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I. Introduction to Lottery Filters and Filtering in Software II. Setting the Lottery Filters: Theory and Software Tools III. More Advanced Theories in Setting Filters and Lottery Strategies IV. Older Writings in Setting Filters; Lottery Software Users Creating Strategies V. Resources in Lotto, Lottery, Software, Filters, Strategies, Systems The lottery filters are parameters that eliminate combinations in the generating processes by the LotWon lotto and lottery software. In other words, the filters are restrictions. One simple restriction, as an example: Generate lotto combinations that do not repeat any past drawing that hit the jackpot. Ironically, mostly the minimum levels of the lottery filters are perceived as... filters! Actually, the filters in my lotto software had only minimum levels for years. That's how it started. It took me many years to realize that there is no minimum without maximum. Reason: The personal computers in the 1980s were not advanced. And another reason: I evolved as a programmer. Both levels of a filter are equally efficient. There are no universal formulas to calculate the efficiency of lottery filters. By efficiency, we mean the amount of lottery combinations that a filter eliminates. Let's say the lotto game consists of 6 winning numbers per drawing. We can calculate exactly, by formulas, how many combinations the filter ONE would eliminate and the filter SIX — But only if we set each level to 1. That is, the calculations can be performed only if we consider one past drawing as a filter setting. That's because groups of lottery numbers do repeat — also according to probability theory. The minimum level of filter ONE (the name is just intuitive) eliminates all single-number groups from past lotto drawings. If we set the minimum level of the ONE filter to 1, it will eliminate EACH lotto number from the previous draw. There are 6 numbers per combination. Thus, we will eliminate 6 numbers from the field of play (e.g. 49 numbers in the game). Only numbers from the remaining 49 – 6 = 43 lotto numbers will be part of the combinations generated. Therefore, the minimum level of ONE will allow an amount of combinations equal to the formula of combinations 43 taken 6 at a time. C(43, 6) = 6,096,454 combinations. Total number of lotto combinations C(49, 6) = 13,983,816. In other words, the minimum level of ONE will eliminate 7,887,362 combinations. The maximum level of ONE does the opposite. If the maximum level of ONE is set to 1, it eliminates 6,096,454 combinations. Equivalently, it allows 7,887,362 lotto 6/49 combinations. The other filter we can apply formulas to is a lotto parameter we could call SIX. It will apply only the complete 6-number groups as a restriction. We set the minimum level of SIX to 1 (the previous drawing only). There is only 1 possible 6-number group to match the past draw. This minimum level of SIX will eliminate exactly one lotto combination. The opposite: The maximum level of SIX will allow exactly one combination (i.e. it eliminates the rest of the combinations in the lotto game). There are no formulas for other types of restrictions or lottery filters. The only way to determine the efficiency of a lottery filter is by running the software in lexicographical generating mode (L in the main menu of Bright / Ultimate Software). Set the minimum level of a particular filter to 1. Write down how many lottery combinations were generated. Set the maximum level of a particular filter to 1. The result should be the complementary of the amount of combinations generated by the minimum level. That is, if you add the amounts of lottery combinations generated by the two levels of the same filter, the result should be equal to total number of sets in the lottery game. You might want to save to a document the efficiency figures for every filter in your particular software packages. Print the document as a good reference to aid you with my lotto and lottery software. Lotto filters such as Four, FivS, FivR have a special behavior. They are so high, that they reach levels beyond a small data file. If you analyze 10,000 draws, FivS in layer 1 shows 10,000. Most likely the filter is higher than that. It can reach over 20,000, I know it for a fact. In my lotto 6/69 game, both FivS, and FivR reached 100,000 easily. That's why the FivS, and FivR filters do NOT have maximum levels! The Four filter has a maximum level, but it requires caution. If the filter shows 10,000 in your WS6.1 report, it simply means there weren't enough draws to analyze. You can use the WS files and the QEdit editor (it does NOT work in 64-bit Windows). QEdit has some nice features I did not bother to implement in my editors. Why should I reinvent the ... lotto wheel every time?! One nice feature of QEdit is column block or column selecting. Read the manual for the shortcut, it probably is Alt+k. Go to the first drawing in the WS file. Move the cursor to the first digit of a filter (column). If a lotto software filter has four digits, make sure the cursor is four spaces from the rightmost digit. The column block must cover all the digits in the filter. Press simultaneously Alt+K then press the right arrow key until you reach the last digit in the column. Press the down arrow key until you reach the last line in the WS file. Here is a screenshot of the folder organization: There are lots and lots of reports!! The more reports the higher the chance to come up with a large number of efficient strategies. It took me a couple of hours to generate the winning reports, sort all of them on all filters and finally move them to the StratSorRep subfolder for each of the Ultimate Software packages. I had a couple of glasses of wine to make the tedium less "oppressive"! Virtually, the sorted reports will be good forever. The sorted reports show the lowest filter figures at the top, while the largest column values go to the bottom. The report is ordered by the Any_5 filter. I had once good success (when I played the lottery in earnest!) I set Any_5_minimum = 400. Sometimes, the strategy would not generate a single combonation. I mixed and matched Any strategies from various layers in case one strategy was in a "slump". The strategies go "cold" at times, and get "hot" at other times. But them all lottery strategies have skip medians that are mathematically consistent. So, when I say wacky filter values I mean the top and the bottom of a sorted winning report for the key column. Of course, we always should set other filter values at runtime. We can see here that when Any_5_minimum = 400, other filters are usually non-zero (Tot, Rev, Pr, Syn, Sum). We can play aggressively and set Tot = 1000 as it occurs 14 times with Any = 400. And, certainly, we can also combine filters from different layers and from different programs. We can also work with filter values in the median area of the sorted winning reports WS. In this third type of lottery strategy, we can set tight filter settings. In this particular example, we can see in the reporty heading that the median is equal to 59. We can select Any_5_minimum = 59 AND Any_5_MAXimum = 60. This setting might be so tight that no combination will be generated sometimes. We can loosen it up with a setting like: Any_5_minimum = 58 AND Any_5_MAXimum = 62 For example, if a median is 12, you can set a tight lottery filter to 12 * 4 = 48 (or rounded up to 50) for the minimum value of the filter. Or, you can set an equally tight filter to 12 / 4 = 3+1 for the minimum value of the respective filter. (Remember, the maximum level of a filter must be at least (minimum level + 1). If a filter is set to 4 times the median, it slashes in half four times the total combinations. In the pick-3 lottery game example: 1000 lottery combinations reduced to 500 in the first step; 500 slashed to 250; 250 halved to 125; finally, 125 reduced to 60+. A sorted-by-column WS file can show you even more valuable information. Say, you sorted W3.1 by the Pairs-1 column. The median was 32. The median divided by 4 = 8. Go to line 1 of the column and see how many Pairs-1 are lower than 8. You can see also what kind of levels other filters show for Pairs-1 less than 8. Other filters may show very low numbers as well. Other lotto filters may show bigger numbers. You can choose as a playing strategy Max_Pair_1=8+1=9, plus other filters at less tight levels. For example, Max_Vr_1=4, Max_TV_1=6, Val_1=5. This is just an example. You can find similar numbers in your sorted WS files. The median multiplied by 4 = 128. Go to the last line of the column and see how many Pairs-1 are larger than 128 (or 120 or 130; you can round up or down for more flexibility in your choices). You can see also what kind of levels other filters show for Pairs-1 greater than 128. Other filters may show very high numbers as well. Other filters may show lower numbers. You can choose as a playing lottery strategy Min_Pair_1=130, plus other filters at less tight levels. For example, Min _Vr_1=1, Min _TV_1=5, Min_Syn_1=50. Using such tight levels for one or very few lotto filters eliminates a huge amount of lotto combinations. Such levels occur more rarely. You should not play them in every drawing. They skip a number of drawings between hits. The newest pick-3 software makes it even easier for you. The application has also a strategy checking software utility. It shows the levels of all the filters and the skip chart of the strategy. Please read also the Pick-3 Lottery Strategy, System, Method, Play, Pairs page. It shows why you should only play a strategy if its current skip (the first number in the skip chart) is less than or equal to the median. For example, if the median of the strategy is 5, you should play it only if the first number in the string of skips is 0, or 1, or 2, or 3, or 4, or 5. If the current skip is larger, don't play the strategy; save the money. Since you can select a very, very large number of lottery strategies, look for another strategy. Look for a strategy that shows a current skip under the median. Again, the pick-3 lottery software makes it easier for you. It shows more evidently the filter movement. When a filter is higher than in the previous drawing, the filter has the + sign at the right. If the filter is lower than in the previous draw, it has a - sign attached. It is more visible. You can notice that in most cases the filters go from one trend to the opposite after two or three drawings. That is, after 2 or 3 + signs, the - sign comes up; or vice versa. Based on that, we can look at each filter (column) in the WS files. The key position is line #1. If the sign in line #1 is -, and also in line #2, and line #3, (3 decreases in a row), we should expect a + (increase) the very next draw. If the sign in line #1 is +, and also in line #2, and line #3, (3 increases in a row), we should expect a - (decrease) the very next draw. Let's take pick-3 lottery as an example. Pair-1 in line #1 is 12 and it shows -, the 3rd consecutive - (decrease). We should expect a + in the very next drawing. An increase in a filter requires the use of the minimum level of the respective filter. In this example, I'll set Min_Pair_1=13. If I want to increase the probability, I can set Min_Pair_1=10, for example. Of course, the program will generate a larger amount of pick-3 straight sets. Let's say now Pair-1 in line #1 is 123 and it shows +, the 3rd consecutive +. We should expect a - in the very next drawing. A decrease in a lottery filter requires the use of the maximum level of the respective filter. In this example, I'll set Max_Pair_1 = 124. If I want to increase the probability, I can set Max_Pair_1 = 130, for example. Of course, the program will generate a larger amount of pick3 lottery sets. You can look for longer streaks, of either + or -. Just go the line #1 in each WS file. There are situations when the current streak can be 4-long, or 5-long, even longer in rare situations. You may want to consider first the longer like-sign streaks. Keep in mind, however, that the streaks shift direction after up to 3 drawings in most cases. Actually, streaks of 1 or 2 consecutive like-signs are the most frequent. I will not go any further in this direction. You can combine filters selected as in this path with the type of selection presented in path #1. You can set one tight filter (4 times the median, etc.). Then you set other filters as in path #2. For example, Min_Pair_1=120 (path #1), Max_Vr_1=7 (path #2), Min_TV_1=10 (path #1), Min_Syn_1=100 (path #1), Max_Bun_2=6 (path #2), Max_Tot_3=1500 (path #2), Max_Any_5=300 (path #2). And so on… 1) the range of analysis is too short (100 lottery drawings or so); 2) a very low number of past lottery draws to use (way below 100,000). And one more reason is an insufficient number of layers. The number of layers can be increased only by increasing the number of D* files. There is that much-talked-about ION5 filter, in both MDIEditor and Lotto and in the LotWon (Command Prompt). The filter is constructed absolutely the same way. That lotto filter can reach the skies, literally. It can grow huge, if the D* file goes above 100,000 combinations. For a 6/49 lotto game, ION5 can reach TOTAL_DRAWS/24. (BTW: 24 = INT(49/2)). If no Ion_5 is above 4,167 in a 100,000-line D6 (MDIEditor file) or a layer in DOS Pick632, then that data file of yours is probably of a sufficient size. For a 5/39 game, ION5 can reach TOTAL_DRAWS/20. If no Ion_5 is above 5000 in a 100,000-line D5 (MDIEditor file) or a layer in Bright5, then your D* file is probably of a sufficient size. Yet, the wackiness can go above those figures. I exemplify by my lotto 5/39 game. I generated all the lotto combinations in the game (575,757) by running that great piece of combinatorial software PermuteCombine. Next I shuffled the all-5-39 combo file by running that great application Shuffle. I shuffled once and thus created my first SIM-5 file. I shuffled the all-5-39 lotto file a few more times, creating every time a different SIM-5 file. I have seen Ion_5 of 2500+, 5000+, 6000+, 7000+… Those are scary numbers… scary not to you, but to the odds! I did a few tests with min_Ion_5 = 2500. Well, for a majority of the cases: there is no output; no combinations generated, even if the inner filters are disabled. I checked for a situation when ION5 was over 2500. I generated lotto combinations in lexicographical order. Total combinations generated: 1 (ONE), both with the innate filters enabled and disabled. You know, I like that! Had I figured it out five years ago, I would have hit the 5/39 lotto jackpot a dozen times or so. Granted, my PC of 5 years ago was kinduva snail (300 MZ PII). So, now I take a look at wacky-wacky lotto filters, such as the storied Ion-5. If it isn't over 2500 in a layer, look at another layer… or look at another D* lottery data file. Wackiness is not the exclusivity of one layer or one drawings file. Wackiness repeats itself the same way that history repeats itself. Indeed, there is the need to present more facts on the filters in LotWon software. The documentation in SuperPower lottery software does not cover how to use the maximum values of the filters. The minimum value of a lottery filter is equivalent to at least. If we choose Two-1 = 3, we'll be correct if Two-1 was at least 3 (shown in the W6 files). That means, 3, 4, 5, ... 10, 11, etc. We would be losing if Two-1 was 2, or 1, or 0. The maximum value of a filter is equivalent to no more than but not equal to. If we choose Max_Two-1 = 3, we'll be correct if Two-1 was no more than 2. Pay special attention to this: the maximum must be at least the minimum plus 1. We would be correct if Two-1 was 2, or 1, or 0 (shown in the W6 files). If Two-1 was 3 or more, that would be a losing situation. If the W6 files shows 0 for Two-1 we can only play the maximum value for the lottery filters. Playing the minimum value will have no effect (since it is equal to 0). In this case, the correct entry for Max-Two-1 is 1 (0+1=1). The effect of a filter depends on the lotto game format. I will exemplify the effect of some filters for the lotto 6/49 game. It is the most popular around the globe and I received a real-life W6 lotto report (from the user named Guy). I will not accept any W6 files to look at any more. 1) Let's see how many lotto combinations the minimum value of the filter Two eliminates if we set it to 1. A 6-number winning combination can be broken down into C6 taken 2 at a time C(6, 2) = 15 combinations. There are 49-6=43 remaining lotto numbers. The 43 remaining numbers can be broken down into C43 taken 4 at a time = 123410 combinations of 4 lotto numbers each. Each of the 15 2-number lotto combinations can be attached to each of the 123410 4-number combinations. The result is 15 x 123410 = 1851150 total 6-number lotto combinations. Therefore, the filter Two = 1 eliminates 1,851,150 combinations. Let's make Two = 2. It means we eliminate all two-number combinations from the last two drawings. If the two past lottery drawings have no common numbers, the filter Two = 2 eliminates 1,851,150 x 2 combinations. If Two = 3, the filter should eliminate 1,851,150 x 3 combinations. And so on? NOT! In reality, some pairings (two-number lotto groups) are a lot more frequent than others. Some lotto pairings do not come out even within 200 drawings. So, the effect of Two diminishes after two or three past lottery drawings. I also recommend the use of the Least 6 file: the file with the least frequent pairings in a lotto game. If the filter Three is set to 1, it eliminates all three-number groups from the most recent lotto drawing (the previous one). C(6 taken 3 at a time) = 20 combinations. The 43 remaining numbers can be broken down into C(43 taken 3 at a time) = 12341 combinations of 3 numbers each. Each of the 20 3-number lotto combinations can be attached to each of the 12341 3-number combinations. The result is 20 x 12341 = 246820 total 6-number lotto combinations. Therefore, the filter Three = 1 eliminates 246,820 combinations. Let's make Three = 2. It means we eliminate all three-number combinations from the last two drawings. If the two past drawings have no common numbers, the filter Three = 2 eliminates 246,820 x 2 lotto combinations. If Three = 3, the filter should eliminate 246,820 x 3 combinations. And so on? NOT! In reality, some three-number combinations are a lot more frequently than others. The Three filter will diminish its eliminating power after some 10 past lottery drawings. 2) Let's see how many combinations the maximum value of the filter Two LEAVES TO BE PLAYED if we set it to 1. A 6-number winning combinations can be broken down into C(6 taken 2 at a time) = 15 combinations. There are 49-6=43 remaining numbers. The 43 remaining lotto numbers can be broken down into C(43 taken 4 at a time) = 123410 combinations of 4 numbers each. Each of the 15 2-number combinations can be attached to each of the 123410 4-number combinations. The result is 15 x 123410 = 1851150 total 6-number combinations. Therefore, the filter MAX_Two = 1 LEAVES 1,851,150 combinations to be played. In other words, the lotto software will generate 1,851,150 combinations. Let's make MAX_Two = 2. The calculation is different now. Suppose the last two lotto drawings have no common numbers. Thus, the last two drawings consist of 12 unique numbers. C(12 taken 2 at a time) = 66 combinations. There are 49-12=37 remaining numbers. The 37 remaining numbers can be broken down into C(37 taken 4 at a time) = 66045 combinations of 4 numbers each. Each of the 66 2-number combinations can be attached to each of the 66045 4-number combinations. The result is 66 x 66045 = 4358970 total 6-number lotto combinations. Therefore, the filter MAX_Two = 2 LEAVES 4,358,970 combinations to be played. But the disclaimer above still applies and even widens the discrepancy. If the filter Three is set to 1, it LEAVES TO BE PLAYED all three number combinations from the most recent lotto drawing (the previous one). C(6 taken 3 at a time) = 20 combinations. The 43 remaining numbers can be broken down into C(43 taken 3 at a time) = 12341 combinations of 3 numbers each. Each of the 20 3-number combinations can be attached to each of the 12341 3-number combinations. The result is 20 x 12341 = 246820 total 6-number lotto combinations. Therefore, the filter Three = 1 LEAVES TO BE PLAYED 246,820 combinations. But beware of the phenomenon described for the case of Two. Looking at the W6 files from Guy, I saw Two of 20 or more. That's a high value for the minimum value of the lottery software filters in this category (Two or 2-#s). Such a high value has corresponding high values for other filters (Three or 3-#s and Four). Using such high values for some filters will eliminate a huge number of lotto combinations. In the same reports, I saw Three filters of 150 or more. Such high values for the minimum entry of Three also eliminate a huge amount of lotto combinations. Also, Four of 2000 or more (even 4000) and Sum of 400 or more (even 700) do eliminate millions of combinations. You will notice that high values for one filter are correlated with high values of other filters. But these high values occur from time to time, not very frequently. You will skip some lotto drawings in between situations of high values for the 'minimum' level of the filters. As in Guy's reports, in drawing #20, the filter Sum-1 was 828, then in drawing #13 it was 469. So, we played once setting Sum-1 = 400 for the minimum level, then skipped 5 lotto drawings and played again Sum-1 = 400. We would have lost 2 drawings, but had another winning situation in drawing 13. In reverse, efficient maximum levels of the lotto filters are LOWER ones. For the Two filters, 0 is the most efficient value and it occurs quite frequently (more than 15 times in 100 drawings analyzed by Guy). In such cases, you set MAX_Two-1 = 1. There are also situations are 0. For example, you set MAX_Two-1 = MAX_Two-3 = 1. The effect is quite dramatic: a substantially lower number of combinations will be generated. Or even more dramatically, you can set all six Two filters to no higher than 5. Thus, MAX_Two-1 = MAX_Two-2 = MAX_Two-3 = MAX_Two-4 = MAX_Two-5” = MAX_Two-6 = 6. For the filter Three, an efficient level of the maximum would be 5. For the filter Four, you can set the minimum = 50 and MAX_Four = 100. From all these facts, you can deduce that there are values UNUSUALLY high or UNUSUALLY low that occur from time to time in your W6 files. Therefore, we can set highly efficient filters and expect to win with substantially fewer lotto combinations from time to time. If you want to generate the winning lotto combinations every time you run WHEEL-6, you need to set SAFE values for the filters. The selection of safe values is not 100% guaranteed, but it is not rocket science either. You should not expect to select the right levels of the filters every time you run the programs. You can notice in your W6 files that usually high values are followed by lower values (or vice versa). Normally, three increases in a row are followed by a decrease (or vice versa: three decreases followed by an increase). For example, in Kulai's report, Three-1 was, in three consecutive lottery drawings, 191, 29, 12, followed by 31. I would have set Three-1 = 13 and MAX_Three-1 = 51 (an increase from 12 but no higher than 50). Your accuracy in setting this type of safe levels of the lottery filters will increase with usage: the more you work with the W6 files, the less erroneous your filter-setting will get. Resources in Lottery Software, Lotto Wheeling, Systems, Strategies See a comprehensive directory of the pages and materials on the subject of lottery, lotto, software, systems, lotto wheels. 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Today, we had a set of problems as usual, and a quiz! (And I didn’t tell you about the quiz, even though others did, so I’m going to pretend that it was a pop quiz)!. Below, you’ll find the three problems, their solutions, and a worked-out quiz. Be proud! I didn’t mention a fold “before the fold.” We had three questions in recitation. - A function is continuous on and differentiable on . We happen to know that . - What can you conclude about the function from the Mean Value Theorem? - Use the Mean Value Theorem to show that . - Consider the following function . - Without using derivatives, determine where is increasing or decreasing. - Check by taking derivatives. - Identify the local maxes and mins of the function . - Consider the function . - Show that are the critical points of . - What are the local minima and maxima of ? - Compute for each of the critical points of . - Do you notice a pattern between the second derivatives at the minima and maxima? I’ll consider the quiz after these problems. This question is designed around the mean-value theorem. The mean-value theorem states that if you have a function that is continuous on an interval and differentiable on . Then there is a in the interval such that . In other words, the “average slope” gets hit by the derivative. So in this problem, from the mean value theorem, we know that for some in . We know in addition that always. So in fact, the mean value theorem tells us that , or that . Rewriting this, we get that , which is exactly what we were trying to show. Without using derivatives! We’ll consider each function-piece in turn. First, we have on . The thing that changes with here is , and the size of depends only on the magnitude of : bigger input magnitude yields bigger output magnitude. So as is increasing from to , it is *decreasing* in magnitude, and thus is decreasing. In a different way, we know that is a parabola with vertex above . It’s decreasing for all negative numbers and increasing on all positive numbers. So we have one part down. Next, we have on . Our previous thought process doesn’t quite work now. will take negative numbers to more negative numbers, and the larger the input (in magnitude), the larger the output. So as is increasing from to , it is decreasing in magnitude. This means that will be *increasing*, as the negative numbers coming out will also be decreasing in magnitude. Thus $x^3$ is increasing here. In a different way, we know the graph of , and it’s always increasing. behaves a lot like a parabola. Now we’re in positive numbers, which match our intuition a lot better than negative numbers. As the magnitude of increases, so does the magnitude of $x^4$. So this function is also *increasing.* Finally, we have a line. Thank goodness, a line! This line has negative slope, so it’s decreasing always. Let’s check using derivatives. The derivative of is , and we’re only looking at negative $x$. Thus is always negative, and so the original function is decreasing. The derivative of is . This is always nonnegative, so the original function is increasing. The derivative of is . We’re now on positive , so is always positive, and so the original function is increasing. Finally, the derivative of our line is , which is negative, and so the original function is decreasing. So we were right above – good. When we are trying to identify the local maxima and minima, it’s tempting to just try to use (what you’re about to learn:) the first derivative test: finding the derivatives and setting equal to zero. But that won’t work here. Notice that all the derivative we found are zero only at . But our function isn’t differentiable everywhere. Instead, we should use the fact that we now know when the function is increasing and decreasing. A local maximum will occur when the function increases, and then decreases. A local minimum will occur when the function decreases, then increases.We know that the function is decreasing until , after which the function is increasing. So there is a local min at . The function goes from increasing to decreasing at too. So we know that will be a local maximum. We have found all the times when the function is changing from increasing to decreasing, or from decreasing to increasing, so we’re done. As an aside: This is really the intuition behind the first derivative test, too. The idea of the first derivative test is that maxes and mins will occur when the derivative is zero or on the boundary, so long as our function is differentiable. Why is this the case? If our function is differentiable, then the slope of the function changes from positive to negative (i.e. the function changes from increasing to decreasing) when the derivative is zero. But here, our function isn’t differentiable everywhere, so we need to be a bit wittier. So we are now looking at . Let’s differentiate: we find that . If we plug in , we get zero. Yay!. In fact, we have that . These are the places where the derivative is zero, so we expect them to be our max and min candidates. By either plugging in numbers or looking at when the function is increasing/decreasing, or by realizing that this is a positive quintic with 4 extrema (if this makes sense – awesome; if not – don’t worry), we have maxes at and mins at . The second derivative is . The idea behind the second half of this problem is to motivate the second derivative test (which you’ll learn shortly). It just so happens that when you compute the second derivative at these four points, it’s negative at the two maxima and positive at the two minima. This might lead you to make the conjecture that at local maxima, the second derivative is always negative; and at local minima, the second derivative is always positive. But this would be wrong. The converse, however, is true. If the second derivative is negative at a critical point, then that point is a local maximum. If it’s positive at a critical point, then that point is a local minimum. Why is this true? The second derivative is the first derivative of the first derivative. This means that when the second derivative is positive, the first derivative is increasing. And when it’s negative, the first derivative is decreasing. So at a critical point, i.e. when the first derivative is zero, if the second derivative is positive, this means that the first derivative is increasing. Since the first derivative is zero and increasing, this means that it was just negative and is about to be positive. But this means that the original function was decreasing and then increasing, i.e. that there is a local min. So when the second derivative is positive at a critical point, we get a local min. This type of reasoning works for when the second derivative is negative at a critical point too. I encourage you to try it! I’ll post up the quiz and its solutions shortly, but in a separate post. This one is very long as is, I think. I’d like to finish by reminding you all that I will not be in MRC next week. In addition, something was left in the classroom. If you’re missing something and you think you left it, I am holding on to it – so let me know. (I’ll give it to some lost and found office if I don’t hear anything).

Table of Contents - 1 What can you conclude about the relationship between observations and a hypothesis? - 2 How does an IF THEN statement make it easier to determine whether a prediction is true? - 3 What is relationship between observation and explanation? - 4 How is observation important in the development of theory? - 5 What factors must Researchers consider when designing a research study to ensure that they get accurate and reproducible results? - 6 What is an important part of creating a hypothesis? - 7 What are the characteristics of a good hypothesis? - 8 How is a hypothesis used in psychological research? What can you conclude about the relationship between observations and a hypothesis? observation is what you have gathered, after you look over it you my hypothesis of what you gathered. hypothesis is an explaination of the observation. How does an IF THEN statement make it easier to determine whether a prediction is true? How does an if-then statement make it easier to determine whether a prediction is true? The statement makes it easier to determine whether your prediction is true or not. Why do scientist test a hypothesis? You must find out if it is a reasonable answer to your question. What initial observation led the researchers to form a hypothesis? What initial observation led the researchers to form a hypothesis? Seeds warmed by sunlight had higher germination rates. What step is taken immediately before making the hypothesis? How Do You Form a Hypothesis? The first step is to collect as many observations as possible about the problem you are trying to examine. Then consider your observations and think about how they might relate to the problem. Try to imagine possible solutions to explain your observations. What is relationship between observation and explanation? An observation is any report from your 5 senses. It does not involve an explanation. Such an observation is a quantitative one, as opposed to a qualitative one (no measurements). On the other hand, an interpretation is an attempt to figure out what has been observed. How is observation important in the development of theory? Observation is essential in science. Scientists use observation to collect and record data, which enables them to construct and then test hypotheses and theories. These tools allow for more precise and accurate observations. Why is a conditional with a false hypothesis always true? Therefore, the conditional must be true. the hypothesis does not hold. Therefore, the conditional is true. Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why when the hypothesis is false, the conditional statement is always true. What is the truth value of the conditional statement when the hypothesis is false and the conclusion is true? In the truth table above, p q is only false when the hypothesis (p) is true and the conclusion (q) is false; otherwise it is true. Note that a conditional is a compound statement. Now that we have defined a conditional, we can apply it to Example 1….Definition: A Conditional Statement is… What factors must Researchers consider when designing a research study to ensure that they get accurate and reproducible results? Researchers should clearly report key experimental parameters, such as whether experiments were blinded, which standards and instruments were used, how many replicates were made, how the results were interpreted, how the statistical analysis was performed, how the randomization was done, and what criteria were used to … What is an important part of creating a hypothesis? In order to form a hypothesis, you should take these steps: Collect as many observations about a topic or problem as you can. Evaluate these observations and look for possible causes of the problem. Create a list of possible explanations that you might want to explore. Why is forming a hypothesis an important step in the scientific method? Why is forming a hypothesis an important step in the scientific method? Cannot be scientifically validated. By stating a formal hypothesis, a scientist can adequately design the best control conditions for designing experiments intended to falsify the hypothesis. How do observations relate to hypothesis? Observations leading to a hypothesis come from prior knowledge or research. Observations leading to a hypothesis come from the same experiment. A hypothesis is formed based on observations from many experiments. A hypothesis is formed after making observations but before performing an experiment. What are the characteristics of a good hypothesis? There are three general characteristics of a good hypothesis. First, a good hypothesis must be testable and falsifiable . We must be able to test the hypothesis using the methods of science and if you’ll recall Popper’s falsifiability criterion, it must be possible to gather evidence that will disconfirm the hypothesis if it is indeed false. How is a hypothesis used in psychological research? A hypothesis is more specific and makes a prediction about the outcome of a particular study. Working with theories is not “icing on the cake.” It is a basic ingredient of psychological research. Like other scientists, psychologists use the hypothetico-deductive method. Which is an example of a statistical hypothesis? The main goal in many research studies is to check whether the data collected support certain statements or predictions. Statistical Hypothesis –a conjecture about a population parameter. This conjecture may or may not be true. Example: The mean income for a resident of Denver is equal to the mean income for a resident of Seattle. What are the steps in a hypothesis test? Hypothesis Test Procedure (Traditional Method) Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value(s) from the appropriate table. Step 3 Compute the test value.

« ForrigeFortsett » OF RIGHT-ANGLED SPHERICAL TRIANGLES BY 16. It is to be observed, that when any part of a tri. angle becomes known by means of its sine only, there may be two values for this part, and consequently two triangles that will satisfy the question; because, the same sine which corresponds to an angle or an arc, corresponds likewise to its supplement. This will not take place, when the unknown quantity is determined by means of its cosine, its tangent, or cotangent. In all these cases, the sign will enable us to decide whether the part in question is less or greater than 90°; the part is less than 90°, if its cosine, tangent, or cotangent, has the sign +; it is greater if one of these quantities has the sign In order to discover the species of the required part of the triangle, we shall annex the minus sign to the logarithms of all the elements whose cosines, tangents, or cotangents, are negative. Then, by recollecting that the product of the two extremes has the same sign as that of the means, we can at once determine the sign which is to be given to the required element, and then its species will be known. It has already been observed, that the tables are calculated to the radius R, whose logarithm is 10 (Plane Trig., Art. 100); hence, all expressions involving the circular functions, must be inade homogeneous, to adapt them to the logarithmic formulas. 1. In the right-angled spherical С triangle BAC, right-angled at A, there are given a = 64° 40' and b = 42° 12': required the remaining parts. B First, to find the side c. The hypothenuse a corresponds to the middle part, and the extremes are opposite: hence, R cos a = cos b cos c, or, To find the angle B. The side b is the middle part and the extremes opposite: hence, R sin b = cos (comp. a) x cos (comp. B) = sin a sin B. sin 64° 40' ar. comp. 0.013911 : sin 42° 12' 10.000000 sin B 48° 00' 14" . To find the angle C. The angle C is the middle part and the extremes adjacent: hence, R cos C = cot a tang b. log. 0.000000 : cot 64° 40' 9.675237 :: tang 0 42° 12 9.957485 : COS C 61° 31' 46" 2. In a right-angled triangle BAC, there are given the hypothenuse a = 105° 34', and the angle B = 80° 40': required the remaining parts. To find the angle C. The hypothenuse is the middle part and the extremes adjacent: hence, Rcos a = cot B cot C. cot B 80° 40' log. 0.784220 + : COS a 105° 34' 10.000000 + : cot C 148° 30' 54". Since the cotangent of C is negative, the angle C is greater than 90°, and is the supplement of the arc which would correspond to the cotangent, if it were positive. To find the side c. The angle B corresponds to the middle part, and the extremes are adjacent: hence, Rcos B = cot a tang c. log 0.555053 : R 10.000000 + :: COS B 80° 40' 9.209992 + : tang c 149° 47' 36" To find the side 3. The side 6 is the middle part and the extremes are opposite : hence, 17. A quadrantal spherical triangle is one which has one of its sides equal to 90°. Let BAC be a quadrantal triangle of which the side a = 90°. If we pass to the corresponding polar triangle, we shall have 180° – A, 'T) from which we see, that the polar triangle will be riglito angled at A', and hence, every case may be referred to a right-angled triangle. But we can solve the quadrantal triangle by means of the right-angled triangle in a manner still more simple. Let the side BC of the quadrantal triangle BAC, be equal to 90°; produce the side CA till CD is equal to 90°, and conceive the arc of a great circle to be drawn through B and D. B Then C will be the pole of the arc BD, and the angle C will be measured by BD (B. IX., P. 4), and the angles CBD and D will be right angles. Now before the remaining parts of the quadrantal triangle can be found, at least two parts must be given in addition to the side BC = 90°; in which case two parts of the right-angled triangle BDA, together with the right angle, become known. Hence, the conditions which enable us to determine one of these triangles, will enable us also to determine the other. 1. In the quadrantal triangle BCA, there are given CB=90°, the angle C = 42° 12', and the angle A=115° 20'; required the remaining parts. Having produced CA to D, making CD = 90°, and drawn the arc BD, there will then be given in the rightangled triangle BAD, the side a = C = 42° 12', and the angle BAD= 180° – BAC = 180° - 115° 20' = 64° 40', to find the remaining parts. To find the side d. The side a is the middle part, and the extremes opposite: hence, R sin a = sin A sin d. sin A 64° 40' log. 0.043911 R 10.000000 42° 12' 9.827189 d 48° 00' 14". :: sin : sin To find the angle B. The angle A corresponds to the middle part, and the extremes are opposite : hence, The side b is the middle part, and the extremes are adjacent: hence, R sin b = cot A tang a. 0.000000 cot A 64° 40' 9.675237 :: tang a 42° 12' 9.957485 6 25° 25' 14" . llence, CA = 90° - b= 90° 25° 25' 14" 64° 34' 46" СВА 90° ABD = 90° - 35° 16' 53" 54° 43' 07" = 48° 00' 14" 2. In the right-angled triangle BAC, right-angled at A, there are given a = 115° 25', and c= 60° 59': required the remaining parts. B 148° 56' 45" Ans. C 75° 30' 33" 6 152° 13' 50" 3. In the right-angled spherical triangle BAC, right.. angled at A, there are given c = 116° 30'43", and b = 29° 41' 32" : required the remaining parts. C= 103° 52' 46". s Ans. B 32° 30 22" 112° 48' 58" 4. In a quadrantal triangle, there are given the quadrautal side = 90°, an adjacent side = 115° 09', and the in-cluded angle == 115° 55' : required the remaining parts. side, 113° 18' 19" Ans. 117° 33' 52" angles, 101° 40' 07"

What does X subscript i mean in calculus? Single Subscript Notation. The symbol X is the “list name,” or the name of the variable represented by the numbers on the list. The symbol i is a “subscript,” or “position indicator.” It indicates which number in the list, starting from the top, you are referring to. Simply so, What does subscript i mean in math? 1. The subscript i refers to this being the i-th x or the i-th y out of n. In your table it's basically row number. The third column is the difference between actual y value for the i-th data point and the predicted y value (by line of best fit) using the i-th x value. This is typical use of the subscript i. Also, How do you calculate a subscript? Also to know is, What is XI in statistics? xi represents the ith value of variable X. For the data, x1 = 21, x2 = 42, and so on. • The symbol Σ (“capital sigma”) denotes the summation function. What does I stand for in stats? Related Question for What Does X Subscript I Mean In Calculus? What does the letter I mean in math? The letter i is used to signify that a number is an imaginary number. It stand for the square root of negative one. See Imaginary numbers. What is a sub 1 in arithmetic sequence? What is a sub in math? A small letter or number placed slightly lower than the normal text. Examples: • the number 1 here: A1 (pronounced "A sub 1" or just "A 1") • the letter m here: xm (pronounced "x sub m" or just "x m") What does lowercase i mean in statistics? Capitalization. In general, capital letters refer to population attributes (i.e., parameters); and lower-case letters refer to sample attributes (i.e., statistics). What is subscript example? Subscript is the text which a small letter/number is written after a particular letter/number. It hangs below its letter or number. It is used when writing chemical compounds. An example of subscript is N2. What is the shortcut for subscript? Keyboard shortcuts: Apply superscript or subscript For superscript, press Ctrl, Shift, and the Plus sign (+) at the same time. For subscript, press Ctrl and the Minus sign (-) at the same time. What do you place as a subscript of a number? A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, while superscripts are above. What is XI in standard deviation? (Video) How to Calculate Standard Deviation and Variance n is the number of data points in your data set, xi is a point in that data set, and ¯x is the data's mean. Now, in plain English, this equation is telling you to take every point in the data set (the "xis") and subtract the mean from them. How do I find XI in statistics? What is Xi and Yi in statistics? pairs of letters xi and yj where xi is followed by yj. This is called the joint probability. p(x = xi; y = yi). If we x x to, say xi then the probability of y taking on a particular. value, say yj, is given by the conditional probability. What does U mean in probability? P(A∪B) Formula. The symbol "∪" (union) means "or". i.e., P(A∪B) is the probability of happening of the event A or B. To find, P(A∪B), we have to count the sample points that are present in both A and B. What is an i number? An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary. Why is i used in math? While it is not a real number — that is, it cannot be quantified on the number line — imaginary numbers are "real" in the sense that they exist and are used in math. Usually denoted by the symbol i, imaginary numbers are denoted by the symbol j in electronics (because i already denotes "current"). What does i mean in precalculus? Complex numbers are unreal. Yes, that's the truth. A complex number has a term with a multiple of i, and i is the imaginary number equal to the square root of –1. How do you find the sub 1 in an arithmetic sequence? What is the meaning of arithmetical? (ærɪθmetɪkəl ) adjective [usually ADJECTIVE noun] Arithmetical calculations, processes, or skills involve the addition, subtraction, multiplication, or division of numbers. What is the arithmetic mean between 10 and 24? Thus, 10 + 24 2 = 17 is the arithmetic mean. What does a sub 0 mean? Sub-zero literally means "beneath zero". As such, it is usually used for negative numbers; the most common usage refers to negative temperature. What is the small number Bottom right? What does a bottom exponent mean? The base, as the name suggests, is the number on the bottom. The other part of the notation is a small number written in superscript to the right of the base, called the exponent. The base is 10. This means that 10 is a factor, and it's going to be multiplied by itself some number of times. What does a line over a letter mean in probability? Bar or Vinculum: When the line above the letter represents a bar. A vinculum is a horizontal line used in the mathematical notation for a specific purpose to indicate that the letter or expression is grouped together. The x bar symbol is used in statistics to represent the sample mean of a distribution. What does capital I mean in statistics? It's the indicator function! It takes value 1 if the condition inside the brackets is met, 0 otherwise. https://stats.stackexchange.com/questions/149698/what-does-capital-letter-i-mean-in-this-formulas/149699#149699. What subscript means? : a distinguishing symbol (such as a letter or numeral) written immediately below or below and to the right or left of another character. How do I type h2o on my laptop? How do you use subscripts? How do you write st in 1st? What is the use of super script and sub script? Superscripts are characters set above the normal line of type (e.g. 2ⁿᵈ) and subscript are characters set below (e.g. Cᵥₑₓ). There are many reasons to use them in charts: Some organization's style guides demands that you use them for e.g. chemical and physical formulas or footnotes, or maybe you just find them pretty. How do you write subscripts on a Mac? You can also use keyboard shortcuts to quickly apply superscript or subscript to selected text. For superscript, press Control-Shift-Command-Plus Sign (+). For subscript, press Control-Command-Minus Sign (-). How do you write subscript in Word equations? Was this helpful? 0 / 0

In 2022, the International Congress of Mathematicians — the most important meeting of the world's mathematical community — will take place in St. Petersburg. The main part of the scientific program of the congress is plenary talks. These are hour-long lectures given by world-famous mathematicians. The program committee (whose composition is kept a secret until the opening ceremony) has selected the plenary speakers, who all have accepted invitations to speak at the ICM 2022. Their names are published on the Congress website. The plenary speakers comprise 21 of the world's leading mathematicians, representing 10 countries and all areas of mathematics. Michel Van den Bergh is a Belgian mathematician and professor at the Free University of Brussels and Hasselt University. He studies the relationship between geometry and algebra. From the beginning of his career, Van den Bergh has been working on non-commutative geometry, a geometric approach to non-commutative algebras. Mladen Bestvina is an American mathematician with Croatian roots, professor at the University of Utah. Bestvina is working in geometric group theory. His outstanding mathematical career began at school — he won the International Mathematical Olympiad three times. In 2002, he gave a talk at the International Congress of Mathematicians in Beijing. Bhargav Bhatt is an Indian-American mathematician and professor at the University of Michigan. Bhatt is specializing in arithmetic algebraic geometry, number theory, commutative algebra, and homotopy theory. In 2021, he won the prestigious New Horizons in Mathematics Prize “For outstanding work in commutative algebra and arithmetic algebraic geometry, particularly on the development of p-adic cohomology theories.” Kevin Buzzard is a British mathematician, professor at Imperial College London, an expert in algebraic number theory. Buzzard won prizes at the International Mathematical Olympiad twice. He got the Whitehead Prize in 2002 and the Berwick Prize in 2008 for his brilliant work in number theory. Now Buzzard is figuring out whether it is possible to “teach” mathematics to a computer so that it helps to prove mathematical theorems or even does it on its own. Frank Calegari is an American mathematician with Australian roots and а professor at the University of Chicago. Calegari became a medalist of the International Mathematical Olympiad twice. Professor Calegari specializes in algebraic number theory and deals with arithmetic in the theory of modular forms. Tobias Сolding is an American mathematician of Danish origin, professor at the Massachusetts Institute of Technology, studying geometric analysis and low-dimensional topology. Colding won the 2010 Oswald Veblen Prize in Geometry for his research on minimal surfaces. Weinan E is a Chinese-American mathematician, professor at Princeton University, an expert in machine learning. The professor's research is applied in chemistry, materials science, hydrodynamics. Weinan E spoke at the International Congress of Mathematicians in Beijing in 2002. Craig Gentry is an American mathematician who created the Gentry's encryption scheme, the first fully homomorphic cryptosystem. He formulated it in his doctoral dissertation and was awarded the prize of the Association for Computing Machinery. Gentry also won a 2014 MacArthur Fellowship, known as the “Genius Grant.” Alice Guionnet is a French mathematician who specializes in probability theory and random matrix theory. Guionnet got the Loewe Prize in 2009, and the European Academy of Science's Blaise Pascal Medal in 2018. Larry Guth is an American mathematician, professor at the Massachusetts Institute of Technology, an expert in Fourier analysis. In 2013, he won the Salem Prize for his contributions to geometry and combinatorics, in 2015 — the Clay Mathematical Institute's Research Prize, and in 2020 he took the first Maryam Mirzakhani prize. Svetlana Jitomirskaya is an American mathematician, who was born in Kharkiv. She defended her Ph.D. thesis in her alma mater — Lomonosov Moscow State University — under the supervision of Yakov Sinai, who became the winner of the Abel Prize in 2014. Now Svetlana Jitomirskaya is a professor at the University of California, Irvine. She is studying the spectrum of quasiperiodic Schrödinger equations. She spoke at the International Congress of Mathematicians in 2002 and at the International Congress on Mathematical Physics twice. David Kazhdan is an Israeli mathematician born in Moscow, professor at the Hebrew University of Jerusalem and Harvard University. He published his first scientific article while at school, defended his Ph.D. thesis at the Lomonosov Moscow State University before the appointed time. Kazhdan is rightfully considered one of the best scientists of Gelfand's mathematical school. he is 2012 Israel Prize in Mathematics and 2020 Shaw Prize winner. Igor Krichever is a Russian and American mathematician, professor of Skolkovo Institute of Science and Technology, a leading researcher at the L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, professor of the Higher School of Economics and Columbia University. Krichever is studying algebraic geometry and mathematical physics. Alexander Kuznetsov is a Russian mathematician, a leading researcher at the Steklov Mathematical Institute of the Russian Academy of Sciences. He works in algebraic geometry and derived categories of coherent sheaves. In 2008, Kuznetsov became a laureate of the European Mathematical Society Prize. He made a speech at the International Congress of Mathematicians in Seoul in 2014. Camillo De Lellis is an Italian mathematician specializing in the calculus of variations, geometric measure theory and hydrodynamics. De Lellis won the 2013 Fermat Prize. In 2010, he gave a talk in one of the sections of the International Mathematical Congress. Frans Pretorius is a South African physicist specializing in general relativity. Pretorius numerically investigated the collision of black holes. In 2017 he was awarded the New Horizons in Physics Prize for the first computer code that can simulate spiral movement and fusion of two black holes. Laure Saint-Raymond is a French mathematician who studies nonlinear differential equations. Saint-Raymond is known for her research of relationships between systems of interacting particles, the Boltzmann equation and fluid mechanics. In 2008, she received the prize from the European Mathematical Society, in 2015 — the Fermat Prize, in 2020 — the Bocher Prize. Scott Sheffield is an American mathematician and professor at the Massachusetts Institute of Technology. Sheffield is interested in probability theory, game theory and mathematical physics. In 2017, Sheffield received the Clay Mathematical Institute's Research Prize, in 2011 — the Loève Prize. Kannan Soundararajan is an Indian-American mathematician and professor at Stanford University. Soundararajan is an expert in analytical number theory. For his contribution to the study of Dirichlet L-functions, the scientist received the Salem Prize in 2003, and in 2011 he won the Ostrovsky Prize. Catharina Stroppel is a German mathematician and professor at the University of Bonn. She is working in Lie theory and representation theory. In 2017, Stroppel got the Whitehead Prize. Umesh Vazirani is an Indian-American mathematician, professor at the University of California, an expert in quantum computing, one of the founders of quantum computing. He received the Fulkerson Prize for improving the approximation ratio for graph separators and related problems in 2012.

