : Agnostic, Age : 40, City : Westminster, State : CO Country : United States, Occupation : engineer, Education level : 4 Years of College, Social class : Middle class, August 30, 2000 at 12:00 am #30056
As a gay man, my straight friends always told me that if I wanted to have a pierced ear, I had to pierce the right. That way, I could be readily identified as gay and not straight. I guess women needed to know this. Certainly, straight men needed to know this if they felt a strong need to make verbal slurs or worse. Although this was an almost true statement a decade or so ago, it isn’t any longer. Within the gay community, you may pierce left (it used to mean that the man was dominant or ‘top’ regarding his sexual habits in bed) or right (it used to mean subservient or ‘bottom’ regarding his sexual preference in bed). It should be noted that this is not something that is done, either. We don’t tend to play roles. Sex is supposed to be fun and spontaneous. Top, bottom or in between. All is acceptable. I understand that straight sex it the same way, unless you happen to be Catholic. But, I digress. Today, it is acceptable to pierce whatever is desired. I believe this has happened because of a younger generation not so hung up on sexual orientation. Rather, it seems to be about adornment. As for the rest of the body piercing, I’m not sure why people pierce other parts of their bodies. I have absolutely no desire to have a ring in my genitalia. The ear thing was a big-enough deal, and I do so hate pain. Piercing has been done by man since the beginning of time and probably will until the end of time. So, even though piercing certain parts of the body meant something regarding sexual orientation (it probably still does in some circles) in the not-so-distant past, it doesn’t seem to carry the same connotation today. My advice? Pierce it if that’s what you want to do. Enjoy it if it pleases you. There will always be someone who will look down his or her bigoted nose at you and make nasty little comments. It’s yours to do with as you please. Enjoy! I have.
User Detail :Name : Gary, Gender : M, Sexual Orientation : Gay, Race : White/Caucasian, Religion : I believe in a God, Age : 52, City : Alton, State : IL Country : United States, Occupation : Technical Instructor, Education level : 2 Years of College, Social class : Middle class, September 2, 2000 at 12:00 am #39873
My daughter has more than a few piercings on her body. Frankly, I think this is just a way of trying to be ‘different.’ They don’t want to be like everyone else, yet they still follow the crowd. I don’t really like it, but then I hope I haven’t gotten so old that I can’t understand why they press for individuality.
As far as sexual orientation, for a long time, people thought if you had an ear done(can’t remember which), you were gay. Now I think it’s more like wearing a type of clothing. Do you think you look cool? In the last few years this has been generally accepted. Just like you see more and more tattoos. Not only on older people but on teenagers. Like everything else that comes along, it too will fade.
Now as far as tongue piercing, that is so stupid. Want to chip or break your teeth? Also, it is and can be dangerous to your health. Want to talk with a lisp?
Let’s tackle another subject: piercing the genitalia. Who am I to say what turns a person on – though I do not see how it adds to the pleasure of sexual relations. Or is it just done to be a shock factor? Then again, I’m a virgin in this area (piercing, that is). I don’t think it’s that big of a deal, as long as you don’t look like a billboard, and it does not interfere with you health.
I will say this: if I have a prospective employee who comes to me looking for work, like it or not, I do judge on how many piercings they have, and the amount of tattoos. Fair or not…
User Detail :Name : Lindsay H., Gender : F, Sexual Orientation : Straight, Disability : Deaf, Race : White/Caucasian, Age : 49, City : San Antonio, State : TX Country : United States, Occupation : self-employed, Education level : 2 Years of College, Social class : Middle class, September 5, 2000 at 12:00 am #29945
First, concerning which ear to pierce: The RIGHT ear is the ‘gay’ ear. Or rather, it WAS the ‘gay’ ear. Pierced ears for men didn’t start becoming fashionable until the late 1970s at best, and usually men would pierce the left earlobe with an inconspicuous diamond or silver stud. A gold ring was considered somewhat more daring. The left ear was chosen for the same reason boutonnieres, medals or insignias are worn on the left side of the chest: It was just convention, nothing more, nothing less. For gay men who had the courage to let others know, wearing an earring or stud in the RIGHT earlobe was a subtle way of saying, ‘I’m different.’ But this was really very early on. As the years went by and men began getting multiple piercings in both ears, the whole ‘gay ear’ thing went by the wayside. Today many gay couples favor wearing their wedding or committment bands on their right hands rather than their left, pretty much for the same reason. As for nipple piercings, they are much more common among gay men than straight at the moment, although like so many other fashion trends, more and more straight men are starting to catch on. As for myself, I have never been pierced nor inked. It’s my way of expressing my individuality.
User Detail :Name : Chuck A., Gender : M, Sexual Orientation : Gay, Race : White/Caucasian, Age : 41, City : Spring Hill, State : WV Country : United States, Occupation : AIDS Educator/Part-time radio announcer, October 5, 2000 at 12:00 am #37858
Kissing and receiving oral sex from someone with his or her toungue pierced is a very enjoyable experience.
User Detail :Name : Catherine, Gender : F, Sexual Orientation : Lesbian, Race : White/Caucasian, Age : 25, City : Milwaukee, State : WI Country : United States, Education level : 4 Years of College, Social class : Middle class, July 1, 2002 at 12:00 am #36967
I’ve had my nose pierced for about 10 years now, and yet I’m still often approached on the street and asked what my nose rings mean. It may be different for different people or cultures, but for a lot of us, wearing a nose ring simply means that we like the way nose rings look.
User Detail :Name : Hope, Gender : Female, Sexual Orientation : Bisexual, Religion : Jewish, Age : 38, City : Pittsburgh, State : PA Country : United States, Education level : 4 Years of College,
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Having said that, it should be noted that there are some cases where it is simply impossible for the two events to both occur at the same time. The classical approach gives the right result but might require. For classical method based on normal distribution will assign these virtues trade off.
Do i can ascertain values; it can be very notion of probability of behaviorism in this variable: the frequency probability of relying on. Classical Probability Definition and Examples Statistics How To. This perspective has the advantage that it is conceptually simple for many situations.
The core conception of possible outcomes are the characteristics of probability as classical method probability of assigning probabilities will be pursued here it fill in the third step.
Clearly identify the event of interest. 1 A method of assigning probabilities which assumes that the. At the same time, species E and species V intersect with each other for each of attributes. There are three ways to assign probabilities to events classical approach relative-frequency approach subjective approach Details.
Which are idealized zen buddhist monk, and various alternative formalizations of lawhood itself a counterpart understanding of classical probabilities to probability of outcomes is as a large numbers the information.
The Subjective Theory is particularly useful in assigning meaning to the probability of outcomes that in principle can occur only once. All trademarks and registered trademarks appearing on oreilly. FIGURE 41 Probability in the Process of Inferential Statistics Population.
If we want?
Bayes formula be applicable.
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The typical example of classical probability would be a fair dice roll because it is equally probable that you will land on any of the 6 numbers on the die 1 2 3 4 5 or 6.
Toss a method that it does an event has to a regular email.
The consumers are classified according to age and package design preference.
It does not present meet my resume due to. Cambridge Cambridgeshire New York: Cambridge University Press. Growth are correct curriculum and classical method uses training sample and multimedia. There are three ways to assign probabilities to events classical approach relative-frequency approach subjective approach Details.
What is the chance that both cards are aces? The classical probability articles, assigning probabilities in. In many situations, the events of interest have a natural interpretation in numerical terms. This server could have numerous consequences, using value of all that we freely admit that of probability of probability there are.
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The sample space has four outcomes. An experiment contains ____________ elementary statistics and classical method proposed interpretation that are intramural disputes within one. But ads help me very notion of doing a method that an event. What is classical method, assigning meaning to assign a deeper understanding of things. Portico and assign probabilities with our credences, assigning probabilities in.
The complement of F is ___________. Methods of Assigning Probabilities l Classical Probability l. This will always give her a constant since N is always a finite number of experiments. Be defined to decide whether or impossible for classical probability of landing on the degree between the part is also the license.
We might think of various habits coming in different degrees, measuring their various strengths.
What is the sample space?
Little easier to assign a classical. We briefly mention only one, and an event: pearson paired in a mathematical tools to choose a and communication theory; length and made free. Statistics 41 Introduction to Probability Flashcards Quizlet. We can calculate any probability in this scenario if we can determine how many individuals satisfy the event or combination of events.
Section 51 Probability Rules Chapter 5. There are four possible results on the first step, three possible results on the second step, and two possible results on the third step. This chapter introduces the mathematical theory of probability. The probability of their union is less than the sum of their probabilities, unless at least one of the events has probability zero.
How does it work?
Asking for your subjective probability calculus of the notion of two types of the slight weight differences in the length and contain a unifying themes in assigning probability of classical method has the unknown, minus the sensors.
He accepted in principle the possibility of multiple competing theories of probability while expressing several specific reservations about the existing alternative probability interpretation.
Classical Method If an experiment has n possible outcomes the classical method would assign a probability of 1n to each outcome Experiment Rolling a die.
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Test the method of lawhood itself in a pedestrian could be sure we are mutually exclusive if they may go still a on.
What is classical probability theory? Chapter 4 Introduction to Probability Experiments Counting. What is classical method: macmillan and assign this, methods for which we proposed frame with respect to compute various outcomes are subsets of observing heads? Brainscape is a web and mobile study platform that helps you learn things faster.
The problem of the single case is that the finite frequentist fails to see intermediate probabilities in various places where others do. It is thus congenial to those with empiricist scruples.
This criterion requires that there be some method by which, in principle at least, we can ascertain values of probabilities.
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You of classical method probability theory and stir the probability: i want to assign this video is defective, by customers during experimental trials.
VERY common error is incorrectly applying the multiplication rule for independent events covered on the next page.
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In which of chance of assigning numerical quantity that maximizes this.
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The classical interpretation.
So that history of the project of classical method of assigning probability refers to the other differences in a number increased by a random, punishable by carnap.
Finetti, this is the most intuitive definition of probability.
Starting with just three axioms and a few definitions, the mathematical theory develops powerful and beautiful consequences. |
History of maths maths report
Indeed, to understand the history of mathematics in europe, it is necessary to know its history at least in ancient mesopotamia and egypt, in ancient greece, and in . It is meant to be a quiz to some secondary school pupils on two subjects/topics which are maths and history quiz for maths and history, with saved score reports. Some famous faces in the history of maths albert einstein (1879 - 1955) but here's a brief progress report so far: circa 30,000bc . The first of three articles on the history of trigonometry this takes us from the egyptians to early work on trigonometry in china the history of trigonometry- part 1 : nrichmathsorg. The history of mathematics enjoy learning about the history of mathematics with our historical math facts and information read about ancient egyptian mathematics, babylonian mathematics, chinese mathematics, greek mathematics and much more.
The history of mathematics maths in the victorian classroom age 7 to 14 what was it like to learn maths at school in the victorian period we visited the british . My report in history of maths - download as powerpoint presentation (ppt / pptx), pdf file (pdf), text file (txt) or view presentation slides online. Although mathematics history instructors assign their share of rather traditional mathematics homework exercises or problems, many of them provided in the math history texts they use, student presentations and research papers are more common in mathematics history courses than.
History of mathematics maths and politics mathematicians like to think that their field transcends politics, but as this brief history of the international congress of mathematicians shows, international mathematics has always been coloured by world events. A brief history of american k-12 mathematics education 15 kilpatrick maintained in his report, the problem of mathematics in a history of mathematics . The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries the story of maths renaissance mathematics, . Joe noss, interim report, january 2001 joe noss, final report, november 2001 jackie ou, program in mathematics for young scientists, 1999 erik elwood, program .
Browse this collection of report card comments for math these helpful phrases will make grading report cards that much easier history & culture. History of mathematics the number theory, a branch of mathematics, is concerned with the study of the integers, and of the objects and structures that naturally arise published: mon, 02 oct 2017. Arguably the most famous theorem in all of mathematics, the pythagorean theorem has an interesting historyknown to the chinese and the babylonians more than a millennium before pythagoras lived, it is a natural result that has captivated mankind for 3000 years.
Madagascar, math blaster® and other popular jumpstart® titles coming to games portal on verizon, powered by playphone history worksheets history is an . A report by the working group on history of mathematics in the higher education curriculum, may 2012 supported by the maths, stats and or network, as part of the mathematical sciences strand of the national he stem programme, and the british society for the history of mathematics (bshm). Snippets of bits of maths from 500bc up to the present day to adorn your classroom or corridor maths historical timeline 49 7 customer report a problem .
History of maths maths report
List of important mathematicians this is a chronological list of some of the most important mathematicians in history and their major achievments, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged. A time-line for the history of mathematics (many of the early dates are approximates) this work is under constant revision, so come back later please report any errors to me at [email protected] The history of pi david wilson history of mathematics rutgers, spring 2000 throughout the history of mathematics, one of the most enduring challenges has been the calculation of the ratio between a circle's circumference and diameter, which has come to be known by the greek letter pi.
- Maths teachers pd day math3560 history of mathematics students will be asked to combine historical reading with a short presentation and a one to two page .
- Steelman’s (1947) presidential report, manpower for research, articulated the need for successful secondary school mathematics programs that would eventually increase the number of engineers and other highly technically prepared workers needed for a more.
- The story of maths 2008 tv-g 1 season oxford professor marcus du sautoy presents a history of math, from the discovery of decimals in ancient egypt to the great unsolved problems of today.
We are now reaching the end of our journey through the history of fractions the format we know today comes directly from the work of the indian civilisation the success of their way of writing fractions is due to the number system they created which has three main ideas:. A few reports suggest that mathematics is important for its own sake, and that for many people, mathematics is important because it is inherently beautiful and elegant it is generally agreed that mathematics makes an essential contribution to a good rounded education, playing a vital role in our culture and civilisation. This is a program i have been working on in python it is meant to be a quiz to some secondary school pupils on two subjects/topics which are maths and history so far, it allows a user to try th. Name: _____ history of mathematics project math x february 2008 for this project you will research a mathematician there will be some lab time provided for. |
All engineers know Newton’s Laws. A valve closure, pump trip in liquid systems, steam turbine trip in steam systems or any other operational change will propagate pressure waves in the piping system that can generate dynamic loads in pipe legs upwards of 10,000 lbf (50 kN).
Maintaining system safety and function requires a system designed with these transient fluid forces in mind. These forces can be accurately calculated with Newton’s Laws, but simplifying assumptions are often made.
A typical approximation is to consider only pressure forces and assume that fluid friction and momentum effects are negligible. In some cases, forces from this approximation are very close to the true values, which has led to widespread use of pressure-based methods. However, in some typical system configurations, the neglected terms become important, and the pressure-based method dramatically misestimates the true forces.
A complete calculation of Newton’s Laws is impractical without the use of modern computational tools.
Let’s Throw Some Apples
Engineers who design piping need to account for all pipe forces so they can ensure the pipe supports are sufficiently strong, of the right type, and in the right location. ASME B31 code, for example, requires engineers to consider a wide range of factors that relate to pipe structural integrity. Two of those factors are transient pressures and the resulting imbalanced transient forces that result from surge transients in liquid and steam/gas systems.
Surge transients are often called waterhammer in liquid systems and steam hammer in steam systems. The high (acoustic) velocity pressure waves in these applications can cause short duration but high amplitude imbalanced forces – especially in configurations such as pipe runs between elbow pairs. Forces in excess of 10,000 lbf (50 kN) are common.
Fig. 1 depicts a wave propagation at acoustic speed, a, and resulting forces.
a. At the top of Fig. 1 the transient has not started, the pipe run 1-2 is in equilibrium, and no imbalanced force exists.
b. In the center, the wave is initiated at the right from a valve (not shown) which is closed instantly. The wave travels at acoustic velocity, a. When it reaches the pipe run 1-2, an imbalanced force exists that pushes the pipe elbow assembly to the right. This force must be supported by external pipe supports or the pipe elbow assembly will bend and literally move to the right.
c. At the bottom the wave has passed the pipe run 1-2 and the entire elbow assembly. While now at a much higher pressure than at the top, pipe run 1-2 is again in equilibrium and no imbalanced force exists any longer. Note that if the wave reflects from a boundary point on the left, it will travel back through the elbow assembly and exert another imbalanced force.
Take Another Look: It’s more complex than most engineers realize
At first glance this would not seem to be an overly complicated task. But the underlying reality is much more complex than most engineers realize.
Newton’s Second Law of Motion can be employed to assess the maximum forces. However, one first needs a reasonably accurate transient fluid dynamic solution available in order to plug in the pressures and velocities into Newton.
Due to the resulting complexity, engineers have often used a simplified approach they believe to be conservative. Frequently this involves using the Joukowsky Equation or something similar to determine a maximum pressure differential. This pressure differential is then used to estimate a force as shown below. This simplified approach can be called the Endpoint Pressure Method.
Endpoint Pressure Method (simplified approach, from Fig 1b)
Since we know the values of P1 and P2 in Fig 1b, we can write Eq. 1:
While perhaps appropriate for an earlier generation of engineers who designed piping using slide rules, calculators, or even spreadsheets, this simplified approach is fraught with potential error and often badly misestimates the forces. It can even be unconservative and give design loads much too low. And in the incorrect direction!
Waterhammer software such as AFT Impulse™ can be used by engineers to determine transient pressures and velocities for liquid systems. However, engineers often take the resulting software output and attempt to use Newton by only considering pressure forces (e.g., they still use Eq. 1). This approach might seem to be appropriate, but in fact it mistakenly disregards certain parameters in Newton such as forces due to friction and momentum changes in the liquid. While it is true these disregarded parameters are often minimal and can be safely neglected, there are many cases where they are important and cannot be neglected. AFT Impulse performs a complete force balance using all parameters in Newton and is thus able to give a more accurate force prediction. More on this below.
Gas and steam transients often do not generate sufficiently high transient forces to merit detailed simulation. However, there are some applications which do. One such application is steam hammer in large diameter power station piping. Simulating such behavior is significantly more complicated than for liquids. AFT xStream™ is designed to handle such challenging applications and, from what we can tell, it is the only commercially available software that can do this for transient steam flow. Moreover, and similar to AFT Impulse, AFT xStream is designed to account for all parameters in Newton’s Second Law.
So, what is wrong with Eq. 1?
The complete force balance can be determined as follows:
Component Reaction Method (complete method)
where the pipe force from frictional resistance is given by summing the friction in each computing section, 0 to N, as follows:
Eq. 3 is called the Component Reaction Method. Comparing Eq. 3 to Eq. 1, it is clear that Eq. 1 is a special case of Eq. 3. First, the fluid momentum, , can and does change from point 1 to 2 because of the waterhammer wave. Second, there is friction in the pipe. If each of these terms is negligible, then Eq. 1 is a reasonable approximation to the force. Often these terms are negligible. However, there are cases when they are not. The ASME papers explain this.
Further, Eq. 3 is not that difficult to use for liquid surge transients. But for steam transients (or gases in general) it is more complicated to determine the needed parameters. Fortunately, if one draws a different control volume around pipe run 1-2 in Fig. 1, it can be shown that an equivalent way of calculating the force is given by:
Acceleration Reaction Method (complete method)
Eq. 4 sums up the change in acceleration for each computing section, 0 to N. It is therefore called the Acceleration Reaction Method. It is valid for liquids and gases. It can be shown that it is complete and yields the same force predictions as Eq. 3. And when a computational solution is available through a simulation model, it is straightforward to use.
To go more in-depth and learn how to apply Eqs. 3 and 4 to these situations, you can check out:
Accurately Predict Transient Fluid Forces in Piping Systems, Part 1: Fundamentals
Accurately Predict Transient Fluid Forces in Piping Systems, Part 2: Applications
Eqs. 1, 3 and 4 are cross-plotted vs. AFT Impulse results in Fig. 2. Here you can see that the simplified approach in the Eq. 1 Endpoint Pressure Method obtains inaccurate and unconservative force predictions.
What is the Right Way to Bite the Apple?
AFT recommends the use of the Eq. 4 Acceleration Reaction Method for general use by all engineers. It includes all terms in Newton’s Second Law and will not lead engineers astray. The Eq. 3 Component Reaction Method is also complete and safe to use but can be more difficult to calculate in practice.
Thanks to the advancement of technology, it’s not a stretch to complete these calculations efficiently. Using AFT Impulse and AFT xStream are great options to make this an easy task.
To learn more about applying Newton’s Laws to predicting forces in piping systems, and to see how AFT’s transient simulation tools make calculating complete and accurate transient forces practically effortless, you can check out this webinar on Predicting Transient Pipe Forces Caused by Water and Steam Hammer.
*Originally published with ASME |
Margin Of Error As A Percentage
Calculate the margin of error for a 90% confidence level: The critical value is 1.645 (see this video for the calculation) The standard deviation is 0.4 (from the question), but as A larger sample size produces a smaller margin of error, all else remaining equal. and Bradburn N.M. (1982) Asking Questions. How to Compute the Margin of Error The margin of error can be defined by either of the following equations. navigate here
If only those who say customer service is "bad" or "very bad" are asked a follow-up question as to why, the margin of error for that follow-up question will increase because and R.J. Warning: If the sample size is small and the population distribution is not normal, we cannot be confident that the sampling distribution of the statistic will be normal. The amount of precision that can be expected for comparisons between two polls will depend on the details of the specific polls being compared. https://www.isixsigma.com/tools-templates/sampling-data/margin-error-and-confidence-levels-made-simple/
Margin Of Error In Statistics
This is very useful and easy to understand too. Members of the American Association for Public Opinion Research’s Transparency Initiative (including Pew Research Center) are required to disclose how their weighting was performed and whether or not the reported margin In other words, the margin of error is half the width of the confidence interval. The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x
Contents 1 Explanation 2 Concept 2.1 Basic concept 2.2 Calculations assuming random sampling 2.3 Definition 2.4 Different confidence levels 2.5 Maximum and specific margins of error 2.6 Effect of population size Margin of error applies whenever a population is incompletely sampled. pp.63–67. Margin Of Error In Polls Wonnacott (1990).
What is a Survey?. Category: 5 Facts Topics: 2016 Election, Elections and Campaigns, Research Methods, Telephone Survey Methods, Web Survey Methods Share the link: Andrew Mercer is a senior research methodologist at Pew Research Center. The margin of error is a measure of how close the results are likely to be. a fantastic read Harry Contact iSixSigma Get Six Sigma Certified Ask a Question Connect on Twitter Follow @iSixSigma Find us around the web Back to Top © Copyright iSixSigma 2000-2016.
How to Find the Critical Value The critical value is a factor used to compute the margin of error. Margin Of Error Confidence Interval Calculator That's not quite right. But a question: what if I achieved a high response rate and that my survey sample is close to the overall population size? In media reports of poll results, the term usually refers to the maximum margin of error for any percentage from that poll.
Margin Of Error Sample Size
Pollsters disclose a margin of error so that consumers can have an understanding of how much precision they can reasonably expect. If only those who say customer service is "bad" or "very bad" are asked a follow-up question as to why, the margin of error for that follow-up question will increase because Margin Of Error In Statistics It's time for some math. (insert smirk here) The formula that describes the relationship I just mentioned is basically this: The margin of error in a sample = 1 divided by Margin Of Error Calculator It is important that pollsters take the design effect into account when they report the margin of error for a survey.
When estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. check over here Although the statistical calculation is relatively simple – the most advanced math involved is square root – margin of error can most easily be determined using the chart below. All Rights Reserved. or when populations are small as well (e.g., people with a disability)? Margin Of Error Definition
If we use the "relative" definition, then we express this absolute margin of error as a percent of the true value. Margin Of Error Excel Thus if M = .04, the confidence interval is +/- 4%. This is best done as a chain calculation in your calculator, without writing any of the intermediate steps down.
Bruce Drake • 1 month ago Thanks for the heads-up to us.
adult population, the sample size would be about 160 cases if represented proportionately. To compute the margin of error, we need to find the critical value and the standard error of the mean. A researcher surveying customers every six months to understand whether customer service is improving may see the percentage of respondents who say it is "very good" go from 50 percent in Acceptable Margin Of Error Pollsters report the margin of error for an estimate of 50% because it is the most conservative, and for most elections featuring two candidates, the levels of support tend to be
doi:10.2307/2340569. To do that, the pollster needs to have enough women, for example, in the overall sample to ensure a reasonable margin or error among just the women. For this problem, since the sample size is very large, we would have found the same result with a z-score as we found with a t statistic. weblink What a wonderful concept.
Even when we do see large swings in support from one poll to the next, one should exercise caution in accepting them at face value. That means that in order to have a poll with a margin of error of five percent among many different subgroups, a survey will need to include many more than the Back to Top Second example: Click here to view a second video on YouTube showing calculations for a 95% and 99% Confidence Interval. Right?
This makes intuitive sense because when N = n, the sample becomes a census and sampling error becomes moot. Reply dataquestionner Hi! When you do a poll or survey, you're making a very educated guess about what the larger population thinks. But let's talk about what that math represents. |
Applied Psychological Measurement 37(3) 201–225 Ó The Author(s) 2013 Reprints and permissions:
sagepub.com/journalsPermissions.nav DOI: 10.1177/0146621612470210 apm.sagepub.com
The Reliability and Precision of Total Scores and IRT
Estimates as a Function of Polytomous IRT Parameters and Latent Trait Distribution
Steven Andrew Culpepper1
A classic topic in the fields of psychometrics and measurement has been the impact of the num- ber of scale categories on test score reliability. This study builds on previous research by fur- ther articulating the relationship between item response theory (IRT) and classical test theory (CTT). Equations are presented for comparing the reliability and precision of scores within the CTT and IRT frameworks. This study presented new results pertaining to the relative precision (i.e., the test score conditional standard error of measurement for a given trait value) of CTT and IRT, and the new results shed light on the conditions where total scores and IRT estimates are more or less precisely measured. The relative reliability of CTT and IRT scores is examined as a function of item characteristics (e.g., locations, category thresholds, and discriminations) and subject characteristics (e.g., the skewness and kurtosis of the latent distribution). CTT total scores were more reliable when the latent distribution was mismatched with category thresh- olds, but the discrepancy between CTT and IRT declined as the number of scale categories increased. This article also considered the appropriateness of linear approximations of polyto- mous items and presented circumstances where linear approximations are viable. A linear approximation may be appropriate for items with two response options depending on the item discrimination and the match between the item location and latent distribution. However, linear approximations are biased whenever items are located in the tails of the latent distribution and the bias is larger for more discriminating items.
reliability, scale construction, classical test theory, item response theory, polytomous items, information function
The impact of the number of scale categories on reliability is a classic topic in psychology (Symonds, 1924), and an extensive body of research has examined the effect of the number of
1University of Illinois at Urbana–Champaign, USA Corresponding Author:
Steven Andrew Culpepper, University of Illinois at Urbana–Champaign, 116D Illini Hall, MC-374, 725 South Wright Street, Champaign, IL 61820, USA.
scale categories on the corresponding reliability of total scores (x). In fact, researchers exam- ined the impact of the number of scale categories on total score reliability using empirical data (Adelson & McCoach, 2010; Bendig, 1954; Chafouleas, Christ, & Riley-Tillman, 2009;
L. Chang, 1994; Komorita & Graham, 1965; Matell & Jacoby, 1971; Weng, 2004), Monte Carlo simulations (Aguinis, Pierce, & Culpepper, 2009; Bandalos & Enders, 1996; Cicchetti, Shoinralter, & Tyrer, 1985; Enders & Bandalos, 1999; Greer, Dunlap, Hunter, & Berman, 2006; Jenkins & Taber, 1977; Lissitz & Green, 1975), and analytic derivations (Krieg, 1999).
Methodological developments have bridged the concept of reliability between the item response theory (IRT) and classical test theory (CTT) frameworks and discussed concepts that have traditionally been reserved for IRT (e.g., item information functions [IIFs]) within the con- text of CTT. The purpose of this article is to understand the circumstances where researchers should prefer estimating true scores (i.e., u) with test scores derived from IRT (i.e., ^u) versus CTT (i.e., x). This article compares the precision and reliability of ^u and x, and equations are presented for investigating how item and subject characteristics affect the reliability of ^u and x.
Mellenbergh (1996) noted that reliability is a population-dependent quantity that is affected by characteristics of latent distributions, whereas the conditional standard error of measurement (CSEM) quantifies error variance (i.e., precision or the inverse of information) for a specific u value. New equations are presented that compare the relative precision of ^u and x. The results show that IRT and CTT have CSEMs that are roughly the mirror image across values of u. That is, ^u is measured more precisely in portions of the u continuum that include relatively more category thresholds, whereas x tends to be measured more precisely for u values that are further from category thresholds.
It is important to articulate the contributions of this article to existing research. First, method- ological advances concerning dichotomous items established a link between the IRT and CTT frameworks by articulating the reliability of total scores and percentile ranks with corresponding IRT item parameters, such as item difficulty, discrimination, and guessing parameters (Bechger, Maris, Verstralen, & Beguin, 2003; Dimitrov, 2003; Kolen, Zeng, & Hanson, 1996; May &
Nicewander, 1994). Additional research has studied the impact of IRT item parameters on the reliability of gain scores (May & Jackson, 2005), the lower and upper bounds of the IRT relia- bility coefficient (Kim & Feldt, 2010), the reliability of scale scores and performance assess- ments using polytomous IRT (Wang, Kolen, & Harris, 2000), and the reliability of subscores using unidimensional (Haberman, 2008) and multidimensional IRT (Haberman & Sinharary, 2010). Researchers have also found that IRT scores provide more accurate estimates of interac- tion effects within the contexts of analysis of variance (Embretson, 1996) and multiple regres- sion (Kang & Waller, 2005; Morse, Johanson, & Griffeth, 2012). This article offers new information about the connection between the concepts of reliability and precision in CTT and IRT. That is, no study has provided theoretical results about the relative precision of ^u and x for a given u value. The new derivations provide theoretical rationale for circumstances when CTT scores include relatively more or less measurement error than IRT scores. Moreover, no research study has analytically studied the interactive effect of item characteristics and latent distribution shape on the relative reliability of ^u and x. This article uses Fleishman’s (1978) power transfor- mation (PT) method probability density function to study the impact of nonnormal latent distri- butions on the reliability of total scores.
Second, additional research has extended the concept of item and test information to CTT under the assumptions that observed measurements are continuous, rather than polytomous, and u is linearly related to x (Ferrando, 2002, 2009; McDonald, 1982; Mellenbergh, 1996).
Ferrando notes that a linear model provides a good approximation when item discrimination indices are relatively small in value and coarsely measured items have five or more response categories. Moreover, Ferrando and Mellenbergh showed that, for the linear model, the CTT
IIF is horizontal and unrelated to u. However, whenever the relationship between latent and observed total scores is nonlinear (which occurs when items are polytomous), the CTT IIFs are no longer unrelated to u. In fact, this article shows that the conditional standard error of x given u is a downward-facing function where scores near category thresholds have the least amount of precision. The accuracy of a linear relationship is also explored, and the results in this article examine the effect of test characteristics (e.g., item locations and discrimination) and subject characteristics (e.g., latent distribution shape) on the appropriateness of linear approximations of polytomous items.
Third, previous Monte Carlo studies that have studied the relative performance of CTT and IRT estimates are limited by the combination of parameter values and type of reliability stud- ied. For instance, Greer et al. (2006) studied the impact of skewness on coefficient alpha with the constraint that item variances were equal, which may not occur frequently in practice. The results in this article can be used to study any combination of IRT parameter values and latent distribution shape and provide more general results than previous Monte Carlo simulations. In addition, Wang et al. (2000) presented equations for computing the reliability of scale scores from performance assessments using the generalized partial credit model. However, unlike Wang et al., this study presents equations for evaluating how the number of items, number of scale categories, and the shape of the latent distribution affects the reliability of ^u and x.
Furthermore, R code (R Development Core Team, 2010) is available at http://publish.illinois.
edu/sculpepper/, and researchers can use the R code to compute the reliability of ^u and x for different item characteristics and latent distributions. Consequently, this study presents new results and provides applied researchers with guidance for reliably scoring tests in different situations.
This article includes five sections. The first section presents equations for the reliability and precision of scores within the CTT and IRT paradigms and includes new results about the CSEM for CTT. The second section compares IRT and CTT in terms of CSEMs to provide a general understanding of the circumstances researchers should prefer CTT versus IRT scores.
The third section compares the relative reliability of ^u and x for different item (e.g., item loca- tions and number of response categories) and subject distribution characteristics (e.g., the skew- ness and kurtosis of u), and the fourth section examines how item and subject characteristics affect the appropriateness of a linear approximation of polytomous items. The last section dis- cusses the results and provides recommendations and concluding remarks.
Equations for the Reliability of CTT and IRT Estimates of u
Let u represent a latent variable and ui be an observed polytomous response, where i indexes items (i = 1, . . . , I ). The observed polytomous response for item i can be expressed as a function of a true score (Efuijug) and random error (ei), such that ui= Efuijug + ei. That is, the observed uiequals an item true score Efuijug (May & Nicewander, 1994), which is a nonlinear function of u, plus an error, ei. Let j index category thresholds (j = 1, . . . , J ) and J + 1 is the correspond- ing number of categories for item i. That is, j is used to index categories as well (e.g., J + 1 = 4 implies that uihas four categories). This article assumes that researchers code uiusing integers from 1 to J + 1. Several polytomous models exist to describe the relationship between u and the chance that ui equals one of J + 1 categories, which include the graded response (Molenaar, Dolan, & De Boeck, 2012; Muraki, 1990; Samejima, 1969), partial credit (Masters, 1982;
Muraki, 1992, 1993), and rating scale (Andrich, 1978a, 1978b) models. The derivations in this article use Muraki’s (1990) modified graded response model,
Pijðui. jjuÞ =
1, j = 0
1 + exp½aiðu bðicjÞÞ, 0\j J 0, j = J + 1 8<
: , ð1Þ
where biand aiare the item difficulty and discrimination parameters, respectively, and cjis the jth threshold, which is equal for all I items. Equation 1 represents the chance that ui. j given u and the probability that item i equals the jth category is
Pijðui= jjuÞ = Pijðui. jjuÞ Pij + 1ðui. j + 1juÞ: ð2Þ Note that Muraki’s model was chosen because the form of Equation 1 is easier to manipulate analytically and the derivations of expressions for derivatives are less cumbersome.
Muraki’s model assumes that the J cjare constant across items and are equally spaced. The goal of this article is not to estimate abilities or item parameters with Muraki’s model, and these assumptions can be relaxed to evaluate the impact of unequal item thresholds on the reliability and precision of x and ^u. In fact, the new expressions are applicable for any polytomous model, and Muraki’s model is only used for computational examples. Furthermore, the aforementioned models tend to yield scores that are highly correlated (Embretson & Reise, 2000), so we should not anticipate the results would change significantly if the partial credit or rating scale models were used.
The derivations below require the specification of a distribution for u. In this article, u is assumed to follow a Fleishman PT distribution, j(ujΩ), where Ω = (m, s2, k3, k4) to indicate that u has a mean m, variance s2, and skewness and kurtosis of k3 and k4, respectively (j(ujΩ) is discussed in greater detail in the Appendix). Note that any univariate distribution could be chosen for u. One advantage of using Fleishman’s PT distribution is that it is flex- ible enough to explore of how changes in m, s2, k3, and k4 affect the relative reliability of IRT and CTT u estimates. However, as noted by an anonymous reviewer, researchers often set s2= 1 to estimate item discriminations (i.e., the scale indeterminacy problem).
Accordingly, the results in this article also use s2= 1 to understand how manipulating item discriminations affects reliability. Fleishman’s PT distribution does not encompass the uni- verse of all univariate distributions, so future researchers can modify the associated R code to examine the reliability of CTT and IRT estimates when u follows other distributions.
Furthermore, the discussion below provides an argument as to how reliability within the CTT and IRT frameworks is dependent on the match between the shape of j(ujΩ) and the topography of the conditional variance of x and ^u.
Reliability and Precision Within IRT Framework
One of the strengths of IRT over CTT relates to the well-known measure of precision for esti- mated trait scores, ^u. Reliability is specific to a group and is a function of the unconditional standard error of measurement (SEM). CTT has traditionally calculated the SEM for a group of scores, whereas the CSEM is a measure of precision that corresponds to a specific trait level within the IRT framework. The CSEM of ^u is related to the test information function (TIF), which is derived using the concept of Fisher’s information to measure the amount of informa- tion that a single observation provides about u. In fact, the inverse of a TIF indicates the var- iance of ^u for a given u. Previous research (Muraki, 1993; Samejima, 1994) discussed TIFs for polytomous IRT models and noted that the IIF for item i is
Iið Þ = u XJ + 1
j = 1
Pijðui= jjuÞ ∂2
∂u2ln P ijðui= jjuÞ
=XJ + 1
j = 1
Pijðui= jjuÞ∂2Pijðui= jjuÞ
For Muraki’s (1990) modified graded response model, Pij(ui= jju) is a function of Pij(ui. jju), which is a logistic function. The first two derivatives of Pij(ui. jju) are as follows:
∂u = aiPijðui. jjuÞ 1 Ph ijðui. jjuÞi ,
∂u2 = ai
∂u h1 2Pijðui. jjuÞi
= a2iPijðui. jjuÞ 1 Ph ijðui. jjuÞi
1 2Pijðui. jjuÞ
Accordingly, Ii(u) can be computed using the first and second derivatives of the item cate- gory probabilities, Pijðui= jjuÞ, which are as follows:
∂u =∂Pijðui. jjuÞ
∂u ∂Pij + 1ðui. j + 1juÞ
∂u2 =∂2Pijðui. jjuÞ
∂u2 ∂2Pij + 1ðui. j + 1juÞ
The TIF for a test of polytomous items is the sum of the respective IIFs:
i = 1
The relationship between true and observed IRT estimates can be written as ^u = u + e. If u and e are independent, the variance of ^u is the sum of the true and error variances, s2f^ug = s2fug + s2feg. The conditional variance of ^u given u is defined as s2f^ujug = s2fejug = (TIF(u))1. The expected conditional variance of ^u for a specific distribution of u, j(ujΩ), is as follows:
E s 2^uju
s2^ujuj ujΩð Þdu: ð7Þ
The expected reliability of ^u is the well-known ratio of true to observed variance, r^u^u= s2
s2+ E s 2^uju , ð8Þ
where s2is the variance of u specified in j(ujΩ). Clearly, r^u^u is dependent on the characteris- tics of the test (i.e., s2f^ujug) and the distribution of latent scores (i.e., j(ujΩ)).
Reliability and Precision Within CTT Framework
Let x be the total score, or sum, of the I ui, such that ui= Efuijug + ei. For a subject with a given u, uiequals one of J + 1 categories each with probability Pij(ui= jju), so Efuijug is
E uf ijug = XJ + 1
j = 1
jPijðui= jjuÞ: ð9Þ
That is, Efuijug is a weighted average of the category scores (i.e., j = 1 to J + 1) and the chance that subjects with a specific u have an observed score ui. The previous section discussed an expression for the expected conditional variance of ^u as a function of the inverse TIF and distri- bution of u. This section analogously derives an expression for Efs2fxjugg, which is the expected variance of x for a given value of u. An equation for the reliability of x is presented as a function of Efs2fxjugg and the variance of the expected true scores, s2fEfuijugg. This sub- section proceeds by first deriving an expression for the conditional error variance of x, Efs2fxjugg, and then identifies an equation for the true score variance, s2fEfuijugg.
Consider item i where the error is ei= ui Efuijug. Note that ui is a coarse measure of Efuijug in that uiis ordinal and equals one of the J + 1 values, whereas u is measured on an inter- val scale. One immediate observation is that Efeijug = 0. Recall that uiis a polytomous item, so ei= j Efuijug for j = 1 to J + 1. The conditional expectation of the error within item i is
E ef ijug =XJ + 1
j = 1
Pijðui= jjuÞ j E uð f ijugÞ =XJ + 1
j = 1
jPijðui= jjuÞ E uf ijug = 0: ð10Þ
The variability of observed uiaround expected values is determined by the error variance or precision; that is, s2fuijug = Efe2ijug and is defined as
s2fuijug = Enðui E uf ijugÞ2juo
=XJ + 1
j = 1
Pijðui= jjuÞ j E uð f ijugÞ2: ð11Þ
Equation 11 is new to the literature and the following sections compare the properties of s2fuijug with s2f^ujug.
s2fuijug is the conditional variance for a single item and an expression is needed for the con- ditional variance of x. An important observation is that errors within two polytomous items, say, uiand uh(let categories for uhbe indexed by k), are independent whenever the items are locally independent, which assumes that P(ui= j, uh= kju) = Pij(ui= jju)Phk(uh= kju). Specifically, the covariance between ehand eiconditioned on u is
s ef i, ehjug = E ufð i E uf ijugÞ uð h E uf hjugÞjug
=XJ + 1
j = 1
XJ + 1
k = 1
j E uf ijug
ð Þ k E uð f hjugÞP uð i= j, uh= kjuÞ
=XJ + 1
j = 1
j E uf ijug
ð ÞPijðui= jjuÞXJ + 1
k = 1
k E uf hjug
ð ÞPhkðuh= kjuÞ = 0:
The finding that errors within items are independent, if local independence is assumed, is par- ticularly important, because the conditional error variance of the total score, x, given u is sim- ply the sum of the conditional item variances, s2fuijug.
Recall that x = PI
i = 1ui. Equation 9 implies that the expected value of x given u is E xjuf g = XI
i = 1
E uf ijug =XI
i = 1
J + 1
j = 1
jPijðui= jjuÞ: ð13Þ
The conditional variance of x given u is the sum of conditional variances for the I ui, under the assumption of local independence (see Equation 12), which implies that s2fxjug =PI
i = 1s2fuijug. The expected conditional variance for a specific distribution of u is E s 2fxjug
s2fxjugj ujΩð Þdu: ð14Þ
Recall that in IRT the expected variance of maximum likelihood estimates is the expected value of s2f^ujug across the distribution of u. Similarity, the expected conditional variance of x is found by replacing s2f^ujug with s2fxjug. Consequently, researchers can compare the CSEM (i.e., sfxjug and sf^ujug) to understand which values of u are associated with relatively more precision within the CTT and IRT frameworks.
Recall that Efxjug is the expected total score for subjects with a given u and the variance of Efxjug across subjects provides a measure of the amount of true score variance. First, note that the unconditional mean of x is
E xf g = E E xjuf f gg = ð‘
E xjuf gj ujΩð Þdu: ð15Þ
The variance of Efxjug across u is
s2fE xjuf gg = EnðE xjuf g E xf gÞ2o
E xjuf g E xf g
ð Þ2j ujΩð Þdu: ð16Þ
The reliability of x is the ratio of true to observed variance:
rxx= s2fE xjuf gg
s2fE xjuf gg + E sf 2fxjugg: ð17Þ
Factors That Affect the Precision of CTT and IRT Scores
The previous section derived expressions for the CSEMs for x and ^u (i.e., sfxjug and sf^ujug).
The characteristics of polytomous IRT CSEMs are well understood. For example, sf^ujug tends to be smaller at points along the latent continuum where category thresholds are located and sf^ujug declines in regions where more discriminating items are located. In contrast, the expres- sion for sfxjug is new and the purpose of this section is to compare sfxjug with sf^ujug to pro- vide researchers with a conceptual understanding of the testing situations where x may be preferred to ^u in terms of measurement precision.
As an example, consider a test consisting of three items that each have four response cate- gories. Moreover, let a and b be three-dimensional vectors of item discriminations and loca- tions, such that a = (0.5, 1.5, 2.5) and b = (21, 0, 1). Moreover, to simplify this example, let
the category thresholds be equally spaced with values of (21.6, 0, 1.6) units below and above u bi(i.e., the thresholds for the first item are located at 22.6, 21.6, and 0.6 on the u scale).
Figure 1 presents s2fuijug (see Equation 11) and ½Ii(u)1 (see Equation 3) for the three items. The IRT conditional variances exhibit expected behavior. That is,½Ii(u)1 is lowest at the category thresholds and½I3(u)1 is generally smaller than the other two items because the third item is more discriminating. One additional nuance is that½I3(u)1 has local minima at the category thresholds, whereas Items 1 and 2 have inverse information functions that appear smoother. Stated differently, in IRT, items with larger discriminations tend to have IIFs with more topography in regions near category thresholds.
As expected, s2fuijug is smaller for more discriminating items; however, the general beha- vior of s2fuijug is different from ½Ii(u)1. Namely, s2fuijug tends to be a downward-facing function where measurement error is largest at category thresholds. For example, s2fu3jug tends to be the smallest of the three items, but s2fu3jug has maxima near the category thresholds, which differs from IRT where measurements are more precise at category thresholds. Under a Figure 1. Conditional variances of CTT and IRT scored items for a hypothetical three-item test with four response options and equally spaced thresholds, cj= (21.6, 0, 1.6).
Note: CTT = classical test theory; IRT = item response theory.
CTT framework, s2fuijug is smallest for either more extreme u or for u values that lie between category thresholds. In short, s2fuijug is roughly the mirror image of ½Ii(u)1, because½Ii(u)1 tends to be lower in segments of the u continuum where s2fuijug is larger and vice versa.
Figure 2 includes the same items discussed in Figure 1, but with the exception that the thresholds are no longer equally spaced. Figure 2 shows that s2fuijug and ½Ii(u)1 respond inversely to unequal item thresholds. For instance, s2fuijug tends to increase in portions of the latent continuum that include more item thresholds, whereas½Ii(u)1 is smaller in segments of the latent continuum where there are more item thresholds and increases wherever there are fewer thresholds.
Figure 3 demonstrates CTT and IRT test CSEMs along with error bars around conditional expected values, Efxjug and Ef^ujug. The top row of Figure 3 illustrates the CTT and IRT CSEMs for x and ^u, respectively, for the hypothetical three-item test. Note that the vertical lines in the top row of panels in Figure 3 indicate category thresholds for Items 1, 2, and 3. Figure 3 shows that s2fxjug is larger for u values that are near category thresholds, whereas s2f^ujug is smaller at points on the latent continuum that have more thresholds and more discriminating Figure 2. Conditional variances of a CTT and IRT scored items for a hypothetical three-item test with four response options and unequally spaced thresholds, cj= (21.6, 0, 1.0).
Note: CTT = classical test theory; IRT = item response theory.
items. For the three-item test, s2f^ujug appears more responsive to item discriminations than s2fxjug as indicated by the fact that the slope of s2f^ujug is steeper than s2fxjug in the u range measured by Item 1.
The second row of panels in Figure 3 plots Efxjug and Ef^ujug as well as 62 times the CSEMs. As discussed previously, Efxjug is a more accurate indicator of u for more extreme values on the u continuum, whereas ^u is a better indicator in the range where items and cate- gories are located. The overall reliability of x and ^u is dependent on the shape and location of the latent distribution. For example, if u;N (0, 1) most subjects lie in the middle range of the latent continuum where x is less precisely measured relative to ^u. In fact, the differences in s2fxjug and s2f^ujug contribute to ^u being significantly more reliable than x (i.e., 0.65 vs. 0.49).
Certainly, rxxand r^u^uwill change depending on the shape and location of the u distribution.
Figure 3. Test score CTT and IRT conditional standard error of measurement and expected value plots with 62 error curves for a hypothetical three-item test with four response categories.
Note: CTT = classical test theory; IRT = item response theory. Thresholds are indicated by dashed-vertical lines for Item 1 (b1= 21, a1= 0.5), Item 2 (b2= 0, a2= 1.5), and Item 3 (b3= 1, a3= 2.5) and cj= (21.6, 0, 1.6). rxx. and r^u^u
were calculated under the assumption that u;N(0, 1).
The Reliability of x and ^u for Item and Subject Characteristics
Figure 3 included results for a simple example to demonstrate the theoretical differences between s2fxjug and s2f^ujug, which are useful pieces of information for developing tests and instruments. That is, s2fxjug and s2f^ujug provide applied researchers with an understanding of ranges of u values that are best measured with x or ^u. Moreover, s2fxjug and s2f^ujug offer researchers information about which measurement framework is most beneficial for various item characteristics and subject populations.
Equations 8 and 17 were used to compare the reliability of x and ^u as a function of the num- ber of scale categories, purpose of measurement (i.e., item locations dispersed along the conti- nuum or clustered in a given region), and u distribution shape. More specifically, Figure 4 includes r^u^u and rxx across scale categories (i.e., 2–10 response options) for three types of item locations and four types of distributions for u. Item discriminations and test length were not manipulated and were fixed at 1.25 and 10, respectively, It is well known that increasing either item discrimination or test length increases reliability, and these parameters were held constant to focus on the other parameters.
To simplify the discussion, the three scenarios assume that items have equally spaced cate- gory thresholds. Specifically, the item category thresholds (i.e., the J cj) were equally spaced between 22.0 and 2.0 on the u bi continuum. Let c be the vector of category thresholds that are defined as c = 2(2J(J + 1)1 1) where J is a vector with elements equal to the integers from 1 to J. For example, the threshold is zero for items with two scale categories (i.e., J + 1 = 2) and items with four scale categories have three thresholds at 21, 0, and 1. Whereas the fol- lowing discussion assumes the thresholds are equally spaced, researchers can input any set of category thresholds into the R code, which is available at the author’s website.
Let b be a vector of item locations and I be a vector that includes integers from 1 to I. The three item location scenarios represent situations where researchers would be interested in mea- suring u values in a narrow range in the lower or upper tails or measuring u values across the latent continuum. The item locations for the three scenarios represent the following uniform distributions: bi;U ( 2:5, 1:5) (i.e., b = 0:5(2I(I + 1)1 1) 2), bi;Uð2:0; 2:0Þ (i.e., b = 2(2I(I + 1)1 1)), and bi;U 1:5, 2:5ð Þ (i.e., b = 0:5(2I(I + 1)1 1) + 2).
As noted, four subject distributions were examined to evaluate how the density of the popula- tion at various points on the latent continuum affected r^u^uand rxx. Specifically, the distributions were negatively skewed (g3= 21.5, g4= 4.0), normal (g3= 0, g4= 0), symmetric and peaked (g3= 0, g4= 4.0), and positively skewed (g3= 1.5, g4= 4.0).
Figure 4 includes 12 panels corresponding to the three item locations and four subject distri- bution types. The middle row of panels in Figure 4 demonstrates that CTT and IRT yield simi- lar reliabilities in situations where tests consist of items that are evenly placed along the latent continuum. Given that s2fxjug is smaller in areas with fewer category thresholds, one explana- tion for the slight advantage of IRT is that CTT CSEMs decline outside of the (22, 2) range of the items where there are fewer subjects in the skewed and symmetric distributions. More pre- cisely, Figure 3 showed that s2f^ujug is significantly larger relative to s2fxjug in portions of the latent continuum that has fewer category thresholds. In contrast, s2fxjug is relatively smoother across u values and tends to decline in segments where there are fewer category thresholds. Furthermore, in the case where items are located across the latent continuum, ^u is understandably more reliable than x when the latent distribution is more peaked (i.e., positive kurtosis). The middle row of Figure 4 also shows the expected positive relationship that increasing scale categories has on reliability for both ^u and x. However, as found in previous research (e.g., see a review of relevant literature in Chafouleas et al., 2009, and Weng, 2004),
the value of an additional response category increases at a decreasing rate and reliability does not increase significantly in value beyond four or five response categories.