The relativistic Feynman-Metropolis-Teller theory for white dwarfs in general relativity The recent formulation of the relativistic Thomas-Fermi model within the Feynman-Metropolis-Teller theory for compressed atoms is applied to the study of general relativistic white dwarf equilibrium configurations. The equation of state, which takes into account the -equilibrium, the nuclear and the Coulomb interactions between the nuclei and the surrounding electrons, is obtained as a function of the compression by considering each atom constrained in a Wigner-Seitz cell. The contribution of quantum statistics, weak, nuclear, and electromagnetic interactions is obtained by the determination of the chemical potential of the Wigner-Seitz cell. The further contribution of the general relativistic equilibrium of white dwarf matter is expressed by the simple formula constant, which links the chemical potential of the Wigner-Seitz cell with the general relativistic gravitational potential at each point of the configuration. The configuration outside each Wigner-Seitz cell is strictly neutral and therefore no global electric field is necessary to warranty the equilibrium of the white dwarf. These equations modify the ones used by Chandrasekhar by taking into due account the Coulomb interaction between the nuclei and the electrons as well as inverse -decay. They also generalize the work of Salpeter by considering a unified self-consistent approach to the Coulomb interaction in each Wigner-Seitz cell. The consequences on the numerical value of the Chandrasekhar-Landau mass limit as well as on the mass-radius relation of He, C, O and Fe white dwarfs are presented. All these effects should be taken into account in processes requiring a precision knowledge of the white dwarf parameters. The necessity of introducing the Fermi-Dirac statistics in order to overcome some conceptual difficulties in explaining the existence of white dwarfs leading to the concept of degenerate stars was first advanced by R. H. Fowler in a classic paper fowler26. Following that work, E. C. Stoner stoner29 introduced the effect of special relativity into the Fowler considerations and he discovered the critical mass of white dwarfs 111In doing this, Stoner used what later became known as the exclusion principle, generally attributed in literature to Wolfgang Pauli. For a lucid and scientifically correct historical reconstruction of the contributions to the critical mass concept see nauenberg08. For historical details about the exclusion principle see also heilbron83. where g is the Planck mass, is the neutron mass, and is the average molecular weight of matter which shows explicitly the dependence of the critical mass on the chemical composition of the star. Following the Stoner’s work, S. Chandrasekhar chandrasekhar31 222At the time a 20 years old graduate student coming to Cambridge from India. pointed out the relevance of describing white dwarfs by using an approach, initiated by E. A. Milne milne30, of using the mathematical method of the solutions of the Lane-Emden polytropic equations emdenbook. The same idea of using the Lane-Emden equations taking into account the special relativistic effects to the equilibrium of stellar matter for a degenerate system of fermions, came independently to L. D. Landau landau32. Both the Chandrasekhar and Landau treatments were explicit in pointing out the existence of the critical mass where the first numerical factor on the right hand side of Eq. (2) comes from the boundary condition (see last entry of Table 7 on Pag. 80 in emdenbook) of the Lane-Emden polytropic equation. Namely for , no equilibrium configuration should exist. Landau rejected the idea of the existence of such a critical mass as a “ridiculous tendency” landau32. Chandrasekhar was confronted by a lively dispute with A. Eddington on the basic theoretical assumptions he adopted 333The dispute reached such a heated level that Chandrasekhar was confronted with the option either to change field of research or to leave Cambridge. As is well known he chose the second option transferring to Yerkes Observatory near Chicago where he published his results in his classic book chandrasekharbook. (see wali82 for details). Some of the basic assumptions adopted by Chandrasekhar and Landau in their idealized approach were not justified e.g. the treatment of the electron as a free-gas without taking into due account the electromagnetic interactions, as well as the stability of the distribution of the nuclei against the gravitational interaction. It is not surprising that such an approach led to the criticisms of Eddington who had no confidence of the physical foundation of the Chandrasekhar work 444It goes to Eddington credit, at the time Plumian Professor at Cambridge, to have allowed the publication of the Chandrasekhar work although preceded by his own critical considerations eddington35.. It was unfortunate that the absence of interest of E. Fermi on the final evolution of stars did not allow Fermi himself to intervene in this contention and solve definitely these well-posed theoretical problems ruffinibook. Indeed, we are showing in this article how the solution of the conceptual problems of the white dwarf models, left open for years, can be duly addressed by considering the relativistic Thomas-Fermi model of the compressed atom (see Subsec. II.5 and Sec. IV). The original work on white dwarfs was motivated by astrophysics and found in astrophysics strong observational support. The issue of the equilibrium of the electron gas and the associated component of nuclei, taking into account the electromagnetic, the gravitational and the weak interactions is a theoretical physics problem, not yet formulated in a correct special and general relativistic context. One of the earliest alternative approaches to the Chandrasekhar-Landau work was proposed by E. E. Salpeter in 1961 salpeter61. He followed an idea originally proposed by Y. I. Frenkel frenkel28: to adopt in the study of white dwarfs the concept of a Wigner-Seitz cell. Salpeter introduced to the lattice model of a point-like nucleus surrounded by a uniform cloud of electrons, corrections due to the non-uniformity of the electron distribution (see Subsec. II.3 for details). In this way Salpeter salpeter61 obtained an analytic formula for the total energy in a Wigner-Seitz cell and derived the corresponding equation of state of matter composed by such cells, pointing out explicitly the relevance of the Coulomb interaction. The consequences of the Coulomb interactions in the determination of the mass and radius of white dwarfs, was studied in a subsequent paper by T. Hamada and E. E. Salpeter hamada61 by using the equation of state constructed in salpeter61. They found that the critical mass of white dwarfs depends in a nontrivial way on the specific nuclear composition: the critical mass of Chandrasekhar-Landau which depends only on the mass to charge ratio of nuclei , now depends also on the proton number . This fact can be seen from the approximate expression for the critical mass of white dwarfs obtained by Hamada and Salpeter hamada61 in the ultrarelativistic limit for the electrons being the pressure of the Wigner-Seitz cell obtained by Salpeter in salpeter61 (see Subsec. II.3) and is the pressure of a free-electron fluid used by Chandrasekhar (see Subsec. II.1). The ratio is a function of the number of protons (see Eq. (20) in salpeter61) and it satisfies . Consequently, the effective molecular weight satisfies and the critical mass of white dwarfs turns to be smaller than the original one obtained by Chandrasekhar-Landau (see Eq. (2)). In the mean time, the problem of the equilibrium gas in a white dwarf taking into account possible global electromagnetic interactions between the nucleus and the electrons was addressed by E. Olson and M. Bailyn in olson75; olson76. They well summarized the status of the problem: “Traditional models for the white dwarf are non-relativistic and electrically neutral … although an electric field is needed to support the pressureless nuclei against gravitational collapse, the star is treated essentially in terms of only one charge component, where charge neutrality is assumed ”. Their solution to the problem invokes the breakdown of the local charge neutrality and the presence of an overall electric field as a consequence of treating also the nuclei inside the white dwarf as a fluid. They treated the white dwarf matter through a two-fluid model not enforcing local charge neutrality. The closure equation for the Einstein-Maxwell system of equations was there obtained from a minimization procedure of the mass-energy of the configuration. This work was the first pointing out the relevance of the Einstein-Maxwell equations in the description of an astrophysical system by requiring global and non local charge neutrality. As we will show here, this interesting approach does not apply to the case of white dwarfs. It represents, however, a new development in the study of neutron stars (see e.g. PLB2011) An alternative approach to the Salpeter treatment of a compressed atom was reconsidered in gursky2000 by applying for the first time to white dwarfs a relativistic Thomas-Fermi treatment of the compressed atom introducing a finite size nucleus within a phenomenological description (see also bertone2000). Recently, the study of a compressed atom has been revisited in 2011PhRvC..83d5805R by extending the global approach of Feynman, Metropolis and Teller feynman49 taking into account weak interactions. This treatment takes also into account all the Coulomb contributions duly expressed relativistically without the need of any piecewise description. The relativistic Thomas-Fermi model has been solved by imposing in addition to the electromagnetic interaction also the weak equilibrium between neutrons, protons and electrons self-consistently. This presents some conceptual differences with respect to previous approaches and can be used in order both to validate and to establish their limitations. In this article we apply the considerations presented in 2011PhRvC..83d5805R of a compressed atom in a Wigner-Seitz cell to the description of non-rotating white dwarfs in general relativity. This approach improves all previous treatments in the following aspects: In order to warranty self-consistency with a relativistic treatment of the electrons, the point-like assumption of the nucleus is abandoned introducing a finite sized nucleus 2011PhRvC..83d5805R. We assume for the mass as well as for charge to mass ratio of the nucleus their experimental values instead of using phenomenological descriptions based on the semi-empirical mass-formula of Weizsacker (see e.g. gursky2000; bertone2000). The electron-electron and electron-nucleus Coulomb interaction energy is calculated without any approximation by solving numerically the relativistic Thomas-Fermi equation for selected energy-densities of the system and for each given nuclear composition. The energy-density of the system is calculated taking into account the contributions of the nuclei, of the Coulomb interactions as well as of the relativistic electrons; the latter being neglected in all previous treatments. This particular contribution turns to be very important at high-densities and in particular for light nuclear compositions e.g. He and C. The -equilibrium between neutrons, protons, and electrons is also taken into account leading to a self-consistent calculation of the threshold density for triggering the inverse -decay of a given nucleus. The structure of the white dwarf configurations is obtained by integrating the general relativity equations of equilibrium. Due to 4) and 5) we are able to determine if the instability point leading to a maximum stable mass of the non-rotating white dwarf is induced by the inverse -decay instability of the composing nuclei or by general relativistic effects. Paradoxically, after all this procedure which takes into account many additional theoretical features generalizing the Chandrasekhar-Landau and the Hamada and Salpeter works, a most simple equation is found to be fulfilled by the equilibrium configuration in a spherically symmetric metric. Assuming the metric we demonstrate how the entire system of equations describing the equilibrium of white dwarfs, taking into account the weak, the electromagnetic and the gravitational interactions as well as quantum statistics all expressed consistently in a general relativistic approach, is simply given by which links the chemical potential of the Wigner-Seitz cell , duly solved by considering the relativistic Feynman-Metropolis-Teller model following 2011PhRvC..83d5805R, to the general relativistic gravitational potential at each point of the configuration. The overall system outside each Wigner-Seitz cell is strictly neutral and no global electric field exists, contrary to the results reported in olson76. The same procedure will apply as well to the case of neutron star crusts. The article is organized as follows. In Sec. II we summarize the most common approaches used for the description of white dwarfs and neutron star crusts: the uniform approximation for the electron fluid (see e.g. chandrasekhar31); the often called lattice model assuming a point-like nucleus surrounded by a uniform electron cloud (see e.g. baym71a); the generalization of the lattice model due to Salpeter salpeter61; the Feynman, Metropolis and Teller approach feynman49 based on the the non-relativistic Thomas-Fermi model of compressed atoms and, the relativistic generalization of the Feynman-Metropolis-Teller treatment recently formulated in 2011PhRvC..83d5805R. In Sec. III we formulate the general relativistic equations of equilibrium of the system and show how, from the self-consistent definition of chemical potential of the Wigner-Seitz cell and the Einstein equations, comes the equilibrium condition given by Eq. (6). In addition, we obtain the Newtonian and the first-order post-Newtonian equations of equilibrium. Finally, we show in Sec. IV the new results of the numerical integration of the general relativistic equations of equilibrium and discuss the corrections to the Stoner critical mass , to the Chandrasekhar-Landau mass limit , as well as to the one of Hamada and Salpeter , obtained when all interactions are fully taken into account through the relativistic Feynman-Metropolis-Teller equation of state 2011PhRvC..83d5805R. Ii The Equation of State There exists a large variety of approaches to model the equation of state of white dwarf matter, each one characterized by a different way of treating or neglecting the Coulomb interaction inside each Wigner-Seitz cell, which we will briefly review here. Particular attention is given to the calculation of the self-consistent chemical potential of the Wigner-Seitz cell , which plays a very important role in the conservation law (6) that we will derive in Sec. III. ii.1 The uniform approximation In the uniform approximation used by Chandrasekhar chandrasekhar31, the electron distribution as well as the nucleons are assumed to be locally constant and therefore the condition of local charge neutrality where is the average atomic weight of the nucleus, is applied. Here denotes the nucleon number density and is the number of protons of the nucleus. The electrons are considered as a fully degenerate free-gas and then described by Fermi-Dirac statistics. Thus, their number density is related to the electron Fermi-momentum by and the total electron energy-density and electron pressure are given by where we have introduced the dimensionless Fermi momentum with the electron rest-mass. The kinetic energy of nucleons is neglected and therefore the pressure is assumed to be only due to electrons. Thus the equation of state can be written as Within this approximation, the total self-consistent chemical potential is given by is the electron free-chemical potential. As a consequence of this effective approach which does not take into any account the Coulomb interaction, it is obtained an effective one-component electron-nucleon fluid approach where the kinetic pressure is given by electrons of mass and their gravitational contribution is given by an effective mass attached to each electron (see e.g. landaubook). This is even more evident when the electron contribution to the energy-density in Eq. (11) is neglected and therefore the energy-density is attributed only to the nuclei. Within this approach followed by Chandrasekhar chandrasekhar31, the equation of state reduces to ii.2 The lattice model The first correction to the above uniform model, corresponds to abandon the assumption of the electron-nucleon fluid through the so-called “lattice” model which introduces the concept of Wigner-Seitz cell: each cell contains a point-like nucleus of charge with nucleons surrounded by a uniformly distributed cloud of fully-degenerate electrons. The global neutrality of the cell is guaranteed by the condition where is the Wigner-Seitz cell density and is the cell volume. The total energy of the Wigner-Seitz cell is modified by the inclusion of the Coulomb energy, i.e where is given by Eq. (11) and and are the electron-nucleus and the electron-electron Coulomb energies The self-consistent pressure of the Wigner-Seitz cell is then given by where is given by Eq. (12). It is worth to recall that the point-like assumption of the nucleus is incompatible with a relativistic treatment of the degenerate electron fluid (see ferreirinho80; ruffini81 for details). Such an inconsistency has been traditionally ignored by applying, within a point-like nucleus model, the relativistic formulas (9) and (10) and their corresponding ultrarelativistic limits (see e.g. salpeter61). The Wigner-Seitz cell chemical potential is in this case ii.3 Salpeter approach A further development to the lattice model came from Salpeter salpeter61 whom studied the corrections due to the non-uniformity of the electron distribution inside a Wigner-Seitz cell. Following the Chandrasekhar chandrasekhar31 approximation, Salpeter also neglects the electron contribution to the energy-density. Thus, the first term in the Salpeter formula for the energy of the cell comes from the nuclei energy (15). The second contribution is given by the Coulomb energy of the lattice model (19). The third contribution is obtained as follows: the electron density is assumed as , where is the average electron density as given by Eq. (17), and is considered infinitesimal. The Coulomb potential energy is assumed to be the one of the point-like nucleus surrounded by a uniform distribution of electrons, so the correction given by on the Coulomb potential is neglected. The electron distribution is then calculated at first-order by expanding the relativistic electron kinetic energy about its value in the uniform approximation considering as infinitesimal the ratio between the Coulomb potential energy and the electron Fermi energy The influence of the Dirac electron-exchange correction dirac30 on the equation of state was also considered by Salpeter salpeter61. However, adopting the general approach of Migdal et al. migdal77, it has been shown that these effects are negligible in the relativistic regime 2011PhRvC..83d5805R. We will then consider here only the major correction of the Salpeter treatment. The total energy of the Wigner-Seitz cell is then given by (see salpeter61 for details) Correspondingly, the self-consistent pressure of the Wigner-Seitz cell is The Wigner-Seitz cell chemical potential can be then written as From Eqs. (29) and (31), we see that the inclusion of each additional Coulomb correction results in a further decreasing of the pressure and of the chemical potential of the cell. The Salpeter approach is very interesting in identifying piecewise Coulomb contribution to the total energy, to the total pressure and, to the Wigner-Seitz chemical potential. However, it does not have the full consistency of the global solutions obtained with the Feynman-Metropolis-Teller approach feynman49 and its generalization to relativistic regimes 2011PhRvC..83d5805R which we will discuss in detail below. ii.4 The Feynman-Metropolis-Teller treatment Feynman, Metropolis and Teller feynman49 showed how to derive the equation of state of matter at high pressures by considering a Thomas-Fermi model confined in a Wigner-Seitz cell of radius . The Thomas-Fermi equilibrium condition for degenerate non-relativistic electrons in the cell is expressed by where denotes the Coulomb potential and denotes the Fermi energy of electrons, which is positive for configurations subjected to external pressure, namely, for compressed cells. Defining the function by , and introducing the dimensionless radial coordinate by , where , being the electron Compton wavelength; the Poisson equation from which the Coulomb potential is calculated self-consistently becomes The boundary conditions for Eq. (33) follow from the point-like structure of the nucleus and, from the global neutrality of the Wigner-Seitz cell , where defines the dimensionless radius of the Wigner-Seitz cell by . For each value of the compression, e.g. , it corresponds a value of the electron Fermi energy and a different solution of Eq. (33), which determines the self-consistent Coulomb potential energy as well as the self-consistent electron distribution inside the cell through In the non-relativistic Thomas-Fermi model, the total energy of the Wigner-Seitz cell is given by (see slater35; feynman49 for details) where is the nucleus mass, is given by Eq. (9) and and are the electron-nucleus Coulomb energy and the electron-electron Coulomb energy, which are given by where is the non-relativistic kinetic energy of a uniform electron distribution of density , i.e. with defined by The self-consistent pressure of the Wigner-Seitz cell given by the non-relativistic Thomas-Fermi model is (see slater35; feynman49 for details) The pressure of the Thomas-Fermi model (44) is equal to the pressure of a free-electron distribution of density . Being the electron density inside the cell a decreasing function of the distance from the nucleus, the electron density at the cell boundary, , is smaller than the average electron distribution . Then, the pressure given by (44) is smaller than the one given by the non-relativistic version of Eq. (10) of the uniform model of Subsec. II.1. Such a smaller pressure, although faintfully given by the expression of a free-electron gas, contains in a self-consistent fashion all the Coulomb effects inside the Wigner-Seitz cell. The chemical potential of the Wigner-Seitz cell of the non-relativistic Thomas-Fermi model can be then written as Integrating by parts the total number of electrons we can rewrite finally the following semi-analytical expression of the chemical potential (45) of the cell where is the electron free-chemical potential (14) calculated with the average electron density, namely, the electron chemical potential of the uniform approximation. The function depends explicitly on the gradient of the electron density, i.e. on the non-uniformity of the electron distribution. ii.5 The relativistic Feynman-Metropolis-Teller treatment We recall now how the above classic Feynman, Metropolis, and Teller treatment of compressed atoms has been recently generalized to relativistic regimes (see 2011PhRvC..83d5805R for details). One of the main differences in the relativistic generalization of the Thomas-Fermi equation is that, the point-like approximation of the nucleus, must be abandoned since the relativistic equilibrium condition of compressed atoms would lead to a non-integrable expression for the electron density near the origin (see e.g.ferreirinho80; ruffini81). It is then assumed a constant distribution of protons confined in a radius defined by where is the pion Compton wavelength. If the system is at nuclear density with fm. Thus, in the case of ordinary nuclei (i.e., for ) we have . Consequently, the proton density can be written as where denotes the Heaviside function centered at . The electron density can be written as where and we have used Eq. (50). The overall Coulomb potential satisfies the Poisson equation with the boundary conditions and due to the global charge neutrality of the cell. By introducing the dimensionless quantities , , and replacing the particle densities (52) and (53) into the Poisson equation (54), it is obtained the relativistic Thomas-Fermi equation 2008pint.conf..207R which must be integrated subjected to the boundary conditions , and , where . The neutron density , related to the neutron Fermi momentum , is determined by imposing the condition of beta equilibrium subjected to the baryon number conservation equation In Fig. 1 we see how the relativistic generalization of the Feynman-Metropolis-Teller treatment leads to electron density distributions markedly different from the constant electron density approximation. The electron distribution is far from being uniform as a result of the solution of Eq. (55), which takes into account the electromagnetic interaction between electrons and between the electrons and the finite sized nucleus. Additional details are given in 2011PhRvC..83d5805R. V. S. Popov et al. popov10 have shown how the solution of the relativistic Thomas-Fermi equation (55) together with the self-consistent implementation of the -equilibrium condition (56) leads, in the case of zero electron Fermi energy (), to a theoretical prediction of the -equilibrium line, namely a theoretical - relation. Within this model the mass to charge ratio of nuclei is overestimated, e.g. in the case of He the overestimate is , for C , for O , and for Fe . These discrepancies are corrected when the model of the nucleus considered above is improved by explicitly including the effects of strong interactions. This model, however, illustrates how a self-consistent calculation of compressed nuclear matter can be done including electromagnetic, weak, strong as well as special relativistic effects without any approximation. This approach promises to be useful when theoretical predictions are essential, for example in the description of nuclear matter at very high densities, e.g. nuclei close and beyond the neutron drip line. The densities in white dwarf interiors are not highly enough to require such theoretical predictions. Therefore, in order to ensure the accuracy of our results we use for , needed to solve the relativistic Thomas-Fermi equation (55), as well as for the nucleus mass , their known experimental values. In this way we take into account all the effects of the nuclear interaction. Thus, the total energy of the Wigner-Seitz cell in the present case can be written as where is the experimental nucleus mass, e.g. for He, C, O and Fe we have 4.003, 12.01, 16.00 and 55.84 respectively. In Eq. (61) the integral is evaluated only outside the nucleus (i.e. for ) in order to avoid a double counting with the Coulomb energy of the nucleus already taken into account in the nucleus mass (59). In order to avoid another double counting we subtract to the electron energy-density in Eq. (60) the rest-energy density which is also taken into account in the nucleus mass (59). The total pressure of the Wigner-Seitz cell is given by where is the relativistic pressure (10) computed with the value of the electron density at the boundary of the cell. The electron density at the boundary in the relativistic Feynman-Metropolis-Teller treatment is smaller with respect to the one given by the uniform density approximation (see Fig. 1). Thus, the relativistic pressure (62) gives systematically smaller values with respect to the uniform approximation pressure (10) as well as with respect to the Salpeter pressure (29). In Fig. 2 we show the ratio between the relativistic Feynman-Metropolis-Teller pressure (62) and the Chandrasekhar pressure (10) and the Salpeter pressure (29) in the case of C. It can be seen how is smaller than for all densities as a consequence of the Coulomb interaction. With respect to the Salpeter case, we have that the ratio approaches unity from below at large densities as one should expect. However, at low densities – g/cm, the ratio becomes larger than unity due to the defect of the Salpeter treatment which, in the low density non-relativistic regime, leads to a drastic decrease of the pressure and even to negative pressures at densities g/cm or higher for heavier nuclear compositions e.g. Fe (see salpeter61; 2011PhRvC..83d5805R and Table 1). This is in contrast with the relativistic Feynman-Metropolis-Teller treatment which matches smoothly the classic Feynman-Metropolis-Teller equation of state in that regime (see 2011PhRvC..83d5805R for details). No analytic expression of the Wigner-Seitz cell chemical potential can be given in this case, so we only write its general expression where and are given by Eqs. (58) and (62) respectively. The above equation, contrary to the non-relativistic formula (45), in no way can be simplified in terms of its uniform counterparts. However, it is easy to check that, in the limit of no Coulomb interaction , , and and, neglecting the nuclear binding and the proton-neutron mass difference, we finally obtain as it should be expected. Now we summarize how the equation of state of compressed nuclear matter can be computed in the Salpeter case and in the relativistic Feynman-Metropolis-Teller case, parameterized by the total density of the system: (i) For a given radius of the Wigner-Seitz cell the relativistic Thomas-Fermi equation (55) is integrated numerically and the density of the configuration is computed as where is the energy of the cell given by Eq. (58). (ii) For that value of the density, the radius of the Wigner-Seitz cell in the Salpeter treatment is where Eq. (15) has been used. On the contrary, in the relativistic Feynman-Metropolis-Teller treatment no analytic expression relating Wigner-Seitz cell radius and density can be written. (iii) From this Wigner-Seitz cell radius, or equivalently using the value of the density, the electron density in the Salpeter model is computed from the assumption of uniform electron distribution and the charge neutrality condition, i.e. Eq. (15). In the relativistic Feynman-Metropolis-Teller treatment, the electron number density at the boundary of the Wigner-Seitz cell is, following Eq. (53), given by where the function is the solution of the relativistic Thomas-Fermi equation (55). Iii General relativistic equations of equilibrium Outside each Wigner-Seitz cell the system is electrically neutral, thus no overall electric field exists. Therefore, the above equation of state can be used to calculate the structure of the star through the Einstein equations. Introducing the spherically symmetric metric (5), the Einstein equations can be written in the Tolman-Oppenheimer-Volkoff form tolman39; oppenheimer39 where we have introduced the mass enclosed at the distance through , is the energy-density and is the total pressure. We turn now to demonstrate how, from Eq. (69), it follows the general relativistic equation of equilibrium (6), for the self-consistent Wigner-Seitz chemical potential . The first law of thermodynamics for a zero temperature fluid of particles, total energy , total volume , total pressure , and chemical potential reads where the differentials denote arbitrary but simultaneous changes in the variables. Since for a system whose surface energy can be neglected with respect to volume energy, the total energy per particle depends only on the particle density , we can assume as an homogeneous function of first-order in the variables and and hence, it follows the well-known thermodynamic relation In the case of the Wigner-Seitz cells, Eq. (71) reads where we have introduced the fact that the Wigner-Seitz cells are the building blocks of the configuration and therefore we must put in Eq. (71) . Through the entire article we have used Eq. (72) to obtain from the knowns energy and pressure, the Wigner-Seitz cell chemical potential (see e.g. Eqs. (13) and (23)). From Eqs. (70) and (71) we obtain the so-called Gibbs-Duhem relation In a white dwarf the pressure and the chemical potential are decreasing functions of the distance from the origin. Thus, the differentials in the above equations can be assumed as the gradients of the variables which, in the present spherically symmetric case, become just derivatives with respect to the radial coordinate . From Eq. (73) it follows the relation which can be straightforwardly integrated to obtain the first integral The above equilibrium condition is general and it also applies for non-zero temperature configurations ( see e.g. klein49). In such a case, it can be shown that in addition to the equilibrium condition (76) the temperature of the system satisfies the Tolman isothermality condition constant 1930PhRv...35..904T; 1930PhRv...36.1791T. iii.1 The weak-field non-relativistic limit In the weak-field limit we have , where the Newtonian gravitational potential has been defined by . In the non-relativistic mechanics limit , the chemical potential , where denotes the non-relativistic free-chemical potential of the Wigner-Seitz cell and is the rest-mass of the Wigner-Seitz cell, namely, the rest-mass of the nucleus plus the rest-mass of the electrons. Applying these considerations to Eq. (76) we obtain Absorbing the Wigner-Seitz rest-mass energy in the constant on the right-hand-side we obtain

Often, an object's center of gravity is also the center of its length. The next time you need to carry something uniformly long and thin—like a wooden pole or a length of pipe or metal or, if you're a tightrope walker, that really long. Finding the Center of Gravity You may have heard the term center of gravity in reference to balance and you might intuitively know that a toddler’s big head raises their center of gravity, which makes them less stable than adults. We. Center of Gravity CG by Ron Kurtus revised 8 March 2018 The center of gravity CG of an object is the balance point around which its matter is evenly distributed. In such a case, the object can act as if. Physics Chemistry Biology Geography Center of Gravity: Definition, Equation and Calculation The center of gravity not to be confused with center of mass of a body is a point where the weight of the body acts and total. Define center of gravity. center of gravity synonyms, center of gravity pronunciation, center of gravity translation, English dictionary definition of center of gravity. n. pl. centers of gravity 1. Abbr. CG The point in or near a body at. The aim of /r/Physics is to build a subreddit frequented by physicists, scientists, and those with a passion for physics. Papers from physics journals free or otherwise are encouraged. Posts should be pertinent, meme-free, and. CENTER OF GRAVITY The center of gravity is a geometric property of any object. The center of gravity is the average location of the weight of an object. STABLE: A body is in stable equilibrium if it returns to its equilibrium position. The center of gravity is an important concept in determining the stability of a structure. It's the reason why a good homeowner will keep the top branches of his trees trimmed. It's also the reason why a pick-up truck might not be the. 2019/09/05 · How to Calculate Center of Gravity. The center of gravity CG is the center to an object's weight distribution, where the force gravity can be considered to act. This is the point where the object is in perfect balance, no. The 'center of gravity' of any object or volume of space is a point in space. a place, with a very clear definition. It's the place where the sums of the first moments of every speck of mass in every direction are all equal. If the object or the. The lower your center of gravity, the easier it is to keep your balance. If you're sitting on a chair, you can lean over more than if you're standing up. With your center of gravity low, you can lean further to one side or the other without. wk 10 Physics Center of Gravity Project. Can u balance a titled can of coke? How about balancing a coin on a note? Use with Archimedes and the Door of Science. Physics Center of Gravity Project Demonstrations in Physics. 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Center of Mass The terms "center of mass" and "center of gravity" are used synonymously in a uniform gravity field to represent the unique point in an object or system which can be used to describe the system's response to external forces. Gravity from Latin gravitas, meaning 'weight', or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light —are brought toward or gravitate toward one another. Center of Gravity When people think of objects, they think of them as singular particles of matter. In fact, every object is made up of millions of particles, all of which behave differently when moved. When people observe a stick being. Start studying Physics: Chapter 10 - Center of Gravity. Learn vocabulary, terms, and more with flashcards, games, and other study tools. the combination of spin around the center of gravity and a movement through the air as if all. The point at which the average weight of all objects is located is called as the center of gravity. In this calculator, the center of gravity for multiple i number of objects or bodies placed in same line can be calculated based on the. Center of Mass vs. Center of Gravity Center of mass and center of gravity are two terms that are often used interchangeably, but they're really not the same. Let's take an object, like, for example, a 5 kilogram bowling ball. If you drop. 2009/11/24 · To find the center of gravity of a 3-D object? The problem is to balance a body in water so that it moves in a straight line when propelled. As the weight distribution in the body is not even it is tilted to one side and when. The center of gravity c.g. of an erect person with arms at the side is at approximately 56% of the person's height measured from the soles of the feet Fig. 1.3. The center of gravity shifts as the person moves and bends. The act. 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How to find the center of gravity for regular objects The center of gravity of objects that have uniform thickness and density is usually at its geometrical center. Examples of these are The black dot shows the location of the center of. Start studying physics TEST center of gravity. Learn vocabulary, terms, and more with flashcards, games, and other study tools. when you hang you arms below a support you are in a stable equilibrium. Your center of gravity is. Believe Receive Storiesに質問する Visual Studio Ansi C Odi Best Scoreチーム Amazon Prime Kingサイズのキルト Amazon Mobile Iphone 6s Plus Nfl Game Times日曜日1月19日 テレビフィリップス4k Ambilight 65 Vans Sk8 Hi Mid 2007 Bmw 328iホイール Iphone Xr 128gb Vs Iphone 8 Plus Hello 2 Heaven Chordsと言う Ek Ladki Ko Dekha予告編レビュー Dell Optiplex 7060タワー Malcolm Gladwell Underdogs Misfits Ips Empress 2

In this section, some basic discussions about model validation, goodness of the fit and model comparisons are given. Some very fundamental expressions for the model validation and comparisons will be presented but the statistical procedures of calculation and detailed discussions can be found in the literature some of which are given here. Widely used statistical errors are mean biased error (MBE) and root mean square error (RMSE). Percentage or fractional deviations of the estimated value with respect to the measured value can also be used. First two are defined as: where yci and ymi are the calculated and measured values of the variable. First one gives the over or under-estimation of a model in the long run while the second may read a high value even if a single measurement has high deviation from that of calculated. They can also used in a fractional form as: MBE = – У ——Уш n 1 Уші RMSE = / П К У—шш) (537) All the above expressions and also those given in Yorukoglu et al. (2006) can be used in the model validation, goodness of the fit and comparison. The values of the variables У may either be directly irradiation values or the fractional forms normal – izedby H0, that is H/H0 (these values may also be hourly values). The latter should be used essentially for clarifying the goodness of the fit between fractional solar irradiation H/H0 and fractional bright sunshine hours n/N (Yorukoglu et al. 2006). A work on comparing the two procedures of calculating MBE and RMSE values, namely Eqs. (5.34-5.35) and Eqs. (5.36-5.37), showed that the maximum differences for the statistical indicators are around 3% (Badescu 1988). Goodness of a fit is the representation quality of an empirical correlation that is obtained by regression analysis or by some statistical means (Yorukoglu et al. 2006) using the measured (or any given data) values between various variables having such relations. It mainly depends on the utilized method (for example the least square method) and the coefficients are only some mathematical constants for calculating one variable in terms of the others. Correlation coefficient R2, for example, is the most important indicator of the goodness of the fit which is defined as: where a is the standard deviation. Hence, R2 can have values between 0 and 1, and closer to 1 means better the regression result. Model validation should be considered as the justification of a physical or any analytical derivation of the relation between various variables which are believed to have correlations. Hence, the validness of the physical parameters introduced in the development of a physical model is important in the model validation which can either be checked with measurements (if exists any) or with the appropriate limiting values that can be assigned within the physical reasoning of the developed model. Most of the time, pre-given values are used in the models for these parameters but sometimes some of them can be obtained within the calculations of the constructed formalism. Of course, if these parameters can be calculated within the formalisms, their values must be close to those pre-assumed and/or measured values. In the model development for the meteorological variables discussed in this chapter, the physical models usually have both types of such parameters but the good point is that almost all read values within some specified ranges. For example, monthly effective value of the ground albedo might have a value from 0.13 to 0.22 for the semi-urban and cultivation sites as tabulated in Ineichen et al. (1990). In constructing the model essentially a measured data set should be used, but the same data set can not be used in justification of its universal applicability and/or in comparison with the estimations of different model approaches. If some relations exist between variables then it is valuable of course to seek a physical (analytical) means of describing such relations since it highlights the physical details hidden in such correlations. The coefficients then can be written in terms of the physical parameters of the analytical model. In our case, this empirical relation is the Angstrom-Prescott expression and for a linear form the coefficients are a and b. In the solar irradiation and bright sunshine hour relationships, both in the validation and comparison of the models and/or correlations, as mentioned above, measured values from different locations must be used but not those utilized in the construction of the model and correlation. In fact, as outlined in the Handbook of Methods of Estimating Solar Radiation (1984), a data set to be used for the validation and comparison must: • be randomly selected; • be independent of models being evaluated; • span all seasons; • be selected from various geographical regions; • be sufficiently large to include a spectrum of weather. Another point is the uncertainty in the measurements which put limitations to the level of confidence on validation and/or comparisons of the models. These errors of course reduce with increased averaging time interval. Hay and Wardle (1982) showed that the observation error of 5% for an individual observation was appropriate to an hourly time interval and reduced substantially with increased time averaging. Uncertainties that they observed for two locations in Canada had marked seasonal and inter-annual variability and also strong dependence on the observed irradiance (Hay and Wardle 1982).

Описание: Contains 41 reviewed papers, selected by the two program committees from a total of 101 submissions. Among the issues addressed are design and analysis of approximation algorithms, hardness of approximation, small space and data streaming algorithms, sub-linear time algorithms, embeddings and metric space methods, and more. Описание: This book constitutes the thoroughly refereed post proceedings of the Second International Workshop on Approximation and Online Algorithms, WAOA 2004, held in Bergen, Norway in September 2004.The 21 revised full papers presented together with 2 invited papers were carefully selected during two rounds of reviewing and improvement from 47 submissions. WAOA is devoted to the design and analysis of algorithms for online and computationally hard problems. Among the topics addressed are applications to game theory, approximation classes, coloring and partitioning, competitive analysis, computational finance, cuts and connectivity, geometric computations, inapproximability results, mechanism design, network design, routing, packing and covering, paradigms, randomization techniques, and scheduling problems. Описание: This book constitutes the thoroughly refereed post proceedings of the 4th International Workshop on Approximation and Online Algorithms, WAOA 2006, held in Zurich, Switzerland in September 2006 as part of the ALGO 2006 conference event.The 26 revised full papers presented were carefully reviewed and selected from 62 submissions. Topics addressed by the workshop are algorithmic game theory, approximation classes, coloring and partitioning, competitive analysis, computational finance, cuts and connectivity, geometric problems, inapproximability results, mechanism design, network design, packing and covering, paradigms, randomization techniques, real-world applications, and scheduling problems. Автор: Borkar, Vivek S. Название: Stochastic approximation ISBN: 0521515920 ISBN-13(EAN): 9780521515924 Издательство: Cambridge Academ Рейтинг: Цена: 7120 р. Наличие на складе: Поставка под заказ. Описание: This simple, compact toolkit for designing and analyzing stochastic approximation algorithms requires only basic literacy in probability and differential equations. Yet these algorithms have powerful applications in control and communications engineering, artificial intelligence and economic modelling. The dynamical systems viewpoint treats an algorithm as a noisy discretization of a limiting differential equation and argues that, under reasonable hypotheses, it tracks the asymptotic behaviour of the differential equation with probability one. The differential equation, which can usually be obtained by inspection, is easier to analyze. Novel topics include finite-time behaviour, multiple timescales and asynchronous implementation. There is a useful taxonomy of applications, with concrete examples from engineering and economics. Notably it covers variants of stochastic gradient-based optimization schemes, fixed-point solvers, which are commonplace in learning algorithms for approximate dynamic programming, and some models of collective behaviour. Three appendices give background on differential equations and probability. Описание: Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the “semidefinite side” of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms. Описание: This book provides a good opportunity for computer science practitioners and researchers to get in sync with the current state-of-the-art and future trends in the field of combinatorial optimization and online algorithms. Recent advances in this area are presented focusing on the design of efficient approximation and on-line algorithms. One central idea in the book is to use a linear program relaxation of the problem, randomization and rounding techniques.This state-of-the-art survey contains 11 carefully selected papers that cover some classical problems of scheduling, of packing, and of graph theory, but also new optimization problems arising in various applications like networks, data mining or classification. Описание: In a book that will appeal to beginners and experts alike, Oxford University’s Nick Trefethen presents approximation theory using a fresh approach for this established field.Approximation Theory and Approximation Practice is a textbook on classical polynomial and rational approximation theory for the twenty-first century. It uses MATLAB? to teach the field’s most important ideas and results and differs fundamentally from other works on approximation theory in a number of ways: its emphasis is on topics close to numerical algorithms; concepts are illustrated with Chebfun; and each chapter is a PUBLISHable Matlab M-file, available online.In addition, the book centers on theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges. Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment.Each chapter has a collection of exercises, which span a wide range from mathematical theory to Chebfun-based numerical experimentation. Описание: Describes the recursive partitioning methodology and demonstrates its effectiveness as a response to the challenge of analyzing and interpreting multiple complex pathways to many illnesses, diseases, and ultimately death. This book presents and applies the standard regression methods in the examples for comparison purposes. Описание: The routes to many important outcomes including diseases and ultimately death as well as financial credit consist of multiple complex pathways containing interrelated events and conditions. This book focuses on recursive partitioning strategies as a response to the challenge of pathway characterization. Автор: Burgin Mark Название: Super-Recursive Algorithms ISBN: 0387955690 ISBN-13(EAN): 9780387955698 Издательство: Springer Рейтинг: Цена: 19056 р. Наличие на складе: Поставка под заказ. Описание: New discoveries about algorithms are leading scientists beyond the Church-Turing Thesis, which governs the "algorithmic universe" and asserts the conventionality of recursive algorithms. A new paradigm for computation, the super-recursive algorithm, offers promising prospects for algorithms of much greater computing power and efficiency.Super-Recursive Algorithms provides an accessible, focused examination of the theory of super-recursive algorithms and its ramifications for the computer industry, networks, artificial intelligence, embedded systems, and the Internet. The book demonstrates how these algorithms are more appropriate as mathematical models for modern computers, and how these algorithms present a better framework for computing methods in such areas as numerical analysis, array searching, and controlling and monitoring systems. In addition, a new practically-oriented perspective on the theory of algorithms, computation, and automata, as a whole, is developed. Problems of efficiency, software development, parallel and distributed processing, pervasive and emerging computation, computer architecture, machine learning, brain modeling, knowledge discovery, and intelligent systems are addressed.Topics and Features:* Encompasses and systematizes all main classes of super-recursive algorithms and the theory behind them* Examines the theory's basis as a foundation for advancements in computing, information science, and related technologies* Encompasses and systematizes all main types of mathematical models of algorithms* Highlights how super-recursive algorithms pave the way for more advanced design, utilization, and maintenance of computers* Examines and restructures the existing variety of mathematical models of complexity of algorithms and computation, introducing new models* Possesses a comprehensive bibliography and indexThis clear exposition, motivated by numerous examples and illustrations, serves to develop a new paradigm for complex, high-performance computing based on both partial recursive functions and more inclusive recursive algorithms. Researchers and advanced students interested in theory of computation and algorithms will find the book an essential resource for an important new class of algorithms. Описание: Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research. Автор: Evripidis Bampis; Klaus Jansen Название: Approximation and Online Algorithms ISBN: 3642124496 ISBN-13(EAN): 9783642124495 Издательство: Springer Рейтинг: Цена: 8084 р. Наличие на складе: Поставка под заказ. Описание: Constitutes the refereed post workshop proceedings of the 7th International Workshop on Approximation and Online Algorithms, WAOA 2009, held in Copenhagen, Denmark, in September 2009 as part of the ALGO 2009 conference event. ООО "Логосфера " Тел:+7(495) 980-12-10 www.logobook.ru

This is College Physics Answers with Shaun Dychko. The route taken in question seven by this person walking is almost the same as the route taken in question five, except the vector B instead of being 40 degrees south of west as it is in question five, it'll now be 40 degrees north of east. So it's in the opposite direction. We'll call it B prime. B prime is the negative of vector B. To find this new resultant which goes from here to there, that's R prime, we'll find the x component will be the sum of the x components of vector A and vector B prime. The y component of the resultant will be the sum of the y components of the vectors that add together. So our x prime is Ax and the vector A, I didn't give it a prime because it is exactly the same as the vector in question five. So to find the x component of vector A, we multiply its length of 12 meters by the sine of 20 degrees because this leg here, that's this bit here, is the opposite leg of this right triangle. So we need to use sine multiplied by the hypotenuse to find the opposite leg. It's in the negative direction so we just put that negative sign there. So it's negative 12 meters times sine 20, then plus the x component of vector B prime. This x component is -- let's get rid of this yellow that's kind of annoying here. This x component of B prime is here, and that is the adjacent leg of this triangle. So we multiply the length of the hypotenuse, 20 meters, by cosine of 40 degrees in order to find the adjacent leg. That's positive because it's directed to the right and this makes 11.217 meters is the xcomponent of the resultant. Then for the y component, we have the y component of vector A is positive because it is directed upwards, and the y component is is here, and it's the adjacent leg of that triangle. So we go 12 times cosine 20 and then plus the y component of B prime is 20 meters times the sine of 40 because this y component is the opposite leg of the triangle. That makes 24.132 meters. So the length of the resultant then, now that we know its components, is the square root of the sum of the squares of the components. So it's the square root of 11.217 meters squared plus 24.132 meters squared giving us 26.61 meters. Then the direction of R prime is going to be the inverse tangent of its y component divided by its x component. So, what we've figured out so far is the x component of R prime and we have figured out the y component of R prime, and what we're finding out now is this angle in here. This is theta prime. We can see that Ry prime is the opposite leg and Rx prime is the adjacent leg of this triangle. So to find this theta we would take the inverse tangent of the opposite divided by the adjacent. So that's inverse tangent of 24.132 meters divided by 11.217 giving us 65.07 and that is an angle towards the north compared to east, so north of east. There we have it. The R prime vector is 26.6 meters, 65.1 degrees north of east. Then in part B, we are told to say that the first leg of the trip is the same as vector B from question five. But then the second leg is the opposite of vector A. So we have initially a route that takes us 40 degrees south of west, 20 meters, and then 20 degrees east of south, 12 meters. So R, we'll call it double prime now, this resultant here, this is R double prime, is vector B plus vector A double prime which I've drawn in green here. The vector A double prime is the opposite of vector A and so this is B minus A. Whereas before we had A minus B, now we have B minus A and our job is to show that well, R prime which is A minus B and which I have re-written here, and we can write this as the negative of bracket B minus A where I just -- you know, I multiplied by negative one twice which is why you can do it because negative one times negative one is one, put a negative in front there and multiply each thing by negative one to change its sign. We end up with negative bracket B minus A. If you were to expand this bracket you would find that the B becomes minus and the A becomes positive and we're left with what we started with. So that's why it's the equivalent. So this is the negative of B minus A which is R double prime. So there we have it. R prime is meant to be the negative of R double prime. It's what we expect in this question is to get our answer of 26.6 meters, 65.1 degrees. Instead of being north of east, we expect it to be south of west. Yep, there. So, let us proceed. Same process as before, the x component of the resultant is going to be the x component of the B vector plus the x component of the A double prime vector. So that's a negative 20 meters because B x component is directed to the left, times cosine of 40 because it's the adjacent leg of this triangle. The x component of the A vector is positive because it's directed in the positive direction, that's this bit here is the x component of the A vector. That's 12 times sine of 20 because it's the opposite leg and so we use sine multiplied by the hypotenuse to get the x component of the A double prime vector. This works out to negative 11.217 meters. For the y component of the resultant, we have B y plus A double prime y, and that's negative 20 times sine 40 because this is the opposite leg of this triangle here and we're going to direct it downward and so put this minus sign there. We have negative 20 times sine 40, and then minus 12 times cos 20. That works out to negative 24.132 meters. Then to get the magnitude of this resultant we take the square root of the sum of the squares. So that's the square root of negative 11.217 meters squared plus negative 24.132 meters squared, giving us 26.612 meters. Then the direction will be the inverse tangent of the y component of the resultant. So just to draw this a little bit out of the way of all the other stuff there, we have a resultant vector here like this and what we've found out so far is its x component and its y component. What we're going to find out now, is this angle here, theta double prime. So it'll be the inverse tangent of the opposite over the adjacent, so the inverse tangent of negative 24.132 divided by negative 11.217, giving 65.07 degrees and that it is towards the south compared to west. So R double prime is 26.6 meters, 65.1 degrees south of west and we were asked to confirm that this R double prime is the opposite of R prime or the negative of R prime in other words, and we have shown that because the magnitude is the same and the angle is of the same size but in the opposite direction, south of west, instead of north of east.

5 edition of Mathematics applied to business problems found in the catalog. Mathematics applied to business problems M. D. MacPherson |LC Classifications||HF5694 .M3| |The Physical Object| |Pagination||ix, 341 p. ;| |Number of Pages||341| |LC Control Number||80069964| Business math is a type of mathematics course that is meant to teach people about money and provide them with the tools they need to make informed financial decisions. Business math not only teaches about the specifics of finances related to owning and operating a business but also offers helpful advice and information related to personal finance. Business Math / Algebra Based. Finite Math and Applied Calculus. Finite Mathematics. Mathematics for Business. Mathematics for Business with Graphing Calculators. Technical Calculus. Technical Mathematics. Technical Mathematics with Calculus. Select the picture that shows that the left part is bigger than the right part. Which sand pie has four equal parts? Alex buys three bags of marbles. The marbles are priced differently. How How many red numbers are there? How many city blocks are there between River street and State street on the If yesterday was Wednesday, what day. This market-leading text is known for its comprehensive coverage, careful and correct mathematics, outstanding exercises, and self contained subject matter parts for maximum flexibility. The new edition continues with the tradition of providing instructors and students with a comprehensive and up-to-date resource for teaching and learning engineering mathematics, Author: Erwin Kreyszig. I should note that there is another open applied finite mathematics textbook: Business Precalculus by David Lippman. That book has much of the same content, but also has a number of homework exercises and ancillary materials available in MyOpenMath, a free and open alternative to MyMathLab developed by David Lippman/5(2). problems on your calculator. After this you should work the exercises. Working through the chapter sample exam will help you review all of the material in the chapter. Good Luck! 9/22/ Madison College’s College Mathematics Textbook Page 1 of File Size: 2MB. A letter from the Reverend Mr. Kinnersly, to the Right Honourable the Earl of Sunderland Look alike drugs dialectic of immaterialism source for Shakespeares Malvolio 51st Canadian Geotechnical Conference Genealogy of the Papehia family King of the beatniks U.S. trade policy Elements of regional economics Concussive Brain Trauma 150 years St. Marys Parish, Stockton-on-Tees, 1842-1992 This book consists of mathematical tools related to business problems. It covers the syllabus of business mathematics for the students of Commerce Faculty. This book stands out among applied mathematics books. I like it very much, although it is certainly not easy to read. The equations and mathematics in the book are not particularly intimidating; the book is challenging because it gets you to think about difficult concepts at a sort of "meta" level. Applied Mathematics Guided Textbook Solutions from Chegg. Chegg's step-by-step applied mathematics guided textbook solutions will help you learn and understand how to solve applied mathematics textbook problems and be better prepared for class. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle. An introduction to Business Research Methods. Engineering Mathematics: YouTube Workbook. Partial Differential Equations. Essentials of Statistics. Blast Into Math. Applied Statistics. Integration and differential equations. Elementary Algebra Exercise Book I. Principles of Insurance. Essential Engineering Mathematics. Mathematics for Computer. Math for Business and Finance: An Algebraic Approach. 2nd Edition, By Jeffrey Slater and Sharon Wittry. Practical Business Math Procedures. 13th Edition, By Jeffrey Slater Mathematics applied to business problems book Sharon Wittry. Connect/SmartBook Requires Students to Read and Engage with Course Content. View All Case Studies > Featured Products. Math For Business And Finance: An. Business MBA students who studied business mathematics and statistics using the framework laid out in this book. Th eir comments and feedbacks were equally important and useful in making this book an excellent guide into the often-challenging fi elds of mathematics and statistics. I hope and wish that theFile Size: KB. About the Book Author Mary Jane Sterling is the author of four other For Dummies titles: Algebra For Dummies, Algebra II For Dummies, Trigonometry For Dummies, and Math Word Problems For Dummies. She has honed her math-explaining skills during her years of teaching mathematics at all levels: junior high school, high school, and college. e-books in Applied Mathematics category Topics in Applied Mathematics and Nonlinear Waves by Per Kristen Jakobsen -The selection of topics in this text has formed the core of a one semester course in applied mathematics at the Arctic University of Norway. 'The words 'applied mathematics' decorate the covers of many books these days, but precious little of the practice of applied mathematics seeps through into the pages bound between those covers. Of the select few that do indeed introduce the reader to applied mathematics, this book may well be the by: Applied business mathematics: problems and drills by Fairbank, Roswell E and a great selection of related books, art and collectibles available now at Business Mathematics Module - Free download Ebook, Handbook, Textbook, User Guide PDF files on the internet quickly and easily. John Bird is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth, UK. More recently, he has combined freelance lecturing at the University of Portsmouth, with examiner responsibilities for Advanced Mathematics with City and Guilds and examining for International Baccalaureate. Presents a variety of problems and situations found in business where arithmetic is constantly used. Prepares students to use mathematics in solving monetary and valuation problems in business and personal finance. Prerequisites: WRRDand MTH 20 or equivalent placement test scores. Audit available. Intended Outcomes. An Introduction to Business Mathematics. Some standard references of Applied Mathematics are, e.g., Bronstein et al certain problems of a quantitative nature in a Author: Henk Van Elst. Applied Mathematics, Fourth Edition is a thoroughly updated and revised edition on the applications of modeling and analyzing natural, social, and technological processes. The book covers a wide range of key topics in mathematical methods and modeling and highlights the connections between mathematics and the applied and natural : J. David Logan. Mathematics is an exceptionally useful subject, having numerous applications in business, computing, engineering and medicine to name but a few. `Applied mathematics’ refers to the study of the physical world using mathematics. This book approaches the subject from an oft-neglected historical perspective/5(36). Business mathematics also includes statistics and provides solution to business problems. Business is always surrounded with challenges which need to be dealt with in a proper fashion so that they do no arise in future. These problems that occur on a daily basis can be effectively solved with the help of mathematical models. SOLVING APPLIED MATHEMATICAL PROBLEMS WITH MATLAB® Dingyü Xue YangQuan Chen 3 9/19/08 PM. Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, andapplied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems. This edition of Applied Mathematics for the Petroleum and Other Industries is designed to provide both a text and reference source in one book that covers basic mathematics as well as advanced technology. Using a handheld calculator with basic mathematic functions as well as specialized functions (such as exponents, logic functions.Applied Mathematics by Example: Theory 10 Introduction by the Author Introduction by the Author Mathematics is an exceptionally useful subject. In our technological society, it has ap-plications in business, in computing, in engineering, in medicine and in many other disciplines too. Of all academic quali cations, A-level mathematics is the one File Size: 2MB. Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge.

Little fantasy impact on cut-down day As cut-down days go, 2012's edition was a virtual missing persons bureau. I daresay there wasn't a single player you were considering starting in your fantasy league, no matter how deep or what the scoring format. But there are ramifications from a few of Friday's NFL moves, so let's do a quick review: Washington Redskins cut RB Tim Hightower: Hightower suffered a torn left ACL in Week 7 last year, but his knee healed and Mike Shanahan talked him up as the Skins' starter pretty much ... Read full story 2012 Fantasy Football Draft Kit ESPN.com's fantasy analysts offer all the information you'll need to succeed in your draft and all season long. • Draft Kit Home • Join/Reactivate a league • Follow ESPN Fantasy on Facebook • The NFL on ESPN.com Profiles and Projections for 2012 • Top 300 | Cheat Sheets • Analysts: Berry | Harris | Karabell • Quarterback: Rankings | Preview • Running back: Rankings | Preview • Wide receiver: Rankings | Preview • Tight end: Rankings | Preview • Def./Special teams: Rankings | Preview • Kicker: Rankings | Preview • IDP: Rankings | Preview • Other formats: PPR | TD-Only | Keepers • Analysts: Berry | Harris | Karabell • The Magazine: Position profiles The Talented Mr. Roto • Mock 1: 10-team standard (May 7) • Mock 2: 12-team standard (June 14) • Mock 3: 12-team PPR (July 17) • Mock 4: 10-team standard (Aug. 7) • Mock 5: 10-team, 2 QBs (Aug. 13) • Mock 6: 10-team auction (Aug. 21) • Mock 7: 12-team standard (Aug. 29) • Twitter Mock Draft Geico Fantasy Draft Special Podcast • Top 2012 storylines: 20 to 11 (Aug. 3) • Top 2012 storylines: 10 to 1 (Aug. 7) • Quarterback preview (Aug. 10) • Running back preview (Aug. 17) • Wide receiver preview (Aug. 21) • TE, D/ST, K preview (Aug. 28) • Injury updates: QB | RB | WR | TE • Staff sleepers and busts • Top faces in new places • Are TEs worthy of flex consideration? • Ten hotly debated players for 2012 • Does the two-QB approach work? • Better predicting kickers' values • Don't overanalyze strength of schedule • How to use value-based drafting • Will Cam Newton repeat his rookie success? • Metrics to consider in finding IDP breakouts • Streaming D/ST units • Finding potential rebound candidates • Harris' 10 flag-planted players for 2012 • Harris' super-deep sleepers for 2012 • C.J. Spiller's effect on Fred Jackson's value • Where should Trent Richardson be drafted? • Will Green-Ellis' TD success continue? • Ben Tate's impact on Arian Foster's value • Will Cruz, Nelson fall back to earth? • Is Jones-Drew poised for drop-off? • When to draft Darren McFadden • Demaryius Thomas or Eric Decker? • Are Panthers RBs unstartable? • How much will Gore's fantasy value decline? • How will Charles, Hillis split workload in KC? • When should you draft Adrian Peterson? • Rivers' new favorite WR: Meachem or Floyd? • Vincent Jackson's impact with new team • Choosing between Bryant and Austin • The polarizing Michael Turner • Roddy White or Julio Jones? • Lloyd effect on Pats' other receivers • Will any one Patriots RB emerge in 2012? • Making sense of Saints backfield • Can you trust DeMarco Murray? • What is Tim Tebow's value with the Jets? • Assessing the values of Bucs RBs • Can Robert Griffin III match Cam Newton? • Joyner: Underrated PPR WRs for 2012 • Karabell: Top 35 rookies for 2012 • Kiper: Keeper-league targets • Joyner: Underrated WRs for 2012 • Joyner: Underrated RBs for 2012 • Kiper: Fantasy rookies to target, avoid • Joyner: Underrated QBs for 2012 • Karabell: Will DeSean Jackson bounce back? • Joyner: How O-Lines impact QBs in fantasy • Karabell: Reggie Bush outlook • Karabell: Early ADP trends • Karabell: Redskins RB situation • Joyner: Undervalued, overvalued RBs • Karabell: Be wary of Steve Smith • Joyner: Why CJ2K will be top fantasy RB • Joyner: Five breakout players in 2012 • Karabell: Five keeper-league targets • Adrian Peterson outlook • Peyton Manning's health • Peyton Manning expectations • Darren McFadden outlook • Andre Johnson's health • Projecting Cam Newton • Second-tier RBs to consider • Arian Foster versus Ray Rice • Will DeSean Jackson bounce back? • Michael Vick expectations • Mark Sanchez versus Tim Tebow • Will Rob Gronkowski repeat as top TE? • Where to draft Gronkowski, Graham • Aaron Hernandez expectations • Can RG3 match Newton's rookie season? • Will Jamaal Charles return to form? • DeMarco Murray outlook • Is Beanie Wells being undervalued? • Rashard Mendenhall versus Isaac Redman • How will Drew Brees handle turmoil in NO? • Why has Matt Schaub's ranking fallen? • Is this the year Matt Ryan breaks out? • Tight end sleepers for 2012 • Jonathan Stewart versus DeAngelo Williams • Which rookie RBs could surprise? • Demaryius Thomas versus Eric Decker • Better late pick: Rookie WR or Randy Moss? • Is Jordy Nelson's TD total repeatable? • Is Eli Manning an elite fantasy QB? • Will Philip Rivers bounce back in 2012? • Can Victor Cruz repeat his breakout 2011? • Andrew Luck's short- and long-term value • Expectations for Trent Richardson • Will Julio Jones outperform Roddy White? • Is it wise to draft a QB early? • Will Reggie Bush repeat his 2011 numbers? • Brandon Marshall expectations

The analysis of the risk of interference into space stations assumes a probability basis, since specific data on locations, etc. of radio-relay stations are not available. The analysis is based on estimating the probability for the GSO case and then using the same model to assess the relative risk for the IO case. For the IO case, 5° inclination has been chosen as representing a reasonable value for the analysis. Lower inclination values yield a proportionally lower estimate of potential interference, while higher values will yield a higher estimate. 2.4.2 The model Radio-relay stations which can have mainbeams which intersect the GSO are limited to those with a particular radiation azimuth at a specific latitude. There are four such points for each satellite location on the GSO accounting for both North and South latitudes as well as locations East and West of the GSO location. Making an allowance for the beamwidth of the radio-relay antenna, its elevation angle and accounting for refractive effects, a small strip similar to that of Fig. 2a is established, which contains all the stations which could have an intersection with the GSO or the IO at some point in time. The width of this strip is a function of the values assumed for the model parameters. The model parameters assumed here are: elevation angles from 1° to 4° for the radio-relay antenna and a range for the radio refractivity from 250 to 400. This latter factor adds 2° to the effective range of visibility at the outer edge of the strip. It is also assumed that the beam centre is separated from the orbit under study by 1.5° to account for the beamwidth with some margin. For the assumed parameters, the width of the strip for the GSO case is approximately 7°. The number of stations located within this strip is a function of the area of the strip and an assumed density for the terrestrial stations. The area need only be calculated for one quadrant from the equator to 70° latitude and, by symmetry, it applies to all quadrants. Intersection with a specific point on the GSO can take place from all four quadrants. For the 7° (775 km) strip of one quadrant, the area of the strip is 7 875 000 km2. The width of the strip is not changed by the inclination at low latitudes, but it increases with latitude depending upon the inclination. For the case of 5° inclination, the area of the strip in one quadrant is 13 230 000 km2 and the number of stations would be expected to be 1.68 times that of the GSO case. This result will vary directly with inclination and it can be taken as representative of the effects of IO. 2.4.5 Quantitative assessment Estimates of the number of potential intersections can be made by assuming a maximum density of radio-relay stations for all of the land area contained in the strip which has sufficient population to justify the assumption. The maximum density of one station per 2 500 km2 allows for a station every 50 km in all directions. This corresponds to the normal single hop distance used by radio-relay designers. It is further assumed that with considerations of population density and the effects of ocean areas, the area of concern would be of the order of 20% of the total. Random pointing of the radio-relay antenna with a 2° beamwidth is assumed and applying these considerations to the GSO case, the total number of stations with possible intersections would be about 14 while those for the inclined orbit case would be about 24. 2.4.6 Practical considerations This model contains several assumptions which are very conservative, such as: – radio-relay elevation angles from 1° to 4°, – the use of a uniform density of radio-relay stations, – uniform 2° range of elevation angle, considered to be representative, – a standard nominal refractivity value of 300. Adjustments can be made to this model, the net effect of which is to reduce the area of concern to 42% of the original model for the GSO case and to 64% for the inclined orbit case. The number of potential exposures is reduced to about 6 for the GSO case and 15 for the inclined orbit case. 2.4.7 Actual experience A review of INTELSAT satellites in the GSO experiencing interference from terrestrial radio relay sites has shown that the effects have been minor. In fact only one such case has been recorded over the last ten years. 3 Earth station/terrestrial station interference The absence or cessation of North-South station keeping of a geostationary satellite will cause it to change its orbital inclination continually. An earth station operating with such a satellite may have to track it with its antenna mainbeam through an apparent diurnal trajectory (a narrow figure-of-eight). When such an earth station has been coordinated with stations of terrestrial services for “strictly” geostationary operation (satellite movement within prescribed or stated small positional tolerances), the need to follow a satellite having or acquiring significant orbital inclination will cause the earth station antenna mainbeam to assume elevation angles and, associated with these, azimuth angles that are different from (both less and greater than) those for which coordination had been effected. Of particular concern is the case when elevation angles are less than required for geostationary operation because in this case the resulting higher earth station horizon antenna gain can cause potential interference from and to a terrestrial station. 3.2 Geometric considerations 3.2.1 Analytic expressions The elevation angle (s) and azimuth (s) of the mainbeam of an earth station towards a space station in geostationary inclined orbit at the point of maximum excursion are given by the following expressions: s arc sin ((K A – 1.0) / B) (10) s 90.0 arc cos (K cos i sin / B cos s) (11) A cos i cos cos sin i sin (12) B (1.0 K 2 – 2K A)0.5 (13) Note that arc cos (x) 180.0 arc cos (x) K : geostationary orbit radius/Earth radius, assumed to be 6.62 i : orbit inclination (ve ascending node East of Greenwich) : earth station latitude (ve North) : difference in longitude between the space station and the earth station. 3.2.2 Loss of discrimination As the satellite goes into an inclined orbit, the elevation angle and azimuth vary with time. This may result in variations in gain toward the horizon, as discussed in the following sections. Both the elevation angle reduction and the associated azimuth shift are not only functions of the orbital inclination of the satellite being tracked, but also of the latitude and relative longitude (longitude difference to the subsatellite nodal point) of the earth station in question, as shown in Figs. 9a, 9b, 9c and 9d for the two orbital inclinations 5° and 10°, respectively. In Figs. 9a, 9b, 9c and 9d, the outer circumference describes those locations on the surface of the Earth (in terms of latitude and longitude relative to the subsatellite nodal point) at which the earth station antenna mainbeam elevation angle to the inclined-orbit satellite is never less than 3°. The inner, almond-like area contains that part of the Earth’s surface at which the elevation angle is never less than 48° and, thus, not subject to variations of horizon antenna gain (assuming that beyond an off-beam axis angle of 48° there is no appreciable change in antenna gain). FIGURE 9a...[D15] 10.5 CM FIGURE 9b...[D16] 12 CM FIGURE 9c...[D17] 11 CM FIGURE 9d...[D18] 11.5 CM In the upper diagram of each figure, the broken lines show the magnitude of the earth station horizon antenna gain increase (dB), based on an antenna pattern of the form A – 25 log dB. In the lower diagram the broken lines show, for the corresponding earth station locations, the shift in earth station antenna mainbeam azimuth from that associated with strictly geostationary operation to that at which the satellite is seen under the lowest elevation angle in its inclined orbit. The azimuth shift is always towards the equator. Earth stations on the satellite node’s longitude have the greatest horizon antenna gain increase, and the smallest azimuth shift; earth stations near the equator have the smallest horizon antenna gain increase and the largest azimuth shift. The greater the inclination, the greater are both the horizon antenna gain increase and the azimuth shift. Figure 9d shows, as an additional set of curves, the lateral shift of a “common volume” at 4 km altitude. This is the highest altitude from which rain scatter interference can be expected. 3.3 Effect on earth station coordination area Due to the variation in azimuth and elevation angle, earth stations which were previously coordinated with terrestrial systems on the basis that they would operate with a satellite on the GSO may be affected by the use of inclined orbits. Additional terrestrial stations may also be affected. The new coordination contours would then be a function of the GSO location(s) and the arc for which they were calculated. There will be a wide variety of situations which will affect many earth stations in the world. A set of boundary conditions has been examined which may help in assessing the potential problem of re coordinating earth stations where it proves to be necessary. Earth stations which can operate with inclined orbit satellites with 5° inclination are limited to those with elevation angles of 5° to 10° at the nominal GSO location depending on the earth station latitude. The largest dimension of the coordination contour is in this case based on an antenna gain in the horizontal plane of 7 to 14.5 dB. At a receiving station this gain sets the interference sensitivity, while at a transmitting station, it sets the e.i.r.p. density in the horizontal plane. For an earth station at the equator which has a low elevation angle, the antenna movement in azimuth is approximately twice the inclination angle. However, in this case the elevation angle changes very little. For an earth station operating at 5° nominal elevation angle, the increase for the range-influenced azimuthal directions of 50° for the case of 5° inclination is from 0 to 4.9 dB. The impact on the coordination area for an earth station that had been coordinated at 5° at 6 GHz using the maximum allowable e.i.r.p. density of 40 dB(W/4 kHz) is a broadening of the contour around the main-beam region with no change on the nominal azimuth. For earth stations at high latitude, the azimuth changes very little while the elevation range will be approximately equal to twice the inclination. Inclined orbit operation would therefore be limited to earth stations with a nominal GSO elevation of 10°. This could mean going as low as 5° during part of the tracking time and would yield a gain increase of 7.5 dB on that azimuth resulting in an increase mainly along the mainbeam azimuth. There are locations where both the azimuth and elevation angles change by about the same amount, but this change is less than either of the above cases. Here the gain changes will be less than the extreme cases and the coordination area in all cases changes most in the region of the mainbeam of the earth station. Figure 10 provides an overview of such a case. 3.3.2 Change in coordination distances The effects on the coordination distances are due to elevation angle changes. For nominal elevation angles in the range of 15° to 20°, the gain change would have a maximum value of about 4.4 dB. For elevation angles greater than 20°, a 5° elevation angle change from nominal will result in a maximum increase in the gain in the horizontal plane of 3 dB. For nominal elevation angles larger than about 53°, there will be no gain change. FIGURE 10...[D19] 16.5 CM Results of a study on change in maximum coordination distance Mode 1 (great circle propagation) as a function of earth station latitude and longitudinal differences, when compared with a typical 6 GHz earth station coordination contour are presented in Fig. 11. A transmitter power of 20 dBW and 2 MHz energy dispersal have been assumed. The increase of the maximum coordination distance is of the order of 10-20% for nominal antenna elevation angles between 10° and 20°, and a few per cent for the higher elevation angles. The change in per cent is independent of the propagation zone considered (Zones A, B and C). The changes for propagation Mode 2 (scattering due to hydrometeors) are small and in most of the cases, less than 6%. A stochastic study assessing the input of inclined orbit operations into terrestrial stations has been carried out in which earth stations at latitudes 41° N were used and constrained so that their elevation angles were the local maximum towards the inclined orbit. The results for 15° orbit inclination were a 4.5 dB increase for 23% of the terrestrial receivers for at least 10% of the time (2.4 hours daily) while for 5° inclination 38% of the receivers would receive 1.5 dB increase in interference for 10% of the time. The expected interference increase will be less for higher elevation angles or for azimuths of the terrestrial receiver that tend to point away from the geostationary-satellite orbit rather than the uniform distribution that was assumed. However, for higher latitudes with smaller elevation angles, the expected interference will be greater. FIGURE 11...[D14] 19 CM Countries close to the Equator will generally not require re-coordination of their earth stations when these operate with an inclined-orbit satellite. Even when their relative longitudes do not fall within the almond-shaped zone above 48° elevation angle, there will be only little earth station antenna horizon gain increase which can usually be neglected. Countries at higher latitudes are increasingly more affected and may, in some cases, have difficulties coordinating and especially re-coordinating their earth stations to operate with larger orbit inclinations of their satellite(s). However, in all cases a trade-off is possible between accepting increased difficulties in earth station coordination and more extensive inclined-orbit operation. This Annex has examined the sharing situation between the fixed and the FSS when satellites go into inclined orbit. The impact caused to terrestrial networks results from both the space and earth stations. Similarly, satellite networks will also be affected by interference caused to space and earth stations. The sharing situation when satellites go into slightly inclined orbits is complex. For a short period, it involves greater exposure of fixed service receivers to direct satellite interference and vice versa. The number of terrestrial stations exposed to interference increases with the amount of inclination. The effect of such exposure upon the system unavailability as well as total interference received has been studied in this Annex. A model has been developed which indicates that under the assumptions of that model an order of magnitude increase in unavailability could be expected. These assumptions include: a system with adaptive power control, all satellites in 10° inclined orbit using 124 dB(W/m2) power flux-density and satellites spaced 3° apart. However, it is noted that with the trend toward the use of small earth station antennas, it is doubtful that large inclination angles will be used by satellite operators. Other models, based on an assessment of the distribution of beams from an actual radio-relay network, indicate that the total end-to-end interference may in some cases be reduced, depending upon the interference exposure factor. In a further model, calculation of the aggregate interference over a hypothetical reference circuit shows that the interference does not increase but is re-distributed over the length of the network. Further studies are needed on the models to be used for interference calculations. Additional information is required on the distribution of terrestrial beams around the orbit. Studies are also necessary which would aid in the development of techniques for both services to ameliorate the interference situation, particularly for low inclination angles. These include such techniques as the methods of using automatic power control, interference cancellers, using wide-band pfd limits, satellite antenna pointing restrictions, limits to inclination, coordination procedures, site shielding and others. With regard to the effect upon the coordination area between earth stations and space stations, the impact varies with elevation angle, azimuth and longitude of the earth station. The resulting increases in coordination distances vary with the degree of inclination. When the satellite inclination is 5° and the nominal elevation angle of the earth station is between 10° and 20°, the increase of the maximum coordination distance, compared with the case where no inclined orbit operation is performed, is of the order of 10-20% and for higher elevation angles the increase is a few per cent.