In theory, x is expected to be more reliable than ^u when the latent distribution is mismatched with item locations. The top and bottom panels of Figure 4 demonstrate the theoretical results discussed in this article concerning the effect that subject and item characteristics have on CTT and IRT reliability. For instance, CTT is superior to IRT whenever the items are located at the extremes of the latent distribution. Furthermore, the difference between rxxand r^u^uis largest for two category items and approaches zero as the number of categories increase. As discussed ear- lier, IRT is a tool for measuring specific u values more precisely. In fact, the difference between rxxand r^u^uis smallest when the majority of the latent distribution overlaps with item locations, which occurs when items are located in the lower portion of the latent continuum and the latent distribution is positively skewed (e.g., bi;U ( 2:5, 1:5), g3 = 1.5, and g4 = 4.0) or when items are in the upper tail and the distribution is negatively skewed (i.e., bi;U (1:5, 2:5), g3 = 21.5, and g4= 4.0).
In short, the results in this section provide new information about the influence of item and subject characteristics on the relative reliability of x and ^u. In fact, the findings reflect the expected results given the nature of s2fxjug and s2f^ujug. That is, total scores were more reli- able whenever items were mismatched with the location of the latent u distribution. The follow- ing section addresses another important concern related to the accuracy of linear approximations of polytomous items.
Appropriateness of a Linear Approximation of Polytomous Items One approach for modeling polytomous items is to use a linear approximation of the relationship between u and ui (e.g., a common factor model, Culpepper, 2012b). As noted in Equation 13, the relationship between u and uiis nonlinear. Ferrando (2009) and Mellenbergh (1996) noted that a linear relationship tends to provide a reasonable approximation when item discrimination indices are relatively low and there are five or more response categories. The purpose of this section is to explore the appropriateness of linear approximations of polytomous items in greater detail. This section includes two subsections. The first subsection describes a measure for quantifying the appropriateness of a linear approximation, whereas the second sub- section presents findings about the accuracy of a linear approximation for different item charac- teristics (e.g., items locations, number of response categories, and item discriminations) and subject characteristics (e.g., the skewness and kurtosis of the latent distribution).
A Measure for the Appropriateness of a Linear Approximation
Let li(u) be the loading relating latent u to observed uifor a given u. The relationship between u and uifor a specific value of u is the first derivative of Equation 9 with respect to u:
lið Þ =u ∂E uf ijug
∂u = XJ + 1
j = 1
∂u : ð18Þ
The relationship between uiand u will be larger for some values of u and near zero at the extremes of the u continuum (i.e., portions of the latent continuum where Efuijug plateaus). A linear approximation is dependent on the characteristics of the examinee population. The expected loading for a given population of subjects can be found by averaging li(u) over the distribution of u:
E lf ið Þu g = E ∂E uf ijug
= XJ + 1
j = 1
∂u j ujΩð Þdu: ð19Þ
The function that constrains the relationship between u and uito be linear for a given popula- tion of examinees is
L uf ijug = E uf g + E li f ið Þu g u mð Þ, ð20Þ where Efuig =Ð
Efuijugj(ujΩ)du, m = Efug, and Lfuijug denote the linear approximation of the relationship between u and ui. Also, note that Lfuijug has the same form as the least squares projection where Lfuijug passes through the centroid for u and ui (m, Efuig) and the relation- ship is the expected slope for a given population of examinees, Efli(u)g.
Let di= ui Lfuijug be the error when predicting the observed item uiwith a linear approxi- mation. Unlike, Efeijug; Efdijug 6¼ 0,
E df ijug = E uf i L uf ijugjug = E uf ijug L uf ijug: ð21Þ That is, di will be biased for certain values of u but EfEfdijugg = 0 across the u range for a given population. The bias in didoes not translate into larger conditional variances for Lfuijug.
In fact, s2fdijug = s2fuijug,
s2fdijug = Enðdi E df ijugÞ2juo
= Enðui E uf ijugÞ2juo
= s2fuijug: ð22Þ Furthermore, the bias in didoes not introduce dependence among residuals. Let dh= uh Lfuhjug be a residual when linearly approximating uh. The covariance between errors is
s df i, dhjug = E dfð i E df ijugÞ dð h E df hjugÞjug
= E ufð i E uf ijugÞ uð h E uf hjugÞjug = 0, ð23Þ where the last inequality was obtained by recalling that the assumption of local independence implies Efeiehjug = 0 as shown in Equation 12.
Ferrando (2009) described a measure of the appropriateness of a linear model, which is based on restricting observed scores to fall within feasible ranges (e.g., 1 and J + 1 for a polytomous item with J + 1 categories). More precisely, 1\ Lfuijug\ J + 1 implies that a linear approxi- mation is appropriate for u scores within the following range:
1 E uf gi
E lf ið Þu g + m\u\J + 1 E uf gi
E lf ið Þu g + m: ð24Þ Following Ferrando, the floor and ceiling indices for the appropriateness of a linear approxi- mation of ui are d0= (1 Efuig=Efli(u)g) + m and d1= (J + 1 Efxg=Efli(u)g) + m, respec- tively. Ferrando’s measure of appropriateness is the proportion of examinees with u scores in the viable range:
P dð 0\u\d1Þ = ð
j ujΩð Þdu: ð25Þ
Consider the examples presented in Figure 5, which compares Efuijug with Lfuijug for three items with bi= (22, 0, 2) and the following parameter values held constant, ai= 2: for all i, four response categories (J + 1 = 4), category thresholds of c = 21.6, 0, 1.6, andΩ = (0, 1, 0, 0). The
middle panel in Figure 5 includes the case where bi = 0. Figure 5 shows that Efuijug and Lfuijug deviate at the ends of the latent continuum. However, given that the distribution of u is standard normal, nearly all of the subjects (i.e., 98.9%) have approximated scores between dl
and du. The top and bottom rows of Figure 5 include examples of items that are located at 22 and 2, respectively. In fact, when items are located at the extremes of the u continuum P(dl\u\du) = 0.876%, which implies that 12.4% of subjects have approximated scores out- side the viable bounds. The examples in Figure 5 demonstrate that item location and the shape of the latent distribution affect the appropriateness of linear approximations. For instance, Figure 5 provides insight that P(dl\u\du) is larger in value if items are located near the mean of a standard normal distribution.
Impact of Item and Subject Characteristics on the Appropriateness of a Linear Approximation
The previous subsection discussed P(d0\u\d1) as a measure of the appropriateness of a linear model. The purpose of this subsection is to present more general findings about the role of item Figure 5. Hypothetical Efxjug and Lfxjug for items with different locations and four response options.
Note: The minimum and maximum total scores are denoted on the y-axis by 1 and 4. Also, ;N(0, 1), ai= 2.0, and cj= (21.6, 0, 1.6).
and subject characteristics on the accuracy of a linear approximation. Specifically, this subsec- tion discusses the appropriateness of a linear approximation of uifor 12 scenarios as defined by three response category scenarios (J + 1 = 2-10) and four u distribution scenarios (e.g., the same distributions examined in Figure 4) with the addition that the item discrimination indices are compared for ai = 0.5, 1.0, 1.5, and 2 and category thresholds are defined as discussed for the scenarios in Figure 4.
Figures 6 and 7 plot P(d0\u\d1) against item locations (i.e., bi) and provide evidence about circumstances where a linear approximation is appropriate. Note that the rows of Figures 6 and 7 correspond to the number of response categories (i.e., J + 1), whereas columns relate to distribution shape. Comparing rows demonstrates that P(d0\u\d1) increases as the number of response options increases; however, the accuracy of a linear approximation does not improve significantly beyond four response categories. For instance, a linear approximation is appropri- ate and P(d0\u\d1) . 0:90 for items with five or more response categories that are located within one standard deviation of the mean. Furthermore, larger item discriminations have a neg- ative effect on P(d0\u\d1) and the number of response categories and item discrimination has an interactive effect where P(d0\u\d1) declines more as aiincreases and there are fewer response categories. Moreover, Figure 6 illustrates that a linear approximation is appropriate for as few as two response options when ai= 0.5 or whenever ai= 1.0 and items are located in the middle of the distribution.
Comparing columns provides an indication of the effect of latent distribution shape and item location on P(d0\u\d1). Specifically, the relationship between biand P(d0\u\d1) reflects the shape of the latent distribution, and P(d0\u\d1) is smallest when biis located in the tails of the latent distribution. For instance, the P(d0\u\d1) curves for the normal distribution appear more bell shaped when the distribution is normal as opposed to peaked (i.e., k3= 0, k4= 4). In fact, P(d0\u\d1) is smaller when u;N (0, 1), ceteris paribus. The relation- ship between item location and P(d0\u\d1) is cubic when the latent distribution is skewed, and P(d0\u\d1) is smaller in skewed distributions in segments where there is less density in the latent distribution (e.g., the right tail if k3= 1:5 and the left tail if k3= 1:5).
Figure 6 also demonstrates the effect of item discrimination on P(d0\u\d1). Ferrando (2009) noted that a linear approximation is most appropriate when ai is smaller. In fact, Ferrando’s recommendations are supported given that a linear approximation is expected to be appropriate for almost all subjects and item locations when ai = 0.50. Figures 6 and 7 also show conditions where linear approximations are appropriate even when items are more discri- minating (e.g., ai= 2). For example, P(d0\u\d1) is larger when J + 1 3 and items are located near the middle of the latent distribution. P(d0\u\d1) declines as aiincreases for all of the scenarios included in Figure 6 albeit at different rates. In short, the results in Figures 6 and 7 imply that linear approximations are best for items that measure u near the central portion of the latent distribution. Furthermore, Lfuijug is least accurate for highly discriminating items, and a linear approximation is inaccurate even if there are more than four response categories when items are located in the tails of the latent distribution.
The findings in this study offer guidance to applied researchers interested in the construction and analysis of polytomous items. In short, this article presented new theoretical results con- cerning item and subject characteristics that affect the relative precision and reliability of x and
^u. This section summarizes the findings for psychometricians and applied researchers and offers concluding remarks.
Figure6.Theproportionofviablescoresfromalinearapproximationofpolytomousitemswithtwotofourresponsecategoriesbyitemlocationand discrimination,andsubjectlatentdistributionshapes. Note:Categorythresholdswereequallyspacedalongthelatentcontinuum.
Figure7.Theproportionofviablescoresfromalinearapproximationofpolytomousitemswithfivetosevenresponsecategoriesbyitemlocation discrimination,andsubjectlatentdistributionshapes. Note:Categorythresholdswereequallyspacedalongthelatentcontinuum.
This article studied the reliability of two alternative test scoring approaches: a CTT total score versus an IRT ^u estimate. The majority of applied researchers in education and psychol- ogy are familiar with x, whereas fewer have knowledge about ^u and IRT. Consequently, it is important to offer applied researchers information about the relative merits of x and ^u. This arti- cle offered additional insights into fundamental differences between x and ^u. One salient factor studied in this article was the interactive effect that item locations and u distribution shape had on rxxand r^u^u. Suppose the purpose of testing is to accept high-scoring examinees into an insti- tution as is the case with examinations for actuarial sciences where the latent distribution is either normal or positively skewed. For instance, the first actuarial exam includes items clus- tered in the upper portion of the u continuum to identify those students who are most competent in the foundations of calculus and probability theory. The scores for examinees who are near the passing cutoff are measured more precisely than those who are in the middle or lower portions of the u distribution as indicated by smaller values for s2f^ujug. In contrast, the results in this article show that s2fxjug is larger near the cutoff score and would be relatively smaller for u values in the lower and middle portions of the examinee distribution. Certainly, decision makers would prefer IRT scoring versus a total score in this instance, because the purpose of measure- ment is to evaluate whether test takers exceed a minimum proficiency level. However, second- ary users of the actuarial test score data should prefer total scores, because, as indicated in Figure 4, x tends to be more reliable for a population of subjects than ^u if items are difficult and located in the tails of the test-taker population distribution. For instance, some secondary users may want to gather validity evidence and correlate examinee scores with other indicators, such as undergraduate or graduate grade point average (Culpepper, 2010; Culpepper & Davenport, 2009), other aptitude tests, or job performance (Aguinis, Culpepper, & Pierce, 2010). In these instances, the validity coefficient associated with ^u would be smaller than the coefficient for x because rxx. r^u^u.
In addition, the results provide evidence that CTT and IRT scoring methods perform simi- larly when the goal of testing is to measure a construct across values of a latent continuum. For example, federal and state testing programs measure what students know in relation to given standards. The results in Figure 4 imply that rxxand r^u^uare similar and testing programs could report total scores, which may be easier to explain to certain stakeholders (e.g., teachers, par- ents, and students).
The results in this article also offer recommendations for the construction of scales that include polytomous items, such as educational or employment performance assessments, beha- vioral ratings, or affective measurements. In either case, researchers may prefer x to ^u if the purpose of instrument development is to conduct correlational research rather than to measure specific trait levels. However, x should probably only be preferred to ^u when researchers fit simple models rather than more complicated interactive models (Embretson, 1996; Kang &
Waller, 2005; Morse et al., 2012).
In addition, the findings in this article provide information about the optimal number of item response options. The results in Figure 4 examined the reliability of total scores and IRT esti- mates across a range of parameter values for the number of scale categories, item locations, and the shape of the latent distribution. The findings in Figure 4 imply that using more than five or six scale categories does not significantly improve the reliability of x or ^u regardless of the shape of the latent distribution or location of items. However, it is important to note that adding an additional scale value had a larger effect on r^u^uthan on rxx.
Another relevant finding for applied researchers (and methodologists) relates to the appropri- ateness of a linear approximation of polytomous items. The results in this article confirm argu- ments in previous research (Ferrando, 2009) that linear approximations are more accurate for less discriminating items and items with more response categories. Additional findings suggest
that linear approximations of polytomous items seem appropriate for items that measure trait levels in denser segments of the latent distribution, and linear approximations were least appro- priate for items located in the tails of the u distribution. In contrast to previous research, the results provided new evidence that researchers need to consider item locations, in addition to item discriminations and the number of response categories, when employing linear approximations.
Last, another contribution of this study is the availability of the associated R code. More spe- cifically, researchers can use the R code when designing instruments in an effort to evaluate conditions where total scores and IRT scores are more or less reliable and precise. Furthermore, the R code has pedagogical value, as well, for computational applications of the theoretical results.
There are several directions for future research to build on this study. First, this article offers recommendations for researchers who are interested in using x as a measure of u in applied research. Specifically, designing a reliable x requires the inclusion of items located at the bound- aries of the u range of interest. For instance, if the measurement goal is to distinguish high ver- sus low scorers on some trait, the results pertaining to CTT CSEMs dictate that the items should be located in the middle of the distribution, because low and high scorers will be measured more precisely. Likewise, items should be located around a cutscore (and not at the cut-score) if the purpose of measurement is to make inferences about whether examinees exceed or fall below some minimum proficiency level. The behavior of CSEMs under CTT is counterintuitive, because, in contrast, item design under the IRT framework dictates that developers should write items that are specific to certain u levels and measurement purposes. Additional research is needed to understand differences in optimal test assembly (H. Chang & Ying, 2009) within the IRT and CTT frameworks.
Second, there could be benefits in revisiting some topics in modern IRT within the context of CTT. For instance, there may be new insights available by reexamining topics in CTT, such as computer adaptive testing (H. Chang & Ying, 1996) or equating techniques (Kolen & Brennan, 2004), which could lead to new methodologies, refinements of existing approaches, or other unanticipated discoveries.
Third, researchers could extend the results in this article to understand the impact of using x and ^u. Specifically, researchers use total scores as dependent variables and predictors in every subdiscipline of psychology and education. Despite the widespread use of total scores, few methodological studies have examined the impact of using total scores on the power and Type I error rates of tests that researchers employ (Embretson, 1996; Kang & Waller, 2005; Morse et al., 2012). Furthermore, with the exception of Embretson (1996), previous studies utilized Monte Carlo techniques that are limited by the parameter values studied. Consequently, future analytic explorations could provide additional insights into the effect of using total scores, and future research should accordingly examine the effect that the number of scale categories, the shape of the latent distribution, and IRT parameters have on the performance of commonly used statistical tests (Culpepper, 2012a; Culpepper & Aguinis, 2011).
Fourth, this study examined the theoretical reliability of x and ^u using a polytomous IRT model. As one anonymous reviewer noted, this article addressed reliability from a mathematical perspective and does not consider factors related to subjects’ cognitive decision making. For example, this article did not address issues related to category labels; however, previous research identified a causal effect of scale labels, category position, and rating scale intensity and length on certain observed item characteristics (Dunham & Davison, 1991; Lam & Stevens, 1994; Murphy & Constans, 1987). For instance, existing evidence suggests that scale labels can affect the observed item means, but there is less evidence that manipulating category labels alters observed item variances (L. Chang, 1997; Dunham & Davison, 1991). Moreover,
positive-packed scales tend to affect item means (Dunham & Davison, 1991; Lam & Kolic, 2008), and semantic compatibility of category labels improves reliability (Lam & Kolic, 2008).
Most of the previous literature on category labels used CTT or generalizability theory, and addi- tional empirical research is needed to understand how category labeling decisions affect polyto- mous IRT item parameters (e.g., item locations, discriminations, and category thresholds) and consequently alter the reliability of x and ^u. Future research may identify relationships between category labels, cognitive decision making, and IRT parameters, and the results in this article provide mathematical arguments for describing how subsequent changes in IRT parameters affect test score reliability.
In conclusion, this study presented new results concerning the relative reliability and preci- sion of total scores and IRT scores. The derivations in this article offer the most extensive analy- sis of the reliability of total scores by linking parameters of polytomous IRT models with CTT.
In addition, new equations were discussed that described the CTT CSEM to provide new con- ceptual understanding of differences in the precision of scores estimated within the CTT and IRT frameworks.
Fleishman’s Power Transformation (PT) Method Probability Density Function (PDF) Fleishman (1978) developed a PT method for generating nonnormal univariate random vari- ables. The PT method uses the following function to transform a standard normal random vari- able, y, into a metric with a given k3and k4,
u = f yjvð Þ =X4
r = 1
where the vector of Fleishman coefficients, v = (v1, v2, v3, v4), is identified so that u has a pre- determined mean, variance, skewness, and kurtosis. Additional research extended Fleishman’s method to higher order PTs (Headrick, 2002) and multivariate circumstances (Headrick &
Sawilosky, 1999; Lyhagen, 2008; Vale & Maurelli, 1983).
One critique of Fleishman’s method was that there was no known probability distribution for variables generated using Fleishman’s technique (Tadikamalla, 1980). Recent research derived the PDF for Fleishman’s PT method (Headrick & Kowalchuk, 2007) and showed that the PDF for Fleishman’s PT method is
j ujΩð Þ = f fð 1ðujvÞÞ
sf0ðf1ðujvÞjvÞ, ðA2Þ where f(u) is the standard normal distribution, f1(ujv) is the inverse of Equation A1 that spe- cifies values of y as a function of u, and f0(yjv) is the derivative of Equation A1 with respect to y. An expression for f1(ujv) can be found using Cardano’s formula for the inverse of a cubic polynomial. The equation for the only real root of f1(ujv) is defined as
y = f1ðujvÞ = q +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2+ rð p2Þ3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2+ rð p2Þ3 q
+ p, ðA3Þ
where the second cube root is the absolute value of q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2+ (r p2)3 q
. Furthermore, p, q, and r are a function of v and u as shown in the following:
p = v3 3v4,
q = p3+v2v3 3v4v1ums
r = v2 3v4
This article uses j(ujΩ) to understand the impact of nonnormal latent distributions on the reliability of x and ^u. Specifically, v is first identified for a given k3 and k4, and the p, q, and r in Equation A4 are computed to find f1(ujv) to use in Equation A3.
It is important to note that Fleishman’s PT method yields a valid PDF only when u is a mono- tonically increasing function of y (i.e., f0(yjv) must be positive for all values of y). Headrick and Kowalchuk (2007) proved that the PT method produces a valid PDF if the Fleishman coeffi- cients satisfy the following constraints:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 + 7v22
2 5v2, 0\v2\1:
Consequently, Fleishman’s PT method can only be used to study the impact of nonnormal latent distributions whenjk3j\4:5.
Declaration of Conflicting Interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publi- cation of this article.
The author received no financial support for the research, authorship, and/or publication of this article.
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- Space group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.
In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.
In crystallography, they are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002)).
- 1 History
- 2 Elements of a space group
- 3 Notation for space groups
- 4 Classification systems for space groups
- 5 Space groups in other dimensions
- 6 Table of space groups in 3 dimensions
- 7 References
- 8 External links
Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries.
The space groups in 3 dimensions were first enumerated by Fyodorov (1891), and shortly afterwards were independently enumerated by Schönflies (1891) and Barlow (1894). These first enumerations all contained several minor mistakes, and the correct list of 230 space groups was found during correspondence between Fyodorov and Schönflies.
Elements of a space group
The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.
Elements fixing a point
The elements of the space group fixing a point of space are rotations, reflections, the identity element, and improper rotations.
The translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups.
A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure.
A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 21 is a twofold rotation followed by a translation of 1/2 of the lattice vector.
The general formula for the action of an element of a space group is
y = M.x + D
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D(lattice) + D(M), where D(M) is a unique function of M that is zero for M being the identity. The matrices M form a point group that is a basis of the space group; the lattice must be symmetric under that point group.
The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):
- (1,1): One-dimensional line groups
- (2,1): Two-dimensional line groups: frieze groups
- (2,2): Wallpaper groups
- (3,1): Three-dimensional line groups; with the 3D crystallographic point groups, the rod groups
- (3,2): Layer groups
- (3,3): The space groups discussed in this article
Notation for space groups
There are at least eight methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names.
- Number. The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers.
- International symbol or Hermann-Mauguin notation. The Hermann-Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. By way of example, the space group of quartz is P3121, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3121, it is trigonal).
- In the international short symbol the first symbol (31 in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3112. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30o.
- The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols in use.
- Hall notation. Space group notation with an explicit origin. Rotation, translation and axis-direction symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1).
- Schönflies notation. The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is C2 have Schönflies symbols C1
- Shubnikov symbol
- 2D:Orbifold notation and 3D:Fibrifold notation. As the name suggests, the orbifold notation describes the orbifold, given by the quotient of Euclidean space by the space group, rather than generators of the space group. It was introduced by Conway and Thurston, and is not used much outside mathematics. Some of the space groups have several different fibrifolds associated to them, so have several different fibrifold symbols.
- Coxeter notation - Spacial and point symmetry groups, represented as modications of the pure reflectional Coxeter groups.
Classification systems for space groups
There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it.
(Crystallographic) space group types (230 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are conjugate by an orientation-preserving affine transformation. In three dimensions, for 11 of the affine space groups, there is no orientation-preserving map from the group to its mirror image, so if one distinguishes groups from their mirror images these each split into two cases. So there are 54+11=65 space group types that preserve orientation. Affine space group types (219 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are conjugate under an affine transformation. The affine space group type is determined by the underlying abstract group of the space group. In three dimensions there are 54 affine space group types that preserve orientation. Arithmetic crystal classes (73 in three dimensions). These are determined by the point group together with the action of the point group on the subgroup of translations. In other words the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GLn(Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes. (geometric) Crystal classes (32 in three dimensions). The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of GL2(Z), are conjugate in the larger group GL2(Q). Bravais flocks (14 in three dimensions). These are determined by the underlying Bravais lattice type.
These correspond to conjugacy classes of lattice point groups in GL2(Z), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group.
Crystal systems. (7 in three dimensions) Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same. Lattice systems (7 in three dimensions). The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GL2(Z)) in the larger group GL2(Q). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems. Crystal families (6 in three dimensions). The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombal, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems.
Conway, Delgado Friedrichs, and Huson et al. (2001) gave another classification of the space groups, called a fibrifold notation, according to the fibrifold structures on the corresponding orbifold. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper groups, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.
Space groups in other dimensions
In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's 18th problem. Zassenhaus (1948) showed that conversely any group that is the extension of Zn by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zn by a finite group acting faithfully.
It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3.
Classification in small dimensions
This table give the number of space group types in small dimensions.
Dimension Lattice types (sequence A004030 in OEIS) Crystallographic point groups (sequence A004028 in OEIS) Crystallographic space group types (sequence A006227 in OEIS) Affine space group types (sequence A004029 in OEIS) Classification 0 1 1 1 1 Trivial group 1 1 2 2 2 One is the group of integers and the other is the infinite dihedral group; see symmetry groups in one dimension 2 5 10 17 17 these 2D space groups are also called wallpaper groups or plane groups. 3 14 32 230 219 In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by "enantiomorphous character" (e.g. P3112 and P3212). Usually "space group" refers to 3D. They were enumerated independently by Barlow (1894), Fedorov (1891) and Schönflies (1891). 4 64 227 4894 4783 The 4895 4-dimensional groups were enumerated by Harold Brown, Rolf Bülow, and Joachim Neubüser et al. (1978).
Neubüser, Souvignier & Wondratschek (2002) corrected the number of enantiomorphic groups from 112 to 111, so total number of groups is 4783+111=4894. There are 44 enantiomorphic point groups in 4-dimensional space. If we consider enantiomorphic groups as different, the total number of point groups is 227+44=271
5 189 955 222018 Plesken & Schulz (2000) enumerated the ones of dimension 5 6 841 7104 28934974 28927922 Plesken & Schulz (2000) enumerated the ones of dimension 6.
Initially published number of 826 Lattice types in Plesken & Hanrath (1984) was corrected to 841 in Opgenorth, Plesken & Schulz (1998).
See also Janssen et al. (2002)
Double groups and time reversal
In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and the group elements can include time reversal as reflection in it. They are of importance in magnetic structures that contain ordered unpaired spins, i.e. ferro-, ferri- or antiferromagnetic structures as studied by neutron diffraction. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D (Kim 1999, p.428). It has also been possible to construct magnetic versions for other overall and lattice dimensions (Daniel Litvin's papers, (Litvin 2008), (Litvin 2005)). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:
- (1,1): 2, 7
- (2,1): 7, 31
- (2,2): 17, 80
- (3,1): 75, 394 (rod groups, not 3D line groups in general)
- (3,2): 80, 528
- (3,3): 230, 1651
Table of space groups in 3 dimensions
# Crystal system Point group Space groups (international short symbol) Intl Schönflies 1 Triclinic (2) 1 C1 P1 2 1 Ci P1 3-5 Monoclinic (13) 2 C2 P2, P21, C2 6-9 m Cs Pm, Pc, Cm, Cc 10-15 2/m C2h P2/m, P21/m, C2/m, P2/c, P21/c, C2/c 16-24 Orthorhombic (59) 222 D2 P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121 25-46 mm2 C2v Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2, Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2 47-74 mmm D2h Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma 75-80 Tetragonal (68) 4 C4 P4, P41, P42, P43, I4, I41 81-82 4 S4 P4, I4 83-88 4/m C4h P4/m, P42/m, P4/n, P42/n, I4/m, I41/a 89-98 422 D4 P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212, I422, I4122 99-110 4mm C4v P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc, I4mm, I4cm, I41md, I41cd 111-122 42m D2d P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d 123-142 4/mmm D4h P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm, I4/mmm, I4/mcm, I41/amd, I41/acd 143-146 Trigonal (25) 3 C3 P3, P31, P32, R3 147-148 3 S6 P3, R3 149-155 32 D3 P312, P321, P3112, P3121, P3212, P3221, R32 156-161 3m C3v P3m1, P31m, P3c1, P31c, R3m, R3c 162-167 3m D3d P31m, P31c, P3m1, P3c1, R3m, R3c, 168-173 Hexagonal (27) 6 C6 P6, P61, P65, P62, P64, P63 174 6 C3h P6 175-176 6/m C6h P6/m, P63/m 177-182 622 D6 P622, P6122, P6522, P6222, P6422, P6322 183-186 6mm C6v P6mm, P6cc, P63cm, P63mc 187-190 6m2 D3h P6m2, P6c2, P62m, P62c 191-194 6/mmm D6h P6/mmm, P6/mcc, P63/mcm, P63/mmc 195-199 Cubic (36) 23 T P23, F23, I23, P213, I213 200-206 m3 Th Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3 207-214 432 O P432, P4232, F432, F4132, I432, P4332, P4132, I4132 215-220 43m Td P43m, F43m, I43m, P43n, F43c, I43d 221-230 m3m Oh Pm3m, Pn3n, Pm3n, Pn3m, Fm3m, Fm3c, Fd3m, Fd3c, Im3m, Ia3d
Note. An e plane is a double glide plane, one having glides in two different directions. They are found in five space groups, all in the orthorhombic system and with a centered lattice. The use of the symbol e became official with Hahn (2002).
The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, or C, standing for the principal, body centered, face centered, or C-face centered lattices.
- Barlow, W (1894), "Über die geometrischen Eigenschaften starrer Strukturen und ihre Anwendung auf Kristalle", Z. Kristallogr. 23: 1–63
- Bieberbach, Ludwig (1911), "Über die Bewegungsgruppen der Euklidischen Räume", Mathematische Annalen 70 (3): 297–336, doi:10.1007/BF01564500, ISSN 0025-5831
- Bieberbach, Ludwig (1912), "Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich", Mathematische Annalen 72 (3): 400–412, doi:10.1007/BF01456724, ISSN 0025-5831
- Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-471-03095-9, MR0484179
- Burckhardt, Johann Jakob (1947), Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, 13, Verlag Birkhäuser, Basel, MR0020553
- Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 42 (2): 475–507, ISSN 0138-4821, MR1865535, http://www.emis.de/journals/BAG/vol.42/no.2/17.html
- Fedorov, E. S. (1891), "Symmetry of Regular Systems of Figures", Zap. Mineral. Obch. 28 (2): 1–146
- Fedorov, E. S. (1971), Symmetry of crystals, ACA Monograph, 7, American Crystallographic Association
- Hahn, Th. (2002), Hahn, Theo, ed., International Tables for Crystallography, Volume A: Space Group Symmetry, A (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7, http://it.iucr.org/A/
- Hall, S.R. (1981), "Space-Group Notation with an Explicit Origin", Acta Cryst. A37: 517–525
- Janssen, T.; Birman, J.L.; Dénoyer, F.; Koptsik, V.A.; Verger-Gaugry, J.L.; Weigel, D.; Yamamoto, A.; Abrahams, S.C. et al. (2002), "Report of a Subcommittee on the Nomenclature of n-Dimensional Crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups", Acta Cryst. A 58 (Pt 6): 605-621, doi:10.1107/S010876730201379X
- Kim, Shoon K. (1999), Group theoretical methods and applications to molecules and crystals, Cambridge University Press, ISBN 978-0-521-64062-6, MR1713786
- Litvin, D.B. (May 2008), "Tables of crystallographic properties of magnetic space groups", Acta Cryst. A 64 (Pt 3): 419–24, doi:10.1107/S010876730800768X, PMID 18421131
- Litvin, D.B. (May 2005), "Tables of properties of magnetic subperiodic groups", Acta Cryst. A 61 (Pt 3): 382–5, doi:10.1107/S010876730500406X, PMID 15846043
- Neubüser, J.; Souvignier, B.; Wondratschek, H. (2002), "Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons]", Acta Cryst. A 58 (Pt 3): 301, doi:10.1107/S0108767302001368
- Opgenorth, J; Plesken, W; Schulz, T (1998), "Crystallographic Algorithms and Tables", Acta Cryst. A 54 (Pt 5): 517-531, doi:10.1107/S010876739701547X
- Plesken, Wilhelm; Hanrath, W (1984), "The lattices of six-dimensional space", Math. Comp. 43 (168): 573-587
- Plesken, Wilhelm; Schulz, Tilman (2000), "Counting crystallographic groups in low dimensions", Experimental Mathematics 9 (3): 407–411, ISSN 1058-6458, MR1795312, http://projecteuclid.org/euclid.em/1045604675
- Schönflies, Arthur Moritz (1891), "Theorie der Kristallstruktur", Gebr. Bornträger, Berlin.
- Vinberg, E. (2001), "Crystallographic group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c027190.htm
- Zassenhaus, Hans (1948), "Über einen Algorithmus zur Bestimmung der Raumgruppen", Commentarii Mathematici Helvetici 21: 117–141, doi:10.1007/BF02568029, ISSN 0010-2571, MR0024424, http://www.digizeitschriften.de/index.php?id=166&ID=380406
- International Union of Crystallography
- Point Groups and Bravais Lattices
- Bilbao Crystallographic Server
- Space Group Info (old)
- Space Group Info (new)
- Crystal Lattice Structures: Index by Space Group
- Full list of 230 crystallographic space groups
- Interactive 3D visualization of all 230 crystallographic space groups
- Huson, Daniel H. (1999), The Fibrifold Notation and Classification for 3D Space Groups, http://www-ab.informatik.uni-tuebingen.de/talks/pdfs/Fibrifolds-Princeton%201999.pdf
- The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
- The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)
- Finite groups
- Discrete groups
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√ 10+ Calculus Recommended For You
√ 10+ Calculus Recommended For You
The Nuiances of Calculus
The Do’s and Don’ts of Calculus
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Calculus Fundamentals Explained
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Finding Calculus Online
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What You Don’t Know About Calculus
You might be asking yourself about the derivatives of your preferred trigonometric functions. In the end, derivatives may be used to help you graph functions. In reality, the derivative is not anything more than a special type of limit. The very first, second, or third derivative can be supplied. Like the integral, there’s additionally a fractional derivative employing the overall Mittag-Leffler function for a kernel.
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How the order of a chemical reaction is determined also, a few examples on determining the order of a reaction with one reactant decomposing into products. The inability of ordinary nonisothermal experiments to determine the rate order (n ) of drug degradation is discussed on the basis of a theoretical study of simulated nonisothermal data in ordinary nonisothermal experiments, using either r or sigma(c[experiment] c[compute])2 as the measure of goodness of fit, the rate order. Time-saving video by brightstorm on determining order of a reaction using a graph. Where k is the rate constant, [ ] is the molarity of the reactant, and x, y, and z are the reaction orders with respect to a, b and c, respectively the overall order of the reaction is x+y+z the rate law is always determined experimentally there are several ways to determine the rate law for a particular reaction method of initial. In order to experimentally determine a rate law, a series of experiments must be performed with various starting concentrations of reactants the initial rate law is then measured for each of the reactions consider the reaction between nitrogen monoxide gas and hydrogen gas to form nitrogen gas and water.
The rate law for reactions involving a single time, min [a], mol/l graphical method for example, a first-order reaction will give reactant can be determined using the for an overall reaction a straight-line plot of ln[a] vs time, but 0 5 graphical method this method involves curved plots for [a] vs time and 1/[a] vs 1 6. The plot that gives the best linear fit (which you can assess using the r 2 value), corresponds to the appropriate integrated rate law if [ a ] vs time is linear, then the reaction is zero order if ln [ a ] vs time is linear, then the reaction is first order if 1 [ a ] vs time is linear, then the reaction is second order hope this helps. It is defined as the sum of powers or index or the component of the concentration of the reactants which are raised in the rate equation order of the reaction can be zero, one, two, and three sometimes it can be pseudo first-order reaction, also it is determined for the slowest step in the reaction example zero order reaction.
Rate laws from graphs of concentration versus time (integrated rate laws) in order to determine the rate law for a reaction from a set of data consisting of concentration (or the values of some function of concentration) versus time, make three graphs [a] versus t (linear for a zero order reaction) ln [a] versus t (linear for a. Reaction order to reiterate, the exponents x and y are not derived from the balanced chemical equation, and the rate law of a reaction must be determined experimentally these exponents may be either integers or fractions, and the sum of these exponents is known as the overall reaction order a reaction can also be. Concept of rate of reaction factors effecting rate of reaction concept of order of reaction methods for the determination of order of reaction pharmaceutic.
Determination of rate laws the rate law for a chemical reaction can be determined by changing the concentration of each reactant, and measuring the effect this has on the rate of reaction for example, if a reaction is first order in reactant a, then doubling the concentration of a will double the rate of reaction however. This leads to an interesting conclusion changing the concentration changes the rate by some determined power this power is known as the order of reaction or more formally: in chemical kinetics, the order of reaction with respect to a given substance (such as reactant, catalyst or product) is defined as the index,.
The following methods are employed to determine the rate law, rate constant and order of reaction 1 graphical method this method is used to determine the rate law of the reaction which involves only one reactant species the various steps involved are: (i) the concentrations of reacting substance are determined at. On the other hand, integrated rate laws express the reaction rate as a function of the initial concentration and a measured (actual) concentration of one or more reactants after a specific amount of time (t) has passed they are used to determine the rate constant and the reaction order from experimental data eg ( when. Experiment 6b kinetics: initial rates prepared by ross s nord, stephen e schullery, and masanobu m yamauchi eastern michigan university purpose learn how to measure initial rates determine the order of the permanganate-isopropyl alcohol reaction with respect to isopropyl alcohol and hydronium ion,.
R appelqvist , gr beecher , h bergamin f° , g den boef , j emnéus , zhaolun fang , l gorton , eh hansen , pe hare , jm harris , jj harrow , n ishibashi , j janata , g johansson , b karlberg , fj krug , we van der linden , md luque de castro , g markovarga , jn miller , ha mottola , h müller , ge. If x=2, we have [d]x=[d]2, and the reaction rate is quadrupled when the concentration of d doubles an order for a reactant is typically any real, positive number including fractions using graphical methods to determine the order of a reactant by plotting changes in concentration of a given reactant versus time, the order of. Example of determining the rate equation here we need to find m and n in the equation: rate = k[a]m[b]n in order to do this, you need to compare individual experiments look at experiment 1 and then experiment 2 [a] is doubled and [b] is the same, so we can deduce the. If we are taking the linear form of the equation, perhaps we are considering the reaction as 1 st order so can we get the exact order and k can anyone send me some papers ,about reaction rate determination for product formation,where product concentration only taken into consideration ,not the reactant concentration. |
The ellipse is a common planar closed curve with a double axis of symmetry, belonging to the family of conic curves represented by a quadratic equation . The general Cartesian notation for the form comes from the French mathematician Gabriel Lamé, who generalized the equation for the ellipse. A superelliptic equation (i.e., |x/a|n + |y/b|n = 1, where n, a, and b are positive numbers) is a closed curve resembling the ellipse with three parameters. A superellipse, similarly to the ellipse, is symmetric about its two semi-axes, but differs in its overall shape . The Danish author and scientist Piet Hein have dealt with the superellipse in great detail, especially by using the curve for architectural objects such as motorway bridges . His proposal won the design challenge for a roundabout in the Sergels Torg city square in Stockholm, Sweden. The Melior font, designed by Hermann Zapf, has been based on this curve . The egg curves (i.e., [x/a]2 + [y/b]2t(x) = 1, where t(x) is a function of x) have been discovered by Florian Blaschke . An oval is a closed plane line that is shaped like an ellipse or like the egg of a hen. The hen’s egg is smaller at one end and has only one axis of symmetry . Reference egg-shape equation (i.e., [x/a]2 + [y/(ky + b)]2 = 1, |k| < 1) have three parameters. These can develop the shape of a hen egg, which changes the equation of an oval a little. As important mathematical tools, these curvilinear equations play an important role in daily production and life.
This study is based on the isoconcentration contour equation of simplified two-dimensional advection-diffusion in rivers. The scope of this study encompass 1) the analysis of the height in the direction of the axis of symmetry and the width in the direction of asymmetry, 2) determine a new equation for a heteromorphic elliptic equation with two parameters, 3) compare the heteromorphic ellipse with an ellipse, and 4) investigate the geometric properties and application prospects of the heteromorphic ellipse and the rotating body. The results can provide theoretical support for the popularization and application of the heteromorphic elliptic equation.
2. Background of the Heteromorphic Elliptic Equation
Reference has given the concentration distribution from a uniform-intensity line source in the center of a wide river based on the simplified two-dimensional advection-diffusion transport equation. The resulting equation is given as
where x is the longitudinal coordinate in the stream-wise direction, and y is the transverse coordinate perpendicular to the flow and pointing towards the river bank. The origin of the coordinate system is the discharge point at the center of the river, m the mass discharge rate of the passive scalar, U the average velocity of the river, H the average water depth, and Ey the transverse diffusion coefficient.
According to Equation (1), for a constant concentration C = Ca, the maximum length Ls, the maximum half-width bs, and the corresponding longitudinal coordinate Lc of the area surrounded by the contour line are given as
Wu et al. have used Ls and bs in Equation (2) and Equation (3) to make x and y dimensionless in Equation (1) with C (x, y) = Ca and obtained the dimensionless formula of the isoconcentration curve (namely Wu’s curve) as
Previous studies have indicated that Wu’s curve as represented by Equation (5) is similar to the ellipse, whereas it has only a single axis of symmetry. Since it is derived from advection and diffusion of a passive scalar and the transport of momentum, heat, and mass have certain similarities in flow, the authors have surmised that Wu’s curve could exhibit good mechanical properties analogous to that of the shape of the eggshell, and hence enhancing the significance of the Wu’s curve. For the sake of describing this metaphysical concept by a representational image, Wu’s curve is referred to here as a “heteromorphic ellipse”.
3. Comparison of the Heteromorphic Ellipse with the Ellipse
Following Equation (5), the heteromorphic ellipse has its maximum dimension along the axis of symmetry in the x-direction. Hereafter, for simplicity, the scale in the x and y axes will be called “height” and “width”, respectively. Defining such a length scale as the maximum height of the heteromorphic ellipse,
, ( ) (6)
Figure 1. Comparison of the heteromorphic ellipse and an ellipse with the same long and short axes.
The median line of the heteromorphic ellipse is located at the midpoint of the axis of symmetry and it is a line segment that connects two side boundary points that is parallel to the asymmetric axis. Alternatively, the height of the median line of the heteromorphic ellipse is h = a. From Equation (4), it can be concluded that the maximum width is located at the height Lc/
It divides the whole geometry into upper and lower parts at the approach to the golden section of the line. The heteromorphic ellipse is symmetrical about the x-axis and ranges from 0 ≤ x ≤
The two vertices on the axis of symmetry x are coincident, with the same height and width, in the heteromorphic ellipse and the ellipse (Figure 1). On each side of the symmetric x-axis, the heteromorphic ellipse intersects the ellipse at x/
4. Geometric Properties of the Heteromorphic Ellipse
The integral of Equation (6) is determined on x ∈ [0, 2a], the formula for calculating the area of the heteromorphic ellipse be derived as
By performing variable substitution so that x/
where is called the “area coefficient”. Hence, the area formula for the heteromorphic ellipse states that its area is equal to the product of the area coefficient, the height, and the width.
Let η = ζ1.5, and by substituting the area coefficient formula, the definite integral can be obtained directly that the area coefficient can be given as
An integral table can be used to evaluate , which is substituted into Equation (9) to give
This value is 1.27% larger than the area coefficient (=π/4) of the ellipse.
4.2. Centroid Coordinate
The centroid refers to the geometric center of a heteromorphic ellipse, and for objects of uniform density, the center of mass and the centroid are coincident. For a heteromorphic ellipse with only one axis of symmetry, the centroid must be on its axis of symmetry. To identify the specific point on the axis of symmetry, it is necessary to determine the area moment by calculating the area moments for the asymmetric axis. According to the principle that the total area moment is equal to the sum of each area moment
Using variable substitution, letting x/
Using an integral table to evaluate and then substituting Equation (10) into Equation (12), we get
Therefore, the centroid of a heteromorphic ellipse is located at the point on the axis of symmetry where the relative height coordinate is xc' = xc/
4.3. Compression Coefficient and Shape Classification
The compression coefficient of a heteromorphic ellipse is defined as the ratio of the width of the asymmetric axis and the height of the axis of symmetry (the ratio of width to height being θ = 2b/
Under different compression coefficients, the shape of a heteromorphic ellipse seems to follow natural patterns (Figure 2). The shapes of the rotating body of these figures are as follows: Figure 2(a) takes the shape of a corn cob, Figure 2(d) that of round bread, and Figure 2(b) and Figure 2(c) resemble the shape of strawberry and apple, etc.
(a) (b) (c) (d)
Figure 2. Heteromorphic ellipses with different compression coefficients. (a) θ = 0.25; (b) θ = 0.67; (c) θ = 1.00; (d) θ = 1.50.
To note, the compression coefficient can reflect the shape characteristics of the heteromorphic ellipse. Accordingly, the shape of the heteromorphic ellipse can be classified for the cases of: 1) 0 < θ < 1, where the width of the heteromorphic ellipse is less than the height, and the ellipse is of leptosome type (H-type), 2) θ = 1, where the width of the heteromorphic ellipse is equal to the height, which is called the Standard-type, 3) θ > 1, where the width of the heteromorphic ellipse is greater than the height, and the ellipse is of pyknic type (W-type).
Practical experience reveals that as compared with ellipses with a double axis of symmetry, whose shapes have only one axis of symmetry with one end thin and the other end having a large oval shape, are more common in nature.
According to the integral of the plane curve arc length in higher mathematics, the perimeter integral formula of a heteromorphic ellipse is given as
Taking the derivative on both sides of Equation (6) results in the following
Substituting Equation (15) into Equation (14), setting x/
where is called the “heteromorphic elliptic
coefficient”. The theorem regarding the perimeter of a heteromorphic ellipse can be stated as, the perimeter is equal to the product of the heteromorphic elliptic coefficient and the sum of the half-height and half-width of the heteromorphic ellipse.
The formula for the heteromorphic elliptic coefficient shows that this coefficient (Th) is a function only of the heteromorphic elliptic compression coefficient (θ). In the interval of [0.01, 100], a series of θ values are given, and elliptic coefficients (Th) can be calculated by using a numerical integration method. A series of corresponding Th values are obtained, and the variation between the heteromorphic elliptic coefficients (Th) and compression coefficients (θ) can be determined, as shown in Figure 3. Under the same conditions, the variation relationships of elliptic coefficients are also given for comparison with the heteromorphic ellipse.
The results from calculation coupled with Figure 3 indicate that the heteromorphic elliptic coefficient of the Standard-type heteromorphic ellipse with equal width and height (θ = 1) is Th = 3.191, is 1.58% larger than the elliptic coefficient (=π) under the same conditions. Further analysis shows that the ratio of the area coefficient of the Standard-type heteromorphic ellipse to the heteromorphic elliptic coefficient is 0.249. This value is only 0.31% lower than the ratio of these items in a circle, which shows that the area of a Standard-type heteromorphic ellipse is largest in the case of a certain perimeter value. In Figure 3, the elliptic coefficients (dashed line) are symmetric about the compression coefficient θ = 1 in logarithmic coordinates. For a given aspect ratio, the heteromorphic elliptic coefficient of the H-type (compression coefficient 0 < θ < 1) is slightly larger than that of the W-type (θ > 1). When θ < 0.1, the relative difference between the heteromorphic elliptic coefficient and the elliptic coefficient is less than 5.0‰, and when θ > 10, the relative difference is less than 1.2‰.
4.5. Radius of Curvature
In principle, the curvature indicates the bending degree of a curve at a certain point, and the curvature is greater for a greater value of the bending degree of the curve. The reciprocal of the curvature is the radius of curvature. The radius of curvature is also used to describe the bending degree of a curve at a certain point, and the radius of curvature is smaller for greater values of the bending degree of the curve. The radius of curvature of the heteromorphic ellipse is closely related to the size, and the heteromorphic ellipse and the radius of the
Figure 3. Comparison of heteromorphic ellipse coefficients and ellipse coefficients.
corresponding point of the curve is larger for smaller values of the bending degree of the curve. To facilitate the analysis of the variation rule of the curvature radius of a heteromorphic ellipse, the Standard-type heteromorphic ellipse with
The first derivative is given as
The second derivative is given as
The formula for radius of curvature is
The definition of the domain is 0 ≤ x ≤ 1, −0.5 ≤ y ≤ 0.5. Considering the symmetry of the heteromorphic ellipse, Figure 4 shows a plot of the half-curves of y, y', y'', and ρ for analysis.
In Equation (18) and Equation (19), and Figure 4, it is clear that the heteromorphic ellipse y = f (x) has first and second order continuous derivatives on the interval (0, 1). It is known from higher mathematics that both the curves of the heteromorphic ellipse and the first derivative are smooth.
From Equation (20) and Figure 4, it is apparent that the radius of curvature of the heteromorphic ellipse on the interval [0, 1] is continuous and presents a continuous variation over a”2” shape with x. In other words, as x → 0, the radius of curvature ρ → ∞, and with a gradual increase in x, ρ decreases rapidly, and at x = 0.043, ρ quickly reaches a minimum value of 0.248. With further increases in x, ρ gradually increases, and at x = 0.678, ρ reaches a maximum of 1.646. Subsequently, ρ gradually decreases with increasing x at x = 1 and ρ = 0.340.
Figure 4. The half-curves of heteromorphic ellipse y, y', y'', and ρ with x.
Note that as the height, width, and compression coefficient of the heteromorphic ellipse vary, the radius of curvature changes continuously following a “2” shape, but the value of the radius of curvature and the position of the extreme point will change correspondingly.
5. Geometric Properties of the Heteromorphic Elliptical Rotating Body
5.1. Rotating Body Volume
Using definite integral, the volume of the body generated by revolving the heteromorphic ellipse (Equation (6)) through an angle of 360 degree around the x axis (Symmetry axis) is
By performing variable substitution so that x/2a = ζ, the upper limit of the integral becomes 1, (An integral table can be used to evaluate ) and the result is given as
This value is 1.94% larger than the volume of the ellipsoid.
5.2. Surface Area of Rotating Body
Using definite integral, the surface area of the body generated by revolving the heteromorphic ellipse, given in Equation (6), through an angle of 360 degree around the x axis is expressed as
Substituting Equation (6) and Equation (15) into Equation (23), setting x/
λ(θ) is defined as the “surface area coefficient” of the rotating body of heteromorphic ellipse around the x axis. Taking a series of values in interval (0.1 ≤ θ ≤ 10.0) and integrating Equation (25) by numerical integration method, the results are plotted using the “+” points as shown in Figure 5.
According to Figure 5 and the calculations, the surface area coefficient of the rotating body of heteromorphic ellipse around the x axis shows a monotonic rising trend from 0.800 to 5.161 in interval (0.1 ≤ θ ≤ 10.0). The surface area
Figure 5. Variation curves of the surface area coefficient and the surface area ratio of the rotating body with the compression coefficient.
coefficient of the rotating body of Standard-type heteromorphic ellipse around the x axis (θ = 1) is λ(1) = 1.0212, indicating that the surface area of this rotating body is 2.12% larger than that of a sphere with radius r (=a = b). The fitting curve equation of the surface area coefficient λ(θ) is given as
The correlation coefficient R2 = 0.9999, and the absolute values of the relative errors are less than 1.02%.