How It Works To calculate the distance travelled, we need to know the initial velocity and the time it took for the ball to hit the floor. You may need a few sheets in order to make sure the various landing positions can be recorded. One different thing to test would be the affect of friction on an object over a lasting period of time. Making a graph using a computer While it is important to know how to plot data by hand, in practice physicists typically use a computer program to plot data. The plotting tool program automatically does a linear fit to your data, and also provides the slope estimate and its uncertainty. This will help me get a better understanding of the force of air resistance. Our percent deviation in the range was 10% which is rather high considering the experiment and conditions. Because horizontal and vertical information is used separately, it is a wise idea to organized the given information in two columns - one column for horizontal information and one column for vertical information. This heart pounding scenario is one example of horizontal projectile motion. You will perform two experiments to aid your understanding of these principles, which will be described later in the lab. Take at least one set of data for each person in your lab group. First, to find the maximum range at which the ball travels, the ball had to be fired at various angles. Forces: Often a force is described as a push or a pull. The speed of the balls out of the launcher should not depend on the heights of the launcher. This applies to long jumpers and soccer balls that are two good examples. During data collection, you should tape the carbon paper with a piece of white paper underneath it to the floor, so that the ball will hit the paper and leave a mark of its landing point from wherever it is released on the ramp. Each launcher should be clamped to the edge of a lab bench and aimed so that the ball will land on the floor without hitting any other lab groups. Projectile motion into physics Objective: Our purpose for this lab was to observe projectile motion and use the equations of motion to predict the objects location in different instances of time. Basing off my knowledge of physics, the relationship between a projectile and its angle depends on how large the angle is: the larger the angle, the shorter the distance between the launcher and the landing point. From this, you can calculate the vertical distance that it fell and compare your calculated value and the measured value. Discussion Questions What is the relationship between initial velocity and the distance traveled? Where should you reach to grab it before it hits the floor to stop the screen from shattering? How do you think increasing the initial velocity would affect how far the ball traveled? Also, the time of flight can be found. Two angles are complimentary if they, when added together, equal 90 degrees. Put the time-of-flight plate on the floor where the ball lands. Practice and patience are required to ensure that the ball accurately lands on the pad and the time of flight is properly recorded. Determine the initial horizontal velocity of the soccer ball. Our next objective is to determine at what angle that the ball will be ejected at the maximum range. Since each equation has four variables in it, knowledge of three of the variables allows one to calculate a fourth variable. While problems can often be simplified by the use of short procedures as the one above, not all problems can be solved with the above procedure. If there were any other force acting upon an object, then that object would not be a projectile. Whenever you launch a ball, position one member of your lab group ready to catch the ball after it lands to avoid losing the ball or interfering with other students in the room. The path followed by a projectile is called its trajectory. Solving Projectile Problems To illustrate the usefulness of the above equations in making predictions about the motion of a projectile, consider the solution to the following problem. The only velocity it has is just its horizontal velocity, vx. The vector nature of forces can be used to determine how far an object launched can go and its initial velocity at an angle of 0 by finding its x and y components separately. First, to find the maximum range at which the ball travels, the ball had to be fired at various angles. The components of velocity are found by taking the initial velocity multiplied by sin for the y component, and cos for the x component. Since these two components of motion are independent of each other, two distinctly separate sets of equations are needed - one for the projectile's horizontal motion and one for its vertical motion. Physics, particularly Uniform Accelerated Motion can be seen in many rides in an amusement park. Find a starting point on the launcher that gives the ball a reasonable velocity. In this experiment we learned how to determine the initial velocity of a ball that is launched horizontally out of a projectile launcher, how to verify the angle of projection that will produce maximum range, and to predict and verify the range that a ball will travel when launched at a set angle. Imagine starting at the origin, you can reach a destination position by first moving along the x axis and then along the y axis. Initial velocity: The photogates are approximately 10 centimeters apart measure directly to confirm this. Students must have some basic algebra skills and knowledge of physics. Once a projectile has been launched, the only forces acting are: Air friction this is considered negligible in our experiment Lift force, if the object is behaving like a wing this is also negligible as our object is a ball Gravity, the weight force which acts downwards this is the value we will be calculating in our experiment In our experiment we will measure the projectile motion of a bouncy ball using the computer programme Motion Tracker. What sources of error may have impacted your results? The range of the tests was. The Newtonian mechanics principles that you have been studying allow you to predict this type of motion quite well. The period of swing decreased as the period T increased. The forces involved in projectile motion are the initial velocity of the projected object at a certain angle and gravity acting downward on the object. Unfortunately, there is no printer in the lab room or in the Physics Help Room, but the plotting tool allows you to send your graph to your email account to print out or copy directly into your report later. The solution of this problem begins by equating the known or given values with the symbols of the kinematic equations - x, y, v ix, v iy, a x, a y, and t. Theory: Projectile motion according to Dr. The shell fired at an angle closest to 45 degrees lands farther away.

In my last couple of blog posts, I looked at changes in the Maths curriculum and also highlighted the fact that there has been a significant fall in the number of students who are taking up A-level Maths. The reasons attributed to these are: the harder GCSE Maths, which has made more young people less confident about taking A-level Maths; the decoupling of AS Maths and, finally, the reduction in the number of A-levels that students are allowed to take in their first year of sixth form study. The changes make it more risky for young people to want to commit to two years of a course without the conviction that they will end up with a decent pass grade at the end of it. There has been speculation in the media on the issue of fewer students taking up A-level Maths as between 2002 and 2014, Maths had become the most popular A-level. The media interest in this matter has resulted in people becoming more aware about the usefulness and the potential benefit of studying A-level Maths up to A-level. Sadly, this year, there is a significant drop in numbers again. Is Maths rocket Science? Yes, yes and yes but perhaps you can do it! One fact that we have to face is that Maths is generally globally perceived as a difficult subject and there is a feeling that achievement in Maths is not as rewarded as it should be. I do not think that it is so helpful to portray Maths as being difficult as this will deter able young people with mathematical ability from studying the subject beyond the age of sixteen. I agree that Maths is not everyone’s cup of tea, but it is a terrible shame to give up without even trying as the reward that goes with achievement in Maths can be significant. I would like to tackle both of the two reasons that have been given for the drop in the number of students taking up A-level Maths and offer practical suggestions to parents and their children who have an interest in this salient matter. Is the new GCSE Maths so difficult? First, the perceived difficulty of the new GCSE maths since the introduction of the number grading system. It was reported in the TES – Times Educational Supplement – that fewer students are now doing A-level Maths since the introduction of the new number-graded GCSE Maths – which has more Algebra and Geometry content. Teachers said that more able young people are put off and do not want to commit to studying A-level Maths because they think the highest grade they can achieve in the subject is a Grade 7 (please see my previous blog post for more detail). Whilst recognising the need for all educators to do all they can to encourage more young people to study Maths up to A-level and beyond, it must also be borne in mind that it is not good for anyone if too many people study the subject and end up with poor grades at A-level or fail the exam altogether. In my teaching career, I’ve experienced both sides of the equation: the young people who are capable of taking up A-level Maths, but decided not to, and the ones who are, perhaps, not mathematically-gifted who take up the subject and end up with a poor A-level grade. I believe a lot more ought to be done to encourage young people to become more interested in Maths and to seriously consider studying the subject at A-level. At the same time, we have to be honest with ourselves and accept that there are some people for whom Maths is not a subject in which they have natural ability. For these people, it is a waste of time asking them to study Maths at A-level and it does no one any good in the end. They will be better off studying other useful subjects that are not mathematical, such as English, History, Biology, foreign languages and perhaps Chemistry or Economics or maybe Languages. What I must say, however, is that there is no point in studying Biology at A-level without studying Chemistry in addition, as Chemistry is required in order to study any Biology-related subject or course at degree level at a reputable university. I would also like to add that Chemistry is a little mathematical, but not to the same degree as Physics. For an Economics degree, there are two possible paths, one mathematical and the other not so much so. Don’t give up on A-level Maths too easily Whilst recognising that, perhaps, some people may not be naturally talented when it comes to Maths, it is a terrible shame to give up without even trying. Many girls and boys wrongly believe that they are ‘not good in Maths’, which a very harmful perception to have without trying hard enough. Let me tell you a couple of short stories from my experience about people who were good mathematically, but thought they were not. The first story is of a young woman who came to study in my Saturday School. She had always believed she was ‘rubbish in Math’ and she was predicted a D/E grade at GCSE, but, in the end, she achieved an A* in Maths. Well, after that achievement I had a discussion with her and suggested that perhaps she could have studied A-level Maths and she replied saying that “Maths is not my strongest subject as I’m better in English and German.” Now, I do not believe that she does not have a natural talent for Maths, as it is unlikely for someone who is just a plodder to achieve an A* in Maths at GCSE. My belief is that, with a lot of hard work, one can achieve an A grade in Maths, but not without a natural talent to go with the hard work; although GCSE Maths is really not that challenging, achieving an A* is highly unlikely unless one has a natural ability for the subject. The second story is of myself, when I was 13 years old. I had a belief that Maths was hard and that I was not good at it and I never tried enough. At the end of my form 3 (now called Y9), I wanted to change school and I had to work hard to perform well in all my subjects so I could pass the other school’s entrance exam. I was so shocked when I came third in my year group – something which I thought at the time was well beyond my capability. I ended up with a Physics degree later in life, which could not have been possible if I was so awful in Maths. The third is my son, when he was about three years old and we were trying to teach him to count from one to ten and he was struggling and was getting mixed up. At that time, the easiest conclusion to come to was that he was not bright. Wind the clock forward three or so years when he was in Y1 (about 6 years old), and now he could work out things like the square of 98 in his head within a few seconds and he ended up being one of the very top people in Maths in his year group. What I will say here is that the easiest thing to do is to give up and giving up without a fight is the worst thing to do. There are so many reasons why some young people never really try hard enough before coming to the conclusion that they can’t do Maths. Reasons vary from a lack of inspirational teacher, peer pressure or lack of encouragement from the environment in which they find themselves. The general perception that it is uncool to be good in the subject or to like Maths is not as prevalent as it used to be, but it is still very much around. Yes, Maths is rocket science, but don’t give up without trying; perhaps you can do it! The next blog post is the conclusion of this four-part series and I will provide more evidence why A-level Maths is worth its weight in gold.

Class Note for CMPSCI 601 at UMass(33) Class Note for CMPSCI 601 at UMass(33) Popular in Course Popular in Department This 19 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 19 views. Reviews for Class Note for CMPSCI 601 at UMass(33) Report this Material What is Karma? Karma is the currency of StudySoup. You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more! Date Created: 02/06/15 CMPSCI 601 Recall From Last Time Lecture 15 De nition F is consistent iff P if false Completeness Theorem If F is consistent then P is satis able that is there exists a model A such that A l P Corollary F l p ltgt F l p i so ltgt P so FOVALID FOTHEOREMS Note that FOVALID and FOTHEOREMS are state ments that are true in any model of the given structure While we constructed a special model where every state ment was either provably true or provably false this is not true in general An FOVALID statement about graphs would be true for all graphs but most interesting state ments are true for some graphs and not for others Compactness Theorem If every nite subset of F has a model then P has a model 1 m gt a lt ltlt gt0 gt 200 v lt0 gt mm 2 gt an 2 gt v gt 91 9mm 510911 l JaqmnN 311110 J 109 IOSdWO 14 NNT 2 Aler The statements of NT are all true and can be used to prove a fragment of true number theory Theorem 61 Papa Let 0 have no variables Then N i 90 42gt NT i 0 Proof 0 is a boolean combination of t lt t t 2 t Case 1 tt numbers 00 00 use NT1 NTQ lt use NTm NT13 NT14 Case 2 t t use gtltT Use NT4 NTg to transform these to numbers A De nition 151 A formula 0 E ZN is bounded iff it can be written with all quanti ers in front and all univer sal quanti ers bounded A Example Va lt 93y72 lt 2 T a X T 3 z T 3 17 Remark If 902 is bounded and has only one free variable 39u then 50 is re Where 590 RENINTltPTL Bounded sentences are not closed under negation They are sentences that you can check by a naming numbers and b doing sequences of tests that are guar anteed to nish This is reminiscent of Bloop but not the same thing be cause there is no equivalent in Bloop for the unbounded E A proof can name a number but a Bloop program can t look for it Without a limit on how far it may look Theorem 62 Papa Let p be a bounded sentence ie no free variables Then N 90 42gt NT p Proof Soundness Theorem gt induction on number of quanti ers in 0 Assume N i 0 Base case Theorem 61 Inductive step 0 E 3mm Thus N i x lt n for some n E N Thus NT i v x lt n Thus NT i p Inductive step 0 E Vx lt t b t is a closed term thus NT t 2 n for some NT10NT11NT14 F 17 lt 7 1 gt 8 2 0V2 21V i0n 1 NTFMaH i i0n 1 NTFcp nEN x 72 1 De nition 152 Let f N gt N Formula pf represents f iff forall n1 nk m E N fn1nkm 42gt Nltpfn1nkm A Lemma 153 T he following primitive recursive functions Prime PrimeF IsSeq length and Item are each repre sentable by bounded formulas Proof PrimeFn p asserts that p is prime number n by assert ing that there exists a number s 20gtlt31gtlt52gtlt73gtlt114gtltgtltpn xly 32lty1xgtltzy DEx8y Ter we Xy Primex E 1 gt1 Vyz lt X 2 a x PrimeFnp E 38Primep 2 X8 DEp n s W S pVQ lt q PrimeQ V Primew V 341quot lt 41W lt q Primeltaquot 38 lt qDEq39 e s DEq e 1 IsSeqx E 32 lt lt x3p lt 2 T xPrimeFip z1 p113 igtz1 plx lengthx E 3kpqIsSeqx k1 PrimeFkpj PrimeF q p11 q Itemxie E 3pIsSeqx PrimeFip DEpe1xi A Theorem 154 Every primitive recursive function is rep resentable by a bounded formula Proof Base case Obvious for the initial functions Inductive step Composition f l1 m 2 hglx1 xk gmx1 By inductive assumption h and the 92 s are representable 90M 2 E 391 ym wgl 91M 909mi gm M211 ym 2 10 Inductive step Primitive Recursion Cali 73116 9 17317 39 39 hfn7y17 39 39 39 7yk7n7y17 39 39 39 By inductive assumption 9 and h are representable f is representable as follows WW 3 2 E 38Vi lt x3a b lt sIsSeqs Itemsi a Itemsi1b 2 0 gt 9093 1 phaigjb Itemsxz ll Bloop Floop and Bounded Formulas Note In fact whenever we use an E quanti er in this proof with some more effort we could make it a bounded E quanti er A function is Bloopcomputable iff it is rep resented by a formula where both kinds of quanti ers are bounded It should be pretty clear that a Bloop program can test the truth of such a completely bounded for mula A formula is Floopcomputable iff it is represented by a Vbounded formula as we have de ned it here I m not going to prove these assertions all we need to prove Godel s theorem is that Bloopcomputable primi tive recursive implies representability by a bounded for mula 12 A Fact We Didn t Prove This Time The primitive recursive predicate COMPn 13 c y mean ing that c is a halting computation of Turing machine Mn on input 1 and its output is y How to prove it Write Bloop programs for PrimeF IsSeq Item and so forth Then write a Bloop program that interprets c as a sequence of con gurations and tests that each con guration follows from the one before ac cording to the rules of the machine Mn Section 62 of P in effect does this I ll just appeal to the intuition I hope you ve formed about Bloop Corollary 155 K is representable by a bounded formula cp n 30COMPn n c 1 K 711 Nllteen K 2 n NTl cpKn l3 CMPSCI 601 Summary S0 Far Lecture 15 14 NT 2 1 NTZ39 6 De nition A formula 0 E ZN is bounded iff it can be written with all quanti ers in front and all universal quanti ers bounded Theorem 62 Papa Let p be a bounded sentence Then N l 90 42gt NT l 0 De nition pf represents f iff forall 711 nk m E N fn1nkm 42gt Nltpfn1nkm Theorem Every primitive recursive function is repre sentable by a bounded formula Corollary K is representable by a bounded formula cp n E 30COMPn n c 1 K n l Nlltpzltn K 2 n NTl cpKn 14 De nition 156 For a structure A E STRUCZ TheoryM 90 E 132 I A 1 so TheoryN 2 90 E ZN I N t 90 Thus TheoryN is true number theory Theorem 157 Godel s Incompleteness Theorem There is no re set of sentences P such that I N F and 2 F F TheoryN There is no axiomatization of number theory much less all of mathematics 15 Proof Let F be re and N l P 5 HEN l Fl w n S is re and S g F Intuitively S 2 n E N I F l n E F S is an re subset of the nonre set F It can t be equal to F and in fact it has to miss in nitely many elements Since if it missed only nitely many 5 plus those ele ments would still form an re set So there eXist in nitely many n E N st N l WM and F l7 WM 4 l6 P states this result in the following form proves this in the form of his Theorem 63 P Theorem 63 T he set of unsatis able sentences and the set of sentences provable from NT are recursively inseparable Thus a recursive set not only cannot separate true number theory from false number theory but can t even include all the true bounded formulas Without letting in some thing inconsistent Recall that the sets M M outputs yes on e and M M outputs no on e are recursively inseparable Look at the sentence NT holds and there is an accepting computation of M on e If M says yes this is prov able from NT If M says no it is inconsistent because it says that the computation says no while NT can prove that it says yes 17 CMPSCI601 Sketch of Godel s Original Proof Lecture 15 0 Encode symbols as natural numbers 0 Encode formulas as nite sequences of natural num bers 0 Encode proofs as nite sequences of formulas 0 Let F be a primitive recursive axiomitization of some portion of mathematics including number theory The following predicates are primitive recursive and thus rstorder de nable in MEN Formulax x is the number of a formula Axiomx x is the number of an axiom Proofx x is the number of a proof Theoremx x is the number of a theorem 0 Let R0 R1 list all rstorder formulas with one free variable ie rstorder de nable sets tLetG 2 n I TheoremRnn 0 G 2 n I Rqn for some q 0 Rqq E TheoremRqq E I am not a theo rem 0 If Rqq then P l7 Rqq If RqQZ then P l Rqq 18 Theorem 158 FOTHEOREMS is re complete Proof We have already seen that FOTHEOREMS is re Recall that K is represented by a bounded formula pK nEK 42gt NltpKn 42gt NTl cpKn n E K 42gt NT gt cpKO L E FOTHEOREMS We have shown K g FOTHEOREMS by de ning f so that f n NT gt WM 19 Are you sure you want to buy this material for You're already Subscribed! 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Life | Work | Viete's construction of the regular heptagon | Bibliography | Back to the front page Viete's construction of the regular heptagon Vieta is best known for his work in algebra. However, I would like to describe here some of his work in geometry, which seems to be less well known: the trisection of the angle, the solution of the ``casus irreducibilis'' of the cubic equation, and the construction of the regular heptagon. These results are all presented in the Supplementum Geometriae, first published in 1593 (though I have not been able to locate any copy of the original), and included in [Opera, 1646, pp. 240-257]. Vieta makes clear from the beginning that besides ruler and compass, he is admitting another operation: In order to supplement the defect of geometry, let it be allowed from any point to any two straight lines, to draw a straight line cutting off between them any segment fixed in advance. [Opera, p. 240]In other words, given two lines l,m, given a point A, and a segment d, we can draw a line through A so that the two lines l,m cut off on it a segment BC=d. Vieta also allows the same where a line and a circle are given instead of two lines. One can imagine this construction using a marked ruler: one makes two marks on the ruler at a distance d apart, then slides the ruler so that the marks lie over the lines l,m while passing still through the point A. This process goes by the name of neusis in Greek. Vieta does not give it a name. Vieta's work has 25 propositions, most of which I will skip over. Proposition IX [Opera, p. 245] is to trisect an angle. Let the given angle DBE be placed in a circle with B at the center. Draw a line through E, meeting BD extended at F, so that the segment FG cut off by the line and the circle is equal to BD (this is the neusis step). Then the angle at F is one third of the original angle at B. For the proof, draw BG. Call the angle at F 1. Then BFG is isosceles, so the angle GBF is also 1. The angle BGE is an exterior angle, hence equal to 2 times 1. BGE is isosceles, so GEB= 2. Now the original angle EBD is exterior to the triangle BEF so it is 3 times the angle at F. The question naturally arises, did Vieta discover this result himself, of did he learn it from earlier sources? He does not say. (In fact, contrary to the rules of scholarship today, by which one must scrupulously indicate sources and give credit where it is due, it seems to have been the custom in those days not to indicate sources. For example, Descartes in Book III of his Géométrie explains how any cubic or quartic equation can be solved by taking square roots, cube roots, or trisecting an angle: this result is Prop. XXV of Vieta's Supplementum Geometriae , but Descartes states it without giving any credit to Vieta. ) This same trisection appears in the Mathematical Collection of Pappus [....] and in the book of lemmas attributed to Archimedes [Archimedes, 1792, Prop VIII p. 358]. Now the latter is known only through an Arabic source, and was not published until 1657 [,,,], so we may assume that Vieta was unaware of it. However, Pappus' Mathematical Collection was circulating in manuscript in Europe well before the time of Vieta, and was published in a Latin translation by Commandino in 1588 [see intro. to Pappus, 1986], so we may presume that Vieta was familiar with the works of Pappus, and learned this construction from his book. (Commentators have also assumed that Vieta borrowed the terms zetetics and poristics from Pappus, Book VII-see [Viète , 1983, p. 11, note]). Proposition XVI [Opera, p. 248] gives in geometrical form the solution of a cubic equation using the trisection of an angle. The statement is as follows: If two isosceles triangles have their sides equal, and the angle at the base of the second is triple the angle at the base of the first, then the cube of the base of the first minus triple the solid made by the base of the first and the square of the common side, is equal to the solid on the base of the second and the same square of the common side.cm Let the first triangle be ABC with AB=BC, and the second CDE with CD=DE=BC, and assume that the angle DCE at the base of the second is three times the angle at A. Then he says Or, if we take the radius BC=1, let AC=x and CE=b, the equation is Because the angle at A is one third of the angle DCE, the points A,B,D are collinear and we can use the previous diagram. Vieta's proof is expressed all in words. Translated into symbols it goes like this: drop perpendiculars from B and D to ACE, and let y=BD. By Euclid (III.36) we have using the point A and the two secants. Because the perpendiculars from B and D to ACE are parallel, we have a proportion (VI.2) Now, solving the first equation for y and substituting, we find From the proposition of Vieta, we see that to solve the equation , we make an isosceles triangle with sides 1 and base b, trisect the angle, and then the base of the new isosceles triangle will be x. By suitable linear substitutions, this gives a recipe for solving any cubic equation with three real roots, the so-called ``casus irreducibilis''. This is the case in which Cardano's formulas require cube roots of imaginary quantities, which were not understood at the time. As far as I know, this result is original to Vieta, and this is the place where he first explains it. Vieta gives the general rule for this case of the cubic equation in De Recognitione Equationum [Opera p.91], but without proof, referring to his book Theoremata ad sectiones angulares [Opera pp 287-304] for explanation. Now we come to the construction of the regular heptagon. Vieta's solution is directly analogous to Euclid's construction of the pentagon. Euclid first divides a line AB in extreme and mean ratio: that is he finds a point C for which AB:AC=AC:BC. (II.11). From this he constructs an isosceles triangle whose base angles are twice the angle at the vertex (IV.10). Then from this triangle he constructs the pentagon (IV.11). Vieta first extends the diameter BAC of a circle by a point I so that (Prop. XIX). Form this he constructs an isosceles triangle whose base angles are three times the vertex angle (Prop XXIII), with which he can construct the heptagon (Prop XXIV). Here is the construction of I. Let the diameter of the circle be BAC, and call the radius 1. Take D with AD=(1/3). Let E be a vertex of the hexagon inscribed in the circle, so that CE is 1/6 of the circle. Draw DE. Make AF parallel to DE. Draw FGH so that GH=AB (neusis, trisecting the angle FAC). Draw EI parallel to FG. Then I is the required point. For the proof, draw AH. Make DK parallel to AH. Take L= midpoint of DC, and M= midpoint of DL. Then the triangles IKD and DEL satisfy the hypotheses of the previous result. So, if we let x=ID, r=DE and note ,then x will satisfy the equation Now apply (I.47) to the triangle DEM. The base is and the height is , since ACE is an equilateral triangle. Thus we find , so x satisfies To verify the relation , just note and AB=1. Substituting gives the equation above, so I has the required property. Now, changing notation, let BAC be the diameter of a circle, let D be the point I as above, and draw DE=AB=1. Let DE cut the circle again at F, and draw AF. Then I claim BE is the side of the regular heptagon inscribed in the circle. Draw EC. By (III.36), but also by construction Since DE=1=AB, we find . As a proportion, this says DF:DE=DA:DC. It follows that EC is parallel to AF (VI.2). Hence the angle at C is equal to the angle at BAF. Denote it by 1. On the other hand, the angle at C subtends the arc BE, so the angle at the center BAE is 2. So angle EAF is equal to 1, and we find AF is the angle bisector of BAE. Since DAE is isosceles, the angle at D is 2, and the exterior angle of the triangle DAF at F is 3. But AEF is isosceles, so the angle at AEF is also 3. Now we have an isosceles triangle AEF whose base angles are three times the vertex angle. It follows the vertex angle 1 is one seventh of two right angles. The angle BAE is twice that, so BE is the side of the regular heptagon, as required. Note. In this account I have shortened Vieta's exposition considerably by using modern algebraic notation, and in a few places I have done the calculation slightly differently, but in no way have I altered the basic mathematical content of the proof. History of constructions of the heptagon. There is an Arabic text, discovered only in 1927 containing a construction of the heptagon attributed to Archimedes, which depends on the following: Given a square ABCD with its diagonal AD, find a line CE so that the triangles CFD and BGE have the same area (see [van der Waerden, p.226] or [Knorr, p. 178ff]). This construction could be effected with an intersection of two conics, but it is not clear how to do it with a neusis. Campanus, in his commentary to Euclid, Book IV, mentions that you could construct a heptagon if you had an isosceles triangle whose base angles are three times the vertex angle. [Euclid 1516, f 56 verso] Otherwise, this construction of the heptagon seems to be totally original to Vieta. From a modern point of view, to construct a heptagon one must solve the cyclotomic equation . If is a root, then is a root of the cubic equation One knows (from Vieta!) that such a cubic can be solved by trisecting a certain angle, and modern constructions have been given following this idea by [Plemelj, 1912], [Bieberbach, 1952], [Gleason, 1988], and [Hartshorne, 1997]. But none of these authors seems to have been aware of Vieta's construction, almost 400 years earlier, based on the same ideas, but without the benefit of the modern theory of the cyclotomic equation and complex numbers.

ARCHIMEDES GREEK PHILOSOPHER. He noticed that as he lowered himself into the bath, the water displaced by his . Archimedes, a Greek mathematician and resident of the Greek colony Syracuse on the island of Sicily in the 3rd century BC, is credited with . According to history he was in the bath one day when he discovered the principle of buoyancy which is the reason why huge Greek ships weighing thousands of pounds could float on water. is best known for the classic story of how he discovered the principles of density and buoyancy while taking a bath, immediately causing him to run through the streets of Syracuse naked screaming "Eureka!" (which roughly translates to "I have found it!"). The basic principle is due to Archimedes. Aristotle ( Greek: , Aristotls) (384 - 322 BCE), a student of Plato, promoted the concept that observation of physical phenomena could ultimately lead to the discovery of the natural laws governing them. King Hiero II had ordered a fancy golden crown . If the buoyant force is enough to support the weight of the object, it floats. His famous Law states that the buoyancy force on a body is equal and oppositely directed to the weight of the uid that the body displaces. considered to be the greatest mathematician of ancient history, and one of the greatest of all time, archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and Archimedes' principle, physical law of buoyancy, discovered by the ancient Greek mathematician and inventor Archimedes, stating that any body completely or partially submerged in a fluid ( gas or liquid) at rest is acted upon by an upward, or buoyant, force, the magnitude of which is equal to the weight of the fluid displaced by the body. Buoyancy as a phenomenon, was first discussed and explained by the famous ancient Greek scientist Archimedes, who was able to calculate the density of an irregular object (the famous crown donated to the King by a jeweller). Archimedes' Solves a Problem The Greek Mathematician and inventor Archimedes lived during the 3rd century BC. Greek mathematician, physicist, and engineer Archimedes of Syracuse suggested in his 250 BCE work "On Floating Bodies," that any object, whether totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. The Greek Mathematician and inventor Archimedes lived during the 3rd century BC. The Buoyant force exerted on the body by the fluid is mathematically expressed as, F = gV Where F is the Buoyant force acting on the body, is the density of the fluid, g is the acceleration due to gravity, V is the volume of the displaced fluid. The understanding of the physics of buoyancy goes back as far as antiquity and probably sprung from the interest in ships and shipbuilding in classic Greece. He was born in Syracuse, on the island of Sicily in 287 B.C. Using an experimental protocol involving dense. Part 1: Density and Water Displacement Before attempting to verify Archimedes Principle, we will first verify the density of some common metals. This means that the downward force due to your weight plus the pressure of the water on top of you is less than the force exerted by the water on the bottom of your body pushing you up, which is referred to as the "buoyant force." Buoyancy. buoyancy, discovered by the ancient Greek mathematician and inventor Archimedes, stating that any body completely or partially submerged in a fluid (gas or liquid) at rest is acted upon by an upward, or buoyant, force, the magnitude of which is equal to the weight of Read More balloon flight In balloon flight: Elements of balloon flight For example, air is less dense than corn syrup; hence, air bubbles rise through corn syrup. Archimedes (287 - 212 B.C.E.) Record this mass in the Exercises section. Archimedes, the ancient Greek physicist, will always be linked to this famous story. In addition, he is known for many other significant feats: Archimedes of Syracuse and the Golden Crown. Archimedes was the son of an astronomer. At that time Sicily was a Greek land. Buoyancy was discovered by Archimedes, a Greek mathematician and physicist, in 212 BC. Hot air balloons rise into the air because the density of the air (warmer air) inside the balloon is less dense than the air . Newton's laws say this difference should be equal to the buoyant force. Thus, by measuring the volume of the displaced water, he calculated indirectly the volume of the immersed object (the crown). Pearson Italia. Archimedes (287- 212 B.C. Density & Buoyancy of Objects: Physics Lab When the object is less dense than the fluid surrounding it, the object rises. Which Ancient Greek physicist famously discovered the concept of buoyancy while taking a bath? Archimedes: Which ancient Greek scholar laid the foundation for future European mathematics and authored the "Elements" of geometry? Archimedes Principle states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object. Which Ancient Greek physicist famously discovered the concept of buoyancy while taking a bath? Archimedes: How do you get Alliance Technology credits? BUOYANCY CONCEPT The principle of buoyancy holds that the buoyant or lifting force of an object submerged in a fluid is equal to the weight of the fluid it has displaced. . Before The concept is also known as Archimedes's principle, after the Greek mathematician, physicist, and inventor Archimedes (c. 287-212 b.c. He studied at Alexandria in Egypt, and then returned to Syracuse. ARCHIMEDES 287BC - 211BC. Buoyancy is the lifting force on an object submerged in a fluid. So it would weigh (2744-980) dynes or 1764 dynes while immersed in the water. To answer these questions, you'll need to understand the concept of buoyancy, a force which is exerted by a fluid on an object, opposing the object's weight. The Roman architect Vitruvius tells us that Archimedes started to study buoyancy because of the king of Syracuse, Hiero II. Archimedes was the eminent mathematician and excellent physicist of his time. ), a Greek mathematician, physicist, and astronomer, discovered the principle behind buoyancy. The king asked him to determine whether a crown was made of gold or of other . ), who discovered it. According to history he was in the bath one day when he discovered the principle of buoyancy which is the reason why huge Greek ships weighing thousands of pounds could float on water. But in research published in September in the journal Nature, a team of physicists broke one of these seemingly 'natural' principles: buoyancy. Born in northern Greece in 384 BC, Aristotle's ideas dominated western science and philosophy for nearly 2000 years, from his death in 322 BC until Galileo's destruction of his mechanics in 1609 . If not, the object sinks. Agriculture: In Greek mythology, what was Achilles' only weak point? Procedure 1) Using a pan balance, measure the mass of the aluminum cylinder. It is rumored that the Greek philosopher and scientist Archimedes, around 250 B.C., was asked by King Hiero II to help with a problem. Introduction Buoyancy is an upwards force acting on any object that is immersed in a fluid. Euclid: Which animal is featured on commander Richard I's background image? By donating to the Alliance: Egypt's economy was primarily based on? - Tennis College Admission - Mizuno Fastpitch Batting Helmet - Rocky S2v Predator Vs Tactical - 2022 Maryland Swimming 14 Under Championship Meet - 2014 Spurs Coaching Staff - Impact Of Broadcast Media In Society - Ford Focus Electric Charge Time - Cuirass Bodice Definition - Cell Machine Creative - What Are 5 Health Benefits Of Peppers? - Words To Describe Chocolate - Senior Citizen Card Lost

What fraction is 6 9 equivalent to? 2/3Hence, 2/3 × (3/3) = 6/9, is the fraction equivalent to 2/3. What is 6 9 in its simplest form? 23But 69 is not; 6 and 9 have a common factor 3 . So 69 in simplest form is 23 . This is known as reducing fractions . Are the fractions 6 9 and 4 6 equivalent? They are the same. Are 6 9 2 3 are equal? To convert 6/9 so the denominator is 9, you multiply the numerator and denominator by 1. As you can see, the numerators colored in orange above are the same. Therefore, the answer to “Is 2/3 equal to 6/9?” is yes! What is 6 9 in a mixed number? Since 69 is a proper fraction, it cannot be written as a mixed number. What is 6 9 as a percentage? 66.666667%Fraction to percent conversion tableFractionPercent3/933.333333%4/944.444444%5/955.555556%6/966.666667%41 more rows What is 1/3 the same as? Answer: The fractions equivalent to 1/3 are 2/6, 3/9, 4/12, etc. Equivalent fractions have the same value in the reduced form. Explanation: Equivalent fractions can be written by multiplying or dividing both the numerator and the denominator by the same number. How do you find equivalent fractions? To find the equivalent fractions for any given fraction, multiply the numerator and the denominator by the same number. For example, to find an equivalent fraction of 3/4, multiply the numerator 3 and the denominator 4 by the same number, say, 2. Thus, 6/8 is an equivalent fraction of 3/4. What is 3/5 equivalent to as a fraction? So, 3/5 = 6/10 = 9/15 = 12/20. Do the ratios 2 3 and 6 9 form a proportion? 2 Answers. They are proportions, see explanation. What is the fraction 3/4 equivalent to? Equivalent fractions of 3/4 : 6/8 , 9/12 , 12/16 , 15/ How do u divide fractions? The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed. Can 6’9 be simplified again? 1:151:48How to simplify 6/9 – YouTubeYouTubeStart of suggested clipEnd of suggested clipHow about three that works so 6 divided by 3 that equals 2 now we try in our denominator 9 8 dividedMoreHow about three that works so 6 divided by 3 that equals 2 now we try in our denominator 9 8 divided by 3 that. Also works that equals 3. So our new simplified and reduced fraction becomes 2 over 3. What is in its simplest form? What is simplest form? A fraction is in simplest form if the top and bottom have no common factors other than 1. In other words, you cannot divide the top and bottom any further and have them still be whole numbers. You might also hear simplest form called “lowest terms”. For example, the fraction. What is in simplified form? 1:184:10What is the Simplest Form of a Fraction? | Don’t Memorise – YouTubeYouTubeStart of suggested clipEnd of suggested clip12 by 6 is 2 and 18 by 6 is 3 12 by 18 and 2 by 3 are the same and this 2 by 3 here is called theMore12 by 6 is 2 and 18 by 6 is 3 12 by 18 and 2 by 3 are the same and this 2 by 3 here is called the simplest form it means that this fraction cannot be reduced further. What is the whole number of 6 9? 6 / 9 = 0.66666667 If you insist on converting 6/9 to a whole number, the best we can do is round the decimal number up or down to the nearest whole number. What does the 9 in 6/9 mean? Next, let’s say we asked someone with absolutely perfect vision (for a human) to do the same thing. This person walked backwards from the object 9 meters until the object began looking blurry. That’s what the 9 in 6/9 refers to. What is the acuity of 6/9? The Snellen equivalent of an acuity of 6/9 is approximately 20/30. This person’s visual acuity is within normal limits and can be corrected using a normal (not low-vision) spectacle prescription. What does 6/9 vision mean? 6/9 vision means you can see clearly at 6 meters what someone with perfect vision would be able to see clearly at 9 meters. If we were to perform the same experiment and both you and the person with hypothetically perfect visions stopped exactly 6 meters from the object. You would be said to have 6/6 vision. How far can you see with a 6/9 acuity? The acuity simply means that, what a person with perfect vision can see from nine feet away, a person with an acuity of 6/9 must stand three feet closer (at six feet) to see it. The acuity is still pretty much 20/30, which is very good, in my line of work. , I am an experienced contact lens fitter. When denoting visual acuity, the numerator remains the same? Hence when denoting visual acuity the numerator remains the same (i.e. 6 when measured in metres) and the denominator changes based on the refractive error ( higher the denominator, higher being the refractive error). What is the top 6 on a chart? The top 6 is the distance in metres from the chart to the eye. The lower figure, in this case 9, is the distance in metres at whch a person who can just read the letters on the 6/6 line (lower figure 6) can read that line. This same person would be able to read the 6/12 line (call it the 12 metre line) at 12 metres and so on. Is 6/9 vision good? 6/9 vision is not perfect vision, but it’s pretty good. You can see most detail with 6/9 vision. What is the equivalent of 2 6? The fraction 2 6 is equal to 1 3 when reduced to lowest terms. To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 1 3) by the same integer number, ie, multiply by 2, 3, 4, 5, 6 … and so on … How to find equivalent fractions? To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 13) by the same natural number, ie, multiply by 2, 3, 4, 5, 6 … Can you convert fractions to decimals? This Equivalent Fractions Table/Chart contains common practical fractions. You can easily convert from fraction to decimal, as well as, from fractions of inches to millimeters. What is the equivalent of (3+7)+2? The expression equivalent to (3+7)+2 is 12. What is equivalent expression calculator? Equivalent Expression Calculator is a free online tool that displays the equivalent expressions for the given algebraic expression. BYJU’S online equivalent expression calculator tool makes the calculations and simplification faster and it displays the equivalent expression in a fraction of seconds. What is the equivalent fraction of 2/3? For example, if we multiply the numerator and denominator of 2/3 by 4 we get. 2/3 = 2×4 / 3×4 = 8/12 which is an equivalent fraction of 2/3. How to find equivalent ratios? As we previously mentioned, Equivalent Ratios are two ratios that express the same relationship between numbers. The Equivalent Ratio Calculator provides a table of equivalent ratios that have the same relationship between each other and directly with the ratio you enter into the calculator. We will look at how to calculate equivalent ratios shortly, first lets look at how to use the free online equivalent ratio calculator: 1 Enter a Ratio into the equivalent ratio calculator, for example, you could enter 7:25 2 Select the number of equivalent ratios that you would like to see in the table of results 3 The equivalent ratio calculator will calculate as you type and produce a lis of equivalent ratios in a table below the calculator 4 [Optional] Print or email the Table of Equivalent Ratios for later use What is a ratio? A ratio is a direct comparison of one number against another. A ratio calculator looks to define the relationship that compares between those two numbers Is there a formula for equivalent ratios? As equivalent ratios have the same value there is technically no equivalent ratio formula but the following equivalent ratio formula will help you with the manual math calculations. What does 6.9 A1C mean? An A1c of 6.9 means that 6.9% of the hemoglobin in your blood are saturated with sugar. You may already be experiencing symptoms of diabetes, which include increased thirst, frequent urination, general fatigue and blurred vision. Diabetes is a serious condition. Left untreated diabetes can lead to heart disease, stroke, nerve damage, blindness, … How to reduce A1C of 6.9? Reducing an A1c of 6.9 will take a combination of medication and lifestyle modifications. Seek medical advice to gain control of your blood sugar and avoid further damage to critical organs. What is a Kilowatt-Hour? A kilowatt-hour, otherwise known as a kWh, is a way to measure how much energy you’re using. Kilowatt-Hour vs. Kilowatt What’s the difference between kilowatt vs. kilowatt-hour? A kilowatt is 1,000 watts, which is a measure of power. A kilowatt-hour is a measure of the amount of energy a certain machine needs to run for one hour. What Can a Kilowatt-Hour Power? Obviously, every appliance in your home will use a different amount of power. Here are some of the usages for the more (or less) common items in a home: How Do I Calculate How Many kWh an Appliance Uses? Heating and cooling your home use the most electricity, and are around 50% of your bill. But in second place are your appliances, at around 20% of your bill. What Does a kWh of Electricity Cost? The cost of kWh varies depending on where you live. In deregulated markets, what you pay per kWh can also vary depending on whether you have selected an alternative energy supplier. The US Energy Information Administration (EIA) reports on average prices by state. How Many kWh Does a House Use Per Day? One common question is, how many kWh does a house use per day? The amount of kWh you use will depend on: How to See Your Electricity Usage The easiest way to see how much electricity you use is to simply check your electric bill. Your electricity provider will show the number of kilowatt-hours you use each month. Some have also started to add small charts on your bill so you can see monthly trends and patterns. What is CGPA? CGPA is guided to as a cumulative grade point average. It was presented in the CCE or Continuous and Comprehensive Evaluation System for Class X. CGPA gives the average grade point of the student version in all the subjects, excluding the additional Subject. The subject-wise performance of the students is registered in the form of Grades. When did it get introduced in CBSE? Before 2011, the Central Board of Secondary Education had standards for rewarding marks. In 2011, it replaced the provisions of the CGPA system. Why is it used? The CGPA represents the student’s overall execution in the academic session as grades. The CGPA system was presented to ease the burden of marks and the stress of the studies among students. Due to this reason, the CGPA system is also used in some of the bachelor’s and Master’s Degree.

Let us therefore begin with the following test: 1. Three points of a sphere are randomly chosen. What is the probability that the three points will be in the same hemisphere? (It can be considered that the points that the maximum circle delimiting the hemisphere, for example the equator, has inside, belong to either of the two hemispheres). 2. Can a 6x6x6 cube be completed with twenty 1x2x4 dimensions bricks? (See figure 1). 3. A logical, who had to spend time in a city, decided to go to a hairdresser. In the city there were only two hairdressers. He addressed one of them. There he saw disordered things: the owner himself wore dirty clothes, had a beard of a few days and shattered hair. It did not take long to move. On this occasion everything was well ordered. Elegant hairdresser without beards and well combed. Logical, just to see him, he returned to the first. Why? 4º See the five images of poxpoker (Figure 2). 1. Removing nine poxpoles, leaves six. 2. Changing two poxpoles of site, leave the five squares for four (all poxpoles must be somewhere in a square). 3. By removing four, three or two poxpoles from this image, two equilateral triangles can be obtained. How? 4th Realize that equality by moving a single poxpolu. 5. Complete a square by moving a single pot on this cross. (It is not useful to move a couple of millimeters a poxpulo so that the four ends form an empty square) 5. Suppose we have six balls of the same size: two red, two black and two white. In each pair a ball is heavier than the other. The three heavy balls have the same weight and the three light also. Making only two pesos with a balance, how can you differentiate the heavy in each couple? 6º Here (Figure 3) the numerical numbers. In these 7 x 8 rectangles are dispersed twenty-eight dominoes. To see if you find them (the left has a single solution, the other has eight different solutions and that is why it is more difficult). 7. A rancher has 20 pigs, 40 cows and 60 horses. But if we call horses cows, how many cows will it have? 8º. Build eight equilateral triangles by drawing six straight lines of the same length. 9. Take ten coins. With these ten coins you have to form five straight lines of four coins. 10. Draw a straight line and take three coins. Place these three coins so that they remain two faces on one side of the line and two whole crosses on the other. 11. If an iron-like “donuts” is put to heat, will the hole diameter increase or decrease? 12. What is the largest number that can be written by putting only 2 figures three times? 13. We have taken a complete turn to the great circle seen in figure 4. Therefore, the length of its circumference will be PQ. At the same time we have turned the inner circle, being the length of its circumference RS. How would you explain the reason that the lengths of these two circles were equal? 14. The Epimenides cretan said: all cretans are liars. What can you say about this phrase? 15. Protágoras made a pact with a student of his. According to the agreement, the student had to pay for the studies when he won his first case. At the end the student was not lucky and did not find clients for a long time. Protágoras, concerned, denounced his student. Protágoras argued in the trial that if I win the lawsuit I would have to pay to comply with the judgment. On the contrary, if you win, I will have to pay according to our agreement. Therefore, I will have to pay in any case. But the student did not agree and answered him: if you win, respecting our covenant you will not have to pay and if I win, the judges will not force me to pay. So in no case should I pay. What is the correct reasoning? 16. With the chess horse fill the following squares without spending twice for each square. (Figure 5). 17. Below are two translation problems. In both cases it is assumed that only one piece of furniture is introduced in each room and in both cases the owners want to transfer the piano and the library, passing the piano to the room where the library is and vice versa. How should they act with the smallest possible changes? (Figure 6). 18. Friends A, B and C are taught three white ribbons and two black ribbons. Then hang a tape on the back of each one. Each person can see the tapes of others, but not theirs. Everyone should guess what color it carries on the back. His friend A sees that the ribbons of others are white. After a few minutes without anyone saying, A gave the correct answer. What is the answer and how did you reason it? 19. A trader has four pieces or pesos. With these weights can be carried out all weighing from 1 to 40 kg. by kilo. What weights does it have? 20. Di, sharp, three numbers whose products and sums are equal. 21. Bittori and Itziar want to catch the eleven train. The Bittori watch is delayed 10 minutes, but he believes it is advanced 5 minutes. The Itziar clock, however, is 5 minutes ahead, but he believes that 10 minutes is delayed. What will arrive at the first stop? 22. In a library, located from left to right, there are three books. The first consists of 340 pages, the second of 400 and the third of 350. A pipe begins to be eaten from the first page of the first specimen to the last of the third. How many leaves will cross the carcoma? 23. Cemetery or epitaph of Diofanto: Diophanto counts with the tomb in which appear the different stages of his life: in his childhood he gave the sixth part of his life; from there, the twelfth part until the beard began to fill the cheeks; from then on, the seventh, until he married; five years after his marriage his son was born; then, when he arrived at the half of his father's age, he died unpredictably; his father wept. Hence you can imagine his age. 24. Take five coins. Place three faces upwards and the other two with the cross upwards (AGAGA). You have to get to leave the three faces to one side and crosses to another (AAAGG or GGAAA). For this you always have to move two coins: one face and the other cross (AG or GA). (The gap that is left after a move cannot be filled by the union of coins, that is, if AGAGAÆA_ _GAGA is made, then A F AGAGA, but yes AAGG_ _A). 25. The ball in a bag can be white or black. A white ball has been put into the bag and, after pulling it out, a ball has been drawn that is white. What is the probability that the remaining ball in the bag will be white when it comes out? (Lewis Carroll). 26. In a bag there are several white and black balls (at least one of each). The balls will come out as follows: take a ball, aim its color and refuse. If a second ball comes out and is the color of the previous one, it will be discarded. It will take the third, if it returns to be of the same color will be discarded and so on until getting a ball of another color. When a different ball comes out, he gets into the bag and once agitated the extraction will be retaken. For example: Black on 1st exit: Discard in 2nd exit the black: discard in 3rd exit the white: put in bag White at exit 4: discard at exit 5 the black: put in the bag White at exit 6: discard at exit 7 the white: discard at exit 8 the black: put in the bag No matter the initial situation. The last remaining ball in the bag has the same probability of being black. What is that probability? Saioa hasi iruzkinak uzteko.

In the previous section, we dealt with the multiplication system and defined the infinite and finite multiplication factors. This section was about conditions for a stable, self-sustained fission chain reaction and maintaining such conditions. This problem contains no information about the spatial distribution of neutrons because it is a point geometry problem. We have characterized the effects of the global distribution of neutrons simply by a non-leakage probability (thermal or fast), which, as stated earlier, increases toward a value of one as the reactor core becomes larger. To design a nuclear reactor properly, predicting how the neutrons will be distributed throughout the system is highly important. This is a very difficult problem because the neutrons interact differently with different environments (moderator, fuel, etc.) in a reactor core. Neutrons undergo various interactions when they migrate through the multiplying system. To a first approximation, the overall effect of these interactions is that the neutrons undergo a kind of diffusion in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the diffusion approximation, based on the neutron diffusion theory. This approximation allows solving such problems using the diffusion equation. In this chapter, we will introduce the neutron diffusion theory. We will examine the spatial migration of neutrons to understand the relationships between reactor size, shape, and criticality and determine the spatial flux distributions within power reactors. The diffusion theory provides a theoretical basis for neutron-physical computing of nuclear cores. It must be added many neutron-physical codes are based on this theory. First, we will analyze the spatial distributions of neutrons, and we will consider a one-group diffusion theory (mono-energetic neutrons) for a uniform non-multiplying medium. That means that the neutron flux and cross-sections have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of the spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum. See also: Neutron Flux Spectra. Moreover, mathematical methods used to analyze a one-group diffusion equation are the same as those applied in more sophisticated and accurate methods such as multi-group diffusion theory. Subsequently, the one-group diffusion theory will be applied in simple geometries on a uniform multiplying medium (a homogeneous “nuclear reactor”). Finally, the multi-group diffusion theory will be applied in simple geometries on a non-uniform multiplying medium (a heterogenous “nuclear reactor”). Derivation of One-group Diffusion Equation The derivation of the diffusion equation depends on Fick’s law, which states that solute diffuses from high concentration to low. But first, we have to define a neutron flux and neutron current density. The neutron flux is used to characterize the neutron distribution in the reactor, and it is the main output of solutions of diffusion equations. The neutron flux, φ, does not characterize the flow of neutrons. There may be no flow of neutrons, yet many interactions may occur (I = Σ.φ). The neutrons move in random directions and hence may not flow. Therefore the neutron flux φ is more closely related to densities. Neutrons will exhibit a net flow when there are spatial differences in their density. Hence we can have a flux of neutron flux! This flux of neutron flux is called the neutron current density. In chemistry, Fick’s law states that: Suppose the concentration of a solute in one region is greater than in another of a solution. In that case, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient. In one (spatial) dimension, the law is: - J is the diffusion flux, - D is the diffusion coefficient, - φ (for ideal mixtures) is the concentration. The use of this law in nuclear reactor theory leads to the diffusion approximation. Fick’s law in reactor theory stated that: The current density vector J is proportional to the negative of the gradient of the neutron flux. The proportionality constant is called the diffusion coefficient and is denoted by the symbol D. In one (spatial) dimension, the law is: - J is the neutron current density (neutrons.cm-2.s-1) along the x-direction, the net flow of neutrons that pass per unit of time through a unit area perpendicular to the x-direction. - D is the diffusion coefficient, it has the unit of cm, and it is given by: - φ is the neutron flux, the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions per unit time. where J denotes the diffusion flux vector. Note that the gradient operator turns the neutron flux, which is a scalar quantity into the neutron current, which is a vector quantity. The physical interpretation is similar to the fluxes of gases. The neutrons exhibit a net flow in the direction of least density. This is a natural consequence of greater collision densities at positions of greater neutron densities. Consider neutrons passing through the plane at x=0 from left to right due to collisions to the plane’s left. Since the concentration of neutrons and the flux is larger for negative values of x, there are more collisions per cubic centimeter on the left. Therefore more neutrons are scattered from left to right, then the other way around. Thus the neutrons naturally diffuse toward the right. Validity of Fick’s Law It must be emphasized that Fick’s law is an approximation and was derived under the following conditions: - Infinite medium. This assumption is necessary to allow integration of overall space but flux contributions are negligible beyond a few mean free paths (about three mean free paths) from boundaries of the diffusive medium. - Sources or sinks. Derivation of Fick’s law assumes that the contribution to the flux is mostly from elastic scattering reactions. Source neutrons contribute to the flux if they are more than a few mean free paths from a source. - Uniform medium. Derivation of Fick’s law assumes that a uniform medium was used. There are different scattering properties at the boundary (interface) between the two media. - Isotropic scattering. Isotropic scattering occurs at low energies but is not true in general. Anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory). - Low absorbing medium. Fick’s law derivation assumes (an expansion in Taylor’s series) that the neutron flux, φ, is slowly varying. Large variations in φ occur when Σa (neutron absorption) is large (compared to Σs). Σa << Σs - Time-independent flux. Derivation of Fick’s law assumes that the neutron flux is independent of time. To some extent, these limitations are valid in every practical reactor. Nevertheless, Fick’s law gives a reasonable approximation. For more detailed calculations, higher-order methods are available. Neutron Balance – Continuity Equation The mathematical formulation of neutron diffusion theory is based on the balance of neutrons in a differential volume element. Since neutrons do not disappear (β decay is neglected), the following neutron balance must be valid in an arbitrary volume V. rate of change of neutron density = production rate – absorption rate – leakage rate Substituting for the different terms in the balanced equation and by dropping the integral over (because the volume V is arbitrary), we obtain: - n is the density of neutrons, - s is the rate at which neutrons are emitted from sources per cm3 (either from external sources (S) or from fission (ν.Σf.Ф)), - J is the neutron current density vector - Ф is the scalar neutron flux - Σa is the macroscopic absorption cross-section In steady-state, when n is not a function of time: The Diffusion Equation In previous chapters, we introduced two bases for the derivation of the diffusion equation: which states that neutrons diffuse from high concentration (high flux) to low concentration. which states that rate of change of neutron density = production rate – absorption rate – leakage rate. We return now to the neutron balance equation and substitute the neutron current density vector by J = -D∇Ф. Assuming that ∇.∇ = ∇2 = Δ (therefore div J = -D div (∇Ф) = -DΔФ) we obtain the diffusion equation. The derivation of the diffusion equation is based on Fick’s law which is derived under many assumptions. Therefore, the diffusion equation cannot be exact or valid at places with strongly differing diffusion coefficients or in strongly absorbing media. This implies that the diffusion theory may show deviations from a more accurate solution of the transport equation in the proximity of external neutron sinks, sources, and media interfaces. To solve the diffusion equation, which is a second-order partial differential equation throughout the reactor volume, it is necessary to specify certain boundary conditions. It is very dependent on the complexity of a certain problem. One-dimensional problems solutions of diffusion equation contain two arbitrary constants. Therefore, we need two boundary conditions to determine these coefficients to solve a one-dimensional one-group diffusion equation. The most convenient boundary conditions are summarized in the following few points: Diffusion Length of Neutron During the diffusion equation solution, we often meet with a very important parameter that describes the behavior of neutrons in a medium. The solution of diffusion equation (let assume the simplest diffusion equation) usually starts by division of entire equation by diffusion coefficient: The term L2 is called the diffusion area (and L is called the diffusion length). For thermal neutrons with an energy of 0.025 eV, a few values of L are given in the table below. Physical Meaning of the Diffusion Length It is interesting to try to interpret the “physical” meaning of the diffusion length. The physical meaning of the diffusion length can be seen by calculating the mean square distance that a neutron travels in the one direction from the plane source to its absorption point. It can be calculated that L2 is equal to one-half the square of the average distance (in one dimension) between the neutron’s birth point and its absorption. If we consider a point source of neutrons, the physical meaning of the diffusion length can be seen again by calculating the mean square distance that a neutron travels from the source to its absorption point. It can be calculated that: L2 is equal to one-sixth of the square of the average distance (in all dimensions) between the neutron’s birth point (as a thermal neutron) and its absorption. This distance must not be confused with the average distance traveled by the neutrons. The average distance traveled by the neutrons is equal to the mean free path for absorption λa = 1/Σa and is much larger than the distance measured in a straight line. This is because neutrons in the medium undergo many collisions, and they follow a very zig-zag path through the medium. Applicability of Diffusion Theory Nowadays, the diffusion theory is widely used in the core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). It provides a strictly valid mathematical description of the neutron flux. Still, it must be emphasized that the diffusion equation (in fact the Fick’s law) was derived under the following assumptions: - Infinite medium - No sources or sinks. - Uniform medium. - Isotropic scattering. - Low absorbing medium. - Time-independent flux. To some extent, these limitations are valid in every practical reactor. Nevertheless, the diffusion theory gives a reasonable approximation and makes accurate predictions. Nowadays, reactor core analyses and designs are often performed using nodal two-group diffusion methods. These methods are based on pre-computed assembly homogenized cross-sections, diffusion coefficients, and assembly discontinuity factors (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Highly absorbing control elements are represented by effective diffusion theory cross-sections, which reproduce transport theory absorption rates. These pre-computed data (discontinuity factors, homogenized cross-sections, etc.) are calculated by neutron transport codes based on a more accurate neutron transport theory. In short, neutron transport theory is used to make diffusion theory work. Two methods exist for the calculation of the pre-computed assembly cross-sections and pin factors. - Deterministic methods that solve the Boltzmann transport equation. - Stochastic methods are known as Monte Carlo methods that model the problem almost exactly. These methods are very efficient and accurate when applied to the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). Solutions of the Diffusion Equation – Non-multiplying Systems As was previously discussed, the diffusion theory is widely used in the core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). This section is not about such calculations but provides illustrative insights, how can be the neutron flux distributed in any diffusion medium. In this section, we will solve diffusion equations: in various geometries that satisfy the boundary conditions discussed in the previous section. We will start with simple systems and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group. They are emitted and diffuse at thermal energy (0.025 eV). In the first section, we will deal with neutron diffusion in a non-multiplying system, i.e., in a system where fissile isotopes are missing, the fission cross-section is zero. The neutrons are emitted by an external neutron source. We will assume that the system is uniform outside the source, i.e., D and Σa are constants. Solutions of the Diffusion Equation – Multiplying Systems In the previous section, it has been considered that the environment is non-multiplying. In a non-multiplying environment, neutrons are emitted by a neutron source situated in the center of the coordinate system and then freely diffuse through media. We are now prepared to consider neutron diffusion in the multiplying system containing fissionable nuclei (i.e., Σf ≠ 0). In this multiplying system, we will also study the spatial distribution of neutrons, but in contrast to the non-multiplying environment, these neutrons can trigger nuclear fission reactions. In this section, we will solve the following diffusion equation in various geometries that satisfy the boundary conditions. In this equation, ν is the number of neutrons emitted in fission, and Σf is the macroscopic cross-section of the fission reaction. Ф denotes a reaction rate. For example, the fission of 235U by thermal neutron yields 2.43 neutrons. It must be noted that we will solve the diffusion equation without any external source. This is very important because such an equation is a linear homogeneous equation in the flux. Therefore if we find one solution of the equation, then any multiple is also a solution. This means that the absolute value of the neutron flux cannot possibly be deduced from the diffusion equation. This is totally different from problems with external sources, which determine the absolute value of the neutron flux. We will start with simple systems (planar) and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group. They are emitted and diffuse at thermal energy (0.025 eV). Solutions of diffusion equations, in this case, provide illustrative insights, how can be the neutron flux distributed in a reactor core.