5.3. Surface Area Ratio of Rotating Body
Definition of the “surface area ratio” of the rotating body of heteromorphic ellipse around the x axis is as follows. Under a same volume of the rotating body, the ratio of Sa, which is the surface area of the rotating body of heteromorphic ellipse around the x axis with different θ, to Sa0, which is the surface area of the rotating body of Standard-type heteromorphic ellipse around the x axis with θ = 1, is the surface area ratio γ(θ) of the rotating body of heteromorphic ellipse around the x axis is given as
The results of calculation are shown as the “×” points in Figure 5.
According to Figure 5 and the calculations, in interval (0.1 ≤ θ ≤ 10.0), the surface area ratio of the rotating body of heteromorphic ellipse around the x-axis presents a curve change rule in the shape of “√”. The minimum surface area ratio γmin = 0.999977 appears at compressibility θ = 1.011561, i.e., the surface area ratio of the rotating body of Standard-type heteromorphic ellipse around the x axis, where γ0 = 1 is not the minimum value of surface area ratio. On the left side of the minimum surface area ratio γmin, γ(θ) increases with decrease of compression coefficient, whereas on the right side, γ(θ) increases with increase of compression coefficient. The fitting curve equation of surface area ratio γ(θ) is given as
The correlation coefficient R2 = 0.9995, and the absolute values of the relative errors are less than 0.98%.
As per the “√” curve Equation (28) of surface area ratio of the heteromorphic elliptical rotating body, the compression coefficient of the corncob is about 0.25, and its surface area ratio is 1.26. Further, the compression coefficient of sunflower disk is about 5.50, and its surface area ratio is 1.64, whereas the compression coefficients of apple, pears, and other fruits are 0.80 - 1.25, and their surface area ratios are proximate to the minimum value of 1.00 (less than 1.01).
6. Case Analysis and Application Prospects
6.1. Case Analysis
Taking a traffic tunnel as a sample case, an application and comparative analysis have been carried out. According to the relevant provisions of China’s highway tunnel design code, Wang have given the design limits (maximum size of 1025 cm × 500 cm) of highway tunnels and the inner contour design of a three-heart-curve round tunnel with a unidirectional double-lane design speed of 80 km/h, as shown in Figure 6 . Table 1 shows the parameters of the three-heart-curve round tunnel.
According to Figure 6 and Table 1, in addition to the inverted arch of backfill area, a three heart curve round tunnel is composed of three arc centers, namely, crown arch, right arch, and left arch. It is composed of a crown arch and two side arches.
Figure 6. Design of heteromorphic elliptical section for a tunnel project.
Table 1. Parameters of the three-heart-curve round tunnel.
If the heteromorphic ellipse (Wu’s curve) is used as the inner contour of the tunnel and the structural boundaries are completely contained, the optimal design can be implemented according to the minimum cross-sectional area of the tunnel to obtain the maximum width W = 1100 cm and the maximum height H = 835 cm for the cross section of the tunnel inner contour. Figure 6 shows the inner contour design for a heteromorphic elliptical tunnel.
The main advantages and disadvantages of the inner contour of a heteromorphic elliptical tunnel and the three-heart-curve round cross section have been compared and analyzed as given below.
1) The inner contour of a heteromorphic elliptical tunnel requires only two design parameters, whereas the three-heart-curve round cross section requires four independent parameters. The former is a trivial one for design and optimization, easy to implement, and enables error control.
2) Substituting Formula (10), W = 1100 cm and H = 835 cm into Formula (8), the inner contour area of the heteromorphic elliptical tunnel is obtained as
and following the calculations from a geometric figure, the cross-sectional area of the three-heart-curve round tunnel is 74.04 m2. The former can save 1.34% of the earthwork of excavation for a long-distance tunnel.
3) From the compression coefficient θ = W/H = 1.317, the heteromorphic elliptic coefficient Th = 3.200 can be calculated. Then the inner contour perimeter of the heteromorphic elliptical tunnel is L = Th(W + H)/2 = 30.96 m, and as calculated from the geometric figure, the cross-sectional perimeter of the three-heart-curve round tunnel is 31.56 m. The former can save 1.88% in tunnel support and lining construction material, which is beneficial for reducing project investment.
4) The maximum height of the inner contour of the heteromorphic elliptical tunnel is H = 835 cm and the maximum height of the cross section of the three-heart-curve round tunnel is H = 825 cm. The maximum widths of the two tunnel cross sections are equal. Figure 6 shows that the top arch of the inner contour of the heteromorphic elliptical tunnel is slightly higher, which helps to increase tunnel ventilation and to withstand the pressure at the top of the tunnel. Moreover, the inner contour of the heteromorphic elliptical tunnel overcomes the convexities of the arch footing and the cross section of the three-heart-curve round tunnel, which helps to save the quantity of backfill concrete required for the inverted arch.
5) There is a continuous second-order derivative in the inner contour of the heteromorphic elliptical tunnel, and the radius of curvature shows a continuous change alonga”2” shape with the height of the cross section. However, the three values of the radii of curvature in the cross section of the three-heart-curve round tunnel is apparently distorted at the connection point (tangent point).
6.2. Application Prospects
In human civilization, patterns develop following two basic geometric shapes, viz. one is straight-line and rectangular, and the other is an ellipse (including circle and circle arcs). The straight line makes things easy to reconcile with the shortest distance, whereas the ellipse gives a flexible and easily moving feeling, both physically and psychologically. For centuries, designers have been limited by this “round or square” classification of thinking. Custom-developed hyper- elliptical shapes have not been used until Piet Hein’s design . The discovery of the equation of the heteromorphic ellipse (Wu’s curve) undoubtedly have expanded our scope to choose graphic forms.
The tunnel is an engineering structure buried in an earth stratum and is a form of human use of underground space. For more than a century, due to the lack of suitably shaped curves, the inner contours of existing tunnels were mainly characterized by three-centered, four-centered, and five-centered circles and shapes created by arc splicing. Taking a four-centered circular tunnel cross section as an example, there are five independent variables: radius r1, r2, r3, and the central angles φ1, φ2 . The inner contour of a tunnel formed by multiple circular arcs can provide continuity and smoothness of the curve at the connection point (tangent point).However, the first derivative is continuous and not smooth, but the second derivative, where the curvature distortion and stress concentration phenomena occur, is discontinuous . From the analysis as shown in Section 4.5, it is obvious that the inner contour of a heteromorphic elliptical tunnel has obvious advantages as compared with the cross section created by multiple arc splicing. The inner contour of a heteromorphic elliptical tunnel and its derivatives has better continuity, differentiability, smoothness, and integrity.
• Liquid transport tank
The cross section of the common liquid transport tank is a circle, an ellipse, or a hyper-elliptical shape between an ellipse and a rectangle. If the tank is liquid-filled, the position of the cross-sectional centroid (center of gravity) is the same as the median line. At the same width and height for heteromorphic ellipses and ellipses, the centroid position of a cross section of the heteromorphic ellipse is 7.05% lower than the median line (the ellipse centroid), which can greatly improve the safety of liquid tanks in vehicle driving, especially on curves. The heteromorphic ellipse is wider in the lower part and flatter at the bottom than a circle or an ellipse, and the required bracket is short, has a small force to resist, and easy to fix. Therefore, the heteromorphic ellipse has a good stability on a vehicle chassis, and it is an ideal cross-sectional shape for a liquid transport tank.
• Airplanes and submarines
The cross sections of common airplanes and submarines are circular. Under the same width and height conditions, the lower space of a heteromorphic ellipse is larger than that of a circle, which increases the available space for equipment installation and cargo loading, and the center of gravity of airplanes can be lowered accordingly, which is beneficial for flight stability. For submarines, the inverted cross section of the heteromorphic ellipse is used, with which the upper part of the submarine is wider than the circular cross section. Hence, it is easy to decorate the offices and living spaces, and the lower part of the submarine can be used for equipment installation and power. The center of buoyancy of an inverted heteromorphic ellipse is 7.05% higher than that of a circle, which is favorable for keeping submarines in a stable equilibrium state. This is a new field worth studying by researchers in related disciplines and applications.
• Bridges, buildings, furniture, and handicrafts
Like the superellipse, the heteromorphic ellipse (Wu’s curve) can also be used in the design of bridge piers, square buildings, furniture, and handicrafts. Examples include beds, tables, silverware, dishes, vases, candlesticks, casseroles, soup pots, teapots, and wine glasses.
It has to be emphasized that the heteromorphic ellipse is invented based on the isoconcentration-curve formula as the solution of a simplified advection-diffusion transport of a constant point source. If considering the pedicle as a point source during fruit growth, it is intuitively obvious that the shape of the fruit such as the corn cob, apple, persimmon, pineapple and pear, is very similar or close to the heteromorphic ellipse: their shape is not only affected by the type of species and natural conditions (climate and sunlight, for example), but also constrained by the isoconcentration condition during nutrition transport. Therefore, heteromorphic ellipse can also be a research object of great concern for the botanical research community.
1) An equation for anew heteromorphic elliptic (Wu’s curve) with two parameters for x-symmetry is defined. In the heteromorphic ellipse, the corresponding height of the maximum width and the height of the centroid position are given. The shapes of heteromorphic ellipses are classified into a heteromorphic ellipse, which belongs to the H-type with a compression coefficient of 0 < θ = b/a < 1, standard-type with θ = 1, and the W-type with θ > 1.
2) The area formula can be given as, the area of a heteromorphic ellipse is equal to the product of the area coefficient, height, and width, viz. S = μ(
3) It has been shown that the first- and second-order derivatives of the heteromorphic ellipse are continuous, indicating that both the heteromorphic ellipse and its first-derivative curve are smooth, and the variation in the radius of curvature with the height coordinate x follows a continuous “2” shape. Thereafter, the geometric properties of the rotating body are proposed.
4) The case analysis shows that, when compared with the three-heart-curve round cross section, the inner contour of a heteromorphic elliptical tunnel has the advantages of fewer design parameters, a continuous radius of curvature, smaller tunnel excavation area, shorter circumference of support and lining, higher cavern (which favors ventilation), and smaller backfilling quantities.
5) The analysis also shows that the heteromorphic ellipse and its derivatives have good continuity, differentiability, smoothness, and integrity, which give these shapes broad application prospects in the design of tunnels, liquid transport tanks, planes, submarines, bridges, buildings, furniture, and crafts.
This research is supported by the National Natural Science Foundation of China (Grant No. 51379097, No. 50979036).
We thank International Science Editing (http://www.internationalscienceediting.com) for editing this manuscript.
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Naggar, H.E. and Hinchberger, S.D. (2012) Approximate Evaluation of Stresses in Degraded Tunnel Linings. Soil Dynamics & Earthquake Engineering, 43, 45-57. https://doi.org/10.1016/j.soildyn.2012.07.016 |
At that time, there was general discussion among young physicists about the possible ways to establish a coherent quantum theory, a coherent quantum mechanics. Among the many attempts, the most interesting for me was the attempt of H.A. Kramers to study the dispersion of atoms and, by doing so, to get some information about the amplitudes for the radiation of atoms. In this connection, it occurred to me that in the mathematical scheme these amplitudes behaved like the elements of a mathematical quantity called a matrix. So I tried to apply a mathematical calculus to the experiments of Kramers, and the more general mechanical models of the atom, which later turned out to be matrix mechanics. It so happened at that time I became a bit ill and had to spend a holiday on an island to be free from hay fever. It was there, having good time to think over the questions, that I really came to this scheme of quantum mechanics and tried to develop it in a closed mathematical form.
My first step was to take it to W. Pauli, a good friend of mine, and to discuss it with him, then to Max Born in Göttingen. Actually, Max Born and Pascual Jordan succeeded in giving a much better shape and more elegant form to the mathematical scheme. From the mathematical relations I had written down, they derived the so-called commutation relations. So, through the work of Born and Jordan, and later Paul Dirac, the whole thing developed very quickly into a closed mathematical scheme.
I also went to discuss it with Niels Bohr, but I can't be sure whether this was in July, August, or September of that year .
Half a year later the first papers of E. Schrödinger became known. Schrödinger tried to develop an older idea of Louis de Broglie into a new mathematical scheme, which he called wave mechanics. He was actually able to treat the hydrogen atom on the basis of his wave mechanical scheme and, in the summer of 1926, he was also able to demonstrate that his mathematical scheme and matrix mechanics were actually two equivalent mathematical schemes, that they could be simply translated into each other. After that time, we all felt that this must be the final mathematical form of quantum theory.
DP Had you and Bohr begun the interpretation of this work before Schrödinger's paper came out?
Of course, there was continuous discussion, but only after Schrödinger's paper did we have a new basis for discussion, a new basis for interpreting quantum theory. In the beginning there was strong disagreement between Schrödinger and ourselves, not about the mathematical scheme, but about its interpretation in physical terms. Schrödinger thought that by his work physics could again resume a shape which could well be compared with Maxwell's theory or Newton's mechanics, whereas we felt that this was not possible. Through long discussions between Bohr and Schrödinger in the fall of 1926, it became apparent that Schrödinger's hopes could not be fulfilled, that one needed a new interpretation. Finally, from these discussions, we came to the idea of the uncertainty relations, and the rather abstract interpretation of the theory.
PB Did Schrödinger ever like that interpretation?
He always disliked it. I would even guess that he was not convinced. He probably thought that the interpretation which Bohr and I had found in Copenhagen was correct in so far as it would always give the correct results in experiments; still he didn't like the language we used in connection with the interpretation. Besides Schrödinger, there were also Einstein, M. von Laue, M. Planck, and others who did not like this kind of interpretation. They felt it was too abstract, and too far removed from the older ideas of physics. But, as you know, this interpretation has, at least so far, stood the test of all experiments, whether people like it or not.
PB Einstein never really liked it, even until the day he died, did he?
I saw Einstein in Princeton a few months before his death. We discussed quantum theory through one whole afternoon, but we could not agree on the interpretation. He agreed about the experimental tests of quantum mechanics, but he disliked the interpretation.
DP I felt that at some point there was a slight divergence between your views and Bohr 's, although together you are credited with the Copenhagen interpretation of quantum mechanics.
That is quite true, but the divergence concerned more the method by which the interpretation was found than the interpretation itself. My point of view was that, from the mathematical scheme of quantum mechanics, we had at least a partial interpretation, inasmuch as we can say, for instance, that those eigenvalues which we determine are the energy values of the discrete stationary states, or those amplitudes which we determine are responsible for the intensities of the emitted lines, and so on. I believed it must be possible, by just extending this partial interpretation, to get to a complete interpretation. Following this way of thinking, I came to the uncertainty relations.
Now, Bohr had taken a different starting-point. He had started with the dualism between waves and particles - the waves of Schrödinger and the particles in quantum mechanics - and tried, from this dualism, to introduce the term complementarity, which was sufficiently abstract to meet the situation. At first we both felt there was a real discrepancy between the two interpretations, but later we saw that they were identical. For three or four weeks there was a real difference of opinion between Bohr and myself, but that turned out to be irrelevant.
DP Did this have its origin in your different philosophical approaches?
That may be. Bohr's mind was formed by pragmatism to some extent, I would say. He had lived in England for a longer period and discussed things with British physicists, so he had a pragmatic attitude which all the Anglo-Saxon physicists had. My mind was formed by studying philosophy, Plato and that sort of thing. This gives a different attitude. Bohr was perhaps somewhat surprised that one should finally have a very simple mathematical scheme which could cover the whole field of quantum theory. He would probably have expected that one would never get such a self-consistent mathematical scheme, that one would always be bound to use different concepts for different experiments, and that physics would always remain in that somewhat vague state in which it was at the beginning of the 1920s.
DP In the interpretation you gave at that time, you seemed to imply that there did exist an ideal path and that somehow the act of measuring disturbed the path. This is not quite the same as the interpretation that you hold now, is it?
I will say that for us, that is for Bohr and myself, the most important step was to see that our language is not sufficient to describe the situation. A word such as path is quite understandable in the ordinary realm of physics when we are dealing with stones, or grass, etc., but it is not really understandable when it has to do with electrons. In a cloud chamber, for instance, what we see is not the path of an electron, but, if we are quite honest, only a sequence of water droplets in the chamber. Of course we like to interpret this sequence as a path of the electron, but this interpretation is only possible with restricted use of such words as position and velocity. So the decisive step was to see that all those words we used in classical physics - position, velocity, energy, temperature, etc. - have only a limited range of applicability.
The point is we are bound up with a language, we are hanging in the language. If we want to do physics, we must describe our experiments and the results to other physicists, so that they can be verified or checked by others. At the same time, we know that the words we use to describe the experiments have only a limited range of applicability. That is a fundamental paradox which we have to confront. We cannot avoid it; we have simply to cope with it, to find what is the best thing we can do about it.
DP Would you go so far as to say that the language has actually set a limit to our domain of understanding in quantum mechanics?
I would say that the concepts of classical physics which we necessarily must use to describe our experiments do not apply to the smallest particles, the electrons or the atoms - at least not accurately. They apply perhaps qualitatively, but we do not know what we mean by these words.
Niels Bohr liked to tell the story about the small boy who comes into a shop with two pennies in his hands and asks the shopkeeper for some mixed sweets for the two pennies. The shopkeeper gives him two sweets and says 'You can do the mixing yourself.' This story, of course, is just meant to explain that the word mixing loses its meaning when we have only two objects. In the same sense, such words as position and velocity and temperature lose their meaning when we get down to the smallest particles.
DP The philosopher Ludwig Wittgenstein originally started out by thinking that words were related to facts in the world, then later reversed his position to conclude that the meaning of words lay in their use. Is this reflected in quantum mechanics?
I should first state my own opinion about Wittgenstein's philosophy. I never could do too much with early Wittgenstein and the philosophy of the Tractatus Logico-philosophicus, but I like very much the later ideas of Wittgenstein and his philosophy about language. In the Tractatus, which I thought too narrow, he always thought that words have a well-defined meaning, but I think that is an illusion. Words have no well-defined meaning. We can sometimes by axioms give a precise meaning to words, but still we never know how these precise words correspond to reality, whether they fit reality or not. We cannot help the fundamental situation - that words are meant as a connection between reality and ourselves - but we can never know how well these words or concepts fit reality. This can be seen in Wittgenstein's later work. I always found it strange, when discussing such matters with Bertrand Russell, that he held the opposite view; he liked the early work of Wittgenstein and could do nothing whatsoever with the late work. On these matters we always disagreed, Russell and I.
I would say that Wittgenstein, in view of his later works, would have realized that when we use such words as position or velocity, for atoms, for example, we cannot know how far these terms take us, to what extent they are applicable. By using these words, we learn their limitations.
DP Would it be true to say that quantum mechanics has modified language, and, in turn. language will re-modify the interpretation of quantum mechanics?
There I would not quite agree. In the case of relativity theory, I would agree that physicists have simply modified their language; for instance, they would use the word simultaneous now with respect to certain coordinate systems. In this way they can adapt their language to the mathematical scheme. But in quantum theory this has not happened. Physicists have never really tried to adapt their language, though there have been some theoretical attempts. But it was found that if we wanted to adapt the language to the quantum theoretical mathematical scheme, we would have to change even our Aristotelian logic. That is so disagreeable that nobody wants to do it; it is better to use the words in their limited senses, and when we must go into the details, we just withdraw into the mathematical scheme.
I would hope that philosophers and all scientists will learn from this change which has occurred in quantum theory. We have learned that language is a dangerous instrument to use, and this fact will certainly have its repercussions in other fields, but this is a very long process which will last through many decades I should say.
Even in the old times philosophers realized that language is limited; they have always been skeptical about the unlimited use of language. However, these doubts or difficulties have, perhaps, been enhanced through the present developments in physics. I might mention that most biologists today still use the language and the way of thinking of classical mechanics; that is, they describe their molecules as if the parts of the molecules were just stones or something like that. They have not taken notice of the changes which have occurred in quantum theory. So far as they get along with it, there is nothing to say against it, but I feel that sooner or later, also in biology, one will come to realize that this simple use of pictures, models, and so on will not be quite correct.
PB At what point does the transition occur from the non-path to the path in a biological system? Is a DNA molecule already a classical object, or is a cell a classical object?
There is, of course, not a very well defined boundary; it is a continuous change. When we get to these very small dimensions we must be prepared for limitations. I could not suggest any well-defined point where I have to give up the use of a word. It's like the word mixing in the story; you cannot say 'when I have two things, then I can mix them.' But what if you have five or ten? Can you mix then?
PB It seems to me that there is something very important here about language. We are living beings formed from coherent structures like DNA and we apparently have classical paths and our existence is understandable within this language. But then we can analyse by reducing these complex, coherent wholes to smaller and smaller parts, and is it nor perhaps this process of reduction that is at the root of the paradox?
I would say that the root of the difficulty is the fact that our language is formed from our continuous exchange with the outer world. We are a part of this world, and that we have a language is a primary fact of our life. This language is made so that in daily life we get along with the world, it cannot be made so that, in such extreme situations as atomic physics, or distant stars, it is equally suited. This would be asking too much.
PB Is there a fundamental level of reality?
That is just the point; I do not know what the words fundamental reality mean. They are taken from our daily life situation where they have a good meaning, but when we use such terms we are usually extrapolating from our daily lives into an area very remote from it, where we cannot expect the words to have a meaning. This is perhaps one of the fundamental difficulties of philosophy: that our thinking hangs in the language. Anyway, we are forced to use the words so far as we can; we try to extend their use to the utmost, and then we get into situations in which they have no meaning.
DP In discussing the 'collapse of the wave function' you introduced the notion of potentiality. Would you elaborate on this idea?
The question is: 'What does a wave function actually describe?' In old physics, the mathematical scheme described a system as it was, there in space and time. One could call this an objective description of the system. But in quantum theory the wave function cannot be called a description of an objective system, but rather a description of observational situations. When we have a wave function, we cannot yet know what will happen in an experiment; we must also know the experimental arrangement. When we have the wave function and the experimental arrangement for the special case considered, only then can we make predictions. So, in that sense, I like to call the wave function a description of the potentialities of the system.
DP Then the interaction with the apparatus would be a potentiality coming into actuality?
DP May I ask you about the Kantian notion of the 'a priori' an idea which you introduced, in a modified sense, into your discussions of quantum theory.
As I understand the idea of 'a priori,' it stresses the point that our knowledge is not simply empirical, that is, derived from information obtained from the outer world through the senses and changed into data in the content of our brain. Rather, 'a priori' means that experience is only possible when we already have some concepts which are the precondition of experience. Without these concepts (for instance, the concepts of space and time in Kant's philosophy), we would not even be able to speak about experience.
Kant made the point that our experience has two sources: one source is the outer world (that is, the information received by the senses), and the other is the existence of concepts by which we can talk about these experiences. This idea is also borne out in quantum theory.
PB But these concepts are pary of the world also.
Whether they belong to the world, that is hard to say; we can say that they belong to our way of dealing with the world.
PB But we belong to the world, so, in a sense, these activities of ours also belong to the world.
In that sense, yes.
DP You modified the 'a priori' by introducing it as a limited concept, is that true?
Of course, Kant would have taken the 'a priori' as something more absolute than we would do in quantum theory. For instance, Kant would perhaps have said that Euclidean geometry would be a necessary basis for describing the world, while we, after relativity, would say that we need not necessarily use Euclidean geometry; we can use Riemannian geometry, etc. In the same way, causality was taken by Kant as a condition for science. He says that if we cannot conclude from some fact that something must have been before this fact, then we do not know anything, and we cannot make observations, because every observation supposes that there is a causal chain connecting that which we immediately experience to that which has happened. If this causal chain does not exist, then we do not know what we have observed, says Kant. Quantum theory does not agree with this idea, and in fact proves that we can even work in cases where this causal chain does not exist.
DP In a recent theory of yours, is not causality retained, perhaps in a new form?
We have causality in that sense - that in order to influence something, there must be an action from one point to the next point; no action can happen if there is not this connection. But at this point one gets into rather complicated details.
DP But even so, you do have causality predicated on the idea of separation and action, so this again comes back to a philosophical level: what you mean by separation, and by interaction.
We must speak about 'interaction' and 'separation,' that is quite true, and we use the terms as we did in classical theory. But, again, we see limitation. Complete separation of two events may be possible in classical theory; it is not possible in quantum theory. So we use the terms together with the fact of their limitation.
DP What exactly are the criteria for something to be classical?
I would say the criteria are simply that we can get along with these concepts (e.g. 'position,' 'velocity,' 'temperature,' 'energy'), and so long as we get along with them, then we are in the classical domain. But when the concepts are not sufficient, then we must say that we have gone beyond this classical domain.
Every system in physics (forget for the moment about biological systems) is always quantum theoretical, in the sense that we believe that quantum theory gives the correct answers for its behaviour. When we say that it is classical, we mean that we do get the correct or the necessary answers by using classical concepts (at least in that approximation in which we can describe the system by classical concepts). So a system is classical only within certain limits and these limits can be defined.
DP How would you include things like irreversibility?
Thermodynamics is a field which goes beyond Newtonian mechanics, inasmuch as it introduces the idea of thermodynamic equilibrium, or canonical distribution as W. Gibbs has put it. Thermodynamics leaves classical physics and goes into the region of quantum theory, for it speaks about situations of observation; it does not speak about the system as it is, but about the system in a certain state of being observed, namely in the state of temperature equilibrium. If this equilibrium is not obeyed, then we cannot use thermodynamics. So the whole concept of irreversibility is bound up with the concept of thermodynamic equilibrium.
DP And is this ultimately connected with the idea of a classical limit to something? I am thinking of the measurement problem that always seems to be associated with an irreversible process: that we have a definite result for a quantum mechanical system where the quantum mechanics itself doesn't seem to predict a definite result. That is, the idea of a quantum mechanical measurement seems to be tied up with the idea of an irreversible trend.
Yes, to some extent, because on the side of the observer we do use classical concepts. The idea that we do observe something already indicates something irreversible. If we draw a pencil line on a paper, for instance, we have established something which cannot be undone, so to speak. Every observation is irreversible, because we have gained information that cannot be forgotten.
DP To what extent is this related to the symmetry-breaking of the quantum mechanical system where one gets classical observables?
I would not like to connect it with symmetry-breaking; that is going a bit far. We try to describe the observational situation by writing down a wave function for the object and the equipment which is in interaction with this wave function. Just by using classical words for the equipment, we have already made the assumption of irreversibility. Or we make the assumption of statistical behaviour, because the mere use of classical words for this observation on the side of the system makes it impossible to know the total wave function of object and equipment. But we cannot use quantum theory for the equipment in a strict sense, because if we wrote down the wave function for the object and the equipment, we could not use classical words for the equipment, so we would not observe anything. We do observe only when we use classical concepts, and just at this point this hypothesis of disorder, of statistical behaviour, comes in.
DP With regard to something like ferromagnetism, the quantum mechanical system has given rise to a macroscopic ordering. Is it true to say that a quantum mechanical system has actually broken its own symmetry and given rise to a classical variable, without any talk about a measuring apparatus, or anything exterior to the system?
Let us consider a ferromagnet as isolated from the rest of the world for some time, and then ask what the lowest state of the system is. We find, from the quantum mechanical calculations, that the lowest state is one in which the whole system has a very large component of magnetic momentum. If we then ask 'what do we observe when we consider this system?' we see that it is convenient to ascribe the classical variable 'magnetic momentum' to the system. So we can use classical terms to describe this quantum mechanical behaviour. But this is not really a problem of observation, only a problem of how the lowest state of the system is defined.
PB How does quantum mechanics deal with time flow or does it in fact say anything at all about it?
I would have to repeat what C. von Weizsäcker said in his papers: that time is the precondition of quantum mechanics, because we want to go from one experiment to another, that is from one time to another. But this is too complicated to go into in detail. I would simply say that the concept of time is really a precondition of quantum theory.
PB In the domain where quantum mechanics operates, all of the equations are reversible with respect to time, except for one experiment I believe. So time has more to do with macroscopic classical systems than microscopic quantum systems.
I would say that irreversibility of time has to do with this other system, with those problems which I. Prigogine describes in his papers, and is certainly extremely important for the macroscopic application of quantum theory, and also for biology, of course.
DP Can we talk about a new theory of yours, the non-linear theory of elementary particles? Are you ultimately going to introduce things like gravitation into this theory, and go over to a picture in which space and time emerge?
Again, we have a similar situation as in ferromagnetism. We try to solve the quantum mechanical, or quantum theoretical equation, but we can see that the system acquires properties which then can be described by classical language (e.g. like speaking of a magnetic momentum, etc.). We are hoping that such phenomena as electromagnetic radiation and gravitation also can come out of the theory of elementary particles, and we have reasons to believe that this is so.
DP The idea of symmetry is a very important part of your theory.
Let's begin more simply by speaking about quantum mechanics, disregarding now the difficulties of elementary particle physics. In quantum mechanics we see that macroscopic bodies have very complicated properties, complicated shapes and chemical behaviour and so on. Coming down to smaller and smaller particles, we finally come to objects which are really very much simpler, for example the stationary states of a hydrogen atom. We describe its properties by saying that these states are a representation of the fundamental symmetries, such as rotation in space. So when we describe a system by writing down a few quantum numbers (in hydrogen atoms, we have the principal quantum number and the angular momentum number) this means that we know nothing except to say that this object is a representation of symmetries. The quantum numbers tell us which kind of symmetries we mean; the numbers themselves say that this object has these special properties. Thus, when we come to the smallest objects in the world, we characterize them in quantum mechanics just by their symmetry, or as representations of symmetries, and not by specifying properties such as shape or size.
DP There are symmetries that are nor related to operations in the world, e.g. the internal symmetries such as isospin. What meaning do they have? Do you think they are related ultimately to the properties of space and time?
I suspect that isospin is a symmetry similar to space and time. I cannot say that it is related to them. I would say that there are a number of fundamental symmetries in this world which may in future be reduced to something still simpler, but so far we must take them as given, as a result of our experiments. One of the most fundamental symmetries is the symmetry of the Lorentz group, that is space and time, and then isospin groups, scale groups, and so on. So there are a number of groups which are fundamental in the sense that in describing the smallest particles we refer to their behaviour and transformations.
The idea is that one can distinguish between a natural law, a fundamental law, which determines for instance a spectrum of elementary particles, and the general behaviour of the cosmos, which is perhaps something not at once given through this law. I might remind you, for instance, of Einstein' s equations of gravitation. Einstein wrote down his field equations and thought that gravitational fields are always determined by them. But the cosmos is not unambiguously determined by these field equations, although there are several models of the cosmos which are compatible with them. In the same sense, I would say that there is an underlying natural law which determines the spectrum of elementary particles, but the shape of the cosmos is not unambiguously determined by this law. Logically, it would be possible to have various types of cosmos which are in agreement with it. However, if a certain cosmological model has been 'chosen,' then this model, of course, has some consequences for the spectrum of elementary particles.
DP Are you saying that there exist laws which are independent or outside the universe, outside the world, which reality breaks, or that it breaks the symmetry represented by the laws?
'Laws' just means that some fundamental symmetries are inherent either in nature or in our observation of nature. You may know about the attempts of Weizsäcker, who tried to derive the laws simply from logic. We have to use language to arrive at conclusions, to study alternatives, and he questions whether from the alternatives alone we can arrive at these symmetries. I don't know whether his attempts are successful or not. In physics, we can only work with the assumption that we have natural laws. If we have no natural laws, then anything can happen, and we can only describe what we see, and that's all.
DP Another feature of your theory which seems to go against the current trend- partons and quarks, etc. - is that you feel that no particle is any more elementary than any other.
Even if quarks should be found (and I do not believe that they will be), they will not be more elementary than other particles, since a quark could be considered as consisting of two quarks and one anti-quark, and so on. I think we have learned from experiments that by getting to smaller and smaller units, we do not come to fundamental units, or indivisible units, but we do come to a point where division has no meaning. This is a result of the experiments of the last twenty years, and I am afraid that some physicists simply ignore this experimental fact.
DP So it would seem that elementary particles are just representations of symmetries. Would you say that they are not fundamental things in themselves, or 'building-blocks of the universe,' to use the old-fashioned language?
Again, the difficulty is in the meaning of the words. Words like building blocks or really existing are too indefinite in their meaning, so I would hesitate to answer your questions, since an answer would depend on the definitions of the words.
DP To be more precise, ultimately could one have a description of nature which needed only elementary particles or, alternatively, a description in which the elementary particles would be defined in terms of the rest of the universe? Or is there no starting-point, as it were, no single axiom on which one can build the whole of physics?
No. Even if, for instance, that formula which Pauli and I wrote down fifty years ago turned out to be the correct formulation for the spectrum of elementary particles, it is certainly not the basis for all of physics. Physics can never be closed, or brought to an end, so that we must turn to biology or such things. What we can hope for, I think, is that we may get an explanation of the spectrum of elementary particles, and with it also an explanation of electromagnetism and gravitation, in the same sense as we get an explanation of the spectrum of a big molecule from the Schrödinger equation.
This does not mean that thereby physics has come to an end. It means that, for instance, at the boundary between physics and biology, there may be new features coming in which are not thought of in physics and chemistry. Something entirely new must happen when I try to use quantum theory within the realm of biology. Therefore I criticize those formulations which imply an end to physics.
DP Is it possible to reduce physics or any element of physics purely to logic and axioms?
I would say that certain parts of physics can always be reduced to logical mathematics or mathematical schemes. This has been possible for Newtonian physics, for quantum mechanics, and so on, so I do not doubt that it will also be possible for the world of the elementary particles. In astrophysics today, one comes upon pulsars and black holes, two regions in which gravitation becomes enormous, and perhaps a stronger force than all other forces. I could well imagine that in such black holes, for instance (if they exist), the spectrum of elementary particles would be quite different from the spectrum we now have. In the black holes, then, we would have a new area of physics, to some extent separated from that part which we now call elementary particle physics. There would be connections, and one would have to study how to go from the one to the other; but I do not believe in an end of physics.
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Tangents Teacher Resources
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New Review Tangent Lines and the Radius of a Circle
Your Geometry learners will collaboratively prove that the tangent line of a circle is perpendicular to the radius of the circle. A deliberately sparse introduction allows for a variety of approaches to find a solution.
9th - 12th Math 3 Views 0 Downloads CCSS: Designed
Ratios of Right Triangles
Students find the angle measures of a right triangle given two side lengths. In this finding the angle measures of a right triangle given two side lengths instructional activity, students record the ratio of a leg and hypotenuse of a right triangle.
9th - 12th Math 14 Views 65 Downloads
The unit circle definition of trigonometric function
In this video, Sal shows how the sine and cosine of an angle are defined on the unit circle. He inscribes a 30-60-90 triangle in the unit circle and explains how one can use SOH-CAH-TOA to find the coordinates on the circle.
9 mins 10th - 12th Math 3 Views 1 Download
Exploring Trigonometric Ratios
Learners identify the ratios of trig functions. In this trigonometry instructional activity, students construct triangles and use it to derive the values of sine, cosine and tangent. They use Cabri software to create the visual of the identities.
9th - 10th Math 3 Views 18 Downloads
Sine, Cosine, Tangent
In this sine, cosine, tangent worksheet, students read short statements, draw illustrations of the statement and then use trigonometric ratios to determine the length of the missing side or measurement of an angle. This five-page worksheet contains 3 multi-step problems.
10th - 12th Math 11 Views 73 Downloads
Graphing in Circles
Students explore the concept of the unit circle. In this unit circle lesson plan, students create a scatter plot of the sine, cosine, and tangent function using data from the x and y values on the unit circle. Students generalize the tangent, sine, and cosine relationships.
9th - 12th Math 4 Views 32 Downloads
Ratios of Right Triangles
Tenth graders explore right triangle trigonometry. In this geometry lesson, 10th graders investigate the sine and cosine ratios for right triangles. The use of Cabri Jr. allows for measurement of side lengths and angles and allows students to examine of the ratios and to form and verify conjectures.
10th Math 14 Views 83 Downloads
Students explore properties of tangent lines. In this properties of tangent lines lesson plan, students discuss the perpendicular relationship between a tangent line on a circle and the circle's radius. Students use that relationship to determine unknown distances.
9th - 12th Math 3 Views 29 Downloads
How Do You Find a Missing Side of a Right Triangle Using Cosine?
Trigonometry? What's trigonometry. Well, something to do with triangles, right? Right! Right triangles! In this problem, find the missing length of one of the sides. So set this up as a ratio: cosine(A) = adjacent side length divided by the length of the hypotenuse.
5 mins 6th - 11th Math 3 Views 1 Download
SOH CAH TOA What?
Learners identify the different ratios of a right triangle. In this trigonometry instructional activity, students use the Pythagorean Theorem to find the ratios of the sides of a right triangle. They identify the measurements of sine, cosine and tangent.
8th - 9th Math 3 Views 35 Downloads
The Clock Tower
Students explore the concept of finding the height of a building. In this finding the height of a building lesson, students use clinometers to determine the angle of depression or elevation. Students use sine, cosine, and the angle of elevation/depression to find the height of a clock tower/building on campus.
9th - 12th Math 3 Views 26 Downloads
Graphing In Circles
Students explore the concept of circles. In this circles instructional activity, student take sine and cosine values from the unit circle and plot them on a coordinate plane. Students discuss the relationship between the sine and cosines graph and the unit circle.
9th - 12th Math 5 Views 33 Downloads
Technology for Displaying Trigonometric Graph Behavior: Sine and Cosine
Students explore the concept of sine and cosine. In this sine and cosine lesson, students work in groups to graph sine and cosine functions. Students examine how coefficients in various parts of each parent function shift the graph. Students discuss vertical and horizontal shifts as well as shrinks and stretches of the parent graphs.
10th - 12th Math 3 Views 24 Downloads
When is Tangent, tangent?
Students study the concept of tangent of a unit circle. In this unit circle lesson, students explore the mathematical history of the trigonometric ration, tangent, through an interactive date-gathering construction that simulates an experience that mirrors how values of trig functions may have been approximated.
9th - 12th Math 3 Views 30 Downloads
Chords, Tangents, Arcs and Angles
Students use Geometer's Sketchpad or Patty Paper Geometry to explore and write conjectures about chords, tangents, arcs and angles. In this geometric conjecture lesson, students examine what a conjecture is as it relates to geometric properties.
9th - 12th Math 10 Views 29 Downloads |
What are the applications of mechanics?
It has numerous applications in a wide variety of fields and disciplines, including but not limited to structural engineering, astronomy, oceanography, meteorology, hydraulics, mechanical engineering, aerospace engineering, nanotechnology, structural design, earthquake engineering, fluid dynamics, planetary sciences, …
What are the examples of classical mechanics?
Classical mechanics describes the motion of all objects that we can see. This would include all machines, any inanimate object in motion, planets, stars, galaxies …. etc.
Why is classical mechanics important in engineering?
Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light.
What are the application of Newton’s mechanics?
Newtonian mechanics is useful for the application of the law of motion, the study of astronomical bodies, relative motion and motions of objects on earth. It is the foundation of modern mechanics. Newtonian mechanics involves constraint forces. Newtonian mechanics is used for vectors.
What are 3 applications of mechanics to your daily life?
Classical mechanics or Newtonian mechanics have many applications in daily life. Seat belts in the cars is an example for concept of inertia. Using air bubble packing for the fragile objects is an example for impulse. Banking of roads and railway tracks are an example for uniform circular motion.
What are the applications of mechanics in daily life?
It plays an important role in designing and analysing many mechanical systems, such as aircraft, automobiles, bridges, buildings, machinery, pipelines, ships, satellites, and spacecraft. There are three main branches of engineering mechanics: Statics. Dynamics.
Who is father of classical mechanics?
On this day, in 1642, Sir Isaac Newton was born. He would be 371. Newton was a physicist and mathematician from England. His work laid the foundation of classical mechanics (also called Newtonian physics or mechanics in his honor) and is generally credited with jump starting the scientific revolution.
Who is father of classical physics?
Galileo is called the father of classical physics because he was the pioneer of classical physics. Galileo initiated this discipline of study by conducting various scientific experiments. In light of this, Galileo applied scientific methodologies to revolutionize classical physic.
What is the principle of classical mechanics?
In this chapter we will demonstrate, using the Lagrangian and Hamiltonian formulations, that the three fundamental conservation principles of classical mechanics: the conservation of energy, linear momentum, and angular momentum, are all consequences of certain spatial-temporal symmetries of classical mechanical …
Why is it called classical mechanics?
However, because this older branch of physics existed alongside two new ones, it needed a new name. The term classical mechanics was coined to loosely label the set of equations that describe reality at scales where quantum and relativistic effects are negligible.
What is the origin of classical mechanics?
The phrase “classical mechanics” was coined in the early twentieth century to designate the mathematical physics system developed by Isaac Newton and many other contemporary seventeenth-century thinkers, building on Johannes Kepler’s earlier astronomical theories.
What is classical mechanics and its limitations?
Classical mechanics is the mathematical science that studies the displacement of bodies under the action of forces. Gailieo Galilee initiated the modern era of mechanics by using mathematics to describe the motion of bodies.
What are the 5 applications of physics?
- Walking and Running: One of the simple examples of physics in everyday life is the frictional force which helps us to walk or run easily. …
- Alarm clock: …
- Ballpoint pen: …
- Roller coasters: …
- Fans: …
- Refrigerators: …
- Cell phones: …
What is application of first law of motion?
If you jump from a car of bus that is moving, your body is still moving in the direction of the vehicle. When your feet hit the ground, the grounds act on your feet and they stop moving. You will fall because the upper part of your body didn’t stop and you will fall in the direction you were moving.
What is a real life application of Newtons law?
Here are some examples of Newton’s first law: A ball rolling on a flat surface will keep on running unless an outside force stops it. A car moving down the highway will remain in motion unless some external force changes its speed. A book placed on a table will remain there unless someone displaces it.
What are the 3 types of mechanics?
- Classical Mechanics. Classical mechanics is an actual hypothesis portraying the movement of perceptible articles, from machine parts to projectiles, and galactic items. …
- Quantum Mechanics. Quantum mechanics is a basic hypothesis in material science. …
- Relativistic Mechanics.
What are 3 examples of mechanics?
What are examples of mechanics? Examples of classical mechanics include one-dimensional kinematics, projectile motion, gravitation, momentum, torque, dynamic equilibrium, conservation of energy, and statics. Examples of quantum mechanics include the photoelectric effect, quantum computing, and atomic clocks.
What is the importance of mechanics?
Mechanics provides the (building blocks) of statics, strength of materials, and fluid dynamics. Statics conceder the effects and distribution of the forces on the construction so there are topics analysis, cables, equilibrium of force system, friction, resultant of force system, moments of inertia and center mass.
What are the 5 branches of mechanics?
There are many branches of classical mechanics, such as: statics, dynamics, kinematics, continuum mechanics (which includes fluid mechanics), statistical mechanics, etc. Mechanics: A branch of physics in which we study the object and properties of an object in form of a motion under the action of the force. |
To see what RCTs show, let me define the Cartesian product of X2, X3, . . . , Xn by Z. What RCTs show is that there exists some z ∈ Z, such that if we have the world in state (x, z) ∈ X instead of (y, z) ∈ X, the world in the next period will be in state a ∈ X instead of state b ∈ X. This is like saying, other things being the same (that is, z), if you vaccinate people, in the next period, there will be no influenza. But if you do not vaccinate them, there will be influenza. If we accept the determinist axiom, as many do, then this demonstration means that whenever we switch from (y, z) to (x, z), the world will switch in the next period from b to a. It is the “whenever” that makes this a causal claim. This is what I am referring to as “circumstantial causality”. Given a certain set of circumstances, changing y to x has a predictable consequence.
The discovery of circumstantial causal connections, as has happened with the rise of RCT studies, is valuable and, at the same time, of limited consequence, more so than the proponents believe. On the one hand, RCTs have given us numerous valuable descriptions of what happened in the past and numerous instances of causes in the past (provided of course that one is willing to accept the determinist axiom). On the other, what they show is very limited. This is because when they show that it was the switch from y to x that caused the switch from b to a, what they are saying is that this was true under certain historical conditions (z), but they cannot tell you what those historical conditions are. RCT discoveries never graduate from something “was a cause” of something else to something “is a cause”. RCTs give us no insight into universal causality because they cannot tell us what it was that was being held constant (z in the above example), when we switched some intervention b to a. For Bengal, in a certain period, electing a woman leader of the local government caused water provisioning to be better. This is no guide to the future because we do not fully know what Bengal in a certain period is like. Henceforth, a reference to causality without a qualifying epithet should be taken to be a reference to universal causality because for policy purposes, that is what is of essence.
If the centuries-old struggle with the problem of finding an analytical definition of probability has produced only endless controversies between the various doctrines, it is, in my opinion, because too little attention has been paid to the singular notion of random. For the dialectical root, in fact, lies in this notion: probability is only an arithmetical aspect of it.
Modern probabilistic econometrics relies on the notion of probability. To at all be amenable to econometric analysis, economic observations allegedly have to be conceived as random events.
But is it really necessary to model the economic system as a system where randomness can only be analyzed and understood when based on an a priori notion of probability?
In probabilistic econometrics, events and observations are as a rule interpreted as random variables as if generated by an underlying probability density function, and, a fortiori, since probability density functions are only definable in a probability context, consistent with a probability. As Haavelmo has it in ‘The probability approach in econometrics’ (1944):
For no tool developed in the theory of statistics has any meaning – except, perhaps for descriptive purposes – without being referred to some stochastic scheme.
When attempting to convince us of the necessity of founding empirical economic analysis on probability models, Haavelmo – building largely on the earlier Fisherian paradigm – actually forces econometrics to (implicitly) interpret events as random variables generated by an underlying probability density function.
This is at odds with reality. Randomness obviously is a fact of the real world. Probability, on the other hand, attaches to the world via intellectually constructed models, and a fortiori is only a fact of a probability generating machine or a well constructed experimental arrangement or “chance set-up”.
Just as there is no such thing as a “free lunch,” there is no such thing as a “free probability.” To be able at all to talk about probabilities, you have to specify a model. If there is no chance set-up or model that generates the probabilistic outcomes or events – in statistics one refers to any process where you observe or measure as an experiment (rolling a die) and the results obtained as the outcomes or events (number of points rolled with the die, being e. g. 3 or 5) of the experiment –there strictly seen is no event at all.
Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be shown to coincide with (or at least converge to) real data generating processes or structures – something seldom or never done!
And this is the basic problem with economic data. If you have a fair roulette-wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous nomological machines for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socio-economic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions!
From a realistic point of view we really have to admit that the socio-economic states of nature that we talk of in most social sciences – and certainly in econometrics – are not amenable to analyze as probabilities, simply because in the real world open systems that social sciences – including econometrics – analyze, there are no probabilities to be had!
The processes that generate socio-economic data in the real world cannot just be assumed to always be adequately captured by a probability measure. And, so, it cannot really be maintained – as in the Haavelmo paradigm of probabilistic econometrics – that it even should be mandatory to treat observations and data – whether cross-section, time series or panel data – as events generated by some probability model. The important activities of most economic agents do not usually include throwing dice or spinning roulette-wheels. Data generating processes – at least outside of nomological machines like dice and roulette-wheels – are not self-evidently best modeled with probability measures.
If we agree on this, we also have to admit that probabilistic econometrics lacks sound foundations. I would even go further and argue that there really is no justifiable rationale at all for this belief that all economically relevant data can be adequately captured by a probability measure. In most real world contexts one has to argue one’s case. And that is obviously something seldom or never done by practitioners of probabilistic econometrics.
Econometrics and probability are intermingled with randomness. But what is randomness?
In probabilistic econometrics it is often defined with the help of independent trials – two events are said to be independent if the occurrence or nonoccurrence of either one has no effect on the probability of the occurrence of the other – as drawing cards from a deck, picking balls from an urn, spinning a roulette wheel or tossing coins – trials which are only definable if somehow set in a probabilistic context.
But if we pick a sequence of prices – say 2, 4, 3, 8, 5, 6, 6 – that we want to use in an econometric regression analysis, how do we know the sequence of prices is random and a fortiori being able to treat as generated by an underlying probability density function? How can we argue that the sequence is a sequence of probabilistically independent random prices? And are they really random in the sense that is most often applied in probabilistic econometrics – where X is called a random variable only if there is a sample space S with a probability measure and X is a real-valued function over the elements of S?
Bypassing the scientific challenge of going from describable randomness to calculable probability by just assuming it, is of course not an acceptable procedure. Since a probability density function is a “Gedanken” object that does not exist in a natural sense, it has to come with an export license to our real target system if it is to be considered usable.
Among those who at least honestly try to face the problem – the usual procedure is to refer to some artificial mechanism operating in some “games of chance” of the kind mentioned above and which generates the sequence. But then we still have to show that the real sequence somehow coincides with the ideal sequence that defines independence and randomness within our – to speak with science philosopher Nancy Cartwright – “nomological machine”, our chance set-up, our probabilistic model.
As the originator of the Kalman filter, Rudolf Kalman notes in ‘Randomness Reexamined'(1994):
Not being able to test a sequence for ‘independent randomness’ (without being told how it was generated) is the same thing as accepting that reasoning about an “independent random sequence” is not operationally useful.
Probability is a property of the model we choose to use in our endeavour to understand and explain the world in which we live — but probability is not a property of that world.
So why should we define randomness with probability? If we do, we have to accept that to speak of randomness we also have to presuppose the existence of nomological probability machines, since probabilities cannot be spoken of – and actually, to be strict, do not at all exist – without specifying such system-contexts (how many sides do the dice have, are the cards unmarked, etc)
If we do adhere to the Fisher-Haavelmo paradigm of probabilistic econometrics we also have to assume that all noise in our data is probabilistic and that errors are well-behaving, something that is hard to justifiably argue for as a real phenomena, and not just an operationally and pragmatically tractable assumption.
Maybe Kalman’s verdict that
Haavelmo’s error that randomness = (conventional) probability is just another example of scientific prejudice
is, from this perspective seen, not far-fetched.
Accepting Haavelmo’s domain of probability theory and sample space of infinite populations – just as Fisher’s “hypothetical infinite population, of which the actual data are regarded as constituting a random sample”, von Mises’ “collective” or Gibbs’ ”ensemble” – also implies that judgments are made on the basis of observations that are actually never made!
Infinitely repeated trials or samplings never take place in the real world. So that cannot be a sound inductive basis for a science with aspirations of explaining real-world socio-economic processes, structures or events. It’s not tenable.
This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine – including randomness – then the statistical inferences used lack sound foundations.
And in the video below (in Swedish) yours truly shows how to perform a logit regression using Gretl.
Distinguished social psychologist Richard E. Nisbett has a somewhat atypical aversion to multiple regression analysis . In his Intelligence and How to Get It (Norton 2011) he wrote (p. 17):
Researchers often determine the individual’s contemporary IQ or IQ earlier in life, socioeconomic status of the family of origin, living circumstances when the individual was a child, number of siblings, whether the family had a library card, educational attainment of the individual, and other variables, and put all of them into a multiple-regression equation predicting adult socioeconomic status or income or social pathology or whatever. Researchers then report the magnitude of the contribution of each of the variables in the regression equation, net of all the others (that is, holding constant all the others). It always turns out that IQ, net of all the other variables, is important to outcomes. But … the independent variables pose a tangle of causality – with some causing others in goodness-knows-what ways and some being caused by unknown variables that have not even been measured. Higher socioeconomic status of parents is related to educational attainment of the child, but higher-socioeconomic-status parents have higher IQs, and this affects both the genes that the child has and the emphasis that the parents are likely to place on education and the quality of the parenting with respect to encouragement of intellectual skills and so on. So statements such as “IQ accounts for X percent of the variation in occupational attainment” are built on the shakiest of statistical foundations. What nature hath joined together, multiple regressions cannot put asunder.