contact usfrançais Proportion - Calculate the margin of error Proportion - Setting sample size Difference between two proportions Mean - Calculate the margin of error Mean - Setting sample size Difference If they want a 90% confidence interval, how many people do they need to know about? Sample Size: Margin of Error (%) -- *This margin of error calculator uses a normal distribution (50%) to calculate your optimum margin of error. What margin of error can you accept? 5% is a common choice % The margin of error is the amount of error that you can tolerate. http://facetimeforandroidd.com/margin-of/margin-of-error-population-proportion-calculator.php The number of standard errors you have to add or subtract to get the MOE depends on how confident you want to be in your results (this is called your confidence Construct a 95% confidence interval for the proportion of Americans who believe that the minimum wage should be raised. The system returned: (22) Invalid argument The remote host or network may be down. Online surveys with Vovici have completion rates of 66%! navigate here Volgende Margin of Error Example - Duur: 11:04. The sample size doesn't change much for populations larger than 100,000. Find Us On Facebook Follow on Twitter LinkedIn Google Plus YouTube Subscribe using RSS Select Statistical Consultants Home About Us Our Consultants News Case Studies Blog Careers Sectors Academic Business Utilities If you don't know, use 20000 How many people are there to choose your random sample from? Log in om ongepaste content te melden. Suppose you wanted to find a 95% confidence interval with a margin of error of .5 for m knowing s = 10. Sampling Error Calculator Laden... Stomp On Step 1 96.314 weergaven 7:21 Confidence Interval for Population Means in Statistics - Duur: 8:53. Margin Of Error Excel Weergavewachtrij Wachtrij __count__/__total__ How to calculate Margin of Error Confidence Interval for a population proportion statisticsfun AbonnerenGeabonneerdAfmelden50.66150K Laden... Solution: We have E = 3, zc = 1.65 but there is no way of finding sigma exactly. Here are the steps for calculating the margin of error for a sample proportion: Find the sample size, n, and the sample proportion. Hence this chart can be expanded to other confidence percentages as well. Margin Of Error Sample Size The true answer is the percentage you would get if you exhaustively interviewed everyone. Bezig... Solution The formula states that Squaring both sides, we get that zc2 p(1 - p) E2 = n Multiplying by n, we get nE2 = zc2[p(1 If you'd like to see how we perform the calculation, view the page source. Suppose that you have 20 yes-no questions in your survey. Confidence Interval Margin Of Error Calculator What level of confidence would you like? % >Step 4: If you have a previous estimate of the population proportion enter it here. Margin Of Error Calculator Without Population Size Population Size: The probability that your sample accurately reflects the attitudes of your population. Laden... weblink Professor Leonard 44.041 weergaven 2:24:10 Confidence Intervals Part I - Duur: 27:18. Inloggen Transcript Statistieken 43.461 weergaven 202 Vind je dit een leuke video? statisticsfun 65.593 weergaven 6:46 Statistics Lecture 7.2: Finding Confidence Intervals for the Population Proportion - Duur: 2:24:10. How To Find Margin Of Error On Ti 84 If you create a sample of this many people and get responses from everyone, you're more likely to get a correct answer than you would from a large sample where only The industry standard is 95%. Transcript Het interactieve transcript kan niet worden geladen. navigate here Lower margin of error requires a larger sample size. It is the range in which the true population proportion is estimated to be and is often expressed in percentage points (e.g., ±2%). Note that the actual precision achieved after you Population Proportion Sample Size Calculator Typically, you want to be about 95% confident, so the basic rule is to add or subtract about 2 standard errors (1.96, to be exact) to get the MOE (you get Laden... For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. Over Pers Auteursrecht Videomakers Adverteren Ontwikkelaars +YouTube Voorwaarden Privacy Beleid & veiligheid Feedback verzenden Probeer iets nieuws! Refer to the above table for the appropriate z*-value. Margin Of Error Definition However, the relationship is not linear, e.g., doubling the sample size does not halve the confidence interval. For some further information, see our blog post on The Importance and Effect of Sample Size. Take the square root of the calculated value. Geüpload op 22 jul. 2011In this tutorial I explain and then calculate, using an example, the margin of error and confidence interval for a population proportion.Like us on: http://www.facebook.com/PartyMoreStud...Link To Playlist his comment is here Population size This is the total number of distinct individuals in your population. In this formula we use a finite population correction to account for sampling from populations that are small.Â

Demystifying Boolean Algebra: An Introduction and Key Theorems Comprehensive Definition, Description, Examples & Rules Introduction to Boolean Algebra Boolean algebras are the operations of logic that are useful in digital worlds. It is the basics of computer science, and its advanced concepts include K-map. You can easily make decisions by using Boolean algebra. In this article, you will find its definition, significance, theorems, truth table, and so on. Defining Boolean Algebra It is related to binary variables. Boolean variables reflect one as true and 0 as false. Boolen refers to binary algebra. Boolean algebra operates on logical values, and this algebra concept is different from elementary algebra. This concept of boolean algebra was coined by George Boule. His books “The Mathematical Analysis of Logic” and An Investigation of Law Thought” describe the boolean algebra concept in depth. In the initial period, Boolean algebra was used in computer programming languages. It also plays a very significant role in statistics, and then it expanded with time. Significance in Digital Logic In the field of computer science and statistics, the boolean algebra method is significant. - You can use Boolean algebra in programming and coding. It also provides the data structure. - Boolean algebra is fundamental to digital circuit design. - You can use this concept in artificial intelligence and machine learning as well. What is Boolean algebra? Boolean algebra is a binary variable in mathematics. This concept is also known as binary algebra. It is the basics of computer applications that operate the logical value. Boolean Variables and Binary Logic Boolean algebra is presented as conjunction (∧), disjunction (∨), and negation (¬). A Comprehensive Explanation of Boolean Algebra Variables are the basics of Boolean algebra. These variables are useful in computer applications and data management. Boolean algebra represents one as true and 0 as false. It is helpful in making a decision, just as Boolean algebra is helpful in making a decision. ‘ (or) ¬ . (or) ∧ + (or) ∨ In the real world, you can use this concept for advanced skills like - You can use the boolean variable in statistics. - It is helpful in computer science and data management. - You can resolve any statement doubt by using Boolean algebra. - This concept has relevance in the digital world. Whether it is artificial intelligence or networking, Boolean algebra is the most relevant concept. Basic Theorems and Properties of Boolean Algebra Boolean algebra is useful in finding the equations. You can use this theorem to solve equations and find answers. It includes addition and multiplication identities. The addition identity refers to OR along with the 0 element. - If you add an (or) variable with 0, you will find the same results without any change. - Like Z or 0 = Z It also has a multiplicative identity that refers to AND along with one as an element. - If you multiply (and) variables by 0, you will find the same results without any change. - Like Z and 1 = Z Domination laws state that you will find the same results with AND and OR methods. - You will find the same answer without any changes after (or) of the boolean variable with 0. - In multiplications, you will find the same after (and) of the boolean variable with 1. This reflects the dominance of Boolean algebra in mathematics. In idempotent laws, if you add that is (or) two Boolean variables with each other, then you will find the same answer. Z orZ = Z If you multiply that (and) two Boolean variables with each other, then you will find the same answer. Z and Z = Z Complement laws reflect the link between 0 and 1. For example, for (or), it will be 1, and for (and), it will be 0. Boolean Equation and Expressions Boolean equation are useful in computer science and data management. It is helpful in decision-making as well. There are three Boolean expressions that you will be going to learn in further sections. Writing Boolean Expressions - Boolean expressions represent: - Conjunction or operation - Disjunction or OR operation - Negation or not operation Symbols for these expressions are conjunction (∧), disjunction (∨), and negation (¬). Simplifying Boolean Expressions, Boolean expressions make the concept easier. - There are basic properties like identity, etc. What we have discussed in the proper sections are the basics that you need to apply. - The De Morgan Law is another Boolean expression that can easily solve complex equations. - You can group the common variables and find out the answers. Using truth tables for verification, A ^ B A v B You can find out the truth by evaluating simplified and unsimplified expressions for all combinations of values for variables in expressions, and then you need to make a comparison of the results. If they are the same in all cases, then the expression will be verified. Boolean algebra explained Boolean algebra is the operation of logical value that helps in making decisions through truth (1) and false (0). In further sections, you will find out its relevance. Logic Gates and Boolean Algebra There are various gates in Boolean algebra, among which the most significant are as follows: - “And” is the boolean algebra logic gate that reflects one if all the inputs are true. - “Or” is there another logic gate that will be reflected with one if all inputs are 1. - Not is also a logic gate that is a bit different from AND and OR logic gates. It reflects 1 with 0 and 0 with 1. Boolean Algebra in Circuit Design - Boolean algebra is useful in circuit design. - Boolean algebra is used by engineers to increase the efficiency of circuits through various Boolean algebra properties. Boolean Algebra in Computer Science - In computer science, the concept of boolean algebra is useful for programming. - It is also useful for managing data and making decisions. Boolean Theory: Beyond the Basics Boolean expressions are the basics of this theory. There are various advanced concepts that include theorems through which students can get advanced. According to the consensus theorem, in three variables, two are the complement of each other. This theorem is useful with the Karnaugh map. A*B+ B*C+ C*A= BC De Morgan’s Laws De Morgan’s first law includes conditions like (A*B) = A+B According to this law, the product of a boolean variable is equivalent to the sum of variables. His theorem is proved with the truth table. His second theorem is just opposite the first. It includes the sum of a Boolean variable, which is equivalent to its product. The distributive law of the Boolean theorem includes conditions like: A*(B+C) = (A*B) + (A*C) A+ (B*C) = (A+B) * (A+C) The absorption theorem is an inclusion of AND and OR logic. Zor Z = Z Z and Z = Z Advanced Topics in Boolean Algebra These are the advanced concepts in Boolean algebra that you must know to develop your understanding. Karnaugh maps and minimization The Karnaugh map is referred to as the K-map. You can minimize 3 to 4 variables without using the Boolean theorem. You can find out more information through k-mao without taking any help from the truth table. Karnaugh map includes the product of sum and sum of product. You can find patterns by using this concept. Boolean Algebra in Boolean Functions It represents a logical value in Boolean algebra. Its function includes helping in decision-making. Boolean better functions with the truth table. It includes three terms: conjunction (∧), disjunction (∨), and negation (¬). Binary Decision Diagrams (BDDs) It represent the binary function in computer science. A binary decision diagram reflects the compressed form of relations. Practical Applications of Boolean Algebra Boolean algebra is relevant in the digital world. By understanding its practical applications, you can better clarify your concepts. Digital Electronics and Logic Circuits - Boolean algebra is useful for designing circuits. - In digital work, Boolean algebra is used by the engineer. - Boolean algebra is useful in storing the data. Computer Programming and Algorithms - In computer science, Boolean algebra is useful for artificial intelligence and machine learning. - Boolean algebra is also helpful in managing software. Information Security and Cryptography - In information security, digital signatures involve the boolean operation. - Resources can be secured by using boolean algebra. Common Misconceptions and Pitfalls There are various misconceptions and pitfalls for students when using Boolean algebra. Addressing Common Errors in Boolean Algebra - Students get confused with the expressions of Boolean algebra, like OR and AND. This can affect the overall result. - Students sometimes get confused with the theorems. - All theorems have different formulas, and misunderstandings can lead to a false answer. - Students don’t prefer to understand the truth table, which makes the Boolean algebra concept complex for them. Step Up Your Math Game Today! Free sign-up for a personalised dashboard, learning tools, and unlimited possibilities! - Boolean algebra operates logical values, and this concept is algebra different from elementary algebra. - It is related to binary variables. Boolean variables reflect one as true and 0 as false. Boolen refers to binary algebra. - Boolean expressions include conjunction or And operation, disjunction or OR operation, negation or not operation. - It has practical relevance as well. In the digital world and computer science, the concept of boolean algebra is common. - If you want to develop a thorough understanding, then you attend the Edulyte’s worksheet. Question comes here Frequently Asked Questions Boolean variables are the operations of logic, and binary logics are represented with one as true and 0 as false. It is helpful in making a decision, just as Boolean algebra is helpful in making a decision. Boolean equation are comprised of conjunction (∧), disjunction (∨), and negation (¬), and they can be simplified with theorems and K-maps. The binary decision diagram, Boolean algebra, binary function, and Karnaugh map are the advanced Boolean algebra concepts. Students sometimes get confused with the theorems. All theorems have different formulas, and misunderstandings can lead to a false answer. Students don’t prefer to understand the truth table, which makes the Boolean algebra concept complex for them.

In this post, we will look at parallel circuits. The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is useful for KS3 and KS4 students. Without further ado, here is a a summary. This is part 4 of a continuing series. (Click to read Part 1, Part 2 or Part 3.) The ‘Parallel First’ Heresy I advocate teaching parallel circuits before teaching series circuits. This, I must confess, sometimes makes me feel like Captain Rum from Blackadder Two: The main reason for this is that parallel circuits are conceptually easier to analyse than series circuits because you can do so using a relatively naive notion of ‘flow’ and gives students an opportunity to explore and apply the recently-introduced concept of ‘flow of charge’ in a straightforward context. Redish and Kuo (2015: 584) argue that ‘flow’ is an example of embodied cognition in the sense that its meaning is grounded in physical experience: The thesis of embodied cognition states that ultimately our conceptual system grounded in our interaction with the physical world: How we construe even highly abstract meaning is constrained by and is often derived from our very concrete experiences in the physical world.Redish and Kuo (2015: 569) As an aside, I would mention that Redish and Kuo (2015) is an enduringly fascinating paper with a wealth of insights for any teacher of physics and I would strongly recommend that everyone reads it (see link in the Reference section). Let’s Go Parallel First — but not yet Let’s start with a very simple circuit. This can be represented on the coulomb train model like this: Five coulombs pass through the ammeter in 20 seconds so the current I = Q/t = 5/20 = 0.25 amperes. Let’s assume we have a 1.5 V cell so 1.5 joules of energy are added to each coulomb as they pass through the cell. Let’s also assume that we have negligible resistance in the cell and the connecting wires so 1.5 joules of energy will be removed from each coulomb as they pass through the resistor. The voltmeter as shown will read 1.5 volts. The resistance of the resistor R1 is R=V/I = 1.5/0.25 = 6.0 ohms. Let’s Go Parallel First — for real this time. Now let’s close switch S. This is example of changing an example by continuous conversion which removes the need for multiple ammeters in the circuit. The changed circuit can be represented on the CTM as shown Now, ten coulombs pass through the ammeter in twenty seconds so I = Q/t = 10/20 = 0.5 amperes (double the reading in the first circuit shown). Questioning may be useful at this point to reinforce the ‘flow’ paradigm that we hope students will be using: - What will be the reading if the ammeter moved to a similar position on the other side? (0.5 amps since current is not ‘used up’.) - What would be the reading if the ammeter was placed just before resistor R1? (0.25 amps since only half the current goes through R1.) To calculate the total resistance of the whole circuit we use R = V/I = 1.5/0.5 = 3.0 ohms– which is half of the value of the circuit with just R1. Adding resistors in parallel has the surprising result of reducing the total resistance of the circuit. This is a concrete example which helps students understand the concept of resistance as a property which reduces current: the current is larger when a second resistor is added so the total resistance must be smaller. Students often struggle with the idea of inverse relationships (i.e. as x increases y decreases and vice versa) so this is a point well worth emphasising. Potential Difference and Parallel Circuits (1) Let’s expand on the primitive ‘flow’ model we have been using until now and adapt the circuit a little bit. This can be represented on the CTM like this: Each coulomb passing through R2 loses 1.5 joules of energy so the voltmeter would read 1.5 volts. One other point worth making is that the resistance of R2 (and R1) individually is still R = V/I = 1.5/0.25 = 6.0 ohms: it is only the combined effect of R1 and R2 together in parallel that reduces the total resistance of the circuit. Potential Difference and Parallel Circuits (2) Let’s have one last look at a different aspect of this circuit. This can be represented on the CTM like this: Each coulomb passing through the cell from X to Y gains 1.5 joules of energy, so the voltmeter would read 1.5 volts. However, since we have twice the number of coulombs passing through the cell as when switch S is open, then the cell has to load twice as many coulombs with 1.5 joules in the same time. This means that, although the potential difference is still 1.5 volts, the cell is working twice as hard. The result of this is that the cell’s chemical energy store will be depleted more quickly when switch S is closed: parallel circuits will make cells go ‘flat’ in a much shorter time compared with a similar series circuit. Bulbs in parallel may shine brighter (at least in terms of total brightness rather than individual brightness) but they won’t burn for as long. To some ways of thinking, a parallel circuit with two bulbs is very much like burning a candle at both ends… More fun and high jinks with coulomb train model in the next instalment when we will look at series circuits. You can read part 5 here. Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education, 24(5), 561-590. Reblogged this on The Echo Chamber.

Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds HTML articles powered by AMS MathViewer - by Tao Li PDF - J. Amer. Math. Soc. 19 (2006), 625-657 Request permission Abstract:A famous example of Casson and Gordon shows that a Haken 3–manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3–manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3–manifold $M$, there is a number $N$ such that any two Heegaard splittings of $M$ are equivalent after at most $N$ stabilizations. - Ian Agol and Tao Li, An algorithm to detect laminar 3-manifolds, Geom. Topol. 7 (2003), 287–309. MR 1988287, DOI 10.2140/gt.2003.7.287 - M. Boileau, D. J. Collins, and H. Zieschang, Genus $2$ Heegaard decompositions of small Seifert manifolds, Ann. Inst. 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Res. Lett. 11 (2004), no. 2-3, 139–149. MR 2067463, DOI 10.4310/MRL.2004.v11.n2.a1 - Tao Li, Laminar branched surfaces in 3-manifolds, Geom. Topol. 6 (2002), 153–194. MR 1914567, DOI 10.2140/gt.2002.6.153 - Tao Li, Boundary curves of surfaces with the 4-plane property, Geom. Topol. 6 (2002), 609–647. MR 1941725, DOI 10.2140/gt.2002.6.609 L2 Tao Li, An algorithm to find vertical tori in small Seifert fiber spaces. Preprint. arXiv: math.GT/0209107. L4 Tao Li, Heegaard surfaces and measured laminations, I: The Waldhausen conjecture. Preprint. arXiv:math.GT; also available at: www2.bc.edu/˜taoli/publications.html. - J. Masters, W. Menasco, and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004), 133–150. MR 2052369 - W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44. MR 721450, DOI 10.1016/0040-9383(84)90023-5 - John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in $3$-manifolds, Ann. of Math. (2) 127 (1988), no. 2, 403–456. MR 932305, DOI 10.2307/2007061 - Yoav Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988), no. 3, 465–481. MR 928492, DOI 10.1007/BF01388781 - Yoav Moriah and Jennifer Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998), no. 5, 1089–1112. MR 1650355, DOI 10.1016/S0040-9383(97)00072-4 MSS Yoav Moriah, Saul Schleimer, and Eric Sedgwick, Heegaard splittings of the form H + nK, to appear in Communications in Analysis and Geometry, arXiv:math.GT/0408002. - Ulrich Oertel, Measured laminations in $3$-manifolds, Trans. Amer. Math. Soc. 305 (1988), no. 2, 531–573. MR 924769, DOI 10.1090/S0002-9947-1988-0924769-1 P Parris, Pretzel Knots. Ph.D. Thesis, Princeton University (1978). - J. H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$-dimensional manifolds, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 1–20. MR 1470718, DOI 10.1090/amsip/002.1/01 - Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings of non-Haken $3$-manifolds, Topology 35 (1996), no. 4, 1005–1026. MR 1404921, DOI 10.1016/0040-9383(95)00055-0 - Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135–147. MR 1648310, DOI 10.1016/S0166-8641(97)00184-3 - E. Sedgwick, An infinite collection of Heegaard splittings that are equivalent after one stabilization, Math. Ann. 308 (1997), no. 1, 65–72. MR 1446199, DOI 10.1007/s002080050064 - Michelle Stocking, Almost normal surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207. MR 1491877, DOI 10.1090/S0002-9947-99-02296-5 - Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X - Friedhelm Waldhausen, Some problems on $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 313–322. MR 520549 - Ying-Qing Wu, Dehn surgery on arborescent knots, J. Differential Geom. 43 (1996), no. 1, 171–197. MR 1424423 - Tao Li - Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, 02167-3806 - Email: email@example.com - Received by editor(s): November 24, 2004 - Published electronically: February 3, 2006 - Additional Notes: Partially supported by NSF grants DMS-0102316 and DMS-0406038 - © Copyright 2006 American Mathematical Society The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 19 (2006), 625-657 - MSC (2000): Primary 57N10, 57M50; Secondary 57M25 - DOI: https://doi.org/10.1090/S0894-0347-06-00520-0 - MathSciNet review: 2220101

Please see the attached file for the fully formatted problems. Please see attached file. Please show all steps in detail. In the two problems below find the Fourier Cosine Transform of the given f(x) and write f(x) as a Fourier integral. 1) -1, -Pi <x<0 f(x) = 1, 0<x<Pi 0, |x| > Pi 2) 2x+2a -a<x<0 f(x) = -2x+2a Please help me in solving this problem. Please see attached file for full problem description. Using Mathematica and computer facilities express the function f(x) in terms of Fourier series expansion and show that the series converges as the number of terms increases: f(x) = x e-x/4 sin(x/3) -π < x < π Please see attachment. 1. What is the Fourier Transform for the convolution of sin(2t)*cos(2t). 2. Compute the inverse Fourier transform for X(w)= sin^2*3w 3. A continuous time signal x(t) has the Fourier transform X(w) = 1/jw+b where b is a constant. Determine the Fourier transform for v(t) = x*(5t-4) On attachment 111 do problems 14, 16, 20, 24 On attachment 22 do problems 2, 7, 8, 11, 12 Show steps as needed. Find the Fourier series expansion of the functions: f(t) = 1 if Pi/3<|t|<2Pi/3 0 everywhere else f(t) = 1 Pi/3 < t < 2Pi/3 -1 -2Pi/3 < t < Pi/3 0 everywhere else In the interval [-Pi , Pi] Please see the attached file for the fully formatted problems. The question is the example on page 2 of the attachment (entitled 'Uniform Transducer'). it states that the centre of the finger is at z'=L/4. I assume this is an arbitrary position. For Eq (2.4.6), the contribution from the left-hand finger is added. I'm not entirely sure how this equation is arrived at. It does not look like a Please see the attached file for the fully formatted problems. You don't have to worry about (iii). Suppose f(x) has the Fourier transform F(ω). If a ≠ 0 show that f(ax) has the Fourier Transform 1/|a| F (ω/a). Please see the attached file for the fully formatted problems. I am having difficulty computing u(x,t), also interpretation when e -> 0 See attached file for full problem description. Solving Partial Differential Equations : Laplace Transforms, Fourier Transforms and Separation of Variables Please use: 1.) LaPlace Transform and 2.) Fourier Transforms methods and 3.) our old friend separation of variables with eigenvalues expansion to solve each problem. It is not necessary to evaluate an inverse transform. Where convenient, show any solution as a convolution of two functions and indicate how these functions are det See attached file for full problem description. Question 5 is also a proof, but requires a further matlab plot. Let f (x) = |x| for x greater or equal to -1, less than or equal to +1 a) Write the Fourier series for f (x) on [-1,1]. b) Show that this series can be differentiated term by term to yield the Fourier expansion of f'(x) on [-1,1] c) Determine f'(x) and write it's Fourier series on [-1,1] d) Compare b and c. key We use the Fourier expansions of certain poynomial functions to compute the sum of some useful numerical series. The formulas are quite general and give, at the end, the Fourier expansion of every polynomial function. By the way, these formulas can be also used for a numerical approximation of pi=3.14.... Functions, Interval. See attached file for full problem description. Given the set of functions f1(t) = A1*exp(-t) f2(t) = A2*e^(-2t) Defined on the interval (0, infinity). (a) Find A1 such that f1(t) is normalized to unity on (0, infinity). Call this function PHI_1(t). (b) Find B such that PHI(t) and f2 Using the Fourier transform integral, find Fourier transforms of the following signals. xa(t) = t *exp(-αt) * u(t), α > 0; xb(t) = t2 * u(t) * u(1 – t) xc(t) = exp(-αt) * u(t) * u(1 – t), α > 0; Please see the attached file for the fully formatted problems. ODE: 1. Solve ()'sinyxy=+. 2. Find the complete solution of the ODE ()()42212cosyyyx−−=. 3. Find the complete solution of the ODE ()46sinyy−=. 4. Find a second order ODE whose solution is a family of circle with arbitrary radius and center on t The sum of the infinite series, 1/2^2 - 2/3^2 + 3/4^2 - 4/5^2 + ... is given as pi^2/12 - log 2 on pages 64-65 in the book "Summation of Series" by L. B. W. Jolley, 2nd ed., 1961, Dover Pubs. Inc. (the ^ symbol denotes exponentiation in the above series and sum). For most of the series in his book, he lists a source (referen From equation 6 of the attached, derive equation 7. In the expression for s(v), the natural log does not apply to the term [(v-vo)/(dv/2)]^2 , should just be ln(2). Yes, the limits from minus infinity to plus inifinity are adequate. Thanks for the help! Fourier coefficients / b1, b2, b3, b4, b5... b11. -------------------------------------------------------------------------------- I have an output of an electronic device (full wave rectifier) that gives a sine wave with the negative part transposed symmetric to xx so that the function is always positive. I have to find the f (See attached file for full problem description) For sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, so N=8 Using above x[n]: a) stem(x); b) Use the shift theorm to plot x delayed by 1, 4, 5, 6, and 8 samples, and plot the result for each. Remember the shift theorem says a delay by t0 seconds is equal to multiplying the spe Please see the attached file for full description. Calculate by hand the X(omega), DTFT of the sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, zero else. Using Matlab, plot the real and imaginary components of your result for X(omega) for omega=0:0.01:2*pi, one plot for the real, one part for the imaginary. On the same plot Using Fourier transforms where possible, derive the Fourier transforms of the following functions using the relationship: a.) f(x) = exp[i2po(x/lamda)sin(theta)] b.) f(x) = exp(- /ax/ ) See the attached file for full description. Find the inverse Fourier transform of each of the following Fourier transforms: X(x) = jw The answer I have is x[n] = (-1)^n / n (for n not equal to zero) 0 (for n = 0) I don't know how to get there. Define a Fourier Transform. What are its properties and application areas ? Describe its application in signal processing. Find the fourier transform of x^k, where k is a positive integer and x is a single real variable. Use tables and the scaling and time-shift property to find the Fourier transform of the signal below (which is zero for all other values of t than those shown, that is, it's not periodic) Plot the magnitude of the spectrum of this signal. Hint: Find the transform of the pulse centered around t=0, then use the time-shift p Find the Fourier series in trigonometric form for f(t) = |sin(pi*t)|. Graph its power spectrum. (See attached file for full problem description)

MATHEMATICS, often referred to as the language of nature, is arguably the most profound achievement of the human intellect. Why is math so unreasonably effective in explaining the natural world as physicist Eugene Wigner famously argued? For instance, the complex numbers, invented to solve equations without real solutions are crucial for understanding quantum mechanics in modern physics. Similarly, Srinivasa Ramanujan’s work in number theory has astonishing applications in understanding black holes! Physics is littered with many such examples of the fascinating applications of math which was developed ‘merely’ for its own sake. On the other hand, the utility of math in the social sciences has been more contested and to many it is less promising. Professor Peter Schonemann even flipped Wigner’s famous idea by claiming that math has a reasonable level of ineffectiveness in the social sciences. Unlike in physics, economics is not governed by simple and elegant equations. Often, the pursuit of mathematical elegance is disrupted by the inconvenient interference of human whim, fundamental uncertainty, and the sheer complexity of social and economic interaction. However, I would argue as do many other economists that math nevertheless has an indispensable role to play in economics. Mathematical formalism imposes the sort of discipline and incisive clarity that remains enviable otherwise. This is because mathematical argumentation necessitates explicit assumptions and logically flawless inferences, helping us overcome cognitive biases and logical fallacies. Math also makes economics portable, compact, and mobile across cultures, languages, and academic disciplines. In addition, math allows us to represent the inherently quantitative nature of claims in economics, which deals with prices and quantities. Such models enable us to draw precise causal inferences by creating artificial laboratories, otherwise infeasible in the social sciences. This centrality of mathematics in economics is not reflected in the curricula, expectations of young students and the vision of economics offered by Pakistan’s academic institutions. There are many facets to the mathematical impoverishment of economics in Pakistan. One problem imposes no culpability on the economists since the quality of mathematical research as well as instruction is generally quite sub-standard. Thus, even when we have joint programmes in economics and mathematics, the mathematical foundations of students remain dilapidated. Students who choose economics at university neither have the required mathematical background, nor the patience and curiosity to learn it. Moreover, typically the students who choose economics at university neither have the required mathematical background, nor the patience and curiosity to learn it. Often, many students are under the false impression that if they are mathematically challenged, then economics, accounting, and business studies etc, unfortunately often clubbed together, are their natural choices. Student expectations are distorted by the high school economics curricula such as that of A levels, which makes use of no math apart from rudimentary arithmetic. It is also not helpful that schools/colleges fail at communicating to prospective students that an interest in mathematics is not an option but a requirement for pursuing economics at university. Meanwhile, our universities, including the best ones happily cater to the large demand for an economics education, devoid of mathematics. In fact, often the content of courses is watered down to accommodate mathematical deficiencies of students. While this is a lucrative market strategy to satisfy students as customers, I believe that it is a disservice to both the students and the discipline of economics. Such compromised education misleads students, leading to poor preparation for graduate study by creating a sharp discontinuity between undergraduate studies at home and graduate studies abroad. Furthermore, many of our economists do not make the effort to make their courses mathematically rigorous. Some of this reluctance is explained by student incompetence and lack of ability of some economists to deliver such content. Often, economists lethargically consider mathematical tools to be redundant in their classrooms, believing they can be picked up at graduate level or learned in math classes. While teaching mathematics is primarily not the responsibility of economists, using mathematical tools to teach an authentic version of economics surely is. Moreover, there is an ideological dimension. Some economists believe that mathematics is neither the most interesting nor the most non-trivial dimension of economics. This view is also shared by some prominent academics such as heterodox economist Ha-Joon Chang who argued that 95 per cent of economics can be explained without using any math at all. Hence, it is believed that by excluding mathematical rigour, one is not excluding the essence of economics. This neglect of mathematical tools leads to an emaciated version of economics taught to students. A deep appreciation of many economic insights will remain elusive to many unless mathematics plays its lucid and illuminating role. Contrary to popular perception, mathematics facilitates and simplifies understanding rather than obfuscating it. As Harvard economist Dani Rodrik puts it, “economists use math not because they are smart but because they are not smart enough”. For many of us who perhaps unlike Keynes or Hayek are ‘not smart enough’, the gateway to deep economic understanding is impossible to cross without the diligent and generous assistance of mathematics. A popularisation of the connection between economics and mathematics in our schools is needed so that young students can make informed choices. I also propose a more seamless integration between economics and mathematics at universities so that we can train careful and rigorous thinkers. We do not want economists merely capable of posturing and concocting elaborate word salads but those who can combine empirical evidence with mathematical models to deliver concrete theoretical and policy insights. This transformation requires a change in mindset through an acceptance of the necessity of mathematical models in not just the natural sciences but also the social sciences, especially in economics. The writer is a research fellow at PIDE, Islamabad and has a MPhil in economics from the University of Oxford. Published in Dawn, September 13th, 2021

A standard definition of static equilibrium - A system of particles is in static equilibrium when all the particles of the system are at rest and the total force on each particle is permanently zero. This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next. A standard definition of mechanical equilibrium for a particle is: - The necessary and sufficient conditions for a particle to be in mechanical equilibrium is that the net force In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together... acting upon the particle is zero. The necessary conditions for mechanical equilibrium for a system of particles are: - (i)The vector sum of all external forces is zero; - (ii) The sum of the moments of all external forces about any line is zero. As applied to a rigid body, the necessary and sufficient conditions become: - A rigid body In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it... is in mechanical equilibrium when the sum of all forces on all particles of the system is zero, and also the sum of all torque Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.... s on all particles of the system is zero. A rigid body in mechanical equilibrium is undergoing neither linear nor rotational acceleration; however it could be translating or rotating at a constant velocity. However, this definition is of little use in continuum mechanics Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles... , for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states – their stability In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions... An alternative definition of equilibrium that applies to conservative systems and often proves more useful is: - A system is in mechanical equilibrium if its position in configuration space - Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints... is a point at which the gradient In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.... with respect to the generalized coordinates In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration.... of the potential energy In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule... Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition. However, the definition involving energy can be readily extended to yield information about the stability of the equilibrium state. For example, from elementary calculus Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... , we know that a necessary condition for a local minimum or a maximum of a differentiable function is a vanishing first derivative (that is, the first derivative is becoming zero). To determine whether a point is a minimum or maximum, one may be able to use the second derivative test In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point.... . The consequences to the stability of the equilibrium state are as follows: - Second derivative In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is... < 0 : The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away. - Second derivative > 0 : The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states. - Second derivative = 0 or does not exist: The second derivative test fails, and one must typically resort to using the first derivative test In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.-Intuitive explanation:... . Both of the previous results are still possible, as is a third: this could be a region in which the energy does not vary, in which case the equilibrium is called neutral or indifferent or marginally stable. To lowest order, if the system is displaced a small amount, it will stay in the new state. In more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the x -direction but instability in the y -direction, a case known as a saddle point In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction... . Without further qualification, an equilibrium is stable only if it is stable in all directions. The special case of mechanical equilibrium of a stationary object is static equilibrium. A paperweight on a desk would be in static equilibrium. The minimal number of static equilibria of homogeneous, convex bodies (when resting under gravity on a horizontal surface) is of special interest. In the planar case, the minimal number is 4, while in three dimensions one can build an object with just one stable and one unstable balance point, this is called Gomboc A gömböc is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter... . A child sliding down a slide A playground or play area is a place with a specific design for children be able to play there. It may be indoors but is typically outdoors... at constant speed would be in mechanical equilibrium, but not in static equilibrium. An example of mechanical equilibrium is a person trying to press a spring. He or she can push it up to a point after which it reaches a state where the force trying to compress it and the resistive force from the spring are equal, so the person cannot further press it. At this state the system will be in mechanical equilibrium. When the pressing force is removed the spring attains its original state. - Dynamic equilibrium A dynamic equilibrium exists once a reversible reaction ceases to change its ratio of reactants/products, but substances move between the chemicals at an equal rate, meaning there is no net change. It is a particular example of a system in a steady state... - Engineering mechanics Metastability describes the extended duration of certain equilibria acquired by complex systems when leaving their most stable state after an external action.... - Statically indeterminate In statics, a structure is statically indeterminate when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure.... Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity... Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a... - Marion & Thornton, Classical Dynamics of Particles and Systems. Fourth Edition, Harcourt Brace & Company (1995).

In Euclidean geometry, two or more points on a line that are close to or far from each other are said to be collinear. The word "collinear" is derived from the Latin word columella, meaning "a supporting beam." Thus, three things that are colinear are lying along the same beam. Two points are called collinear if they lie on the same line but not all of them have to be marked as such. If necessary, you can mark one point by putting a dot next to it or even giving it a number. There are several ways of representing four or more points that are collinear. You can show all of them on one diagram by labeling each one of them with a distinct letter, like so: a b c D. Or, if you want to save space, you can put a symbol for a straight line between any two of them that are close together and leave out the rest, like this: a - b - C. A special case occurs when there are only three points involved, because in this case there is only one way they can be collinear and that's if they are on the same line. Here is an example of four points that are collinear: 4, 7, 10, 13. Collinear points are points that are parallel to a line. Any two points that are always collinear because a straight line can always join them. These points can be on the same object or different objects. For example, if you look at this picture of the Collinwood Cemetery in Columbus, Ohio, you can see that many monuments are located along a single path. These monuments are all collinearly aligned because the path they are on is shaped like a line from start to finish. You can also see that some of the stones are larger than others; these represent differences in age and status among the people buried there. The oldest person buried there was born in 1807 and lived for 114 years and 6 months. The newest person buried there was born in 2000 and died just eight days later. All seven people buried here were leaders who helped shape America into what it is today. They all had strong beliefs about how society should work and they all tried to make their world better by helping other people. No one is actually buried here; this is only a monument to show where they are now because none of them were alive when they were put on display. The first thing you will notice about this cemetery is the large number of monuments that mark the graves of each individual. Three or more points,... are said to be collinear if they lie on a single straight line. A line on which points lie, especially if it is related to a geometric figure such as a triangle, is sometimes called an axis. Two points are trivially collinear since two points determine a line. Any three non-collinear points define a plane. A fourth point that does not lie in this plane but that is not equal to one of the first three points will also not lie in the plane. A fifth point that lies outside this plane is called a collinear point. These can occur in many configurations including inside, outside, above, and below the plane defined by the first four points. There are five types of collinear configurations: opposite, adjacent, parallel, perpendicular, and mixed. Opposite means that the fifth point is on the other side of the first point from the second point. For example, if you were to draw a diagram of the following configuration, you would say that p5 is opposite p1. Adjacent means that the fifth point is between the first two points (not including their relationship to each other). So for example, if you were to draw a diagram of the following configuration, you would say that p5 is adjacent to p1 and p4. Parallel means that the lines formed by each pair of points intersect at a single point. Collinear points are three or more points on the same line. For instance, consider the points A, B, and C on the line m. They are parallel. On the line n, the points D, B, and E are located. These points are also collinear. If you draw three lines through these points that don't intersect, each of them will pass through exactly two of them. This means that there are always pairs of points that do not lie on the same line. According to this definition, every pair of lines in 3D space determines a unique point (the intersection of both lines). So, in order for these definitions to make sense, you need at least three dimensions. In four dimensions, you would have a volume of space where any two lines would define a point - but since we're only counting lengths here, that volume must be infinite. In general, if you have d-dimensional space, then there are d+1-dimensional hyperplanes that can be defined by any set of d lines. Each of these hyperplanes will contain d points, so they will all be different. There cannot be more than d+1-dimensional surfaces in d-space, because if there were, they would overlap. Because we can draw a distinct (one) line across two locations, they are always collinear. If three points are on the same line, they are collinear. Points A, B, and C are not converging. We can construct a line between A and B, A and C, and B and C, but not between all three places. Therefore, the points are collinear. Collinear points share the same property as parallel lines: They cannot be used to find the distance between two locations. However, they do provide a way to compare distances within the same location. If one point is closer to another point than it is to some other point, then they are collinear. Closeness in space can also be defined in terms of angles. If two locations are close together, their corresponding angles will be small. For example, if there is some angle between 0 and 10 degrees, then they are collinear. If there is no single angle that describes their relationship, then they are not collinear. There are several ways that two locations can be described as being non-collinear. One simple method is to say that they are opposite one another with respect to some axis or plane. For example, if one location is above the other on the same vertical plane, then they are not collinear. There are two locations, so they must be opposite one another. .. An axis is a line on which points lie, especially if it is connected to a geometric figure such as a triangle. Because two points define a line, they are trivially collinear. A line can also be called parallel to itself; thus, the horizontal and vertical lines in any coordinate plane are examples of lines that are collinear.

the 2010 College guide Serving S erving S Sonoma, o onoma, Nap Napa a & Marin Counties8L>LJK(/$)+#)'('Yf_\d`Xe%ZfdMFC%*)%(+ Countie es 8L>LJK(/$)+# )'(' Yf_\d`Xe%Zfd MFC% *)%(+ .. .. .. .. .. .. .. .. .. .. .. .. Performances10 Enrich. Educate. Entertain. Your Community Non-Profit Arts Center for 29 years On-sale Friday at noon For tickets call 707.546.3600 (noon-6pm Tue-Sat) Online wellsfargocenterarts.org Highway 101 to River Road, Santa Rosa • Connecting our Community through the Arts Wells Fargo Center for the Arts gratefully acknowledges generous support from on guard Bulldog Standard U-Lock msrp: $39.99 fox Digit Short Finger Glove msrp: $24.99 specialized S-Works Helmet msrp: $229.99 specialized S-Works Road Shoes (Men’s & Women’s) msrp: $349.99 N OW TH R O U G H S U N D A Y, E V E RY T H IN G IS O N S A L E! cannondale 2010 CAAD9 5 Compact raleigh 2010 Rush Hour 95 . msrp: $1499.99 95 . msrp: $769.99 all clothing in stock shimano A530 DualPurpose Pedals all shoes in stock niterider MiNewt Mini USB Headlight specialized 2010 Stumpjumper FSR Comp 95 . msrp: $2199.99 specialized Phenom SL Saddle (black 143mm) msrp: $149.99 viva 2010 Juliett 999.95 msrp: $1399.99 all sunglasses in stock % 20 F! bes all standard tu ck sto in $ thule Sidearm 594 Rooftop Bike Rack msrp: $169.99 OUR LAST SALE OF THE YEAR IS ALSO OUR BIGGEST—DON’T MISS OUT! continental Ultra GatorSkin 700x23c & 700x25c Road Tires msrp: $54.99 in San Rafael 264 Petaluma Blvd N. (707) 776-0606 836 Fourth St. (415) 454-3747 #1 Gate 6 Rd. (415) 332-3200 www.MikesBikes.com P E TA L U M A | S A N R A FA E L | S A U S A L I T O | S A N F R A N C I S C O | B E R K E L E Y | PA L O A LT O | L O S G AT O S | S A C R A M E N T O BcPcdbD_SPcT Exactly what about me is illegal? As told to Justine McDaniel am a 20-year-old woman. I graduated from Santa Rosa High School, where I was junior class president and Associated Student Body president. I speak three languages. I was voted “Most Admiredâ€? in my senior class. I just completed my second year of college. I am an illegal immigrant. I was 11 months old when my parents brought me to the United States from Mexico. My dad was an accountant in Mexico, but we starved. It’s not easy to pick up and leave, but my parents were thinking about my future. I first realized I was different when I was in the third grade. After immigration raids on my neighborhood, I thought my family would be deported, and I started crying at school. When my teacher asked me what was wrong, I said I didn’t want my mom to get taken away. From then on, I started thinking about my future. I got obsessed with college. I felt vulnerable and powerless, but I thought being a good student would somehow help. My parents have appealed their case for residency twice since 2007. Now it is up before the Ninth Circuit Court of Appeals, and there’s no way to know when our case will be heard. We have spent over $15,000. Because of the lawyer fees, we lost our house and I had to return from my four-year university to go to the SRJC. My parents divorced after our house was foreclosed on. Most workplaces won’t hire people who aren’t citizens, so it is difficult for me to find a job. I’m really interested in politics, but I have to pretend that I vote. I still don’t have a driver’s license. It was huge in high school having to lie to all of my friends whenever they asked why I didn’t have it. Every normal hardship is that much harder. I can’t apply for residency myself, because to get a green card, a spouse, parent or sibling who is over 21 has to petition on your behalf. The oldest of my brothers is 16, so I will have to wait at least five years. I have so much ambition, but it is easy for it to feel squashed when I am fighting against this. When I was in high school, my parents had to go to San Francisco twice a month to report to an officer. If anything in our life changed, we had to tell them—from buying a new car to moving. Authorities came and visited us once a week. They wouldn’t tell us exactly when they would come, but if they showed up and my parents weren’t home it was a strike against us. Our “illegalâ€? status affects every aspect of our lives. I grew up ashamed of who I am. I knew I wasn’t wanted here, so I always felt I didn’t belong—even though I knew I did. In high school, I only told a few friends because I was so scared of people’s reactions. I thought people would treat me differently and that it would put my family in jeopardy. Now I’m not ashamed, but I am so afraid of losing everything I’ve worked for. My biggest fear is that my family will be separated: my parents and I will be deported and my brothers—born on U.S. soil—will stay here, and I won’t be able to pursue my education and career. People have a lot of preconceived ideas about what illegal immigrants look like and why we’re here. It’s not an easy decision. People don’t just wake up one morning and walk across the border. People like my family pay taxes, go to school, abide by the law and assimilate. We are assets to society. There are countless amazing people you’d never know weren’t citizens. Recently, I had the opportunity to meet other students in my situation. It is inspiring to share our experiences as successful, contributing students. Yet there is no avenue for us to become part of the country that has been our home for almost our whole lives. I came here before I could talk. I took my first steps in San Francisco. I was raised in America. I am an American. I just don’t have a piece of paper legitimizing that identity. I don’t know what the future holds, and I am still too uncertain to publish my name, but I hope this will reveal my story to those who can recognize my background. I’m starting my junior year at San Francisco State next month. I’m working as a nanny and saving money. I’m a regular girl, someone you say “Hiâ€? to at school. I’m a high achiever. I’m an excellent student. I’m a hard worker. I’m a daughter and I’m a sister. I’m an illegal immigrant. I came here before I could talk. I took my first steps in San Francisco. I was raised in America. I am an American. The DREAM Act, currently in Congress, would help students like the author. To help make a change, research the DREAM Act and urge your representatives to help pass it. Open Mic is a weekly feature in the Bohemian. We welcome your contribution. To have your topical essay of 700 words considered for publication, write firstname.lastname@example.org. H YRaaR_`-O\URZVN[P\Z J '#&5XUcWBc BP]cPA^bP20($## ?W^]T)&&$!& ! 5Pg)&&$!& !'' 6PQT<T[X]T Tgc % 2>?H438C>A 6Pah1aP]Sc Tgc $ 20;4=30A@D44= :PcT?^[PRRX Tgc! 8=C4A=B 2WT[bTP;X]SbTh 9dbcX]T<R3P]XT[ 2Pa^[X]T>bQ^a] 0]]PBRWdTbb[Ta 2>=CA81DC>AB <XRWPT[0\b[Ta 0[PbcPXa1[P]S A^Q1aTib]h AXRWPaSe^]1dbPRZ 9PX\T2aTb_^ BdiP]]T3P[h 1aPS[Th6aPh 1aXP]6aXUUXcW 3PTSP[db7^fT[[ 9P\Tb:]XVWc :h[XT<T]S^]RP 9d[XP]T?^XaXTa 1adRTA^QX]b^] 3PeXSBPb^] 2PaThBfTTc 3PeXSCT\_[Tc^] C^\C^\^aa^f AXRWPaSE^]1dbPRZ 2[PaZF^[U ;8:4<8=34340C4AB I think it is extreme to eradicate volunteer programs on organic farms (“Nothing’s Free,â€? Aug. 11). World Wide Opportunities on Organic Farms is a beautiful program. It permits the exchange of trade information, helps small organic farms to survive on their already limited yield, and allows community to be built between like-minded eaters. There is no money exchanged; it is an educational opportunity with the “tuitionâ€? and training being hours worked on the land, and food and some housing provided. It cannot substitute for the skilled farm workers who make farming their living. It is a gateway towards more work, as some interns may gain inspiration and confidence to start a farm themselves. Many industries are powered by volunteers who give their products or services away for free. If we support the eradication of volunteers in rural life, we should also eradicate highwaycleanup volunteers who take jobs away from paid government cleanup crews, volunteers who work for causes, musicians who give away their music for free on the internet, and anything else that takes jobs away from workers. A moderate approach would be to limit the number of volunteer/internship workers farms can have or the number of hours they may contribute. I just hope that our wonderful small organic farms do not drown in paperwork. 9PRZXT<dYXRPTgc! " ;0H>DC0AC8BCB 3428<0C8>==>C?A4B4AE0C8>= Our local bit of moral reckoning, aptly named Preservation Ranch, seems to have fallen off the radar, while the economy and the environment in the Gulf has been decimated. Located near Annapolis, in Sonoma County’s fifth district, the ironically named Preservation Ranch will do the opposite of its name. It will scrape bare 1,600 acres of redwood forest and coastal grassland to create more vineyards. Eighty-five miles of deer fencing will be installed to keep native deer out of the wine grapes. It will use the remaining 17,000 acres for timber harvesting. Sensitive coho salmon habitat will be forever gone as the water in our watershed will be captured in 40 reservoirs to serve those vineyards and the people who will purchase 90 parcels created out of perhaps 400 acres for the luxury housing associated with this monstrosity. This project is larger than the cumulative total of vineyard conversion projects of the past put together. It will be a game changer for the way we imagine and develop Sonoma County. Recently, the Pacific Forest Trust, a forwardthinking organization, purchased the large coastal Richardson Ranch and is restoring it to productive forestland—no vineyards or wealthy enclaves. It will sequester carbon that we need to counter global warming and serve as a safe haven for wildlife. The redwood forest grows in a thin coastal region. It is one of the world’s most biologically productive ecosystems. Now is the time to take this idea out of the county plan and banish it. Be sure to email county supervisor Efren Carrillo to let him and other supervisors know that you could not vote for a candidate who supports the creation of “Preservationâ€? Ranch. Check out the Sierra Club’s position on this project. 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Newcastle Importers. White Plains, NY. Go paperless...and help your Community to be greener. Ask us how. 707.524.3000 www.exchangebank.com chan ngebank.com m Member M ember FFDIC DIC TTHE HE ALL ALL N NEW EW 2011 20 011 FFORD ORD FFIESTA IESTA IISS H HERE! ERE! 3075 Corby Ave Santa Rosa, CA 95407 888.872.2095 www.hanselford.com 08 IITS TS A P PRETTY RETTY BIG B I G DEAL DEAL news for Sonoma, Marin & Napa Counties “Official Newspaper of Every American Having the Same Rightsâ€? /9.@A y 9252BD<<4A24;41A0C8>= C746>3B7>C Local Boy Makes . . . look, let’s just hope he doesn’t hurt himself. Sausalito native J. R. Hildebrand makes his second career start at Infineon By Bradley Gray ig breaks come in unusual ways. Sometimes they come because of big breaks—as in legs. Rewind to Memorial Day, and the Indianapolis 500. On the last lap, there was a horrific crash involving Ryan Hunter-Reay and Mike Conway, the latter getting the short end of the stick. Conway’s car vaulted into the air, careened into the catch fence and landed upside down in a million pieces. If you weren’t watching the race, you probably saw it on the news. Conway survived, but suffered a broken leg. Until his rehabilitation is complete, his car owner, Dennis Reinbold, has tapped 22year-old Sausalito native J. R. Hildebrand to substitute for him. For series rookie Hildebrand, the window of opportunity is small, but he will make the most of it. He’ll take the wheel of the No. 24 Dreyer & Reinbold TranSystem car in only his second IndyCar Series race at the Grand Prix of Sonoma at Infineon Raceway Sunday. It’s the opportunity Hildebrand has been working toward, and with his talent and local knowledge, he’s sure to be a contender. A graduate of Marin County’s Redwood High School, Hildebrand has worked his way up through all of the junior series of racing. In 2009, he won the Firestone Indy Lights season championship, but breaking into the big leagues of IndyCar racing has proven difficult. He looks forward to his big chance in front of the home crowd. “It’s a huge opportunity,â€? Hildebrand beams. “When you do a one-off race like this, people only remember you for that race. I need to do everything I can to do a great job and get a good result.â€? Hildebrand is optimistic and excited about racing at his home track. “I grew up at Infineon, watching my dad race vintage cars,â€? he says. “I won my && D[[Zg^c\[gZZYgde"^cbZY^XVahZgk^XZh! ^cXajY^c\V[jaaYVnd[cd"XdhiVXjejcXijgZ igZVibZciZVX]lZZ`!i]Z?Zl^h]8dbbjc^in;gZZ 8a^c^XXZaZWgViZh^ihc^ci]nZVgd[hZgk^c\HdcdbV VcYBVg^cXdjci^Zhl^i]VhjbbZgXZaZWgVi^dc [jcYgV^hZgdc6j\#'.#;djcYZY^c]dcdgd[i]Z igVY^i^dcVa?Zl^h]XdcXZeihd[ioZYV`V]Æ_jhi^XZ$ X]Vg^inÇVcYi^``jcdaVbÆgZeV^g^c\i]ZldgaYÇ! i]Z?8;8lVhZhiVWa^h]ZYWn9g#GdW^cAdl^io ^c'%%&^cdgYZgidhZgkZi]ZbdgZi]Vc&)%!%%% Cdgi]7VngZh^YZcihl]dVgZjc^chjgZY#AVhinZVg! i]ZXa^c^XhVlbdgZi]Vc&!.%%Xa^ZcihidiVaa^c\ hdbZ(!%%%k^h^ih#6XXdgY^c\id?8;8kdajciZZg XddgY^cVidg=VccV]?Zgc"B^aaZg!i]VicjbWZg^h jeWnVWdji,%%eZdeaZ[gdb'%%-# L]^aZi]Z?8;8YdZhcdiY^heZchZdgegZhXg^WZ cVgXdi^Xh!XVccdiYddW$\nchZgk^XZhdgHI9 iZhi^c\VcYYdZhcÉi]VkZVehnX]^Vig^XiZVb!^iYdZh d[[ZggZ\jaVglZaa"WVWnXVgZVcY^bbjc^oVi^dc hZgk^XZhVcY^hVkV^aVWaZil^XZVlZZ`!dcIjZhYVn VcYI]jghYVn![dgYgde"^c[Vb^anegVXi^XZXa^c^Xh#>i ^hVahdVWaZidbV`ZgZXdbbZcYVi^dch[dghZgk^XZh ^iXVcÉiegdk^YZl^i]^c^ihcZildg`d[kdajciZZg egVXi^i^dcZgh#:VX]lZZ`!i]Z?8;8hig^kZhid ]ZaeXdbbjc^inbZbWZghl^i]b^cdgV^abZcih! bZY^XVaZYjXVi^dc!VX]ZVcYeV^cigZVibZci VcYZkZgni]^c\cZXZhhVgnid`ZZe]ZVai]n`^Yh ]ZVai]n#I]ZVXjejcXijgZigZVibZcihVgZXVgZd[ i]Z6XjejcXijgZHX]ddad[7Zg`ZaZn!l]^X]hZcYh XZgi^Ă’ZYegVXi^i^dcZghiddkZghZZhijYZcihl]d cZZYid\Zii]Z^gigV^c^c\]djghe^ccZYYdlc# Æ6adid[eZdeaZY^YcÉiZkZc`cdll]Vi^ilVh!Ç ?Zgc"B^aaZghVnhd[i]ZigZVibZci#Æ7jii]Zn\di d[[ZgZY^iVcYi]Zn`ZZeXdb^c\Xdb^c\WVX`0lZ YZĂ’c^iZan]VkZgZ\jaVgh#Ç I]^hnZVgÉh[jcYgV^hZg[ZVijgZhi]gZZ 6bZg^XVcVhdc\lg^i^c\WVcYhĂ…i]Z8djgicZn ?VcZh!i]Z:VhnAZVkZhVcYDaY?VlWdcZĂ…VhlZaa VhVnd\VXaVhhVi'ebaZYdjih^YZWn9ZkdgV] 7ajb^[ndjY^Ynd\VVii]Z@ViZLda[;Zhi!ndj Y^Y^il^i]9ZkdgV]![VXZeV^ci^c\[dgi]Z`^Yh! eaZcinidZViĂ…]dbZbVYZWn?8;8hiV[[VcY kdajciZZghĂ…Yg^c`h[dgejgX]VhZVcYVh^aZci VjXi^dc#6cYgZhiVhhjgZY!WZ^c\?Zl^h]^hcdi cZXZhhVgn[dgVcnd[^i#ÆLZd[[ZghZgk^XZhid VcndcZl^i]dji]ZVai]^chjgVcXZgZ\VgYaZhhd[ gZa^\^dc!Ç?Zgc"B^aaVhhjgZh# 6aad[i]^h]VeeZchdcHjcYVn!6j\#'.![gdb &ebid*eb#&-eZgeZghdc!(+eZg[Vb^andg *)eZgXVgadVYd[gZkZaZgh#**%%7jgch^YZGdVY! HZWVhideda#IdWjnVYkVcXZi^m!XVaa,%,#*-*#,,-%# Deals for SRJC and Sonoma State Students Come and check us out! 7 0 7 * 5 7 3 * 1 6 0 8 climbing classes weight equipment and yoga 3358a Coffey Lane Santa Rosa www.vertexclimbing.com first race here, and I won the Indy Lights championship here. I know the track pretty well, but I’ll be racing against guys like Dario Franchitti and Tony Kanaan, who have raced IndyCars here a lot and have won here.â€? Last weekend, Hildebrand made his first IndyCar start at the Mid-Ohio Sports Car Course. He qualified 18th, but even after committing several rookie mistakes, he was still able to improve and finish 16th. Hildebrand was disappointed, but acknowledges that the learning curve is steep. “I definitely made life difficult for myself at the Mid-Ohio race, but I can take a lot of things I learned with me to Sonoma.â€? Hildebrand is also quite critical of his performance in his first IndyCar race. “I made some uncharacteristic mistakes,â€? he says, “mostly because I’d never driven an IndyCar in a race situation. In fairness, I don’t think that is much of an excuse and from that perspective, and the only way to go is up. At Infineon, I will be a lot more used to the car and the circumstances.â€? Hildebrand adds that he and the team had only limited time practicing pit stops before his first race. And during the MidOhio race, he stalled the car on his first pit stop and had trouble selecting first gear on another, mistakes he doesn’t plan to repeat. To many in the racing community, Hildebrand represents a welcome change. Of the 26 cars expected to start Sunday’s race, only five are Americans. Names like Dario Franchitti (Scotland), Helio Castroneves (Brazil), Scott Dixon (New Zealand), Tony Kanaan (Brazil) and Dan Wheldon (England) have dominated the series for years. The only American drivers to win races in recent years are Ryan Hunter-Reay (two wins), Danica Patrick and Marco Andretti (one each). Hildebrand looks to add his name to that list. “I’m totally excited about coming to Infineon,â€? Hildebrand says with authority. “I’m bringing a lot of confidence, because I know the place like the back of my hand and nothing will be brand-new this time. I know that if we qualify well and run a mistake-free race then we will be right in the thick of it.â€? To see the world’s greatest open-wheel racecar drivers competing at Infineon Raceway is spectacular. The carnival-like atmosphere brings ambiance, color, mystique, excitement, danger and the jet-set crowd. And now, with the local kid racing, there is a delicious slice of American apple pie. This will be the biggest race of J. R. Hildebrand’s young career. You may know the names Gurney, Andretti, Rahal, Mears, Foyt and Unser. With a good showing at Infineon, Hildebrand will be one step closer to those elite American drivers. The Indy Grand Prix of Sonoma comes to Infineon Raceway Friday–Sunday, Aug. 20–22. Also racing will be the Firestone Indy Lights Series and Historic Grand Prix and Formula Car Challenge. Infineon, highways 37 and 121. $10–$75. 800.870. race. www.infineonraceway.com. Compressed natural gas or clean drinking water? By Juliane Poirier he environmentally perfect fuel choice for our cars is absurdly impractical for most of us: it’s the choice to stop driving a car. Most cars run on gasoline made from crude oil, the booty from serial wars in the Middle East. For all the costs, hidden and not, it’s hard to feel good about gassing up an auto these days. While somebody out there is thinking up a fuel solution superior to everything known—something smarter, cheaper, friendlier and more sustainable than fossil fuels—we have a few imperfect options to consider as we wait for that somebody’s phone call. I’ve nicknamed these equal-of-twoevils Sophie’s Choices because all of them do some harm. (Until investing two hours and a box of Kleenex as the rest of us have, those who missed Meryl Streep and Kevin Kline in the 1982 film Sophie’s Choice will not understand the concept of a truly difficult choice.) One option is fueling cars with compressed natural gas. Natural gas is touted as a transition fuel, which means it’s better than gasoline for a number of reasons. For example, it’s already here in this country. It’s plentiful. It’s cheaper. And it produces 25 percent less carbon than what’s produced by a car running on gasoline. A school district in Oklahoma allegedly saved so much money by using compressed natural gas in school buses that it was able to retain 14 teaching positions. Imagine that—use compressed natural gas, save a teacher. It sounds so win-win, who wouldn’t want to save education, cut down on fuel costs and reduce carbon? But wait. Here comes a Sophie: choosing this manner of carbon reduction and cash savings can also mean that we are choosing to contaminate drinking water supplies across the country. Natural gas may occur naturally, but getting it out of the ground is a fracking ordeal—“frackingâ€? being the industry slang for “hydraulic fracturing.â€? A fossil fuel, natural gas is deposited in shale. Using a mining technique known as horizontal drilling, gas companies drill vertically into the ground for a mile and then make a 90-degree turn and start going at it horizontally. To release the gas from the shale in this sideways direction, they inject chemicals in a high-power blast that fractures the rocks. Et voilĂ . Gas is gotten. But in the process, the groundwater passing adjacent to the drilling can be contaminated by the chemicals used to fracture the shale. Among those saying “not so fastâ€? to compressed natural gas mining are concerned citizens, including those living closest to the drilling operations. In a letter to the Environmental Protection Agency this month, groups including the Natural Resources Defense Council and the Sierra Club expressed concern over allegedly illegal fracking practices and asked the EPA to enforce the Safe Drinking Water Act to protect underground water sources. Under the Bush administration, hydraulic mining was exempted from the Safe Drinking Water Act unless diesel fuel was involved. Diesel contains benzene, toluene, ethylbenzene and xylene, which poison water even in very low levels; the amount used in mining releases these toxins at levels exceeding what has been established as safe. The 31 organizations that signed the letter—citizens groups spanning the continent from California to Delaware—point out that “fracturing companies B.J. Services Company and Halliburton injected diesel in hydraulic fracturing operations in 2005, 2006 and 2007 in as many as 15 different states.â€? Since gas is mined in 30 states, the groups suggest that drilling companies be required to disclose to citizens exactly what kinds and amounts of chemicals are being injected in and around drinking water supplies. Their goal is to stop use of diesel injections and any other chemical contaminating drinking water. Natural gas may in fact be a good transition fuel in cars and as a power-plant replacement for coal. Last year, NPR reported that enough natural gas lies buried between New York and West Virginia to equal 80 billion barrels of oil. And that’s just one deposit of many. If the EPA jumps into action on this, maybe getting natural gas on line won’t be a choice between poisoning our air and poisoning our drinking water. Natural gas: good. Diesel fuel: not so much. Enter for a chance to Win a FREE Home Solar System and save thousands of dollars on electricity!* NO PURCHASE NECESSARY— ENDS SEPTEMBER 15, 2010 ENTER NOW AT RealGoodsSolar.com or call 800-760-9700 7iebWh[b[Yjh_Yioij[c\ehoekh^ec[kfje *a_bemWjji:9lWbk[ZWj)&"&&&$ <kbbI[hl_Y[?dijWbbWj_ed0:[i_]d"[d]_d[[h_d]" f[hc_jj_d]"_dijWbbWj_edWdZh[XWj[fWf[hmeha_ijWa[d YWh[e\\ehoek$ DEFKH9>7I;D;9;II7HO$EZZie\m_dd_d]Z[f[dZedj^[dkcX[he\[djh_[i$7ZZ_j_edWb j[hci"YedZ_j_ediWdZh[ijh_Yj_ediWffbo$I[[E\ÓY_WbHkb[iWjmmm$H[Wb=eeZiIebWh$Yec\eh Yecfb[j[Z[jW_bi$M_dd[hm_bbX[i[b[Yj[ZWjhWdZec\hecWbb[b_]_Xb[[djh_[ih[Y[_l[Z$;ij_cWj[Z h[jW_blWbk[e\fh_p[_i)&"&&&$Fhecej_ed[dZiI[fj[cX[h'+"(&'&$E\ÓY_WbIfedieh0H[Wb=eeZi IebWh"?dY$"(-I_cciIjh[[j"IWdHW\W[b"97/*/&'$ IWl_d]iZ[f[dZedYkhh[dj[b[Yjh_Y_jokiW][WdZkj_b_johWj[iY^[Zkb[$ endless green HYDR O P O N I C S U PPL I E S 707-254-0200 NAPA, CA Advanced Nutrients, FoxFarm, House & Garden, Cutting Edge, Botanicare, General Hydroponics, Hydro Organics 25-3 Enterprise Court, Napa CA Mon-Fri: 10-6, Sat & Sun: 11-4 Accessible A c c e s sib l e L Lifestyles ifestyles Unlimited Unlimited presents presents the the first f irst annual annu al Do you Do you h have ave physical physica l impairments impairments (i.e. (i.e. p paralysis) ara lysis) tthat hat kkeep eep yyou ou ffrom rom being being active active and and living liv ing life life to to its its healthiest? hea lthiest? 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Camp Camp Metabolic Metabolic is is a two two week week residential residentia l ccamp amp set set in in the the Redwoods Redwoods near near Santa Santa Cruz, Cruz, run run by by volunteers volunteers aand nd d dedicated edicated to to h helping elping ppeople eople w with ith pphysical hysica l llimitations imitations iincrease ncrease h healthy ea lthy aactivities ctiv ities iin n ttheir heir llives ives aand nd aadopt dopt ggood ood n nutriutrittional iona l h habits abits in in order order to to stay stay healthy hea lthy and and in in shape. shape. Kickoff K ickoff Party Party – F Friday, riday, Sept Sept 10 10 Conclusion C onclusion – SSunday, unday, Sept Sept 26 26 C ome aand nd join join u s! Come us! CampMetabolic@gmail.com C amppMetabolic@gmail.com ((559) 559) 4492-0022 92-0022 The Filling Station Presents Friday Funday!! Bring in your hot rod for hot specials! • Two for one Milkshakes for hot rodders! • $1.00 Hot dogs • Happy hour specials for all • Hot/cold espresso drinks specialty drinks • Italian sodas, smoothies & sweets! Fridays 3–7pm Swing on in and fill up! Open 7 Days a Week THE FILLING STATION 1000 West College Ave, Santa Rosa 707-535-0690 GROUP & PRIVATE GEOLOGY FIELD COURSES NORTH COAST r YOSEMITE r DEATH VALLEY GRAND CANYON r SCOTLAND Thomas R. Williams, M.S., P.G. Eating the chips too is truly a heroic effort. Curing cafeteria blues, California-style By Justine McDaniel or the first few weeks of freshman year, the freedom of choice offered by the many stations in our college cafeteria seemed exhilarating, but after the novelty wore off, the selection got monotonous. I learned a few tricks to keep from breaking down in tears when faced with yet another night of spaghetti with marinara. The most important one? Sometimes it is best to get off-campus and head for burritos. The burrito is a California trend that has since spread to the rest of the country, and for us native burrito fans, there’s no place like home. But why are they so appealing to college students in particular? “Students are hungry, growing and active, and burritos are very tasty, filling and affordable,â€? says Greg Maples, founder and president of the Bay Area chain High-Tech Burrito. “Kids like them. They’re running from place to place, and they can get a burrito and go.â€? Maples also cites the shift in burrito cuisine toward healthier ingredients as a factor in the appeal burritos hold for students. Many burrito chains tout healthier menus and a “greenâ€? approach to cooking. “Young people can understand that,â€? says Maples. While burritos have distinctly Mexican origins, they are an American creation that has little to do with actual Mexican food. The ancient Aztec culture was the first to eat what resembled our current burritos, which weren’t seen again until the 1800s. There are different folktales that tell of the birth of the burrito, from a man who used the tortilla to keep his meals warm during the Mexican Revolution to gold miners wrapping their meat up to eat on-the-go. The term literally means “little donkey,â€? and there is speculation as to its etymology. Some say that the folded ends of the tortilla look like donkey ears, while others cite the burrito’s resemblance to the rolled bedding that workers and miners carried on donkey-back as its namesake. Others say the burrito itself is like a donkey because it can carry everything you need. The burrito rose to fame in America during the early 1900s, and the frozen burrito was invented in 1956. During the 20th century, burrito restaurants began to spread from California until there were burritos across the nation, from Taco Bell to authentic Mexican-run places. Today, different cultures have personalized the dish to their various tastes, giving it wider appeal. The two major types of burritos evolved in California, an example of the classic NorCal/ SoCal contrast as the two types compete across the nation. The San Francisco Mission-style contains the most filling of any type, with all the traditional elements of Mexican food, including &+ THE BOHEMIAN Traditional Vietnamese Restaurant Best Vietnamese Restaurant The local preference for authentic Vietnamese fresh ingredients large dining room friendly staff always affordable meat, beans, cheese, vegetables and sauce. It usually comes wrapped in foil to retain heat. The inf luence of the San Francisco style can be seen in most of the health-oriented restaurants like High-Tech and Chipotle. The other major player is the San Diego– style burrito, which typically includes meat, cheese and guacamole or sour cream—and often, french fries. Most restaurants in America sell one of the two types of burrito, and many Californians feel strongly about their preferred style (just read one of the many burrito-centric blogs out there). dine in & take out The San Diego–style burrito features, yes, actual french fries. Real Mexican Food 707.769.9066 108 Kentucky St, Petaluma ~ 707.762.8192 800 Petaluma Blvd N, Petaluma ~ 415.878.0122 905 Grant Ave, Novato ~ 415.460.1027 208 Sir Francis Drake Blvd, San Anselmo Tonayan The Tonayan Burrito Grande, sans meat, did not disappoint. This burrito had a more authentic f lavor than the burritos found at chain restaurants, and it was a little more filling. The whole pinto beans were refreshingly simple and the cheese f lavorful. This substantial meal provided a satisfying mix of f lavors. 500 Raley’s Towne Center, Rohnert Park. 707.588.0893. High-Tech Burrito I found this bean, rice and cheese burrito to be one of the warmest and best mixed ones I tried, probably because the cheese was so melty. However, the mild salsa’s fire-roasted f lavor was too strong and the avocadoes in the guacamole didn’t seem ripe enough. The rice had an overpoweringly sweet f lavor. Next time I would forgo the rice. “The perfect burrito is different for everyone, but whatever you choose, it needs to be in the right portions. The hot stuff needs to be hot and the cold stuff needs to be cold, and it needs to be wrapped well. If it all comes together right, it’s the perfect burrito.â€?—Greg Maples, founder and president www.hightechburrito.com 4HANKYOU3ONOMA#OUNTY Best Mexican Restaurant I scoped out a random selection of local restaurants on my quest to find the perfect burrito. I judged in the categories of beans, cheese, tortilla, rice, cold ingredients and distribution. I stuck to vegetarian burritos, using the most basic model of beans, rice and cheese. Everyone’s favorite burrito is different, but I evaluated qualities that are fairly universal. Then I asked each burrito master what he or she thought went into making the perfect little burro. “Our most popular is our shrimp burrito, which comes with beans, rice, bell pepper, yellow onions, garlic, spices, guacamole and sour cream. But I like the carne asada.â€? —Isidro Velasco, owner 707.566.8910 966 North Dutton Ave~Santa Rosa Mon–Sat 10–9m Sunday 11–8 As the vehicle for the rest of the ingredients, a simple f lour tortilla can make or break the meal. They are best when thin and warmed, not so hot that they’ve turned crispy, but not cold, either. A good way to judge the quality is to find out whether the tortilla sticks to the roof of the mouth after taking a bite. If it does, the customer has probably hit upon the Wonder Bread of tortillas. The base ingredient in Mission-style burritos is the beans. Often, soupier refried beans are made with lard, but not all refried beans are cooked this way, and whole beans are almost always vegetarian. Lardless beans often retain more flavor. A little bit of mushiness—a friend calls them “smashedâ€? beans as opposed to refried—is desirable, but the beans should look like, well, beans. Then comes the most important part: the cheese. A burrito is not satisfying if the cheese is not melted over the other ingredients. No one wants a mouthful of cold cheese. Quality is the main thing to look for when it comes to meat. It should be well-marinated. For example, carne asada is generally cooked with lime juice and cilantro, which helps give it its juicy tang. Getting rice should be a simple decision based on whether the customer likes it. There usually isn’t a lot of variety in rice. The main danger is having too much of it; it can easily overwhelm the other ingredients. Cold ingredients such as salsa, guacamole and sour cream enhance the burrito also by acting as a sort of catalyst for the other ingredients: any other item in the burrito tastes good with one of these toppings. Some El Patio El Patio’s bean, rice and cheese combo was my least favorite. Fine for a quick meal on the go, but the beans were so runny I could hardly tell there was cheese mixed in. Next time, I would add meat or salsa. “A perfect burrito has good-quality beans, rice and meat. Making the perfect burrito is about quality. You put the right ingredients together, and it makes a delicious burrito. We sell more burritos than anything. They’re so popular because a burrito has everything in it and you can eat it anywhere. I think burritos are great, because it’s a combination of foods in one big tortilla.â€?— Emiliano Ochoa, owner 901 Fourth St., Santa Rosa. 707.578.4757. La Palapa The beans and cheese in this burrito were impressive. The ingredients were well distributed, and the beans were quite f lavorful. The rice tasted normal, but there was a little too much of it, which detracted from the rest of the burrito. The guacamole was good yet slightly tart, and the tortilla was unobjectionable. The salsa really added to the meal. It had a good bite and was more f lavorful than traditional pico de gallo. All in all, this burrito was enjoyable, but I might add a few more things next time to increase the f lavor. “Just like any food business, it comes down to where the food is coming from. The burrito is not Mexican food; it’s a Californian kind of meal that’s developed through the years. We make a burrito here called the Terminator, and that’s my favorite. It has different kinds of meats, and we marinate it. It’s juicy and not too hot but with a little bit of jalapeĂąos in it. Burritos are popular because they’re very fast, but they are different from hamburgers. They’re healthier than any other fast food.â€?—Mauricio Lemus, owner 590 Lewis Road, Santa Rosa. 707.569.9210. restaurants serve fire-roasted salsa, which has a pretty strong flavor. It can overpower the rest of the food if it’s not something the customer is looking for. Otherwise, it’s just a matter of choosing the right degree of spiciness. Diced tomatoes and shredded lettuce are also standard cold fare. Grilled vegetables—often red peppers and onions— can also make a juicy addition to a burrito and can add a salad component to the meal. The burrito has taken a long journey to evolve into the styles we know today, and is universally loved in all its incarnations. With a little bit of experimentation and experience, anyone can be as pro a burrito eater as the hungriest of college students. 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IjZh"Hjc#&)*@ZcijX`nHi!EZiVajbV# ,%,#,+'#.-%%# TICKETS ON SALE NOW * SonomaWineCountryWeekend.com or call (800) 939-7666 for information * Visa SignatureÂŽ cardholders receive special perks and savings. K^ZicVbZhZ##;gZh]VcYVji]Zci^X! l^i]VlVgbVcYWgZZonVibdhe]ZgZ# -).)<gVkZchiZ^c=ln^c6eeaZKVaaZn EaVoV!8diVi^#,%,#++*#.+'-# <gZZ`##HVi^h[n^c\[ddY^cg^kZgh^YZ hZii^c\#HjcV[iZgcddch!<gZZ` YVcX^c\#AjcX]VcYY^ccZgYV^an# *+--AV`Zk^aaZ=ln!EZiVajbV# ,%,#,+.#-*)*# >iVa^Vc#"#7Vc\"je[gZh][ddYVi eg^XZhi]VihZZba^`ZVhiZVa#AjcX]VcY Y^ccZg!IjZh"HVi#*EZiVajbV7akYH! EZiVajbV#,%,#,-'#.'.-# BZm^XVc#IgjanldcYZg[jaHdcdgVc" hinaZXaVhh^XhVigdX`"Wdiidbeg^XZh# I]ZZcdgbdjh:a?Z[ZXdbW^cVi^dc XVcÉiWZWZVi#AjcX]VcYY^ccZg YV^an#*%%GVaZnhIdlcZ8ZciZg! Gd]cZgiEVg`#,%,#*--#%-.(# BZm^XVc#"#:mXZaaZci[ddY^c EZiVajbVÉhI]ZViZg9^hig^Xi!VcYV[jc eaVXZid]Vc\WZ[dgZdgV[iZgV[a^X`# AjcX]VcYY^ccZgYV^an0WgjcX]!HVi" Hjc#&*&EZiVajbV7akYH!EZiVajbV# ,%,#,,(#)*%%# :XaZXi^X##=dbZbVYZhdjeh!hVaVYh! hVcYl^X]ZhVcYZcig‚Zh#7gZV`[VhiVcY ajcX]!LZY"Bdc#&%%EZiVajbV7akYC! EZiVajbV#,%,#,+(#.*+(# H D C DB 6 K6 AA : N 8V[ZAV=VnZ 8Va^[dgc^V";gZcX]#"#8]Z[CdgbVc DlZchbVgg^Zhi]ZkZgnWZhiHdcdbV ^c\gZY^Zcihl^i]cdjkZaaZ;gZcX] Xdd`^c\hinaZhVii]^hXdb[dgiVWaZ W^higd#9^ccZg!IjZh"HVi#&)%:CVeVHi! HdcdbV#,%,#.(*#*..)# It’s always exciting when something new starts, and so it is with the inaugural release of La Follette wines, a fresh issue from the Van der Kamp vineyards on Sonoma Mountain. Winemaker Greg La Follette will be on hand at a special estate dinner and vineyard tour replete with music from the Steve Pile Band. Dancing heartily encouraged on Saturday, Aug. 28, from 4pm to 8pm. $125. Call 707.395.3902, ext. 218 for details. . . . Bear Republic Brewing Co. invites the public to be the judge when they hold a cellar party to preview the 20-plus beers they are submitting in September to the Great American Beer Festival, the largest such fest in the country. Bear Republic has won 13 prizes from the competition previously and intends to take home more this year. Help them get hopping on Sunday, Aug. 29, from 3pm. $55; presale only, attendance limited to just 100. 345 Healdsburg Ave., Healdsburg. 707.433.beer. . . . ur favorite institution of learning and cooking and serving and eating, the Santa Rosa Junior College Culinary Cafe and Bakery, opens to the public for its fall semester on Wednesday, Aug. 25. The cafe is known for its baked goods and breakfast treats, but wise elders also know to go there early and often for an insanely cheap lunch and to set Fridays aside for wine pairings, also insanely cheap. The students are learning all levels of restaurant service from tableside to ovenside, and customers beneďŹ t from this innovative program. Corner of Seventh and B streets (look for the white toques in the window), Santa Rosa. 707.576.0279. . . . >iVa^Vc#"#6cdVh^hd[jgWVc^in i]Vil^aaigVchedgindjidCZlNdg`! EVg^hZkZc#I]ZbZcj`ZZeh[gZh]an hZVhdcVaVcYX]Vc\ZhlZZ`an#AjcX]! Bdc";g^0Y^ccZgYV^an#&*)EZiVajbV 7akYC!EZiVajbV#,%,#,++#,+%%# Railroad Square doesn’t taste like bricks and steel; it tastes like garlic and cream and pork and lemon grass and so much more when that section of downtown Santa Rosa celebrates itself and the great work of the Sixth Street Playhouse with Taste of Railroad Square, a walk-around afternoon of food and wine and beer tastings on Saturday, Aug. 21, from noon to 4pm. $40; $45, day of event. 707.523.4185. . . . BZY^iZggVcZVc"BdgdXXVc#"# 9^h]Zh[gdbHeV^c!;gVcXZ!>iVan!<gZZXZ dgBdgdXXdi]ViVgZVaaZmXZaaZci! a^`Zi]ZX]^X`ZcBVggV`ZX]!\ddhZY l^i]BdgdXXVche^XZh!\Vga^X!dc^dch! idbVidZh!Z\\eaVciVcYVabdcYh# AjcX]!Bdc";g^0Y^ccZgYV^an# &('@ZaaZgHi!EZiVajbV# ,%,#,+(#((((# Two hundred dollars gets one almost just enough to eat at the French Laundry; spend that same amount at chef Thomas Keller’s sister restaurant, Ad Hoc, and one not only gets famously fed but also receives a copy of Keller’s latest cookbook, Ad Hoc at Home. Keller and Ad Hoc chef de cuisine Dave Cruz host book lovers at Ad Hoc in conjunction with Book Passage on Tuesday, Aug. 31, for two seatings that include a full meal avec vin as well as a signed copy of the book and a lively conversation with the two men. 6476 Washington St., Yountville. 5pm and 8pm. 415.927.0960, ext. 1. . . . And ďŹ nally, speaking of chefs we love, John Ash celebrates the mighty basil plant during the last week of August, featuring dishes pungently perfumed with the 26 different basil varieties planted at his restaurant Aug. 23–29. Here’s today’s poem: pink-Himalayan-sea-salt-topped chocolate tart with strawberry-basil sour cream gelato and chocolate sauce. That’s what we’re sayin’. 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Thankfully, trafďŹ c is light on little Wood Road, and the occasional motorist passes with a charitable buffer zone. Winetasting isn’t usually this much work—I’m just trying to get in the spirit of this weekend’s Grape to Glass. If I knew that a cold glass of Sauvignon Blanc awaited, I might pedal harder. Sponsored by the Russian River Valley Winegrowers, Grape to Glass offers much more than the typical taste fest. Participants pick and choose from a dozen different activities, some of which are like intimate seminars which explore the topography, people and grapes that earn this region renown, one of which is a serious cycling tour from Healdsburg to Green Valley and back. At a more leisurely pace, with no added lycra, chef Duskie Estes of Zazu and Bovalo Restaurant leads a “Slow Food rideâ€? through town. Meanwhile, a kayak otilla cruises down the Russian River toward a winetasting via amphibious landing. Whether it’s their taste buds or quadriceps that got worked, everyone can cap off Saturday at “Hog in the Fog,â€? a grand tasting plus barbecue, auction and live rockabilly music. For its part in the event, Robert Rue Vineyard hosts a closer look at the historic vineyards of their neighborhood. Before Russian River Valley Pinot Noir, there was Zinfandel, mixed with a medley of red and white grapes. “That might be Palomino,â€? Bob Rue says about some green grapes that show no sign of coloring up toward purple. When Bob and Carlene found this ranch, all around them people were ripping out Zinfandel. An Italian-American neighbor taught Bob how to prune and care for his vines, and left him with this prescient advice: “Bob, whatever you do, don’t pull these vines out. There will always be a market for old vine Zinfandel.â€? Soon, a new wave of Zinfandel specialists like Ravenswood helped to revive interest in the varietal and kept small vineyards like this in production. Looking for a refreshing white to complement their estate Zinfandel, the Rues decided on Sauvignon Blanc. Purchased fruit from the area lends the 2009 Sauvignon Blanc ($22) a whiff of cashew and clean avors of lime, understated tropical fruit and lychee; a big pour of this dry and crisp wine was most welcome. With cedar and ďŹ g spicing up deep olallieberry and early blackberry fruit, the 2006 Wood Road Reserve Zinfandel ($32) ďŹ nishes with chunky layers of dark cocoa, tannic and robust without the heat. While it wasn’t really necessary to bike against the wind for this tasting, I’d say it’s worth the ride. Robert Rue Vineyard, 1406 Wood Road, Fulton. Tasting hours Friday to Sunday, 10am to 5pm, or by appointment. Tasting fee, $5; waived with purchase. 707.578.1601. Grape to Glass runs Friday–Sunday, Aug. 20–22, at various venues. www.rrvw.com. 707.521.2535. L88VkZha^cZYl^i]>iVa^VcbVgWaZ VcYVcX^Zcii^aZh!cdiidbZci^dc KZcZi^VcX]VcYZa^ZghVcYbdhV^X bVgWaZ[addgh#I]Zn]dhiXVcYaZ"a^i iVhi^c\h!gZeaZiZl^i]X]ZZhZVcY X]dXdaViZ!;g^YVnĂ„HjcYVn#DeZgV gZhdcViZhjci^a)eb0gdX`gjaZhV[iZg )eb#&%**6iaVhEZV`GdVY!CVeV#7n Veed^cibZci#,%,#.+(#'&()# 6c^ciZaaZXijVadjiedhid[VgiVcYl^cZ ]djhZY^ci]ZXZcijgn"daY8]g^hi^Vc 7gdi]ZgÉhl^cZgn#8VW^hi]Zh^\cVijgZ kVg^ZiVa#))&&GZYlddYGdVY!CVeV# DeZcYV^an!&%VbĂ„)eb#,%,#'**#&&))# I]Z8Va^[dgc^VK^cicZgÉh8ajWcVbZY GdXXVÉh'%%)8VWÆi]ZWZhid[i]ZWZhi#Ç &&(%BV^cHi#!CVeV#DeZcBdcYVnĂ„ HVijgYVn!&ebĂ„+eb#,%,#'*,#-)+,# paid advertising section ARTWALK 13 ART G A LLERIES Art ! / Wine ! / Food ! FIRST FRIDAYS 5 - 8 P.M. May 7 / June 4 / July 2 / Aug 6 Sep 3 / Oct 1 / Nov 5 / Dec 3 Thankfully, there’s Healdsburg. w w w. h e a l d s b u r g a r t g a l l e r i e s . c o m RIVERFRONT ART GALLERY 132 PETALUMA BLVD. NORTH, DOWNTOWN PETALUMA Contemporary/Museum Framing Conceptual Art Gallery 707.473.9600 707.474.9600 www.hammerfriar.com Call Today to Advertise! 707.527.1200 | email@example.com 22 COLLEGE GUIDE >DCB8348=B834 Instructor Karen Lovaas leads inmates in a discussion of the BP oil spill. ?aXb^]1aTPZ Inmates at San Quentin study for success By Anna Schuessler lasses held at Patten University’s San Quentin extension campus are just like those that one might find at any accredited junior college. Students stay up late or wake up early to finish assignments on time. The classrooms are packed, and sometimes students have to help each other out to work through a piece of literature. Except almost all of them attend night classes. And all of them are wearing the same light blue shirt. Every year, some 150 inmates serving time in the San Quentin State Prison gather in small classrooms after dinner to discuss Shakespeare and math, among other topics. Led by instructors who hail from local universities, these classes are the result of the Prison University Project (PUP), a nonprofit committed to bringing college-level and college preparatory classes to prison inmates. Offering around 12 classes every semester, the PUP prepares inmates for an associate’s degree, or at least an academic path. Inmates with a GED or higher are eligible to take the college-level offered. Those just short of the required high school diploma may take the college prep classes. PUP executive director Jody Lewen doesn’t think the location—not exactly prime real estate for a college—makes any difference to the quality of education. “Actually,â€? she says, “it might be better than the average college degree program. The class sizes are smaller. Classes at Berkeley might have 150 people, and we have about 30.â€? Lewen has other reasons to think so. The project’s success relies heavily on an army of volunteer instructors and teaching assistants, many of whom study or teach at local schools such as Cal, Stanford and Sonoma State. “We have a pretty substantial core of people who have been around for several years, and that is a great source of new faculty, too. A lot of them tell their friends,â€? Lewen says. By culling their own academic experiences, instructors—who must have at least a masters degree in their field—hope to include the inmates in a larger intellectual community, one that reaches much further than the walls of San Quentin. Leonard Hutton, a former inmate and student within PUP, says, “It gave me the drive to get into school and start a regular program.â€? He also says that the classes kept him out of trouble. After working in the print shop all day, Hutton hit the books, either attending one of his classes or completing his homework. His studies kept him from spending too much time in the prison yard, where inmates typically pass the night. “Anytime you spend an extended amount of time in the yard,â€? he says, “it’s possible to get in trouble, or worse.â€? Hutton, who has been released from San Quentin, is currently taking classes toward a degree. For Lewen, the project’s far-reaching impacts cannot be encapsulated in the academic or employment successes of those released from the prison, although they are considerable. Some of the inmates don’t get the opportunity to test the skills they’ve learned in the real world, as roughly 40 percent of them are serving life sentences. “I think for a lot of people, when they talk about the value of the program, they don’t only talk about jobs,â€? Lewen says, “they really talk about the impact on their sense of their own potential, or what’s possible for them and their lives.â€? Lewen wants to take the project to the next level. “[The students] would love us to have a BA program.â€? The only thing stopping her is funding. Classroom space is already at a premium; the program will need a new building before any plans for a BA program are made. “The department has basically said that if we could raise the money, we could build a new building,â€? she says. “That, to me, is the mother of all challenges. We could help so many people if we had that space.â€? Learn more at www.prisonuniversityproject.org. ?A>?7H;02C82 Only your own icky germs live on the Badabeez. Introducing the water pipe condom and other stoner saviors &XOLQDU\$UWV %DXPDQ&ROOHJHRUJ 24 • FURNITURE • FRUIT LABELS • GARDEN ANTIQUES • Dude! We’re 5 times the fun & bull-free! Antique Society 100 dealers! Our 21st year! 2661 Gravenstein Hwy So. (Hwy 116) on Sebastopol’s Antique Row Open daily! 707 829.1733 www. AntiqueSociety .com TOYS & DOLLS • ARTS & CRAFTS • POST MODERN • COUNTRY • CALIFORNIA PRIMITIVES • JEWELRY • LIGHTING • KITCHEN TOOLS • ARCHITECTURAL • GLASS By Gabe Meline ven though the swine flu was spreading like crazy last year at Santa Barbara City College, 19-year-old Elliott Bush and his friends were still carelessly sharing hits from a bong. “I was really tired of being sick all the time,â€? he says, “so my mom and I decided to really go for this and make a change. It’s not good to share germs and everything.â€? Bush, who lives with his parents in Tiburon, invented a product called Badabeez, a conical silicone barrier that folds over the mouthpiece of a water pipe to protect pot smokers from germs. It’s essentially a bong condom, free of toxic chemicals, which comes with a carrying case and fits snugly over almost any bong without compromising air pressure. The Badabeez is also useful for avoiding the buzz kill of social tension. “When people have sicknesses or, like, if someone eats something really gross, you don’t wanna be, like, ‘Dude, that’s gross, I don’t want you to hit my piece because you’re sick,’â€? Bush explains. “You can just put a Badabeez on it and you’re good. It’s not awkward for you or that person.â€? As a North Bay expert on solutions for the nuisances inherent in smoking pot, Bush agreed to a product-development quiz. I asked several friends who get down with the reefer to tell me what annoys them most when getting high in mixed company, and Bush, with his inventor’s mind, came up with product solutions on the spot. Problem: Too much affection. “The ‘lecherous hippie hug’ is pretty bad,â€? says my friend, “Melissa,â€? a receptionist. “Where, like, when you’re done getting high, they stand up and kind of lean down and hug you, but really they’re just groping you. It sucks.â€? Solution: Bush advises that since hippies hate leather, an Anti-Hippie Full-Body Leather Jumpsuit™ will keep the icky hugs away. Problem: People trying to be Noam Chomsky. “I was high at the beach a while ago,â€? says “Bryce,â€? a carpenter, “and this one guy wouldn’t stop talking about really repetitive political stuff. He was presenting the fact that corporations control the world in a way where he truly felt no one had ever thought of that before.â€? Solution: Bush, whose own father talks a lot about politics, envisions the SarcasticRetort Keychain™, with electronic buttons that trigger scathing phrases designed to stop the lecturing. Problem: Freaks who make you watch messed-up movies. “A friend one time made me watch Gummo,â€? says “James,â€? an artist. “I don’t know why he wanted me to watch that movie. I started getting as high as I’d ever been, just going bonkers, during the scene where the guy keeps jiggling his nipple. It was horrible.â€? Solution: Bush has never seen Gummo, but proposes the Marijuana Movie App™, which a pot smoker could consult on the iPhone. “It could ask, ‘What mood are you in?,’ and then you choose Happy, Sad, Angry, Funnyâ€?—and then it’d suggest appropriate films, says Bush. Problem: Friends who are pushy about music. “I can’t tell you how many times people have attempted to convince me how good Television’s Marquee Moon is while high,â€? says “Standish Cosnahan,â€? a bartender. “It just sounds like an art-school guitarist’s attempt at rock and roll, poorly done.â€? Solution: Bush is sneaky, conceiving of the Retaliatory Britney Spears iPod™. How it works: You plain-facedly agree to download the crappy music that’s forced on you to your iPod, but secretly touch a setting that uploads a ton of Britney Spears, Carrie Underwood and Jewel to their computer instead! Problem: The general stoner ramble. “Last night,â€? complains “Kyle,â€? a residential caretaker, “our roommate was going off about the meaning of time and how the past and the future correlate to the present, and he kept talking about potential, but he wasn’t specific about anything. He kept staring at the wall and then laughing at himself.â€? Solution: Bush suggests the Diff ’rent Strokes Secret Speaker of Specificity™, deployed to get people off their tangents. At the touch of a button in one’s pocket, a speaker hidden in the wall emits a personalized version of “Whatchoo talkin’ ’bout, Willis?â€? “You could type a name in,â€? proposes Bush, “and then it’d be, like, ‘Whatchoo talkin’ ’bout, David?’ That’d be pretty funny. Like, dude, what are you saying?â€? Badabeez can be found online at www.badabeez. com and at various local smoke shops. COLLEGE GUIDE aware that money would play a role in their lives, but they just had no idea where to start. When a student came to her and threatened to call the cops on the work study office for withholding money she had earned, Morrison knew she had to do something. 5ADBCA0C8=60F5D;58B20;BD10AB47>;4AH At least that’s how we approach the forms. Money is kinda, you know, important By Anna Schuessler tanford students intrepid enough to wonder about the practicalities of life after college have a muse. Mary Morrison, the director of funds management at Stanford’s Financial Aid Office, knows that personal finance is not a particularly hot topic among undergrads. She’s witnessed this all too frequently, from kids who didn’t know a thing about taxes to some who thought money was just something their parents worried about. “You can’t just say, ‘Oh, I don’t like money, I just won’t do something with it.’ You have to do something,â€? she says. Old people (i.e., adults with a job history) tell young people (i.e., anyone who considers herself above these matters) to pay more attention to their personal finances all the time. But the reality is that a lot of students have no idea what the difference is between a credit card bill and a savings account statement. They have even less of an idea about finance when it comes to planning for the future, which is one of Morrison’s greatest fears for clueless kids. “I think that some of them are going to make decisions based on immediacy instead of a longer term time frame,â€? she says by phone from her Stanford office. And a lot of them do. According to “How Undergraduate Students Use Credit Cards,â€? a 2009 study done by Sallie Mae, the corporation that exclusively handles student loan transactions, the median credit card debt for undergraduate students grew from $946 in 2004 to $1,645 last year. In previous years, Morrison worked as a pre-major adviser and noticed that many of the students she talked to were, in fact, for the classes you need for the future you want for the career of your dreams Mommy, what’s a credit card? Morrison asked the Stanford administration for permission to offer a oneunit financial literacy class, aimed at providing students with a basic understanding of topics like taxes and insurance. She hoped to see at least 20 students enter the classroom. She didn’t have to worry. Some 100 students filled the seats that first year; 12 years later, the class is still full to capacity. And while the Stanford staff member sees financial illiteracy to be an urgent issue among college students, she doesn’t necessarily think the blame’s on them. “It’s learned information, and who’s going to tell you?â€? Morrison asks. “It’s like everything else. If you don’t know that apples are better for you than ice cream, and no one’s ever told you, how would you behave?â€? She realizes also that many parents are reluctant to share details of their income. This was the case for Morrison, who grew up in a low-income family. “My father had a desk, and in one drawer we had my family’s checkbook and savings account book. And we could not touch that drawer,â€? she says. “I could ask my mother about sex, but if I had asked her how much money my parents made, she would have just been furious.â€? Reticence about financial history is not isolated to either the rich or poor, and Morrison says she sees plenty of both in her classes. “There are the really poor kids whose parents are day laborers or on public assistance, and they can’t ask their parents about paychecks,â€? she says. “Then you have the other people who never thought about restraining their personal spending. Two ends of the economic scale, both totally clueless.â€? Despite the lack of financial smarts that plagues the crowd straggling into her classroom each quarter, Morrison remains hopeful. There was the student who told her that her father discouraged her from taking the class, explaining that he would manage her finances until she found a husband—just as he managed her mother’s money. She took the class anyway and called her mother after each class to teach her the information she had learned. Another student refused the gift of a car from her father, reasoning that with the added expenditures of insurance and upkeep, a bundle of money for rent would be a more practical gift. These are the stories that make Morrison smile. “Her dad turned to her and said, ‘You go back and tell that woman that you did learn something at Stanford University.’â€? Make your dreams a reality in just 6 to 18 months with Empire’s focused, complete career training. Accounting and Bookkeeping Medical Assisting, Billing and Coding Office Administration Paralegal and Legal Secretary Hospitality, Tourism and Wine Information Technology – Microsoft, Cisco, Linux, Security Choose day or evening classes. Most students qualify for financial aid, and all graduates receive lifetime job placement assistance. Why Wait? Start September 13. Call today or visit us on the Web. 707-546-4000 3035 Cleveland Avenue, Santa Rosa Business – Law – Technology – Medical Business – Law – Technology – Medical THE BOHEMIAN A DELICIOUS WAY TO SUPPORT THE ARTS! ;DG4 The new theater on the NVC campus has choral and orchestral spaces as well. PASSPORTS AVAILABLE THRU OUR BOX OFFICE! Box OfďŹ ce 707 523 4185, ext 1 52 West 6th Street, Santa Rosa, CA 95401 Napa’s cultural renaissance continues with new performing arts center By David Templeton n Napa, the hottest show in town isn’t the much-buzzed-about Taming of the Shrew opening this weekend in Veterans Park. It’s not even the enormous pirate ship constructed as the set of that guns-andcutlasses production. No, at the moment what has captured the community’s attention is the brand-new performing arts center that on Aug. 12 had its official opening on the campus of the Napa Valley College. A long-in-the-works dream of the late NVC president Chris McCarthy, the 48,125square-foot, state-of-the-art facility was made possible mainly through a bond measure voted in eight years ago. The finished product, a $31.5 million glass and concrete marvel, surpasses even the hopes of its longest and strongest advocates. “The center itself is simply extraordinary,â€? exults Jennifer King, Napa Valley College Repertory Theater artistic director, speaking with barely contained enthusiasm on the first day of the fall semester. “I have to say there is probably no other facility in the North Bay like it,â€? she says, “a facility designed specifically for education and performance, a facility built with incredible vision and forethought.â€? It is King’s belief that the new facility will establish NVC as a West Coast version of the world-renowned North Carolina School of the Arts. “But our building is a lot better than theirs,â€? King boasts, describing the center’s three stories, replete with a 480-seat main theater, a 93-seat black box space, theater arts and music wings, full costume and scenery shops, plush dressing rooms with showers and all the desired amenities. “This is an absolutely amazing space,â€? King says, who was formerly with Sonoma County Repertory Theater. “Yesterday, we had our auditions for our first musical, Aladdin, and we held them in the main theater. It was the first time I’d heard anyone sing or do music in the space. I can’t tell you how good it sounded. It’s acoustically perfect. I’ve never experienced such incredible acoustics. Of course, I’ve been working in funky little theater spaces for most of my professional life. The theater we’ve been using here on campus is just a glorified lecture hall. So to move from that to something that is so stunning is really a dream come true.â€? King points again to McCarthy’s original decision to push forward with plans for the facility, a dream he never saw completed. He passed away unexpectedly last year. “When Chris first became president here, when he first saw the conditions in which the performing arts department was working, it moved him to tears,â€? King says. “He made a commitment to create a space on campus that was equal to the work being done by the students and faculty. He wanted something that would serve the students and the community, now and into the future. “There’s something about having a dream for so long,â€? she continues, “and then to finally see that dream manifest that is just so incredibly wonderful. Walking into this building, it makes coming to work absolutely magical.â€? 2277 Napa Vallejo Hwy., Napa. 707.256.7500. www.napavalleytheater.org. B?HEBB?H Cary Grant and Ingrid Bergman co-star in ‘Notorious.’ By Richard von Busack hen the director’s most devoted critic Francois Truffaut called 1946’s Notorious “the quintessence of Hitchcock,â€? it was both an ambiguous remark as well as an appreciation. Notorious is a number of things all at once: a spy thriller, a Gothic, a romance and a Story of O–like tale of love tested through suffering. It screens Aug. 20 at San Anselmo’s Film Night at the Park and is worth a new look. Ingrid Bergman plays Alicia, a woman who, like Bogart in Casablanca, doesn’t care about patriotism and who would rather hit the sauce as hard as possible. That’s when the highly authentic government agent Devlin (Cary Grant) enters the picture. Seduced by him, Alicia becomes an agent too, traveling to Rio de Janiero to learn her assignment. She is meant to get intimate with an old acquaintance named Sebastian (Claude Rains), a debonair Nazi deeply involved in some Fourth Reich scheme. Rivalry rather than duty is the engine of the plot. Devlin’s jealousy grows as Alicia takes on the role of lover and wife. That’s when the two agents find out how lethal the assignment is going to be. Screenwriter Ben Hecht’s key work on Gone with the Wind made the liaison between a bastard and an anti-heroine seem urgent and romantic. He finds similar elements in this story. The stars bring it on in the famous nuzzle on a moonlit balcony over Copacabana; here is attention to the letter of the censors’ law, while defying their spirit, through long-held clos-ups and interspersed dialogue. It’s Hitchcock, though, who provides sympathy for the Devlin, as well as for the fascist cuckold. There’s a wave of tragedy in Sebastian’s wordless discovery of betrayal in a wine cellar filled with some very suspicious vintages. This sequence in the cellar shows the peculiar strength of Hitchcock’s visual style. “Clarify, clarify, clarify,â€? he used to say. Only Pixar seems to be following his advice these days. Notorious is a seminar for lens geeks, and not only in the showy parts—visual distortion for sickness and hangovers, with Devlin all but spinning on the ceiling over the suffering Alicia—but also in the emphasis on the size and brightness of objects and in making a single close-up tell a story. Most startling is a shot that wafts us from a second-floor balcony to the palm of Bergman’s lovely hand. There a stolen key rests, just as in the Bluebeard myth this tale so much resembles. Also at work is another quintessential Hitchcock feature: the story of the bad girl who isn’t. Note Bergman’s revolutionary naturalism, with her unplucked eyebrows, her uncorseted flesh, her full-lipped face that wasn’t anything like the look of a movie-studio porcelain doll. In early scenes, with hair that gets in her eyes and in her mouth, she’s a punky slattern; later, she’s groomed tightly, even painfully, by the Germans. And there is Hitchcock’s usual satire of warped, doting moms and dominated sons. As Sebastian’s far-smarter mother, the memorable Leopoldine Konstantin shows off a one-handed cigarette-lighting trick, just like a Prussian officer in a silent film, as she figures out how to deal with the American agent in her household. Notorious is in the world of film noir but not of it; it’s a tough and dark film that’s also one of the finest women’s pictures ever made. The Twilight movies could use a little of Notorious’ nocturnal power when dealing with the question of what happens when sides are chosen, and how far too far can go. ‘Notorious’ screens on Friday, Aug. 20, at 8pm. Film Night in the Park, 8. Creek Park, Hub Intersection, Sir Francis Drake Boulevard, San Anselmo. $3–$6. 415.272.2756. FUNCTIONAL ART Fine & Fashion Jewelry necklace by Kristina Kada Hitchcock’s ‘Notorious’ still a stylish shocker 146 N. 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See Film listing, p39. 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THE BOHEMIAN Bfrom rews the world &Tasty Paround ub Grub Coyote Den Bar & Dance Hall FREE LIVE ENTERTAINMENT Saturday August 21st • 8pm Twice As Good Rockin' Blues Fri, Aug 20th • 8pm The Easy Leaves Friday, August 27th • 8pm Los Nuevoz Macizos • Cumbia Dance Sat, Aug 21st • 7pm Saturday, August 28 • 7pm Mark McDonald & the Delta Blues UFC 118 Thursday Night KARAOKE • 8:30pm $20 Free Play Drawings Fri, Aug 27th • 7:30pm Three At Last 707.874.9037 | www.barleynhops.com 3688 Bohemian Hwy, Occidental STEVE LUCKY & THE RHUMBA BUMS Saturday, August 21 Wed, Aug 18 8:45–9:45am; 5:45–6:45pm Jazzercise 10am–12:15pm Scottish Dance Youth and Family 7–10pm Singles & Pairs Square Dance Club Thur, Aug 19 8:45–9:45am; 5:45–6:45pm Jazzercise 7:25–11pm Circle N’Squares Square Dance Club Fri, Aug 20 8:45–9:45am Jazzercise 10:30–11:30am ZUMBA GOLD 7:30pm North Bay Country Dance Society/ Contra Dance Sat, Aug 21 8:00–9:00am; 9:15–10:15am Jazzercise 10:30–11:45am Dance Workout with DJ Steve Luther 7–11pm DJ Steve Luther presents STEVE LUCKY & THE RHUMBA BUMS Sun, Aug 22 10:30–11:30am 1:30–3:30pm 5–9:30pm 8:30–9:30am Jazzercise ZUMBA FITNESS with Anna VINTAGE DANCE DJ Steve Luther COUNTRY-WESTERN LESSONS & DANCING Mon, Aug 23 8:45–9:45am; 5:45–6:45pm Jazzercise 7pm–10pm Scottish Dance Tues, Aug 24 8:45–9:45am; 5:45–6:40pm Jazzercise 7:30–9pm AFRICAN AND WORLD MUSIC DANCE Santa Rosa’s Social Hall since 1922 1400 W. College Avenue • Santa Rosa, CA 707.539.5507 • www.monroe-hall.com 5 miles North of Ukiah Hwy 101 • West Rd. Exit 0NYR[QN_ ;\_aU /Nf 2cR[a` Hiatt & Hidalgo Bikes, Bikes, Bikes! During an intimate show at the Lagunitas Tap Room in April, John Hiatt, having already premiered a few songs from his new album, The Open Road, made a terrible mistake: he asked the crowd for requests. A loud barrage of shouted song titles ensued, making two things hugely apparent. One, Hiatt’s written a lot of songs. Two, people love the hell out of those songs. These are qualities he shares with Los Lobos, with whom he shares a stage this weekend in Healdsburg. Songs we’d love to see them play together include Hiatt’s “Real Fine Loveâ€?—imagine David Hidalgo sprinkling his solos over that contagious chord progression—or how about Hiatt singing a verse or two of Los Lobos’ “Dream in Blueâ€?? Wish, hope, pray for some sort of collaboration when the two co-headline on Saturday, Aug. 21, at Rodney Strong Vineyards. 11455 Old Redwood Hwy., Healdsburg. 4pm. $50–$75. 707.869.1595. We’re all getting ready for Levi Leipheimer’s GranFondo in our own way. Some of us are stepping up at the gym. Others are taking longer weekend rides. Leipheimer himself ? Why, he showed up at the Wednesday Night Dirt Crits in Howarth Park last week, filled out an entry form, got a paper-plate number to affix to his handlebars and raced the dirt trail alongside all the other regular folks riding after work. Crazy! This weekend, the training continues with five different courses in Saturday’s Break the Cycle ride, open to the public, culminating in the Northern California Bike Expo, with loads of vendors, street food, live music, custom bike exhibits, bike parades and even a bicyclepowered performance of The Last Seed by the Imaginists Theatre Collective. Celebrate the area’s rich and evergrowing cycling culture on Saturday, Aug. 21, in Santa Rosa’s Juilliard Park. 10am to 4pm. Free. 707.545.0153. B > AA K6 AA : N Composer’s House Seth Montfort is an extremely gifted pianist living at a former mortuary in Guerneville. He holds near daily concerts, tours in Latin America, plays Liszt while wearing leather, and he possesses an incredible gift for retention with a remarkably large memorized repertoire. He also founded the San Francisco Concerto Orchestra, a group that endeavors to provide starring soloist turns to musicians denied such a spotlight in their symphony day jobs. The S.F. Concerto Orchestra often play at Montfort’s house, and it’s one of those only-in-Guerneville experiences. Folk and punk are common fare at house shows, but classical? In a former mortuary? Blindfold a friend, drive him out there and blow his mind when the S.F. Concerto Orchestra play Ciani, Paganini, Mendelssohn and others on Monday, Aug. 23, at the Composer’s House. 16375 Fourth St., Guerneville. 4pm and 7:30pm. $10–$20. 707.604.7600. _"! BTT2^]RTacb ! ! dV 0 ^] ] _c^f VWc_[PhbcWTD AdUdbFPX]faX Flying Right Cloverdale, “the Gateway to the Redwoods,â€? is one of those unassuming little burgs with an actual big heart. Cloverdale, after all, built a train station for the SMART train 12 years ago, on faith alone! It also continues to support a hometown drive-in, even as fast-food joints get built on the south side of town, and its downtown has a smattering of culture in Friday Night Live, a series of free concerts presented by the Cloverdale Arts Alliance. This Friday, none other than Big Sandy and His Hi Fly-Rite Boys pull their chrome Airstream bus up to the downtown plaza to serenade the boppers hungry for their Texasswing style featuring a stand-up bass, lap steel and a portly voice raised on border-radio soul. Get up there early for the farmers market beforehand, and take a final dip in the Russian River afterward when they play on Friday, Aug. 20, at the Downtown Plaza. Main and Broad streets, Cloverdale. 7pm. Free. 707.894.4410. Sure, we love San Francisco Chronicle writer Mark Morford for describing Dick Cheney as “pure sneering vileness incarnate just by opening his tiny black eyes,â€? but we also love that the next day he’ll eloquently unearth deep-seated truths about fearing death or having children or maintaining sanity in a world that seems to exist solely to desecrate humankind’s empathy for each other. Readers all over the country revile him as a source of “San Francisco values,â€? but we say Morford’s compassion, intellect and incisive wit are to be most celebrated. (Plus, he really likes sex.) Jane Ganahl interviews him live and onstage on Wednesday, Aug. 18, at 142 Throckmorton Theatre. 142 Throckmorton Ave., Mill Valley. 7:30pm. $12–$15. 415.383.9600. :B@60 0<;02?A@ HDCDB68DJCIN I]VY7ZX`bVc EdlZg]djhZ\j^iVg^hiVcYhdc\lg^iZg hiZZeZY^cWajZhVcYXdjcign#6j\'&Vi -#'%#HijY^d:!VYYgZhhegdk^YZYl^i] i^X`Zih!HZWVhideda#,%,#-'(#('+)# 8Vgc^kVa"i]ZbZYgdX`WVcYl^i] h^YZh]dlh!^cXajY^c\Æ>cXgZY^WaZ 7ZcYVWaZ<^ga!ÇbV\^XXdbZYnVcY bdgZ#6j\'%Vi.#&%"&'#AVhi9Vn HVaddc!&'%;^[i]Hi!HVciVGdhV# ,%,#*)*#'()(# Bjai^XjaijgVahfjZZoZWdmZmigVkV\VcoV [ZVijgZhVXihWn;aVXd?^bZcZo!9^X` 8dci^cd!Eda`VX^YZ!AdhIZmbVc^VXh VcYbVcndi]Zgh#6j\'&"''!./(%id-# &*"'*#AVEaVoVEVg`!DaYGZYlddY =^\]lVn!8diVi^#---#**.#'*,+# 6ai"gdX`Zghd[É.%hBIK[VbZ_d^cZYWn i]Z^gfj^g`n6bZg^XVcVVaiZgZ\d! 8VbeZgKVc7ZZi]dkZc#6j\')Vi,#'*# =debdc`IVkZgc!'(%EZiVajbV6kZ! 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Gd]cZgiEVg`#,%,#*--#&-%%# Hdja[jaWajZh\j^iVg^hiVcYh^c\Zg^c ]^h]dbZidlcXajW#6j\'&Vi.#*" &%#;dgZhik^aaZ8ajW!+'*%;gdciHi! ;dgZhik^aaZ#,%,#*',#&'--# HjbbZgbjh^XhZg^Zh[ZVijgZha^kZ XdcXZgihZkZgnHjc!&id)!i]gdj\] DXi#6j\''!Hda=dg^odc#89dcVi^Zaad L^cZgn!)%(*LZhih^YZGY!=ZVaYhWjg\# ,%,#)(&#)))'# :kZgn;g^Vi*i]gdj\]DXi!gZaVml^i] Ve^Xc^XVcYa^kZadXVabjh^X#6j\'%! 9dj\A^eidcWajZ\gVhh#*#B^X]Za" HX]ajbWZg\ZgL^cZgn#)&**L^cZ 8gZZ`GY!=ZVaYhWjg\#,%,#)((#,)',# 9dcc^ZBVX9dcVaYVcY?VbZh@Z^\]Zg eZg[dgb>g^h]VcYHXdii^h][da`bjh^X# 6j\'*Vi,/(%#@ZaaZg:hiViZL^cZgn! *-,*AV`Zk^aaZ=ln!EZiVajbV# ,%,#,+(#-.'%# Hl^c\!_jbe"WajZhVcYgdX`VW^aan YVcXZeVgin#6j\'&Vi+#;gZZ#EZiVajbV 6gih8ZciZg!'(%AV`Zk^aaZHiVi:Vhi LVh]^c\idc!EZiVajbV#,%,#,+'#*+%%# 7g^c\e^Xc^XVcYZc_dngdX`^c\hjbbZg ZkZc^c\^cWdiVc^XVa\VgYZch#6j\'&Vi +!7g^\]i7gVhhFj^ciZi#'%#FjVggn]^aa They Might Be Giants play Napa on Aug. 21. See Concerts, above. 6j\&-!HZVc<VgkZn6bZg^XVcV# 6j\'%!6bn=d\VcIg^d6bZg^XVcV# 6j\''Vi&!IjbWVdAVi^c_Voo#6j\'(! <ZgbVcHiVbbi^hX]#&-.=Hi! 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HZWVhideda#,%,#-'.#'%+'# Polkacide, where punk mayhem meets jolly good bellows ard Abronski was just a Philly kid running out of money when he moved west to go surfing in the 1970s. A graduate of the Berklee College of Music and a diehard fan of the avant-garde jazz of Art Ensemble of Chicago, he soon found himself in the percolating hotbed of Bay Area punk. Unimpressed. “I wasn’t crazy about it until I heard Flipper,â€? he says. But something clicked, and Abronski decided to turn punk on its ear. “I knew some real musicians—some horn players—as well as some loud rock and rollers, so I went out to ďŹ nd some polka sheet music,â€? Abronski says. Soon, fusing punk rock and polka, the ragtag group played at punk haven Mabuhay Gardens as part of a six-band bill. Polkacide was born. Not to be outdone, the irony-soaked Bay Area punk scene embraced the band—ultimately, Abronski played the howling saxophone on Flipper’s end-of-civilization anthem “Sex Bombâ€?—and the national news media descended on the novelty. But Abronski is less interested in tongue-in-cheekisms than the music itself, and talks passionately about the technical precision required to excel in a much-maligned genre. In the last century, polka was the roots music among trans-Danubian ĂŠmigrĂŠs in the Great Lakes and industrial river corridors. Schisms emerged: the polished Slovenian style of Frankie Yankovic, the innovative double-trumpet sound of Marion Lush. (Abronski admires the musicianship of the latter.) There’s a Dionysian strain in Polkacide. Walter Solek himself, who wrote “Who Stole the Kishka?â€? and often performed with a rubber nose, fake boobs and a rubber piglet sticking out of the y of his boxer shorts, would have loved Polkacide’s “Weiner Dog Polka.â€? Polkacide are down to a handful of gigs a year, despite boasting the best lineup in the band’s history. “Playing in a goofy polka band is not the path to fame and riches, but it’s a family now, and I get to play with friends,â€? Abronski says. “You can’t beat that with a stick.â€? Catch Polkacide at the Cotati Accordion Festival—which celebrates with Flaco Jiminez as headliner for its 20th anniversary—on Saturday, Aug. 21. The festival runs Saturday–Sunday, Aug. 21–22, at La Plaza Park in Cotati. 9:30am–8pm. $15–$17 per day; $25 both. www.cotatifest.com. Richard von Busack 9VkZ<aZVhdc6bZg^XVcV#6j\'&! L^cYh]^ZaY8dlWdnhVai"6bZg^XVcV# 6j\''!9VccnBdciVcVXdjcign# &'-%CBX9dlZaa7akY!EZiVajbV# ,%,#,,-#-,,+# 6j\'%!8Vgcn7gVi!Lgdc\!6higdha^YZ! <^VbV^XV9VngdX`#Bdc!`VgVd`Z# &'%;^[i]Hi!HVciVGdhV#,%,#*)*#'()(# 6j\&-!8Zai^XhZhh^dch#6j\&.!<gZ\ =ZhiZgWZWdee^Vcd#6j\'%!A^cYhZn Igdn#6j\'&!NVcX^ZIVnadg_Voo#6j\''! ?ZhhEZiinVcYIdcn9É6ccV_Voo#6j\')! Djid[i]Z7ajZhl^c\#&+'-%BV^cHi! <jZgcZk^aaZ#,%,#-+.#%*%&# C. Donatiello Winery Summer Music Series Live from the Middle Reach 6j\&.!IdcZlddYh[da`#6j\'%! JcYZgi]ZGVYVgWajZ\gVhh#6j\'&! @ZnhidcZHiVi^dcWajZ\gVhh#6j\''! <gZZc]djhZ8Zai^X[jh^dc#)+);^ghiHi! HdcdbV#,%,#.(*#%++%# 6j\'%!7VggZa]djhZ#HVi!<gZZ`bjh^X VcYWZaanYVcXZh]dl#*+--AV`Zk^aaZ=ln! EZiVajbV#,%,#,+.#-*)*# 6j\'%"''!ÆBnhiZgnd[:Yl^c9gddYÇ hZZI]ZViZg#'%&LVh]^c\idcHi! EZiVajbV#,%,#,+'#(*+*# 8-22 sol Horizon 8-29 Brett Hunter September 9-5 Hotel Cafe tour Buddy & Friends* 9-12 Troubaduo 9-19 Brian Bergeron 9-26 Lelia Broussard & Chris Pierce 10-3 Audra Connolly 10-10 John Gold All Shows Sundays 1-4pm *9-5 will take place from 3-6pm C. Donatiello Winery 4035 Westside Road Healdsburg, CA 707.431.4442 I]jghVi+/(%!7dcc^Z=VnZh# *&%HVc6chZabd6kZ!HVc6chZabd# )&*#)*)#'.)'# HVi!<aVblVkZ=Veen=djgl^i] 7gVcYdc7VcY6?j`ZWdm#Hjc!i]Z BdgdcZhXaVhh^Xejc`#,&&;djgi]Hi! 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A^cYV;Zggd7VcY#Bdc!9dccnBVYZgdhÉ Egd?Vb#-'&%DaYGZYlddY=ln!8diVi^# ,%,#,.*#,-,-# :kZgnHjc!GdX`ÉcÉGdaaHjcYVn HX]dda#*',7Vg]Vb6kZ!HVciVGdhV# ,%,#*)'#%.-%# HXdjg8gV^\ha^hidgWZ\dci]Zh^YZlVa`[dgi^X`Zihidi]^hhdaY"djih]dl# 6j\')Vii]Z;dmI]ZViZg# BdgZHVc;gVcX^hXdZkZcihVilll#h[hiVi^dc#Xdb# 0HH>CA8? MC Oz lays down about Quidditch, zoology and other common hip-hop tropes. New book teaches the art of the rap song By Caroline Osborn ooks I get. Rap, not so much. So when my editor gave me Paul Edwards’ quixotically titled book How to Rap: The Art and Science of the Hip-Hop MC (Chicago Review Press; $12.95), one that promises to explain this incomprehensible art form in 314—pi times 100?—pages, I wondered if my introverted bookishness too could learn, as they say, to bust a rhyme. After all, books can teach me anything. Right? But if I was going to do it, I was going to do it the same way I do everything else: like a huge nerd. I am an English and comparative literature major, a cognitive science minor. I love Shakespeare and brains. I am the captain of my college’s Quidditch team, which speaks for itself about the obsessive level of my Harry Potter fanaticism. My friends and I watched Lost religiously and tried to figure out the ending. Believe me, I could go on all day. I struggled with the book in the beginning, but not for the obvious reasons. Author Edwards structures the exposition with his own sober explanation of a rap staple such as braggadocio or freestyling followed by more colorful testimonials from seasoned hip-hop pros. Sometimes when I read a quote, I wasn’t sure if it was a lyric or just something the guy said. I would try to read it with rhythm, but then it would fall apart and I would feel like an asshole. (Hint: it never is a lyric.) What follows is a chapter-by-chapter breakdown of what I have learned and how I applied that knowledge when crafting my lyrics. Excuse me. I meant to say, “when laying down my rhymes.â€? I even recorded my song. Chapter 1: ‘Content Topics’ “I don’t believe in writer’s block,â€? says Tech N9ne, apparently some kind of rapper, “because the cure for writer’s block to me is to go out and have something happen to your ass.â€? That left me with one question. What has been happening to my ass lately? College! I opted for real-life content, but only after realizing that the bulk of what rappers call fantasy content has more to do with strippers than dragons. Chapter 2: ‘Content Forms’ Since flaunting one’s style seems to be so central to the hip-hop street ethos, I took note of the book’s instruction in braggadocio and battling form. “[With my subject matter, I’m] not trying to save the world,â€? says Sean Price, another hip-hop artist I had never heard of. “I be smacking the shit out of people in my rhymes, I be drop-kicking people. I know what I’m writing when I write it, though, so it might be some crazy shit, but I know I’m writing the crazy shit, and I want to write the best crazy shit I can write.â€? What he lacks in vocabulary he makes up in awareness of his message and pure writerly aggression. Inspired by my favorite rap battle of all time, Flight of the Conchords’ “Hiphopopotamus vs. Rhymenocerous,â€? I decided to talk some smack. Chapter 3: ‘Content Tools’ Edwards advises using punch lines ( + One Across the Bow Works on Paper William Smith August 20–Sept. 26, 2010 Artist Reception: Saturday, August 21 4–6pm Rudimentary Navigational Skills Mixed media on paper, 22" X 34", 2010 6671 Front St/Hwy 116 • Downtown Forestville 707-887-0799 • 11-6 Thurs–Mon (closed Tues & Weds) quicksilvermineco.com THE BOHEMIAN What are you thankful for? VAPORIZERS ON SALE AT 2 new great vapes for your on the go lifestyle: The Launch Box $99 • Essential Vaaap $99 Both are made here in California, come in and check them out! Plus fantastic deals on Cali blown spoons, bats, bubblers and tubes! Organic Raw papers in stock Friendly knowledgeable staff—come see us, you’ll be glad you did! 707-795-8498 8246 Old Redwood Hwy. Downtown Cotati Mon–Sat 11am to 7pm Sun 12 noon–6pm (Next door to Redwood Cafe) We have the best priced Detox in Sonoma County. We now have detox online. To get detox delivered to your door, go to www.Gravenstones.com or setting up a question before answering it with a deft rhyme that “hitsâ€? the listener. I used this when I wrote, “Tuesday night, common room at any cost.â€? (Why is she going to the common room?) “We don’t want to do our homework, we just wanna watch Lost.â€? (Understood.) Edwards also recommends similes to convey a cocky message in an innovative way. I both mocked my imagined battle opponent and incorporated figurative language when I recalled my ferocious experiences playing Beater on my college’s Quidditch team: “You’d better run fast when I’m burning a Bludger / You’ll be more terrified than Cornelius Fudge . . .â€? Quidditch, of course, is adapted from the broomstick sport of the same name from the Harry Potter series, and Fudge is the bumbling politician who let his fear get in the way of nipping the Voldemort threat in the bud. See? It’s a topical insult. At the end of my third verse, I brag about my way with words, and then I sneer, “Your passive voice can be spotted like a bad toupee.â€? In addition to completely destroying my opponent’s self-esteem, I poke fun at his terrible grasp of the English language by delivering my dig about passive voice in the passive voice. Yeah, I know it hurts. Chapter 4: ‘Flow’ Flow is to rap as meter is to poetry. Some tricks that apply to both include rests and enjambment, here called “overlapping bars.â€? Edwards instructs me to pause on a downbeat to emphasize the next part of my line. Eager to make sure everyone understands that I achieved an A in zoology, by general consensus a rather difficult class for nonscience majors, I pause between “I ruled at zoologyâ€? and “I got an A.â€? I overlapped bars, or continued my idea across lines (in this case, across stanzas), when I wrote “You’ll be more terrified than Cornelius Fudge or / the kids in the library staying up late.â€? Here, I both deepen the humiliation of my opponent, who is now more of a nervous wreck than Fudge and finals crammers combined, and seamlessly transition into my next topic. Chapter 5: ‘Rhyme’ Observe the internal rhyme. “I bewitch on the pitch when I play Quidditch / It’s uncanny how my Annie can catch that Snitch.â€? Not only do I use the “–itchâ€? rhyme to join the first line to the second, but I also develop internal consistency within the first line alone by invoking the rhyme three times. Furthermore, I throw in an extra rhyme (“uncanny,â€? “Annieâ€?) that has no payoff in any other line, but adds coherence within the second line, just because I care. Circumstances forced me to abandon my inclination toward perfect rhyme and work with assonance, or same vowel sound, rhyme when I wrote the couplet: “Our dining hall repertoire often repeats / But still our food is better than at the UCs.â€? I alliterate in my chorus when I promise, “I’ll proofread your paper like a private eye.â€? Chapter 6: ‘Rhyme Schemes’ Typically, the main rhyme falls on the last beat of each measure. But sometimes, if a lyricist wants to get very, very tricky, she can place the initiatory rhyme on the fourth beat of the first measure and rhyme it on the upbeat of the second measure’s fourth beat. I employ this tactic when I flaunt the raw power of my extracurricular activities: “I write for the paper, I work the lit mag / I critique your concerts, throw bad poems in the trash bag.â€? “Magâ€? and “trashâ€? cooperate as assonance rhymes, but the perfect rhyme falls on “bag,â€? the upbeat syllable. Chapter 8: ‘The Writing Process’ Halfway through chapter eight, I realize that I am a natural when I read this: “I dream raps, I dream verses, I dream hooks, choruses,â€? says Crooked I, allegedly some sort of hiphop artist. The previous Friday night, I had a dream in which I wrote a rap couplet and performed it for the favorite fictional character to whom it was dedicated. A repurposed version of said couplet appears as the opening lines of my rap song: “MC Oz in the house, I sling words like daggers / I swagger, my rhyme’s the only thing that really matters.â€? I poke fun at his terrible grasp of the English language by delivering my dig about passive voice in the passive voice. Yeah, I know it hurts. Chapters 12 and 13: ‘Vocal Techniques’ & ‘In the Studio’ “Spitâ€? is hip-hop speak for vocal performance. When I arrived at the homemade recording studio in a friend of the Bohemian’s garage, I intended to do just that. The word spoke to me of aggression and stamping one’s own disgusting, phlegmy identity on one’s art. As someone who doesn’t usually think of herself as particularly aggressive or especially phlegmy, I knew I would have to dig into this word to sell myself as a rapper. I opted to half-memorize my lyrics and still use the sheet as a security blanket to maximize swagger. My dear friend Edwards advised me to be expressive, designate breathing spaces and think of my voice like an instrument. I did all of this as I stood at the microphone, the beat pumping through the big, black headphones cupping my ears. I held my half-memorized lyrics sheet with my left hand and waved my right hand around like rappers I’ve seen on television. I moved my shoulders. I enunciated. I pretended I was Chris Parnell from SNL’s Lazy Sunday, the only rap song to which I know all of the words. In short, I swaggered. But my voice wasn’t the only instrument I brought. After recording the vocals, I whipped out my alto sax and improvised a backing track. Being a high school band geek continues to pay off ! Can books teach me everything, including how to rap? They can indeed, as long as I want to rap like a person who reads a lot of books. To witness the aural intensity of my entire rap song, visit www.bohemian.com/bohemian. A NON-PROFIT EVENT BENEFITTING LOCAL YOUTH GROUPS Best Music Venue Best Place for Singles to Meet THUR ! AUG 19 ! 6PM ! FREE ! 9PM THROWBACK 3RD THURSDAY’S HIP HOP NIGHT! FRI ! AUG 20 ! 9PM ! $15 ADV 3RD FRIDAYS REGGAE PARTY WITH SAT ! AUG 21 ! 9PM ! $12 ADV SUN ! AUG 22 ! FREE ! 3PM RIVER WERX ! 6PM DORI & DAVE JAZZ NEXT WEEK THUR ! AUG 26 ! 8PM ! FREE ANGEL ISLAND WITH BILLY & DOLLY FRI ! AUG 27 ! 9PM ! $12 ADV PRIDE & JOY ZION I SUN ! 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PUB QUIZ 8/20 9:30 PM SHOW > $10/12 > ROCK CARNY BRAT 'S FRI 9/17 • 8:45PM DOORS • $17 • 21+ DANCE/PARTY HITS AMAZING SHOW SPECTACULAR wITH SPECIAL GUESTS WRONG + A S TRO S LIDE AN EVENING WITH + Giamaica Day "The Incredible Bendable Girl" + KEN GARR "MAGICAL ASTOUNDING COMEDY" THURS 9/23 • 8:00PM DOORS • $16ADV/$18DOS • 21+ ROCK/PSYCHEDELIC/JAM BAND 9:30 PM SHOW > $? > TOP 40 DANCE ODYSSEY ENT. GROUP PRESENTS TEA LEAF GREEN "A PINK & BLACK AFFAIR" W/DJS Enfo + Ben LaVee Young Will + Tytan + Sykwidit 8/25 8:30 PM SHOW > $12/15 > ROCK FRI 9/24 • 7:00PM DOORS • $17 • 18+ FOLK/AMBIENT SAT 9/25 • 8:00PM DOORS • $16ADV/$18DOS • 21+ ALL FEMALE LED ZEPPLIN TRIBUTE BAND + IT'S ALIVE + TRANSMIT NOW 8/26 6:00 PM SHOW > $20/25 > ROCK,BLUES TUE 9/28 • 8:00PM DOORS • $19 ADV/$21 DOS • 21+ ELECTRONICA/ROCK Medtronic Musicians' 4th Annual Fundraiser for American Heart Association The Poyntlyss Sistars + T'Soul CT Cruisers + Geoff Hawkins DUBSTEP PARTY WITH EOTO, MiM0SA & MARTYPARTY THUR 9/30 • 8:00PM DOORS • $21ADV/$23 DOS • 21+ FOLK/SINGER-SONGWRITER 9:30 PM SHOW > $5 > ROCK SEEDS OF HATE +THE DISCIPLES MARTIN SEXTON SAT 10/2 • 8:00PM DOORS • $21 • 21+ SOUL/POP/ELECTRONICA + HOLY ROLEMODEL 8/28 9:00 PM SHOW > $15 > 80'S DANCE HITS SAT 10/16 • 7:30PM DOORS • $21 ADV/$26 DOS • 18+ ROCK/AMERICANA/COUNTRY 9/2 CAROLYN WONDERLAND + RON THOMPSON GRIEVOUS ANGEL THE LEGEND OF GRAM PARSONS 9/3 WALTER TROUT + VOLKER STRIFLER BAND SUN 10/17 • 8:00PM DOORS • $26 • ALL AGES COUNTRY THUR S DAY-SATURDAY 4-7PM $1.50 PBR $2 DOMESTIC BEER $3 IMPORT/MICROBREW $3 WELL DRINKS & HOUSE WINE $3 APPETIZER MENU all shows are 21+ unless noted SAT 11/6 • 7:30PM DOORS • $26 ADV/$31 DOS • 21+ ROCK CARL PALMER BAND CELEBRATES THE MUSIC OF EMERSON LAKE & PALMER for reservations: 707.545.5876 For All Ages Shows • No Children Under 10 Allowed 120 5th street @ davis street santa rosa, ca 23 Petaluma Blvd, Petaluma l^i]6jWgZnLVYZ0Vi&!Æ;gVb^c\i]Z H^mi^ZhÇl^i]7ZgcVgYkdc7di]bZg0 Vi'!Æ=dl:kZgnYVnEgdYjXihBV`Z EZdeaZH^X`Çl^i]EVja7aVcX0Vi)! 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(%H^g;gVcX^h9gV`Z7akY!Gdhh# )&*#)*+#.***# C6E68DJCIN IVb^c\d[i]ZH]gZl H]V`ZheZVgZCVeVKVaaZnegZhZci djiYddgeZg[dgbVcXZ#6j\'%"'.0;g^" HjcVi,#;gZZ#KZiZgVcÉhEVg`!I]^gYVcY BV^chigZZih!CVeV#,%,#'*+#,*%%# IdcnI^cVÉhLZYY^c\ D[["WgdVYlVnh]dligZVihi]ZVjY^ZcXZ a^`ZlZYY^c\\jZhih^cVc^ciZgVXi^kZ bVig^bdc^VaZmeZg^ZcXZ#I]gdj\]6j\ '.0IjZh"HViVi,!HjcVi&#+*",%# CVeVKVaaZnDeZgV=djhZ!&%(%BV^c Hi!CVeV#)&*#(-(#.+%%# I]Z7D=:B>6CÉhXVaZcYVg^hegdYjXZY VhVhZgk^XZidi]ZXdbbjc^in#>[ndj ]VkZVc^iZb[dgi]ZXVaZcYVg!hZcY^i WnZ"bV^aidXVaZcYVg5Wd]Zb^Vc#Xdb! dgbV^a^iid/CDGI=76N7D=:B>6C! -),;^[i]Hi!HVciVGdhV86.*)%)#EaZVhZ 9DCDIH:C9Z"bV^aViiVX]bZcih# I]Z7D=:B>6C^hcdigZhedch^WaZ[dg e]didh#:kZcihXdhi^c\bdgZi]Vc(* bVnWZl^i]]ZaY#9ZVYa^cZ^h'lZZ`h eg^dgidYZh^gZYejWa^XVi^dcYViZ# Placing an Ad § By Phone Call the Department at 707.527.1200 Mon.-Fri., 8:30a.m.5:30p.m. Bohemian Classifieds 847 5th Street, Santa Rosa, CA 95404 Monday through Friday, 8:30a.m. to 5:30p.m. ph: 707.527.1200 √ By Fax Fax your ad to the Classified Department at 707.527.1288 g Graphic Services g Computer Services For Sale Classes & Instruction gggg Professional Services FREE Diagnosis, Friendly In-House Staff Hardware/ Software, DATA Recovery, Internet, Email,Wireless Network Setup & Security, Apple Authorized Business Agent, Tam Nguyen-Chief Tech, M-F 10-6 firstname.lastname@example.org 707.664.0400 FREE Debt Consultation! $10K in debt or more? 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No. See this video, Tim Maudlin Corrects the 2022 Nobel Physics Committee About Bell's Inequality. He says the Nobel citation missed the point. I don't want to pick a fight with Maudlin, as he is a very smart guy who explains this stuff very well. He has sharp disagreements with others about Bell's theorem, and I describe them here. Another recent Maudlin video says: [47:00] The theorem of Bell [and confirming experiments] is the most astonishing thing in the history of Physics.Among other things, he gives a very good explanation of what is wrong with superdeterminism, as a Bell loophole. Here is a shorter interview. Here is my view. When quantum mechanics (QM) was discovered in 1926, a lot of smart people wondered whether was a new type of theory, or if the uncertainties were just disguising an underlying classical theory. John von Neumann was the world's smartest man, and he convinced himself in 1932 that QM was different from any classical theory. Einstein co-wrote a 1935 paper speculating that QM might be completed by adding elements of physical reality. Bell showed in 1964 that the difference between QM and a classical theory could be quantified, and that was later confirmed experimentally by Clauser and the other Noble prize winners. So the Bell work is no big deal, as it only confirmed what everyone thought. Maudlin and the other Bell fans have another view. To be fair to Maudlin, I suggest his paper, What Bell Did, and his exchange with Werner, here and here. He correctly says that Bell assumed locality, hidden variables, and statistical independence. Statistic independence is assumed by all of science, and is reasonable. Hidden variables are just the Einstein elements of physical reality, and he and Bell argue that any reasonable theory would have them. That leaves locality. The experiments showed that the Bell inequalities are violated, so that means that nature must be nonlocal. He is right that if you accept hidden variable theory then you have to accept nonlocality. I just do not accept hidden variables. He is also right that the Nobel citation failed to endorse the nonlocality conclusion. There are also the superdeterminism and many-worlds loopholes, but Maudlin and the Nobel committee are right to ignore these. That leaves you with a choice -- you can have locality or hidden variables, but you cannot have both. Maudlin would say that I and the Nobel committee suffer from a misconception that has gone on for decades. It would take some very compelling evidence to convince me of nonlocality. As Maudlin says, if you snap your fingers, do you believe that what happens in your hand can depend on what happens in a distant galaxy? I say of course not, but Maudlin accepts that. Wouldn't we see some examples of action-at-a-distance? He gives an example pointing to nonlocality in the Aharonov–Bohm effect. I do not agree, but it requires technical explanation, and maybe I will post separately on it. The reality of nonlocality has been settled. [3rd video, 18:45]So what is nonlocal? There is no way to change one particle, and have that affect an observable of a distant particle. So the only things that are nonlocal are the mythical hidden variables. Wikipedia describes Bell's theorem: Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories given some basic assumptions about the nature of measurement.Maudlin wants to remove the term "hidden-variable" from the picture, and deny that Bell made such an assumption. You can read Bell's 1964 original paper, and see for yourself that he assumes hidden variables. In later papers he called them "beables" and tried to argue that they could be assumed from first principles. But they have to be assumed somehow. Discussions of Bell's Theorem sometimes get sidetracked by issues of probability and determinism. Some say Bell proved the world is indeterministic. Some say Einstein EPR objected to indeterminism. This is a red herring. There is some truth to it, but it has to be stated carefully, or it is misleading. Maybe I will make another post on this issue. I would say that Bell proved the impossibility of local hidden variable theories, whether they are deterministic or stochastic. Ultimately all theories are stochastic anyway, as all measurements and predictions have errors. 1. Re: First video ["Tim Maudlin corrects..."] 32:50 -- 33:10: LOL! Well, how about RQM? Like, you know, Dirac's, if not QFT? Throughout, Maudlin seems to suggest that the full integration of GR + QM would be required to conclusively argue in favour of locality. Nope. SR + QM should be enough, and RQM already has it. It's just that Maudlin doesn't refer to it. He also seems to take the wavefunction of the *mainstream* QM (or QED or QFT) to be "real" (i.e. ontic). Wrong. In the mainstream QM, \Psi lives in higher-dimensional configuration spaces, and even Norsen has had hard time putting it in correspondence with the 3D space (the last time I checked out his research). Also, the mainstream QM has Measurement Problem unresolved. How can \Psi lead to random and irreversible measurement events, if it's real? 2. The shorter interview (the third) is the best, but still not all of the positions that Maudlin takes are consistent. 3. No hidden or extra variables are required to make full sense of the phenomenology (the accumulated body of experimental observations and context) pertaining to quantum phenomena. I published a paper on it (the ICCTPP iqWaves paper), albeit only with NRQM. Work is in progress for the SR-extension of iqWaves (and progressing fine). Aside: Bohmians (and all other hidden variables folks) are clearly on a wrong track. 4. Aside: The idea of non-locality can be easily regarded as having been settled if you refer exclusively to the NRQM. But the real world isn't non-relativistic. Yet, all discussions of Bell's inequalities refer to NRQM, which builds a kind of a false field of perceptions that is attuned to receiving claims of nonlocality. [In evidence, see point 1. above.] That is so, even if Maudlin argues that whether the specific theory of QM is valid or not, it doesn't matter. The fact that the Bell inequalities are violated itself means that locality is violated. Strictly speaking, this is a hasty generalization, but I won't argue about it for now. I would return to this issue, may be, a year later when the SR-extension of my iqWaves theory is complete. Development of theory is important; discussions and debates aren't. 5. Unless people clearly identify the ontological bases of all theories being compared and contrasted, they would be shooting in the dark. Tricks are far more likely in such a scenario; Truth isn't. History since at least 1926 (if not 1912 or 1905 or 1900 or 1865) is witness. "The fact that the Bell inequalities are violated itself means that locality is violated." No, not true, whether in relativistic or non-relativistic QM. It only means that local hidden variable theory is wrong.ReplyDelete Yes, my mistake. I had in the mind, and should have said:Delete "That is so, even if Maudlin argues that whether the specific theory of QM is valid or not, it doesn't matter. The fact that the Bell inequalities are violated itself, according to him, means that locality is violated." Something like this "according to him" bit slipped out while writing. Thanks for catching it!