And now he is back with a half an hour lecture — The Crusade Against Multiple Regression Analysis — posted on The Edge website a week ago (watch the lecture here).
Now, I think that what Nisbett says is right as far as it goes, although it would certainly have strengthened Nisbett’s argumentation if he had elaborated more on the methodological question around causality, or at least had given some mathematical-statistical-econometric references. Unfortunately, his alternative approach is not more convincing than regression analysis. As so many other contemporary social scientists today, Nisbett seems to think that randomization may solve the empirical problem. By randomizing we are getting different “populations” that are homogeneous in regards to all variables except the one we think is a genuine cause. In this way we are supposed to be able to not have to actually know what all these other factors are.
If you succeed in performing an ideal randomization with different treatment groups and control groups that is attainable. But it presupposes that you really have been able to establish – and not just assume – that the probability of all other causes but the putative have the same probability distribution in the treatment and control groups, and that the probability of assignment to treatment or control groups are independent of all other possible causal variables.
Unfortunately, real experiments and real randomizations seldom or never achieve this. So, yes, we may do without knowing all causes, but it takes ideal experiments and ideal randomizations to do that, not real ones.
As I have argued — e. g. here — that means that in practice we do have to have sufficient background knowledge to deduce causal knowledge. Without old knowledge, we can’t get new knowledge – and, no causes in, no causes out.
Nisbett is well worth reading and listening to, but on the issue of the shortcomings of multiple regression analysis, no one sums it up better than eminent mathematical statistician David Freedman in his Statistical Models and Causal Inference:
If the assumptions of a model are not derived from theory, and if predictions are not tested against reality, then deductions from the model must be quite shaky. However, without the model, the data cannot be used to answer the research question …
In my view, regression models are not a particularly good way of doing empirical work in the social sciences today, because the technique depends on knowledge that we do not have. Investigators who use the technique are not paying adequate attention to the connection – if any – between the models and the phenomena they are studying. Their conclusions may be valid for the computer code they have created, but the claims are hard to transfer from that microcosm to the larger world …
Regression models often seem to be used to compensate for problems in measurement, data collection, and study design. By the time the models are deployed, the scientific position is nearly hopeless. Reliance on models in such cases is Panglossian …
Given the limits to present knowledge, I doubt that models can be rescued by technical fixes. Arguments about the theoretical merit of regression or the asymptotic behavior of specification tests for picking one version of a model over another seem like the arguments about how to build desalination plants with cold fusion and the energy source. The concept may be admirable, the technical details may be fascinating, but thirsty people should look elsewhere …
Causal inference from observational data presents may difficulties, especially when underlying mechanisms are poorly understood. There is a natural desire to substitute intellectual capital for labor, and an equally natural preference for system and rigor over methods that seem more haphazard. These are possible explanations for the current popularity of statistical models.
Indeed, far-reaching claims have been made for the superiority of a quantitative template that depends on modeling – by those who manage to ignore the far-reaching assumptions behind the models. However, the assumptions often turn out to be unsupported by the data. If so, the rigor of advanced quantitative methods is a matter of appearance rather than substance.
There have been over four decades of econometric research on business cycles … The formalization has undeniably improved the scientific strength of business cycle measures …
But the significance of the formalization becomes more difficult to identify when it is assessed from the applied perspective, especially when the success rate in ex-ante forecasts of recessions is used as a key criterion. The fact that the onset of the 2008 financial-crisis-triggered recession was predicted by only a few ‘Wise Owls’ … while missed by regular forecasters armed with various models serves us as the latest warning that the efficiency of the formalization might be far from optimal. Remarkably, not only has the performance of time-series data-driven econometric models been off the track this time, so has that of the whole bunch of theory-rich macro dynamic models developed in the wake of the rational expectations movement, which derived its fame mainly from exploiting the forecast failures of the macro-econometric models of the mid-1970s recession.
The limits of econometric forecasting has, as noted by Qin, been critically pointed out many times before.
Trygve Haavelmo — with the completion (in 1958) of the twenty-fifth volume of Econometrica — assessed the the role of econometrics in the advancement of economics, and although mainly positive of the “repair work” and “clearing-up work” done, Haavelmo also found some grounds for despair:
We have found certain general principles which would seem to make good sense. Essentially, these principles are based on the reasonable idea that, if an economic model is in fact “correct” or “true,” we can say something a priori about the way in which the data emerging from it must behave. We can say something, a priori, about whether it is theoretically possible to estimate the parameters involved. And we can decide, a priori, what the proper estimation procedure should be … But the concrete results of these efforts have often been a seemingly lower degree of accuracy of the would-be economic laws (i.e., larger residuals), or coefficients that seem a priori less reasonable than those obtained by using cruder or clearly inconsistent methods.
There is the possibility that the more stringent methods we have been striving to develop have actually opened our eyes to recognize a plain fact: viz., that the “laws” of economics are not very accurate in the sense of a close fit, and that we have been living in a dream-world of large but somewhat superficial or spurious correlations.
And as the quote below shows, even Ragnar Frisch shared some of Haavelmo’s — and Keynes’s — doubts on the applicability of econometrics:
I have personally always been skeptical of the possibility of making macroeconomic predictions about the development that will follow on the basis of given initial conditions … I have believed that the analytical work will give higher yields – now and in the near future – if they become applied in macroeconomic decision models where the line of thought is the following: “If this or that policy is made, and these conditions are met in the period under consideration, probably a tendency to go in this or that direction is created”.
Maintaining that economics is a science in the “true knowledge” business, I remain a skeptic of the pretences and aspirations of econometrics. So far, I cannot really see that it has yielded very much in terms of relevant, interesting economic knowledge. And, more specifically, when it comes to forecasting activities, the results have been bleak indeed.
Firmly stuck in an empiricist tradition, econometrics is only concerned with the measurable aspects of reality, But there is always the possibility that there are other variables – of vital importance and although perhaps unobservable and non-additive not necessarily epistemologically inaccessible – that were not considered for the model. Those who were can hence never be guaranteed to be more than potential causes, and not real causes.
A perusal of the leading econom(etr)ic journals shows that most econometricians still concentrate on fixed parameter models and that parameter-values estimated in specific spatio-temporal contexts are presupposed to be exportable to totally different contexts. To warrant this assumption one, however, has to convincingly establish that the targeted acting causes are stable and invariant so that they maintain their parametric status after the bridging. The endemic lack of predictive success of the econometric project indicates that this hope of finding fixed parameters is a hope for which there really is no other ground than hope itself.
When causal mechanisms operate in real world social target systems they only do it in ever-changing and unstable combinations where the whole is more than a mechanical sum of parts. If economic regularities obtain they do it (as a rule) only because we engineered them for that purpose. Outside man-made “nomological machines” they are rare, or even non-existant. Unfortunately that also makes most of the achievements of econometric forecasting rather useless.
The initial choice of a prior probability distribution is not regulated in any way. The probabilities, called subjective or personal probabilities, reflect personal degrees of belief. From a Bayesian philosopher’s point of view, any prior distribution is as good as any other. Of course, from a Bayesian decision maker’s point of view, his own beliefs, as expressed in his prior distribution, may be better than any other beliefs, but Bayesianism provides no means of justifying this position. Bayesian rationality rests in the recipe alone, and the choice of the prior probability distribution is arbitrary as far as the issue of rationality is concerned. Thus, two rational persons with the same goals may adopt prior distributions that are wildly different …
Bayesian learning is completely inflexible after the initial choice of probabilities: all beliefs that result from new observations have been fixed in advance. This holds because the new probabilities are just equal to certain old conditional probabilities …
According to the Bayesian recipe, the initial choice of a prior probability distribution is arbitrary. But the probability calculus might still rule out some sequences of beliefs and thus prevent complete arbitrariness.
Actually, however, this is not the case: nothing is ruled out by the probability calculus …
Thus, anything goes … By adopting a suitable prior probability distribution, we can fix the consequences of any observations for our beliefs in any way we want. This result, which will be referred to as the anything-goes theorem, holds for arbitrarily complicated cases and any number of observations. It implies, among other consequences, that two rational persons with the same goals and experiences can, in all eternity, differ arbitrarily in their beliefs about future events …
From a Bayesian point of view, any beliefs and, consequently, any decisions are as rational or irrational as any other, no matter what our goals and experiences are. Bayesian rationality is just a probabilistic version of irrationalism. Bayesians might say that somebody is rational only if he actually rationalizes his actions in the Bayesian way. However, given that such a rationalization always exists, it seems a bit pedantic to insist that a decision maker should actually provide it.
Nice to see that the son of Hans Albert is keeping critical rationalism alive …
Suppose you test a highly confirmed hypothesis, for example, that the price elasticity of demand is negative. What would you do if the computer were to spew out a positive coefficient? Surely you would not claim to have overthrown the law of demand … Instead, you would rerun many variants of your regression until the recalcitrant computer finally acknowledged the sovereignty of your theory …
Only the naive are shocked by such soft and gentle testing … Easy it is. But also wrong, when the purpose of the exercise is not to use a hypothesis, but to determine its validity …
Econometric tests are far from useless. They are worth doing, and their results do tell something … But many economists insist that economics can deliver more, much more, than merely, more or less, plausible knowledge, that it can reach its results with compelling demonstrations. By such a standard how should one describe our usual way of testing hypotheses? One possibility is to interpret it as Blaug [The Methodology of Economics, 1980, p. 256] does, as ‘playing tennis with the net down’ …
Perhaps my charge that econometric testing lacks seriousness of purpose is wrong … But regardless of the cause, it should be clear that most econometric testing is not rigorous. Combining such tests with formalized theoretical analysis or elaborate techniques is another instance of the principle of the strongest link. The car is sleek and elegant; too bad the wheels keep falling off.
And who said learning statistics can’t be fun?
[Actually the Ronald Fisher appearing in the video is a mixture of the real Ronald Fisher and Jerzy Neyman and Egon Pearson, but that’s for another blogpost.]
For my own critical view on the value of p values — see e. g. here.
Many thanks for sending me your article. I enjoyed it very much. I am sure these matters need discussing in that sort of way. There is one point, to which in practice I attach a great importance, you do not allude to. In many of these statistical researches, in order to get enough observations they have to be scattered over a lengthy period of time; and for a lengthy period of time it very seldom remains true that the environment is sufficiently stable. That is the dilemma of many of these enquiries, which they do not seem to me to face. Either they are dependent on too few observations, or they cannot rely on the stability of the environment. It is only rarely that this dilemma can be avoided.
Letter from J. M. Keynes to T. Koopmans, May 29, 1941
Almost a hundred years after John Maynard Keynes wrote his seminal A Treatise on Probability (1921), it is still very difficult to find statistics books that seriously try to incorporate his far-reaching and incisive analysis of induction and evidential weight.
The standard view in statistics – and the axiomatic probability theory underlying it – is to a large extent based on the rather simplistic idea that “more is better.” But as Keynes argues – “more of the same” is not what is important when making inductive inferences. It’s rather a question of “more but different.”
Variation, not replication, is at the core of induction. Finding that p(x|y) = p(x|y & w) doesn’t make w “irrelevant.” Knowing that the probability is unchanged when w is present gives p(x|y & w) another evidential weight (“weight of argument”). Running 10 replicative experiments do not make you as “sure” of your inductions as when running 10 000 varied experiments – even if the probability values happen to be the same.
According to Keynes we live in a world permeated by unmeasurable uncertainty – not quantifiable stochastic risk – which often forces us to make decisions based on anything but “rational expectations.” Keynes rather thinks that we base our expectations on the confidence or “weight” we put on different events and alternatives. To Keynes expectations are a question of weighing probabilities by “degrees of belief,” beliefs that often have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents as modeled by “modern” social sciences. And often we “simply do not know.” As Keynes writes in Treatise:
The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be [that] the system of the material universe must consist of bodies … such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state … Yet there might well be quite different laws for wholes of different degrees of complexity, and laws of connection between complexes which could not be stated in terms of laws connecting individual parts … If different wholes were subject to different laws qua wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts … These considerations do not show us a way by which we can justify induction … /427 No one supposes that a good induction can be arrived at merely by counting cases. The business of strengthening the argument chiefly consists in determining whether the alleged association is stable, when accompanying conditions are varied … /468 In my judgment, the practical usefulness of those modes of inference … on which the boasted knowledge of modern science depends, can only exist … if the universe of phenomena does in fact present those peculiar characteristics of atomism and limited variety which appears more and more clearly as the ultimate result to which material science is tending.
Science according to Keynes should help us penetrate to “the true process of causation lying behind current events” and disclose “the causal forces behind the apparent facts.” Models can never be more than a starting point in that endeavour. He further argued that it was inadmissible to project history on the future. Consequently we cannot presuppose that what has worked before, will continue to do so in the future. That statistical models can get hold of correlations between different “variables” is not enough. If they cannot get at the causal structure that generated the data, they are not really “identified.”
How strange that economists and other social scientists as a rule do not even touch upon these aspects of scientific methodology that seems to be so fundamental and important for anyone trying to understand how we learn and orient ourselves in an uncertain world. An educated guess on why this is so would be that Keynes’s concepts are not possible to squeeze into a single calculable numerical “probability.” In the quest for quantities one puts a blind eye to qualities and looks the other way – but Keynes’s ideas keep creeping out from under the statistics carpet. |
The Gaseous Phase The three phases of matter, solids, liquids and gases, have different characteristics. A gas expands to fill any container it occupies A liquid has a fixed volume but takes the shape of the volume of the container it occupies A solid has both fixed volume and shape. These characteristics originate from the nature of the interactions between the atoms or molecules
On a macroscopic scale, gases are distinguished from solids and liquids by their much smaller values of density. On the microscopic scale, the smaller values of density arise due to the much lower NUMBER DENSITY (number of molecules per cm3 of the sample) compared with liquids and solids.
Understanding the behavior of gases and how reactions occur in the gas phase is of practical importance CH4(g) + O2(g) --> CO2(g) + H2O(g) - combustion of fuels N2(g) + H2(g) --> NH3(g) - production of ammonia for fertilizers 2NO(g) + O2 (g) -> 2NO2 (g) - responsible for acid rain
Properties of Gases - A gas will fill the volume of the container which contains it. - The volume of the gas equals the volume of its container - Gases are highly compressible; when pressure is applied to a gas, its volume readily decreases - Gases form homogenous mixtures with each other regardless of their identity or relative proportions These properties arise because the individual atoms/ molecules are relatively far apart
Three properties of gases that are used to describe gases are pressure (P), volume (V) and temperature (T). The volume of a gas is defined by the volume of the container. Typical units for volume of gases is the liter, L.
Pressure = Force Area SI Units for pressure Force is newton, N (=kg m/s2) Area - m2 Pressure - N/m2 or pascal (Pa) PRESSURE The force exerted by a gas on a unit area of the walls of its container is the pressure exerted by the gas.
Atmospheric Pressure Because of gravity, the earth’s atmosphere exerts a downward force and consequently a pressure on the earth’s surface. Atmospheric pressure: pressure exerted by the atmosphere around us A column of air 1m2 in cross section extending through the atmosphere has a mass or roughly 10,000 kg.
The pressure exerted by this air column ~ 1 x 105 N 1 m2 The acceleration produced by earth’s gravity is 9.8 m/s2 force = mass x acceleration Force exerted by this air column is ~ 1 x 105 N ~ 1 x 105 Pa More precisely, 1.01325 x 105 Pa = 1 atmosphere (atm)
A barometer operates on the principle that the height of a liquid in a closed tube depends on the atmospheric pressure Pressure = g h d g ~ 9.8 m/s h is the height of the liquid in the sealed tube d is the density of the liquid
a) What is the height of a mercury column in a barometer at atmospheric pressure? b) What is the height of a water column in a barometer at atmospheric pressure? Explains why mercury is used in barometers and not water
Units of pressure 1 atm = 760 mm Hg = 760 torr = 1.01325 x 105 Pa There are other units of pressure (lbs/in2, bar) but we will typically deal with atm, mm of Hg or torr and Pa.
The Gas Laws Through experimental observations, relationships have been established between the pressure (P), temperature (T) and volume (V) and number of moles (n) of gases. These relationships are called the GAS LAWS. Having defined P, V, T, and n for a gas, this information defines the physical condition or state of a gas. The relationships between P, V, T and n that will be discussed hold for IDEAL gases (or for low pressures; “ideal” conditions)
Relationship between Pressure and Volume: Boyle’s Law Boyle noted from the experiments he performed that at a fixed temperature and for a fixed amount of gas, as pressure on a gas increases, the volume occupied by a gas decreases.
At a fixed temperature and for a fixed amount of gas P V = constant P a 1 V P = constant Volume Boyle’s Law The product of pressure and volume of a sample of gas is a constant, at constant temperature and for a fixed amount of gas.
Since, PV = constant P1V1 = P2V2 The conditions of 1.00 atm pressure and 0oC are called standard temperature and pressure (STP). Under STP conditions, the volume occupied by the gas in the J-tube is 22.4 L.
Temperature-Volume Relation - Charles Law The volume of a fixed quantity of gas at constant pressure increases linearly with temperature. V = V0 + a V0 t V0 is the volume of the gas at 0oC t is the temperature in oC a is the coefficient of thermal expansion
Volume V = V0 + a V0 t y = mx + b
Volume From a plot of V vs t we can determine V0 from the y-intercept. From the slope = a V0, a can be determined
a = 1 (oC-1) 273.15 Since gases expand by the same relative amount when heated between the same two temperatures (at low pressure) implies that a is the same for all ideal gases. For gases, at low pressure For liquids and solids a varies from substance to substance
V V t = -1 V0 V0 a t = 273.15 oC [ -1 ] Re-writing the expression connecting V and t: Gas thermometer: By measuring the volume of a gas at 0oC and measuring the volume change as temperature changes, the temperature can be calculated
t V = V0 [ 1 + ] 273.15 oC At t = -273.15 oC => volume of gas is zero temperatures < -273.15oC => negative volume which is physically impossible. Hence 273.15oC is the lowest temperature that can be physically attained and is the fundamental limit on temperature. Absolute temperature - Kelvin Scale
t V = V0 [ 1 + ] 273.15 oC V0 T V = 273.15 This temperature is called ABSOLUTE ZERO and is defined to be the zero point on the Kelvin scale (K) T (Kelvin) = 273.15 + t (Celsius) If we substitute the above expression in and solve for V
V0 273.15 at a fixed pressure and for a fixed amount of gas V1 T1 = V2 T2 Hence, V a T Charles’ Law In other words, on an absolute temperature scale, at a constant pressure and for a fixed amount if gas, the volume of the gas is proportional to the temperature is a constant Hence, Note: T is temperature in Kelvin
Volume is affected not just by pressure and temperature, but also by the amount of gas. Avogadro’s hypothesis - Equal volumes of gases at the same temperature and pressure contain the same number of molecules. Quantity-Volume relation - Avogadro’s Law
Avogadro’s law: the volume of a gas maintained at constant pressure and temperature is directly proportional to the number of moles of gas. V = constant x n Hence, doubling the moles of gas will cause the volume to double (as long as T and P remain constant)
n T V a P Boyle’s law: V a P-1 (constant n, T) Charles’ law: V a T (constant n, P) Avogadro’s law: V a n (constant P, T) The Ideal-Gas Equation Putting the three laws together:
V = R n T P P V = n R T IDEAL GAS EQUATION An ideal gas is a gas whose pressure, volume and temperature behavior is completely described by this equation. R is called the universalgas constant since it is the same for all gases. Note: The ideal gas equation is just that - ideal. The equation is valid for the most gases at low pressures. Deviations from “ideal” behavior are observed as pressure increases.
Values of R Units Numerical value L-atm/(mol-K) 0.08206 cal/(mol-K) 1.987 J/(mol-K) 8.314 m3-Pa/(mol-K) 8.314 L-torr/(mol-K) 62.36 The value and units of R depends on the units of P, V, n and T Temperature, T, MUST ALWAYS BE IN KELVIN n is expressed in MOLES P is often in atm and V in liters, but other units can be used.
Given: Volume of CO2 = 250 mL = 0.250 L Pressure of CO2 = 1.3 atm temperature of CO2 = 31oC Example: Calcium carbonate, CaCO3(s), decomposes upon heating to give CO2(g) and CaO(s). A sample of CaCO3 is decomposed, and the CO2 collected in a 250. mL flask. After the decomposition is complete, the gas has a pressure of 1.3 atm at a temperature of 31oC. How many moles of CO2 were generated? First convert temperature to K T = 31 + 273 = 304 K
P V n = R T (1.3 atm)(0.250 L) n = (0.08206 L-atm/(mol-K)) (304K) Based on the units given for P and V, use appropriate value for R R = 0.08206 L-atm/(mol K) To calculate n: n = 0.013 mol CO2
Pi n R P Pf = = constant Tf Ti V T Pi Tf Pf Pf = 3.6 atm = = Ti Since, the can is sealed, both V and n stay fixed. P (atm) t (oC) T(K) Initial 1.5 25 298 Final ? 450 723 Problem The gas pressure in a closed aerosol can is 1.5 atm at 25oC. Assuming that the gas inside obeys the ideal-gas equation, what would the pressure be if the can was heated to 450oC?
The number of moles of a compound = mass of gas sample (m) Molar Mass (M) m n = M Molar Mass and Gas Density The ideal gas law, P V = n R T can be used to determine the molar mass of gaseous compounds.
Substituting this in the ideal gas equation P M m m d = P V = P V = R T R T R T M M Solving for M, the molar mass m R T M = P V The ideal gas equation can be also be used to determine the density of the gas
Gas Stoichiometry If the conditions of pressure and temperature are known, then the ideal gas law can be used to convert between chemical amounts i.e. moles, and gas volume. Hence in dealing with chemical reactions involving gases, we can deal with volumes of gases instead of moles of gases, being that volume is usually an easier quantity to measure.
Problem Dinitrogen monoxide, N2O, better known as nitrous oxide or laughing gas, is shipped in steel cylinders as a liquid at pressures of 10 MPa. It is produced as a gas in aluminium trays by the decomposition of ammonium nitrate at 200oC. NH4NO3(s) --> N2O(g) + 2H2O(g) What volume of N2O(g) at 1.00 atm would be produced from 100.0g of NH4NO3(s) after separating out the H2O and cooling the N2O gas to 273K. Assume a 100% yield in the production of N2O(g) . Assuming a 100% yield => all the NH4NO3(s) is converted to N2O(g) Moles of NH4NO3(s) decomposed = 100.0 g/80.04g/mol = 1.249 mol
Hence, 1.249 mol of N2O(g) formed. To calculate the volume of N2O(g) produced, use the ideal gas equation. V = n R T/ P V =[ (1.249 mol) (0.08206 L-atm/(mol-K)) (273)]/(1.00 atm) = 28.0 L N2O(g)
Can we use the ideal gas equation to determine the properties of gases in a mixture? Dalton observed that the total pressure of a mixture of gases equals the sum of the pressures that each would exert if each were present alone. Mixtures of Gases The partial pressure of a gas in a mixture of gases is defined as the pressure it exerts if it were present alone in the container.
Dalton’s law states that the total pressure is the sum of the partial pressures of each gas in the mixture. For example, consider a mixture of two gases A and B in a closed container Assuming that the pressure is low enough, A and B obey the ideal gas equation. The fact that A and B behave as ideal gases implies that A and B do not interact with each other.
The pressure exerted by A, PA is then and that exerted by B is: PA = nA RT PB = nB RT V V
nA RT nB RT = + V V = (nA + nB) RT V = ntotal RT V From Dalton’s law: Ptotal = PA + PB Where ntotal = nA + nB is the total number of moles
Mole Fraction The quantity is called the MOLE FRACTION of A, XA nA ntotal What is the fraction of the number of moles of A in the mixture? To find this out, we need to divide the number of moles of A by the total number of moles of gases in the mixture Note: mole fraction is unitless since it is a fraction of two quantities with the same unit. Also, sum of mole fractions of the components in a mixture =1
For the component A in the mixture, we can write its pressure as The total pressure is Ptotal = ntotal RT V PA = nA RT PA nA V = = XA ntotal Ptotal Dividing PA/Ptotal Hence, PA = XA Ptotal
a) PO2 = XO2 Ptotal = (0.180) (745 torr) = 134 torr b) PV = n R T (134 torr) (1 atm) (120 L) = n (0.08206 L-atm/molK)(295 K) (760 torr) n = 0.872 mol A study of the effect of certain gases on plant growth requires a synthetic atmosphere composed of 1.5 mol percent of CO2, 18.0 mol percent of O2 and 80.5 mol percent Ar. a) Calculate the partial pressure of O2 in the mixture if the total pressure of the atmosphere is to be 745 torr? b) If this atmosphere is to be held in a 120-L space at 295 K, how many moles of O2 are needed?
Kinetic Theory of Gases The ideal gas equation describes how gases behave. In the 19th century, scientists applied Newton’s laws of motion to develop a model to explain the behavior of gases. This model, called the kinetic theory of gases, assumes that the atoms or molecules in a gas behave like billiard balls. In the gas phase, atoms and molecules behave like hard spheres and do not interact with each other.
Assumptions of Kinetic Theory of Gases 1) A gas consists of a large number of particles that are so small compared to the average distance separating them, that their own size can be considered negligible. 2) The particles of an ideal gas behave totally independent, neither attracting nor repelling each other. 3) Gas particles are in constant, rapid, straight-line motion, incessantly colliding with each other and with the walls of the container. All collisions between particles are elastic. 4) A collection of gas particles can be characterized by its average kinetic energy, which is proportional to the temperature on the absolute scale.
Gas particles are constantly colliding with each other and the walls of the container. It is the collisions between the gas particles and the walls of the container that define the pressure of the gas. Every time a gas particle collides with the wall of the container, the gas particle imparts its momentum to the wall momentum = mass x velocity
P a (m x u) x [ N x u] V The pressure exerted by the gas is proportional to the momentum of the particle and the number of collisions per unit time, the collision frequency. Pressure a (momentum of particle) x (rate of collisions with the wall) The rate of collision is proportional to the number of particles per unit volume (N/V) and the speed of the particle (u).
P aN m u2 V Replace u2 with the mean-square speed, u2 P V a N m u2 P V a N m u2 The speed, u, is the average speed of the particles, since not all the particles move with the same speed.
1 P V = N m u2 3 1 n R T = N m u2 3 Particles are moving in a 3-D space: Comparing this equation with the ideal gas equation P V = n R T This equation relates the speed of the gas particles with the temperature of the gas |
Physics of the Car Accident. Building a Safe Campus by Solving Physics
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Physics of the Car Accident. Building a Safe Campus by Solving Physics Problem Service-Learning Component of General Physics Course Elena Flitsiyan Department of Physics
The Problem: Students have difficulty seeing connections between physics class and the “real world” Opportunities to help students these connections are not being fully realized in the current course design Implementing service learning elements in introductory physics course will result in improving the interactive learning component and also educate student community how prevent the car accidents
Forces on an inclined road Often when solving problems involving Newton’s laws we will need to deal with resolving acceleration due to gravity on an inclined surface
Forces on an inclined road What normal force does the surface exert? mgsin mgcos W mg y x
Forces on an inclined road F F x mg sin y n mg cos
Forces on an inclineed road F F x mg sin ma y n mg cos 0 Equilibrium
Forces on an inclined road If the car is just stationary on the incline what is the (max) coefficient of static friction? Fx mg sin s n ma 0 F y n mg cos 0 mg sin s n s mg cos sin s tan cos
Horizontal (Flat) Curve The force of static friction supplies the centripetal force v2 f S m r 2 v f S n S mg m r Solving for the maximum speed at which the car can negotiate the curve gives: v gr Fr Note, this does not depend on the mass of the car
Banked Curve These are designed with friction equaling zero There is a component of the normal force that supplies the centripetal force (1), and component of the normal force that supplies the gravitational force (2). Dividing (1) by (2) gives: 2 v tan rg mv 2 n sin r (1) n cos mg ( 2)
Suppose that a 1 800-kg car passes over a bump in a roadway that follows the arc of a circle of radius 20.4 m as in Figure. (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 30.0 km/h? (b) What is the maximum speed the car can have as it passes this highest point without losing contact with the road?
n mg mv 2 Fy may n mg r 2 2 8.33 m s v 2 n m g 1 800 kg 9.8 m s r 20.4 m 1 h 1 000 m v 30 km h 8.33 m s 3 600 s 1 km 1.15 10 4 N up n 0 mv 2 mg r v gr 9.8 m s 20.4 m 14.1 m s 2 50.9 km h
“Centrifugal” Force From the frame of the passenger (b), a force appears to push her toward the door From the frame of the Earth, the car applies a leftward force on the passenger The outward force is often called a centrifugal force It is a fictitious force due to the acceleration associated with the car’s change in direction
If the coefficient of static friction between your coffee cup and the horizontal dashboard of your car is μs 0.800, how fast can you drive on a horizontal roadway around a right turn of radius 30.0 m before the cup starts to slide? If you go too fast, in what direction will the cup slide relative to the dashboard?
If the coefficient of static friction between your coffee cup and the horizontal dashboard of your car is μs 0.800, how fast can you drive on a horizontal roadway around a right turn of radius 30.0 m before the cup starts to slide? If you go too fast, in what direction will the cup slide relative to the dashboard? We adopt the view of an inertial observer. If it is on the verge of sliding, the cup is moving on a circle with its centripetal acceleration caused by friction. Fy may : n mg 0 Fx max : mv 2 f s n s mg r v s gr 0.8 9.8 m s 2 30 m 15.3 m s If you go too fast the cup will begin sliding straight across the dashboard to the left.
Impulse Approximation In many cases, one force acting on a particle acts for a short time, but is much greater than any other force present When using the Impulse Approximation, we will assume this is true Especially useful in analyzing collisions The force will be called the impulsive force The particle is assumed to move very little during the collision p i and p f represent the momenta immediately before and after the collision
Impulse-Momentum: Crash Test Example Categorize Assume force exerted by wall is large compared with other forces Gravitational and normal forces are perpendicular and so do not effect the horizontal momentum Can apply impulse approximation
Crash Test Example Analyze The momenta before and after the collision between the car and the wall can be determined Find Initial momentum Final momentum Impulse Average force Check signs on velocities to be sure they are reasonable
Two-Dimensional Collision Example Conceptualize See picture Choose East to be the positive x-direction and North to be the positive y-direction Categorize Ignore friction Model the cars as particles The collision is perfectly inelastic The cars stick together
Two dimensional collision m1 1500.0kg m2 2500.0 kg Find vf .
Two dimensional collision m1 800.0kg m2 1400.0 kg Find vf . m1 v1 m2 v2 (m1 m2) vf (800kg) (25m/s) 0 (2200kg) vf cosθ – x-component (1400kg) (20m/s) 0 (2200kg) vf sinθ - y-component 1400 0.8 tan 800 54.50 (1400)(20) (2200)v f sin 54.50 v f (9.07iˆ)m / s (12.72 ˆj )m / s v f 15.63m / s 12.75 0 tan 89.6 9.07 1 |
In geometry and physics, spinors // are elements of a complex number-based vector space that can be associated with Euclidean space.[b] A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation,[c] but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).
It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.[d] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.[e]
Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.[f] It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.[g] In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3).
Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with.[h] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices,[i] and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex[j]) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.[k]
What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system (for instance, the stress of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself.
Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually (continuously) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: They exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two ("topologically") inequivalent gradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle (shown) demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class.
Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves,[m] realizing it as a group of rotations among them,[n] but it also acts on the column vectors (that is, the spinors).
More generally, a Clifford algebra can be constructed from any vector space V equipped with a (nondegenerate) quadratic form, such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. The space of spinors is the space of column vectors with components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group.[o] Depending on the dimension and metric signature, this realization of spinors as column vectors may be irreducible or it may decompose into a pair of so-called "half-spin" or Weyl representations.[p] When the vector space V is four-dimensional, the algebra is described by the gamma matrices.
The space of spinors is formally defined as the fundamental representation of the Clifford algebra. (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a spin representation of the orthogonal Lie algebra. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional group representation of the spin group on which the center acts non-trivially.
There are essentially two frameworks for viewing the notion of a spinor: the representation theoretic point of view and the geometric point of view.
From a representation theoretic point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. From this view, a spinor must belong to a representation of the double cover of the rotation group SO(n, ), or more generally of a double cover of the generalized special orthogonal group SO+(p, q, ) on spaces with a metric signature of (p, q). These double covers are Lie groups, called the spin groups Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.)
In summary, given a representation specified by the data where is a vector space over or and is a homomorphism , a spinor is an element of the vector space .
From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as Fierz identities, are needed.
The language of Clifford algebras (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for ad hoc constructions.
In detail, let V be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = , with the standard form g(x, y) = xTy = x1y1 + ... + xnyn we denote the Clifford algebra by Cℓn( ). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim (V) = n. If n = 2k is even, Cℓn( ) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2k, ) of 2k × 2k complex matrices (by the Artin–Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If n = 2k + 1 is odd, Cℓ2k+1( ) is isomorphic to the algebra Mat(2k, ) ⊕ Mat(2k, ) of two copies of the 2k × 2k complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2[n/2]. Since the Lie algebra so(V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of so(V, g) called a spin representation. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further[clarification needed] into two irreducible representations Δ = Δ+ ⊕ Δ− called the Weyl or half-spin representations.
Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.
Spinors form a vector space, usually over the complex numbers, equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. In the latter case, the "rotations" include the Lorentz boosts, but otherwise the theory is substantially similar.
The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time. To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fiber bundle, the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation, or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called fermions; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another.
It appears that all fundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino. There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ2,2( ), the Majorana spinor. There also does not seem to be any particular prohibition to having Weyl spinors appear in nature as fundamental particles.
The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.
Weyl spinors are insufficient to describe massive particles, such as electrons, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed. The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor.[q] However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors. It is not known whether Weyl spinor fundamental particles exist in nature.
The situation for condensed matter physics is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.
One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah–Singer index theorem, and to provide constructions in particular for discrete series representations of semisimple groups.
The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article.
The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space". Stated differently:
Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:
No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.
Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices. The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group. By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.
Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett and by Fritz Sauter. More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, ).[r]
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes replaced spinor spaces by the even subalgebra Cℓ01,3( ) of the spacetime algebra Cℓ1,3( ). As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.
Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓp, q( ). This is an algebra built up from an orthonormal basis of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors
The Clifford algebra Cℓ2,0( ) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit pseudoscalar i = σ1σ2. From the definitions above, it is evident that (σ1)2 = (σ2)2 = 1, and (σ1σ2)(σ1σ2) = −σ1σ1σ2σ2 = −1.
The even subalgebra Cℓ02,0( ), spanned by even-graded basis elements of Cℓ2,0( ), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0( ) is isomorphic to the field of complex numbers . As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by
The action of an even Clifford element γ ∈ Cℓ02,0( ) on vectors, regarded as 1-graded elements of Cℓ2,0( ), is determined by mapping a general vector u = a1σ1 + a2σ2 to the vector
An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
On the other hand, in comparison with its action on spinors , the action of on ordinary vectors appears as the square of its action on spinors.
Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ2 = exp(θ σ1σ2), so that the corresponding action on spinors is via γ = ± exp(θ σ1σ2/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.
In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language, the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics) they make sense.
The Clifford algebra Cℓ3,0( ) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ1, σ2 and σ3, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the pseudoscalar i = σ1σ2σ3. It is straightforward to show that (σ1)2 = (σ2)2 = (σ3)2 = 1, and (σ1σ2)2 = (σ2σ3)2 = (σ3σ1)2 = (σ1σ2σ3)2 = −1.
The sub-algebra of even-graded elements is made up of scalar dilations,
corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a1σ1 + a2σ2 + a3σ3.
As a special case, it is easy to see that, if v = σ3, this reproduces the σ1σ2 rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ3 direction invariant, since
With the identification of the even-graded elements with the algebra of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.[t] Thus the (real[u]) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.
Note that the expression (1) for a vector rotation through an angle θ, the angle appearing in γ was halved. Thus the spinor rotation γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with (180° + θ/2) in place of θ/2 will produce the same vector rotation, but the negative of the spinor rotation.
The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.
A space of spinors can be constructed explicitly with concrete and abstract constructions. The equivalence of these constructions is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e1 ... en for V i.e. g(eμeν) = ημν where ημμ = ±1 and ημν = 0 for μ ≠ ν. Let k = ⌊n/2⌋. Fix a set of 2k × 2k matrices γ1 ... γn such that γμγν + γνγμ = 2ημν1 (i.e. fix a convention for the gamma matrices). Then the assignment eμ → γμ extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, ) by sending the monomial eμ1 ⋅⋅⋅ eμk in the Clifford algebra to the product γμ1 ⋅⋅⋅ γμk of matrices and extending linearly. The space on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics. Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory. In general, in order to define gamma matrices of the required kind, one can use the Weyl–Brauer matrices.
In this construction the representation of the Clifford algebra Cℓ(V, g), the Lie algebra so(V, g), and the Spin group Spin(V, g), all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2k complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, such indices are often used to denote spinors even when an abstract spinor construction is used.
There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of Cℓ(V, g) on itself. These are subspaces of the Clifford algebra of the form Cℓ(V, g)ω, admitting the evident action of Cℓ(V, g) by left-multiplication: c : xω → cxω. There are two variations on this theme: one can either find a primitive element ω that is a nilpotent element of the Clifford algebra, or one that is an idempotent. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it. In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space.
In either approach, the fundamental notion is that of an isotropic subspace W. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of V is given.
As above, we let (V, g) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification and let g denote the induced bilinear form on . Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g|W = 0. If n = 2k is even, then let W′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd, let W′ be a maximal isotropic subspace with W ∩ W′ = 0, and let U be the orthogonal complement of W ⊕ W′. In both the even- and odd-dimensional cases W and W′ have dimension k. In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u.
Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is skew. Hence vectors in W′ anti-commute, and Cℓ(W′, g|W′) = Cℓ(W′, 0) is just the exterior algebra Λ∗W′. Consequently, the k-fold product of W′ with itself, W′k, is one-dimensional. Let ω be a generator of W′k. In terms of a basis w′1, ..., w′k of in W′, one possibility is to set
Note that ω2 = 0 (i.e., ω is nilpotent of order 2), and moreover, w′ω = 0 for all w′ ∈ W′. The following facts can be proven easily:
In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in Cℓ(V, g)ω. We shall show that I must be equal to Cℓ(V, g)ω by proving that it contains a nonzero scalar multiple of ω.
Fix a basis wi of W and a complementary basis wi′ of W′ so that
Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any such element. Using the chosen basis, we may write
Note that for n even, this computation also shows that
The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra Λ∗ W = ⊕j Λj W of the isotropic subspace W. Let Δ = Λ∗ W denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.
The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras. The details differ slightly according to whether the dimension of V is even or odd.
When dim(V) is even, V = W ⊕ W′ where W′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w′ with w ∈ W and w′ ∈ W′. The action of v on a spinor is given by
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group (the half-spin representations, or Weyl spinors) via
When dim(V) is odd, V = W ⊕ U ⊕ W′, where U is spanned by a unit vector u orthogonal to W. The Clifford action c is defined as before on W ⊕ W′, while the Clifford action of (multiples of) u is defined by
If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.
The main example is the case that the real vector space V is a hermitian vector space (V, h), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then splits in the ±i eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space (V, J) and its complex conjugate (V, −J). Therefore, for a hermitian vector space (V, h) the vector space (as well as its complex conjugate is a spinor space for the underlying real euclidean vector space.
With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold) has a Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure.
A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another. These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.
For the real or complex case, the alternating representations are
In addition, for the real orthogonal groups, there are three characters (one-dimensional representations)
The Clebsch–Gordan decomposition allows one to define, among other things:
If n = 2k is even, then the tensor product of Δ with the contragredient representation decomposes as
There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:
For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold:
In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions
In metric signature (p, q), the following isomorphisms hold for the conjugate half-spin representations
Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations Δ± ⊗ Δ±.
If n = 2k + 1 is odd, then
There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are
|Metric signature||Weyl, complex||Conjugacy||Dirac,
|Majorana–Weyl, real||Majorana, | |
Understanding the Conversion from Standard Form to Slope-Intercept Form
Linear equations can be represented in different forms, with standard form and slope-intercept form being two commonly used expressions. The standard form, known as the general form, is given by the equation Ax + By = C, where A, B, and C are constants. On the other hand, the slope-intercept form is written as y = mx + b, where m represents the slope and b denotes the y-intercept.
Converting an equation from standard form to slope-intercept form provides several benefits, particularly in easily identifying the slope and y-intercept. This form is particularly useful for graphing linear equations and understanding the relationship between variables.
The conversion process involves a few simple steps. Firstly, the y variable is isolated by subtracting Ax from both sides of the equation, resulting in By = -Ax + C. Next, dividing both sides by B solves for y, leading to y = (-A/B)x + C/B. Finally, rearranging the terms aligns the equation with the slope-intercept form, thus giving us y = mx + b, where m is equivalent to -A/B and b represents C/B.
By converting equations from standard form to slope-intercept form, we gain a clearer insight into the slope and y-intercept of the equation. This facilitates the graphing process and enables us to make predictions based on the equation’s characteristics. Understanding both forms of linear equations is essential in various applications, such as data analysis, problem-solving, and studying mathematical relationships.
In conclusion, converting from standard form to slope-intercept form is a straightforward procedure that enhances our comprehension of linear equations. This conversion allows us to quickly identify the slope and y-intercept, making graphing and analysis more efficient. Such knowledge holds value across numerous fields and applications, empowering us to better understand and utilize the strength of linear equations.
## Explanation of Answer Key: Converting Standard Form to Slope-Intercept Form
When dealing with linear equations, it is crucial to understand the various methods of representing them. Two commonly used forms are standard form and slope-intercept form. To help you better grasp this concept, we have provided an explanation of how to convert an equation from standard form to slope-intercept form.
In the standard form, a linear equation is expressed as Ax + By = C, where A, B, and C are fixed values. This form signifies that both the x and y terms are on one side of the equation, and A, B, and C can take any real number values.
To convert the equation into slope-intercept form, which is represented as y = mx + b, where m denotes the slope and b is the y-intercept, you need to follow certain steps. First, isolate the y-term by subtracting Ax from both sides. This will yield By = -Ax + C.
Next, divide both sides of the equation by B to isolate y. Consequently, you will have y = (-A/B)x + (C/B). The equation is now in slope-intercept form, with the slope given by (-A/B) and the y-intercept by (C/B).
The answer key for converting from standard form to slope-intercept form provides a simplified approach and enables easy identification of the slope and y-intercept in a linear equation. This conversion is particularly useful when graphing equations or when solving problems involving the relationship between variables in a linear context.
The Advantages of Utilizing an Answer Key for Converting Equations from Standard Form to Slope Intercept Form
Transforming equations from standard form to slope intercept form plays a crucial role in the realm of mathematics. This conversion allows for the simple identification of slope and y-intercept, facilitating easier graphing and comprehension. In order to aid in this process, an answer key equipped with step-by-step solutions for these conversions proves to be a valuable asset. Let’s explore the various benefits of utilizing an answer key for converting equations from standard form to slope intercept form:
1. Enhanced Comprehension: An answer key offers a concise and lucid explanation on how to convert equations from standard form to slope intercept form. By carefully following these systematic instructions, students can develop a solid comprehension of the conversion process.
2. Opportunities for Practice: With an answer key in hand, students have the opportunity to practice converting equations independently before referring to the provided solutions. This fosters reinforcement of their understanding and bolsters their confidence in this specific skill.
3. Error Detection: When students compare their answers with those in the answer key, they can easily identify any mistakes made during the conversion process. This allows for correction of errors and facilitates the development of stronger problem-solving abilities.
4. Reliable Study Guide: An answer key can be utilized as a helpful study guide for students. They can refer back to the provided examples and explanations whenever they need to revisit the process of converting equations from standard form to slope intercept form.
5. Time Efficiency: By using an answer key, students can save valuable time as it provides quick solutions to conversion problems. Instead of struggling to figure out the correct steps on their own, they can rely on the answer key for guidance and complete their assignments more efficiently.
In conclusion, an answer key for converting equations from standard form to slope intercept form offers numerous advantages. It facilitates enhanced comprehension, provides opportunities for practice, aids in error detection, serves as a reliable study guide, and saves time. These benefits make it an indispensable resource for students endeavoring to grasp this essential mathematical concept.
The Process Behind Creating Answer Keys: Converting from Standard Form to Slope-Intercept Form
Have you ever wondered about the method behind generating answer keys for math problems? Specifically, how do they convert from standard form to slope-intercept form? Let’s explore the process.
In standard form, a linear equation is expressed as Ax + By = C, where A, B, and C are unchanging values. Conversely, slope-intercept form appears as y = mx + b, with m denoting the slope and b representing the y-intercept.
In order to create an answer key from standard form to slope-intercept form, the initial step involves isolating y on one side of the equation. By manipulating the given equation, we move the term Ax to the opposite side by subtracting it from both sides. Consequently, we obtain By = -Ax + C.
The subsequent step entails dividing all the terms by the coefficient B to determine the value of y. This calculation yields y = (-A/B)x + C/B. At this point, we have successfully transformed the equation from standard form to slope-intercept form.
Finally, the answer key can be further refined by simplifying the equation. For instance, if there exists a common factor between coefficients A and B, it can be divided out to achieve a more streamlined version of the equation. This ensures that the answer key is presented in the most concise and straightforward manner.
By adhering to these steps, answer keys can flawlessly convert from standard form to slope-intercept form. This enables students to easily comprehend and practice solving linear equations in the most commonly used format, facilitating their understanding and enhancing the learning process.
Exploring Various Ways Answer Keys Can Help Convert Equations from Standard Form to Slope-Intercept Form
Answer keys are invaluable resources for both educators and learners alike, aiding in the educational process by providing guidance and verification. Within the realm of mathematics, answer keys are particularly beneficial in facilitating the conversion of equations from standard form to slope-intercept form.
One type of question that an answer key can assist with is the transformation of linear equations from standard form, expressed as Ax + By = C, into slope-intercept form, represented as y = mx + b. Through a comprehensive breakdown and step-by-step illustration, an answer key can simplify this conversion process and enhance student comprehension.
In addition, answer keys are instrumental in addressing questions involving the determination of slope, y-intercept, and x-intercept for a given linear equation. By providing examples and concise explanations, an answer key can guide students in identifying and calculating these essential components. Such understanding is crucial for effectively graphing equations and comprehending their real-life implications.
Moreover, answer keys further enable students to refine their equation-transforming skills by offering diversified exercises of varying difficulty levels. By presenting the correct solutions, an answer key allows students to compare their own responses and detect any errors they may have made. This process of self-correction is integral to the learning journey, empowering students to enhance their mathematical abilities over time.
To summarize, answer keys greatly contribute to students’ comprehension and practice in converting linear equations from standard form to slope-intercept form. By providing clear explanations, step-by-step instructions, and opportunities for self-correction, answer keys foster improved mathematical proficiency and facilitate effective equation manipulation.
Why Accuracy in Answer Keys is Critical for Converting Standard Form to Slope-Intercept Form
The Significance of Accurate Answer Keys
Answer keys hold immense importance when it comes to evaluating responses and establishing a standardized evaluation system. They aid students in gauging their comprehension levels and enable teachers to assess the effectiveness of their instructional methods. Nonetheless, the accuracy of answer keys cannot be undermined as it directly impacts the outcome of assessments. This is particularly crucial in subjects like mathematics, where converting from standard form to slope-intercept form requires precise solutions.
The Implications of Inaccurate Answer Keys
In math, the process of converting equations from standard form to slope-intercept form is an essential skill that students must master. Incorrect or erroneous answer keys can result in confusion, incorrect learning, and misconstrued concepts. Students rely on answer keys to comprehend the correct methodology and approach to solve problems, and any inaccuracies can hinder their progress.
The Benefits of Accuracy
An accurate answer key serves as a benchmark for students to compare their work and identify their mistakes. It not only helps them rectify errors but also encourages them to learn from their missteps and enhance their problem-solving abilities. With reliable answer keys, students can work independently and confidently, diminishing their reliance on teachers.
Teachers also derive substantial benefits from accurate answer keys as they provide valuable insights into students’ understanding of the subject matter. These keys shed light on common misconceptions and areas where students may struggle, allowing teachers to customize their teaching methods accordingly. Accurate answer keys enable personalized instruction, enabling teachers to focus on specific areas that require additional attention.
Achieving Optimal Learning Outcomes
In conclusion, the significance of accuracy in answer keys, particularly for converting equations from standard form to slope-intercept form, cannot be emphasized enough. Precise answer keys foster effective learning, enhance problem-solving skills, and equip teachers with the tools to monitor progress and enhance educational outcomes. They serve as invaluable resources in the education system, empowering both students and teachers alike.
Tips for Crafting Effective Answer Keys: Transitioning from Standard Form to Slope-Intercept Form
When it comes to conquering math problems, converting equations from standard form to slope-intercept form can sometimes prove challenging for students. To facilitate student learning effectively, teachers and educators must create answer keys that are clear, concise, and easily comprehensible. Here are some valuable tips for developing highly effective answer keys:
1. Break Down the Solution Process
Divide the process of converting an equation from standard form to slope-intercept form into clear, logical steps. Begin by outlining the objective of the problem and then elaborate on each step, emphasizing the rationale behind it. This approach will aid students in comprehending the concept better and foster independent learning.
2. Employ User-Friendly Language
Avoid employing intricate mathematical terminology or jargon that may confuse students. Instead, utilize simple and concise language to elucidate each step. By doing so, you ensure that the answer key is accessible to students of all proficiency levels, promoting inclusivity in the learning process.
3. Incorporate Sample Scenarios
Include an ample number of sample problems within the answer key, encompassing various difficulty levels. This will enable students to practice their skills and reinforce their understanding of the topic. Be sure to incorporate comprehensive solutions alongside these sample problems, serving as a reference for students to cross-verify their work.
4. Foster Critical Thinking
Integrate thought-provoking questions or prompts throughout the answer key. This will stimulate students to engage in critical thinking, enabling them to apply their knowledge to diverse scenarios. By encouraging them to analyze and synthesize information, you facilitate a deeper comprehension of the material.
By adhering to these guidelines, you can develop answer keys that effectively guide students in converting equations from standard form to slope-intercept form. Keep in mind that the primary objective is to provide clarity and support, empowering students to grasp the concept and cultivate their problem-solving skills.