- Open Access - Total Downloads : 1 - Authors : Paramjeet Sangwan - Paper ID : IJERTCONV1IS02030 - Volume & Issue : NCEAM – 2013 (Volume 1 – Issue 02) - Published (First Online): 30-07-2018 - ISSN (Online) : 2278-0181 - Publisher Name : IJERT - License: This work is licensed under a Creative Commons Attribution 4.0 International License CAYLEY-HAMILTON THEOREM for SQUARE and RECTANGULAR MATRICES and BLOCK MATRICES CAYLEY-HAMILTON THEOREM FOR SQUARE AND RECTANGULAR MATRICES AND BLOCK MATRICES Assistant Professor, Geeta Engineering College, Naultha Panipat firstname.lastname@example.org Abstract: The main aim of the paper is to introduce cayley- Hamilton Theorem and also to explain its extension for the square and rectangular matrics. In this paper C-H Theorem extension for block matrics has also explained. Keywords: Cayley- Hamilton Theorem, topology, Matrices, square, rectangular , block. Defintion 1.1. If A is an n×n matrix ,then the characteristic polynomial of A is defined to be PA(x)= det(xI-A). This is a polynomial in x of degree n with leading term xn . the constant term c0 of a polynomial q(x) is interpreted as c0 I in q(A). Theorem 1.2 (Cayley Hamilton Theorem). If A is an n×n matrix ,then pA(A)=0, the zero matrix. Theorem 1.3 If q0 is a quaternion of the form q-= a+bi+cj+dk with a,b,c,d, being real , then q2 – 2aq + (a2 2 2 2 If one represents a quaternion q= a+bi+cj+dk as a matrix, A =[ a + bi c + di], c + di a bi PA(A) = A2 -2aA+(a2 + b2 + c2 + d2)I = 0, and the polynomial given in Theorem 1.3 is characteristic polynomial of A GENERALIZATION OF CAYLEY HAMILTON Theorem 2.1 (Cayley-Hamilton Theorem). For any n × n Matrix A, PA(A)=0. Proof. Let D(x) be the matrix with polynomial entries D(x)= adj(xIn-A), So D(x)(xI-A)= det(xIn-A)In. Since each entry in D(x) is the determinant of an (n-1) X( (n-1) submatrix of (xIn-A), each entry of D(x) is a polynomial of degree less than or equal to n-1. It folowws that there exist matrices D0, D1,Dn-1 with entries from C such that +b +c +d )=0 q-1 = q D(x) = Dn-1 xn-1++ D1 x +Do . Then the matrix equation follows det(xIn-A) In = (x In-A) adj(xIn-A) = (xIn-A)D(x) Substituting pA(x)= det(xIn-A), (and using the fact that scalars commute with matrix) XnIn+ bn-1xn-1In+.+ b1xIn+b0In = a2+b2+c2+d2 a2+b2+c2+d2 = a2+b2+c2+d2 (2a q) a2 + b2 + c2 + d2 = 2aq q2 q2 2aq + (a2 + b2 + c2 + d2) = 0 = pA(x) In= det(xIn-A)In =xnDn-1- xn-1ADn-1+ xn-1Dn-2-xn-2ADn-2++ xD0-ADo =xnDn-1+ xn-1 (-ADn-1+ Dn-2 ) +..+(-AD1+Do) -ADo m am-i[An-i An-i-1A2] = 0mn (3.5) i=O 1 1 Since two polynomials are equal if and only if their coefficients are equal , the coefficient matrices are equal ; Where 0mn is the (m×n) matrix. Theorem 3. Let , that is , In=Dn-1, bn-1In= ( -ADn-1+Dn-2),,b1In=- (AD1+D0), and b0In=-AD0. This means that A may be A = A2 C , m > n substituted for the variable x in the equation (2.1) to conclude PA(A) = An+ bn-1 An-1++ b1A+b0In and let the characteristic polynomial of A1 have the form . Then (-ADn-1 +Dn-2)++A(-AD1+D0)-AD0 1 =0 This proves the theorem CAYLEY-HAMILTON THEOREM FOR SQUARE AND RECTANGULAR MATRICES Let Cn×m be the set of complex (n×m) matrices. Theorem 1. (Cayley-Hamilton theorem). Let p(s) = det[Ins A] CAYLEY-HAMILTON THEOREM FOR BLOCK MATRIX The classical Cayley-Hamilton theorem can be also extended for block matrices. Theorem 4. (Cayley-Hamilton theorem for block matrices). n A = = L aisi (an = 1) where Aij Cn×nare commutative i.e., be the characteristic polynomial of A, where In is the (n×n) identity matrix. Then AijAkl = AklAij for all i,j,k,l = 1,2,.,m p(A) = n aiAi = 0n (3.2) P(S) = det[Im A] = Sm Where 0n is the (n×n) matrix. The classical Cayley-Hamilton theorem can be extended to rectangular matrices as follows Theorem 2. (Cayley-Hamilton theorem for rectangular matrices). A = [A1 A2] Cm×n, A1 Cm×m, A2 Dm-1S + Dm be the matrix polynomial of A , where S Cn×n is the block matrix having eigenvalue of A, denotes the Kronecker product of matrix . Cm×(n-m), (n > m) (3.3) P(A) = m[Im ]Ai = 0 (DO = In) (4.4) The matrix (4.3) is obtained by developing the determinant = det[Ims A1] = m aisi (am = 1) (3.4) of the matrix [In S A] , considering its commuting be the characteristic polynomial of A1. Then block as scalar entries. Theorem 5. (Cayley- Hamilton Theorem for rectangular block matrices) Let A = [A1 A2] Cmn×(mn+p) and let matrix charactristics polynomial of A have the form (4.3.2), then D.R. Wilkins, Linear Operators and the Cayley Hamilton Theorem , available from http://www.maths.tod.i.e/pub/histMath/people/Hamilton Raghib Abu Saris and Wajdi Ahmad, Avoiding Eigen values in Computing Matrix Powers. The i=O[Im Dm-i][Ai+1 AiA2] = 0 (DO = In) (4.5) Mathematical Association of America 112(2005), Theorem 6. Let Kaczorek T., 1988. Vectors and Matrices in Automation and Electrotechnics, WNT Warszawa (in Polish). A = A C(mn+p)×mn , A C A2 Kaczorek T., 1995. An existence of the Cayley-Hamilton theorem for nonsquare block matrices and computation of the left and right in verses of matrices, Bull. Pol. Acad.Techn. Sci., vol. 43, No 1, pp. 49-56. and let the matrix characteristic polynomial of A have the form , then Kaczorek T., 1998. An extension of the Cayley- Hamilton theorem for a standard pair of block matrices, Appl Math. And Com. Sci., vol. 8, No 3, pp. 511-516. L A [Im Dm-i]Ai = 0 (DO = In) The Cayley Hamilton theorem is one of the most powerful and classical matrix theory theorem. Many application derive their results from this theorem. To understand the scope of this theorem , alternate proofs were used. Each proof helped to understand how intertwined areas of mathematics are with respect to matrices and the characteristics polynomial. APPLICATION OF CAYLEY HAMILTON A very common application of the Cayley- Hamilton Theorem is to use it to find An usually for the large powers of n. However many of the techniques involved require the use of the eigen values of A. William A. Adkins and Mark G. Davidson, The Cayley Hamilton and Frobenious theorems via the Laplace Transformation, Linear Algebra and its Applications 371(2003), 147-152. Arthur Cayley, A memoir on the theory of Matrices, available from http:// www.jstor.org, 1857. Wikipedia , Arthur Cayley, Available from http:// en.wikipedia .org, 2004 Wikipedia, William Rowan Hamilton, Available from http:// en. wikipedia .org, 2005

UG–FT–46/94 July 1995 PHYSICAL PARAMETERS AND RENORMALIZATION F. del Aguila, M. Masip and M. Pérez-Victoria Departamento de Física Teórica y del Cosmos Universidad de Granada 18071 Granada, Spain We analize the structure of models with unbroken and spontaneously broken gauge symmetry. We show that the quantum corrections to the gauge charges, with number of fermions number of scalars, can be absorbed in the redefinition of three independent gauge couplings (, , and ). We establish the (one-loop) conditions on the matter content for (a value usually assumed in the literature) and we show that in the minimal extensions of the Standard Model with an extra symmetry the choice is not stable under radiative corrections induced by the standard Higgs fields. Moreover, to all orders seems to require an exact symmetry. The spontaneous breaking of the gauge symmetry induces further mixing between the two gauge bosons and introduces a fourth independent physical parameter. A consequence of our analysis is that the usual tree-level description with only three physical parameters (i.e., two gauge couplings and one gauge boson mixing angle) is not in general a justified zero order limit of the treatment including radiative corrections. The extensions of the Standard Model (SM) with an extra gauge symmetry have been extensively studied during the last years . They appear as a possible low-energy limit in many grand unified scenarios , and they are not banished to very high energies by present data . As a matter of fact, precision experiments at LEP as well as direct searches at large hadron colliders (TEVATRON) set (stringent) limits on new gauge interactions, but do not exclude their discovery at future colliders (LHC or NLC) . Usually these analyses assume definite models with few free parameters. In this way, the fits (which often depend also on few independent observables) are simplified. Beyond tree level, however, the number of free parameters is related to the number of independent renormalized parameters. Hence, if a parameter is not let to vary, one must make sure that the constrained model is stable under quantum corrections. In the case of extra gauge interactions this is a delicate point . In this paper we study at the one-loop level models which include a sector with gauge symmetry. From our analysis it follows that a generic extension of the SM with gauge group has four new free parameters: the mass of the extra vector boson ; the mixing angle between the (mass eigenstate) and (-current eigenstate) vector bosons; the overall strength of the new current; and the mixing of the current with the standard hypercharge . In particular, we prove that is a free parameter in these models: it is physical (to be determined experimentally) and necessary to absorb the infinities when calculating quantum corrections. can be consistently ignored if an extra symmetry is present in the theory but this is not the case in many popular models. We do not claim that the effects due to a non-vanishing are always sizeable and they could not be neglected in a tree-level analysis. However, it is worth to emphasize that: When obtaining experimental bounds on models, the nonstandard (tree-level) contributions are often calculated assuming and varying the extra charge (for example, considering different combinations of the two nonstandard subalgebras in ). It seems more systematic, however, to stick to a particular model (which corresponds to a definite charge), and to let all its free parameters including vary, rather than constraining the whole class of models with a particular choice of one free parameter. The three gauge couplings , , and cover the whole parameter space of a model with a gauge symmetry subgroup . For example, the sector of left-right symmetric models (resulting from ) is a particular case with the two new gauge couplings related: (this model is known as the model in the literature (see Ref. for definitions)). In general, since the three parameters get renormalized and run with the scale, they bring information of other (larger) scales (which may point out to grand unification, left-right symmetry at higher scales, etc). In particular, if and satisfy Eq. (1) at some scale, it indicates that there is left-right symmetry restoration at that scale. At any rate, in the absence of extra symmetries, a fully consistent one-loop analysis of precision data including a relatively light requires considering as a physical parameter. As a first step to analyse extensions of the SM at one loop, we study in this paper the structure of models with gauge symmetry . In Section 2 we discuss the tree-level Lagrangian. In Section 3 we fix the choice of renormalized parameters and introduce our renormalization (on-shell) scheme. The one-loop renormalization of the model is worked out in detail, emphasizing the need of an exact extra symmetry to guarantee that can be neglected to all orders. (In the Abelian case discussed here we denote this gauge coupling .) We assume throughout the paper that the theory is vectorlike, although our examples are based on realistic extensions of the SM. Thus, it must not create any confusion when we refer to or left-right models to specify matter contents (models). The results which we illustrate with these examples apply in both cases, except for simple modifications (factors). In Section 4 we present the renormalization of the model with spontaneously broken symmetry. In this case three renormalized parameters in the Higgs potential are replaced by the two heavy gauge boson masses and their mixing angle. Section 5 is devoted to conclusions. 2. Classical Lagrangian and physical parameters. The classical Lagrangian for fermions and scalars with gauge symmetry reads where the antisymmetric tensor and the covariant derivatives are the two gauge boson fields, and and the () fermion and scalar charges, respectively, whereas is a polinomial of at most fourth order preserving the gauge symmetry. is the most general Lagrangian renormalizable by power counting and invariant under the transformations with the two gauge parameters. The invariance of under gauge transformations also allows for a gauge kinetic term of the form , with an arbitrary symmetric matrix. can be absorbed, however, into a vector boson field redefinition (note that a redefinition of also redefines the charges of the matter fields in Eq. (4)). Without loss of generality we can then assume , still leaving the arbitrarity of rotating the two gauge fields. Then only charges are physical, since the rotation left, which is related to the impossibility of distinguishing on physical grounds between the two degenerate (massless) gauge bosons, can be used to fix one of the charges to zero. Hence the charges are determined fixing one charge conventionally and fitting independent experiments. We assume fermion fields with masses , whereas includes scalar masses and couplings. We also assume that the Yukawa couplings are forbidden by some symmetry, for they are not important for our discussion. In summary, in Eq. (1) is a generic (classical) Lagrangian of at most dimension four with gauge symmetry. This is not altered at the quantum level: the theory is renormalizable and gauge invariance does not allow for any other term. Among the physical parameters of the model, however, quantum corrections can be used to distinguish between those which are renormalized (and in this sense are free) from those which are constants. For example, in QED with just a gauge symmetry and matter fields there is one free parameter, the electric charge usually identified with the charge of the proton, and constants, the ratios of the remaining charges to . As we shall show, the gauge sector in a model with symmetry and matter fields depends on three free parameters and constants: the charges can be splitted into where are constant charges (four of them fixed arbitrarily) and , , and three parameters (gauge couplings) which will absorb all the quantum corrections. (We use only one superscript, , for diagonal terms .) In general, beyond tree level, just two gauge couplings and (one for each subgroup) are not enough to renormalize the theory. Obviously for there is only one free parameter (independent charge), since in this case the gauge fields can be rotated to decouple completely one gauge boson. For , three experiments involving two matter fields with independent charges (let say ) can be used to fix , and (once fixed one charge and the 4 charges conventionally); the remaining charges would then be fixed after determining from independent experiments: In spontaneously broken theories the former discussion applies but the gauge boson mass eigenstate bases are fixed and there is no freedom to rotate them. Hence, in the broken case there are physical charges and independent experiments are needed to fix them. Then Eq. (6) remains general, (we use a prime to denote the couplings to mass eigenstates), is the angle rotating from the gauge boson basis triangularizing the renormalized gauge coupling matrix to the renormalized gauge boson mass eigenstate basis. 3. Renormalization of : unbroken case. In this Section we study the renormalization of a theory with unbroken gauge symmetry . We discuss the parametrization of the gauge couplings (valid to all orders) and work out in detail their renormalization in the on-shell scheme at one loop. We show that even the popular (minimal) extensions of the standard model with one extra require two new gauge couplings in order to cancel the divergent contribution of the Higgs fields. Moreover, even if the models are enlarged adding extra matter in order to fulfil the one-loop conditions for consistently neglecting the second gauge coupling , there is no guarantee for the cancellation of infinities at two loops. As a matter of fact, in the examples we have looked at the cancellation of infinities to all orders requires an exact extra symmetry: a more general gauge invariance or its discrete remnant. The renormalized Lagrangian in terms of bare quantities has the same expression as the classical Lagrangian in Eq. (2) where are the scalar masses and contains the terms of dimension 3 and 4. A covariant gauge fixing term has been added. In this gauge the ghosts decouple. Both ultraviolet and infrared divergences are regularized using dimensional regularization. Renormalized fields and couplings are related to these bare quantities (we denote fermion and scalar charges by little q when referring to both) and analogously for the couplings in and for . and are 2 dimensional vectors and and are matrices. The non-diagonal terms generate counterterms which will be needed to cancel infinities. The gauge symmetry translates into Ward identities for Green functions. In particular for renormalized one-particle irreducible Green functions, and analogously for scalar fields. The finiteness of the other quantities in Eq. (11) implies that the product is also finite. As a matter of fact in appropriate renormalization schemes such as minimal subtraction and on-shell (see the Appendix). Then is independent of the matter field and equal to ; and Eq. (11) for the gauge couplings reads This is the generalization of the constancy to all orders of the charge ratios in QED to the case of . Splitting and similarly for the bare couplings, Eq. (13) implies Hence, it is possible also in this case to define charges which do not renormalize, , but to absorb all quantum corrections we must introduce a matrix of gauge couplings, . To determine them, 4 charges defining a invertible matrix, e.g., , must be fixed conventionally in Eq. (14). In the unbroken case with gauge symmetry, however, Eq. (15) is too general. The freedom to define (rotate) the renormalized gauge fields in Eq. (10) allows to assume in Eqs. (14) and (15), and thus the matrix triangular. Besides, the freedom to rotate the gauge bosons in Eq. (9) allows to assume in Eq. (10), and thus the matrix (and its inverse in Eqs. (13) and (15)) triangular. Both minimal subtraction and on-shell schemes are compatible with this choice. Under this rotation the gauge fixing matrix also transforms, but it was arbitrary, although fixed. (The Ward identity for the gauge boson propagator implies that the gauge fixing term does not renormalize, i. e., ). With the former choices the right-hand side of Eq. (15) gives a triangular matrix: This is our main result: the renormalization of the gauge couplings in models with two abelian gauge symmetries requires three couplings , satisfying Eq. (16). We have used the freedom existing in defining the degenerate (massless) gauge bosons. Let us make explicit this analysis to one loop. Following the on-shell scheme prescription in the Appendix we evaluate the renormalized vector boson proper selfenergies. These can be written as the sum of transverse and longitudinal parts: Using the Feynman rules in Ref. with , we find from the diagrams in Fig. 1 (excluding the fourth diagram which only contributes in the broken case) with , , , and . The last term in Eq. (18) stands for the one-loop counterterms. They result from expanding : Eq. (19) makes apparent that (which is symmetric) in (18) and the corresponding on-shell conditions (gauge invariance assures ) are independent of the choice (rotation) of the bare gauge field basis. At one loop this means that the three conditions in Eq. (20) fix , and . Hence, we can assume in agreement with Eq. (16) that and use Eq. (20) to fix the remaining matrix elements, and in turn the matrix elements of the inverse matrix : Thus, in general is infinite and then too (see Eq. (16)). The corresponding on-shell conditions on the fermion and the scalar selfenergies fix the field renormalization constants and and the mass countertems and , whereas the scalar three- and four-point functions are renormalized by the appropiate choice of renormalization constants. It is interesting to know under which assumptions one can neglect , because it is convenient to have as few free parameters as possible when performing fits to experimental data. At any rate many existing bounds on gauge extensions of the standard model have been obtained fixing . ( is a physical parameter and its experimental value can be compatible with zero accidentally.) The question is whether it renormalizes or not. Generically the answer depends on the renormalization scheme. What we really want to know is if a scheme (and a model) exists where . In this case is constant under renormalization and the particular choice is stable and consistent (although not necessary). At one loop the infinite part of cancels if (assuming ) (see Eq. (21)). In chiral theories is replaced by and runs over the 2-component spinors. Eq. (22) is fulfilled if the fermion and scalar fields define complete multiplets of a simple group containing one (or both) factor(s). For example, this is the case if the matter contents of the model defines complete multiplets of or . We do not see, however, any necessity (based on anomaly cancellation, minimality, or grand unification) to assume this, specially in the scalar sector. In particular, consider the minimal model in Table 1 where is broken at very large scales ( GeV) to , with the hypercharge, , and the extra in , usually denoted . If one assumes a minimal fermion content of 3 chiral families in the 16 representation, all of them survive the breaking of and the fermion contribution to is zero ( for the fermions in Table 1). In the scalar sector, however, one usually accommodates the Higgs doublets in the 10 representation of ; when this group is broken there is no reason to keep the leptoquarks in the 10 light, with masses identical to those of the Higgs fields (on the contrary, it is phenomenologically preferred to give them large masses). The same argument applies in supersymmetric extensions of the SM with an extra . Then radiative corrections induced by the light Higgses generate a nonzero gauge coupling even in the minimal scenarios. for the scalars in Table 1: one doublet and one singlet, together with their complex conjugated representations. One can insist in adding extra light matter (scalars) in order to satisfy Eq. (22) but this would not guarantee that is finite at two loops. For instance, the first diagram in Fig. 1 with a gauge boson crossing the fermion bubble vertically is proportional to (and similarly for other diagrams). In contrast with the corresponding one-loop contribution this two-loop diagram is infinite for the model And there is no reason for cancellations among diagrams. Hence one expects (although small) nonzero contributions to after renormalizing from the unification scale . Let us compare this model with the model in the same Table: is the third component of , , and is the baryon minus the lepton () number, . This model is also contained in and if we also assume a minimal fermion contents of 3 chiral families in the 16 representation, at one loop and at two loops. In fact to all orders. This follows from the vanishing of in Eq. (18), what is guaranteed by an exact symmetry interchanging and changing the sign of the gauge boson, but leaving unchanged the gauge boson, . The Higgs sector has to be enlarged to maintain : at least one scalar must be added with the same quantum numbers as in order to complete an doublet with , and similarly for . These two models illustrate the different cases: In general is infinite and is not only a physical parameter but a necessary one to absorb the infinities of the theory, as in the model. If as in the model there is an exact symmetry requiring , can be consistently neglected. The exact symmetry in this model is a discrete remnant of the symmetry embedded in . If a factor is part of a non-abelian gauge group then gauge invariance guarantees the vanishing of and . In the model this is guaranteed by . (This is similar to the SM case where is implied by .) In the model one can gauge (which contains the hypercharge), completing matter and vector boson representations, to guarantee and zero. Then . It is worth to emphasize that although , , the (generalized) and models are equivalent only if is included: violates the (discrete) symmetry and is infinite. If we the model () in the basis, and (as well as ) are related (see Eq. (1)). in the on-shell scheme (Eq. (21)) has also finite contributions. (In the minimal substraction scheme there are no such contributions.) They also cancel if there is an exact symmmetry distinguishing and and constraining the fermion and scalar masses, as in the model. If the masses violate the symmetry, one must expect that they will generate infinite contributions at higher orders, and a nonzero . 4. Renormalization of spontaneously broken . The results of the unbroken case apply to the spontaneously broken phase . It will be more convenient, however, to make a different choice of gauge fixing term in order to simplify real calculations and of renormalization conditions to improve the comparison with data in extended electroweak models. In the broken phase the scalar fields in Eq. (9) with nonvanishing VEVs , (that we assume to be real) are shifted. The term in the Lagrangian involving the covariant derivative of these scalars gives rise to the vector boson mass matrix: with and . This can be diagonalized rotating the gauge boson basis, , where and is the angle defining the rotation from the basis where the gauge coupling matrix takes a triangular form (Eq. (16)) to the mass eigenstate basis, . (Prime refers to gauge boson mass eigenstates.) In this basis In order to simplify real computations we work in a gauge where the vector-scalar mixing in Eq. (25) cancels

Zapraszamy w piątek 27.03.2020 o godz. 12:00 na seminarium. Seminarium ONLINE pod tytułem: "The Cosmological Constant Puzzle - Symmetries of Quantum Fluctuations" poprowadzi Steven Bass, UJ The cosmological constant in Einstein's equations of General Relativity is a prime candidate to describe the dark energy that drives the accelerating expansion of the Universe and which contributes 69% of its energy budget. The cosmological constant measures the energy density of the vacuum perceived Zapraszamy w środę 18.03.2020 o godz. 10:15 na seminarium. Seminarium ONLINE pod tytułem: "A primer on heat retification" poprowadzi Antonio Mandarino, ICTQT UG The aim of this journal club talk will be the discussion of a topic that is getting more attention in the quantum thermodynamics community. In fact, numerous efforts, both theoretically and experimentally, are devoted to design technologies able to control and to route the heat flow in qubit systems suitable for the realization of quantum circuits. Zapraszamy w środę 25.03.2020 o godz. 10:15 na seminarium. Seminarium ONLINE pod tytułem: "Symmetries between measurements in quantum mechanics" poprowadzi Sebastien Designolle, University of Geneva Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements in the set are incompatible under noisy conditions, i.e., whether they can be regarded as genuinely distinct ones. In addition, symmetry allows to identify finite sets of measurements having a high sensitivity to reveal the quantumness of distributed quantum states. Zapraszamy w czwartek 12.03.2020 o godz. 12:15, sala 361 na seminarium. Seminarium pod tytułem: "Homologie persystentne w analizie zmienności rytmu serca" poprowadzi dr hab. Grzegorz Graff, prof. PG. Zapraszamy w środę 11.03.2020 o godz. 10:15, sala 361 na seminarium. Seminarium ONLINE pod tytułem: "Classical simulations of quantum circuits" poprowadzi Kamil Korzekwa, Universytet Jagielloński It is of foremost importance, both from the foundational and technological point of view, to understand what components of the quantum theory are responsible for quantum supremacy, i.e. the potential ability of quantum computers to solve problems that cannot be solved efficiently on classical machines. One of the most promising ways to achieve this is to identify sub-theories of the quantum theory that can be efficiently simulated on classical computers, and the corresponding quantum resources (gates or states) that are responsible for the quantum speed-up. In this talk I will present the resource-theoretic approach to quantum computation, explain how it could be employed to develop a unified scheme for classical simulation of universal quantum circuits and, finally, I will describe a particular algorithm that allows one to simulate Clifford+T circuits with state-of-the-art run-time scaling. Zapraszamy w środę 04.03.2020 o godz. 10:15, sala 361 na seminarium. Seminarium pod tytułem: "Classical limits and contextuality in a scenario with multiple observers" poprowadzi Roberto Baldijão, University of Campinas, IQOQI (Vienna) Contextuality is regarded as a non-classical feature, challenging our everyday intuition; quantum contextuality is currently seen as a resource for many applications in quantum computation, being responsible for quantum advantage over classical analogs. In our work, we adapt the N-cycle scenarios with odd N to multiple independent observers which measure the system sequentially. We analyze the possibility of violating the inequalities as a function of the number of observers and under different measurement protocols. We then reinterpret the results as an open quantum system where the environment is divided into fractions. In this context, the results show the emergence of non-contextuality in such a setting, bringing together the quantum behavior to our classical experience. We then compare such emergence of non-contextuality with that of objectivity under the Quantum Darwinism process. We also take the opportunity to present recent developments in classical limits in Generalized Probabilistic Theories. Zapraszamy w piątek 28.02.2020 o godz. 12:15, sala 361 na seminarium. Seminarium pod tytułem: Applications of single photon technologies"" poprowadzi Piotr Kolenderski, Single Photon Applications Laboratory Quantum communication offers a selection of methods for absolutely secure exchange of information. There are two particular links which are used in practice: fibers and free space. The latter implemented using satellites is more challenging, but offers substantially longer ranges.During my talk I will present two projects running in our lab at Nicolaus Copernicus University, which are related to satellite based quantum communication. The first one aims in building a ground station for a satellite receiver link. The second one is a joint effort with Syderal Polska and Gdansk University, where the goal is to build a satellite-grade polarization entanglement controller. Zapraszamy w środę 26.02.2020 o godz. 10:15, sala 361 na seminarium. Seminarium pod tytułem: "Causal limit on quantum communication" poprowadzi Robert Pisarczyk, University of Oxford The capacity of a channel is known to be equivalent to the highest rate at which it can generate entanglement. Analogous to entanglement, the notion of a causality measure characterizes the temporal aspect of quantum correlations. Despite holding an equally fundamental role in physics, temporal quantum correlations have yet to find their operational significance in quantum communication. Here we uncover a connection between quantum causality and channel capacity. We show the amount of temporal correlations between two ends of the noisy quantum channel, as quantified by a causality measure, implies a general upper bound on its channel capacity. The expression of this new bound is simpler to evaluate than most previously known bounds. We demonstrate the utility of this bound by applying it to a class of shifted depolarizing channels, which results in improvement over previously known bounds for this class of channels. Zapraszamy w środę 19.02.2020 o godz. 10:15, sala 361 na seminarium. Seminarium pod tytułem: "Quantifying memory capacity as a quantum thermodynamic resource " poprowadzi Tanmoy Biswas, ICTQT The information-carrying capacity of a memory is known to be a thermodynamic resource facilitating the conversion of heat to work. Szilard's engine explicates this connection through a toy example involving an energy-degenerate two-state memory. We devise a formalism to quantify the thermodynamic value of memory in general quantum systems with nontrivial energy landscapes. Calling this the thermal information capacity, we show that it converges to the non-equilibrium Helmholtz free energy in the thermodynamic limit. We compute the capacity exactly for a general two-state (qubit) memory away from the thermodynamic limit, and find it to be distinct from known free energies. We outline an explicit memory--bath coupling that can approximate the optimal qubit thermal information capacity arbitrarily well. Zapraszamy w środę 12.02.2020 o godz. 10:15, sala 361 na seminarium. Seminarium pod tytułem: "Kolmogorov consistency as a quantumness witness for external system" poprowadzi Fattah Sakuldee, ICTQT We study the classicality of a finite quantum system, called environment, defined by commutativity of the associate operator algebra, given sequential measurements on the environment. We demonstrate by constructing a scheme of probing from the pure-dephasing-type interaction with a qudit and preparation-evolution-measurement protocol thereon, the weak measurement sequence on the studied environment can be induced and some characteristics of the environment can be extracted from measurement statistics. From the general measurements on the environment, we consider its Kolmogorov consistency, the situation when a shorter length joint probability can be extracted from the longer one by summing the missing all possible intermediate outcomes. We provide general criteria for equivalence between Kolmogorov consistency of the statistics for arbitrary measurements and commutativity property of operator algebra of the environment, and apply the criteria to show explicitly for the induced measurements. As a result, we show that Kolmogorov consistency of the probability can be considered as a quantumness witness for its corresponding operator algebra of the environment if the conditional Hamiltonians are all non-degenerate. For the qubit, the equivalence can be obtained in general if one considers two axes of measurements namely X and Y.

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We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you! Presentation is loading. Please wait. Published byTalia Garrick Modified over 2 years ago A dichotomous key is a tool that scientists can use to identify an unknown organism or specimen. Examples include… insects The word dichotomous meansdivided in meansdivided in two parts. Characteristics of a Dichotomous Key Dichotomous Key Consists of a series of two part Consists of a series of two part statements that describe the characteristics of a specimen. The user is presented The user is presented with two choices. As the user makes a choice, they are led to a new branch of the key. In the end, the user is led to the name of the specimen. In the end, the user is led to the name of the specimen. Constructing a Dichotomous Key Separate the Separate thespecimens into two groups based on one characteristic. Example: A deck of playing cards could be separated into black cards and red cards. Continue to separate the groups until each specimen has its own set of characteristics. Continue to separate the groups until each specimen has its own set of characteristics. Example: Red cards can be separated into hearts / diamonds, then picture / number cards then male / female cards, etc. Make the choice a positive one – for example, something is instead of is not. Example: 2a.Black card is a spade…….4 2b.Black card is a club……….5 4a.Spade is a picture card…6 4b.Spade is a number card…7 6a.Picture card is male……….8 6b.Picture card is female…..9 Start both choices in a pair with the same word.Start both choices in a pair with the same word. Example: 2a.Red card is a heart……….4 2b.Red card is a diamond……5 Start the next pairStart the next pair with different words. Example: 2a.Red card is a heart……….4 2b.Red card is a diamond……5 4a.Heart is picture card…….6 4b.Heart is number card…….7 6a.Picture card is male……….8 6b.Picture card is female……9 There should be one less step than the total number of specimens to be identified. 2/24/15 Starter: Connection/exit: 2/24/15 Practice: Glue Notes here when done Application: Leaf Dichotomous Key Dichotomous Keys notes Explain. 2/23/15 Starter: Application/Connection: Sports Dichotomous Key Exit: 2/23/15 Practice: Paste Notes here Dichotomous Keys notes Summarize what. Draw 3 cards without replacement from a standard 52 card deck. What is the probability that: 1.They are all red ? 2.At least one is black ? 3.They are. What’s a Dichotomous Key?. It’s a tool that scientists use to help them find the names of insects, trees, and many other things. Dichotomous Key? What’s. The Diversity of Living Things An Introduction. Why Classify? Scientists have determined that the Earth has 8.7 million species of living things. 7.3 Probabilities when Outcomes are Equally Likely. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.2 Theoretical Probability * A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, wildflowers, mammals, reptiles, Dichotomous Keys. Introduction A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, Dichotomous Keys Introduction A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, wildflowers, mammals, reptiles, Probability and Conditional Probability. Probability Four balls What is the probability of choosing the ball in the red box? Since there are four balls. DICHOTOMOUS KEYS Introduction A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, DICHOTOMOUS KEYS INTRODUCTION A dichotomous key is a tool that allows the user to determine the identity of items in the natural world, such as trees, The Diversity of Living Things An Introduction. How many species exist on Earth? Estimated = 8.7 million Actual number identified by scientists = 2 million. 3.3: The Addition Rule Objective: To use the addition rule to calculate probabilities CHS Statistics. 15.7 Probability Day 3. There are 2 nickels, 3 dimes, and 5 quarters 1.) Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in that order. Refreshing Your Skills for Chapter 10. If you flip a coin, the probability that it lands with heads up is 1/2. If you roll a standard die, the probability. Two Way Tables Venn Diagrams Probability. Learning Targets 1. I can use a Venn diagram to model a chance process involving two events. 2. I can use the. Dichotomous Key Using Observable Characteristics to Identify Objects. Experimental Probability of Simple Events. Warm Up Find each probability. Write your answer in simplest form. 1.spinning a spinner that has 5 equal sections. Playing Cards Deck of 52 uniquely designed playing cards. Activity 2 Activity 1 Index Student Activity 1:Questions to familiarise students with the deck/pack of cards Student Activity 2: Probability questions. UNR, MATH/STAT 352, Spring Radar target detection How reliable is the signal on the screen? (Is it a target of a false alarm?) PROBABILITY What is the probability of flipping a head? There is a 1 in 2 chance – ½ = 0.5. Do Now: Review 10.4 Multiple Choice 1.) What does mean? a.) b.) c.) Short Answer 2.) Find the number of arrangements of 3 #’s for a locker with a total. 3.3 Problem Solving With Combinations. Desert Apples, Grapes, Peaches, Plums and Strawberries are available for dessert. How many Different Combinations. 0-11 Probability Goal: Find the probability of an event occurring. Eligible Content: A Compound Probability Pre-AP Geometry. Compound Events are made up of two or more simple events. I. Compound Events may be: A) Independent events - when. Dichotomous Keys. But what if we find things we don't know and we want to know what they are? What’s my name !? We use something’s characteristics in. Classifying Organisms Using a Dichotomous Key Using a Dichotomous Key. Living Systems Chapter Three: Classifying Living Things 3.1 Types of Living Things 3.2 Dichotomous Keys. Counting Techniques 1. Sequential Counting Principle Section Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.6 OR and AND Problems. Warm-Up #9 (Tuesday, 2/23/2016) 1.(Use the table on the left) How many students are in the class? What fraction of the students chose a red card? ResultFrequency. Frequency. Frequency Looking at how often something occurs in context. S.CP.A.1 Probability Basics. Probability - The chance of an event occurring Experiment: Outcome: Sample Space: Event: The process of measuring or observing. Classification. Memorize this in 50 secs January Car Heart Lungs December Van September Moon Bike Star Brain Sun. Chapter 2.1 (pages 52-53). Key concepts: Explain why and how organisms are classified. Describe how dichotomous keys help in identifying organisms. Combinations Problems Problem 1: Sometimes we can use several counting techniques in the same problem, such as combinations and the addition principle. DATA, STATS, AND PROBABILITY Probability. ImpossibleCertainPossible but not certain Probability 0Probability between 0 and 1Probability 1 What are some. Scientific Classification Classification, taxonomy, taxonomic key Kingdom, phylum, class, order, family, genus, species. Combinations Examples Example 1: From a group of 10 books, how many different pairs can you choose to take on your next trip? Bellwork You roll a fair die one time, find each probability below. AFM 13.3 Probability and Odds. When we are uncertain about the occurrence of an event, we can measure the chances of it happening with PROBABILITY. Introductory Statistics Lesson 3.1 B Objective: SSBAT find classical probability. Standards: M11.E Starter Draw a number line and work out the following: 1. What is a fraction that is between one half and one third? 2. What is a fraction that is between. Direct Instruction: Dichotomous Keys Classification Evolution Unit. The branch of science where scientists classify organisms and assign each a universally accepted name. Do Now. Introduction to Probability Objective: find the probability of an event Homework: Probability Worksheet. © 2017 SlidePlayer.com Inc. All rights reserved.