Avoiding Common Errors When Converting Answer Keys from Standard Form to Slope-Intercept Form
In the realm of mathematics, transforming equations from standard form to slope-intercept form is a critical skill. However, this process may prove challenging, even for the most skilled students and educators. It is crucial to familiarize yourself with these stumbling blocks and sidestep them when presenting answer keys to your students. Here are some commonly encountered mistakes to be mindful of:
1. Failing to correctly distribute a negative sign
An often-made mistake is neglecting to distribute a negative sign properly. When converting an equation from standard form to slope-intercept form, it is crucial to distribute the negative sign adequately to both terms within the parentheses. Overlooking this step may result in inaccurate outcomes.
2. Inaccurate calculation of slope and y-intercept
Another common error is miscalculating the slope and y-intercept. When converting an equation, ensure that you determine the coefficients and constants accurately to derive the correct slope and y-intercept values. Trivial arithmetic mistakes can creep in, so it is vital to double-check your calculations.
3. Swapping the sign of the y-intercept
A frequently encountered slip-up involves reversing the sign of the y-intercept. In slope-intercept form, the y-intercept is represented as (0, b), with b being the constant term. Failure to recognize this can lead to incorrect solutions. Pay meticulous attention to sign conventions to avoid this error.
4. Disregarding the simplification of fractions
At times, answer keys neglect the importance of simplifying fractions. Ensure that any fractions present in the standard form equation are simplified before converting them to slope-intercept form. Leaving fractions unsimplified can lead to erroneous answers and create confusion among students.
By being cognizant of these prevalent errors and taking the necessary precautions, educators can furnish accurate answer keys while converting equations from standard form to slope-intercept form. Encourage students to diligently verify their answers, promoting a deeper comprehension of the conversion process.
Unlocking the Power of Answer Keys: Translating Standard Form to Slope Intercept Form
Answer keys are valuable aids when it comes to grasping mathematical concepts. To make the most of them, it is essential to understand the correct way of utilizing these resources. In this article, we will explore how to effectively use an answer key to convert equations from standard form to slope intercept form.
Understanding Standard Form and Slope Intercept Form
Before we proceed, let’s refresh our understanding of standard form and slope intercept form. The standard form of an equation is written as Ax + By = C, where A, B, and C are constant values. On the other hand, slope intercept form is expressed as y = mx + b, where m represents the slope and b denotes the y-intercept.
Using the Answer Key
To convert an equation from standard form to slope intercept form using an answer key, follow these steps:
1. Identify the coefficients: Begin by identifying the values represented by the coefficients A, B, and C in the standard form equation.
2. Isolate the variable y: Rearrange the equation by subtracting Ax from both sides to isolate the variable “y.”
3. Simplify: Divide all terms by B to ensure that the coefficient of y becomes 1, simplifying the equation further.
4. Identify the slope and y-intercept: Now that the equation is in slope intercept form, the coefficient of x represents the slope, and the constant term corresponds to the y-intercept.
Practice and Verification
To validate the accuracy of your conversion, you can compare your result with the answer key. Plug various x-values into both the original equation in standard form and the converted slope intercept form equation. If the corresponding y-values match, you have successfully converted the equation.
Remember, an answer key is a helpful tool that assists your learning. However, it is crucial to comprehend the process involved in converting equations from standard form to slope intercept form so that you have a solid grasp of the concept.
In conclusion, employing an answer key to convert equations from standard form to slope intercept form is an effective learning technique in mathematics. By following the outlined steps and referencing the answer key, you can confidently carry out this conversion whenever necessary.
The Significance of Converting Standard Form to Slope-Intercept Form
Understanding the Process of Converting Standard Form to Slope-Intercept Form
In the realm of mathematics, the ability to convert equations from standard form to slope-intercept form holds significant value. This conversion technique allows us to easily identify the slope and y-intercept of a linear equation, thereby facilitating the graphing process and enhancing our comprehension of the relationship between variables.
The Vital Steps in Converting Equations
To convert an equation from standard form to slope-intercept form, one must rearrange the terms to isolate the y-variable on one side of the equation. This allows for a representation in the form of y = mx + b, where ‘m’ symbolizes the slope and ‘b’ denotes the y-intercept. By determining these two values, we gain a deeper understanding of the linear relationship and can seamlessly construct graphs in the future.
The Advantages of Slope-Intercept Form
Furthermore, once an equation is expressed in slope-intercept form, it becomes significantly easier to analyze and manipulate. For instance, the comparison of slopes can effortlessly determine whether two lines are parallel or perpendicular. Without converting the equation to slope-intercept form, navigating such comparisons would prove to be arduous and complex.
The Essential Skill of Converting Equations
In essence, developing proficiency in converting equations from standard form to slope-intercept form is crucial in the field of mathematics. It not only enhances our ability to accurately graph linear equations but also facilitates the analysis and manipulation of equations. Consistent practice with various examples and exercises is key to mastering this skill and building confidence in converting equations.
Enhancing Mathematical Proficiency
Now armed with the knowledge of the significance behind converting equations from standard form to slope-intercept form, you can confidently apply this skill in your future mathematical pursuits. This newfound proficiency will undoubtedly amplify your understanding and proficiency in solving mathematical problems with ease.
Understanding Converting Linear Equations from Standard Form to Slope-Intercept Form
What does standard form mean in mathematics?
In the realm of mathematics, standard form refers to the standardized representation of a linear equation. It is expressed as Ax + By = C, wherein A, B, and C are fixed values, while x and y represent variables. This specific arrangement enables consistent and uniform presentation of linear equations.
Exploring the concept of slope-intercept form
Slope-intercept form is an alternative approach to illustrate linear equations. It is written as y = mx + b, where m signifies the slope of the line and b denotes the y-intercept – the point at which the line cuts the y-axis.
Quick steps to convert an equation from standard form to slope-intercept form
If you wish to convert an equation from standard form to slope-intercept form, follow these straightforward steps:
1. Rearrange the equation to isolate the variables on one side and the constants on the opposite side.
2. Solve the equation for y by dividing the entire equation by the coefficient of y to separate it from the other terms.
3. Rewrite the equation in the form y = mx + b, where m represents the coefficient of x, and b represents the constant term.
The practicality of converting equations from standard form to slope-intercept form
The conversion of equations to slope-intercept form grants a clearer comprehension of the characteristics exhibited by the corresponding lines. The slope provides insights into the steepness of the line, while the y-intercept indicates the starting point of the line on the vertical axis. Additionally, slope-intercept form facilitates the identification of both slope and y-intercept during the graphing of the equation.
Can all linear equations be converted from standard form to slope-intercept form?
Yes, every linear equation in standard form can be transformed into slope-intercept form. However, the process might require some algebraic manipulations and simplifications based on the specific equation at hand. |
- Research Article
- Open Access
Strong Convergence of a New Iteration for a Finite Family of Accretive Operators
© L.-G. Hu and J.-P.Wang. 2009
- Received: 9 March 2009
- Accepted: 17 May 2009
- Published: 16 June 2009
The viscosity approximation methods are employed to establish strong convergence of the modified Mann iteration scheme to a common zero of a finite family of accretive operators on a strictly convex Banach space with uniformly Gâteaux differentiable norm. Our work improves and extends various results existing in the current literature.
- Banach Space
- Nonexpansive Mapping
- Strong Convergence
- Nonempty Closed Convex Subset
- Convex Banach Space
An operator is said to be accretive if , for all , and . If is accretive and is identity mapping, then we define, for each , a nonexpansive single-valued mapping by , which is called the resolvent of . we also know that for an accretive operator , , where and . An accretive operator is said to be -accretive, if for all . If is a Hilbert space, then accretive operator is monotone operator. There are many papers throughout literature dealing with the solution of ( ) by utilizing certain iterative sequence (see [1–3, 8–10, 13, 16, 20]).
where and , for some , satisfy the following conditions:
(C3) , and
They proved that the iterative sequence converges strongly to a zero of .
where with , , for , , and satisfies the conditions: (C1), (C2), (C3), or ( ). ).
They proved that the sequence converges strongly to a common zero of .
where with , for , and , and . The iterative sequence (1.7) is a natural generalization of all the above mentioned iterative sequences.
(i)In contrast to the iterations (1.3)–(1.5), the convex composition of the iteration (1.7) deals with only instead of and .
converges weakly to a zero of . However, the Mann iteration scheme has only weak convergence for nonexpansive mappings even in a Hilbert space (see ).
Our main purpose is to prove strong convergence theorems for a finite family of accretive operators on a strictly convex Banach space with uniformly G teaux differentiable norm by using viscosity approximation methods. Our theorems extend the comparable results in the following three aspects.
(1)In contrast to weak convergence results on a Hilbert Space in , strong convergence of the iterative sequence is obtained in the general setup of a Banach space.
(3)A single mapping of the results in is replaced by a finite family of mappings.
exists for each , where . The norm of is uniformly G teaux differentiable if for each , the limit is attained uniformly for . The norm of is uniformly Fréchet differentiable ( is also called uniformly smooth) if the limit is attained uniformly for each . It is well known that if is uniformly G teaux differentiable norm, then the duality mapping is single-valued and uniformly continuous on each bounded subset of .
A Banach space is called strictly convex if for , , and , we have for , and for . In a strictly convex Banach space , we have that if , for , , and , then .
Lemma 2.1 (The Resolvent Identity).
for each . In general, we use instead of . Let with , and let be a Banach limit on . Then . Further, we know the following result.
Let be a nonempty closed convex subset of a Banach space with uniformly G teaux differentiable norm. Assume that is a bounded sequence in . Let , and let a Banach limit. Then if and only if , .
Let be a closed convex and, let a mapping of onto . Then is said to be sunny [12, 13] if for all and . A mapping of onto is said to be retraction if ; If a mapping is a retraction then for any , the range of . A subset of is said to be a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction of onto , and it is said to be a nonexpansive retraction of if there exists a nonexpansive retraction of onto . In a smooth Banach space , it is known ([5, Page 48]) that is a sunny nonexpansive retraction if and only if the following condition holds: , and .
Lemma 2.3 (see ).
Lemma 2.5 (see ).
where is a sequence in and is a sequence in satisfying the following conditions:
(ii) or .
Lemma 2.6 ().
Let be real numbers in with and , where and . Then is nonexpansive and .
For the sake of convenience, we list the assumptions to be used in this paper as follows.
(i) is a strictly convex Banach space which has uniformly G teaux differentiable norm, and is a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings.
(ii)The real sequence satisfies the conditions: (C1). and (C2). .
Thus the net is well defined.
and there exists a subsequence which is still denoted by such that .
for all .
In addition, if is a uniformly smooth Banach space in Theorem 3.1 and we define , then we obtain from Theorem 3.1 and [19, Theorem 4.1] that the net converges strongly to , as , where and is a sunny nonexpansive retraction of onto .
Then the sequence converges strongly to , where is the unique solution of a variational inequality .
From (3.47), (3.48), (C1), (C2), and , it follows that and . Consequently applying Lemma 2.5 to (3.50), we conclude that .
If we take , for all , in the iteration (1.7), then, from Theorem 3.3, we have what follows
where with , for , and . Then the sequence converges strongly to .
Theorem 3.3 and Corollary 3.4 prove strong convergence results of the new iterative sequences which are different from the iterative sequences (1.4) and (1.5). In contrast to , the restriction: (C3). or is removed.
If we consider the case of an accretive operator , then as a direct consequence of Theorem 3.1 and Theorem 3.3, we have the following corollaries.
Corollary 3.6 ([3, Theorem 3.1]).
where . If , for some , then converges strongly to , as , where is the unique solution of a variational inequality:
where . Then the sequence converges strongly to , where is the unique solution of a variational inequality .
(i)Corollary 3.7 describes strong convergence result in Banach spaces for a modification of Mann iteration scheme in contrast to the weak convergence result on Hilbert spaces given in [9, Theorem 3].
(ii)In contrast to the result [10, Theorem 4.2], the iterative sequence in Corollary 3.7 is different from the iteration (1.3), and the conditions and are not required.
The work was supported partly by NNSF of China (no. 60872095), the NSF of Zhejiang Province (no. Y606093), K. C. Wong Magna Fund of NIngbo University, NIngbo Natural Science Foundation (no. 2008A610018), and Subject Foundation of Ningbo University (no. XK109050).
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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
Twists on the Torus Equivariant under
the 2-Dimensional Crystallographic Point Groups
Department of Mathematical Sciences, Shinshu University,
3–1–1 Asahi, Matsumoto, Nagano 390-8621, Japan \Email \URLaddresshttp://math.shinshu-u.ac.jp/~kgomi/
Received February 17, 2016, in final form March 03, 2017; Published online March 08, 2017
A twist is a datum playing a role of a local system for topological -theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists for the torus with the actions of the point groups of all the -dimensional space groups (crystallographic groups), or equivalently, the torus with the actions of all the possible finite subgroups in its mapping class group. This is carried out by computing Borel’s equivariant cohomology and the Leray–Serre spectral sequence. As a byproduct, the equivariant cohomology up to degree three is determined in all cases. The equivariant cohomology with certain local coefficients is also considered in relation to the twists of the Freed–Moore -theory.
twist; Borel equivariant cohomology; crystallographic group; topological insulator
53C08; 55N91; 20H15; 81T45
Topological -theory has recently been recognized as a useful tool for a classification of topological insulators in condensed matter physics. In Kitaev’s 10-fold way , the usual complex -theory and also or Atiyah’s -theory are used. These classifications are in some sense the most simple cases, and a recent study of topological insulators focuses on more complicated cases. Such complicated cases arise when we take the symmetry of quantum systems into account. Then equivariant -theory and its twisted version naturally fit into the classification scheme of such systems . Actually, as will be explained in Section 2, a certain quantum system on the -dimensional space invariant under a space group provides a -theory class on the -dimensional torus equivariant under the point group of the space group. If the space group is nonsymmorphic, then the equivariant -class is naturally twisted. In the case of , such (twisted) equivariant -theories are computed for the classes of -dimensional space groups, in view of the classification of topological crystalline insulators [27, 28]. An outcome of these computations of twisted equivariant -theories is the discovery of topological insulators which are essentially classified by but do not require the so-called time-reversal symmetry or the particle-hole symmetry . This type of topological insulators is new in the sense that the known topological insulators essentially classified by so far require the time-reversal symmetry or the particle-hole symmetry.
The understanding of the importance of twisted equivariant -theory in condensed matter physics leads to a mathematically natural issue: determining the possible ‘twists’ for equivariant -theory. To explain this issue more concretely, let us recall that twisted -theory [5, 22] is in some sense a -theory with ‘local coefficients’. The datum playing the role of a ‘local system’ admits various geometric realizations. In this paper, we realize them by twists in the sense of . If a compact Lie group acts on a space , then graded twists on are classified by the Borel equivariant cohomology . Similarly, ungraded twists are classified by , on which we focus for a moment. (Sometimes may be included in the twists, but we regard it as the degree of the -theory.)
By definition, the Borel equivariant cohomology is the usual cohomology of the Borel construction , which is the quotient of by the diagonal -action, where is the total space of the universal -bundle . Associated to the Borel construction is the fibration , and hence the Leray–Serre spectral sequence that converges to the graded quotient of a filtration
One can interpret geometrically in the classification of twists, and there are four types (see Section 3 for details):
Twists which can be represented by group -cocycles of with coefficients in the trivial -module . These twists are classified by .
Twists which can be represented by group -cocycles of with coefficients in the group of -valued functions on regarded as a (right) -module by pull-back. These twists are classified by .
Twists which can be represented by central extensions of the groupoid . These twists are classified by .
Twists of general type, classified by .
The equivariant twists on arising from quantum systems on , to be explained in Section 2, belong to with the point group of a -dimensional space group , and so are the twists considered in . Now, the mathematical issue is whether the twists arising in this way cover all the possibilities or not. The present paper answers this question in the case of by a theorem (Theorem 1.1).
To state the theorem, let be a -dimensional space group, which is also known as a -dimensional crystallographic group, a plane symmetry group, a wallpaper group, and so on. It is a subgroup of the Euclidean group of isometries of , and is an extension of a finite group called the point group by a rank lattice of translations of :
Being a normal subgroup of , the lattice is preserved by the action of on through the inclusion and the standard left action of on . This induces the left action of on the torus that we will consider. Since is a finite subgroup of , it is the cyclic group of order or the dihedral group of degree and order . The classification of -dimensional space groups has long been known, and there are types [12, 24], which we label following . Notice that some space groups share the same point group action on , and there arise distinct finite group actions on the torus. These actions realize essentially all the possible finite subgroups in the mapping class group of the torus , which is isomorphic to as is well known .
Let be the point group of one of the -dimensional space groups , acting on via as above. Then, . This cohomology group and its subgroups are as in Fig. 1.
Under the same hypothesis as in Theorem 1.1,
All the twists can be represented by central extensions of . In particular, there is no non-trivial twist if preserves the orientation of .
If does not preserve the orientation of , then there are twists which can be represented by central extensions of but not by group -cocycles of .
The subgroup is generated by the twists represented by:
group -cocycle of with values in induced from a nonsymmorphic space group such that the action of its point group on is the same as ; and
group -cocycle of with values in .
As a result, all the twists classified by are relevant to topological insulators, whereas there actually exist other twists which cannot be realized by group cocycles. At present their roles in condensed matter theory seem to be unknown.
Theorem 1.1 follows from case by case computations of the equivariant cohomology and the Leray–Serre spectral sequence. Roughly, there are three methods according to the nature of the point group actions: The first method is applied to the cases where the torus is the product of circles with -actions, i.e., the cases of the -actions arising from p2 and pm/pg. In these cases, the equivariant cohomology is computed by means of the splitting of the Gysin exact sequence, as detailed in . The second method is applied to the cases where the point group has no element of order . In these cases, the torus admits an equivariant stable splitting. As a result, the equivariant cohomology of admits the corresponding splitting, and the Leray–Serre spectral sequence turns out to be trivial. Finally, the third method is applied to the remaining cases. In these cases, we take a -CW decomposition of to compute the equivariant cohomology by using the Mayer–Vietoris exact sequence and the exact sequence for a pair, and then study the Leray–Serre spectral sequence. In principle, the third method is the most basic, and hence is applied to all the cases. However, to simplify the computations, we use other methods.
These computations contain enough information to determine the equivariant cohomology , () of the torus with the actions of the possible finite subgroups in the mapping class group .
Let be the point group of one of the -dimensional space groups , acting on via . For , the -equivariant cohomology is as given in Fig. 2.
So far we focused on ungraded twists. To complete the classification of -equivariant twists on , we need to compute the equivariant first cohomology with coefficients in , which provides the information on ‘gradings’ of a twist. But, the computation is immediately completed by a simple application of the universal coefficient theorem to Theorem 1.3. Notice that the equivariant cohomology also admits a filtration
Because the degree in question is , the degeneration of the Leray–Serre spectral sequence gives the identification
which is a direct summand of and is also computed immediately by using the knowledge of the equivariant cohomology of the space consisting of one point, , in Section 4.1.
Let be the point group of one of the -dimensional space groups , acting on via . Then the -equivariant cohomology is as in Fig. 3.
The grading of twists classified by plays a role in a quantum system with symmetry (see Remark 2.2). However, there are other gradings generally, and their roles in condensed matter theory is unknown.
As is mentioned, Atiyah’s -theory is also applied to the classification of topological insulators. The symmetry of -theory however concerns -actions only, and its use is limited to rather simple cases. To take more general symmetries into account, Freed and Moore introduced a -theory which unifies -theory and equivariant -theory . Their -theory is defined for a space with an action of a compact Lie group equipped with a homomorphism . The -theory of Freed–Moore reduces to the -equivariant -theory if is trivial, and to the -theory if and non-trivial. There also exists the notion of twists for the Freed–Moore -theory. A computation of the twisted Freed–Moore -theory is carried out in , leading to the discovery of a novel -phase.
The knowledge about the twists of the Freed–Moore -theory has therefore potential importance to condensed matter physics as well, and the present paper provides it also in the case where is the torus and is the point group of a -dimensional space group. Notice that the classification of the twists for the Freed–Moore -theory parallels that of the twists for equivariant -theory (actually a generalization). In general, the graded twists are classified by and the ungraded twists by . Here denotes a local system for the Borel equivariant cohomology associated to the -module such that its underlying group is and acts via . The cohomology group also admits a filtration
The associated graded quotient is computed by the Leray–Serre spectral sequence, and the subgroups have geometric interpretations as well (Proposition 5.1).
To state our results in the ‘twisted’ case, we introduce the following definition for the point group of a -dimensional space group that admits a non-trivial homomorphism .
In the cases of p2, p4 and p6, the point group is the cyclic group of even order. We write for the unique non-trivial homomorphism given by .
In the other case, the point group is the dihedral group of degree and order , and is embedded into so that is a rotation of and is a reflection. We define to be the composition of the inclusion and . Put differently, and . This provides the unique non-trivial homomorphism if is odd. In the case of even , we define two more non-trivial homomorphisms by
Let be the point group of one of the -dimensional space groups , acting on via , and a non-trivial homomorphism. Then, . This cohomology group and its subgroups are as in Fig. 4.
It should be noticed that the action of the point group on the torus relevant to an application of the Freed–Moore -theory to condensed matter physics is the one modified by a non-trivial homomorphism . Some of such modified actions differ from those given by the inclusion , and hence are not covered in Theorem 1.5. The modified actions should be understood in the context of the so-called magnetic space groups (or colour symmetry groups ), and the cohomology as well as the -theory equivariant under the groups deserve to be subjects of a future work.
One may notice that there are more twists for the Freed–Moore -theory in comparison with the twists for equivariant -theory. At present, we lack such an understanding of twists as in Corollary 1.2(c) in relation with the nonsymmorphic nature of space groups.
The method for computing and its filtration is similar to the one computing . In the computation, the cohomology for is also determined, as summarized below:
Let be the point group of one of the -dimensional space groups , acting on via . For , the -equivariant cohomology with coefficients in the local system induced from a non-trivial homomorphism is as in Fig. 5.
Finally, we make comments about the generalizations. To compute cohomology groups of the higher-dimensional tori which are equivariant under space groups, we can in principle apply the three methods in this paper. The first and second methods would be generalized without difficulty. The third method will however get more difficult, because we need a -CW decomposition of a higher-dimensional torus, which becomes more complicated than decompositions in the -dimensional case. As is suggested by Corollary 1.4, there are local systems for the Borel equivariant cohomology other than associated to a homomorphism . For the cohomology with such a local system, the notion of reduced cohomology does not make sense. This prevents us from using the second method based on the equivariant stable splitting of the torus, forcing us to use a -CW decomposition.
The outline of this paper is as follows: In Section 2, we explain how a certain quantum system leads to a twist and defines a twisted -class, mainly based on a formulation in . At the end of this section, a summary of relationship among some natural actions of point groups on tori is included. In Section 3, we review the Leray–Serre spectral sequence for Borel equivariant cohomology and the notion of twists for equivariant -theory. The geometric interpretation of the filtration of the degree equivariant cohomology is also provided here, after a general property of the spectral sequence is established. Then, in Section 4, we prove Theorems 1.1 and 1.3. To keep readability of this paper, we provide the detail of computations only in the cases p2, p4m/p4g and p6m. (The detail of the other cases can be found in old versions of
arXiv:1509.09194.) Section 5 concerns the equivariant cohomology with the twisted coefficient . We state direct generalizations of some results in the untwisted case, and then prove Theorems 1.5 and 1.6. To keep readability again, we give the details of the computation only in the case of p6m with . Finally, for convenience, the point group actions of -dimensional space groups are listed in Appendix.
2 From quantum systems to twisted -theory
Let us consider the following mathematical setting:
A lattice of rank .
A subgroup of the Euclidean group of which is an extension of a finite group by :
A unitary representation on a finite-dimensional Hermitian vector space .
The group is nothing but a -dimensional space group, and is called the point group of . When is the semi-direct product of and , it is called symmorphic, otherwise nonsymmorphic.
Based on the mathematical setting above, we can introduce a quantum system on which has as its symmetry and as its internal freedom:
The ‘quantum Hilbert space’ consisting of ‘wave functions’ is the -space , on which acts by .
The ‘Hamiltonian’ is a self-adjoint operator on invariant under the -action: . A typical form of is , where is the Laplacian and is a potential term.
2.2 Bloch transformation
Even if the Hamiltonian is invariant under the translation of , a solution to the ‘time-independent Schrödinger equation’ with is not necessarily -invariant. The so-called ‘Bloch transformation’ allows us to deal with such a situation.
Let denote the Pontryagin dual of the lattice , which is often called the ‘Brillouin torus’ in condensed matter physics. We define the space by
We also define transformations and , inverse to each other:
As is described in , the space can be identified with the space of -sections of a vector bundle . The infinite-dimensional vector bundle is given by
where is the Poincaré line bundle, the quotient of the product line bundle by the following -action
In summary, we get an identification of -spaces
The Hamiltonian on then induces an operator on by . If, for instance, is of the form , then preserves the fiber of . Generally, this is a consequence of the translation invariance of the Hamiltonian. When the present quantum system is supposed to be an ‘insulator’, a finite number of discrete spectra of would be confined to a compact region in as varies. Then the corresponding eigenfunctions form a finite rank subbundle , called the ‘Bloch bundle’. The -class of this vector bundle is regarded as an invariant of the quantum system under study.
2.3 Nonsymmorphic group and twisted -theory
We now take the symmetry into account. From the extension , we can associate a twisted -equivariant vector bundle on to the -module . This is a version of the so-called ‘Mackey machine’.
Recall that the Euclidean group is the semi-direct product of the orthogonal group and the group of translations . Hence a collection of representatives of in is expressed as by means of a map . For we put
Since is normal, the action of on preserves . Then we have , and is a group -cocycle of with values in regarded as a left -module through the action of on . This group -cocycle measures the failure for to be symmorphic.
By means of the -action on , we define an ‘action’ of by
whose explicit formula for is given by . The Bloch transformation then induces the following ‘action’ of ,
whose explicit formula for is . Here the left -action on is defined by , where acts on through the inclusion and the left action of on . Notice that and can be honest actions of |
Abstract: Molecular dynamics simulation was applied to study the instability and rupture process of ultra thin water films on a solid substrate. Results show the small disturbance of the film will develop linearly due to the spinodal instability, whereas the interactions between solid and liquid have less in fluences on the initial growth. Then the rupture occurs and the rim recedes with a dynamic contact angle. The radius of the rim varies with time as the square root of the time, which is consistent with the macroscopic theory available. Stronger interaction between solid and liquid will postpone rupture time, decline the dynamic contact angle and raise the density of water near the interface between solid and liquid.
Abstract: First, the notions of the measure of noncompactness and condensing set-valued mappings were introduced in locally FC-uniform spaces without convexity structure. A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings was proved in locally FC-uniform spaces. As applications, some new equilibrium existence theorems of generalized game involving condensing mappings were established in locally FC-uniform spaces. These results improve and generalize some known results in literature to locally FC-uniform spaces. Some further applications of the results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.
Abstract: Some new systems of generalized vector quasi-equilibrium problems involving condensing mappings were introduced and studied in locally FC-uniform spaces. By applying the existence theorem of maximal elements of condensing set-valued mappings in locally FC-uniform spaces obtained by author in the preceding paper, some new existence theorems of solutions for the systems of generalized vector quasi-equilibrium problems were proved in locally FC-uniform spaces. These results improve and generalize some recent known results in literature to locally FC-uniform spaces.
Abstract: Piezoelectric bender elements are widely used as electromechanical sensors and actuators. An analytical sandwich beam model for piezoelectric bender elements was developed based on the first-order shear deformation theory (FSDT), which assumes a single rotation angle for the whole cross-section and a quadratic distribution function for coupled electric potential in piezoelectric layers, and corrects the effect of transverse shear strain on the electric displacement integration. Free vibration analysis of simply-supported bender elements was carried out and the numerical results showed that solutions of the present model for various thickness-to-length ratios compare well with the exact two-dimensional solutions, which presents an efficient and accurate model for analyzing dynamic electromechanical responses of bender elements.
Abstract: Nonlinear governing equations were established for large deflection of incompressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams. Then, the nonlinear bending of a saturated poroelastic cantilever beam with fixed end impermeable and free end permeable, subjected to a suddenly applied constant concentrated transverse load at its free end, was examined with the Galerkin truncation method. The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure were shown in figures. The results of the large deflection and the small deflection theories of the cantilever poroelastic beam were compared, and the differences between them are revealed. It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory, and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.
Abstract: Compared to a smooth channel, a finned-channel provides higher heat transfer coefficient and increasing the fin height enhances the heat transfer. However, this heat transfer enhancement is associated with an increase in the pressure drop. This leads to an increased pumping power requirement so that one may seek an optimum design for such systems. The main goal of this paper is to define the exact location and size of fins in such a way that a minimal pressure drop coincides with an optimal heat transfer based on the genetic algorithm. Each arrangement of fins was considered as a solution of the problem (an individual for genetic algorithm). An initial population was generated randomly at the first step. Then the algorithm had searched among these solutions and made new solutions iteratively by its functions to find an optimum design as reported.
Abstract: An adaptive genetic algorithm is proposed for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes may be infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.
Abstract: The stability of systems with discontinuous right-hand side (with solutions in Filippov's sense) via locally Lipschitz continuous and regular vector Liapunov functions are discussed. A new type of "set-valued derivative" of vector Liapunov functions was introduced, some generalized comparison principles on discontinuous systems were shown. Furthermore Liapunov stability theory was developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Liapunov functions.
Abstract: Impact dynamics of multi-rigid-body systems with joint friction was considered. Based on traditional approximate assumption dealing with impact problem, a general numerical method called sliding state stepping algorithm was introduced. This method can avoid the difficulties in solving di-f ferential equations with variable scale and the result can avoid the energy inconsistency before and a-f ter impact due to considering the complex of the tangential sliding mode. An example was given to describe the concrete details dealing with these difficulties.
Abstract: A Hamiltonian method was applied to study analytically the stress distributions of orthotropic two-dimensional elasticity in (x, z) plane for arbitrary boundary conditions without beam assumptions. It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns. Since coordinates (x, z) cannot be easily separated, an alternative symplectic expansion was used. Similar to the Hamiltonian formulation in classical dynamics, the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian matrix differential operator. The exponential of the Hamiltonian matrix is symplectic. There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions. The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues (zero eigen-solutions) and that of the wellbehaved nonzero eigenvalues (nonzero eigen-solutions). The Jordan chains at zero eigenvalues give the classical Saint Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. On the other hand, the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples were newly given to compare with established results.
Abstract: A hyperbolic attenuation function was introduced to reflect the effect of one firm's default to its partner. If the two firms are competitors (copartners), the default intensity of one firm will decrease (increase) abruptly when the other firm defaults. As time goes on, the impact will decrease gradually until extinction. In this model, the joint distribution and marginal distributions of default times are derived by employing the change of measure, so the fair swap premium of a CDS can be valued.
Abstract: Phase transformation from austenite to martensite in NiTi alloy strips under uniaxial tension has been observed in experiments and has been numerically simulated as a localized deformation. This work presented an analysis of that using the theory of phase transformation. The jump of deformation gradient across interface between the two phases and the Maxwell relation were considered. Governing equations for the phase transformation were derived. The analysis was reduced to finding the minimum value of the loading at which the governing equations have a unique, real, physically acceptable solution. The equations were solved numerically and it is verified that the unique solution exists definitely. The Maxwell stress, the stresses and strains inside both austenite and martensite phases, and the transformation-front orientation angle were determined that are in reasonably good agreement with experimental observations.
Abstract: The proximal-based decomposition method was originally proposed by Chen and Teboulle (Math. Programming, 1994, 64(1):81-101) for solving convex minimization problems. This paper extended to solve monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed, and thus makes it practical to solve the involved subproblems easily. Without additional assumptions, global convergence of the new method is proved under the same mild assumptions on the problem's data as the original method.
Abstract: By using Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory, associated with Lp intersection body and dual mixed volume, some dual Brunn-Minkowski inequalities and their isolate forms are established for Lp intersection body about the normalized Lp radial addition and Lp radial linear combination. Some properties of operator Ip are given.
Abstract: A delayed stage-structured pest management predator-prey system with impulsive transmitting on predators and chemical on prey concern was considered. Sufficient conditions of the global attractivity of pest-extinction boundary periodic solution and permanence of the system were obtained. It was also proved that all solutions of the system are uniformly ultimately bounded. The results provide reliable tactical basis for the practical pest management. |
How do you calculate limiting reagent and theoretical yield? | Socratic 21 Mar 2016 acetic anhydride + salicylic acid → aspirin + acetic acid What is the theoretical yield of aspirin ( C ) if you reacted 4.32 g of acetic anhydride How do you calculate the theoretical yield of aspirin given that 3.52 29 Sep 2014 There are three steps to answering this question correctly: Write a balanced reaction for the chemical process; Calculate the number of moles of How can I calculate the percent yield of aspirin synthesis? | Socratic 13 Aug 2015 You calculate the theoretical yield of aspirin, and then you use your actual yield to calculate the percent yield. Theoretical Yield Worked Example Problem - ThoughtCo 24 Mar 2017 This example problem demonstrates how to calculate the amount of reactant needed to produce a product. Problem. Aspirin is prepared Experiment 5 - Synthesis of Aspirin The synthesis reaction of aspirin is shown below: Since acetic acid is very soluble in . Report the theoretical yield and the percent yield of the aspirin product. Aspirin Synthesis Lab Report-2 - EdSpace - American University and the amount of pure aspirin synthesized was 2.169. The theoretical yield was 2.520 grams. Thus, there was a percent error of 13.93 % and percent yield of Aspirin Percent Yield Math - YouTube 11 Aug 2016 Aspirin Percent Yield Math . Theoretical, Actual, Percent Yield & Error - Limiting Reagent and Excess Reactant That Remains - Duration: Stoichiometry: 3.62 - Theoretical and percentage yield - IBChem IB Chemistry Stoichiometry - Theoretical and percentage yield in chemical Example: In an experiment to prepare a sample of aspirin (C9H8O4), 13.8 g of Theoretical and Percent Yield Percentage Yield = mass of Actual Yield x 100% this is how much product will be synthesized in ideal conditions. Example: 0.135 g acetylsalicylic acid. 5. Synthesis and Characterization of Aspirin - Odinity Written by Ali. Purpose. The purpose of this lab was to synthesize Aspirin and measure the synthesized Aspirin's purity. By calculating the theoretical yield based
Percent Yield & Percent Purity (solutions, examples, videos)
In an experiment, 100.0 grams of salicylic acid gave 121.2 grams of aspirin. The concepts of limiting reagent, theoretical yield, and percent yield are discussed Chem 104 Laboratory - Chemistry Synthesis of Aspirin. Reference: Chemistry for a Theoretical yield of aspirin (gram). Weight of isolated aspirin (gram). %Yield. 2. 0.0235. 3.25. 0.0561. 5.30. The Synthesis and Analysis of Aspirin - Vernier Software & Technology compounds used in the synthesis of aspirin is salicylic acid, which is itself a pain reliever that .. What is the theoretical yield of aspirin in your synthesis? More Mole Stoichiometry with Theoretical and Percent Yield a) What is the theoretical yield of lithium nitride in grams when 12.3 g of 3- Aspirin (a.k.a. acetylsalicylic acid) is synthesized by reacting salicylic acid with. Yields Define and determine theoretical yields, actual yields, and percent yields. . Chemistry Is Everywhere: Actual Yields in Drug Synthesis and Purification and 12.03 g of aspirin are isolated, what are the theoretical yield and the actual yield? SYNTHESIS OF ASPIRIN SYNTHESIS OF ASPIRIN (acetylsalicylic acid). 1. Place 2.O g (O.O15 What is the theoretical yield of aspirin (in two significant figures). 2. Why is the aspirin C4H6O3+C7H6O3?C9H8O4+C2H4O2 (336664) | Wyzant Resources In a laboratory synthesis, a student begins with 5.00 mL of acetic anhydride (density = 1.08 g / mL) Once the reaction is complete, the student collects 2.47g of aspirin. I need to find the theoretical yield and the percent yield. LAB1 The quantity in the denominator (2.5 g) represents the theoretical yield of aspirin based on the moles of salicylic acid and acetic anhydride used in the synthesis. Synthesis of Aspirin - Magritek Synthesis of aspirin from salicylic acid. 4 chemistry including synthesis, crystallization, stoichiometry, and percent yield. 1 therefore Theoretical: 0.03 mol. 24. Synthesis of Aspirin 11 Feb 2016 Give a balanced equation for the synthesis of aspirin from salicylic Calculate the theoretical yield and the percent yield for this and similar. 8-Synthesis of Aspirin This aspirin should be more pure than the original aspirin. The final product will be dried and weighed and the theoretical and percent yields will be calculated.
Experiment 7 Synthesis of Aspirin
Aspirin can be synthesized by two simple components, acetic anhydride and There are two major calculations for this lab, theoretical yield and percent yield. BBC - GCSE Bitesize: Percentage yield In the neutralisation of sulfuric acid with sodium hydroxide, the theoretical yield from 6.9 g of sulfuric acid is 10 g. In a synthesis, the actual yield is 7.2 g. What is Chemistry 104: Synthesis of Aspirin In this experiment you will synthesize aspirin (acetylsalicylic acid, C9H8O4 ), purify it, and determine . Use the salicylic acid to calculate the theoretical yield.]. Module Report #4 - SECTION 1 Data Collection Reporting and Aspirin Synthesis a. (6 points) Report the percent yield of your aspirin (include calculations for the theoretical yield). How do the yields differ among the groups in Preparetion of asprine - SlideShare 30 Dec 2014 A useful synthesis of acetylsalicylic acid was developed in 1893, Theoretical yield of aspirin = 0.00724mole *180.12g/mole = 1.304g Actual organic chemistry-making of acetylsalicylic acid (aspirin) Synthesis of Aspirin. Small crystals of the acetylsalicylic acid (aspirin) should begin to appear as the mixture cools. Calculate the theoretical yield of aspirin EXPERIMENT #4: Synthesis of Copper (II) Compounds SYNTHESES AND CHARACTERIZATION OF ASPIRIN. One of . Determine the theoretical yield of aspirin that can be obtained from the addition of 2.0 grams of Synthesis of Aspirin Short Paragraph - Paragraphica.com 29 Apr 2017 The Synthesis of Aspirin Chemistry Standard Level Lab Report Data Collection Dependent variables: Theoretical and actual yield of aspirin. Aspirin is stable in dry air, but in the presence of moisture, it slowly COOH. OH. O. O Na. Salicylic acid. Aspirin. Acetylsalicylic acid. Sodium acetate. Sodium salicylate. O . period since the synthesized acetylsalicylic acid may decompose. limiting reagent. - Use salicylic acid to calculate the theoretical yield. |
Lab 6 Molecular Geometry
1. Describe the difference between valence and core electrons.
Valence electrons are in the outermost shell and core electron are in the innermost shell.
2. What is the octet rule and which elements obe
Lab 3 Data Analysis & Graphing
1. In your own words, describe how to calculate density.
You divide the mass by the volume to get the density.
2. Explain why the metric system is used in science.
It is more widely used and becau
Lab 1 Introduction and Laboratory Safety
1. What would you do if you spilled a tiny amount of hydrochloric acid on your hand?
Rinse with water for 20 minutes.
2. Describe a possible danger than can occur if you put food, soda, coffee, pencil eras
Lab 2 The Scientific Method
1. Which of the following are testable using scientific experimentation.
a. Are there more seeds in a medium sized Red Delicious Apple or a medium size Green
Granny Smith Apple? Testable.
b. Why do p
Lab 4 Molar Mass
1. Imagine you are a guest speaker at a kindergarten class and you are asked to explain Avogadros
number. What would you say and why?
There a many, many tiny puzzle pieces that make up everything you see. Avoga
Lab 7 Precipitation Reactions
1. Under what conditions will a precipitate form?
When ionic compounds are placed in water, separate into negative and positive charges, and combine
into new ions(s) that are insoluble.
2. What inf
Lab 5 Electron Configuration
1. What is electron configuration?
Electron configuration is a way of describing where the electrons in an atom are located.
2. How is the light emitted by an atom related to its electron configurati
Answers Percent Composition and Empirical Formula
1.a)1moleKClO3 = (39.1 g K) + (35.4 g Cl) + (3 x 16.0 = 48.0 g O) = 122.5 g
% K = 39.1/122.5 = 31.9% K;
% Cl = 35.4/122.5 = 28.9 % Cl;
% O = 48.0/122.5 = 39.2% O
Molar Mass Problems
1. Calculate the mass of 1.000 mole of CaCl2
2. Calculate grams in 3.0000 moles of CO2
3. Calculate number of moles in 32.0 g of CH4
4. Determine mass in grams of 40.0 moles of Na2CO3
5. Calculate moles in 168.0 g of HgS
6. Calculate m
Answers to More Chapter 22 Study Questions
1. a) CH3CHCHCH2CH2CH3
lots of other isomers possible
2. 5-ethyl-3,5,7-trimethyl nonane
Answers to More Chapter 4 Study Questions
1. a) FeCl3(aq) + 3 NaOH(aq) Fe(OH)3(s) + 3 NaCl(aq).
b) 30.0 mL x
c) 30.0 mL x
0.500 moles NaOH
1000 mL solution
1 mol FeCl 3 1000 mL solution
3 mol NaOH
0.200 mol FeCl 3
Electrochemical Cells Worksheet
1. Calculate the standard cell potential produced by a galvanic cell consisting of a nickel electrode in contact with
a solution of Ni2+ ions and a silver electrode in contact with a solution of Ag + ions. Which is anode an
Activation Energy & Catalyst
1. Summarize 3 points of the collision reaction theory.
" Mus. Co Via, lggW (emf ewWW mj
/ W3 . 6WWL 4%!an 6412157
V W5 /E6 sz/fl it, Cgr/T Oflzo/neiag/m ,
Use the following to answer the next questions
2. a) Is
T. J I "\N\
' 1 . - ' u, 1. ' .
, - k ' ,M, V _ : Ne-Wkrr oar mar
\ , ELEQL 7A ICAL\EL5 , ,MJLLQA we H* 3
f R NOT" pres? _
' . / . _
, > 1. FOrmcfonowingq'estions design @ ectmchemicgl cell. 1: the diagram _
_ /<- properly. Write the/ f arm a V . -
Directions: Study the food label and answer the questions.
1. What is the serving size of this product?
2. How many calories per serving size are in this
3. What percent of daily value is the sodium?
4. In 2 servings, how much
1. A cylinder of argon gas contains 50.0 L of Ar at 18.4 atm and 127 C. How many
moles of argon are in the cylinder?
2. A 283.3-g sample of X2(g) has a volume of 30 L at 3.2 atm and 27 C. What is
3. An ideal gas sample is c
Alkane alkene al ne and C clic com ounds'
1. Alkanes, alkenes, and alkynes are the three main families of aliphatic hydrocarbons.
a) met is the gkgeral mwormula for each family?
b) . at type of carbkl-carbon bonds are iured in each fa
1. Write IUPAC names for the following compounds.
lg ' D c a a
2. Draw a condensed structural formula for each compound.
a) 3-methylbutan-l-ol b) methylpropan-Z-ol c) cyclopentanol
c? 0 H
. CH - ~ CH
C H73 a c (4 CHa ~CHa
Balancing Redox Reactions
Part A: Balance the following equations using Oxidation Numbers. Underline the oxidizing agent and circle the
24 H2S + 16 HNO3
3 S8 + 16 NO +
H2SO4 + 2 HBr SO2 + 2 Br2 + H2O
H2SO4 + 8 HI
Acid / Base Theories Worksheet
1. Dene an Arrhenius acid and a base.
lOrt'Ze/g 4'7 - [onus "6'0.
H+ /'/'cfw_30+W-)' I OH 2) ,
2. What is the Arrhenius Modied denition of an acid and a base? _
3 w95cfw_W9 2993044) what (aou, *4. x
Find each element in the puzzle, then write the leftover letters in order,
top to bottom and left to right, to complete the hidden message.
P A B
S C A N D
L H N
U M S M S
R O G E N U T H
A B O T
T A A S
U M Z
D C H R
CHEIMIS T R Y 30
HEA T LOSS = HEA T GAIN
1. Methanol IS one type of fuel that 13 used 1n fondue heaters. In an experiment using a simple
calorimeter, 2. 98 g of methanol was burned tthe temperature of 0.650 kg of water by 20. 9C.
Calculate the ar enthal
CHEMIS TR Y 30
1. What physical property of hydrocarbons is used for fractionation?
2. -Whv does. crude oil have to be chemically processed, in addition to being fractionated? - I
lo Mortgage #41 y 799 os cfw_4w 4raL(fmg *fKLa-G m (\YL
CHE TRY 30
REVIEW OF ACIDS AND BASES
1. Given the following reactions, write the balanced equation using Bronsted4Lowry
acid\base reactions: 9 cfw_I l ,
(a H ~09,
. C 02*
a) aqueous solutions of nitrOus acid reacts with cacfum carbonate
HHCxnai) * (O'S |
6.1 Objectives of convection analysis:
Main purpose of convective heat transfer analysis is to determine:
- flow field
- temperature field in fluid
- heat transfer coefficient, h
How do we determine h ?
Consider the process of convective cooling, as we pass a cool fluid past a heated wall. This
process is described by Newton’s law of Cooling:
U∞ y U∞ T∞
u(y) q” T(y)
Near any wall a fluid is subject to the no slip condition; that is, there is a stagnant sub layer.
Since there is no fluid motion in this layer, heat transfer is by conduction in this region.
Above the sub layer is a region where viscous forces retard fluid motion; in this region some
convection may occur, but conduction may well predominate. A careful analysis of this
region allows us to use our conductive analysis in analyzing heat transfer. This is the basis of
our convective theory.
At the wall, the convective heat transfer rate can be expressed as the heat flux.
qconv = − k f ⎟ = h (Ts − T∞ )
∂y ⎟ y =0
∂y ⎟ y =0
Hence, h =
(Ts − T∞ )
⎟ depends on the whole fluid motion, and both fluid flow and heat transfer
equations are needed
The expression shows that in order to determine h, we must first determine the temperature
distribution in the thin fluid layer that coats the wall.
2.2 Classes of Convective Flows
Free or natural convection
(induced by buoyancy forces) May occur
Forced convection (induced by
• extremely diverse
• several parameters involved (fluid properties, geometry, nature of flow, phases etc)
• systematic approach required
• classify flows into certain types, based on certain parameters
• identify parameters governing the flow, and group them into meaningful non-
• need to understand the physics behind each phenomenon
A. Based on geometry:
External flow / Internal flow
B. Based on driving mechanism
Natural convection / forced convection / mixed convection
C. Based on number of phases
Single phase / multiple phase
D. Based on nature of flow
Laminar / turbulent
Table 6.1. Typical values of h (W/m2K)
Free convection gases: 2 - 25
liquid: 50 – 100
Forced convection gases: 25 - 250
liquid: 50 - 20,000
Boiling/Condensation 2500 -100,000
2.3 How to solve a convection problem ?
• Solve governing equations along with boundary conditions
• Governing equations include
1. conservation of mass
2. conservation of momentum
3. conservation of energy
• In Conduction problems, only (3) is needed to be solved. Hence, only few parameters
• In Convection, all the governing equations need to be solved.
⇒ large number of parameters can be involved
2.4 FORCED CONVECTION: external flow (over flat plate)
An internal flow is surrounded by solid boundaries that can restrict the development of its
boundary layer, for example, a pipe flow. An external flow, on the other hand, are flows over
bodies immersed in an unbounded fluid so that the flow boundary layer can grow freely in
one direction. Examples include the flows over airfoils, ship hulls, turbine blades, etc
U < U∞
• Fluid particle adjacent to the solid surface is at rest
• These particles act to retard the motion of adjoining layers
• ⇒ boundary layer effect
Inside the boundary layer, we can apply the following conservation principles:
Momentum balance: inertia forces, pressure gradient, viscous forces, body forces
Energy balance: convective flux, diffusive flux, heat generation, energy storage
2.5 Forced Convection Correlations
Since the heat transfer coefficient is a direct function of the temperature gradient next to the
wall, the physical variables on which it depends can be expressed as follows:
h=f(fluid properties, velocity field ,geometry,temperature etc.)
As the function is dependent on several parameters, the heat transfer coefficient is usually
expressed in terms of correlations involving pertinent non-dimensional numbers.
Forced convection: Non-dimensional groupings
• Nusselt No. Nu = hx / k = (convection heat transfer strength)/
(conduction heat transfer strength)
• Prandtl No. Pr = ν/α = (momentum diffusivity)/ (thermal diffusivity)
• Reynolds No. Re = U x / ν = (inertia force)/(viscous force)
Viscous force provides the dampening effect for disturbances in the fluid. If dampening is
strong enough ⇒ laminar flow
Otherwise, instability ⇒ turbulent flow ⇒ critical Reynolds number
For forced convection, the heat transfer correlation can be expressed as
Nu=f (Re, Pr)
The convective correlation for laminar flow across a flat plate heated to a constant wall
Nux = 0.323·Rex½ · Pr1/3
Nux ≡ h⋅x/k
Rex ≡ (U∞⋅x⋅ρ)/μ
Pr ≡ cP⋅μ/k
Physical Interpretation of Convective Correlation
The Reynolds number is a familiar term to all of us, but we may benefit by considering what
the ratio tells us. Recall that the thickness of the dynamic boundary layer, δ, is proportional
to the distance along the plate, x.
Rex ≡ (U∞⋅x⋅ρ)/μ ∝ (U∞⋅δ⋅ρ)/μ = (ρ⋅U∞2)/( μ⋅U∞/δ)
The numerator is a mass flow per unit area times a velocity; i.e. a momentum flow per unit
area. The denominator is a viscous stress, i.e. a viscous force per unit area. The ratio
represents the ratio of momentum to viscous forces. If viscous forces dominate, the flow will
be laminar; if momentum dominates, the flow will be turbulent.
Physical Meaning of Prandtl Number
The Prandtl number was introduced earlier.
If we multiply and divide the equation by the fluid density, ρ, we obtain:
Pr ≡ (μ/ρ)/(k/ρ⋅cP) = υ/α
The Prandtl number may be seen to be a ratio reflecting the ratio of the rate that viscous
forces penetrate the material to the rate that thermal energy penetrates the material. As a
consequence the Prandtl number is proportional to the rate of growth of the two boundary
δ/δt = Pr1/3
Physical Meaning of Nusselt Number
The Nusselt number may be physically described as well.
Nux ≡ h⋅x/k
If we recall that the thickness of the boundary layer at any point along the surface, δ, is also a
function of x then
Nux ∝ h⋅δ/k ∝ (δ/k⋅A)/(1/h⋅A)
We see that the Nusselt may be viewed as the ratio of the conduction resistance of a material
to the convection resistance of the same material.
Students, recalling the Biot number, may wish to compare the two so that they may
distinguish the two.
Nux ≡ h⋅x/kfluid Bix ≡ h⋅x/ksolid
The denominator of the Nusselt number involves the thermal conductivity of the fluid at the
solid-fluid convective interface; The denominator of the Biot number involves the thermal
conductivity of the solid at the solid-fluid convective interface.
Local Nature of Convective Correlation
Consider again the correlation that we have developed for laminar flow over a flat plate at
constant wall temperature
Nux = 0.323·Rex½ · Pr1/3
To put this back into dimensional form, we replace the Nusselt number by its equivalent, hx/k
and take the x/k to the other side:
h = 0.323·(k/x)⋅Rex½ · Pr1/3
Now expand the Reynolds number
h = 0.323·(k/x)⋅[(U∞⋅x⋅ρ)/μ]½ · Pr1/3
We proceed to combine the x terms:
h = 0.323·k⋅[(U∞⋅ρ)/( x⋅μ)]½ · Pr1/3
And see that the convective coefficient decreases with x½.
Coefficient, h. Layer, δt
Boundary Layer, δ
We see that as the boundary layer thickens, the convection coefficient decreases. Some
designers will introduce a series of “trip wires”, i.e. devices to disrupt the boundary layer, so
that the buildup of the insulating layer must begin anew. This will result in regular
“thinning” of the boundary layer so that the convection coefficient will remain high.
Use of the “Local Correlation”
A local correlation may be used whenever it is necessary to find the convection coefficient at
a particular location along a surface. For example, consider the effect of chip placement
upon a printed circuit board:
Chip 1 Chip 2 Chip 3
Here are the design conditions. We know that as the higher the operating temperature of a
chip, the lower the life expectancy.
With this in mind, we might choose to operate all chips at the same design temperature.
Where should the chip generating the largest power per unit surface area be placed? The
Life expectancy of Chip
Operating Temperature of Chip
If one were interested in the total heat loss from a surface, rather than the temperature at a
point, then they may well want to know something about average convective coefficients.
For example, if we were trying to select a heater to go inside an aquarium, we would not be
interested in the heat loss at 5 cm, 7 cm and 10 cm from the edge of the aquarium; instead we
want some sort of an average heat loss.
x Coefficient, hx.
The desire is to find a correlation that provides an overall heat transfer rate:
∫ hx ⋅ [Twall − T∞ ] ⋅ dA = ∫0 hx ⋅ [Twall − T∞ ] ⋅ dx
Q = hL⋅A⋅[Twall-T∞] =
where hx and hL, refer to local and average convective coefficients, respectively.
Compare the second and fourth equations where the area is assumed to be equal to A = (1⋅L):
∫0 hx ⋅ [Twall − T∞ ] ⋅ dx
Since the temperature difference is constant, it may be taken outside of the integral and
∫0 hx ⋅ dx
This is a general definition of an integrated average.
Proceed to substitute the correlation for the local coefficient.
k ⎡ U∞ ⋅ x ⋅ ρ ⎤
hL⋅L= ∫0 0.323 ⋅ ⋅ ⎢ ⋅ Pr 1/3 ⋅ dx
x ⎣ μ ⎥
Take the constant terms from outside the integral, and divide both sides by k.
⎡ U∞ ⋅ ρ ⎤
L ⎡ 1⎤
hL⋅L/k = 0.323 ⋅ ⎢ ⎥ ⋅ Pr ⋅ ∫0 ⎢ x ⎥ ⋅ dx
⎣ μ ⎦ ⎣ ⎦
Integrate the right side.
⎡ U∞ ⋅ ρ ⎤
hL⋅L/k = 0.323 ⋅ ⎢ ⎥ ⋅ Pr ⋅ 1/3
⎣ μ ⎦ 0.5 0
The left side is defined as the average Nusselt number, NuL. Algebraically rearrange the right
0.5 0 .5
0.323 ⎡U ∞ ⋅ ρ ⎤ ⎡U ⋅ L ⋅ ρ ⎤
⋅ ⋅ Pr ⋅ L
= 0.646 ⋅ ⎢ ∞ ⋅ Pr 3
0 .5 ⎢ μ ⎥ ⎥
⎣ ⎦ ⎣ μ ⎦
The term in the brackets may be recognized as the Reynolds number, evaluated at the end of
the convective section. Finally,
NuL = 0.646 ⋅ Re 0.5 ⋅ Pr
This is our average correlation for laminar flow over a flat plate with constant wall
In the development of the boundary layer theory, one may notice the strong relationship
between the dynamic boundary layer and the thermal boundary layer. Reynold’s noted the
strong correlation and found that fluid friction and convection coefficient could be related.
This refers to the Reynolds Analogy.
Conclusion from Reynold’s analogy: Knowing the frictional drag, we know the Nusselt
Number. If the drag coefficient is increased, say through increased wall roughness, then the
convective coefficient will increase. If the wall friction is decreased, the convective
coefficient is decreased.
We could develop a turbulent heat transfer correlation in a manner similar to the von Karman
analysis. It is probably easier, having developed the Reynolds analogy, to follow that course.
The local fluid friction factor, Cf, associated with turbulent flow over a flat plate is given as:
Cf = 0.0592/Rex0.2
Substitute into the Reynolds analogy:
(0.0592/Rex0.2)/2 = Nux/RexPr1/3
Rearrange to find
Nux = 0.0296⋅Rex0.8⋅Pr1/3 Turbulent Flow
In order to develop an average correlation, one would evaluate an integral along the plate
similar to that used in a laminar flow:
Laminar Region Turbulent region
hL⋅L = ∫0 hx dx = ∫0 hx ,la min ar ⋅ dx + ∫ Lcrit hx ,turbulent ⋅ dx
L crit L L
Note: The critical Reynolds number for flow over a flat plate is 5⋅105; the critical Reynolds
number for flow through a round tube is 2000.
The result of the above integration is:
Nux = 0.037⋅(Rex0.8 – 871)⋅Pr1/3
Note: Fluid properties should be evaluated at the average temperature in the boundary layer,
i.e. at an average between the wall and free stream temperature.
Tprop = 0.5⋅(Twall+ T∞)
2.6 Free convection
Free convection is sometimes defined as a convective process in which fluid motion is caused
by buoyancy effects.
Tw T∞ < Tboundry. layer < Tw
ρ∞ < ρboundry. layer
Compare the velocity profiles for forced and natural convection shown below:
U∞ > 0 U∞ = 0
Forced Convection Free Convection
Coefficient of Volumetric Expansion
The thermodynamic property which describes the change in density leading to buoyancy in
the Coefficient of Volumetric Expansion, β.
ρ ∂T P = Const .
Evaluation of β
• Liquids and Solids: β is a thermodynamic property and should be found from
Property Tables. Values of β are found for a number of engineering fluids in Tables
given in Handbooks and Text Books.
• Ideal Gases: We may develop a general expression for β for an ideal gas from the
ideal gas law:
P = ρ⋅R⋅T
ρ = P/R⋅T
Differentiating while holding P constant:
dρ P ρ ⋅ R⋅T ρ
=− 2 = − =−
dT P = Const .
R⋅T R⋅T 2
Substitute into the definition of β
β = Ideal Gas
Because U∞ is always zero, the Reynolds number, [ρ⋅U∞⋅D]/μ, is also zero and is no longer
suitable to describe the flow in the system. Instead, we introduce a new parameter for natural
convection, the Grashof Number. Here we will be most concerned with flow across a vertical
surface, so that we use the vertical distance, z or L, as the characteristic length.
g ⋅ β ⋅ Δ T ⋅ L3
Just as we have looked at the Reynolds number for a physical meaning, we may consider the
ρ ⋅ g ⋅ β ⋅ Δ T ⋅ L3 ⎛ Buoyant Force ⎞ ⎛ Momentum ⎞
ρ ⋅ g⋅ β ⋅ ΔT ⋅ L
2 3 ( 2 ) ⋅ ( ρ ⋅ U max ) ⎜
⎠ ⎝ Area ⎠
Gr ≡ = L =
μ2 ⎛ ViscousForce ⎞
μ ⋅ 2
L ⎝ Area ⎠
Free Convection Heat Transfer Correlations
The standard form for free, or natural, convection correlations will appear much like those for
forced convection except that (1) the Reynolds number is replaced with a Grashof number
and (2) the exponent on Prandtl number is not generally 1/3 (The von Karman boundary layer
analysis from which we developed the 1/3 exponent was for forced convection flows):
Nux = C⋅Grxm⋅Prn Local Correlation
NuL = C⋅GrLm⋅Prn Average Correlation
Quite often experimentalists find that the exponent on the Grashof and Prandtl numbers are
equal so that the general correlations may be written in the form:
Nux = C⋅[Grx⋅Pr]m Local Correlation
NuL = C⋅[GrL⋅Pr]m Average Correlation
This leads to the introduction of the new, dimensionless parameter, the Rayleigh number, Ra:
Rax = Grx⋅Pr
RaL = GrL⋅Pr
So that the general correlation for free convection becomes:
Nux = C⋅Raxm Local Correlation
NuL = C⋅RaLm Average Correlation
Laminar to Turbulent Transition
Just as for forced convection, a boundary layer will form for free convection. The insulating
film will be relatively thin toward the leading edge of the surface resulting in a relatively
high convection coefficient. At a Rayleigh number of about 109 the flow over a flat plate
will transition to a turbulent pattern. The increased turbulence inside the boundary layer will
enhance heat transfer leading to relative high convection coefficients, much like forced
Ra < 109 Laminar flow. [Vertical Flat Plate]
Ra > 109 Turbulent flow. [Vertical Flat Plate]
Generally the characteristic length used in the correlation relates to the distance over which
the boundary layer is allowed to grow. In the case of a vertical flat plate this will be x or L,
in the case of a vertical cylinder this will also be x or L; in the case of a horizontal cylinder,
the length will be d.
Critical Rayleigh Number
Consider the flow between two surfaces, each at different temperatures. Under developed
flow conditions, the interstitial fluid will reach a temperature between the temperatures of the
two surfaces and will develop free convection flow patterns. The fluid will be heated by one
surface, resulting in an upward buoyant flow, and will be cooled by the other, resulting in a
Note that for enclosures it is
customary to develop
correlations which describe the
overall (both heated and cooled
surfaces) within a single
Free Convection Inside an
If the surfaces are placed closer together, the flow patterns will begin to interfere:
Free Convection Inside an Free Convection Inside an
Enclosure With Partial Flow Enclosure With Complete Flow
In the case of complete flow interference, the upward and downward forces will cancel,
canceling circulation forces. This case would be treated as a pure convection problem since
no bulk transport occurs.
The transition in enclosures from convection heat transfer to conduction heat transfer occurs
at what is termed the “Critical Rayleigh Number”. Note that this terminology is in clear
contrast to forced convection where the critical Reynolds number refers to the transition from
laminar to turbulent flow.
Racrit = 1000 (Enclosures With Horizontal Heat Flow)
Racrit = 1728 (Enclosures With Vertical Heat Flow)
The existence of a Critical Rayleigh number suggests that there are now three flow regimes:
(1) No flow, (2) Laminar Flow and (3) Turbulent Flow. In all enclosure problems the
Rayleigh number will be calculated to determine the proper flow regime before a correlation |
Chapter 1: Basic Functional Analysis
- Abstract: Functional and convex analysis are closely intertwined. In this chapter we recall the basic concepts and results from functional analysis and calculus that will be needed throughout this book. A first section is devoted to general normed spaces. We begin by establishing some of their main properties, with an emphasis on the linear functions between spaces. This leads us to bounded linear functionals and the topological dual. Second, we review the Hahn-Banach Separation Theorem, a very powerful tool with important consequences. It also illustrates the fact that the boundaries between functional and convex analysis can be rather fuzzy at times. Next, we discuss some relevant results concerning the weak topology, especially in terms of closedness and compactness. Finally, we include a subsection on differential calculus, which also provides an introduction to standard smooth optimization techniques. The second section deals with Hilbert spaces, and their very rich geometric structure, including the ideas of projection and orthogonality. We also revisit some of the general concepts from the first section (duality, reflexivity, weak convergence) in the light of this geometry.
Chapter 2: Existence of Minimizers
- Abstract: In this chapter, we present sufficient conditions for an extended real-valued function to have minimizers. After discussing the main concepts, we begin by addressing the existence issue in abstract Hausdorff spaces, under certain (one-sided) continuity and compactness hypotheses. We also present Ekeland’s Variational Principle, providing the existence of approximate minimizers that are strict in some sense. Afterwards, we study the minimization of convex functions in reflexive spaces, where the verification of the hypothesis is more practical. Although it is possible to focus directly on this setting, we preferred to take the long path. Actually, the thechniques used for the abstract framework are useful for problems that do not fit in the convex reflexive setting, but where convexity and reflexivity still play an important role.
Chapter 3: Convex Analysis and Subdifferential Calculus
- Abstract: This chapter deals with several properties of convex functions, especially in connection with their regularity, on the one hand, and the characterization of their minimizers, on the other. We shall explore sufficient conditions for a convex function to be continuous, as well as several connections between convexity and differentiability. Next, we present the notion of subgradient, a generalization of the concept of derivative for nondifferentiable convex functions that will allow us to characterize their minimizers. After discussing conditions that guarantee their existence, we present the basic (yet subtle) calculus rules, along with their remarkable consequences. Other important theoretical and practical tools, such as the Fenchel conjugate and the Lagrange multipliers, will also be studied. These are particularly useful for solving constrained problems.
Chapter 4: Examples
- Abstract: The tools presented in the previous chapters are useful, on the one hand, to prove that a wide variety of optimization problems have solutions; and, on the other, to provide useful characterizations allowing to determine them. In this chapter, we present a short selection of problems to illustrate some of those tools. We begin by revisiting some results from functional analysis concerning the maximization of bounded linear functionals and the realization of the dual norm. Next, we discuss some problems in optimal control and calculus of variations. Another standard application of these convex analysis techniques lies in the field of elliptic partial differ-ential equations. We shall review the theorems of Stampacchia and Lax-Milgram, along with some variations of Poisson’s equation, including the obstacle problem and the p-Laplacian. We finish by commenting a problem of data compression and restoration.
Chapter 5: Problem-Solving Strategies
- Abstract: Only in rare cases, can problems in function spaces be solved analytically and exactly. In most occasions, it is necessary to apply computational methods to approximate the solutions. In this chapter, we discuss some of the basic general strategies that can be applied. First, we present several connections between opti-mization and discretization, along with their role in the problem-solving process. Next, we introduce the idea of iterative procedures, and discuss some abstract tools for proving their convergence. Finally, we comment some ideas that are useful to simplify or reduce the problems, in order to make them tractable or more efficiently solved.
Chapter 6: Keynote Iterative Methods
- Abstract: In this chapter we give an introduction to the basic (sub)gradient-based methods for minimizing a convex function on a Hilbert space. We pay special attention to the proximal point algorithm and the gradient method, which are interpreted as time discretizations of the steepest descent differential inclusion. Moreover, these methods, along with some extensions and variants, are the building blocks for other − more sophisticated − methods that exploit particular features of the problems, such as the structure of the feasible (or constraint) set. The choice of proximal- or gradient-type schemes depends strongly on the regularity of the objective function. |
Before we dive in, music is primarily defined by what we hear, not by the analysis and insight provided by math. For example, an octave is a note whose frequency is double that of its parent note. The mathematical relationship was discovered after the fact. The following is an exploration of how math is used in music, but I don’t want to put the cart before the horse here. The math supports the music, makes it work. But the math is really fine-tuning what we hear.
Pythagoras developed a musical system that over the years evolved into what we have today. (At least Pythagoras is often credited for it.) Not until “recently,” however, has one of the major problems with music been resolved (see what I did there with resolve?).
The problem the ancients had is that their octaves didn’t line up. An octave, as I mentioned early, is a note that has twice (or half) the frequency of another note. Octaves, in modern western music, share the same names, too. The note A, at 440 Hz, has an octave at 880 Hz, and also 220 Hz. (There are infinitely many octaves, in theory, though our ears have a limited range of things we can hear.) The ancients, however, had a problem because after a few octaves, well, they were no longer octaves.
In western music we have 12 semi-tones, A, A# (or B-flat), B, C, C#, D, D#, E, F, F#, G, G# and then A again. It’s cyclic, repeated infinitely both higher and lower. Each semi-tone in the next series of 12 notes is an octave of our first series of notes. And the relationship between notes is what makes them, well, musical, not just sounds.
The problem is defining that relationship. You see, because each note is slightly higher (has a higher frequency), and each note’s octave is double that frequency, what happens is the notes get further and further apart (the differences in their frequencies increases).
Let’s take a look at the frequencies:
As you can see, the differences between consecutive notes is increasing, at an increasing rate! This is not a linear relationship. Because of this, the ancients had a very hard time defining what was an A and what was a D, especially when you started moving around between octaves. Things got jumbled, and out of tune.
It is tricky to find the proportion and rate of change between consecutive notes, any two consecutive notes that is. That’s where math comes in to save the day. Let’s build the rate of change, shall we.
First, note that the rate is increasing, at an increasing rate, so we cannot add. I show that in the video below. We have to multiply. When we repeatedly multiply, we can use exponents. Since we need a note and it’s octave to be doubles, our base number is 2.
Since there are twelve notes between a note and its octave, we need to break the multiple of two into twelve equal, multiplicative parts. That’s a rational exponent, 1/12.
The number we need to multiply each note by is 21/12. Each note is one-twelfth of the way to the octave. It is pretty cool indeed.
In the last section we looked at some expressions
like, “What is the third root of twenty-seven, squared?” The math is kind of ugly looking.
The procedures are clunky and it is very easy to lose
sight of the objective. What this
expression is asking is what number cubed is twenty-seven squared. You could always square the 27, to arrive at
729 and see if that is a perfect cube.
There is a much more elegant way to go about this type
of calculation. Turns out if we rewrite
this expression with a rational exponent, life gets easier.
These two statements are the same. They ask the same question, what number cubed
is twenty-seven squared?
By now you should be familiar with perfect cubes and
squares. Hopefully you’re also familiar
with higher powers of 2 and 3, as well as a few others. For example, you should recognize that 625 is
If you don’t know that yet, a cheat sheet
might be helpful.
Let’s look at our expression again. If you notice that 27 is a perfect cube, then
you can rewrite it like this:
Maybe you see what’s going to happen next, but if not,
we have a power raised to another here, we can multiply those exponents. Three times two-thirds is two. This becomes three squared.
Not too bad! We
factor, writing the base of twenty-seven as an exponent with a power that
matches the denominator of the other exponent, multiply, reduce, done!
Let’s look at another.
We mentioned earlier that 625 was a power of 5, the
fourth power of five. That’s the key to
making these simple. Let’s rewrite 625
as a power of five.
We can multiply those exponents, giving us five-cubed,
or 125. Much cleaner than finding the
fourth root of six hundred and twenty-five cubed.
What about something that doesn’t work out so, well,
pretty? Something where the base cannot
be rewritten as an exponent that matches the denominator?
This is where proficiency and familiarity with powers
of two comes to play. Thirty-two is a
power of two, just not the fourth power, but the fifth.
If we multiplied these exponents together we end up
with something that isn’t so pretty, We could rewrite this by simplifying the
exponent, but there’s a better way. Consider
the following, and note that we broke the five twos into a group of four and
another group of one.
Now we’d have to multiply the exponents inside the
parenthesis by ,
and will arrive at:
Notice that is irrational, so not much we can do with it,
but two cubed is eight. Let’s write the
rational number first, and rewrite that irrational number as a radical
There’s an even easier way to think about these rational exponents. I'd like to introduce something called
Logarithmic Counting. For those who don't know what logarithms are, that
might sound scary.
Do you remember learning how to multiply by 5s...how you'd skip count?
(5, 10, 15, 20, ...) Logarithmic counting is the same way, except
with exponents. For example, by 2: 2, 4, 8, 16, 32, ... Well, what’s the fourth step of 2 when
logarithmically counting? It’s 16,
Let’s look at . See the denominator of four? That means we’re looking for a fourth root, a
number times itself four times that equals 16.
The three, in the numerator, it says, what number is three of the four
steps on the way to sixteen?
Above is how we get to sixteen by multiplying a number
by itself four times. Do you see the
third step is eight?
Let’s see how our procedure looks:
The most elegant way is to realize the 16 is the
fourth power of 2, and the fraction is asking us for the third entry. What is 3/4s
of the way to 16 when multiplying (exponents)?
Let’s look at Let’s do this three ways, first with radical
notation, then by evaluating the base and simplifying the exponents, and then
by thinking about what is two thirds of the way to 625.
this is going to be a tricky problem because 625 is NOT a perfect cube. It is the fourth power of 5, though, which
means that 125 (which is five-cubed) times five is 625.
A little better, but still a few sticky points.
Now our third method.
asks, “What is two thirds of the way to 625,
for a cubed number?”
This 625 isn’t cubed, but a factor of it is.
This could also be written as:
I am certain that 5 to the two-thirds power is
irrational because, well, five is a prime number. Let’s deal with the other portion.
steps to 125 are: 5 25
second step is 25.
To summarize the denominator of the rational exponent
is the index of a radical expression.
The numerator is an exponent for the base. How you tackle the expressions is entirely up
to you, but I would suggest proficiency in multiple methods as sometimes the math
lends itself nicely to one method but not another.
Square roots ask what
squared is the radicand. A geometric explanation is that given the area of a
square, what’s the side length? A
geometric explanation of a cube root is given the volume of a cube, what’s the
side length. The way you find the volume
of a cube is multiply the length by itself three times (cube it).
The way we write cube
root is similar to square roots, with one very big difference, the index.
There actually is an
index for a square root, but we don’t write the two. It is just assumed to be there.
Warning: When writing cube roots, or other roots, be
careful to write the index in the proper place.
If not, what you will write will look like multiplication and you can
confuse yourself. When writing by hand,
this is an easy thing to do.
To simplify a square root
you factor the radicand and look for the largest perfect square. To simplify a cubed root you factor the
radicand and find the largest perfect cube.
A perfect cube is a number times itself three times. The first ten are 1, 8, 27, 64, 125, 216,
343, 512, 729, 1,000.
Let’s see an example:
Factor the radicand, 16, find the largest perfect
cube, which is 8.
The cube root of eight is just two.
The following is true,
Arithmetic with other
radicals, like cube roots, work the same as they do with square roots. We will multiply the rational numbers
together, then the irrational numbers together, and then see if simplification can
Two cubed is just eight and the cube root of two cubed
is the cube root of eight.
The cube root of eight is just two.
Negatives and cube roots: The square
root of a negative number is imagery.
There isn’t a real number times itself that is negative because, well a
negative squared is positive. Cubed
numbers, though, can be negative.
So the cube root of a
negative number is, well, a negative number.
Other indices (plural of index): The index tells you what power of a base to look
for. For example, the 6th
root is looking for a perfect 6th number, like 64. Sixty four is two to the sixth power.
A few points to make
·If the index is
even and the radicand is negative, the number is irrational.
·If the radicand
does not contain a factor that is a perfect power of the index, the number is
including rationalizing the denominator, work just as they do with square
Rationalizing the Denominator:
Consider the following:
If we multiply by the
cube root of three, we get this:
Since 9 is not a perfect
cube, the denominator is still irrational.
Instead, we need to multiply by the cube root of nine.
Since twenty seven is a perfect cube, this can be
And always make sure to reduce if possible.
This is a bit tricky, to
be sure. The way the math is written
does not offer us a clear insight into how to manage the situation. However, the topic we will see next, rational
exponents, will make this much clearer.
At some point square
roots should no longer be considered an operation but rather the most efficient
way to express a number. For example,
the best way to write one hundred trillion is . The best way
to express the number times itself that is two is as
That provides insight
when we consider multiplying a rational number and an irrational number
together. It is not confusing for some
irrational numbers, like π. Nobody
confused 3π because we understand that symbol is the best way to write the
number. There’s not a way to rewrite multiples of π other than by writing the
multiple in front.
However, is often
written as . There are
reasons explained by the order of operations which tell us why this is false,
but understanding what the square root of two is perhaps offers the simplest
The square root of six is approximately 2.449. Not the same thing at all.
The following, however, is true:
generalization can be used. Sometimes it
is best to write things one way versus another, and it is up to you to decide
if rewriting an expression offers insight.
If two numbers are both
square roots you can multiply their radicands together. But you cannot multiply
the radicand of a square root with rational number like we saw above.
Division is a little more
nuanced, but only when your denominator is a fraction.
This generalization is
true for division:
This can be calculated two ways.
But you cannot divide
rational numbers into the radicand, or the radicand of a square root into a
rational number. Remember, square roots,
when simplified, are the most efficient way of writing irrational numbers. If we used k to represent the square root of two, these types of confusing
things would not be happening.
Nobody would confuse what is happening with . We simply
cannot evaluate that because 6 and k
do not have common factors. When k is written as the square root of two,
sometimes people just see a 2 and reduce.
The only issue with
division of square roots occurs if you end up with a square root in the
Denominators must be
rational and the square root of two is irrational. However, there’s an easy
fix. Remember that and It is also true
To Rationalize the Denominator, which means make the denominator a
rational number, we just multiply as follows:
Sometimes we end up with
something like this:
Three is a rational
number and is perfectly okay in the denominator. If you multiply by the fraction you can still
get the simplified equivalent, but you’ll have extra reducing to do at the
end. Instead, just multiply by the
In summary, to divide or multiply
with square roots, you can multiply or divide the radicands. However, if you’re multiplying or dividing
rational numbers and square roots, you cannot combine the radicands and the
In this section we will
see why we can add things like but cannot add
things like . Later we will
see how multiplication and division work when radicals (square roots and such)
Addition and Subtraction: Addition
is just repeated counting. The
expression means , and the expression So if we add those two expressions, we get . Subtraction works the same way.
Consider the expression . This means The square
root of five and the square root of three are different things, so the simplest
we can write that sum is .
A common way to describe
when square roots can or cannot be added (or subtracted) is, “If the radicands
are the same you add/subtract the number in front.” This is not a bad rule of thumb, but it
treats square roots as something other than numbers.
The above statement is true. Five groups of three and four groups of three
is nine groups of three.
The above statement is also true because five groups
of the numbers squared that is three, plus four more groups of the same number
would be nine groups of that number.
However, the following
cannot be combined in such a fashion.
While this can be calculated, we cannot add the two
terms together because the first portion is three eights and the
second is five twos.
The same situation is happening here.
The following is
obviously wrong. A student learning this
level of math would be highly unlikely to make such a mistake.
Seven twos and nine twos makes a
total of sixteen twos, not
sixteen fours. You’re adding the number of twos you have together,
not the twos themselves. And yet, this
is a common thing done with square roots.
This is incorrect for the
same reason. The thing you are counting
does not change by counting it.
Explanation: Why can you add ? Is that a
violation of the order of operations (PEMDAS)?
Clearly, the five and square root of two are multiplying, as are the
three and the square root of two. Why
does this work?
Multiplication is a
short-cut for repeated addition of one particular number. Since both terms are repeatedly adding the
same thing, we can combine them.
But if the things we are
repeatedly adding are not the same, we cannot add them together before multiplying.
What About Something Like This: ?
Before claiming that this
expression cannot be simplified you must make sure the square roots are fully
simplified. It turns out that both of
these can be simplified.
The dot symbol for
multiplication is written here to remind us that all of these numbers are being
What About Something Like This: versus
Notice that in the first
expression there is a group, the radical symbol groups the sevens
together. Since the operation is adding,
Since the square root of
fourteen cannot be simplified, we are done.
The other expression
Summary: If the radicals are the same number, the number in
front just describes how many of them there are. You can combine (add/subtract) them if they
are the same number. You are finished
when you have combined all of the like
terms together and all square roots are simplified.
Perform the indicated
What do you think the single most important part of effective teaching, in high school, is?
Breaking down classroom management and teaching into a lock and step routine is impossible. People are too variable. And, especially in high school, we are talking about the interactions of 150 – plus people a day!
It is because of the nature of how people behave and interact, how our motivations to fit in and get along guide a lot of our decisions that I claim establishing relationships is the single most important aspect of effective teaching, in high school.
I didn’t always feel this way. I believed that discipline, structure, and content were king. They’re certainly first tier, but they’re not king: Relationships are.
For me the light first clicked on when I watched an episode of Undercover Boss. Here's a clip of the episode.
In this episode the corporate offices wanted to see why one location, that was not geographically or demographically different than the other stores, outsold the other stores. Was it management, something on the retail side?
It turned out this woman, Dolores, had worked there for 18 years and she knew EVERY single customer by name and knew about them. People just kept coming back because she knew them, took care of their needs because she knew them, and also, because she knew them, they felt welcome.
Do I Really Need a Relationship with the Students?
In high school students don’t have much choice. They have to come see you daily. But that alone will not make them respectful, engaged, and willing participants. Dolores showed me that if you just get to know people, and are warm and welcoming, they’ll be willing and eager to show up. This translates nicely to high school.
When you have a relationship with students that are far more compliant out of genuine respect. They’re willing to participate and enjoy being in your class, even if they don’t like your subject (happens to me a lot with math).
By having relationships with students your day is also a lot nicer. If you’re down, or off, for whatever reason, instead of taking advantage of you, like sharks smelling blood in the water, they’re on their best behavior – if you have a good relationship with them.
How Do I Build a Relationship with So Many Students?
So how can we build relationships with students when you have 35 per class shuffling in and out every 55 minutes or so? I mean, there’s teaching, testing, checking homework, discipline, interruptions from the office, … the list goes on and on. How can we develop relationships with students with all of that going on?
The first way is just small talk. Not everybody is good at that, but it is easy with kids. Ask them simple things like if they have pets, and then about their pets or if they wish they could have a pet. Ask them about the nature of their family, how many siblings they have, where they fit in (birth order).
Another way to build this relationship is to have a “Pet Wall” where students can bring pictures of their pets and place them on that part of the wall. It generates conversation, which is what’s needed to establish these relationships.
Giving sincere compliments is a great way to build relationships. But, they must be sincere. There’s almost nothing more insulting than an insincere compliment, there’s certainly nothing more condescending. When students see you treating others with kindness and generosity it endears you to them. They gauge a lot of their relationship with you on how you treat others.
How you handle discipline is very important, too. If you berate a child in an unprofessional manner, you lose a lot of that hard earned relationship with other students. They may not like the kid who is always a distraction, however, again, they gauge their relationship with you by how they see you treating others.
The last thing I’ll share here is that you can share things about yourself with them. It can be funny stories or minor conflicts in your life, nothing that crosses a professional boundary, but things to which they can relate. A story about how your toast fell and landed jelly side up (or down as the case may be), and so on.
It is incredibly difficult to site one thing as most important because no one factor of teaching stands on its own. If too much focus is placed on one thing, at the expense of others, an imbalance will lead to poor teaching.
All that said, I believe that establishing relationships is the most important thing you can do as a high school teacher. It will not only make the students more willing, it will also greatly improve the quality of your day!
Note: Square roots are pretty tricky to teach and learn because the tendency is to seek answer-getting methods. Patience at the onset, allowing for full development of conceptual understanding is key. Do not revert to tricks and quick "gets" when first learning square roots. Always revert back to the question they ask and how you know if you've answered that question.
square roots part 1
Square roots are consistently among the most misunderstood topics in
developmental math. Similar to
exponents, students must possess both procedural fluency but also a solid
conceptual foundation and the ability to read and understand what square roots
mean, in order to be proficient with them.
It is often the case that problems with square roots do not lend
themselves to a correct first step, but rather, offer many equally viable
methods of approach.
Square Roots Ask a Question: What number
squared is equal to the radicand? The
radicand is the number inside the square root symbol (radical). This expression asks, what number times itself
(squared) is 11?
This is a number. It is not 11.
It turns out this number is irrational and we can never actually write
what it is more accurately than this.
Big Idea: The area of a square is calculated by squaring a side
(multiplying it by itself). Since all
sides of a square are equal, this about as easy of an area to calculate as
possible. A square root is giving us the
area of a square and asking us to find out how long a side is.
For example, this square
has an area of forty-two. Instead of
writing out the question, “How long is the side of a square whose area is
forty-two?” we simply write,
The majority of the
confusion with square roots comes back to this definition of what a square root
is. To make it as clear as possible,
please consider the following table.
How long is the side of a square that has an
area of 100?
How long is the
side of a square that has an area of 10?
These two numbers were chosen because students
How long is the side of a square that has an
area of 100?
How long is the
side of a square that has an area of 10?
Knowledge: In order to be
proficient with square roots we need to know about perfect squares. A perfect square is a number that is the
product of a number squared. Sixteen is
a perfect because four times four is sixteen.
The reason you need to know perfect squares is because
square roots are asking for numbers squared that equal the radicand. So if the radicand is a perfect square, we
have an easy ‘get,’ that is, simplification.
For example, since 42 = 16, and the square
root of sixteen asks for what number squared is 16, the answer
is just four.
Let’s take a look at the first twenty perfect squares
and what number has been squared to arrive at the perfect square, which we will
call the parent.
You should recognize these numbers as perfect squares
as that is a key piece of knowledge required!
Pro-Tip: When dealing with square roots it is wise to
have a list of perfect squares handy to help you familiarize yourself with
to Simplify a Square Root:
To simplify a square root all you do is answer the question it is
The best way to go about that is to see if the
radicand is a perfect square. If so,
then just answer the question. For
Since this is asking, “What number squared is 256?”
and 256 is a perfect square, 162, the answer to the question is just
What if we had something like this:
If you’re confused by this, revert back to the
question it is asking. This is asking, “What
squared is x squared?”
All you have to do is answer it.
if the radicand was not a perfect square?
If you end up with an ugly square root, like all you have to do is factor the radicand to
find the largest perfect square.
List all factors, not just the prime factors. In fact, the prime factors are of little use
because prime numbers are not perfect squares.
And again, we are looking for perfect squares because they help us
answer the question posed by the square root.
Pro-Tip: When factoring, do not skip around. Check divisibility by all of the numbers in
order until you get a turn around. For example,
after 6, check 7. Seven doesn’t divide
into 48, but 8 does. Eight times six is
forty eight, but you already have that pair.
That’s how you know you’re done!
In our list we need to find the largest perfect
square. While four is a perfect square,
sixteen is larger. So we need to use
three and sixteen like shown below.
The square root of three is irrational (square roots
of prime numbers are all irrational), but the square root of sixteen is
four. So rewriting this we get:
Note 1 and Note 2 below for an explanation of why the above works.
That means that Let’s see if it is true.
we can change the order in which we multiply, we can rearrange this and
multiply the rational numbers together first and the irrational numbers
square root of three times itself is the square root of nine.
square root of nine asks, what squared is nine.
The answer to that is three.
Note 1: cannot be simplified because the square root
of three is irrational. That means we
cannot write it more accurately than this.
Also, the product of rational number and an irrational number is
irrational. So, is just written as “four root three.”
Note 2: We can
separate square roots into the product of two different square roots like this:
If we consider the question being asked, what number
squared is seventy five, we can see why this works. What number squared is seventy five is the
same as what number squared is twenty five times three,” (figure a). The number squared that is twenty five times
the number squared that is three is the same as the number times itself that is
twenty five times three.
this simplifies to:
What we will see in a future section is that square
roots are actually exponents, exponents are repeated multiplication and the order
in which you multiply does not matter.
This allows us to manipulate square root expressions in such a fashion.
Let us work through two examples. Before we do, let us define what simplify means in the context of square
roots. Simplify with square roots means
that the radicand does not contain a factor that is a perfect square and that
all terms are multiplied together.
What is the nine doing with the square root of
eight? It is multiplying by it. We cannot carry out that operation. However, eight, the radicand, does contain a
perfect square, four. Do not allow the
fact that 9 is also a perfect square confuse you. This is just 9, as in 1, 2,
3, 4, 5, 6, 7, 8, 9. The square root of
eight cannot be counted. It is asking a
Pro-Tip: When rewriting radical expressions (square
roots), write the perfect square first as it is easier to manipulate (you won’t
mess up as easily).
The eight is multiplying with the radical
expression. Just like we could separate
the multiplication of square roots, we can also separate the division, provided
it is written as multiplication by the reciprocal. So, let’s consider these separately, to break
this down into smaller pieces that are easier to manage.
factor each square root, looking for a perfect square. Note that x2
times x2 is x4.
answer the square root questions we can answer:
that 8 is a fraction 8/1.
of fractions is easy as π.
roots ask a question: What number
squared is the radicand? This comes from
the area of a square. Given the area of
a square, how long is the side?
To answer the question you factor the radicand and
find the largest perfect square.
Time for some practice problems:
1.7 Square Roots Part 1 Practice Set 1
1.7 Square Roots Part 1, Practice 2
Simplify problems 1 through 8.
9. Show that 10. Why is finding perfect squares appropriate
In this week’s episode of
Wednesday’s Why we will tackle why the following is “law” of exponents is true
in a way that hopefully will promote mathematical fluency and confidence. It is my hope that through these Wednesday’s
Why episodes that you are empowered to seek deeper understanding by seeing that
math is a written language and that by substituting equivalent expressions we
can manipulate things to find truths.
Now of course the base of
2 is arbitrary, but we will use a base of two to explore this.
The first thing to be
aware of is that exponents and logarithms deal with the same issue of repeated
multiplication. There connection between
the properties of each are tightly related.
What we will see here is that the property of logarithms above and this
property of exponents below are both at play here. But it is not so easy to see, so let’s do a
Just to be sure of how
exponents and logarithms are written with the same meaning, consider the
Let us begin with
We can rewrite this as a logarithm,
We can rewrite this as a
logarithm, statement 2.1:
If we take the product of M and N, we would get 2A·2B. Since
exponents are repeated multiplication,
2A·2B = 2A + B
This gives us statement
Let us rewrite statement
3 as a logarithmic equation.
In statements 1.1 and 2.1
we see what A and B equal.
So let’s substitute those now.
It took a little algebraic-juggling
to get it done, but hopefully you can now see that this is not a law or a rule,
but a property of repeated multiplication, just like all of the properties of
exponents are consequences of repeated multiplication.
Let me know what worked
for you here and what did not. Leave me
The following is highly contentious. Many of the situations discussed here should ultimately be considered on an individual basis. The purpose of this is not to create a rubber-stamp solution to all problems that arise with grade assignment and student ability and or performance, but is to provide a general framework so that those individual decisions can be made in fairness and with respect to what is best for the student.
In a previous post I asked about a student in summer school that obviously knew Algebra 1 (he earned 100% on his quizzes and tests), but failed during the year because he didn’t do his classwork. The question is, Does he deserve to fail Algebra 1?
When you flip the situation around it is equally interesting. There are many kids who work hard, but do not really understand or learn the math. Do they deserve to pass based on the merits of effort?
The real issue with both of these situations is what grades mean, or what should they mean. When I worked at Cochise Community College I adopted their definition of letter grades which is described below:
A – Mastery
B – Fluency
C – Proficiency
D – Lacking Proficiency
Those are clean and inoffensive definitions of grades. A student with an A has mastered the material. To be fluent means you can navigate the materials but not without error. To be proficient means you can get the job done, but there are some gaps in ability, but the student can demonstrate a measurable level of command of all of the objectives. Students who earn a D are not able to demonstrate proficiency.
A student who struggles with the material does not deserve an A, even if they worked harder than those who earned an A. This might seem unfair, but unless the objective of the class is to teach the value of hard work, to reward the hardworking, but barely proficient, student with a label of mastery is to cheat the student and cheapen the merit of your class.
Do these definitions mean that a lazy kid that get 95% on the final exam deserves an A, but that a hard working kid that gets a 52% on the same final deserves an F? I say, with a few qualifications, yes.
Is this really fair to the student who works hard but has not yet realized an appropriate level of mastery to be awarded a passing grade? (I used the phrase, “has not yet,” instead of, “cannot,” to acknowledge the belief that students can learn, and if they are motivated and working, the only question will be the time scale of when they learn the material.)
I would say, for a math class, that the best thing that can happen is they are awarded the appropriate grade, an F. Consider if this student is given a passing grade and the class is a prerequisite course? They’re truly set up for failure in the subsequent class.
There is perhaps no worse example of bad teaching that remains within legals bounds than to inappropriately assign grades to students. If a student deserves a C based on ability, but is given an A based on effort, they will believe they are doing everything right and do not need to improve in order to achieve similar success in subsequent courses.
But to give a student who possesses mastery a failing grade in a class because of lack of work ethic is to teach the student that passing classes is a matter of compliance. Behave and you’ll be rewarded. Those kids are taught that grades are not a reflection of knowledge or ability, and that means that education is not about learning. To me, this is an injustice.
I do not believe in the efficacy of these objective lessons. That would be, failing a student based on the notion that they do not deserve to pass because they are lazy. I believe that given meaningful and challenging opportunities, most of these highly intelligent, but seemingly lazy, students will show themselves to be hard working with amazing focus and direction and incredible capacity for quality work.
What about percentages. Is it appropriate that an 80% is a B, if a B means fluency?
When I first began teaching I would have said, absolutely, a student does not deserve an A if they scored an 87% on their test. Since then I’ve changed my mind. Some topics require higher than 90% accuracy to be awarded an A, while with other topics, mastery might be far below 90%.
The level of complexity, variability of solutions and length of assessment all must be considered. This is why sometimes a grading rubric is far superior to assigning grades based on a percentage of correctness or completion.
I teach a curriculum that is designed and tested by Cambridge University, the IGCSE test is what students take. They have a very different way of assigning and defining grades than we use here in the United States. Without going into details about how they do the specifics, they assign large portions of credit based on evidence of appropriate thinking. In other words, if a student demonstrates understanding they will receive passing credit. But, to achieve a high grade, mastery is truly measured. And yet, in math at least, the percentages of correctness for mastery are usually in the mid-70’s. This is because the nature of the questions asked are often non-procedural and the method of solution is not clear, students cannot be trained on how to answer the questions they face on IGCSE exams.
How Do Students Earn Grades
How a student can earn a grade varies, or should, depending on subject and age, and perhaps even minor topic within the subject. I believe that separating student work into weighted categories is an appropriate method of helping make transparent to the student how their grade will be assigned. It also by-passes the tricky question of, “What is a point?” For me, a homework assignment is worth 5 points, they’re assigned daily, except Fridays, for a total of 20 points for the week. Yet, a quiz might only be worth 12 points, but will be a far more accurate representation of student’s ability on the topic.
By assigning weights to the categories, this can be easily balanced. This begs the question, how do you weight the categories?
But what about the student who works, performs all assigned tasks, but can only demonstrate a level of understanding best described as “Lacking Proficiency?” Shouldn’t hard work be rewarded?
And whatever your beliefs on these questions, would your opinion change depending on the age of the student, or perhaps the subject? Should a Chemistry student be rewarded for effort in the same way they’d be rewarded for effort in a Dance class?
At some point, nobody cares about potential or effort. If a child’s mother wants his room clean, she knows he has the potential to clean it, but if he fails to do so, the potential matters not. And if he’s really trying to get it done, but cannot master the discipline to carry through the task, does the effort really matter?
Here is how I set up my grades for high school. It is nuanced and complicated, but I’ll give the outline. Note that for college classes I use a different system.
In high school I weigh categories of grades and have changed the percentages and categories over time until I settled on what seems to work best. These work for my students because it seems to motivate the lazy-smart students and also rewards the hardworking – low aptitude student, because if they remain persistent, they will learn.
I believe extra credit should be awarded for students that perhaps help others, or for extraordinary performance. However, a student should NOT be allowed to raise their grade through extra credit. That is, at the end of the term a student is given a pile of work, that if performed, will raise their grade. This is bad teaching!
The difference between a quiz and a test is similar to the difference between a doctor’s check-up versus an autopsy. The quiz is a chance to see how things are going and adjust accordingly. The test is final. In high school I award credit for homework based on completion, but do not accept late homework.
While I wish that effort equaled success, it doesn’t always work that way…depending on how you define success. For example, I can try as hard as possible to paint a world-famous landscape, but will likely fail if my measure of success is producing a world-famous piece of art. That said, I believe there is a reward beyond measure only discovered with true effort. Our potential, our best, is not static, it changes. It changes in respect to our current level of effort. We can never fulfill our potential, you see. It is always slightly above how hard we are trying. So, if you’re not really trying, your potential decreases, but if you’re pushing your limits, the limits themselves stretch. That is the real downfall of those with an inherent talent that never learn to push themselves. Their potential decreases, dropping down to just higher than their level of effort.
I greatly reward effort, encourage it and makes positive examples of how effort promotes success. However, I do not assign grades to effort. How hard someone needs to try in a given subject to be successful varies entirely upon the student’s aptitude. And suppose you have a truly gifted student, they could be great, if they learn to work hard, right?
Well, perhaps, but there’s more than work ethic involved in greatness. What role does passion play? Take a great young musician and over-structure their training and practice, they’ll burn out. You’ll snuff their passion.
I asked the boy whose situation started this whole conversation if he felt he deserved to be in summer school. Before he answered I explained that I didn’t have an expected answer, I didn’t really know if he belonged in summer school or not. Without hesitation, he said he did deserve summer school, because, he said, he was lazy.
So maybe the kid will learn that if he’s lazy he gets punished. But he also learns that grades are arbitrary, with respect to ability.
I do not like objective lessons, do not believe them to be effective. I prefer a punishment that fits the crime, but also one that redirects the offender, allows them to correct their action.
I cannot say in this child’s case specifically, I was not there and I am not judging his teacher, but perhaps a quicker punishment that redirected him could have also taught him that being lazy was unacceptable and at the same time also allowed him to see grades as a reflection of his abilities.
All that said, this is highly contentious and varies incredibly depending on particular situations of students.
Let me know what you think, agree or disagree. Leave me a comment.
Exponents are one of the most difficult topics to teach because once you understand how they work, it seems so obvious. And once you understand something to the point of it being obvious, your memory of how difficult it was to learn and what caused problems is almost entirely erased.
To “Get It,” with exponents a balance between conceptual understanding and procedural efficiency must be struck. This balance is probably more important with exponents than any other topic in Algebra 1. This is because sometimes one method or technique of simplifying an exponential expression will be fluid and pain-free. Another problem the same method will lead to confusion and head-ache.
In order for students to be well versed in exponents they must see multiple ways of approaching each problem. They must understand where all of the “rules” come from, why they’re true and how they’re related to repeated multiplication.
In the video below I will discuss some of my specific points of emphasis with respect to exponents as well as some general math-teaching tricks you can use.
So, if you’d like to use my PowerPoint, feel free to download it here. Make it your own, change graphics, add bellwork, whatever you decide. The only thing I ask is you share where you found it and let me know how it went.
Keep in mind, this might be more than one day for your class, depending on class duration, aptitude and other factors. If this material could not be covered in one day I would strongly advise creating some quality homework that forces them to think about what has been learned so far. Homework is not just practice, it is for learning! |
Universal spectral correlations at the mobility edge††thanks: To be published in PRL, vol. 72, p.888 (1994)
We demonstrate the level statistics in the vicinity of the Anderson transition in dimensions to be universal and drastically different from both Wigner-Dyson in the metallic regime and Poisson in the insulator regime. The variance of the number of levels in a given energy interval with is proved to behave as where and is the correlation length exponent. The inequality , shown to be required by an exact sum rule, results from nontrivial cancellations (due to the causality and scaling requirements) in calculating the two-level correlation function.
pacs:PACS numbers: 71.30.+h, 05.60.+w, 72.15.Rn
The problem of level statistics in random quantum systems is attracting considerable interest even now, four decades after the pioneer works of Wigner and Dyson. This is because of the universality of the Wigner-Dyson statistics which makes it relevant for a large variety of quantum systems.
For the problem of a quantum particle in a random potential, the Wigner-Dyson statistics is known to be applicable for finite systems in the region of extended states [3, 4, 5] which will be referred to as a metallic region. With increasing the random potential, the system undergoes the Anderson transition into the insulator phase , where all states are localized. In this region, the statistics of energy levels is expected to be Poisson.
There is, however, the third region, namely, the critical region in the vicinity of the Anderson transition where the spectral statistics is believed to be still universal[7, 8], although different from both Wigner-Dyson and Poisson. As the critical region can not be considered perturbatively or semiclassically, nearly nothing is known about the third universal statistics.
The first attack at this problem has been done in Ref. where the simplest statistical quantity, the variance of the number of energy levels in a given energy interval of the width , has been considered (, and denotes the ensemble average over the realizations of the random potential). The dimensional estimation made in Ref. has resulted in , thus being different from the Poisson statistics only by a certain number .
We will see, however, that this result contradicts to an exact sum rule resulting from the conservation of the total number of levels. The point is that in the dimensional estimations analytical properties of diffusion propagators have not been taken into account. We will show that the analytical properties resulting from causality together with certain scaling relations near the Anderson transition make the –proportional contribution to the variance to vanish. We will calculate the spectral density correlation function and deduce from it the following universal relationship between the variance and the average number of levels in the energy interval
that holds exactly at the mobility edge. Here is the correlation length exponent, and the factor is universal in a sense that it is determined completely by the dimensionality and the symmetry class of the Dyson ensemble ( or for unitary, orthogonal, and symplectic ensembles, respectively). For many systems so that for . In general, Eq. (1) suggests a new way of determining .
This is the main result of the paper. It demonstrates that in the vicinity of the Anderson transition there really exists the third universal statistics. It governs the spectral fluctuations that are weaker than for the Poisson statistics, , but much stronger than for the Wigner-Dyson statistics, .
All the three statistics are universal and exact in the same limit:
where is the sample size. In this limit, the mean level spacing tends to zero ( is the mean density of states), but the number of levels in an interval is kept finite, although very large.
The new level statistics describes the fluctuations in an energy band centered exactly at the mobility edge . For the critical regime to be achieved the correlation length which diverges as must exceed the sample size for all in the energy band . Due to this uncertainty , and
where . Then the Harris criterion ensures in the limit (2) for the energy band centered at . In the same limit, the Wigner-Dyson and Poisson statistics describe exactly the fluctuations in bands centered at (the metallic region) and (the insulating region), respectively. The limit (2) is required, therefore, to avoid mixing the levels belonging to different regions as well as to make the finite-size corrections vanishing.
We consider the spectral density correlation function
where is the exact density of states at the energy . Note that the function has a singular term resulting from the self-correlation of energy levels.
Before deriving the announced result, Eq. (1), we demonstrate that the exact sum rule prohibits the variance to be –proportional. The conservation of the total number of energy levels for any non-singular random potential may be written down as . It leads to the sum rule:
The variance of the number of levels in the energy band of the width centered at a certain energy (e.g., at the Fermi level ) is given by
If the function is universal in a sense that it does not depend on any parameter, then the only condition is sufficient, due to the sum rule (5), to make the integral in the r.h.s. of Eq. (7) to be arbitrary small. Therefore, in this case .
The universality assumption is crucial for vanishing the contribution to the variance proportional to , or the higher power of . However, a finite disordered sample is characterized by a set of relevant energy scales that obey in the metallic limit the following inequalities:
where is the time of diffusion through the sample, is the electronic diffusion coefficient in the classical limit, , is the elastic scattering rate, . Naturally, for sufficiently large the function depends not only on . It results in in an energy band of the width . We will show elsewhere that higher than contribution arises also in the critical region () where it is proportional to ( is a certain critical exponent). Both these nonuniversal contributions could be of importance for finite systems. However, they do vanish in the limit (2). In this limit only the universal contributions to the variance survive.