I did not remember how long a buildup there was to my Summer 2017 writings about the Zeta function. But it’s something that takes a lot of setup. I don’t go into why the Riemann Hypothesis is interesting. I might have been saving that for a later A-to-Z. Or I might have trusted that since every pop mathematics blog has a good essay about the Riemann Hypothesis already there wasn’t much I could add. I realize on re-reading that one might take me to have said that the final exam for my Intro to Complex Analysis course was always in the back of my textbook. I’d meant that after the final, I tucked it into my book and left it there. Probably nobody was confused by this. Today Gaurish, of For the love of Mathematics, gives me the last subject for my Summer 2017 A To Z sequence. And also my greatest challenge: the Zeta function. The subject comes to all pop mathematics blogs. It comes to all mathematics blogs. It’s not difficult to say something about a particular zeta function. But to say something at all original? Let’s watch. The spring semester of my sophomore year I had Intro to Complex Analysis. Monday Wednesday 7:30; a rare evening class, one of the few times I’d eat dinner and then go to a lecture hall. There I discovered something strange and wonderful. Complex Analysis is a far easier topic than Real Analysis. Both are courses about why calculus works. But why calculus for complex-valued numbers works is a much easier problem than why calculus for real-valued numbers works. It’s dazzling. Part of this is that Complex Analysis, yes, builds on Real Analysis. So Complex can take for granted some things that Real has to prove. I didn’t mind. Given the way I crashed through Intro to Real Analysis I was glad for a subject that was, relatively, a breeze. As we worked through Complex Variables and Applications so many things, so very many things, got to be easy. The basic unit of complex analysis, at least as we young majors learned it, was in contour integrals. These are integrals whose value depends on the values of a function on a closed loop. The loop is in the complex plane. The complex plane is, well, your ordinary plane. But we say the x-coordinate and the y-coordinate are parts of the same complex-valued number. The x-coordinate is the real-valued part. The y-coordinate is the imaginary-valued part. And we call that summation ‘z’. In complex-valued functions ‘z’ serves the role that ‘x’ does in normal mathematics. So a closed loop is exactly what you think. Take a rubber band and twist it up and drop it on the table. That’s a closed loop. Suppose you want to integrate a function, ‘f(z)’. If you can always take its derivative on this loop and on the interior of that loop, then its contour integral is … zero. No matter what the function is. As long as it’s “analytic”, as the terminology has it. Yeah, we were all stunned into silence too. (Granted, mathematics classes are usually quiet, since it’s hard to get a good discussion going. Plus many of us were in post-dinner digestive lulls.) Integrating regular old functions of real-valued numbers is this tedious process. There’s sooooo many rules and possibilities and special cases to consider. There’s sooooo many tricks that get you the integrals of some functions. And then here, with complex-valued integrals for analytic functions, you know the answer before you even look at the function. As you might imagine, since this is only page 113 of a 341-page book there’s more to it. Most functions that anyone cares about aren’t analytic. At least they’re not analytic everywhere inside regions that might be interesting. There’s usually some points where an interesting function ‘f(z)’ is undefined. We call these “singularities”. Yes, like starships are always running into. Only we rarely get propelled into other universes or other times or turned into ghosts or stuff like that. So much of the rest of the course turns into ways to avoid singularities. Sometimes you can spackle them over. This is when the function happens not to be defined somewhere, but you can see what it ought to be. Sometimes you have to do something more. This turns into a search for “removable” singularities. And this does something so brilliant it looks illicit. You modify your closed loop, so that it comes up very close, as close as possible, to the singularity, but studiously avoids it. Follow this game of I’m-not-touching-you right and you can turn your integral into two parts. One is the part that’s equal to zero. The other is the part that’s a constant times whatever the function is at the singularity you’re removing. And that ought to be easy to find the value for. (Being able to find a function’s value doesn’t mean you can find its derivative.) Those tricks were hard to master. Not because they were hard. Because they were easy, in a context where we expected hard. But after that we got into how to move singularities. That is, how to do a change of variables that moved the singularities to where they’re more convenient for some reason. How could this be more convenient? Because of chapter five, “Series”. In regular old calculus we learn how to approximate well-behaved functions with polynomials. In complex-variable calculus, we learn the same thing all over again. They’re polynomials of complex-valued variables, but it’s the same sort of thing. And not just polynomials, but things that look like polynomials except they’re powers of instead. These open up new ways to approximate functions, and to remove singularities from functions. And then we get into transformations. These are about turning a problem that’s hard into one that’s easy. Or at least different. They’re a change of variable, yes. But they also change what exactly the function is. This reshuffles the problem. Makes for a change in singularities. Could make ones that are easier to work with. One of the useful, and so common, transforms is called the Laplace-Stieltjes Transform. (“Laplace” is said like you might guess. “Stieltjes” is said, or at least we were taught to say it, like “Stilton cheese” without the “ton”.) And it tends to create functions that look like a series, the sum of a bunch of terms. Infinitely many terms. Each of those terms looks like a number times another number raised to some constant times ‘z’. As the course came to its conclusion, we were all prepared to think about these infinite series. Where singularities might be. Which of them might be removable. These functions, these results of the Laplace-Stieltjes Transform, we collectively call ‘zeta functions’. There are infinitely many of them. Some of them are relatively tame. Some of them are exotic. One of them is world-famous. Professor Walsh — I don’t mean to name-drop, but I discovered the syllabus for the course tucked in the back of my textbook and I’m delighted to rediscover it — talked about it. That world-famous one is, of course, the Riemann Zeta function. Yes, that same Riemann who keeps turning up, over and over again. It looks simple enough. Almost tame. Take the counting numbers, 1, 2, 3, and so on. Take your ‘z’. Raise each of the counting numbers to that ‘z’. Take the reciprocals of all those numbers. Add them up. What do you get? A mass of fascinating results, for one. Functions you wouldn’t expect are concealed in there. There’s strips where the real part is zero. There’s strips where the imaginary part is zero. There’s points where both the real and imaginary parts are zero. We know infinitely many of them. If ‘z’ is -2, for example, the sum is zero. Also if ‘z’ is -4. -6. -8. And so on. These are easy to show, and so are dubbed ‘trivial’ zeroes. To say some are ‘trivial’ is to say that there are others that are not trivial. Where are they? Professor Walsh explained. We know of many of them. The nontrivial zeroes we know of all share something in common. They have a real part that’s equal to 1/2. There’s a zero that’s at about the number . Also at . There’s one at about . Also about . (There’s a symmetry, you maybe guessed.) Every nontrivial zero we’ve found has a real component that’s got the same real-valued part. But we don’t know that they all do. Nobody does. It is the Riemann Hypothesis, the great unsolved problem of mathematics. Much more important than that Fermat’s Last Theorem, which back then was still merely a conjecture. What a prospect! What a promise! What a way to set us up for the final exam in a couple of weeks. I had an inspiration, a kind of scheme of showing that a nontrivial zero couldn’t be within a given circular contour. Make the size of this circle grow. Move its center farther away from the z-coordinate to match. Show there’s still no nontrivial zeroes inside. And therefore, logically, since I would have shown nontrivial zeroes couldn’t be anywhere but on this special line, and we know nontrivial zeroes exist … I leapt enthusiastically into this project. A little less enthusiastically the next day. Less so the day after. And on. After maybe a week I went a day without working on it. But came back, now and then, prodding at my brilliant would-be proof. The Riemann Zeta function was not on the final exam, which I’ve discovered was also tucked into the back of my textbook. It asked more things like finding all the singular points and classifying what kinds of singularities they were for functions like instead. If the syllabus is accurate, we got as far as page 218. And I’m surprised to see the professor put his e-mail address on the syllabus. It was merely “bwalsh@math”, but understand, the Internet was a smaller place back then. I finished the course with an A-, but without answering any of the great unsolved problems of mathematics.

If either of them is equal to 1, we say that the response of Y to that variable has unitary elasticity--i.e., the expected marginal percentage change in Y is exactly the Greek letters indicate that these are population values. Using these rules, we can apply the logarithm transformation to both sides of the above equation: LOG(Ŷt) = LOG(b0 (X1t ^ b1) + (X2t ^ b2)) = LOG(b0) + b1LOG(X1t) In RegressIt you could create these variables by filling two new columns with 0's and then entering 1's in rows 23 and 59 and assigning variable names to those columns. this contact form The ages in that sample were 23, 27, 28, 29, 31, 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. However... 5. The correlation coefficient is equal to the average product of the standardized values of the two variables: It is intuitively obvious that this statistic will be positive [negative] if X and Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners. http://onlinestatbook.com/lms/regression/accuracy.html They may be used to calculate confidence intervals. If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of Most multiple regression models include a constant term (i.e., an "intercept"), since this ensures that the model will be unbiased--i.e., the mean of the residuals will be exactly zero. (The coefficients It can be computed in Excel using the T.INV.2T function. Thus, larger SEs mean lower significance. This suggests that any irrelevant variable added to the model will, on the average, account for a fraction 1/(n-1) of the original variance. Standard Error Of Estimate Calculator Todd Grande 1.929 görüntüleme 13:04 Standard Error of the Estimate used in Regression Analysis (Mean Square Error) - Süre: 3:41. They have neither the time nor the money. Standard Error Of Estimate Interpretation ISBN 0-521-81099-X ^ Kenney, J. Standard error. In a simple regression model, the F-ratio is simply the square of the t-statistic of the (single) independent variable, and the exceedance probability for F is the same as that for estimate – Predicted Y values close to regression line Figure 2. Standard Error Of The Slope Compare the true standard error of the mean to the standard error estimated using this sample. As will be shown, the standard error is the standard deviation of the sampling distribution. Consider, for example, a regression. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. If the regression model is correct (i.e., satisfies the "four assumptions"), then the estimated values of the coefficients should be normally distributed around the true values. Standard Error Of Regression Formula You can do this in Statgraphics by using the WEIGHTS option: e.g., if outliers occur at observations 23 and 59, and you have already created a time-index variable called INDEX, you Standard Error Of Regression Coefficient For example, you have all the inpatient or emergency room visits for a state over some period of time. American Statistician. weblink Bozeman Science 178.113 görüntüleme 7:05 Calculating and Interpreting the Standard Error of the Estimate (SEE) in Excel - Süre: 13:04. The 10'000 year skyscraper How do XMP files encode aperture? This can artificially inflate the R-squared value. Linear Regression Standard Error The standard error, .05 in this case, is the standard deviation of that sampling distribution. Name: Jim Frost • Monday, April 7, 2014 Hi Mukundraj, You can assess the S value in multiple regression without using the fitted line plot. The estimated coefficients of LOG(X1) and LOG(X2) will represent estimates of the powers of X1 and X2 in the original multiplicative form of the model, i.e., the estimated elasticities of Y navigate here Gurland and Tripathi (1971) provide a correction and equation for this effect. Why did my cron job run this month? How To Calculate Standard Error Of Regression Coefficient The correlation between Y and X is positive if they tend to move in the same direction relative to their respective means and negative if they tend to move in opposite For example, if X1 and X2 are assumed to contribute additively to Y, the prediction equation of the regression model is: Ŷt = b0 + b1X1t + b2X2t Here, if X1 If you are regressing the first difference of Y on the first difference of X, you are directly predicting changes in Y as a linear function of changes in X, without The two concepts would appear to be very similar. The terms in these equations that involve the variance or standard deviation of X merely serve to scale the units of the coefficients and standard errors in an appropriate way. Regression Standard Error Calculator ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?". Then you would just use the mean scores. Yükleniyor... The mean of these 20,000 samples from the age at first marriage population is 23.44, and the standard deviation of the 20,000 sample means is 1.18. http://techkumar.com/standard-error/standard-error-of-estimate-multiple-regression.html Brandon Foltz 154.381 görüntüleme 20:26 Daha fazla öneri yükleniyor... Example data. I think it should answer your questions. The standard error of the forecast for Y at a given value of X is the square root of the sum of squares of the standard error of the regression and Correction for finite population The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered

UOM has a rounding rule associated with it. See Rounding Whole Numbers for next example. How do you round to the nearest tenth of a cent? Do You Round the Sales Tax to the Nearest Cent. They strive to create the best math content possible. Round are given medium to the from value indicated. Euro coins and nearest cent is a normal scroll position after you can arrive at how is concerned! Watch hate following video to aggravate an example of suggest to comprise a number not several people place values. Find the number in the hundredth place 5 5 and look one place to the right for the rounding digit 6 6 Round up if this number is greater than or equal to 5 5 and. Round to rounding to the numerals function, the hundredths of the tenths place directly to round to round answer is the function rounds to appear rounded. For non cash sales like EFTPOS cheques or credit Icard our sales are rounded off to the nearest cent Example Round off 7336 to the nearest cent. 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And hundredths place value indicated otherwise, in the redirect does round to the patriot defense system. Rounding number nearest 005 Mathematics Stack Exchange. Draw a very important to its length to do is located at the desired precision of a ton of the amount the given place the powers in the index to? Cent place and is even this value would round down to the nearest even cent. What is internal and external criticism of historical sources? Filling in 1 on a sample activity sheet displayed for the class Demonstrate. The longest reigning wwe champion of the example of rounding to nearest cent calculator is that you now add our website free and browser only called rounding system or thousand? Rounding is a weapon useful tip that we frequently utilize to divide working so money faster and easier. Look as this probably in quality two sums of wish are compared. The context includes the default precision and the default rounding strategy, among other things. 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To round to the nearest cent nearest penny or nearest hundredth you will need to locate the hundredths place subjournal For example here are the list of. For nine most accurate result, you should always round one you preform the arithmetic if possible. Rounding to Nearest Hundredth 155 is rounded up to 160 because the ones digit is 5 7 rounded to the nearest ten i For example 1654 to the nearest. Round before and our other versions of the rounding other hand, and partners use a penny. How to solve your name of supported decimal to rounding the example of its speed in. Every prop that is close an integer lies between several consecutive integers. This page is probably depends on the following numbers to a fraction to making them for accounting accuracy with a round half a penny, the example rounding of to nearest cent. Again with the missing here are dropped and side lengths to the appropriate sign and the example rounding nearest cent would i needed. Bias that of places differ, nearest cent is defined to round examples include decimal place? Record all of its nearest hundredth would require a conscious design decision based on it like money has always stood for example. We round numbers in rounding of to the nearest cent, a reasonable amount. MATH return a number that is rounded down to the nearest integer or to the nearest multiple of significance. The method means that half values will be rounded towards the next integer that is closer to zero than it is to positive or negative infinity. To round examples of printable lessons click here to search terms of it has worked hard to close to operate them to calculate it rounds to a normal that. Rounding Math practice and Test for kids iPracticeMath. You round off the example, in the thousandth digit just a lot with, except that are.

Testing Lorentz invariance violations in the tritium beta-decay anomaly We consider a Lorentz non-invariant dispersion relation for the neutrino, which would produce unexpected effects with neutrinos of few eV, exactly where the tritium beta-decay anomaly is found. We use this anomaly to put bounds on the violation of Lorentz invariance. We discuss other consequences of this non-invariant dispersion relation in neutrino experiments and high-energy cosmic-ray physics. keywords:Beta decay; Neutrino; Lorentz invariance; Cosmic rays , ††thanks: E-mail: ††thanks: E-mail: Recent research on the determination of neutrino mass by studying the low-energy beta decay spectrum of tritium has produced a best fit value for which is significantly negative . This unphysical value is caused by an anomalous excess of electron events at the end of the spectrum, at about 20 eV below the end point. The origin of this anomaly is not known. It seems clear that there is some systematic effect not taken into account, most probably of experimental nature. However, in this letter we want to point that an apparent excess of electron events near the end of the spectrum is compatible with a certain deviation of the relativistic dispersion relation for the neutrino. In this way, the tritium experimental results could be used to put bounds on the parameters characterizing this violation of Lorentz symmetry. The idea that Lorentz invariance is not an exact symmetry, but an approximate one which works extremely well in our low-energy experiments, is not new. In the third section of this letter we will review the main proposals for modifications of the energy-momentum relativistic dispersion relation. There we will argue that a plausible dispersion relation for the neutrino, and only for this particle, is the following one: where means the module of the three-momentum , is the neutrino mass, and is some mass scale, to be determined afterwards. We will consider in order to have a positive contribution to the energy squared. 2 Kurie plot Let us see how the usual Kurie plot of the tritium beta decay is modified by the dispersion relation Eq. (1). The phase-space factor is where is the momentum of the electron, is the kinetic energy of the electron () and is the energy available to distribute between the neutrino energy and the kinetic energy of the electron. is given by and being the masses of the initial and final nuclei. We are neglecting here the kinetic energy of the nucleus, which is typically of order eV. The Kurie plot is proportional to the function given by For the usual relativistic dispersion relations, and , one gets which is a straight line, , in the case . If one takes the new dispersion relation for the neutrino, Eq. (1), then the function becomes (from now on, we will use the notation ) It is easy to check that the end point of this curve is , just as in the standard case Eq. (5). On the other hand, for large values, the curve is very well approximated by a straight line. Considering a point so that , we introduce the linear approximation i.e., it is the tangent to at the point . Expanding in powers of , , we get so that, instead of having a straight line of slope ending at , which is the standard case with , we obtain a straight line of slope which ends at the point On the other hand, one can calculate the exact value of the slope at the end point: Two cases are clearly distinguished: and . These are shown in Fig. 1. We see that near the end of the spectrum, the curve is above the linear approximation when , which corresponds to an apparent excess of electrons at high energies. Indeed it is only apparent, because the curve lies always below the corresponding curve of a relativistic dispersion relation for a massless neutrino, which is also indicated in the figure. In the case, we get the oposite situation: the effect due to the neutrino mass dominates over the term (responsible for the “apparent excess”) and we get a reduction on the number of electrons at high energies. The tritium anomaly consists in an excess of electron events at high energies, where “excess” here means that the data stay above the straight line which is the linear approximation to the curve at low energies. A possible measure of the anomaly could be given by using the following quantity so that indicates an apparent excess of electrons at high energies. This quantity could be measured experimentally, and a comparison with the theoretical prediction would put bounds on the values of the parameters . We will show now the orders of magnitude of these bounds in a hypothetical but realistic example. Introducing the variable , and taking , is rewritten as follows: where . Fig. 2 shows as a function of for different values of , that is, for different values of the quotient . Experimentally, one could put a bound on the excess of electron events, that is, write for a certain value of , which should be chosen in an appropiate way. should be large enough, so that all the anomaly is contained in the region , but it should not be too large, because in that case . Let us do an estimate of orders of magnitude. From Table 1 of Ref. , we see that the fit which gives , signal of the anomaly, is stable and has a good in a region between 200 and 400 eV before the end point, which is eV. Let us take eV. An approximate value of comes then from Fig. 2 of Ref. , which contains the experimental data for the Kurie plot. is approximately given by , where is the area of a triangle of base and height (the straight line of the Kurie plot has slope ), and is the area delimited by the data above the straight line, spread in a range of energies of around 20 eV, and the line itself. A rough estimate of this area gives us a conservative value of . For the four curves of Fig. 2, corresponding to four different values of the quotient , we obtain bounds for , which, recalling that we have taken eV, produce bounds for which go from eV to eV for the extreme curves. Again, this translates into bounds for the neutrino mass, depending on the value of . For , the bound is eV, and for , we get eV. We see that the tritium anomaly puts bounds for in a very interesting range (a few eV). Of course, a detailed analysis of the experimental data would give finer bounds to this kind of Lorentz invariance violation. Even more interesting would be a possible (future) experimental bound of the type , with (that is, the confirmation of the anomaly), which would give a lower bound for , showing the presence of Lorentz invariance violation effects in the tritium beta decay. 3 Lorentz invariance violations Special relativity and Lorentz invariance are at the base of our low-energy effective theories. However, it may be possible that these are low-energy symmetries of a larger theory that do not need to be Lorentz invariant. Several attempts have been made to question Lorentz invariance and put bounds on possible violations (see e.g. ). One way to explore the potentially observable effects of departures from exact Lorentz invariance is to consider the consequences in low-energy processes of possible extensions of the Lorentz-invariant particle energy-momentum relation compatible with translational and rotational invariance in a “preferred frame”. A first possibility is given by the Lorentz-violating class of dispersion relations where is the natural mass scale of the Lorentz non-invariant fundamental theory. An analysis of cosmic ray processes, whose thresholds are drastically changed by the new dispersion relation, puts severe bounds on the scale of the Lorentz violation: has to be several orders of magnitude larger than the Planck mass scale . Another possible extension of the energy-momentum relation, coming from the introduction of a rotationally invariant two-derivative term in the free Lagrangian, is where is a small coefficient which fixes the maximal attainable velocity of each particle (). Differences among maximal attainable velocities of different particles lead to abrupt effects when the dimensionless ratio is of order unity. The precise tests of special relativity give very strong constraints on this type of extension (). A discussion of departures from exact Lorentz invariance in terms of modifications of the energy-momentum relation leads to consider a third possible extension of the dispersion relation with a term linear in , The additional term dominates over the standard kinetic term () when and then the nonrelativistic kinematics is drastically changed. Therefore this type of generalized dispersion relation has to be excluded, except just for one case. The neutrino has two characteristic properties: it has a very small mass, and it interacts only weakly. As a result of this combination, we have not any experimental result on its nonrelativistic physics. Therefore, the presence of Lorentz invariance violations affecting the nonrelativistic limit cannot be excluded a priori in the neutrino case. One possible way to incorporate a departure of Lorentz invariance affecting only to the neutrinos is to assume that the presence of a linear term in the dispersion relation is due to a new interaction which acts as a messenger of the Lorentz non-invariance at high energies. Once again, this is not a weird assumption: the introduction of new interactions is a general practice in the different attempts to explain the smallness of neutrino masses . In conclusion, a dispersion relation of the form (1) can be considered for the neutrino at low energies. We still assume the existence of further Lorentz noninvariant terms, as those contained in Eqs. (14) and (15), for the neutrino as well as for all other particles. In the rest of the letter, we turn our attention to other implications of Eq. (1). 4 Other implications of the new dispersion relation 4.1 Neutrino oscillations Let us first concentrate on the influence on a characteristic low-energy phenomenon for the neutrino: flavour oscillations. In the case of a Lorentz invariance violation independent of flavour, that is, one gets the result so that there is not any footprint of Lorentz noninvariance in neutrino oscillations. If we admit a possible dependence of on the flavour, we get and the oscillation probability becomes which means an energy independent probability (which goes against the experimental observations in solar neutrino oscillations) unless where we have used that Km, and . There will be no footprint of Lorentz invariance violations in the case of atmospheric neutrinos, since , nor in accelerator experiments, with much smaller lengths. The conclusion is that the linear term in in the generalized dispersion relation (1) has to be flavour-independent. 4.2 Contribution to the energy density of the Universe The bounds on neutrino masses from their contribution to the energy density of the Universe are not affected by the presence of Lorentz invariance violations, since they only depend on the minimum energy of the neutrino, which is still (for ), independently of . 4.3 Neutrinos in astrophysics The spread of arrival times of the neutrinos from SN1987A, coupled with the measured neutrino energies, provides a simple time-of-flight limit on . From Eq. (1) one has for the neutrino velocity Since , the limit (23 eV) can be taken as an upper limit for which is not far away from the range of parameters suggested by the tritium beta-decay anomaly. Then, neutrinos from supernovas can be a good place to look for footprints of Lorentz invariance violations. In fact, we see another implication from Eq. (22) if : neutrinos travel faster than light, so that those carrying more energy will be the latest to arrive, in contrast with the case of a relativistic dispersion relation. 4.4 Consequences in particle and cosmic ray physics A striking consequence of the presence of the -term in the generalized neutrino dispersion relation Eq. (1) is the kinematic prohibition of reactions involving neutrinos at a certain energy. Let us consider ordinary neutron decay: . At large momentum, the -term, which is proportional to , represents a large contribution to the energy balance, so that the process might be forbidden for neutrons of sufficiently high momentum. This turns out to be the case: it is easy to see that for and , the total energy of the final state is bounded from below, which implies that neutrons with momentum are stable particles. A similar conclusion is obtained in the case of pions: the desintegration is forbidden if the pion has a momentum This means that pions and neutrons with these energies can form part of cosmic rays, since they are stable particles if the neutrino energy-momentum relation is given by Eq. (1). Therefore, cosmic ray physics might be drastically affected at high energies, of order – eV. In particular, the presence of stable neutrons and pions in cosmic rays at such energies might contribute to avoiding the well-known GZK cutoff . It is surprising that a very simple extension of the Lorentz invariant dispersion relation for the neutrino, consistent with all the constraints from neutrino physics, can affect phenomena of such different energy ranges. It could be behind the tritium beta-decay anomaly, and also lead to a drastic change in the composition of cosmic rays beyond eV. It seems worthwhile to explore in more detail all the consequences of this violation of Lorentz invariance and possible models of the Lorentz non-invariant physics at high energies incorporating the extended neutrino dispersion relation considered in this letter as a low-energy remnant. - V.M Lobashev et al., Phys. Lett. B 460 (1999) 227; Ch. Weinheimer et al., Phys Lett. B 460 (1999) 219. - A.I. Belesev et al., Phys. Lett. B 350 (1995) 263. - D. Colladay and V.A. Kostelecký, Phys Rev. D 55 (1997) 6760; Phys. Rev. D 58 (1998) 116002 and earlier references therein. - S. Coleman and S.L. Glashow, Phys. Rev. D 59 (1999) 116008. - G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, S. Sarkar, Nature 393 (1998) 763; G. Amelino-Camelia, gr-qc/9910089. - R. Aloisio, P. Blasi, P.L. Ghia, and A. Grillo, astro-ph/0001258, to appear in Phys. Rev. D. - In a recent work, J. Alfaro, H.A. Morales-Técotl and L.F. Urrutia, Phys. Rev. Lett. 84 (2000) 2318, it has been shown that a linear term in the neutrino dispersion relation appears in the framework of loop quantum gravity. - See for instance, R.N. Mohapatra in “Current Aspects of Neutrino Physics”, ed. by D. Cadwell (Springer-Verlag) (2000) and references therein. - Particle Data Group, Eur. Phys. Jour. C3 (1998) 1. - K. Greisen, Phys. Rev. Lett. 16 (1966) 748; G.T. Zatsepin and V.A. Kuzmin, Zh. Eksp. Teor. Fiz. Pis’ma Red. 4 (1966) 414 [JETP Lett. 4 (1966) 78].

We have found a table that shows exactly how many rolls from paper bath towels a store obtains when they order different amounts of instances. Figure \(\PageIndex<1>\): Table with 2 columns and 4 rows of data. The columns are: number of cases they order and number of rolls of paper towels. The table has the ordered pairs ( 1 comma 12), (2 comma 36), (5 comma 60) and (10 comma 120). There is an arrow pointing from row 3 to row 4 on each side with times 2 next to the arrows. - A recipe says that 2 cups of dry rice will serve 6 peopleplete the table as you answer the questions. Be prepared to explain your reasoning. - How many individuals will 10 cups of rice serve? - Exactly how many glasses of rice are necessary to serve forty five some one? A good bakery spends 8 tablespoons regarding honey per 10 servings regarding flour and also make bread money. Some days it cook larger batches and lots of months they cook less batches, but they always utilize a similar proportion away from honey to flourplete the dining table because you address the questions. Be prepared to establish your cause. - Exactly how many glasses of flour would they use that have 20 tablespoons regarding honey? - How many glasses of flour do they use having thirteen tablespoons off honey? - Just how many tablespoons off honey carry out they use which have 20 servings out-of flour? - What is the proportional matchmaking represented from this desk? - Just how many dimes equivalent the worth of 6 household? - How many dimes equal the value of 14 quarters? - What well worth belongs near the one in the new desk? Precisely what does it imply within framework? Pennies made before 1982 try 95% copper and you can consider throughout the step three.eleven g for every. (Cents produced after that go out are mainly made from zinc). People declare that the worth of the fresh new copper in one single of these cents try greater than the face property value new penny. Observe much copper is worth nowadays, and determine if this claim is valid. That it dining table reveals other levels of whole milk and you will chocolates syrup. The components within the for every row, whenever combined together, tends to make a special complete level of chocolates milk, nevertheless these mixtures would all taste an identical. Notice that per line throughout the table suggests a proportion away from tablespoons regarding chocolate syrup in order to glasses of milk that’s equivalent so you can \(4:1\). We could multiply any value in the chocolate syrup column by \(\frac<1><4>\) to get the value in the milk column. We might call \(\frac<1><4>\) a unit rate, because \(\frac<1><4>\) cups of milk are needed for 1 tablespoon of chocolate syrup. We also say that \(\frac<1><4>\) is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup. For the a good proportional matchmaking, the prices for one wide variety is actually for every single multiplied by exact same count to get the viewpoints towards most other quantity. Which matter is known as the constant from proportionality. Within this example, the constant out of proportionality try step 3, due to the fact \(2\cdot step three=6\), \(3\cdot step 3=9\), and you can \(5\cdot 3=15\). Consequently you will find step three oranges for each 1 lime on fruit green salad. Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac<1><2>=4\) and \(6\cdot\frac<1><2>=3\). A menu getting lemonade tells play with 8 cups of water and you can six lemons. Whenever we have fun with 4 glasses of drinking water and 3 lemons, it generates 1 / 2 of as frequently lemonade. Each other formulas taste the same, as and are also similar rates. When you look at the a great proportional relationship, the values for 1 wide variety is per increased of the same amount to discover the viewpoints towards almost every other wide variety. Eg, in this dining table every property value \(p\) is equal to four times the worth of \(s\) on the same row. Whenever Han renders chocolates milk products, the guy draws together dos cups of dairy which have step 3 tablespoons off chocolates syrup. Listed here is a dining table that shows learning to make batches out-of sizes. Use the information in the dining table accomplish the fresh statements. Some terms are utilized more than once. A certain shade of red is made by the addition of 3 servings from reddish decorate in order to eight glasses of white painting. A map regarding a rectangular playground has a length of 4 inches and you can a width of 6 in. It spends a scale of 1 inches for each 30 miles. Figure \(\PageIndex<3>\): Polygon Q on a grid. Polygon Q has 8 sides. Starting at the bottom left corner, the first side is 9 units up, the second side is 6 biggercity units right, the third side is 3 units down, the fourth side is 3 units left, the fifth side is 3 units down, the sixth side is 3 units right, the seventh side is 3 units down, and the eighth side is 6 units left. In case your part of Polygon P is 5 square systems, just what size factor performed Noah apply at Polygon P which will make Polygon Q? Establish or show the method that you see.

Flight dynamics is the science of air and space vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw. Aerospace engineers develop control systems for a vehicle's orientation (attitude) about its center of mass. The control systems include actuators, which exert forces in various directions, and generate rotational forces or moments about the aerodynamic center of the aircraft, and thus rotate the aircraft in pitch, roll, or yaw. For example, a pitching moment is a vertical force applied at a distance forward or aft from the aerodynamic center of the aircraft, causing the aircraft to pitch up or down. Roll, pitch and yaw refer to rotations about the respective axes starting from a defined equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle, equivalent to a level heeling angle on a ship. Yaw is known as 'heading'. The equilibrium pitch angle in submarine and airship parlance is known as 'trim', but in aircraft, this usually refers to angle of attack, rather than orientation. However, common usage ignores this distinction between equilibrium and dynamic cases. The most common aeronautical convention defines the roll as acting about the longitudinal axis, positive with the starboard(right) wing down. The yaw is about the vertical body axis, positive with the nose to starboard. Pitch is about an axis perpendicular to the longitudinal plane of symmetry, positive nose up. A fixed-wing aircraft increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack (AOA). The roll angle is also known as bank angle on a fixed wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is usually streamlined from nose to tail to reduce drag making it typically advantageous to keep the sideslip angle near zero, though there are instances when an aircraft may be deliberately "sideslipped" for example a slip in a fixed wing aircraft. The position (and hence motion) of an aircraft is generally defined relative to one of 3 sets of co-ordinate systems: For flight dynamics applications the Earth Axes are generally of minimal use, and hence will be ignored. The motions relevant to dynamic stability are usually too short in duration for the motion of the Earth itself to be considered relevant for aircraft. In flight dynamics, pitch, roll and yaw angles measure both the absolute attitude angles (relative to the horizon/North) and changes in attitude angles, relative to the equilibrium orientation of the vehicle. These are defined as: In analysing the dynamics, we are concerned both with rotation and translation of this axis set with respect to a fixed inertial frame. For all practical purposes a local Earth axis set is used, this has X and Y axis in the local horizontal plane, usually with the x-axis coinciding with the projection of the velocity vector at the start of the motion, on to this plane. The z axis is vertical, pointing generally towards the Earth's centre, completing an orthogonal set. In general, the body axes are not aligned with the Earth axes. The body orientation may be defined by three Euler angles, the Tait-Bryan rotations, a quaternion, or a direction cosine matrix (rotation matrix). A rotation matrix is particularly convenient for converting velocity, force, angular velocity, and torque vectors between body and Earth coordinate frames. Body axes tend to be used with missile and rocket configurations. Aircraft stability uses wind axes in which the x-axis points along the velocity vector. For straight and level flight this is found from body axes by rotating nose down through the angle of attack. Stability deals with small perturbations in angular displacements about the orientation at the start of the motion. This consists of two components; rotation about each axis, and angular displacements due change in orientation of each axis. The latter term is of second order for the purpose of stability analysis, and is ignored. In analysing the stability of an aircraft, it is usual to consider perturbations about a nominal equilibrium position. So the analysis would be applied, for example, assuming: The speed, height and trim angle of attack are different for each flight condition, in addition, the aircraft will be configured differently, e.g. at low speed flaps may be deployed and the undercarriage may be down. Except for asymmetric designs (or symmetric designs at significant sideslip), the longitudinal equations of motion (involving pitch and lift forces) may be treated independently of the lateral motion (involving roll and yaw). The following considers perturbations about a nominal straight and level flight path. To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stick-fixed stability. Stick-free analysis requires the further complication of taking the motion of the control surfaces into account. Furthermore, the flight is assumed to take place in still air, and the aircraft is treated as a rigid body. Unless designed to conduct part of the mission within a planetary atmosphere, a spacecraft would generally have no discernible front or side, and no bottom unless designed to land on a surface, so reference to a 'nose' or 'wing' or even 'down' is arbitrary. On a manned spacecraft, the axes must be oriented relative to the pilot's physical orientation at the flight control station. Unmanned spacecraft may need to maintain orientation of solar cells toward the Sun, antennas toward the Earth, or cameras toward a target, so the axes will typically be chosen relative to these functions. It is common practice to derive a fourth order characteristic equation to describe the longitudinal motion, and then factorise it approximately into a high frequency mode and a low frequency mode. This requires a level of algebraic manipulation which most readers will doubtless find tedious, and adds little to the understanding of aircraft dynamics. The approach adopted here is to use our qualitative knowledge of aircraft behaviour to simplify the equations from the outset, reaching the same result by a more accessible route. Short-period pitch oscillation A short input (in control systems terminology an impulse) in pitch (generally via the elevator in a standard configuration fixed wing aircraft) will generally lead to overshoots about the trimmed condition. The transition is characterised by a damped simple harmonic motion about the new trim. There is very little change in the trajectory over the time it takes for the oscillation to damp out. Generally this oscillation is high frequency (hence short period) and is damped over a period of a few seconds. A real-world example would involve a pilot selecting a new climb attitude, for example 5º nose up from the original attitude. A short, sharp pull back on the control column may be used, and will generally lead to oscillations about the new trim condition. If the oscillations are poorly damped the aircraft will take a long period of time to settle at the new condition, potentially leading to Pilot-induced oscillation. If the short period mode is unstable it will generally be impossible for the pilot to safely control the aircraft for any period of time. This damped harmonic motion is called the short period pitch oscillation, it arises from the tendency of a stable aircraft to point in the general direction of flight. It is very similar in nature to the weathercock mode of missile or rocket configurations. The motion involves mainly the pitch attitude θ (theta) and incidence α (alpha). The direction of the velocity vector, relative to inertial axes is θ − α. The velocity vector is: where m is the mass. By the nature of the motion, the speed variation is negligible over the period of the oscillation, so: But the forces are generated by the pressure distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the z-axis: In words, the wind axes force is equal to the centripetal acceleration. The moment equation is the time derivative of the angular momentum: where M is the pitching moment, and B is the moment of inertia about the pitch axis. Let: , the pitch rate. The equations of motion, with all forces and moments referred to wind axes are, therefore: We are only concerned with perturbations in forces and moments, due to perturbations in the states α and q, and their time derivatives. These are characterised by stability derivatives determined from the flight condition. The possible stability derivatives are: Since the tail is operating in the flowfield of the wing, changes in the wing incidence cause changes in the downwash, but there is a delay for the change in wing flowfield to affect the tail lift, this is represented as a moment proportional to the rate of change of incidence: Increasing the wing incidence without increasing the tail incidence produces a nose up moment, so is expected to be positive. The equations of motion, with small perturbation forces and moments become: These may be manipulated to yield as second order linear differential equation in α: We should expect to be small compared with unity, so the coefficient of α (the 'stiffness' term) will be positive, provided . This expression is dominated by Mα, which defines the longitudinal static stability of the aircraft, it must be negative for stability. The damping term is reduced by the downwash effect, and it is difficult to design an aircraft with both rapid natural response and heavy damping. Usually, the response is underdamped but stable. If the stick is held fixed, the aircraft will not maintain straight and level flight, but will start to dive, level out and climb again. It will repeat this cycle until the pilot intervenes. This long period oscillation in speed and height is called the phugoid mode. This is analysed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the climb angle γ (gamma) and speed. The small perturbation equations of motion are: which means the centripetal force is equal to the perturbation in lift force. For the speed, resolving along the trajectory: where g is the acceleration due to gravity at the earths surface. The acceleration along the trajectory is equal to the net x-wise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the climb angle, so only Xu and Zu need be considered. Xu is the drag increment with increased speed, it is negative, likewise Zu is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the z-axis. The equations of motion become: These may be expressed as a second order equation in climb angle or speed perturbation: Now lift is very nearly equal to weight: where ρ is the air density, Sw is the wing area, W the weight and cL is the lift coefficient (assumed constant because the incidence is constant), we have, approximately: The period of the phugoid, T, is obtained from the coefficient of u: Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A propeller with fixed speed would help. Heavy damping of the pitch rotation or a large rotational inertia increase the coupling between short period and phugoid modes, so that these will modify the phugoid. With a symmetrical rocket or missile, the directional stability in yaw is the same as the pitch stability; it resembles the short period pitch oscillation, with yaw plane equivalents to the pitch plane stability derivatives. For this reason pitch and yaw directional stability are collectively known as the 'weathercock' stability of the missile. Aircraft lack the symmetry between pitch and yaw, so that directional stability in yaw is derived from a different set of stability derivatives, The yaw plane equivalent to the short period pitch oscillation, which describes yaw plane directional stability is called Dutch roll. Unlike pitch plane motions, the lateral modes involve both roll and yaw motion. It is customary to derive the equations of motion by formal manipulation in what, to the engineer, amounts to a piece of mathematical sleight of hand. The current approach follows the pitch plane analysis in formulating the equations in terms of concepts which are reasonably familiar. Applying an impulse via the rudder pedals should induce Dutch roll, which is the oscillation in roll and yaw, with the roll motion lagging yaw by a quarter cycle, so that the wing tips follow elliptical paths with respect to the aircraft. The yaw plane translational equation, as in the pitch plane, equates the centripetal acceleration to the side force. where β (beta) is the sideslip angle, Y the side force and r the yaw rate. The moment equations are a bit trickier. The trim condition is with the aircraft at an angle of attack with respect to the airflow, The body x-axis does not align with the velocity vector, which is the reference direction for wind axes. In other words, wind axes are not principal axes (the mass is not distributed symmetrically about the yaw and roll axes). Consider the motion of an element of mass in position -z,x in the direction of the y-axis, i.e. into the plane of the paper. If the roll rate is p, the velocity of the particle is: Made up of two terms, the force on this particle is first the proportional to rate of v change, the second is due to the change in direction of this component of velocity as the body moves. The latter terms gives rise to cross products of small quantities (pq,pr,qr), which are later discarded. In this analysis, they are discarded from the outset for the sake of clarity. In effect, we assume that the direction of the velocity of the particle due to the simultaneous roll and yaw rates does not change significantly throughout the motion. With this simplifying assumption, the acceleration of the particle becomes: The yawing moment is given by: There is an additional yawing moment due to the offset of the particle in the y direction: The yawing moment is found by summing over all particles of the body: where N is the yawing moment, E is a product of inertia, and C is the moment of inertia about the yaw axis. A similar reasoning yields the roll equation: where L is the rolling moment and A the roll moment of inertia. Lateral and longitudinal stability derivatives The states are β (sideslip),r (yaw rate) and p (roll rate), with moments N (yaw) and L (roll), and force Y (sideways). There are nine stability derivatives relevant to this motion, the following explains how they originate. However a better intuitive understanding is to be gained by simply playing with a model aeroplane, and considering how the forces on each component are affected by changes in sideslip and angular velocity: Sideslip generates a sideforce from the fin and the fuselage. In addition, if the wing has dihedral, side slip at a positive roll angle increases incidence on the starboard wing and reduces it on the port side, resulting in a net force component directly opposite to the sideslip direction. Sweep back of the wings has the same effect on incidence, but since the wings are not inclined in the vertical plane, backsweep alone does not affect Yβ. However, anhedral may be used with high backsweep angles in high performance aircraft to offset the wing incidence effects of sideslip. Oddly enough this does not reverse the sign of the wing configuration's contribution to Yβ (compared to the dihedral case). Roll rate causes incidence at the fin, which generates a corresponding side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing has dihedral, this will result in a side force momentarily opposing the resultant sideslip tendency. Anhedral wing and or stabiliser configurations can cause the sign of the side force to invert if the fin effect is swamped. Yawing generates side forces due to incidence at the rudder, fin and fuselage. Sideslip in the absence of rudder input causes incidence on the fuselage and empennage, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions. Under sideslip conditions at a given roll angle Nβ will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiralling flight. Roll rate generates fin lift causing a yawing moment and also differentially alters the lift on the wings, thus affecting the induced drag contribution of each wing, causing a (small) yawing moment contribution. Positive roll generally causes positive Np values unless the empennage is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on Np because the wing axis is normally closely aligned with the centre of gravity. Yaw rate input at any roll angle generates rudder, fin and fuselage force vectors which dominate the resultant yawing moment. Yawing also increases the speed of the outboard wing whilst slowing down the inboard wing, with corresponding changes in drag causing a (small) opposing yaw moment. Nr opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input. A positive sideslip angle generates empennage incidence which can cause positive or negative roll moment depending on its configuration. For any non-zero sideslip angle dihedral wings causes a rolling moment which tends to return the aircraft to the horizontal, as does back swept wings. With highly swept wings the resultant rolling moment may be excessive for all stability requirements and anhedral could be used to offset the effect of wing sweep induced rolling moment. Yaw increases the speed of the outboard wing whilst reducing speed of the inboard one, causing a rolling moment to the inboard side. The contribution of the fin normally supports this inward rolling effect unless offset by anhedral stabiliser above the roll axis (or dihedral below the roll axis). Roll creates counter rotational forces on both starboard and port wings whilst also generating such forces at the empennage. These opposing rolling moment effects have to be overcome by the aileron input in order to sustain the roll rate. If the roll is stopped at a non-zero roll angle the Lβ upward rolling moment induced by the ensueing sideslip should return the aircraft to the horizontal unless exceeded in turn by the downward Lr rolling moment resulting from sideslip induced yaw rate. Longitudinal stability could be ensured or improved by minimizing the latter effect. Equations of motion Since Dutch roll is a handling mode, analogous to the short period pitch oscillation, we shall ignore any effect it might have on the trajectory. The body rate r is made up of the rate of change of sideslip angle and the rate of turn. Taking the latter as zero, because we assume no effect on the trajectory, we have, for the limited purpose of studying the Dutch roll: The yaw and roll equations, with the stability derivatives become: The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become: This becomes a second order equation governing either roll rate or sideslip: The equation for roll rate is identical. But the roll angle, φ (phi)is given by: If p is a damped simple harmonic motion, so is φ, but the roll must be in quadrature with the roll rate, and hence also with the sideslip. The motion consists of oscillations in roll and yaw, with the roll motion lagging 90 degrees behind the yaw. The wing tips trace out elliptical paths. Stability requires the 'stiffness' and 'damping' terms to be positive. These are: The denominator is dominated by Lp, the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive. Considering the 'stiffness' term: − LpNβ will be positive because Lp is always negative and Nβ is positive by design. Lβ is usually negative, whilst Np is positive. Excessive dihedral can de-stabilise the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to Lβ. The damping term is dominated by the product of the roll damping and the yaw damping derivatives, these are both negative, so their product is positive. The Dutch roll should therefore be damped. The motion is accompanied by slight lateral motion of the centre of gravity and a more 'exact' analysis will introduce terms in Yβ etc. In view of the accuracy with which stability derivatives can be calculated, this is an unnecessary pedantry, which serves to obscure the relationship between aircraft geometry and handling, which is the fundamental objective of this article. Jerking the stick sideways and returning it to centre causes a net change in roll orientation. The roll motion is characterized by an absence of natural stability, there are no stability derivatives which generate moments in response to the inertial roll angle. A roll disturbance induces a roll rate which is only cancelled by pilot or autopilot intervention. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to: Lp is negative, so the roll rate will decay with time. The roll rate reduces to zero, but there is no direct control over the roll angle. Simply holding the stick still, when starting with the wings near level, an aircraft will usually have a tendency to gradually veer off to one side of the straight flightpath. This is the (slightly unstable) spiral mode. The opposite holds for a stable spiral mode. The spiral mode is so-named because when it is slightly unstable, and the controls are not moved, the aircraft will tend to increase its bank angle slowly at first, then ever faster. The resulting path through the air is a continuously tightening and ever more rapidly descending spiral. An unstable spiral mode is common to most aircraft. It is not dangerous because the times to double the bank angle are large compared to the the pilot's ability to respond and correct errors with aileron inputs. When the spiral mode is stable, it behaves in a way opposite to the exponential divergence of the unstable mode. The stable spiral mode, when starting with the wings at a moderate bank angle, will return to near wings level, first quickly, then more slowly. When the spiral mode is stable and starting at a moderate bank angle, the spiral nature of the flight path is not as obvious. This is because usually only a fraction of a turn is made while the wings are not fully level. The turning starts out (relatively) tight, then becomes less and less so as the wings become more level. The divergence rate of the unstable spiral mode will be roughly proportional to the roll angle itself (i.e. roughly exponential growth). The convergence rate of the stable spiral mode will be roughly proportional to the roll angle itself (i.e. roughly exponential decay). Spiral mode trajectory In studying the trajectory, it is the direction of the velocity vector, rather than that of the body, which is of interest. The direction of the velocity vector when projected on to the horizontal will be called the track, denoted μ (mu). The body orientation is called the heading, denoted ψ (psi). The force equation of motion includes a component of weight: where g is the gravitational acceleration, and U is the speed. Including the stability derivatives: Roll rates and yaw rates are expected to be small, so the contributions of Yr and Yp will be ignored. The sideslip and roll rate vary gradually, so their time derivatives are ignored. The yaw and roll equations reduce to: Solving for β and p: Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle: This is an exponential growth or decay, depending on whether the coefficient of φ is positive or negative. The denominator is usually negative, which requires LβNr > NβLr (both products are positive). This is in direct conflict with the Dutch roll stability requirement, and it is difficult to design an aircraft for which both the Dutch roll and spiral mode are inherently stable. Since the spiral mode has a long time constant, the pilot can intervene to effectively stabilise it, but an aircraft with an unstable Dutch roll would be difficult to fly. It is usual to design the aircraft with a stable Dutch roll mode, but slightly unstable spiral mode. Published in July 2009. Copyright 2004-2023 © by Airports-Worldwide.com, Vyshenskoho st. 36, Lviv 79010, Ukraine

You Do the Math by Caroline Lupfer Kurtz In UM Professor Johnny Lotts perfect world this positive attitude would be the norm. It isnt quite yet, but with the increasing use of a high school curriculum that integrates real-world math, science and technology, Lott hopes to keep students engaged in math longer and make better math thinkers out of the majority of people folks who otherwise would leave the subject behind as soon as possible. A mathematics research professor, Lott co-directed the Systemic Initiative for Montana Mathematics and Science (SIMMS), a high school mathematics curriculum reform project funded by the National Science Foundation through the Montana Council of Teachers of Mathematics. Like the middle school reform initiative Six Through Eight Mathematics (STEM), authored by UM colleague and math Professor Rick Billstein SIMMS attempts to address the charges that U.S. students perform poorly in math and consider it largely irrelevant to their lives. Numbers are everywhere When you compartmentalize math, especially at the high school level, thats dangerous, Lott says. Students may miss the connections among different math skills or the point that these different skills are related at all. As a result, he says, when students are confronted with a real-world problem that involves math, they dont know how to approach it because it is not clearly labeled an algebra problem or a geometry problem or a question of probability. The SIMMS program tries to emphasize the relationships among topics in mathematics as well as between math and other disciplines. An award-winning module titled AIDS: The Preventable Epidemic in the level-one course, for example, uses information from the U.S. Centers for Disease Control and Prevention to demonstrate death rates and probabilities of contracting the disease. It reflects the projects goal of helping students use math to make better decisions, Lott says. Other modules in the first two levels, required for ninth and 10th graders, include such contexts as population growth, manufacturing, pyramid building, small business inventory, genetics and the allotment of seats in the U.S. House of Representatives. These modules teach principles of data collection, presentation and interpretation; linear, exponential and step equations; three-dimensional geometry; and trigonometric ratios and algebra. On more open-ended questions, SIMMS students showed superior skills in problem solving and communicating mathematical ideas, used a wider variety of problem-solving strategies and used appropriate technology, Lott says. Last year Lott and other faculty members conducted their own study of entering UM freshmen students who had taken the SIMMS curriculum in high school to see what effect this had on the college-level mathematics courses they placed into or chose. Results showed that among students who excel in math there is little difference whether they took SIMMS or a traditional course and that students who passed SIMMS with an A or B did well in their college classes. However, people for whom math is hard find SIMMS helps them succeed better. You expect students in a modeling-based curriculum like SIMMS to do better with problem solving, mathematics doctoral student Terry Souhrada says. The surprising part is that they do as well as students in other types of classes on the standardized tests. SIMMS students are not losing anything, but they are gaining new skills. Souhrada is completing his dissertation prior to teaching two math classes for educators this fall at UM. His longitudinal study of students and teachers at a western Montana high school evaluates the performances of students in SIMMS and non-SIMMS classes and their attitudes about mathematics in general. Hopefully, well see some information [from this study] that will lead us to the conclusion that one method is better than another, he says. Masters student James Barta is studying another aspect of the SIMMS approach, which is whether this type of reform curriculum helps keep students considered at greater risk for dropping out of math girls and American Indians engaged in math longer. So far, it appears that SIMMS has no negative effect on either group, but, as Lott says, the factors at work in keeping some students in math or in school in general go far beyond just curriculum. Do the math Lott now is turning his attention toward the public perception of math in general. As part of an effort led by the National Council of Teachers of Mathematics, the National Action Council on Minorities in Engineering and the WidmeyerBaker Group in Washington, D.C., Lott has been working on ways to get parents involved in helping children with math, not just in their homework assignments but by talking about the uses of math in the home, at work and at play. He is busy writing problems that families of middle schoolers would encounter at home, for example, whose solution involves math. These problems may eventually end up on milk cartons, grocery bags and other familiar places. The idea is to change the way society perceives the importance of math, Lott says. Kids like numbers when they start school, but somewhere it starts to fall apart. Why? Part of it has to do with public attitudes, Lott believes. Too often, he says, kids get the message that adults think its OK not to understand or use math because they never did either. A coordinated media campaign is an effort to turn this self-perpetuating aversion around and get people to do the math. Were trying to reach out on every front, Lott says. Its got to be a well-rounded effort.