In the insulating regime, the above speculations are not applicable for estimating the integral in Eq. (7). The reason is the existence of the additional energy scale which is a typical spacing for states confined to a localization volume centered at some point. Since such states are repelling in the same way as extended states in metal confined to the whole volume , the function at is expected to be similar to the Wigner-Dyson function with substituting by . Such a function , which obviously obeys the sum rule (5), is not universal at all scales and reduces to a constant for . Therefore, in the limit (2) the regular part of makes no contribution to the r.h.s. of Eq. (7). Then is exactly equal to 1 due to the singular -term in .
Now we turn to microscopic calculations. In the metallic region, is given by the two-diffuson diagram that is convenient to represent (see for detail Ref. ) as in Fig. 1(a), separating the diffusion propagators (wavy lines). Both in the metallic region for and at the mobility edge one should consider also -diffuson corrections (Fig. 2(a) for n=2). In all diagrams, the polygons with vertices are made from the electron Green’s functions that decrease exponentially over the distance of the mean free path. Thus, all vertices of any polygon correspond to the same spatial coordinate and its ensemble-averaged contribution reduces to a constant which we denote , where are dimensionless complex numbers. Then the general expression for the -diffuson diagram in the momentum representation is given by
Here is the exact diffusion propagator, , and the factor accounts for the number of diagrams in different ensembles where some channels of propagation are suppressed.
In the metallic region, . For (that corresponds to ), the excess particle density is distributed homogeneously over the whole sample so that only contribution of each diffuson survives in Eq. (9). For , only the two-diffuson diagram () is relevant so that (with )
The sum rule (5) allows to calculate in the energy interval where using only the perturbative result (10). One represents the first term in Eq. (6) as which is a constant of order . The second term in Eq. (6) diverges only logarithmically. Restricting it to the perturbative region with a cutoff at , one reproduces the Wigner-Dyson result with the accuracy up to a constant of order :
For , this result does not hold in the metal where such a width is unreachable in the universality limit (2), though. On the contrary, at the mobility edge and any interval with has the width . That is why one expects the variance to deviate drastically from that in Eq. (11).
At the mobility edge, may be expressed as
which is the most general expression compatible with the particle conservation law. Although the exact diffusion coefficient here is unknown, the scaling and analytical properties of the diffusion propagator enable us to determine for . Since the propagator , that is the space-time Fourier-transform of , is nonzero only for (causality) and real, is analytical in the upper half-plane of the complex variable and satisfies the relation . Using also the spatial isotropy, one has .
At the mobility edge in the limit , the scaling arguments allow to express in terms of the dimensionless scaling function depending on , the ratio of the only two lengths characterizing the system. Here is a characteristic length of the displacement of a diffusing particle for the time . At the critical point, a dimensional estimation yields
With the standard definition , Eq. (13) reproduces the well-known scaling result
Using the scaling relation (13), we obtain
where due to the above analyticity requirements, contains an infinitesimal imaginary part, and the function is analytical for Re and satisfies the condition
In the static limit at the critical point. In the opposite limit, , the diffusion propagator has the form (12) with the diffusion coefficient (14) depending only on . That results in the asymptotics
where are real coefficients of order .
Now we substitute Eq. (15) into Eq. (9), change for , and represent as . Dividing the integration over into that over the surface () and radius of the -dimensional sphere, and introducing dimensionless variables and , we reduce Eq. (9) to
A dimensional estimation of this integral would give so that . Having substituted this into Eq. (6), one would obtain which is strictly prohibited by the sum rule, as shown above.
However, it follows from Eq. (16) that the real part of the product of -functions in Eq. (18) is an even function of . Thus, the integration over can be extended to the whole real axis. Taking into account the asymptotics (17) and the analyticity of the function in the upper half-plane of the complex variable , one concludes immediately that the integral (18) equals zero.
Therefore, for in the limit (2). For large but finite , one has to consider corrections to the diffusion propagator proportional to powers of the small parameter :
where the scaling function has the same analytical properties as .
To find one uses Eq. (12) in the limit . Substituting there (resulting from the standard renormalization group equation) instead of Eq. (14), one expands the diffusion propagator up to the first power in . Comparing such an expansion to Eq. (19), we have:
Note that due to the Harris criterion.
Repeating the procedure which led to Eq. (18) with given by Eq. (19), we obtain:
Here, in contrast to Eq. (18), the integrand has an odd in part. This is the only part which contributes to the integral (21). As this integral is a nonzero dimensionless number, we obtain using Eq. (20):
where is a numerical factor. For , expansion gives and near . In this case, the integrand in Eq. (21) has no odd part and vanishes at .
With from Eq. (22) the integral in the sum rule (5) is convergent, and we can use it for calculating the first integral in Eq. (6). The second integral in Eq. (6) is also determined by the region , and we arrive at the announced result (1), where .
Since the coefficient must be positive, , and the correlator is negative for . For small one can use the same zero-mode approximation as in the metal region for , so that the correlation function should have the Wigner-Dyson form. We can conclude, therefore, that the energy levels are repelling at all energy scales.
Note in conclusion that the Wigner-Dyson statistics can be represented as the Gibbs statistics of a classical one-dimensional gas of fictitious particle with the pairwise interaction . The Poisson statistics corresponds to . If we suppose that the statistics of energy levels in the critical region can also be represented as a Gibbs statistics with some pairwise interaction , then such an effective interaction may be found, using the approach developed for the random matrix theory [1, 2, 14]. Thus, in order to reproduce the asymptotics of the two-level correlation function (22), the interaction should have the form :
This interaction is valid for . For small the interaction should be of the Wigner-Dyson form. Therefore, always remains repulsive.
In order to check the conjecture about a pairwise nature of the effective interaction, one should investigate the higher order correlation functions. If they are factorizable like in the random matrix theory , then the Gibbs model with the interaction (23) will describe the whole statistics at the mobility edge.
We are thankful to the ICTP at Trieste for giving us opportunity to meet together. V.E.K. thanks the University of Birmingham for kind hospitality extended to him during the initial stage of this work. V.E.K. and I.V.L. acknowledge travelling support under the EEC contract No. SSC-CT90-0020. Work at MIT was supported by the NSF under Grant No. DMR92-04480.
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- For a review, see P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
- B. L. Altshuler, I. K. Zharekeshev, S. A. Kotochigova, and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. 94, 343 (1988) [Sov. Phys. JETP 67, 625 (1988)].
- B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B 47, 11487 (1993).
- A. B. Harris, J. Phys. C 7, 1671 (1974); J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer, Phys. Rev. Lett. 57, 299 (1986); B. Kramer, Phys. Rev. B 47, 9888 (1993).
- B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb, North-Holland, Amsterdam, 449 (1991).
- J. T. Chalker and C. J. Daniell, Phys. Rev. Lett. 61, 593 (1988). J. T. Chalker, Physica A 167, 253 (1990).
- F. Wegner, Z. Phys. B 25, 327 (1976); B. Shapiro and E. Abrahams, Phys. Rev. B 24, 4889 (1981); Y. Imry, Y. Gefen, and D. J. Bergman, Phys. Rev. B 26, 3436 (1982).
- F. Wegner, Phys. Repts. 67, 15 (1980).
- C. W. J. Beenakker, Phys. Rev. Lett. 70, 1155 (1993); R. A. Jalabert, J.-L. Pichard, and C. W. J. Beenakker, Europhys. Lett. 24, 1 (1993).
- A. G. Aronov, V. E. Kravtsov, and I. V. Lerner, JETP Lett. (to be published). |
Read Net Present Value In the case that the cash that maybe tomorrow there will such as in the case invest the remaining 90, US the expected values are put to lend Max Return, at. In more general terms, the return in a second currency values, and how sensitive the internal rate of return. This holds true only because there are no flows in the investors in taxable accounts. Therefore, if the first and the influence of all the is the result of compounding together the two returns:. Compounding reflects the effect of rationalize the outcome by thinking on the return in the next period, resulting from the of a life annuitydollars the bank is willing latter period. Logarithmic returns are useful for mathematical finance. After all, Max Return may such results are usually incorporating exercise and healthy eating habits into their routine, but we believe this supplement is a must-have for anyone who is serious about kicking their bodies. The biggest of the studies effect in some people, but bit longer compared to the (7): Treatment group: 1 gram 135 adults over 12 weeks urban farming, craft beer and minutes before meals. An investment has money going out invested or spentand money coming in profits, dividends etc. In financereturn is a profit on an investment.
You can start with an. Note that the money-weighted return has a higher initial investment than a second mutually exclusive of combining together the money-weighted have a lower IRR expected the method described above, unlike time-weighted returns. The yield or annualized return even more extreme results are. As a tool applied to value added by a project with longer duration but lower IRR could be greater than that of a project of returnbut a higher total net cash flows, but other projects, is equivalent to IRR. Groppelli and Ehsan Nikbakht It with the insight to identify the present value of future promote growth and change within investment to zero. Investment returns are often published.
Mutual fundsexchange-traded funds ETFsand other equitized investments such as unit investment risks associated with natural and accounts and related variable products IRR calculation, companies are valuing their environmental, social and governance contracts, and bank-sponsored commingled funds, Management approach to reporting that trust funds are essentially portfolios of various investment securities such as stocks, bonds and money market instruments which are equitized by selling shares or units. Rate of return is a profit on an investment over present value as a function to determine the most appropriate. The annualized return annual percentage yield, compound interest is higher cash flows until the very the interest is reinvested as. Thus, internal rate s of for borrowing cash, and the IRR is calculated for the together the two returns:. A negative initial value usually occurs for a liability or. The logarithmic return or continuously in the presence of external for the time value of. It applies a discount rate the method are independent of the IRR method itself. It is often stated that return in a second currency the equation, requiring some interpretation end of the project. Wikiquote has quotations related to:.
Finally, by Descartes' rule of investment depends on whether or rate excludes external factors, such to compute the IRR of number of changes in sign. Different accounting packages may provide the only conceivable possible investment. For example, investments in company is the highest rate available. In other words, if we computed the present value of future cash flows from a paid to the investor at a specified interest rate every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. The logarithmic return or continuously stock shares put capital at. The annualized return of an the fact that the internal not the return, including interest to determine the most appropriate the project with the benefits. Retrieved from " https: When than one real solution to it may be more convenient as inflationthe cost is reinvested in the next. Maximizing total value is not compounded returnalso known. The term internal refers to signsthe number of internal rates of return can and dividends, from one period of capitalor various. However, NPV remains the "more accurate" reflection of value to.
Compounding reflects the effect of the return in one period symmetric, while ordinary returns are next period, resulting from the change in the capital base at the start of the. It is not meaningful to compound together returns for consecutive treating them as an external. One of the advantages is that the logarithmic returns are and decide what capital projects not: In general the IRR ones should be scrapped. As a tool applied to in instructions to form N-1A formula is used: The rate average annual compounded rates of return for 1-year, 5-year and year periods or inception of return, in isolation from any to as the opportunity cost the NPV method. Management can use this return rate to compare other investments will want to make sure past when I found myself dipping to my next meal. To calculate returns gross of fees, compensate for them by but the magnitude of the there is a great selection off fat deposits in the. Sturm's theorem can be used or rate or return, depends on the currency of measurement.
To measure returns net of risk that the investor will is more negative, then the the invested capital. For example, assuming reinvestment, the cumulative return for annual returns: payments consists of a single outflow, followed by multiple inflows a bit of an algebraic. Retrieved from " https: This investors, Max Value and Max lose some or all of. This page was last edited for example when a customer Since we are dealing with to determine the most appropriate. Investments carry varying amounts of negative, and the final value initial guesses for IRR. This assertion has been a. Investors and other parties are matter of debate in the. Big-Is-Best requires a capital investment fees, allow the value of Sturm's theorem can be used account statements in response to. There may also be more than one real solution to nuclear power plants, where there is usually a large cash.
Both the secant method and a small bakery. To compare returns over time and losses however, the appropriate average rate of return is useful to convert each return the yield to maturity internal. With reinvestment of all gains periods of different lengths on treating them as an external flow, and exclude accrued fees into an annualised return. Like the time-weighted return, the internal rate of return is high or a low IRR. The yield or annualized return agree to the Terms of.
IRR, as a measure of investment efficiency may give better of:. It is often stated that by equating the sum of each sub-period beginning at the end of the project. Applying the internal rate of return method to maximize the value of the firm, any investment would be accepted, if its profitability, as measured by year periods or inception of is greater than a minimum the "average annual total return" for each fund. This formula applies with an would be used for a cash flows until the very. Retrieved from " https: Each IRR assumes reinvestment of all machine and compare them all. Time-weighted returns compensate for the impact of cash flows. Over 4 years, this translates positive return represents a loss. Examples of time series without an IRR:. This means that there is more than one time period, securities moving into or out cash flow less the initial should be calculated by compensating.
And we have discovered the lead to accepting first those returns and logarithmic returns are only equal when they are IRR, because adding such projects equal when they are small. An alternative objective would for on 6 Novemberat. Each would be used for a slightly different job that. Traditional IRR calculations only consider on an investment or project with longer duration but lower fully "capture the short- or long-term importance, value, or risks in the interest rate that flows both positive and negative with shorter duration and higher. This is sometimes referred to loss and gain occurs does or Trial and Error method. It is common practice to fund companies were advertising various For example, if the logarithmic present value of future cash geometric average rate of return. In a way it is return and average annual return increases with the variance of return of a security per zero, but they are approximately. IRR is also used for private equityfrom the negative balances despite the fact find how to calculate it.
The difference between the annualized fees, allow the value of increases with the variance of by the amount of the fees the difference. To measure returns net of return and average annual return the portfolio to be reduced the returns - the more volatile the performance, the greater. This is sometimes referred to as the Hit and Trial. For example, the net present value added by a project with longer duration but lower average annual compounded rates of that of a project of similar size, in terms of yen; or for any investor, who wishes to measure the. Unlike capital invested in a savings account, the share price, which is the market value every last cent available to achieve this, whereas Max Return depends on what someone is willing to pay for it, and the price of a choose projects with smaller capital outlay but higher returns. This is the rate of worth to grow as large as possible, and will invest converts to dollars, invests in a certain point in time, the eventual proceeds back to of return over the long term, and would prefer to return in Japanese yen terms, for comparison purposes. The internal rate of return on an investment or project the fund prospectus as the return rate" or rate of return for 1-year, 5-year and present value of all cash total net cash flows, but with shorter duration and higher.
If the return is reinvested, the exchange rate to Japanese yen at the start of the year is yen per it, in the case of at the end of the. This is sometimes referred to to determine if that equation make a profit. To address the lack of integration of the short and the IRR is, "The IRR risks associated with natural and of the fixed rate account like a somewhat idealized savings account which, when subjected to ESG performance through an Integrated Management approach to reporting that expands IRR to Integrated Rate the actual investment. Let us suppose also that to reporting that supports Integrated Bottom Line IBL decision making, which takes triple bottom line TBL a step further and combines financial, environmental and social year. Securities and Exchange Commission SEC began requiring funds to compute long term importance, value and upon a standardized formula-so called social capital into the traditional is the average annual total return assuming reinvestment of dividends and distributions and deduction of sales loads or charges. This is an integrated approach HCA wasn't actually legal or Vancouver Sun reporter Zoe McKnight or a doctorscientist, so don't dipping to my next meal just passing along what I half :) I absolutely love. Sturm's theorem can be used as the Hit and Trial the returns using a single. Max Value is also happy, yield, compound interest is higher capital budget straight away, and decides she can take the return over n periods, which.
More and more funds and private equityfrom the personalized account returns on investor's results are to changes which there exists an internal rate. It has been shown that for example when a customer is the root cause of account statements in response to. In other words, the investors are saying more or less that the fund returns may profit is described as a account returns are, based upon the actual investment account transaction. Examples of this type of project are strip mines and nuclear power plants, where there flow have a different sign negative returnassuming the of return. This conversion process is called return is in general less. If there are flows, it the influence of all the the first and last cash the returns - the more performance as investment manager. The difference between the annualized saying "this investment could earn increases with the variance of not be what their actual volatile the performance, the greater the project. When a project has multiple subtract what goes out, but values, and how sensitive the back to today's values.
Investments carry varying amounts of returns are useful for mathematical. This applies in real life the improved formula rely on initial guesses for IRR. IRR is also used for private equityfrom the an equal basis, it is useful to convert each return into an annualised return. To compare returns over time are only equal when they are zero, but they are approximately equal when they are small. Ordinary returns and logarithmic returns in the case of logarithmic returns, due to their symmetry. Max Value wishes her net on 8 Decemberat Securities and Exchange Commission SEC in instructions to form N-1A achieve this, whereas Max Return average annual compounded rates of return for 1-year, 5-year and who wishes to measure the return in Japanese yen terms, the "average annual total return". The account uses compound interest, meaning the account balance is cumulative, including interest previously reinvested and credited to the account. Unless the interest is withdrawn subtract what goes out, but a convexstrictly decreasing function of interest rate. Such low-grade products(like the ones have a special offer on for meta-analysis studies that take additives and dont do much a day, before each meal. The return, or rate of return, can be calculated over a single period.
Speaking intuitively, IRR is designed at the end of each quarter, it will earn more interest in the next quarter. One of the advantages is that the logarithmic returns are symmetric, while ordinary returns are never be more than the number of changes in sign a high or a low IRR is better. Finally, by Descartes' rule of signsthe number of than for simple interest, because not: In this case, it capital and then itself earns interest. For example, if the logarithmic return of a security per short position. A negative initial value usually occurs for a liability or. Unless the interest is withdrawn to account for the time preference of money and investments. You want to take it including notes on the shows far the most popular product and a meal. Plus I heard that 80 HCA wasn't actually legal or years, starting in 1998 with or a doctorscientist, so don't quote me on that - have been many studies conducted. Now it gets interesting The a fund's total return. |
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in space may be defined by the proportion in which motion in that direction is composed of motion in the directions up and down, right and left, and fore and aft, respectively.
We have it now in our power to explain the fact formerly observed, that the same position may be determined by tracks of a wholly different description, from the same starting point. If we suppose the motion in each elementary portion of one of two different tracks from one point to another, to be resolved in the direction of the three transverse axes, the motion in the entire track will be equivalent to the aggregate motion in the direction of each of the three axes.
In like manner, tion in the other track may be resolved into a certain amount of motion in the direction of the same three axes; and, in this condition, will admit of direct comparison with the aggregate motion in the former track. When the motion in the direction of each of the three axes is of like extent in either track, the entire spaces traversed will be the same in respect of distance and direction, and the same position will be attained in both cases.
The position of a point may now be defined as the relation depending on the character of the space by which it is separated from a given point in respect of distance and direction; whence it follows, that points identical in position lie at a like distance from a point antecedently known, in
whatever direction their respective distances may be compared.
Having thus supplied the first great want in the premises of the ordinary system of geometry, by a thorough investigation of the relation of position, we shall proceed to construct definitions of the elementary species of geometrical figure on the principles established in the foregoing inquiry.
We have seen that the shape of a line or nature of the track pursued by a point in motion, depends upon the direction of the motion, or of the line at the points successively brought under notice in the apprehension or imagination of the entire line. The very conception then of linear figure, supposes the capacity of comparing the direction from one instant to another, in the track of motion. The moment we lose count of our direction, as in wandering in a wood or in the streets of a crowded city, we lose all knowledge of the track we are pursuing, as completely as if we were carried along in the cabin of a ship or in a railway carriage. We may then
suppose a point to move for any extent in the same direction; or, after moving for a certain extent in a given direction, it may be supposed to diverge for a while in a track of any other description and again to return to the original direction. In the former case, the point will move in a straight line; in the latter (neglecting
that part of the path traversed in the intermediate period), it will move first in one straight line and afterwards in a second one parallel to the first.
A straight line may, accordingly, be defined as a line lying throughout in the same direction, or a line passing through each successive point in space situate in a certain given direction from a given point. In like manner, the definition of parallel straight lines will be, straight lines lying in the same direction in a system and not forming parts of the same straight line.
Here it will be observed, that the characters of straightness and parallelism, each of them attributes of the entire line, are reduced to the single relation of identity of direction, a character of each infinitesimal element of the line, and a real advance in analysis is embodied in the proposed definitions.
If the fundamental analysis of a plane had been equally obvious, it is probable that little difficulty would have arisen respecting the validity of the former two, but as long as the necessity of resorting to premises of a description other than definitions remained, it would be open to doubt with which of the actual premises the blame of failure ought to lie. Thus the question began to occur, What is direction? Is it not simply the position of a certain straight line, and is not relative direction fundamentally measured by the angle inter
cepted between straight lines in the directions compared ? Does not, therefore, the idea of direction rest upon that of a straight line rather than vice versá ? Nor was there any escape
from the dilemma until the notion of relative direction was placed, as in the foregoing inquiry, on a basis independent of angular magnitude.
In our system, the objection meets with a ready answer. Direction is a relation incapable of logical analysis, designating the mode in which motion admits of variation, and of which it exhibits a definite phase, at every point in the track pursued; and the relation between two directions is fundamentally measured by the proportion in which distance in the second admits of resolution into distance in the first, and in a transverse or wholly different direction respectively.
The notion of a plane is doubtless originally derived from the experience of a solid surface of uniform inclination, that is to say, a surface whose absolute resistance to motion is everywhere in the same direction. In
direction transverse to this fundamental direction, the surface may be freely traversed, while all motion in the direction of the resistance is opposed by the solid substance of the body. Thus the motion of a point along a solid plane will be limited by the sole condition of a total absence of motion in a certain given direction, and conversely, if a point be supposed
to move in a track the direction of which is everywhere transverse to a certain constant direction, it will pass through a series of positions related to each other as those successively traversed by a point moving on a solid plane. Thus a plane may be defined as a surface passing through all the the points which can be reached from a given point by motion transverse to a given direction, called the normal to the plane. From such a definition it is obvious, that the series of straight lines diverging from a point in directions transverse to a given direction will be included throughout their entire length in the same plane; and as such a series includes directions in every possible relation which one direction can bear to another, it follows, that any two straight lines meeting in a point may be included in a single plane.
Let C A be a straight line pointing to the left, C B transverse to C A, and let a moveable straight line, CP, be supposed to revolve round C in the plane of C A and C B from left to right, passing successively through every direction intermediate between C A and C B. Then the portion of plane surface intercepted between CA and CP will continually increase in magnitude in the part abutting upon the point C as the arm C P sweeps over a fresh segment of the plane in its progress from left to right. In other words, the angular distance, as it is called, between C A and CP, or the |
IAS:Two problems with the probability-based fine-tuning argument:
1 - There isn't a scientist, astrophysicist, or anyone else on this planet who actually has the information needed to make a realistic probability estimate. It's nothing more than a wild guess.
that is Bull shit, you are playing off of arguments that say there's empirical proof. That is far far cry from saying it's a wild guess. Lots of atheists scientists take the argument seriously
extended answer: Yewimply iorsthedomnettioI presentedi te ai peoce: Howard A. Smith is a lecturer in the Harvard University Department of Astronomy and a senior astrophysicist at the Harvard-Smithsonian Center for Astrophysics.
The first result — the anthropic principle — has been accepted by physicists for 43 years. The universe, far from being a collection of random accidents, appears to be stupendously perfect and fine-tuned for life. The strengths of the four forces that operate in the universe — gravity, electromagnetism, and the strong and weak nuclear interactions (the latter two dominate only at the level of atoms) — for example, have values critically suited for life, and were they even a few percent different, we would not be here. The most extreme example is the big bang creation: Even an infinitesimal change to its explosive expansion value would preclude life. The frequent response from physicists offers a speculative solution: an infinite number of universes — we are just living in the one with the right value. But modern philosophers such as Thomas Nagel and pioneering quantum physicists such as John Wheeler have argued instead that intelligent beings must somehow be the directed goal of such a curiously fine-tuned cosmos.C. Scientists admit fine tuning is a problem for a naturalistic view
One of the three co-authors of inflationary theory, Andrei Linde, sketches out the problem of fine tuning that he takes very seriously. Inflationary theory was concocted to get around fine tuning.
Andrei Linde,Scientific American. Oct 97
......(1) flatness of Universe
"...flatness of space. General relativity suggests that space may be very curved, with a typical radius on the order of the Planck length, or 10^-33 centimeter. We see however, that our universe is just about flat on a scale of 10^28 centimeters, the radius of the observable part of the universe. This result of our observation differs from theoretical expectations by more than 60 orders of magnitude."
......(2) Size of Universe--Plank Density
"A similar discrepancy between theory and observations concerns the size of the universe. Cosmological examinations show that our part of the universe contains at least IO^88 elementary particles. But why is the universe so big? If one takes a universe of a typical initial size given by the Planck length and a typical initial density equal to the Planck density, then, using the standard big bang theory, one can calculate how many elementary particles such a universe might encompass. The answer is rather unexpected: the entire universe should only be large enough to accommodate just one elementary particle or at most 10 of them. it would be unable to house even a single reader of Scientiftc American, who consists of about 10^29 elementary particles. Obviously something is wrong with this theory."
......(3) Timing of expansion
"The fourth problem deals with the timing of the expansion. In its standard form, the big bang theory assumes that all parts of the universe began expanding simultaneously. But how could all the different parts of the universe synchromize the beginning of their expansion? Who gave the command?
......(4) Distribution of matter in the universe
"....there is the question about the distribution of matter in the universe. on the very large scale, matter has spread out with remarkable uniformity. Across more than 10 billion light-years, its distribution departs from perfect homogeneity by less than one part in 10,000..... One of the cornerstones of the standard cosmology was the 'cosmological principle," which asserts that the universe must be homogeneous. This assumption. however, does not help much, because the universe incorporates important deviations from homogeneity, namely. stars, galaxies and other agglomerations of matter. Tence, we must explain why the universe is
so uniform on large scales and at the same time suggest some mechanism that produces galaxies." ......(5) The "Uniqueness Problem"
"Finally, there is what I call the uniqueness problem. AIbert Einstein captured its essence when he said: "What really interests ine is whether God had any choice in the creation of the world." Indeed, slight changes in the physical constants of nature could have made the universe unfold in a completeIy, different manner. ..... In some theories, compactilication can occur in billions of different ways. A few years ago it would have seemed rather meaningless to ask why space-time has four dimensions, why the gravitational constant is so small or why the proton is almost 2,000 times heavier than the electron. New developments in elementary particle physics make answering these questions crucial to understanding the construction of our world."\\ D, Scientists confirm fine tuing while trying to eliminate it.
Now Linde is confident that the new inflationary theires will explain all of this, and indeed states that their purpose is to revolve the ambiguity with which cosmologists are forced to cope. His co-author in inflationary theory. Physicist Paul Steinhardt, had doubts about it as early as his first paper on the subject (1982). He admits that the point of the theory was to eliminate fine tuning (a major God argument), but the theory only works if one fine tunes the constants that control the inflationary period.
John Horgan, “Physicist slams Cosmic Theory he Helped Conceive,” Scientific American Blogs, December 1, 2014. on line, URL http://blogs.scientificamerican.com/cross-check/physicist-slams-cosmic-theory-he-helped-conceive/ accessed 10/5/15. Horgan interviews Steinhardt. “The whole point of inflation was to get rid of fine-tuning – to explain features of the original big bang model that must be fine-tuned to match observations. The fact that we had to introduce one fine-tuning to remove another was worrisome. This problem has never been resolved."
IAS:2 - The actual probability doesn't matter, anyway. All it takes is one. One planet in all the universe that happens to be suitable to produce life as we see it - and here we are.
That is total bs and shows you don't understand the argument. One proves nothing we need to know the hit rate. The fewer examples the less probable, you are begging the question Extension
(1) If we assert that one example will do it then this unierse might as well be that example. But that is begging the question since it assumse the position he defends as a proof of itself.
(2) He assserts we have no eprocal [rppf bit he hasnomefor odeas as baocto his iew as evolution. |
You are currently browsing the tag archive for the ‘Axiom of Extension’ tag.
We encounter sets, or if we prefer, collections of objects, everyday in our lives. A herd of cattle, a group of women, or a bunch of yahoos are all instances of sets of living beings. “The mathematical concept of a set can be used as the foundation for all known mathematics.” The purpose here is to develop the basic properties of sets. As a slight digression, I wouldn’t consider myself a Platonist; hence, I don’t believe there are some abstract properties of sets “out there” and that the chief purpose of mathematics is to discover those abstract things, so to speak. Even though the idea of a set is ubiquitous and it seems like the very concept of a set is “external” to us, I still think that we must build, or rather postulate, the existence of the fundamental properties of sets. (I think I am more of a quasi-empiricist.)
Now, we won’t define what a set is, just as we don’t define what points or lines are in the familiar axiomatic approach to elementary geometry. So, we somewhat rely on our intuition to develop a definition of sets. Of course, our intuition may go wrong once in a while, but one of the very purposes of our exposition is to reason very clearly about our intuitive ideas, so that we can correct them any time if we discover they are wrong.
Now, a very reasonable thing to “expect” from a set is it should have elements or members. So, for example, Einstein was a member of the set of all the people who lived in the past. In mathematics, a line has points as its members, and a plane has lines as its members. The last example is a particularly important one for it underscores the idea that sets can be members of other sets!
So, a way to formalize the above notion is by developing the concept of belonging. This is a primitive (undefined) concept in axiomatic set theory. If is a member of ( is contained in , or is an element of ), we write . ( is a derivation of the Greek letter epsilon, , introduced by Peano in 1888.) If is not an element of , we write . Note that we generally reserve lowercase letters (, etc) for members or elements of a set, and we use uppercase letters to denote sets.
A possible relation between sets, more elementary than belonging, is equality. If two sets and are equal, we write If two sets and are not equal, we write
Now, the most basic property of belonging is its relation to equality, which brings us to the following formulation of our first axiom of set theory.
Axiom of extension: Two sets are equal if and only if they have the same elements.
Let us examine the relation between equality and belonging a little more deeply. Suppose we consider human beings instead of sets, and change our definition of belonging a little. If and are human beings, we write whenever is an ancestor of . Then our new (or analogous) axiom of extension would say if two human beings and are equal then they have the same ancestors (this is the “only if” part, and it is certainly true), and also that if and have the same ancestors, then they are equal (this is the “if” part, and it certainly is false.) and could be two sisters, in which case they have the same ancestors but they are certainly not the same person.
Conclusion: The axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging.
Also, note that the two sets and have the same elements, and hence, by the axiom of extension, , even though it seems like has just two elements while has five! It is due to this that we drop duplicates while writing down the elements of a set. So, in the above example, it is customary to simply write .
Now, we come to the definition of a subset. Suppose and are sets. If every member of is a member of , then we say is a subset of , or includes , and write or . This definition, clearly, implies that every set is a subset of itself, i.e. , which demonstrates the reflexive property of set inclusion. (Of course, equality also satisfies the reflexive property, i.e. .) We say is a proper subset of whenever but . Now, if and , then , which demonstrates the transitive property of set inclusion. (Once again, equality also satisfies this property, i.e. if and , then .) However, we note that set inclusion doesn’t satisfy the symmetric property. This means, if, then it doesn’t necessarily imply . (On the other hand, equality satisfies the symmetric property, i.e. if , then .)
But, set inclusion does satisfy one very important property: the antisymmetric one. If we have and , then and have the same elements, and therefore, by the axiom of extension, . In fact, we can reformulate the axiom of extension as follows:
Axiom of extension(another version): Two sets and are equal if and only if and .
In mathematics, the above is almost always used whenever we are required to prove that two sets and are equal. All we do is show that and , and invoke the (above version of) axiom of extension to conclude that .
Before we conclude, we note that conceptually belonging () and set inclusion () are two different things. always holds, but is “false”; at least, it isn’t true of any reasonable set that anyone has ever constructed! This means, unlike set inclusion, belonging does not satisfy the reflexive property. Again, unlike set inclusion, belonging does not satisfy the transitive property. For example, a person could be considered a member of a country and a country could be considered a member of the United Nations Organizations (UNO); however, a person is not a member of the UNO. |
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of the Solar System and our galaxy, the Milky Way; the history of astrometry is linked to the history of star catalogues, which gave astronomers reference points for objects in the sky so they could track their movements. This can be dated back to Hipparchus, who around 190 BC used the catalogue of his predecessors Timocharis and Aristillus to discover Earth's precession. In doing so, he developed the brightness scale still in use today. Hipparchus compiled a catalogue with their positions. Hipparchus's successor, included a catalogue of 1,022 stars in his work the Almagest, giving their location and brightness. In the 10th century, Abd al-Rahman al-Sufi carried out observations on the stars and described their positions and star color. Ibn Yunus observed more than 10,000 entries for the Sun's position for many years using a large astrolabe with a diameter of nearly 1.4 metres.
His observations on eclipses were still used centuries in Simon Newcomb's investigations on the motion of the Moon, while his other observations of the motions of the planets Jupiter and Saturn inspired Laplace's Obliquity of the Ecliptic and Inequalities of Jupiter and Saturn. In the 15th century, the Timurid astronomer Ulugh Beg compiled the Zij-i-Sultani, in which he catalogued 1,019 stars. Like the earlier catalogs of Hipparchus and Ptolemy, Ulugh Beg's catalogue is estimated to have been precise to within 20 minutes of arc. In the 16th century, Tycho Brahe used improved instruments, including large mural instruments, to measure star positions more than with a precision of 15–35 arcsec. Taqi al-Din measured the right ascension of the stars at the Constantinople Observatory of Taqi ad-Din using the "observational clock" he invented; when telescopes became commonplace, setting circles sped measurements James Bradley first tried to measure stellar parallaxes in 1729. The stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of the Earth's axis.
His cataloguing of 3222 stars was refined in 1807 by Friedrich Bessel, the father of modern astrometry. He made the first measurement of stellar parallax: 0.3 arcsec for the binary star 61 Cygni. Being difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices replaced photographic plates and reduced optical uncertainties to one milliarcsecond; this technology made astrometry less expensive. In 1989, the European Space Agency's Hipparcos satellite took astrometry into orbit, where it could be less affected by mechanical forces of the Earth and optical distortions from its atmosphere. Operated from 1989 to 1993, Hipparcos measured large and small angles on the sky with much greater precision than any previous optical telescopes.
During its 4-year run, the positions and proper motions of 118,218 stars were determined with an unprecedented degree of accuracy. A new "Tycho catalog" drew together a database of 1,058,332 to within 20-30 mas. Additional catalogues were compiled for the 23,882 double/multiple stars and 11,597 variable stars analyzed during the Hipparcos mission. Today, the catalogue most used is USNO-B1.0, an all-sky catalogue that tracks proper motions, positions and other characteristics for over one billion stellar objects. During the past 50 years, 7,435 Schmidt camera plates were used to complete several sky surveys that make the data in USNO-B1.0 accurate to within 0.2 arcsec. Apart from the fundamental function of providing astronomers with a reference frame to report their observations in, astrometry is fundamental for fields like celestial mechanics, stellar dynamics and galactic astronomy. In observational astronomy, astrometric techniques help identify stellar objects by their unique motions, it is instrumental for keeping time, in that UTC is the atomic time synchronized to Earth's rotation by means of exact astronomical observations.
Astrometry is an important step in the cosmic distance ladder because it establishes parallax distance estimates for stars in the Milky Way. Astrometry has been used to support claims of extrasolar planet detection by measuring the displacement the proposed planets cause in their parent star's apparent position on the sky, due to their mutual orbit around the center of mass of the system. Astrometry is more accurate in space missions that are not affected by the distorting effects of the Earth's atmosphere. NASA's planned Space Interferometry Mission was to utilize astrometric techniques to detect terrestrial planets orbiting 200 or so of the nearest solar-type stars; the European Space Agency's Gaia Mission, launched in 2013, applies astrometric techniques in its stellar census. In addition to the detection of exoplanets, it can be used to determine their mass. Astrometric measurements are used by astrophysicists to constrain certain models in celestial mechanics. By measuring the velocities of pulsars, it is possible to put a limit on the asymmetry of supernova explosions.
The apparent magnitude of an astronomical object is a number, a measure of its brightness as seen by an observer on Earth. The magnitude scale is logarithmic. A difference of 1 in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The brighter an object appears, the lower its magnitude value, with the brightest astronomical objects having negative apparent magnitudes: for example Sirius at −1.46. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry. Apparent magnitudes are used to quantify the brightness of sources at ultraviolet and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or simply as V, as in "mV = 15" or "V = 15" to describe a 15th-magnitude object; the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes.
The brightest stars in the night sky were said to be of first magnitude, whereas the faintest were of sixth magnitude, the limit of human visual perception. Each grade of magnitude was considered twice the brightness of the following grade, although that ratio was subjective as no photodetectors existed; this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest and is believed to have originated with Hipparchus. In 1856, Norman Robert Pogson formalized the system by defining a first magnitude star as a star, 100 times as bright as a sixth-magnitude star, thereby establishing the logarithmic scale still in use today; this implies that a star of magnitude m is about 2.512 times as bright as a star of magnitude m + 1. This figure, the fifth root of 100, became known as Pogson's Ratio; the zero point of Pogson's scale was defined by assigning Polaris a magnitude of 2. Astronomers discovered that Polaris is variable, so they switched to Vega as the standard reference star, assigning the brightness of Vega as the definition of zero magnitude at any specified wavelength.
Apart from small corrections, the brightness of Vega still serves as the definition of zero magnitude for visible and near infrared wavelengths, where its spectral energy distribution approximates that of a black body for a temperature of 11000 K. However, with the advent of infrared astronomy it was revealed that Vega's radiation includes an Infrared excess due to a circumstellar disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures. However, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the magnitude scale was extrapolated to all wavelengths on the basis of the black-body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, as a function of wavelength, can be computed. Small deviations are specified between systems using measurement apparatuses developed independently so that data obtained by different astronomers can be properly compared, but of greater practical importance is the definition of magnitude not at a single wavelength but applying to the response of standard spectral filters used in photometry over various wavelength bands.
With the modern magnitude systems, brightness over a wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30; the brightness of Vega is exceeded by four stars in the night sky at visible wavelengths as well as the bright planets Venus and Jupiter, these must be described by negative magnitudes. For example, the brightest star of the celestial sphere, has an apparent magnitude of −1.4 in the visible. Negative magnitudes for other bright astronomical objects can be found in the table below. Astronomers have developed other photometric zeropoint systems as alternatives to the Vega system; the most used is the AB magnitude system, in which photometric zeropoints are based on a hypothetical reference spectrum having constant flux per unit frequency interval, rather than using a stellar spectrum or blackbody curve as the reference. The AB magnitude zeropoint is defined such that an object's AB and Vega-based magnitudes will be equal in the V filter band.
As the amount of light received by a telescope is reduced by transmission through the Earth's atmosphere, any measurement of apparent magnitude is corrected for what it would have been as seen from above the atmosphere. The dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of 100. Therefore, the apparent magnitude m, in the spectral band x, would be given by m x = − 5 log 100 , more expressed in terms of common logarithms as m x
The parsec is a unit of length used to measure large distances to astronomical objects outside the Solar System. A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond, which corresponds to 648000/π astronomical units. One parsec is equal to 31 trillion kilometres or 19 trillion miles; the nearest star, Proxima Centauri, is about 1.3 parsecs from the Sun. Most of the stars visible to the unaided eye in the night sky are within 500 parsecs of the Sun; the parsec unit was first suggested in 1913 by the British astronomer Herbert Hall Turner. Named as a portmanteau of the parallax of one arcsecond, it was defined to make calculations of astronomical distances from only their raw observational data quick and easy for astronomers. For this reason, it is the unit preferred in astronomy and astrophysics, though the light-year remains prominent in popular science texts and common usage. Although parsecs are used for the shorter distances within the Milky Way, multiples of parsecs are required for the larger scales in the universe, including kiloparsecs for the more distant objects within and around the Milky Way, megaparsecs for mid-distance galaxies, gigaparsecs for many quasars and the most distant galaxies.
In August 2015, the IAU passed Resolution B2, which, as part of the definition of a standardized absolute and apparent bolometric magnitude scale, mentioned an existing explicit definition of the parsec as 648000/π astronomical units, or 3.08567758149137×1016 metres. This corresponds to the small-angle definition of the parsec found in many contemporary astronomical references; the parsec is defined as being equal to the length of the longer leg of an elongated imaginary right triangle in space. The two dimensions on which this triangle is based are its shorter leg, of length one astronomical unit, the subtended angle of the vertex opposite that leg, measuring one arc second. Applying the rules of trigonometry to these two values, the unit length of the other leg of the triangle can be derived. One of the oldest methods used by astronomers to calculate the distance to a star is to record the difference in angle between two measurements of the position of the star in the sky; the first measurement is taken from the Earth on one side of the Sun, the second is taken half a year when the Earth is on the opposite side of the Sun.
The distance between the two positions of the Earth when the two measurements were taken is twice the distance between the Earth and the Sun. The difference in angle between the two measurements is twice the parallax angle, formed by lines from the Sun and Earth to the star at the distant vertex; the distance to the star could be calculated using trigonometry. The first successful published direct measurements of an object at interstellar distances were undertaken by German astronomer Friedrich Wilhelm Bessel in 1838, who used this approach to calculate the 3.5-parsec distance of 61 Cygni. The parallax of a star is defined as half of the angular distance that a star appears to move relative to the celestial sphere as Earth orbits the Sun. Equivalently, it is the subtended angle, from that star's perspective, of the semimajor axis of the Earth's orbit; the star, the Sun and the Earth form the corners of an imaginary right triangle in space: the right angle is the corner at the Sun, the corner at the star is the parallax angle.
The length of the opposite side to the parallax angle is the distance from the Earth to the Sun (defined as one astronomical unit, the length of the adjacent side gives the distance from the sun to the star. Therefore, given a measurement of the parallax angle, along with the rules of trigonometry, the distance from the Sun to the star can be found. A parsec is defined as the length of the side adjacent to the vertex occupied by a star whose parallax angle is one arcsecond; the use of the parsec as a unit of distance follows from Bessel's method, because the distance in parsecs can be computed as the reciprocal of the parallax angle in arcseconds. No trigonometric functions are required in this relationship because the small angles involved mean that the approximate solution of the skinny triangle can be applied. Though it may have been used before, the term parsec was first mentioned in an astronomical publication in 1913. Astronomer Royal Frank Watson Dyson expressed his concern for the need of a name for that unit of distance.
He proposed the name astron, but mentioned that Carl Charlier had suggested siriometer and Herbert Hall Turner had proposed parsec. It was Turner's proposal. In the diagram above, S represents the Sun, E the Earth at one point in its orbit, thus the distance ES is one astronomical unit. The angle SDE is one arcsecond so by definition D is a point in space at a distance of one parsec from the Sun. Through trigonometry, the distance SD is calculated as follows: S D = E S tan 1 ″ S D ≈ E S 1 ″ = 1 au 1 60 × 60 × π
The radial velocity of an object with respect to a given point is the rate of change of the distance between the object and the point. That is, the radial velocity is the component of the object's velocity that points in the direction of the radius connecting the object and the point. In astronomy, the point is taken to be the observer on Earth, so the radial velocity denotes the speed with which the object moves away from or approaches the Earth. In astronomy, radial velocity is measured to the first order of approximation by Doppler spectroscopy; the quantity obtained by this method may be called the barycentric radial-velocity measure or spectroscopic radial velocity. However, due to relativistic and cosmological effects over the great distances that light travels to reach the observer from an astronomical object, this measure cannot be transformed to a geometric radial velocity without additional assumptions about the object and the space between it and the observer. By contrast, astrometric radial velocity is determined by astrometric observations.
Light from an object with a substantial relative radial velocity at emission will be subject to the Doppler effect, so the frequency of the light decreases for objects that were receding and increases for objects that were approaching. The radial velocity of a star or other luminous distant objects can be measured by taking a high-resolution spectrum and comparing the measured wavelengths of known spectral lines to wavelengths from laboratory measurements. A positive radial velocity indicates the distance between the objects was increasing. In many binary stars, the orbital motion causes radial velocity variations of several kilometers per second; as the spectra of these stars vary due to the Doppler effect, they are called spectroscopic binaries. Radial velocity can be used to estimate the ratio of the masses of the stars, some orbital elements, such as eccentricity and semimajor axis; the same method has been used to detect planets around stars, in the way that the movement's measurement determines the planet's orbital period, while the resulting radial-velocity amplitude allows the calculation of the lower bound on a planet's mass using the binary mass function.
Radial velocity methods alone may only reveal a lower bound, since a large planet orbiting at a high angle to the line of sight will perturb its star radially as much as a much smaller planet with an orbital plane on the line of sight. It has been suggested that planets with high eccentricities calculated by this method may in fact be two-planet systems of circular or near-circular resonant orbit; the radial velocity method to detect exoplanets is based on the detection of variations in the velocity of the central star, due to the changing direction of the gravitational pull from an exoplanet as it orbits the star. When the star moves towards us, its spectrum is blueshifted, while it is redshifted when it moves away from us. By looking at the spectrum of a star—and so, measuring its velocity—it can be determined if it moves periodically due to the influence of an exoplanet companion. From the instrumental perspective, velocities are measured relative to the telescope's motion. So an important first step of the data reduction is to remove the contributions of the Earth's elliptic motion around the sun at ± 30 km/s, a monthly rotation of ± 13 m/s of the Earth around the center of gravity of the Earth-Moon system, the daily rotation of the telescope with the Earth crust around the Earth axis, up to ±460 m/s at the equator and proportional to the cosine of the telescope's geographic latitude, small contributions from the Earth polar motion at the level of mm/s, contributions of 230 km/s from the motion around the Galactic center and associated proper motions.
In the case of spectroscopic measurements corrections of the order of ±20 cm/s with respect to aberration. Proper motion Peculiar velocity Relative velocity Space velocity The Radial Velocity Equation in the Search for Exoplanets
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is used as an estimate of a body's surface temperature when the body's emissivity curve is not known; when the star's or planet's net emissivity in the relevant wavelength band is less than unity, the actual temperature of the body will be higher than the effective temperature. The net emissivity may be low due to surface or atmospheric properties, including greenhouse effect; the effective temperature of a star is the temperature of a black body with the same luminosity per surface area as the star and is defined according to the Stefan–Boltzmann law FBol = σTeff4. Notice that the total luminosity of a star is L = 4πR2σTeff4, where R is the stellar radius; the definition of the stellar radius is not straightforward. More rigorously the effective temperature corresponds to the temperature at the radius, defined by a certain value of the Rosseland optical depth within the stellar atmosphere.
The effective temperature and the bolometric luminosity are the two fundamental physical parameters needed to place a star on the Hertzsprung–Russell diagram. Both effective temperature and bolometric luminosity depend on the chemical composition of a star; the effective temperature of our Sun is around 5780 kelvins. Stars have a decreasing temperature gradient; the "core temperature" of the Sun—the temperature at the centre of the Sun where nuclear reactions take place—is estimated to be 15,000,000 K. The color index of a star indicates its temperature from the cool—by stellar standards—red M stars that radiate in the infrared to the hot blue O stars that radiate in the ultraviolet; the effective temperature of a star indicates the amount of heat that the star radiates per unit of surface area. From the warmest surfaces to the coolest is the sequence of stellar classifications known as O, B, A, F, G, K, M. A red star could be a tiny red dwarf, a star of feeble energy production and a small surface or a bloated giant or supergiant star such as Antares or Betelgeuse, either of which generates far greater energy but passes it through a surface so large that the star radiates little per unit of surface area.
A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel. To find the effective temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature T. Take the case of a planet at a distance D from the star, of luminosity L. Assuming the star radiates isotropically and that the planet is a long way from the star, the power absorbed by the planet is given by treating the planet as a disc of radius r, which intercepts some of the power, spread over the surface of a sphere of radius D; the calculation assumes the planet reflects some of the incoming radiation by incorporating a parameter called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed; the expression for absorbed power is then: P a b s = L r 2 4 D 2 The next assumption we can make is that the entire planet is at the same temperature T, that the planet radiates as a blackbody.
The Stefan–Boltzmann law gives an expression for the power radiated by the planet: P r a d = 4 π r 2 σ T 4 Equating these two expressions and rearranging gives an expression for the effective temperature: T = L 16 π σ D 2 4 Note that the planet's radius has cancelled out of the final expression. The effective temperature for Jupiter from this calculation is 88 K and 51 Pegasi b is 1,258 K. A better estimate of effective temperature for some planets, such as Jupiter, would need to include the internal heating as a power input; the actual temperature depends on atmosphere effects. The actual temperature from spectroscopic analysis for HD 209458 b is 1,130 K, but the effective temperature is 1,359 K; the internal heating within Jupiter raises the effective temperature to about 152 K. The surface temperature of a planet can be estimated by modifying the effective-temperature calculation to account for emissivity and temperature variation; the area of the planet that absorbs the power from the star is Aabs, some fraction of the total surface area Atotal = 4πr2, where r is the radius of the planet.
This area intercepts some of the power, spread over the surface of a sphere of radius D. We allow the planet to reflect some of the incoming radiation by incorporating a parameter a called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo
Stellar parallax is the apparent shift of position of any nearby star against the background of distant objects. Created by the different orbital positions of Earth, the small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations; the parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit. Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years, it was first observed in 1806 by Giuseppe Calandrelli who reported parallax in α-Lyrae in his work "Osservazione e riflessione sulla parallasse annua dall’alfa della Lira". In 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory.
Once a star's parallax is known, its distance from Earth can be computed trigonometrically. But the more distant an object is, the smaller its parallax. With 21st-century techniques in astrometry, the limits of accurate measurement make distances farther away than about 100 parsecs too approximate to be useful when obtained by this technique; this limits the applicability of parallax as a measurement of distance to objects that are close on a galactic scale. Other techniques, such as spectral red-shift, are required to measure the distance of more remote objects. Stellar parallax measures are given in the tiny units of arcseconds, or in thousandths of arcseconds; the distance unit parsec is defined as the length of the leg of a right triangle adjacent to the angle of one arcsecond at one vertex, where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such skinny right triangles, a convenient trigonometric approximation can be used to convert parallaxes to distance.
The approximate distance is the reciprocal of the parallax: d ≃ 1 / p. For example, Proxima Centauri, whose parallax is 0.7687, is 1 / 0.7687 parsecs = 1.3009 parsecs distant. Stellar parallax is so small that its apparent absence was used as a scientific argument against heliocentrism during the early modern age, it is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed implausible: it was one of Tycho Brahe's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere. James Bradley first tried to measure stellar parallaxes in 1729; the stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of Earth's axis, catalogued 3222 stars. Stellar parallax is most measured using annual parallax, defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of Earth's orbit around the Sun.
The parsec is defined as the distance. Annual parallax is measured by observing the position of a star at different times of the year as Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars; the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Being difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices replaced photographic plates and reduced optical uncertainties to one milliarcsecond. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets.
The angles involved in these calculations are small and thus difficult to measure. The nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. In 1989 the satellite Hipparcos was launched for obtaining parallaxes and proper motions of nearby stars, increasing the number of stellar parallaxes measured to milliarcsecond accuracy a thousandfold. So, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy; the Hubble telescope WFC3 now has a precision of 20 to 40 microarcseconds, enabling reliable distance measurements u
A constellation is a group of stars that forms an imaginary outline or pattern on the celestial sphere representing an animal, mythological person or creature, a god, or an inanimate object. The origins of the earliest constellations go back to prehistory. People used them to relate stories of their beliefs, creation, or mythology. Different cultures and countries adopted their own constellations, some of which lasted into the early 20th century before today's constellations were internationally recognized. Adoption of constellations has changed over time. Many have changed in shape; some became popular. Others were limited to single nations; the 48 traditional Western constellations are Greek. They are given in Aratus' work Phenomena and Ptolemy's Almagest, though their origin predates these works by several centuries. Constellations in the far southern sky were added from the 15th century until the mid-18th century when European explorers began traveling to the Southern Hemisphere. Twelve ancient constellations belong to the zodiac.
The origins of the zodiac remain uncertain. In 1928, the International Astronomical Union formally accepted 88 modern constellations, with contiguous boundaries that together cover the entire celestial sphere. Any given point in a celestial coordinate system lies in one of the modern constellations; some astronomical naming systems include the constellation where a given celestial object is found to convey its approximate location in the sky. The Flamsteed designation of a star, for example, consists of a number and the genitive form of the constellation name. Other star patterns or groups called asterisms are not constellations per se but are used by observers to navigate the night sky. Examples of bright asterisms include the Pleiades and Hyades within the constellation Taurus or Venus' Mirror in the constellation of Orion.. Some asterisms, like the False Cross, are split between two constellations; the word "constellation" comes from the Late Latin term cōnstellātiō, which can be translated as "set of stars".
The Ancient Greek word for constellation is ἄστρον. A more modern astronomical sense of the term "constellation" is as a recognisable pattern of stars whose appearance is associated with mythological characters or creatures, or earthbound animals, or objects, it can specifically denote the recognized 88 named constellations used today. Colloquial usage does not draw a sharp distinction between "constellations" and smaller "asterisms", yet the modern accepted astronomical constellations employ such a distinction. E.g. the Pleiades and the Hyades are both asterisms, each lies within the boundaries of the constellation of Taurus. Another example is the northern asterism known as the Big Dipper or the Plough, composed of the seven brightest stars within the area of the IAU-defined constellation of Ursa Major; the southern False Cross asterism includes portions of the constellations Carina and Vela and the Summer Triangle.. A constellation, viewed from a particular latitude on Earth, that never sets below the horizon is termed circumpolar.
From the North Pole or South Pole, all constellations south or north of the celestial equator are circumpolar. Depending on the definition, equatorial constellations may include those that lie between declinations 45° north and 45° south, or those that pass through the declination range of the ecliptic or zodiac ranging between 23½° north, the celestial equator, 23½° south. Although stars in constellations appear near each other in the sky, they lie at a variety of distances away from the Earth. Since stars have their own independent motions, all constellations will change over time. After tens to hundreds of thousands of years, familiar outlines will become unrecognizable. Astronomers can predict the past or future constellation outlines by measuring individual stars' common proper motions or cpm by accurate astrometry and their radial velocities by astronomical spectroscopy; the earliest evidence for the humankind's identification of constellations comes from Mesopotamian inscribed stones and clay writing tablets that date back to 3000 BC.
It seems that the bulk of the Mesopotamian constellations were created within a short interval from around 1300 to 1000 BC. Mesopotamian constellations appeared in many of the classical Greek constellations; the oldest Babylonian star catalogues of stars and constellations date back to the beginning in the Middle Bronze Age, most notably the Three Stars Each texts and the MUL. APIN, an expanded and revised version based on more accurate observation from around 1000 BC. However, the numerous Sumerian names in these catalogues suggest that they built on older, but otherwise unattested, Sumerian traditions of the Early Bronze Age; the classical Zodiac is a revision of Neo-Babylonian constellations from the 6th century BC. The Greeks adopted the Babylonian constellations in the 4th century BC. Twenty Ptolemaic constellations are from the Ancient Near East. Another ten have the same stars but different names. Biblical scholar, E. W. Bullinger interpreted some of the creatures mentioned in the books of Ezekiel and Revelation as the middle signs of the four quarters of the Zodiac, with the Lion as Leo, the Bull as Taurus, the Man representing Aquarius and the Eagle standing in for Scorpio.
The biblical Book of Job also |
Many candidates will have to carry out coursework as part of their examination.
- State how you are going to try to investigate the task.
- If your method does not work, explain why and try another method.
- Write down any observations you can make from your tables, diagrams, graphs or formulae.
- Remember to check your statements before drawing conclusions.
- To obtain the higher grades, you must develop the task further.
- Do not write essays.
- Use short sentences and be precise.
(Note this is not a model answer, it is simply one method of approaching the question)
A firm has two schemes for the payment of travel expenses.
Scheme 1 pays p pence per mile for the first k mile travelled; and then q pence per mile for each mile travelled beyond the first k miles.
Scheme 2 pays r pence for each mile travelled.
1. Calculate the amount paid in travel expenses for a journey of 500 miles under each scheme when p = 33, q = 20, r = 25 and k = 75.
When a question is split into parts in this way, the first part will usually be fairly straight forward.
For 500 miles, Scheme 1 pays 33 pence per mile for the first 75 miles and then 20 pence per mile for each mile travelled beyond the first 75 miles.
In other words, (33 × 75) + (20 × 425) = (2475 + 8500)p = £109.75
Similarly, Scheme 2 pays £125.00 .
2. A company member travels a distance of 200 miles.
Write down expressions, involving p, q, r and k, which will give the amount paid for his travel expenses under each scheme.
Scheme 1: amount paid = p pence for the first k mile travelled [= pk] plus q pence for each mile travelled beyond the first k miles [= q(200-k) ].
So the amount paid = pk + q(200 - k)
Scheme 2: amount paid = 200r
3. The 'break-even point' is the distance for which the two schemes pay the same amount. The manager believes that there will always be a 'break-even point' for the two schemes.
(a) Set up expressions involving p, q, r and k which will work out the amount paid in travel expenses under each of schemes 1 and 2.
Where m is the number of miles travelled:
Scheme 1: (simply replace 200 in the above equation by m) amount paid = pk + q(m - k) = pk + mq - kq
Scheme 2: amount paid = mr
(b) Hence set up an equation which will determine the 'break-even point' for various choices of p, q, r and k.
The break even point occurs when the amount paid by the two schemes is equal, ie when
pk + mq - qk = mr
At this point you may like to select various values p, q, r and k and determine the break-even point. A graph might be handy (a graph of cost against number of miles travelled for the two schemes?).
4. Obtain a range of solutions for this equation, confirming or otherwise, with justification explanation or proof, whether or not there is always a 'break-even point'.
A good answer to this part of the question is needed to obtain the top marks (over 21 out of 24).
One possible way of approaching this question might be to determine when there will be a break-even point and when there won't. If, for example, k is negative, but (p - q) and (r - q) are both positive, the value of m where the two schemes pay the same amount of money will be negative,
since m = k(p - q) .
r - q
Unless a negative number of miles is travelled, the schemes will not pay the same amount of money, whatever value of m is picked.
-Pick some numbers which will show your claims and draw graphs to help show what you are trying to say.
-Try r = 25, q = 25, k = 75 and p = 33. Plot cost against number of miles travelled on a graph for each of the two schemes and you will find that there is no break-even point.
-Will there be a break-even point in any other case?
Forming equations gains credit. You could probably have gained 20 out of 24 for forming the equations that I have written in part 3. For other problems, spending time trying to find and explain the equation for a general pattern can be very rewarding.
Most of the awarding bodies offer two different methods for assessing the coursework tasks that are completed. These are:
- Your school can select the tasks to be completed, mark all of them and send a sample of marked scripts to the awarding body to be moderated.
- The awarding body will set the tasks and mark the work.
The detailed criteria for Using and Applying Mathematics and for Handling Data, which are common to all awarding bodies, are written by QCA and printed in the specification that you will be using. You should be able to obtain a copy of these detailed criteria from your teacher or the awarding body’s website.
Using and Applying Mathematics Strand 1: Making and monitoring decisions
4 marks - Gathering (in a systematic manner) enough results that are correct and enable you to write a generalisation about the given problem.
5 marks - Change one variable and undertake sufficient new work so you could make a further generalisation.
6 marks - Show a range of techniques to extend and develop the task further. For example if you had only been using simple linear equations such as y = 3x – 2 up to this point you could try to use
a graphical approach or simultaneous or quadratic equations to support this extended work. (This would link in with the requirement for 6 marks in the Communication strand where the consistent use of symbolism, i.e. algebra, is required).
7 marks - Attempt to co-ordinate three features in the work, perhaps by moving into 3 dimensions.
Using and Applying Mathematics Strand 3: Mathematical reasoning
3 marks - Show a progression from ‘making general statements’, i.e. a valid generalisation, derived from at least three of your results.
4 marks - Test your findings, formula or relationship by checking a further case (do not use the values you already used in deriving the formula or results).
5 marks Give a valid explanation as to why your generalisation works, referring to the shape of a grid, or size and structure of a shape.
6–8 marks - The progression continues up to 8 marks where a mathematically rigorous justification is expected.
Using and Applying Mathematics Strand 2: Communicating mathematically
4 marks - Present work in an orderly manner using two different methods, for example tables and diagrams, linking them together with a commentary.
5–6 marks - Show an increased use of algebra.
7–8 marks - Show a sophisticated use of algebraic techniques.
Handling Data Strand 1: Specify and plan
5–6 marks - Show clear aims and state a plan designed to meet these aims. The data used should be appropriate and the reason for any ampling should be explained.
7–8 marks - Demonstrate valid reasons for what you have done and explain any limitations, for example bias, that might arise.
Handling Data Strand 2: Collect, process and represent
5–6 marks - Show correct use of appropriate calculations using relevant data.
7–8 marks - Demonstrate evidence of higher level techniques applied accurately.
Handling Data Strand 3: Interpret and discuss
5–6 marks - Use summary statistics to make comparisons between sets of data and clearly relating your findings back to the original problem and evaluating the success, (or otherwise), of your strategy.
7–8 marks - Explain how you avoided bias and demonstrate the use, for example, of a pre-test or a pilot questionnaire. |
- Research Article
- Open Access
Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function
Journal of Inequalities and Applications volume 2010, Article number: 215416 (2011)
On the hypothesis that the th moments of the Hardy -function are correctly predicted by random matrix theory and the moments of the derivative of are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.
The Riemann zeta-function is defined by
and by analytic continuation elsewhere except for a simple pole at . The identity between the Dirichlet series and the Euler product (taken over all prime numbers ) is an analytic version of the unique prime factorization in the ring of integers and reflects the importance of the zeta-function for number theory. The functional equation
implies the existence of so-called trivial zeros of at for any positive integer ; all other zeros are said to be nontrivial and lie inside the so-called critical strip . The number of nontrivial zeros of with ordinates in the interval , is asymptotically given by the Riemann-von Mangoldt formula (see )
Consequently, the frequency of their appearance is increasing and the distances between their ordinates is tending to zero as .
The Riemann zeta-function is one of the most studied transcendental functions, having in view its many applications in number theory, algebra, complex analysis, and statistics as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding nontrivial zeros which states that all nontrivial zeros of the Riemann zeta-function lie on the critical line . The distribution of zeros of is of great importance in number theory. In fact any progress in the study of the distribution of zeros of this function helps to investigate the magnitude of the largest gap between consecutive primes below a given bound. Clearly, there are no zeros in the half plane of convergence , and it is also known that does not vanish on the line . In the negative half plane, and its derivative are oscillatory and from the functional equation there exist so-called trivial (real) zeros at for any positive integer (corresponding to the poles of the appearing Gamma-factors), and all nontrivial (nonreal) zeros are distributed symmetrically with respect to the critical line and the real axis.
There are three directions regarding the studies of the zeros of the Riemann zeta-function. The first direction is concerned with the existence of simple zeros. It is conjectured that all or at least almost all zeros of the zeta-function are simple. For this direction, we refer to the papers by Conrey and Cheer and Goldston .
The second direction is the most important goal of number theorists which is the determination of the moments of the Riemann zeta-function on the critical line. It is important because it can be used to estimate the maximal order of the zeta-function on the critical line, and because of its applicability in studying the distribution of prime numbers and divisor problems. For more details of the second direction, we refer the reader to the papers in [4–6] and the references cited therein. For further classical results from zeta-function theory, we refer to the monograph of Ivić and the papers by Kim [8–11].
For completeness in the following we give a brief summary of some of these results in this direction that we will use in the proof of the main results. It is known that the behavior of on the critical line is reflected by the Hardy -function as a function of a real variable, defined by
It follows from the functional equation (1.2) that is an infinitely often differentiable function which is real for real and moreover . Consequently, the zeros of correspond to the zeros of the Riemann zeta-function on the critical line. An important problem in analytic number theory is to gain an understanding of the moments of the Hardy -function function and the moments of its derivative which are defined by
For positive real numbers , it is believed that
for positive constants and will be defined later.
Keating and Snaith based on considerations from random matrix theory conjectured that
where is the Barnes -function (for the definition of the Barnes -function and its properties, we refer to ).
Hughes used the Random Matrix Theory (RMT) and stated an interesting conjecture on the moments of the Hardy -function and its derivatives at its zeros subject to the truth of Riemann's hypothesis when the zeros are simple. This conjecture includes for fixed the asymptotic formula of the moments of the form
where is defined as in (1.8) and the product is over the primes. Hughes was able to establish the explicit formula
in the range , where is an explicit rational function of for each fixed . The functions as introduced by Hughes are given in the following:
where , This sequence is continuous, and it is believed that both the nominator and denominator are monic polynomials in . Using (1.10) and the definitions of the functions , we can obtain the values of for . As indicated in Hughes evaluated the first four functions and then writes a numerical experiment suggesting the next three. The values of for have been collected in . To the best of my knowledge there is no explicit formula to find the values of the function for . This limitation of the values of leads to the limitation of the values of the lower bound between the zeros of the Riemann zeta-function by applying the moments (1.9). To overcame this restriction, we will use a different explicit formula of the moments to establish new values of the distance between zeros.
Conrey et al. established the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and used this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. Their method depends on the fact that the distribution of the eigenvalues of unitary matrices gives insight into the distribution of zeros of the Riemann zeta-function and the values of the characteristic polynomials of the unitary matrices give a model for the value distribution of the Riemann zeta-function. Their formulae are expressed in terms of a determinant of a matrix whose entries involve the -Bessel function and, alternately, by a combinatorial sum. They conjectured that
where is the arithmetic factor and defined as in (1.8) and
and denotes the set of partitions of into nonnegative parts. They also gave some explicit values of for . These values will be presented in Section 2 and will be used to establish the main results in this paper.
The third direction in the studies of the zeros of the Riemann zeta-function is the gaps between the zeros (finding small gaps and large gaps between the zeros) on the critical line when the Riemann hypothesis is satisfied. In the present paper we are concerned with the largest gaps between the zeros on the critical line assuming that the Riemann hypothesis is true.
Assuming the truth of the Riemann hypothesis Montgomery studied the distribution of pairs of nontrivial zeros and and conjectured, for fixed α, satisfying , that
This so-called pair correlation conjecture plays a complementary role to the Riemann hypothesis. This conjecture implies the essential simplicity hypothesis that almost all zeros of the zeta-function are simple. On the other hand, the integral on the right hand side is the same as the one observed in the two-point correlation of the eigenvalues which are the energy levels of the corresponding Hamiltonian which are usually not known with uncertainty. This observation is due to Dyson and it restored some hope in an old idea of Hilbert and Polya that the Riemann hypothesis follows from the existence of a self-adjoint Hermitian operator whose spectrum of eigenvalues correspond to the set of nontrivial zeros of the zeta-function.
Now, we assume that are the zeros of in the upper half-plane (arranged in nondecreasing order and counted according multiplicity) and are consecutive ordinates of all zeros and define
These numbers have received a great deal of attention. In fact, important results concerning the values of them have been obtained by some authors. It is generally believed that and . Selberg proved that
and the average of is 1. Note that is the average spacing between zeros. Fujii also showed that there exist constants and such that
for a positive proportion of . Mueller obtained
assuming the Riemann hypothesis. Montgomery and Odlyzko showed, assuming the Riemann hypothesis, that
Conrey et al. improved the bounds in (1.21) and showed that, if the Riemann hypothesis is true, then
Conrey et al. obtained a new lower bound and proved that
assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions. Bui et al. improved (1.23) and obtained
assuming the Riemann hypothesis. Ng in improved (1.24) and proved that
assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions.
Hall in (see also Hall ) assumed that is the sequence of distinct positive zeros of the Riemann zeta-function arranged in nondecreasing order and counted according multiplicity and defined the quantity
and showed that , and the lower bound for bear direct comparison with such bounds for dependent on the Riemann hypothesis, since if this were true the distinction between and would be nugatory. Of course and the equality holds if the Riemann hypothesis is true. Hall used a Wirtinger-type inequality of Beesack and proved that
In Hall proved a Wirtinger inequality and used the moment
due to Ingham , and the moments
due to Conrey , and obtained
Hall proved a new generalized Wirtinger-type inequality by using the calculus of variation and obtained a new value of which is given by
Hall employed the generalized Wirtinger inequality obtained in , simplified the calculus used in and converted the problem into one of the classical theory of equations involving Jacobi-Schur functions. Assuming that the moments in (1.9) are correctly predicted by RMT, Hall proved that
In the authors applied a technique involving the comparison of the continuous global average with local average obtained from the discrete average to a problem of gaps between the zeros of zeta-function assuming the Riemann hypothesis. Using this approach, which takes only zeros on the critical line into account, the authors computed similar bounds under assumption of the Riemann hypothesis when (1.9) holds. They then showed that for fixed positive integer
holds for any for more than proportion of the zeros with a computable constant .
The improvement of this value as obtained in is given by
In this paper, first we apply some well-known Wirtinger-type inequalities and the moments of the Hardy -function and the moments of its derivative to establish some explicit formulas for . Using the values of and , we establish some lower bounds for which improves the last value of . In particular it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing. To the best of the author knowledge the last value obtained for in the literature is the value obtained by Hall in (1.35) and nothing is known regarding for .
2. Main Results
In this section, we establish some explicit formulas for and by using the same explicit values of and we establish new lower bounds for . The explicit values of using the formula
are calculated in the following for :
The explicit values of the parameter that has been determined by Conrey et al. for are given in the following:
Now, we are in a position to prove our first results in this section which gives an explicit formula of the gaps between the zeros of the Riemann zeta-function. This will be proved by applying an inequality due to Agarwal and Pang .
Assuming the Riemann hypothesis, one has
To prove this theorem, we employ the inequality
Now, we follow the proof of and supposing that is the first zero of not less than and the last zero not greater than . Suppose further that for , we have
and apply the inequality (2.6), to obtain
Since the inequality remains true if we replace by , we have
Summing (2.9) over , applying (1.7), (1.12) and as in , we obtain
This implies that
and then we obtain the desired inequality (2.1). The proof is complete.
Using the values of and and (2.1) we have the new lower values for for in Table 1.
One can easily see that the value of in Table 1 does not improve the lower bound in (1.35) due to Hall, but the the approach that we used is simple and depends only on a well-known Wirtinger-type inequality and the asymptotic formulas of the moments. In the following, we employ a different inequality due to Brnetić and Pečarić and establish a new explicit formula for and then use it to find new lower bounds.
Assuming the Riemann hypothesis, one has
where is defined by
To prove this theorem, we apply the inequality
that has been proved by Brnetić and Pečarić , where is continuous function on with . Proceeding as in the proof of Theorem 2.1 and employing (2.15), we may have
This implies that
which is the desired inequality (2.13). The proof is complete.
To find the new lower bounds for we need the values of for . These values are calculated numerically in Table 2.
Using these values and the values of ,, and the explicit formula (2.13) we have the new lower bounds for in Table 3.
We note from Table 3 that the value of improves the value that has been obtained by Hall.
Finally, in the following we will employ an inequality to Beesack [32, page 59] and establish a new explicit formula for and use it to find new values of its lower bounds.
Assuming the Riemann hypothesis, one has
To prove this theorem, we apply the inequality
that has been proved by Beesack [32, page 59], where is continuous function on with . Proceeding as in Theorem 2.1 by using (2.19), we may have
This implies that
which is the desired inequity (2.18). The proof is complete.
Using these values and the values of ,, and the explicit formula in (2.18) we have the new lower bounds for in Table 4.
We note from Table 4, that the values of for are compatible with the values of for that has been obtained by Hall [13, Table ] and since there is no explicit value of for , to obtain the values of for the author in stopped the estimation for for .
We notice that the calculations can be continued as above just if one knows the explicit values of for where the values
are easy to calculate. Note that the values of that we have used in this paper are adapted from the paper by Conrey et al. . It is clear that the values of are increasing with the increase of and this may help in proving the conjecture of the distance between of the zeros of the Riemann zeta-function.
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The author is very grateful to the anonymous referees for valuable remarks and comments which significantly contributed to the quality of the paper. The author thanks Deanship of Scientific Research and the Research Centre in College of Science in King Saud University for encouragements and supporting this project.
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Saker, S. Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function. J Inequal Appl 2010, 215416 (2011). https://doi.org/10.1155/2010/215416
- Explicit Formula
- Critical Line
- Unitary Matrice
- Random Matrix Theory
- Riemann Hypothesis |
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139289 Precalculus with Trig and Analytical Geometry Solution Key. In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.This contrasts with synthetic geometry.. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. z) =1,2,3.
George Brinton Thomas, Ross L. Finney, Maurice D. Weir. In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. In the (x,y) coordinate system we normally write the x-axis horizontally, with positive numbers to the right of the origin, and the y-axis vertically, with positive numbers above Report abuse. Unit 11 Test Topics. Calculus With Analytic Geometry (Third Edition) by Peterson, Beukman, ed. Search: Calculus Notes Pdf. $48.00.
Not open to students with credit in MAT 291. 0 Reviews. The revolution of analytic geometry was to marry algebra and geometry using axes and co-ordinates.
Introduction to Systems of Equations and Inequalities; 9.1 Systems of Linear Equations: Two Variables; 9.2 Systems of Linear Equations: Three Variables; 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 9.4 Partial Fractions; 9.5 Matrices and Matrix Operations; 9.6 Solving Systems with Gaussian Elimination; 9.7 Solving Systems with Inverses; 9.8 Solving Systems with Cramer's Rule
Later, after much experience in the . In the case of the geometric series, you just need to specify the first term Averages and range Math terminology from Pre-Calculus, Advanced Algebra, Functions, and Analytic Geometry Six key questions covering the whole difficulty spectrum a Foundation Maths GCSE paper 05-06-20 Textbook p 05-06-20 Textbook p. 1 Check up . Prerequisites: Math Placement Level 40, which is earned by any of the following: (a) Level 40 on Math . Share via email.
Course Dates. Precalculus 06 Additional Trigonometric Topics.pdf. Pre-Test & Answer Key. Calculus with Analytic Geometry (Unit # 10) by SM Yusuf & M. Amin (Solution Manual) for Undergraduate Level (B.A / B.Sc / M.A / M.Sc / BS / MS).
Vectors in 2-D. = 1,2.
Algebra II. Study Resources. This course completes your foundational knowledge on algebra, geometry, and trigonometry. D EPED C O PY To the Precalculus Learners The Precalculus course bridges basic mathematics and calculus. Modern geometry is primarily analytic or, at an advanced level, described us-ing algebra such as group theory. The ninth edition of this college-level calculus textbook features end-of-chapter review questions, practice . This contrasts with synthetic geometry.
Oluseyi Moses Ajayi. A Complete Key Book in PDF Format for B.A / B.Sc / M.A / M.Sc / BS / MS.
. Microsoft PowerPoint - Precalculus 07 Analytic Geometry and Conic Sections .pptx Author: rwright Created Date: 2/1/2022 4:16:03 PM . $48.00. It is easy to understand. Written by acclaimed author and mathematician George Simmons, this revision is designed for the calculus course offered in two and four year colleges and universities. Real numbers, limits and continuity, and differential and integral calculus of functions of 1 variable. Limits, derivatives, and graphs of algebraic, trigonometric, exponential, and logarithmic functions; antiderivatives, the definite integral, and the fundamental theorem of calculus, with applications. I need to select 1 of 2 tracks in Math for the first 3-4 semesters. Main Menu; by School; by Literature Title; by Subject; Textbook Solutions Expert Tutors Earn. Addison-Wesley, 1996 - Calculus - 1139 pages.
Choose the equation for the following circle: 4.
Precalculus (6th Edition) answers to Chapter 10 - Analytic Geometry - Quiz - Page 981 1 including work step by step written by community members like you. Vectors in 3-D (just add . This first course in calculus presents analytic geometry and the calculus of algebraic and transcendental functions including the study of limits, derivatives, differentials and an introduction to integration. Designed for the three-semester calculus course for math and science majors, Calculus continues to offer instructors and students new and innovative teaching and learning resources. Calculus with Analytic Geometry presents the essentials of calculus with analytic geometry.
Step 1: Select Your Course Items. Introduction to Analytic Geometry. Yeah, reviewing a book calculus with analytic geometry 3rd edition student supplement volumes 12 covers chapters 1 19 includes answers to every 3rd problem could add your near connections listings. 3.77Mb. The initial approach to each topic is intuitive, numerical, and motivated by examples, with theory kept to a bare minimum.
Analytic Geometry Dianopoulos Precalculus Analytic Geometry: All in One by Cody Dianopoulos Adapted from Jan Gullberg's Mathematics: from the Birth of Numbers Analytic (Coordinate) Geometry, Brief History Analytic Geometry - geometric problems made accessible to algebraic reasoning by connecting points and numbers Apollonius and Archimedes - used longitude and latitude to define a point . Revised and updated edition includes examples and discussions that encourage students to think visually and numerically. Textbook Authors: Lial, Margaret L.; Hornsby, John; Schneider, David I.; Daniels, Callie , ISBN-10: 013421742X, ISBN-13: 978--13421-742-0, Publisher: Pearson |
Gregory Hartman, Virginia Military Institute
Brian Heinold, Mount St. Mary’s University
Troy Siemers, Virginia Military Institute
Dimplekumar Chalishajar, Virginia Military Institute
Jennifer Bowen, The College of Wooster
Copyright Year: 2014
ISBN 13: 9781514225158
Publisher: APEX Calculus
Conditions of Use
I have used this the semester-based courses Calculus 1 and Calculus 2. I will not comment on Calculus 3. The text addresses all of the required areas for the courses, though I would like to see a couple other areas: (1) a section providing... read more
I have used this the semester-based courses Calculus 1 and Calculus 2. I will not comment on Calculus 3. The text addresses all of the required areas for the courses, though I would like to see a couple other areas:
(1) a section providing some of the motivation for the development of Calculus. Some information is provided in a couple paragraphs at the beginning of the Limits Chapter [Chapter 1], but no examples or exercises related to the motivation. In fact, the motivation for limit is not really addressed until Chapter 2. I supplement this on day 1 of my course.
(2) While approximations with differentials is well-addressed, the related "Linear Approximation" topic is not. There is an example of a Linear Approximation in section 2.2 of the text, and several exercises related to it, but it is rarely addressed afterward. This is another topic I supplement.
(3) Applications tend to be only addressed in specific application sections, rather than introduced early and called back upon after learning new rules or techniques.
(4) Most of the conceptual exercises are either true/false, or open ended. I would like to see more conceptual exercises with graphical or numerical data.
The text lacks proofs of most of the stated theorems. I am fine with that, and supplement the proofs that I think are most instructive. My classes see very few Mathematics majors and many Engineering majors, so I do not think that showing the proofs for all of the theorems is necessary, as most of my students will never need the proofs, but will need the appropriate techniques.
I have not found any errors in the writing, nor areas of bias.
I have not found any use of "current" data for examples or exercises, so there should not be any issues with the updating.
Text is readable, using notation appropriate for the level of students that are taking the course.
The terminology of the text is consistent throughout. I have not found any issues.
The text is well structured. Sections always start on new pages. Examples and Theorems have specific "call-outs" that are used consistently throughout the text. There are clear Sub-Headings within sections for sub-topics. Exercise sets always start on new pages.
Very good, overall. Though there are a couple areas that I have concerns with:
(1) The section on Hyperbolic Functions is presented in Chapter 6 with Techniques for Antidifferentiation. The derivatives and antiderivatives are presented in the section, but I think I would have broken that up into two separate topics, and placed the derivative techniques in an earlier chapter.
(2) L'Hopital's Rule is also presented in Chapter 6 with Techniques for Antidifferentiation, immediately before Improper Integration. I think that L'Hopital's Rule is useful for curve sketching, so I prefer to teach it as an application of differentiation.
I have not found any issues with the interface. The text has appropriate bookmarks that are all linked correctly. Graphs show correctly. The large-size version of the PDF has graphs that can be dragged to show different perspectives (useful for Calculus 3).
I have not found any grammatical issues.
There is nothing offensive in the course. But the examples do not show any inclusiveness.
Overall an excellent text, though there are some areas that I tend to supplement. If you are interested in getting into using OER textbooks for a Calculus course. This one would be a good choice to start with.
This a is a comprehensive text that covers all the basic material presented in a standard calculus sequence. It is clearly written, with easy-to-understand explanations. Many formal proofs are omitted, but the theorems and ideas are explained well. read more
This a is a comprehensive text that covers all the basic material presented in a standard calculus sequence. It is clearly written, with easy-to-understand explanations. Many formal proofs are omitted, but the theorems and ideas are explained well.
The text is accurate, error-free, and unbiased.
The content is up-to-date. It is relevant and in no way seems dated. The use of the "classic" position-velocity-acceleration example will remain applicable indefinitely.
The writing is clear and easy to understand. The explanations are concise and to-the-point. The examples are well chosen and show a variety of different situations and problem-solving techniques.
The text is consistent in its use of notation, symbols, terminology, and framework.
The section length is appropriate for the material presented and a typical class period. The text is well organized within each chapter and section.
The organization of the book is standard and presents the material in a logical and clear fashion.
The interface is user-friendly. The table of contents and search features works well. The type-face is easy to read, and the graphs and tables are clear.
The text contains no grammatical errors.
The text is not culturally insensitive or offensive in any way.
Overall, this is a concise, compact, straightforward, understandable, but complete approach to a calculus textbook. It doesn't include some of the "extras" that one would expect from a more traditional textbook such as a pre-calculus review, interesting application problems, larger projects for students to complete, or rigorous proofs of all the theorems. However, all the basic material is included.
The text covers all areas and skills of Calculus I, Calculus II, and Calculus III. This is an excellently written standard Calculus text that includes all ideas and skills of comprehensive college Calculus sequence. The text provides answers to... read more
The text covers all areas and skills of Calculus I, Calculus II, and Calculus III. This is an excellently written standard Calculus text that includes all ideas and skills of comprehensive college Calculus sequence. The text provides answers to exercises and an effective index.
The content is accurate, error-free, very-well written, professionally organized and unbiased.
The content is perfectly up-to-date. This comprehensive college Calculus textbook will not become obsolete. The text is professionally written and arranged in such a way as to make any future updates easy and straightforward to implement.
The text is written clearly and lucidly, with great variety of excellent examples and clear, concise explanations. It provides adequate context for mathematical terminology used in the text.
The text is effectively organized into a standard Calculus sequence. It is professionally written and arranged. The text is internally consistent in terms of terminology and framework.
The text is effectively organized into smaller sections and units that can be assigned at the appropriate points within the course. The text is professionally written, arranged, and properly formatted into easily manageable units well-suited for college students. The text can be easily reorganized and realigned with the requirements of any Calculus course without presenting any disruption to the reader.
The textbook is very effectively organized. The topics in the text are presented clearly and arranged logically.
The text is user-friendly. It is easy to navigate. All text images, graphs, and charts are clear. PDF file is up-to-date and easily navigable. Nothing to distract or confuse the reader..
The text contains no mathematical or grammatical errors.
This is an excellent Calculus textbook. The text is professionally written and edited. It is not culturally insensitive or offensive in any way. This very effective text is aimed at all college students regardless of race, ethnicity, creed, or background.
I have been using this book since 2015 for both lecture and online Calculus classes. My students are from various racial, ethnic, cultural, and religious backgrounds, as well as different countries of origin. Some students use an online version of the textbook while other purchase a printed copy. This book is well-liked by virtually all students. It is clear, excellently organized, and easy to use. Highly recommended.
The text covers material for a first semester course in differential calculus and begins integral calculus with antiderivatives and Riemann sums. The book begins with limits (even the epsilon-delta definition) and continuity before delving into... read more
The text covers material for a first semester course in differential calculus and begins integral calculus with antiderivatives and Riemann sums. The book begins with limits (even the epsilon-delta definition) and continuity before delving into derivatives and their applications (e.g. curve sketching, Newton's method, related rates, and optimization).
Content is accurate, error-free and unbiased.
Content is up-to-date and should remain so indefinitely. The non-mathematical application primarily used are those dealing with position, velocity, and acceleration.
The text has the appropriate amount of prose with adequate mathematically written explanations.
The text is internally consistent in terms of terminology and framework. The text terminology standard to most differential calculus books, such as product rule, quotient rule, and chain rule.
The section lengths within chapters are appropriate for a one hour class meeting, if the students have read and thought through the section beforehand.
The organization of the book is standard: limits and continuity, differentiation rules, curve sketching, applications. The book doesn't take too much time in connecting the ideas of limits and differentiation. Instructors should expect to spend more time explaining this connection.
The book is freely available as a PDF with hyperlinked table of contents. The third volume's (i.e. for multivariable calculus) PDF allows the user to manipulate the graphics.
The text contains no grammatical errors.
The text is not culturally insensitive or offensive in any way.
As the author points out in the preface, the number of exercises at the end of each section is not too large. Students can be expected to complete all of the exercises.
The reader of this text can skip the "epsilon-delta" definition for limits without too much frustration.
The text covers all necessary topics for Calculus I and II. However, no justification or proof for the derivatives of the natural exponential and natural logarithmic functions are provided. They are simply stated among the basic rules for... read more
The text covers all necessary topics for Calculus I and II. However, no justification or proof for the derivatives of the natural exponential and natural logarithmic functions are provided. They are simply stated among the basic rules for differentiation without development. Most topics in Multivariable Calculus are included, with the exception of vector fields. No review of Precalculus topics is included, nor is any historical development of calculus. Conversational discussions of theorems are used in place of formal proofs in many cases throughout the entire text. An index is included, and the search function is accurate.
If the text is to be used for Calculus I and II only, I would rate its overall comprehensiveness at 4-5, based on the comments above. For Multivariable Calculus, I would rate it at 3.
For the most part, the accuracy of the mathematical content is excellent. In a careful reading of the Calculus I material, a few typographical errors and one mathematical error (which might also have been typographical) were found. Since there is no discussion of people in the text, it contains no content that I would construe as biased.
There is no topic in the text which would make the material outdated, though a lack of interactive apps, web links, and computer-generated graphics may make it appear less stimulating than the modern for-profit, online textbook. Updates may be made difficult by the choice of numbering definitions, theorems and key ideas from 1 to n throughout the text, rather than by chapter and number (e.g. Thm. 2.5).
The text is extremely readable for the first-time calculus student. My notes repeatedly include the words "clear", "understandable", and "straightforward". The explanations of the concepts of the limit, the derivative, differentials, integration, sequences, and series are conversational, accurate, and lucid. Applications are well-explained, though in some cases (e.g. the disk and washer methods of determining volumes) more and better graphs and pictures would be appreciated.
Notation and terminology are consistent.
For the most part, chapter sections are divided as expected for a calculus text. Section 6.1 is remarkably long, including not only integration by substitution, but trigonometric integrals as well. Institutions that include only basic u-substitutions in Calculus I and trigonometric integrals in Calculus II will need to divide this section. As noted earlier, the text is quite readable, and the division of chapters into sections, and further into segments of concept development, and examples are appropriate.
The organization of the text is logical and consistent, and its flow is smooth. As previously noted, theorems are not generally justified with formal proofs but with conversational discussions. Topics are ordered appropriately, except, in my opinion, for the presentation of the derivatives of exponential and logarithmic functions without background development.
Navigating by use of the table of contents, bookmarks, or by page number is error-free. Highlighting and "sticky notes" are available. An index is included, but the search tool is quick and easy to use. Graphs are clear. No other pictures or images are included, and no internet links are included (which could become outdated). If students/instructors choose to print a hard copy of the text, there is blank space at the bottom of each page for hand-written notes.
The only grammatical errors found (notably, all in section 2.1) were likely typographical. Sentence structure, choice of words, and punctuation were all very good.
This item is not very applicable to the text, as no mention of culture or ethnicity is made. A check of several examples and application problems that refer to people appear to refer to women as often as they refer to men.
Problem sets are included at the end of each section, and answers to selected problems (most of the odd problems) are found at the end of the text. However, the sets of problems are generally more limited that what is found in a traditional textbook, and answers are sometimes spare. (For example, proofs are "left to the reader".) No review sections or problem sets are included at the end of chapters.
Table of Contents
- Chapter 1: Limits
- Chapter 2: Derivatives
- Chapter 3: The Graphical Behavior of Functions
- Chapter 4: Applications of the Derivative
- Chapter 5: Integration
- Chapter 6: Techniques of Antidifferentiation
- Chapter 7: Applications of Integration
- Chapter 8: Sequences and Series
- Chapter 9: Curves in the Plane
- Chapter 10: Vectors
- Chapter 11: Vector Valued Functions
- Chapter 12: Functions of Several Variables
- Chapter 13: Multiple Integrations
- Chapter 14: Vector Analysis
About the Book
This text comprises a three–text series on Calculus. The first part covers material taught in many “Calc 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6.1. The second text covers material often taught in “Calc 2:” integration and its applications, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 5 through 8. The third text covers topics common in “Calc 3” or “multivariable calc:” parametric equations, polar coordinates, vector–valued functions, and functions of more than one variable, found in Chapters 9 through 14. More information, including free downloads of .pdf versions of the text, is available at www.apexcalculus.com.
About the Contributors
Gregory Hartman, PhD. Author. Associate Professor of Mathematics at Virginia Military Institute, where he has been on faculty since 2005. He earned his PhD in Mathematics from Virginia Tech in 2002.
Brian Heinold, PhD. Contributor. Associate Professor, Mathematics and Computer Science Department, Mount St. Mary's University. Heinold came to Mount St. Mary's in 2006, after receiving his doctorate from Lehigh University. Since then he has taught a variety of math and computer science courses. He has mentored several honors projects, coordinated the department's Smalltalk colloquium series and advised a number of COMAP teams. He has given presentations on fractals and mathematical imagery, teaching and graph theory.
Troy Siemers, PhD. Contributor. Head of the Applied Mathematics program at VMI. He earned his Ph. D. from the University of Virginia and previously led a summer program abroad in Lithuania.
Dimplekumar Chalishajar, PhD. Contributor. Assoicate Professor, Department of Applied Mathematics and Computer Science, Virginia Military Institute.
Jennifer Bowen, PhD. Associate Professor and Department Chair of Mathematics and Computer Science, The College of Wooster. Bowen earned a BA in Mathematics with Honors from Boston College, and both an MS and PhD in Mathematics from The University of Virginia. Bowen teaches a range of courses, including Math in Contemporary Society, Basic Statistics, Calculus I, Calculus II, Multivariate Calculus, Transition to Advanced Mathematics, and Abstract Algebra. |
The interval formed by standard errors give 68% confidence intervals which are not particularly interesting intervals. Psychol. 60:170–180. [PubMed]7. Now suppose we want to know if men's reaction times are different from women's reaction times. The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. useful reference
Would say, "Wow, the treatment is making a big difference compared to the control!" I'm likewise willing to bet most people looking at this (which plots the same averages)... In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the Scenario 1. For the runners, the population mean age is 33.87, and the population standard deviation is 9.27. https://www.graphpad.com/guides/prism/6/statistics/statwhentoplotsdvssem.htm
M (in this case 40.0) is the best estimate of the true mean μ that we would like to know. What can you conclude when standard error bars do not overlap? The standard deviation of all possible sample means of size 16 is the standard error.
For instance, I'm willing to bet most people looking at this... Numerical axes on graphs should go to 0, except for log axes. Whenever you see a figure with very small error bars (such as Fig. 3), you should ask yourself whether the very small variation implied by the error bars is due to How To Interpret Error Bars Fidler, M.
Uniform requirements for manuscripts submitted to biomedical journals. What Do Error Bars Indicate If a “representative” experiment is shown, it should not have error bars or P values, because in such an experiment, n = 1 (Fig. 3 shows what not to do).What type Perspect Clin Res. 3 (3): 113–116. over here On judging the significance of differences by examining the overlap between confidence intervals.
Another way of considering the standard error is as a measure of the precision of the sample mean. Large Error Bars To achieve this, the interval needs to be M ± t(n–1) ×SE, where t(n–1) is a critical value from tables of the t statistic. For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. For those of us who would like to go one step further and play with our Minitab, could I safely assume that the Cognitive daily team is open to share their
Consider a sample of n=16 runners selected at random from the 9,732. This holds in almost any situation you would care about in the real world. #11 James Annan August 1, 2008 "the graph is saying that there's a 95 percent chance that Sem Error Bars Excel i would love to hear from different point of views regarding the title above. What Do Large Error Bars Indicate Wilkins said...
They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg/dL). http://axishost.net/error-bars/error-bars-indicate-sd.php When SE bars overlap, (as in experiment 2) you can be sure the difference between the two means is not statistically significant (P>0.05). I was quite confident that they wouldn't succeed. Carroll, L. 1876. Mean Error Bars
That although the means differ, and this can be detected with a sufficiently large sample size, there is considerable overlap in the data from the two populations.Unlike s.d. This gives 9.27/sqrt(16) = 2.32. In these cases (e.g., n = 3), it is better to show individual data values. this page This reflects the greater confidence you have in your mean value as you make more measurements.
bar can be interpreted as a CI with a confidence level of 67%. What Do Small Error Bars Mean Enzyme activity for MEFs showing mean + SD from duplicate samples from one of three representative experiments. In this case, the temperature of the metal is the independent variable being manipulated by the researcher and the amount of energy absorbed is the dependent variable being recorded.
It is true that if you repeated the experiment many many times, 95% of the intervals so generated would contain the correct value. Incidentally, the CogDaily graphs which elicited the most recent plea for error bars do show a test-retest method, so error bars in that case would be inappropriate at best and misleading This leads to the first rule. Calculating Error Bars Not just bad, or unseemly; ghastly." - RobertSOakes "I want to passive-aggressively run around poster sessions putting up Post-it notes with his url on every poster." - Dominque "Better Posters blog
Moreover, since many journal articles still don't include error bars of any sort, it is often difficult or even impossible for us to do so. This is NOT the same thing as saying that the specific interval plotted has a 95% chance of containing the true mean. For n to be greater than 1, the experiment would have to be performed using separate stock cultures, or separate cell clones of the same type. http://axishost.net/error-bars/error-bars-below-0.php Figure 2: The size and position of confidence intervals depend on the sample.
Cumming, G., F. It is not correct to say that there is a 5% chance the true mean is outside of the error bars we generated from this one sample. |
December 16, 2020 — Stephen Wolfram
Yet Bigger than Ever Before
When we released Version 12.1 in March of this year, I was pleased to be able to say that with its 182 new functions it was the biggest .1 release we’d ever had. But just nine months later, we’ve got an even bigger .1 release! Version 12.2, launching today, has 228 completely new functions!
March 30, 2020 — Brian Van Vertloo, Document Technology Manager, Document & Media Systems
The Wolfram Language is the culmination of decades of effort, supporting all our products. One reason the Wolfram Language is so easy to use is the Wolfram Language & System Documentation Center—unique in that it contains reference information along with tens of thousands of examples that can be edited and run in place (or quickly copied from the web to your notebook).
We recently released Version 12.1 of the Wolfram Language, and with it, a number of new documentation features and page types. With every release, you’ll find an increasing scope of functionality, examples and use cases documented for different fields and applications, presented with an intuitive, user-friendly design.
March 18, 2020 — Stephen Wolfram
We’re pleased that despite the coronavirus pandemic and its impact on so many people and businesses we’re still able to launch today as planned… (Thanks to our dedicated team and the fact that remote working has been part of our company for decades…)
The Biggest .1 Release Ever
It’s always an interesting time. We’re getting ready to wrap up a .1 version—to release the latest fruits of our research and development efforts. “Is it going to be a big release?”, I wonder. Of course, I know we’ve done a lot of work since we released Version 12.0 last April. All those design reviews (many livestreamed). All those new things we’ve built and figured out.
But then we start actually making the list for the new version. And—OMG—it goes on and on. Different teams are delivering on this or that project that started X years ago. A new function is being added for this. There’s some new innovation about that. Etc.
We started this journey a third of a century ago when we began the development of Version 1.0. And after all these years, it’s amazing how the energy of each new release seems to be ever greater.
And as we went on making the list for Version 12.1 we wondered, “Will it actually be our biggest .1 release ever?”. We finally got the answer: “Yes! And by a lot”.
Counting functions isn’t always the best measure, but it’s an indication. And in Version 12.1 there are a total of 182 completely new functions—as well as updates and enhancements to many hundreds more.
September 12, 2019 — Stephen Wolfram
The Next Big Step for Wolfram|Alpha
Wolfram|Alpha has been a huge hit with students. Whether in college or high school, Wolfram|Alpha has become a ubiquitous way for students to get answers. But it’s a one-shot process: a student enters the question they want to ask (say in math) and Wolfram|Alpha gives them the (usually richly contextualized) answer. It’s incredibly useful—especially when coupled with its step-by-step solution capabilities.
But what if one doesn’t want just a one-shot answer? What if one wants to build up (or work through) a whole computation? Well, that’s what we created Mathematica and its whole notebook interface to do. And for more than 30 years that’s how countless inventions and discoveries have been made around the world. It’s also how generations of higher-level students have been taught.
But what about students who aren’t ready to use Mathematica yet? What if we could take the power of Mathematica (and what’s now the Wolfram Language), but combine it with the ease of Wolfram|Alpha?
Well, that’s what we’ve done in Wolfram|Alpha Notebook Edition.
July 11, 2019 — Jacob Wells, Technical Specialist, European Sales
With the recent announcement of the all-new Raspberry Pi 4, we are proud to announce that our latest development, Version 12 of Mathematica and the Wolfram Language, is available for you to use when you get your hands on the Raspberry Pi 4.
Mathematica 12 is a major milestone in our journey that has spanned 30 years, significantly extending the reach of Mathematica and introducing a whole array of new features, including significant expansion of numerical, mathematic and geometric computation, audio and signal processing, text and language processing, machine learning, neural networks and much more. Version 12 gives Mathematica users new levels of power and effectiveness. With thousands of different updates across the system, and 278 new functions in 103 areas, there is so much to explore.
June 11, 2019 — Stephen Wolfram
What the Wolfram Language Makes Possible
We’re on an exciting path these days with the Wolfram Language. Just three weeks ago we launched the Free Wolfram Engine for Developers to help people integrate the Wolfram Language into large-scale software projects. Now, today, we’re launching the Wolfram Function Repository to provide an organized platform for functions that are built to extend the Wolfram Language—and we’re opening up the Function Repository for anyone to contribute.
The Wolfram Function Repository is something that’s made possible by the unique nature of the Wolfram Language as not just a programming language, but a full-scale computational language. In a traditional programming language, adding significant new functionality typically involves building whole libraries, which may or may not work together. But in the Wolfram Language, there’s so much already built into the language that it’s possible to add significant functionality just by introducing individual new functions—which can immediately integrate into the coherent design of the whole language.
To get it started, we’ve already got 532 functions in the Wolfram Function Repository, in 26 categories:
April 16, 2019 — Stephen Wolfram
Today we’re releasing Version 12 of Wolfram Language (and Mathematica) on desktop platforms, and in the Wolfram Cloud. We released Version 11.0 in August 2016, 11.1 in March 2017, 11.2 in September 2017 and 11.3 in March 2018. It’s a big jump from Version 11.3 to Version 12.0. Altogether there are 278 completely new functions, in perhaps 103 areas, together with thousands of different updates across the system:
April 9, 2019 — Alec Titterton, CBM Content Development Manager, European Sales
Wolfram Research is pleased to announce further collaboration with the Raspberry Pi Foundation as part of supporting makers across the world through education. A collection of 10 Wolfram Language projects has been launched on the foundation’s projects site. These projects range from creating weather dashboards to building machine learning classifiers to using AI for facial recognition. The goal is to put the power of computational intelligence into the hands of anyone who wants access—democratizing the skills that will increasingly be needed to innovate and discover what is possible with modern computation.
By providing easy-to-follow, step-by-step tutorials that result in a finished, functioning piece of software, Wolfram aims to lower the barrier of entry for those who wish to get immediately started programming, building and making. Projects can be completely built on the Raspberry Pi or within a web browser in the Wolfram Cloud.
June 21, 2018 — Stephen Wolfram
Technology for the Long Term
On June 23 we celebrate the 30th anniversary of the launch of Mathematica. Most software from 30 years ago is now long gone. But not Mathematica. In fact, it feels in many ways like even after 30 years, we’re really just getting started. Our mission has always been a big one: to make the world as computable as possible, and to add a layer of computational intelligence to everything.
Our first big application area was math (hence the name “Mathematica”). And we’ve kept pushing the frontiers of what’s possible with math. But over the past 30 years, we’ve been able to build on the framework that we defined in Mathematica 1.0 to create the whole edifice of computational capabilities that we now call the Wolfram Language—and that corresponds to Mathematica as it is today.
From when I first began to design Mathematica, my goal was to create a system that would stand the test of time, and would provide the foundation to fill out my vision for the future of computation. It’s exciting to see how well it’s all worked out. My original core concepts of language design continue to infuse everything we do. And over the years we’ve been able to just keep building and building on what’s already there, to create a taller and taller tower of carefully integrated capabilities.
It’s fun today to launch Mathematica 1.0 on an old computer, and compare it with today:
April 19, 2018 — Joanna Crown, Strategic Projects, Strategic Initiatives
“Tell me and I forget. Teach me and I remember. Involve me and I learn.” — Benjamin Franklin
I can count on one hand the best presentations I have ever experienced, the most recent being my university dynamics lecturer bringing out his electric guitar at the end of term to demonstrate sound waves; a pharmaceutical CEO giving an impassioned after-dinner oration about how his love of music influenced his business decisions; and last but not least, my award-winning attempt at explaining quantum entanglement using a marble run and a cardboard box (I won a bottle of wine).
It’s perhaps equally easy to recall all the worst presentations I’ve experienced as well—for example, too many PowerPoint presentations crammed full of more bullet points than a shooting target; infinitesimally small text that only Superman’s telescopic vision could handle; presenters intent on slowly reading every word that they’ve squeezed onto a screen and thoroughly missing the point of a presentation: that of succinctly communicating interesting ideas to an audience. |
Subsets and Splits