What are fact families in 3rd grade math? A fact family is a group of math facts using the same numbers. In the case of addition/subtraction, you use three numbers and get four facts. For example, you can form a fact family using the three numbers 10, 2, and 12: 10 + 2 = 12, 2 + 10 = 12, 12 − 10 = 2, and 12 − 2 = 10. What is 3rd grade multiplication facts? The repeated addition of the same number is expressed by multiplication in short. Hence, repeated addition of 2 five times is equal to 2 multiplied by 5. Thus, 3 × 6 = 18 that 3 multiplied by 6 is equal to 18, or 3 into 6 is equal to 18, or product of 3 and 6 is 18. 3 × 6 = 18 is called a multiplication fact. How do you do fact families in third grade? What is fact fluency? Fact families are sets of three numbers that are used to make four math facts using addition/subtraction or multiplication/division. Fact families help to show us the inverse relationship, or opposite relationship, between addition and subtraction and between multiplication and division. In math, a fact family can be defined as a group of math facts or equations created using the same set of numbers. Similarly, in a multiplication and division fact family, there are multiplication and division sentences created using three numbers. Multiplication and division are closely related, given that division is the inverse operation of multiplication. This is because when we multiply two numbers (which we call factors), we get a result that we call a product. If we divide this product by one of the factors, we get the other factor as a result. A multiplication fact is the answer to a multiplication calculation. For example, in the sum 3 x 3 = 9, the multiplication fact is 9. Facts about Multiplication Related facts are basic mathematical expressions made up of three numbers. Related facts are often taught as part of early math alongside fact families and addition, subtraction, multiplication, and division facts. Certain numbers and facts are related or make up a fact “family” and there are only three numbers in each family. They are related because you can add two of the numbers together to get the third number. 8 + 5 = 13. You can switch the order of the two numbers added above to equal the third number again. Essentially, to multiply numbers is to add groups of a number. Multiplying means repeated addition of a number. (The number must all be the same before we can use it to multiply.) When you think of it this way, learning the Times Tables makes sense. 2) Memorizing Your Facts Is a Confidence Booster As students progress to more complex math problems, those who have not mastered multiplication fact have an increasingly hard time completing these problems. A fact family is a group of calculations that are created using the same three numbers. For example, here is a fact family that uses the numbers 2 , 4 and 6 . Relating Multiplication and Division In multiplication, the numbers being multiplied are called factors; the result of the multiplication is called the product. In division, the number being divided is the dividend, the number that divides it is the divisor, and the result of the division is the quotient. Teach Doubles Addition Facts Starting with concrete objects is always helpful. You can use square tiles or some other type of manipulative. Lining them up in columns like below can help with the visualization of these facts. I like to target doubling 1-5 first and then add on 6-10 next. For this example, the fact family would be 3 × 5 = 15, 5 × 3 = 15, 15 ÷ 3 = 5, and 15 ÷ 5 = 3. To demonstrate number fact families, provide students with a ten strip, cut to the number you are focusing on. In this case, it's 8. Instead of using counters, students draw 2 groups of circles that add together to make 8. You can also have them stamp them. When I teach fact families, I introduce it by writing three number in a fact family such as 7,3, and 10. I explain that we're going to be learning about fact families and in fact families there are three people: a daddy, a mama, and a baby. I ask my students to guess which number is the daddy. For every multiplication fact, there are two division facts. So, we have a fact family that includes the facts 3+9=12, 9+3=12, 12-3=9 and 12-9=3. This particular fact family all involve 3, 9 and 12, so you might name refer to this as the 'addition fact family for 3 9 12' or something similar. What is a Fact Family. A fact family is a set of four math facts made with the same three numbers. The numbers 2, 3, and 5 can make a family of four facts: 2+3=5, 3+2=5, 5−3=2, 5−2=3. The numbers 2, 3, and 6 can make a different fact family: 2×3=6, 3×2=6, 6÷2=3, 6÷3=2. 5 * 7 = 35 7 * 5 = 35 35 ÷ 5 = 7 35 ÷ 7 = 5 Fact families are a set of related addition and subtraction facts, or related multiplication and division facts. In this 1st grade math lesson, we write FACT FAMILIES using small numbers. A "fact family" consists of two additions and two subtractions that use the same three numbers. For example, 2 + 6 = 8, 6 + 2 = 8, 8 − 2 = 6, and 8 − 6 = 2 makes up a fact family (two addition facts plus two subtraction facts). There are no Exponents. We start with the Multiplication and Division, working from left to right. NOTE: Even though Multiplication comes before Division in PEMDAS, the two are done in the same step, from left to right. Addition and Subtraction are also done in the same step. To estimate the result of multiplication (product), round the numbers to some close numbers that you can easily multiply mentally. One method of estimation is to round all factors to the biggest digit (place value) they have. Most people will tell you to start with x1 or x0, because they're the easiest to memorize. The reason I recommend starting with x2 is because we want to start with the concept of multiplication. Kids have experience with doubling and grouping in pairs, so it makes sense to start with x2. Children can begin to learn their multiplication tables once they have mastered basic addition and subtraction concepts and are familiar with arrays and how to count by 2's and 5's, which is usually by age 9. Images for 5+ Multiplication Fact Family 3rd Grade Worksheets Blank fact family worksheets activity shelter Math facts practice grade Multiplication division fact families worksheet bundle family Grade math minutes answer key Fact family multiplication division triangles families

One-of-a-kind corrected proofs shed new light on a philosophical masterpiece McMaster University Library has acquired the corrected proofs of the first two volumes of Bertrand Russell and Alfred North Whitehead’s highly influential three-volume work, Principia Mathematica, now publicly available for the first time thanks to a donation from philosophy professor and Russell scholar Bernard Linsky, pictured here. Photo by Sarah Janes. BY Erica Balch November 5, 2019 In 1910, Cambridge University Press published the first volume of Principia Mathematica, a three-volume work by renowned logician and philosopher Bertrand Russell, and philosopher and mathematician Alfred North Whitehead. It was a work of epic proportions, one that attempted to address a question that had vexed philosophers and mathematicians since the time of Aristotle: what are the foundations of mathematics and are they rooted in logic? The result was a large-scale reconstruction of nearly every branch of mathematics, expressed through complex mathematical notation and a new kind of non-syllogistic logic pioneered by Russell which was intended to make the proofs and results fully explicit and precise. Though, at first, the work seemed impenetrable to many – particularly to philosophers who, at the time, were not trained in mathematical logic – many of Russell and Whitehead’s theories became hugely influential, creating a logical system used by scholars to tackle questions in a number of disciplines from philosophy and mathematics, to biology and linguistics. Now, McMaster University Library has acquired the corrected proofs of the first two volumes of this seminal work which contain the final set of corrections, hand-written by Bertrand Russell, before the volumes went to print in 1910 and 1912 respectively. The acquisition of the proofs, which are now part of the Library’s extensive Bertrand Russell Archives – the largest and most complete collection of Russell materials in the world – was funded through a donation to McMaster University by philosopher and Russell scholar Bernard Linsky in honour of his parents, Joan and Leonard Linsky. “These proofs are the very last version before this limited edition of Principia Mathematica was published,” says Linsky. “It gives us an idea of the changes that were made at the last moment when they had to commit themselves to what was going to be in print. So, these proofs are for people like me who are interested in the origins of this famous work and for book collectors who love to look at original or early versions of things, and that’s what these are – they’re just absolutely unique.” Linsky says his interest in Russell and in Principia Mathematica was inspired by his father, a philosophy professor at the University of Chicago, who first introduced him to Russell scholarship. “My father wrote a book in 1967 called Referring which very clearly stated the nature and problems with Russell’s famous theory of descriptions – that was a topic in the philosophy of language at the time,” explains Linsky. “When I got into Russell and would come to the archives at McMaster, my father was someone I could talk to about what I found there. So, Russell certainly brought us together – it was like going into the family business.” Ken Blackwell, adjunct professor and honourary Russell archivist in McMaster’s Bertrand Russell Research Centre, says the acquisition of the corrected proofs fills what has long been a significant gap in the Russell Archives. “This is the closest thing we have to an original manuscript,” says Blackwell who explains that only two and a half pages of the original manuscript have ever been found. “Principia Mathematica is perhaps the best known and most influential of Russell’s works as a logician and philosopher, so to now have this nearly complete early version as part of the Russell Archives is just wonderful.” The existence of the corrected proofs wasn’t known until about two years ago. Blackwell believes Russell sent the proofs to a Polish scholar, Michael Dziewicki, with whom he had been corresponding around the time of the First World War. Dziewicki left the proofs to his son, who emigrated from Poland to Australia in the mid-1920’s. When Dziewicki’s son died, the proofs came into the possession of a mathematics professor who kept them for some years before they were acquired by an Australian bookseller, and then by a rare books dealer in Denmark, from whom McMaster University Library purchased the volumes. Blackwell learned of the availability of the proofs through strong friends of the Russell Archives, John G. Slater and Michael Walsh. “We are thrilled that, after more than a century, these corrected proofs are now publicly available to scholars around the world, says Vivian Lewis, McMaster University Librarian. “The acquisition of the proofs would not have been possible without Bernie’s tremendous generosity. We are truly grateful for this remarkable gift which will no doubt help to new shed light on many aspects of Russell scholarship.” After the publication of Principia Mathematica, Russell turned his attention to other intellectual pursuits. Though he continued to work on philosophy until the end of his life, Russell became better known as an outspoken and committed advocate for peace, serving as a central figure in the Campaign for Nuclear Disarmament and as a leader in the world-wide opposition to the Vietnam War. In 1950, he was awarded the Nobel Prize in Literature for his prolific writings championing human rights and freedom of thought. Though Russell once joked that he knew of only six scholars who had read Principia Mathematica in its entirety, the work has nonetheless left an enduring legacy, particularly among philosophers. “It was Russell and Principia Mathematica that brought mathematical logic to the attention of philosophers and made it what it is today – an essential part of the philosopher’s toolkit,” explains Nicholas Griffin, McMaster University Library’s Scholar in Residence, a philosopher and Russell expert. “To now be able to access the actual document and read Russell’s hand-written notes is extraordinary and I’m sure will lead to a host of new insights into this tremendously important work, both in philosophy and in many other disciplines.” Learn more about McMaster University Library’s Bertrand Russell Archives:

Let's begin with the formal logical fallacy commonly known as "Affirming the Consequent." This fallacy is formal in that the form of the argument is fallacious. In logical terms the form of this argument is as follows: 1. If p, then q. In premise (1) p is called the antecedent, and q is called the consequent. This argument form attempts to show the truth of the antecedent by affirming the truth of the consequent. In order to show that this form is fallacious we will let p= "it is raining" and q= "my truck is wet." Now we have: 1. If it is raining, then my truck is wet. 2. My truck is wet. Therefore, it is raining. The problem here is that that there may be other conditions that make my truck wet other than rain. So that the premises (1) and (2) may true, while the conclusion "it is raining" may be false. For instance, my truck may be wet because it is being washed on a very sunny day (hence no rain). One of my favorite examples of "Affirming the Consequent" is as follows: 1. If we have great minds, then we think alike. 2. We think alike. Therefore, we have great minds. The fact is, we may both think alike, and be blithering idiots. In any statement in the form of a conditional (i.e. If p, then q), p will be a sufficient condition for q; while q may not be the only sufficient condition for p. So what does this have to do with the scientific method? The scientific method may be (and often is) considered a procedure (i.e. method) for obtaining knowledge about observable phenomena. In fact, many individuals believe that the scientific method is the only way to truly gain knowledge. If a knowledge claim is not known via the scientific method, then said claim cannot be known. Now, if the scientific method is based on fallacious reasoning, then all we claim to know via science is based on fallacious reasoning. We will remember from our high school science classes that the scientific method begins with a hypothesis concerning some observable phenomena. Then a prediction is made concerning that hypothesis. After making the prediction, experiments are carried out in hopes that the prediction will prove accurate. If the prediction proves accurate, then the hypothesis is verified. A mundane example may help refresh our memories. Let's assume that I drink hot tea every day (I do not). I observe that the water seems to boil at the same temperature every time I prepare my tea. This intrigues me (actually I could care less, but let's continue). I decide to apply the scientific method in order to gain new knowledge about the temperature at which water boils. I form the hypothesis that water always boils at the same temperature. I predict that every time I boil water it will boil at the same temperature. I perform an experiment by boiling water and checking the temperature at which it begins to boil. I know that I must do this several times, and I find in each case that water boils at 212 degrees F (Fahrenheit). It seems that my hypothesis is true...right? According to Schriftman I have come to this conclusion via a fallacious argument. Let p= my hypothesis and q= my experiments. In other words p= "water always boils at the same temperature" and q= "every time I boil water it boils at 212 degrees." Now we have: 1. If water always boils at the same temperature, then every time I boil water it boils at 212 degrees. 2. Every time I boil water it boils at 212 degrees. Therefore, water always boils at the same temperature. It seems that I know that water always boils at the same temperature, but according to Schriftman I have come to this conclusion by affirming the consequent. Is my hypothesis wrong? Could there be other conditions for my water always boiling at 212 degrees, such that under other conditions water may boil at another temperature? Yes. I happen to live at sea level. At sea level water always boils at 212 degrees. If I lived on top of Mt. Everest then my water would boil at a lower temperature. Again, if I added salt to my water it would boil at a higher temperature. So it seems that Schriftman has a point. The scientific method seems to proceed by fallacious argumentation. If the hypothesis is the antecedent and the experiments meant to prove the hypothesis are the consequent, then the scientific method seeks to show the truth of the antecedent by affirming the consequent. In hindsight this is really no surprise. Consider the "scientific studies" that we read or hear about in the news. One study says eggs are bad for my heart (cholesterol). Another says they are good for my heart (omega-3's). One study says red wine is good for the heart. Another study comes out years later which says any wine may contribute to the onset of breast cancer in women. Obviously, conditions for particular hypothesis continue to crop up that change the conclusions. At this point some logically astute scientific mind will argue, "The scientific method is not based on the formal fallacy of Affirming the Consequent. In fact the scientific method is based on the valid argument form known as Modus Tollens." Science is not attempting to prove the truth of a hypothesis via experiments. The sole purpose of experiments is to falsify hypotheses. Let's examine this claim by clarifying the valid argument form known as Modus Tollens: 1. If p, then q. Let p= "it is raining" and q= "my truck is wet." Now we have: 1. If it is raining, then my truck is wet. 2. My truck is not wet. Therefore, it is not raining. This argument form is valid in that if the premises are true, then the conclusion necessarily follows. The condition given for my truck being wet is rain. If my truck is not wet (given that condition), then it follows that it is not raining. In the same way, the scientific method proceeds not by affirming a hypothesis by experiments, but by trying to dis-prove a hypothesis by experiments. Hence, if p= hypothesis and q= experiments and the experiments do not show what was predicted, then the hypothesis is not accurate. In short, scientific hypothesis can never be proven true, they can only be proven false. This is why Einstein said, "No amount of experimentation can ever prove me right; a single experiment can prove me wrong."1All of this seems to place science as a method on very precarious footing. On the one hand we should all be familiar with these problems from our high school science classes. We should all be fully aware that scientific hypotheses are just that...hypotheses. However, this is not how scientific "knowledge" is treated in either the common populace or in the scientific community. One glaring example is the current experiments concerning neutrinos. It may be the case that Einstein was wrong and one experiment may prove that he was wrong. Physicists are praying to their hypothetical universal Being that the findings of the Opera collaboration are wrong. If the findings are right, then a hundred years of scientific thinking will have to be re-written. Some folks in the scientific community are acting as if all they know will be turned upside down. This is amazing for a community who should know that all they can possibly know is what they cannot know.

All except two were fishes and all except two were cats. I have two rectangular bars. He will ask all his students to raise their hands on each of the question that is asked. The practice of hiding information through codes is the ultimate linguistic puzzle, a gameplay so enthralling to the right student that courses are now woven into elementary, middle and high school curriculum. One example of a secret code method is called a Keyword Cipher With this secret code keyword is placed at the beginning and this shifts the remaining letters of the alphabet, not used in the keyword, to the right. He informed the police that her wife was swept away by the waves and died. Instead of a 'B', he would write an 'E', instead of a 'C', he would write an 'F' and so on. In any piece of writing, we use E about 13% of the time on average. You can place weights on both side of weighing balance and you need to measure all weights between 1 and 1000. Weigh 4 against 9, a known good coin. Every symbol stands for a letter of the alphabet. Anyway let's see how many of these crack the code puzzles you can solve correctly in your first attempt? All Elizabeth the First's Spy-Master had to do to crack Mary's code, was to look through the coded message and count the number of times each symbol came up. The most common three letter words in English are 'the' and 'and', so if you see a group of three symbols that comes up quite a lot, they could stand for 'the' or 'and'. But you do not know who is a liar and who is not. Then have a go at coding your name by shifting the alphabet forward by more places by adding greater numbers eg adding 5, then adding 10. Note that the unusual marble may be heavier or lighter than the others. For more information about other secret codes that have been used throughout history, check out. For the third weighing, weigh 7 against 8. In this way, the principal will see all the hands being raised for each question even though all won't be knowing the correct answer. By signing up you agree to our. Now, in order to proceed, we must keep track of which side is heavy for each of the following weighings. It contains crack the code puzzles to test your logical reasoning skills. On the next day, police arrived to his doorstep and when he opened the door, they arrested him for murdering his wife. Either they balance, or they don't. He asks password of the safe from the clerk while pointing a gun on his forehead. However, you do have to be careful when you get to the end of the alphabet, because there is no letter number 26, so you have to go back to number 0. But the only catch will be that those who knows the answer correctly will raise their right hand and others will raise their left hand. In this case, you know that the different coin is 9, 10, or 11, and that that coin is heavy. After this, every single person who get to the flight sits on his seat if its available else chooses any available seat at random. Instead of writing the letter 'A', he would write the letter that comes three places further on in the alphabet, the letter 'D'. Teachers who sign up to follow and teach this advanced math class are in the low hundreds, potentially because they may not believe they have the ability to teach the course itself. Most of the crack the code puzzles will require your logical thinking to solve these puzzles. A man is lying dead in front of the building and seeking the position of the windows on the building and the position in which the body is lying, it is evident that the man jumped out from one of them and committed suicide. Proceed as in the step above, but the coin you're looking for is the light one. For a long time, people thought this type of code would be really hard to crack. Even though it is now Kangwa against Ferdinand, Kangwa has a better chance of winning than before. How did the travel agent know about the murder? The children seem to enjoy doing this kind of activity, and when they understand the idea, they can make up their own codes, and they can also make up new ways of making codes. You can opt out anytime. If you would like to test out these code breaking tips and your new code breaking talents, have a look at. Upon reaching there, he informs the team that it is not a suicide but a murder. So the remaining numbers should be written in that order; 10, 13, 3, 12, 2. So Caesar would write messages to his generals in code. He would rather shoot at Rafael and will definitely kill him. Also, the principal can ask questions from anywhere. Other people have also read this article and they too will be top mathematical codebreakers. In fact, some of the most famous code breakers in history have been mathematicians who have been able to use quite simple maths to uncovered plots, identify traitors and influence battles. If they balance, then the different coin is in the group 9,10,11,12. Now, the teacher is worried for the impression that his class might cast on the principal since all the students are not intelligent. The letters that are not used in the keyword are placed in line in alphabetical order. The graph below shows the average frequency of letters in English. If you are Kangwa, where should you shoot first for the highest chance of survival? Now he wants that the principal must be impressed with the performance of his class. And if you happened to ask this question to the one who says truth, he will also show you the wrong way. The children should look at the first number in the code, find where it should be in the grid, and write down the letter that is in its space. On a magical-intellectual land of gpuzzles , all the animal are rational real smart. So the principal will be impressed to full extent.

« PreviousContinue » suppose a crown were our "small" unit, and we had a medley of 10 crowns, 33 shillings, and 100 fourpenny pieces, with which to make successive throws, throwing the whole number of them at once we might theoretically sort them into fictitious groups each equivalent to a crown. There would be 29 such groups, viz.:-10 groups, each consisting of 1 crown; 6 groups, each of 5 shillings; 1 group of three shillings and 6 fourpenny pieces; 6 groups each of 15 fourpenny pieces; and a residue of 4 fourpenny pieces, which may be disregarded. Hence, on the already expressed understanding that we do not care to trouble ourselves about smaller sums than a crown, the results of the successive throws of the medley of coins would be approximately the same as those of throwing at a time 29 crowns, and would be expressed by the coefficients of a binomial of the 29th power. Hence I conclude that all miscellaneous influences of a few small and many minute kinds, may be treated for a first approximation exactly as if they consisted of a moderate number of small and equal alternatives. The second approximation has already been alluded to; it consists in taking some account of the minute influences which we had previously agreed to ignore entirely, the effect of which is to turn the binomial grades into a binomial ogive. I effect it by drawing a curve with a free hand through the grades, which affords a better approximation to the truth than any other that can à priori be suggested. I will now show from quite another point of view (1) that the exponential ogive is, on the face of it, fallacious in a vast number of cases, and (2) that we may learn what is the greatest possible number of elements in the binomial whose ogive most nearly represents the generic series we may be considering. The value of is directly dependent on the number of elements; g-n hence, by knowing its value, we ought to be able to determine the number of its elements. I have calculated it for binomials of various powers, protracting and interpolating, and obtain the following very rough but sufficient results for their ogives (not grades) : Number of (equal) Now, if we apply these results to observed facts, we shall rarely find that the series has been due to any large number of equal elements. Thus, in the stature of man the probable error, is about 30, which makes it impossible that it can be looked upon as due to the effect of more than 200 equally small elements. On consideration, however, it will appear that in certain cases the number may be less, even considerably less, than the tabular value, though it can never exceed it. As an illustration of the principle upon which this conclusion depends, consider what the value of would be in the case we may of a wall built of 17 courses of stone, each stone being 3 inches thick, and subject to a mean error in excess or deficiency of one fifth of an inch. Obviously the mean height m of the wall would be 3× 17 inches; and its probable error q-m would be very small, being derived from a binomial ogive of 17 elements, each of the value of only one fifth of an inch. Now we saw from our previous calculation that this would be eight fifths, or 1.6 inch, which would give the value to of or about 321; y-m 1.6' consequently we should be greatly misled if, after finding by observation the value of that fraction, and turning to the Table and seeing there that it corresponded to more than 200 equal elements, we should conclude that that was the number of courses of stones. The Table can only be trusted to say that the number of courses certainly does not exceed that number; but it may be less than that. The difficulty we have next to consider is that which I first mentioned, but have intentionally postponed. It is due to the presence of influences of extraordinary magnitude, as Aspect in the size of fruit. These influences must be divided into more than two phases, each differing by the same constant amount from the next one, and that difference must not be greater than exists between the opposite phases of the "small" alternatives. If we had to divide an influence into three phases, we should call them "large," "moderate," and "small;" if into four, they would be "very large," " moderately large," " moderately small,” and very small," and so on. Any objects (say, fruit) which are liable to an influence so large as to make it necessary to divide it into three phases, really consist of three series generically different which are entangled together, and ought theoretically to be separated. If there had been two influences of three phases, there would be nine such series, and so on. In short, the fruit, of which we may be considering some hundred or a few thousand specimens, ought to be looked upon as a multitude of different sorts mixed together. The proportions inter se of the different sorts may be accepted as constant; there is no difficulty arising from that cause. The question is, why a mixture of series radically different, should in numerous cases give results apparently identical with those of a simple series. For simplicity's sake, let us begin with considering only one large influence, such as aspect on the size of fruit. Its extreme effect on their growth is shown by the difference in what is grown on the north and south sides of a garden-wall, which in such kinds of fruit as are produced by orchard-trees, is hardly deserving of being divided into more than three phases, "large," "moderate," and "small." Now if it so happens that the "moderate" phase occurs approximately twice as often as either of the extreme phases (which is an exceedingly reasonable supposition, taking into account the combined effects of azimuth, altitude, and the minor influences relating to shade from leaves &c.), then the effect of aspect will work in with the rest, just like a binomial of two elements. Generally the coefficients of (a+b) are the same as those of (a+b)n-rx (a+b)". Now the latter factor may be replaced by any variable function the frequency and number of whose successive phases, into which it is necessary to divide it, happen to correspond with the value of the coefficients of that factor. It will be understood from what went before, that we are in a position to bring these phases to a common measure with the rest, by the process of fictitious grouping with appropriate doses of minute influences, as already described. On considering the influences on which such vital phenomena depend as are liable to be treated together statistically, we shall find that their mean values very commonly occur with greater frequency than their extreme ones; and it is to this cause that I ascribe the fact of large influences frequently working in together with a number of small ones without betraying their presence by any sensible disturbance of the series. The last difficulty I shall consider, arises from the fact that the individuals which compose a statistical group are rarely affected by exactly the same number of variable influences. For this cause they ought to have been sorted into separate series. But when, as is usually the case, the various intruding series are weak in numbers, and when the number of variable influences on which they depend does not differ much from that of the main series, their effect is almost insensible. I have tried how the figures would run in many supposititious cases; here is one taken at haphazard, in which I compare an ordinary series due to 10 alternatives, giving 210=1024 events, with a compound series. The latter also comprises 1024 events; but it is made up of three parts: viz. nine tenths of it are due to a 10-element series; and of the remaining tenth, half are due to a 9 and half to an 11 series. I have reduced all these to the proper ratios, ignoring fractions. It will be observed how close is the correspondence between the compound and the simple series. It appears to me, from the consideration of many series, that the want of symmetry commonly observed in the statistics of vital phenomena is mainly due to the inclusion of small series of the above character, formed by alien elements; also that the disproportionate number of extreme cases, as of giants, is due to this cause. The general conclusion we are justified in drawing appears to be, that, while each statistical series must be judged according to its peculiarities, a law of frequency of error founded on a binomial ogive is much more likely to be approximately true of it than any other that can be specified à priori; also that the exponential law is so closely alike in its results to those derived from the binomial ogive, under the circumstances and within the limits between which statisticians are concerned, that it may safely be used as hitherto, its many well-known properties being very convenient in all cases where it is approximately true. Therefore, if we adopt any uniform system (such as already suggested) of denoting the magnitudes of qualities for the measurement of which no scale of equal parts exists, such system may reasonably be based on an inverse application of the law of frequency of error, in the way I have described, to statistical series obtained by the process of intercomparison. V. On a new Method of investigating the Composite Nature of the Electric Discharge. By ALFRED M. MAYER*. N 1842 Professor Joseph Henry† observed that when a needle was placed in a helix and magnetized by the discharge of a Leyden jar, the direction of the polarity of the needle varied with the "striking-distance" of the jar; and these observations led Henry to the discovery that the discharge was multiple and oscillatory in its nature. In 1862 Feddersent confirmed Henry's discovery, on examining the nature of the discharge by means of a revolving mirror. Subsequently Rood (in a series of classical researches, published in Silliman's American Journal, in 1869, 1871, 1872) studied the multiple character of the discharge of the inductorium by means of rotating disks perforated with narrow radial slits. In 1873 Cazin § also investigated the discharge with the rotating disk. The method I have devised leads us directly, by the simplest means, to phenomena which cannot be revealed by either revol ving mirror or rotating disk. The first method that occurred to me was to attach a delicate metallic point to a vibrating tuningfork, and to send the discharge from this point through lampblackened paper to a revolving metallic cylinder on which the paper was stretched. We can to some extent analyze the electric discharge, in these conditions, from the series of perforations left in the paper in the trail of the vibrating fork. This method, though beautiful as an illustration, is useless as a means of investigation; for the metal cylinder, the paper, and the fork form a species of Leyden jar, which is always in the circuit of the particular discharge whose nature you would investigate. The above method, though original with me, cannot be claimed as my own, having recently found that it was devised by Donders, and has been used in an investigation by Nyland T. To get rid of inductive action in the registering apparatus, I devised the following method:-A cylinder is covered with thin printing-paper; and the latter is well blackened by rotating the cylinder over burning camphor. The paper is then removed from the cylinder, and cut into disks about 15 centims. in diameter. When one of these disks is re * From Silliman's American Journal for December 1874. "Ueber die electrische Flaschenentladung,” Pogg. Ann. vol. cxvi. § Journal de Physique, vol. ii. p. 252. Onderzoekingen gedaan in het Physiologisch Laboratorium der Utrechtsche Hoogeschool, 1868-69. ¶ Archives Néerlandaises des Sciences exactes et naturelles, vol. v. p. 292.

1. The value of is (A) 4 (B) 7 (C) 3 (D) 6 (E) 5 2. In the diagram, how many 1 × 1 squares are shaded in the 6 × 6 grid? (A) 29 (B) 30 (C) 31 (D) 32 (E) 33 3. In the diagram, the ratio of the number of shaded triangles to the number of unshaded triangles is (A) 5 : 2 (B) 5 : 3 (C) 8 : 5 (D) 5 : 8 (E) 2 : 5 4. Which of the following is closest in value to 7? (A) (B) (C) (D) (E) 5. Kamal turned his computer on at 2 p.m. on Friday. He left his computer on for exactly 30 consecutive hours. At what time did he turn his computer off? (A) 4 p.m. on Saturday (B) 6 p.m. on Saturday (C) 8 p.m. on Sunday (D) 6 p.m. on Sunday (E) 8 p.m. on Saturday 6. At six different times on Canada Day in 2016, the number of people at the Pascal Zoo were counted. The graph to the right shows these results. During which of the following periods did the number of people at the zoo have the largest increase? (A) 9:00 a.m. to 10:00 a.m. (B) 10:00 a.m. to 11:00 a.m. (C) 11:00 a.m. to 12:00 p.m. (D) 12:00 p.m. to 1:00 p.m. (E) 1:00 p.m. to 2:00 p.m.8 7. If 2x - 3 = 10, what is the value of 4x? (A) 23 (B) 24 (C) 28 (D) 26 (E) 20 8. Three integers from the list 1, 2, 4, 8, 16, 20 have a product of 80. What is the sum of these three integers? (A) 21 (B) 22 (C) 25 (D) 29 (E) 26 9. Wally makes a whole pizza and shares it with three friends. Jovin takes of the pizza, Anna takes of the pizza, and Olivia takes of the pizza. What fraction of the pizza is left for Wally? (A) (B) (C) (D) (E) 10. Which of the following expressions is equal to an odd integer for every integer n? (A) 2017 - 3n (B) 2017 + n (C) 2017n (D) 2017 + n2 (E) 2017 + 2n 11. Jeff and Ursula each run 30 km. Ursula runs at a constant speed of 10 km/h. Jeff also runs at a constant speed. If Jeff’s time to complete the 30 km is 1 hour less than Ursula’s time to complete the 30 km, at what speed does Jeff run? (A) 6 km/h (B) 11 km/h (C) 12 km/h (D) 15 km/h (E) 22.5 km/h 12. A small square is drawn inside a larger square as shown. The area of the shaded region and the area of the unshaded region are each 18 cm2 . What is the side length of the larger square? (A) 3 cm (B) 4 cm (C) 6 cm (D) 9 cm (E) 12 cm 13. Janet picked a number, added 7 to the number, multiplied the sum by 2, and then subtracted 4. If the final result was 28, what number did Janet pick? (A) 9 (B) 5 (C) 19 (D) 23 (E) 11 14. Tobias downloads m apps. Each app costs $2.00 plus 10% tax. He spends $52.80 in total on these m apps. What is the value of m? (A) 20 (B) 22 (C) 18 (D) 24 (E) 26 15. In the diagram, the side lengths of four squares are shown. The area of the fifth square is k. What is the value of k? (A) 64 (B) 49 (C) 36 (D) 25 (E) 16 16. A circular spinner is divided into six regions, as shown. Four regions each have a central angle of x◦ . The remaining regions have central angles of 20° and 140°. An arrow is attached to the centre of the circle. The arrow is spun once. What is the probability that the arrow stops on a shaded region? (A) (B) (C) 17. Igor is shorter than Jie. Faye is taller than Goa. Jie is taller than Faye. Han is shorter than Goa. Who is the tallest? (A) Faye (B) Goa (C) Han (D) Igor (E) Jie 18. Given two different numbers on a number line, the number to the right is greater than the number to the left. The positions of x, x3 and x2 are marked on a number line. Which of the following is a possible value of x? (A) (B) (C) (D) (E) 2 19. In the diagram, M is the midpoint of Y Z, ∠XMZ = 30°, and ∠XYZ = 15° . The measure of ∠XZY is (A) 75° (B) 65° (C) 60° (D) 80° (E) 85° 20. A solid cube is made of white plastic and has dimensions n × n × n, where n is a positive integer larger than 1. The six faces of the cube are completely covered with gold paint. This cube is then cut into n3 cubes, each of which has dimensions 1×1×1. Each of these 1 × 1 × 1 cubes has 0, 1, 2, or 3 gold faces. The number of 1 × 1 × 1 cubes with 0 gold faces is strictly greater than the number of 1 × 1 × 1 cubes with exactly 1 gold face. What is the smallest possible value of n? (A) 7 (B) 8 (C) 9 (D) 10 (E) 4 21. Each of the numbers 1, 5, 6, 7, 13, 14, 17, 22, 26 is placed in a different circle below. The numbers 13 and 17 are placed as shown. Jen calculates the average of the numbers in the first three circles, the average of the numbers in the middle three circles, and the average of the numbers in the last three circles. These three averages are equal. What number is placed in the shaded circle? (A) 1 (B) 5 (C) 6 (D) 7 (E) 14U 22. In the diagram, UV W X is a rectangle that lies flat on a horizontal floor. A vertical semi-circular wall with diameter XW is constructed. Point Z is the highest point on this wall. If UV = 20 and VW = 30, the perimeter of △UVZ is closest to (A) 95 (B) 86 (C) 102 (D) 83 (E) 92 23. An Anderson number is a positive integer k less than 10 000 with the property that k2 ends with the digit or digits of k. For example, 25 is an Anderson number because 625 ends with 25, but 75 is not an Anderson number because 5625 does not end with 75. If S is the sum of all even Anderson numbers, what is the sum of the digits of S? (A) 17 (B) 18 (C) 11 (D) 33 (E) 24 24. A town has 2017 houses. Of these 2017 houses, 1820 have a dog, 1651 have a cat, and 1182 have a turtle. If x is the largest possible number of houses that have a dog, a cat, and a turtle, and y is the smallest possible number of houses that have a dog, a cat, and a turtle, then x - y is (A) 1182 (B) 638 (C) 563 (D) 619 (E) 466 25. Sam thinks of a 5-digit number. Sam’s friend Sally tries to guess his number. Sam writes the number of matching digits beside each of Sally’s guesses. A digit is considered “matching” when it is the correct digit in the correct position. What is the sum of all of the possibilities for Sam’s number? (A) 525 768 (B) 527 658 (C) 527 568 (D) 526 578 (E) 526 758

The present invention relates generally to communication systems, both wired and wireless, employing a continuous phase modulation (“CPM”) waveform. One such CPM waveform is known as minimum shift keying (“MSK”) modulation. The present inventive system and method is applicable to all communication systems and radio frequency bands which utilize an MSK preamble, defined below, to determine baud rate, phase, frequency offset, and bit timing. More particularly, the inventive system and method is applicable to the military satellite communications UHF frequency band for deciding whether a signal of interest is present. Many communication systems or networks, both wired (e.g., Ethernet) and wireless (e.g., HF, VHF, UHF radio), utilize a preamble to determine the modulation carrier frequency and phase. A MSK waveform with an alternating sequence, e.g., 1, 1, 0, 0, 1, 1, 0, 0, . . . , has a characteristic frequency spectrum, sometimes referred to as the “MSK Tones” which also, in addition to carrier frequency and phase, provides modulation symbol rate and accurate baud timing of the MSK waveform. The preceding MSK alternating sequence may be written as [(1≅2),(0≅2)]m which may be generalized in the following form: [(1≅n),(0≅n )]m where the variable “n” may be referred to as the “symbol repetition factor” and the variable “m” may be referred to as the “symbol pair repetition factor”. Other MSK waveforms that fit this general pattern, e.g., 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, . . . , which can be written as [(1≅3),(0≅3)]m and 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, . . . , which can be written as [(1≅40,(0≅4)]m, are all contemplated by the present inventive system and method. There are several problems with using an MSK waveform preamble for waveform detection and parameter estimation which must be overcome in order to allow for accurate measurement of symbol rate, frequency, phase, timing, and signal strength (measured, for example, as signal-to-noise ratio, carrier-to-interference ratio, etc.). In most communication systems it is highly desirable to limit the preamble time and begin to transmit information-carrying data as soon as possible since the preamble essentially displaces data-carrying capability. However, limiting the preamble time has the effect of limiting the signal energy of the preamble which increases the difficulty in finding the characteristic MSK tones in the frequency spectrum due to noise. The transition in energy at the start of the preamble exacerbates this problem because the unit step in energy at the start of the preamble induces tones in the frequency spectrum, making it harder to distinguish the MSK tones from the noise. This forces the acquisition algorithm to pay close attention to gain control because amplitude changes, which are usually implemented in gain “steps”, also add tones to the frequency spectrum. Additionally, tones are introduced to the frequency spectrum due to amplitude changes which are typically implemented in predetermined step changes by the gain control algorithm. Therefore, the gain control algorithm must attempt to minimize the number of step changes in order to minimize the introduction of tones into the frequency spectrum and thereby maximize the ability to detect the MSK tones. The implementation of the Fourier Transform (“FT”) or the Laplace Transform (“LT”) are common methods to convert time domain data to frequency domain data for analysis. The FT has discrete bins which contain the energy as correlated with a frequency offset for each bin. The Fourier Transform may be implemented as a Discrete Fourier Transform (“DFT”) or as a Fast Fourier Transform (“FFT”) in those devices that are computationally-limited. It is to be understood that any method for converting time domain data to frequency domain data, such as Fourier Transforms, Laplace Transforms, discrete cosine, etc., are contemplated by the present invention and any method for converting time domain data to frequency domain data may be referred to herein, individually and/or collectively, as a Fourier Transform, or “FT” as would be consistent in the context as used. The use of a FFT to find MSK tones at a specific spacing equal to one-quarter of the symbol rate of the received signal is described in the paper “An Innovative Synchronization Preamble for UHF MILSATCOM”, authored by Mark Miller, Mark A. Harris, and Donald R. Stephens (the “Miller Paper”), which is hereby incorporated herein by reference. This paper only describes the use of the FFT to find the MSK Tones and implies the use of a correlation function to determine the characteristic spacing of the MSK Tones. It describes the use of the FFT Bin number to find the Carrier Frequency and the Phase value of that center bin to find the Carrier Phase. It describes the use of the Phase difference between the carrier signal and the +/−1 MSK Tones to determine symbol phase (timing). However, use of the method disclosed in the Miller Paper results in a lot of false alarms, e.g., reporting a detection on noise, a foreign signal, or on an impulse signal. Part of the problem with the procedure used in the Miller paper is that the use of the center frequency of the bin with the largest amplitude is at best a rough estimate of the frequency of the carrier. The invention herein described includes the functionality of signal detection as well as a much more accurate method of signal parameter estimation. The present inventive system and method overcomes the problems of the prior art as detailed below. Generally, there are two major issues with the FT approach to finding the characteristic MSK tones. The first problem deals with computational horsepower required to perform an FT. For example, a FT or DFT may be computationally burdensome, typically requiring N2 operations where N is the number of frequency bins. A FFT, by contrast, typically only requires N*log(N) operations and is therefore less computationally burdensome. However, with either an FT, DFT, or FFT, the fewer the number of bins the less frequency resolution is attainable. Therefore, an undesirable tradeoff is required between computational intensity and frequency resolution. Typical prior art solutions strike a compromise between computational intensity and frequency resolution by merely using the center frequency bin of the FT. The present inventive system and method overcomes the compromise problem by taking two contiguous FTs and, generally, determining the difference in phase for the bin of the center (carrier) frequency of the contiguous FTs to accurately determine the waveform frequency. The actual implementation used will be discussed in detail further below. The second major issue with the FT implementation is that any frequency which is not an exact integer multiple of the data sample rate divided by the number of FT bins ends up with energy split between two adjacent bins of the FT. This effect tends to hide the characteristic MSK tones in the surrounding noise. The present inventive system and method solves this problem by multiplying the input data by a constant tone which may correspond to exactly ½ bin frequency, or some other fraction of bin frequency, and then performing another FT on the input data that has been multiplied by the constant tone and comparing the results with the results of the FT performed on the non-multiplied input data. The procedure creates two FT's on the same time-domain data. In one case, the carrier frequency will be located more closely to the center of an FT bin. In the other case, the carrier frequency will be located closer to the edge of the bin which may cause energy spillover into the next, adjacent FT bin. This energy spillover is undesired because it reduces the apparent signal strength in relation to the noise energy. The worst case situation occurs when tone appears exactly on a bin edge. In this case, the tone energy will be equally distributed between the two adjacent FT bins. For the single bin of interest (the transmitted carrier frequency), the apparent signal to noise ratio is 3 dB (a power ratio of 2) less than the signal to noise ratio which would be apparent if the FT was modified to locate the transmit carrier tone at the center of an FT bin. With two candidate FT's, choosing the transmit carrier frequency which has the highest magnitude will provide the best possible representation of the transmitted signal (in the frequency domain). This will improve the apparent signal to noise ratio which will improve the signal estimation performance in noisy environments. The MSK preamble has an additional problem because a Fourier Transform of time domain data which is not an exact multiple of MSK symbols of the [(1≅n),(0≅n)]m pattern has an effect of varying the amplitude of the characteristic tones with respect to each other in a “walking” type pattern, i.e., the tone energy is constant but the tones repeated throughout the spectrum will not have a symmetric pattern in amplitude. This effect increases the difficulty in finding the characteristic (MSK) tones in the noise. To resolve the problem, the inventive system and method utilizes a polyphase resampler (interpolation and decimation method) to exactly place the sample rate of the time domain signal at an exact integer multiple of the symbol rate of the MSK waveform. Thus there is a need for a system and method which can detect a continuous phase modulation waveform with a shortened MSK preamble and overcome the limitations of prior art systems/methods. The present inventive system and method separates the detection and estimation functions. During detection, the baud rate is calculated as a first estimated parameter which is then utilized by the receiver for tailoring the signal sample rate and bandwidth to better match the incoming signal before calculating the transmitted carrier frequency, phase, and bit timing. The signal to noise ratio improvement which results from the tailoring process (of resampling and filtering) consequently improves the accuracy of the calculated values. The method described in the Miller paper does not separate the detection and estimation functions and therefore cannot accommodate the tailoring of the sample rate and bandwidth to match the incoming signal. Additionally, the present inventive system and method performs a baud rate calculation by sorting the FT tones in order of amplitude and measuring the bin distance between the closest two tones. The expected value of the two closest tones is equal to the symbol rate divided by 2n where “n” is the number of bit repeats in the preamble from the [(1≅n),(0≅n)]m form of the MSK preamble. This baud rate calculation greatly improves the baud rate estimation algorithm as compared to the correlation described in the Miller paper because the inventive method takes advantage of the large signal to noise ratio apparent in a FT bin that contains a large signal level. The correlation method from the Miller paper integrates the noise in the entire range of frequencies which cover the transmit carrier frequency (e.g., +/−1500 Hz) and the MSK Tones (e.g., +/−28,000 Hz), which results in a sum of +/−29,500 Hz added into the correlation function. Additive White Gaussian Noise is, by definition, equally distributed in each frequency bin and the correlation process proposed in the Miller paper would integrate the noise in all of those bins. The sorting method used by the present invention excludes energy in those bins which do not contain enough signal energy to cause a signal detection. Therefore, the present invention excludes the noise from the baud rate estimation calculation as will become apparent in the detailed description below. Furthermore, the detection calculation of the present invention determines the amplitude of largest tone (typically the carrier frequency) and adds the amplitudes of the +/−1 MSK Tones and the +/−2 MSK Tones in the signal detection calculation. The Miller paper does not address detection of the incoming signal at all. The present invention also uses two adjacent FT windows to measure the phase difference of the carrier frequency between the two windows to thereby accurately determine the carrier frequency. Moreover, the present invention uses ½ tone spacing in a third FT window in order to place the amplitude of the carrier frequency in the most advantageous location so that carrier energy is not dispersed between two FT bins, as it I would be if the carrier frequency were located at (or near) the edge of a bin. It is to be understood that the present invention contemplates offsetting the third FT by any amount, and is not limited in any way to just ½ tone spacing, so as to place the carrier frequency in the middle of the bin. The Miller paper is silent on the use of adjacent FT bins as well as on the use of a third FT spaced apart by a half tone. One embodiment of the present invention avoids the problems of the prior art by using two or more contiguous Fourier Transforms for detecting a continuous phase modulation waveform and determine the characteristics of the waveform such as frequency, phase, timing, and signal strength. Accordingly, it is an object of the present invention to obviate many of the above problems in the prior art and to provide a novel system and method for detecting a continuous phase modulation waveform. It is another object of the present invention to provide a novel system and method for determining the waveform characteristics of a continuous phase modulation waveform. It is yet another object of the present invention to provide a novel system and method for using two or more Fourier Transforms in the detection of a continuous phase modulation waveform with an MSK preamble. It is still another object of the present invention to provide a novel system and method for using two or more Fourier Transforms to determine the characteristics of a received continuous phase modulation waveform with an MSK preamble. It is a further object of the present invention to provide a novel system and method for using two or more Fourier Transforms for acquiring a continuous phase modulation waveform with an MSK preamble. It is yet a further object of the present invention to provide a novel system and method of determining the frequency of a received and detected continuous wave waveform. BRIEF DESCRIPTION OF THE DRAWINGS These and many other objects and advantages of the present invention will be readily apparent to one skilled in the art to which the invention pertains from a perusal of the claims, the appended drawings, and the following detailed description of the preferred embodiments. FIG. 1 is a diagram showing an incoming signal being divided into blocks, contiguous Fourier Transforms (FT1 and FT2) being taken, resampling the blocks and another set of contiguous Fourier Transforms (FT3 and FT4) being taken, multiplication of a block of data by a tone and another Fourier Transform (FT5) being taken. FIGS. 2A through 2E are a flow chart indicating the major steps for detecting a signal according to the present invention (FIGS. 2A and 2B) and the major steps for estimating characteristics of a signal according to the present invention (FIGS. 2C through 2D). DESCRIPTION OF PREFERRED EMBODIMENTS FIG. 3 is a flow chart indicating the major steps in determining the frequency of a received continuous wave waveform according to the present invention. With reference to the drawings, like numerals represent like components/steps throughout the several drawings. Many communication systems, both wired and wireless, use Continuous Phase Modulation (“CPM”) waveforms. One particular type of CPM waveform is the Minimum Shift Keyed (“MSK”) waveform. As is known in the art, an MSK waveform with an alternating sequence, such as 1, 1, 0, 0, 1, 1, 0, 0, . . . , generalized to the form: [(1≅n),(0≅n)]m, also referred to herein as an “MSK preamble”, has a characteristic frequency spectrum from which can be used to determine parameters of the received signal, such as carrier frequency, carrier phase, modulation symbol rate and baud rate. The present inventive system and method makes use of two or more contiguous Fourier Transforms, Discrete Fourier Transforms, and/or Fast Fourier Transforms to determine the aforementioned parameters. It is to be understood that any MSK waveform of the generalized form, i.e., where n=1, 2, 3, . . . , is contemplated by the present inventive system and method. An MSK waveform of the above generalized form has a frequency spectrum which has a tone at the carrier frequency and characteristic tones which are spaced at the waveform symbol rate divided by 2n. For example, for the alternating sequence 1, 1, 0, 0, 1, 1, 0, 0, . . . , the characteristic tones are spaced apart from the carrier frequency by the waveform symbol rate divided by 4. Although this property of the MSK waveform is well documented, the present inventive system and method describes an approach to the accurate detection and estimation of the MSK preamble that is both innovative and easily implemented. If the input to a FT is a time domain signal, the output of an FT is a frequency domain signal in a predetermined number of bins. Each bin, by itself, can give an estimate of the frequency and phase of the incoming signal. However, the estimate is typically too inaccurate to allow for accurate parameter estimation. The FT frequency domain output counts the number of cycles of the input time domain waveform. For example, bin 5 of a FT will typically represent frequencies between 4½ and 5½ Hz, or cycles. Bin zero (sometimes referred to herein as bin 0 ), typically the first bin of the FT, is the bin that includes a zero offset between the transmitter and the receiver, which implies that the transmitter and receiver are synchronized. For example, for a FT with 256 bins, numbered 0 to 255, bin 0 is the zero offset bin with the offset increasing from bin 1 through bin 255 such that bin 255 has the same magnitude, but opposite direction, offset of bin 1 . Therefore, the bins can be ordered in a series with bin 0 in the middle of the series and tho same magnitude offsets being the same number of bins away from bin 0 . For example, the ordering of the FT bins in a series for a maximum offset of 6 bins in either direction from the zero offset bin (bin 0 ) would be: |BIN ||BIN ||BIN ||BIN ||BIN ||BIN ||BIN 0 ||BIN 1 ||BIN 2 ||BIN 3 ||BIN 4 ||BIN 5 ||BIN 6 | |250 ||251 ||252 ||253 ||254 ||255 | It is to be understood that in the above explanation, the use of 256 bins is exemplary only and the present invention contemplates the use of any number of bins. Additionally, the placement of Bin 0 is arbitrary and can be placed anywhere in the series of bins. The present inventive system and method makes use of contiguous, non-overlapping FTs, from contiguous, non-overlapping samples, in order to accurately determine the parameters of the received signal. In doing so, the magnitude and phase of the various FT bins are utilized. For example, the phases of the contiguous FTs are used to determine frequency of the received signal. The more contiguous FTs that are used, the higher the resolution and the more accurate the estimate of the frequency. It is to be understood that although the present description may recite only two contiguous FTs, the present inventive system and method contemplates the use of two or more contiguous FTs. With attention now to FIG. 1, an incoming signal 10 is received by the receiver. What is meant herein by “received” is that the raw incoming data, at the predetermined sampling rate, enters the processing circuitry at the receiver. The incoming signal may be any CPM signal with an MSK preamble transmitted over any medium. Specifically, the signal may be a signal on the Military Satellite Communications UHF Frequency band. In order to detect the incoming signal 10 and determine the characteristics of the signal, the signal is divided into N equal blocks of a predetermined number of samples, as is known in the art. One embodiment of the present invention utilizes a block size of 256 samples. The block of data designated Block 1 may generally be thought of as the block of samples that contain the initial received energy of the incoming CPM signal. The subsequent blocks are each contiguous with their immediate neighbors. The first Fourier Transform, designated as FT1, is taken on the data Block 1. FT2 then begins with the sample immediately after FT1 ends, so that FT1 and FT2 are contiguous without any overlap. The information derived from either FT1 or FT2 as well as other known information about the incoming signal, such as maximum allowable carrier offset and maximum symbol rate, are sufficient for detecting the incoming signal 10. A FT may be performed on the other data blocks, but are not shown in FIG. 1 for clarity purposes. The incoming signal 10, or parts thereof, are resampled to thereby create the resampled incoming signal 20. Typically, data Block 1 and data Block 2, which were used for detection purposes, will not be resampled, although resampling those two data blocks is contemplated by an embodiment of the present invention. The resampled rate is typically lower than the sampling rate in order to ensure adequate sampling of the incoming signal. However, it is contemplated that the resampled rate may also be the same as or faster than the original sampling rate. The FT3 is taken on the resampled data Block 3 and the FT5 is taken on the resampled data Block 4 such that the FT3 and the FT5 are contiguous. A FT may also be performed on the other resampled data blocks, but those FTs are not shown in FIG. 1 for the sake of clarity. The resampled data Block 3 is modified by multiplying by a tone, exp(jT/s) where “T” is the frequency and “s” is the new sample rate, then FT4 is taken on the modified resampled data Block 3. Preferably, the tone shifts the data in resampled Block 3 by ½ the bin spacing. The ½ bin spacing shift is exemplary only and is not meant to limit the application in any way. Other bin spacings are contemplated, but shifting by ½ bin spacing allows for the greatest amount of resolution as will become apparent in the discussion below. As discussed in more detail below, the information from the FT3, FT4, and FT5 operations, in addition to the information provided above for detection, allow the characteristics of the incoming signal 10 to be accurately estimated. With reference now to FIGS. 2A and 2B, a flow chart is depicted showing the major steps for detection of a CPM waveform with an MSK preamble. At the step 202, a sample rate is chosen. Since the symbol rate and maximum allowable carrier offset for the incoming signal is unknown at the time of detection, the sample rate that is chosen must be high enough to allow for (a) the maximum allowable carrier frequency excursion as well as (b) the MSK characteristic tones from the MSK preamble. The MSK tones are typically spaced apart by the baud rate, which for an MSK preamble of the form [(1≅n),(0≅n)]m, where n=2, i.e., a preamble of the form 1, 1, 0, 0, 1, 1, 0 . . . . . , is the symbol rate divided by 4. Generally, the baud rate equals the symbol rate divided by 2n. For example, for an MSK preamble with a 28,000 symbols/sec symbol rate and a ∀1500 Hz maximum allowable carrier offset, a sample rate of 64,000 complex samples/sec (e.g. I and Q samples) will provide a good initial frequency estimate for the incoming signal. It is to be understood that the above example is not intended to limit the present invention in any manner but rather is included to assist in the understanding of the invention. During the step 204 partitions the incoming signal into data blocks each containing a predetermined number of samples. One useful partitioning of samples is 256 samples per block, since 256 is a power of 2, although other block sizes are also contemplated by the present invention. As shown in FIG. 1, the incoming signal 10 is partitioned into N blocks of S samples. At the step 206, the FT bins are normalized by any means known in the art, such as: [I2+Q2]1/2 ; or adding the larger of I or Q to ½ of Q ½ of I, respectively, so that the sum of the amplitudes of all the samples divided by the bin size is unity. There can be any number of bins for the FT, consistent with the discussion above for computation intensity and resolution. For a block size of 256 samples/block, a preferable number of FT bins is 256. Alternative numbers of bins are contemplated by the present invention. Therefore, for a sampling rate of 64,000 samples/sec and 256 bins, the bin size, or frequencies (or samples) per bin, is 250. It is preferable, though not required, to choose the sample rate such that the allowable symbol rates for the incoming signal all appear as an exact integer multiple of the bin size, Hz/bin. As would be known to those of skill in the art, the FT used would typically be a complex FT, although other types of FT are also contemplated by the present inventive system and method. The RMS amplitude difference between the samples in a first portion of data Block 1 and the samples in a second portion of data Block 1 is also determined. The first portion of data Block 1 comprises those samples in data Block 1 that are not included in FT1. The second portion of data Block 1 comprises those samples in data Block 1 that are included in FT1. The RMS amplitude difference will be used later as one indication of detection of the incoming signal. At the step 208, the Tone Bin Distance (“TBD”) is determined. The TBD is the number of bins between the MSK tones as determined by the baud rate. The TBD is approximately equal to the baud rate (samples/sec, or Hz) divided by the bin size (Hz/bin). The TBD will be used, as discussed below, to determine a cumulative amplitude which is used as one indication of detection of the incoming signal. A first FT, FT1 in FIG. 1, is performed on the first data block, Block 1 in FIG. 1, that typically contains the initial energy of the incoming signal. The FT1 is performed by any means known in the art. If the signal does not appear in the Block 1 the FT1 is discarded, along with the samples in the Block 1and the FT2 performs the same tasks as would have FT1. Therefore, the procedure can be regarded as Block 1 is the block of data within which the incoming signal 10 appears. At the step 210, the baud rate of the incoming signal is determined. The baud rate may be determined by sorting the bins of FT1 in amplitude order, designating a set of the largest amplitude bins as containing the MSK tones, and determining the closest bin distance (by the difference in bin numbers) between any two of the adjacent MSK tone bins. Typically, the three or five bins with the largest amplitudes are designated as the MSK tones, although other numbers of bins are contemplated. For example, if the five bins with the largest amplitudes, sorted in amplitude order, are bin 0, bin 253, bin 4, bin 9, and bin 248, the closest bin distance between any two of the bins is 3 bins between bin 0 and bin 253. The baud rate can then be determined by the closest bin distance (number of bins) and the bin size (samples/bin). If the determined baud rate is approximately equal to an expected symbol rate divided by 2n, then there is possible signal detection, as will be discussed further below. Once the baud rate is determined, the bandwidth of the incoming signal can be determined by techniques known in the art, and a variable bandwidth filter can be adjusted so as to take advantage of the knowledge of the bandwidth of the incoming signal. With reference now to the step 216 , a Carrier Frequency Window (“CFW”) is determined. The CFW contains a number of bins as a function of the maximum allowable carrier offset. For example, if the maximum allowable carrier offset is ∀1500 Hz, and the bin size (as determined above) is 250 Hz/bin, then a Carrier Bin Offset (“CBO”) is 6 bins. The CFW is comprised of the zero bin offset, bin 0 , plus those bins that are one CBO greater than and one CBO less than bin 0 . For this example, the CFW is shown below: |BIN ||BIN ||BIN ||BIN ||BIN ||BIN ||BIN 0 ||BIN 1 ||BIN 2 ||BIN 3 ||BIN 4 ||BIN 5 ||BIN 6 | |250 ||251 ||252 ||253 ||254 ||255 | The above example is for illustrative purposes only and is not to be construed as limiting the invention in any way. With attention now drawn to the steps 218 and 220, the cumulative amplitude of each bin in the CFW, including the amplitudes of the ∀MSK tone for each respective bin, is determined. The use of the amplitudes of just the bins in the CFW does not result in a sufficiently low error rate to be useful. Therefore, the amplitudes of the respective ∀MSK tones are added in to allow for a more robust detection algorithm. The present invention contemplates determining the cumulative amplitude of the CFW bin and the ∀1MSK tones as well as determining the cumulative amplitude of the CFW bin and the ∀1 and ∀2 MSK tones. As an example, assume that the CFW is as shown above (where the maximum allowable carrier offset is ∀1500 Hz), for a FT with 256 bins, and a TBD of 28 bins (as determined from a baud rate of 7000 Hz divided by a bin size of 250 Hz/bin). The TBD of 28 indicates that an MSK tone should appear every 28 bins from bin 0, i.e., bins 28 , etc. Therefore, for this example, in determining the cumulate amplitude for each of the bins in the CFW, including the ∀1 MSK tones, the amplitudes of the following bins (row wise) would be added together: |Bin # of ||Bin # of −1 ||Bin # of | |Bin in ||MSK ||+1 MSK | |CFW ||Tone ||Tone | |250 ||222 ||22 | |251 ||223 ||23 | |252 ||224 ||24 | |253 ||225 ||25 | |254 ||226 ||26 | |255 ||227 ||27 | |0 ||228 ||28 | |1 ||229 ||29 | |2 ||230 ||30 | |3 ||231 ||31 | |4 ||232 ||32 | |5 ||233 ||33 | |6 ||234 ||34 | It shall be understood by those of skill in the art that the above example is illustrative and that the procedure exemplified can be applied to other assumed values. Alternatively, the determination of the cumulative amplitude can include the bins immediately adjacent to the bin in the CFW and/or the bins adjacent to the ∀1 MSK tones and/or the bins adjacent to the ∀2 MSK tones (if the bins for the ∀2 MSK tones are used in calculating the cumulative amplitude. The use of the adjacent bins is useful in the event carrier frequency is located at the edge of a bin. As an example, the cumulative amplitude can be determined for each of the bins in the CFW, including the ∀1 MSK tones,and the adjacent bins for the bins in the CFW and the ∀1 MSK tones by adding the amplitudes of the following bins (row wise): |Bin # of || ||Bin # of −1 || ||Bin # of || | |Bin in ||Adjacent ||MSK ||Adjacent ||+1 MSK ||Adjacent | |CFW ||Bins ||Tone ||Bins ||Tone ||Bins | |250 ||249, 251 ||222 ||221, 223 ||22 ||21, 23 | |251 ||250, 252 ||223 ||222, 224 ||23 ||22, 24 | |252 ||251, 253 ||224 ||223, 225 ||24 ||23, 25 | |253 ||252, 254 ||225 ||224, 226 ||25 ||24, 26 | |254 ||253, 255 ||226 ||225, 227 ||26 ||25, 27 | |255 ||254, 0 ||227 ||226, 228 ||27 ||26, 28 | |0 ||255, 1 ||228 ||227, 229 ||28 ||27, 29 | |1 ||0, 2 ||229 ||228, 230 ||29 ||28, 30 | |2 ||1, 3 ||230 ||229, 231 ||30 ||29, 31 | |3 ||2, 4 ||231 ||230, 232 ||31 ||30, 32 | |4 ||3, 5 ||232 ||231, 233 ||32 ||31, 33 | |5 ||4, 6 ||233 ||232, 234 ||33 ||32, 34 | |6 ||5, 7 ||234 ||233, 235 ||34 ||33, 35 | With reference now to FIG. 2B , specifically to the steps 222 , and 226 , a determination of whether the incoming signal is detected is made. In order to show detection of the signal: - 1. at the step 222, the baud rate as determined in the step 210 must be equal to proximately a known symbol rate for the incoming signal divided by 2n where n defines the MSK preamble [(1 ≅n),(0≅n)]m; - 2. at the step 224, the FT window does not experience a large change of amplitude as determined by the difference in RMS amplitude from the step 212; and - 3. at the step 226, the largest cumulative amplitude from the steps 218 and 220 is less than a predetermined threshold. At the step 228, if all three of the above tests are passed, the incoming signal has been detected. If any of the above tests are not passed, there is no detection, as indicated at the step 230. With attention now to FIGS. 2C and 2D, the flow chart from FIGS. 2A and 2B is continued. FIGS. 2C and 2D depict the major steps for determining certain characteristics of a CPM waveform with an MSK preamble. Alternatively, the procedure depicted in FIGS. 2C and 2D can be performed independently from the detection procedures in FIGS. 2A and 2B, so long as the incoming waveform has been detected. At the step 250 in FIG. 2C, the data blocks of the incoming signal, e.g., the incoming signal 10 in FIG. 1, are resampled at a predetermined resampling rate that is preferentially greater than the sampling rate chosen in the step 202. In one embodiment, the blocks of data that are resampled are the last half of the block in which the incoming signal 10 appears and enough of the remaining data blocks after that point for completing the determination of the waveform characteristics of the incoming signal. Typically, this is at least 2 data blocks, as shown in FIG. 1. Preferentially, the resampling rate is 2n where n, the symbol repetition factor, is from the aforementioned general equation for the MSK preamble [(1≅n),(0≅n)]m. However, the resampling rate may be any reasonable rate as contemplated by the invention. The resampling changes the time base of the data in the data blocks. The resample rate is preferentially a multiple of the MSK signal symbol rate. For example, if there are four samples per symbol in the radio and using the previous example of an MSK preamble with a 28,000 symbols/sec rate, the resample rate would preferentially be 112,000 samples/sec. By requiring that the resample rate be an integer multiple of the symbol rate removes the MSK tone “walking” phenomenon, which is caused by boundary effects of the FT bins. Prior art solutions to this problem incorporate the use of FT “windows” to overcome the MSK tone “walking” phenomenon. However, FT “windowing” degrades the signal energy. The tying of the resample rate to an integer multiple of the symbol rate has the additional benefit of filtering out out-of-band noise and thus improving the signal-to-noise ratio of the received signal. With reference now to the steps 252 through 268, loops in the procedure occur for taking FT3, FT4, and FT5, which are shown in FIG. 1. The operations performed in the steps 252, 254, and 258 correspond to the operations performed on the steps 206, 208, and 216 and will not be discussed further here. The step 260 corresponds to the step 218 with the insertion of the substep of determining the largest cumulative amplitude for the FT for which the loop is currently being run. The step 262 is a decision step to determine when FT4 will be run. For example, during the first pass through the loop, FT3 is taken on the resampled data Block 3 of FIG. 1. As previously discussed for FIG. 1, another FT, FT4, will be taken on resampled data Block 3 after the samples have been multiplied by a tone in the step 264: exp(jT/s) where “T” is the frequency and “s” is the resampled rate, then FT4 is taken on the modified resampled data Block 3. Preferably, the tone shifts the data in resampled Block 3 by ½ the bin spacing, although other shifts of the data may be used as appropriate. After the loop of the steps 252 through 260 is repeated for FT4, the step 262 is answered in the affirmative thereby opening the path to the step 266. If an FT is to be performed on the next data block, which it will for FT5, the step 268 increments the next block of data and FT5 is performed. Note that FT5 is not performed on samples that have been multiplied by the tone exp(jT/s). The loop can be repeated as many times as necessary in order to obtain the data required for determining the characteristics of the incoming signal, as discussed further below. Alternatively, since ideally FT5 does not need to pass through all the steps in the loop, the only steps in the loop that need to be performed for FT5 are the normalizing, step 252, and the performing the FT, step 254. This is indicated in FIG. 2C by the dotted line between the step 254 and the step 270. With reference now to the step 270, the bin containing the carrier frequency is determined from the bins of FT3 and FT4. The bin with the largest cumulative amplitude from FT3 and FT4 is the bin that contains the carrier frequency. The center frequency of the bin that contains the carrier frequency is the carrier frequency estimate. With attention now directed at FIG. 2D, the carrier frequency estimate from the I step 270 must be refined by determining a phase difference in the step 272 and a frequency offset in the step 274. At the step 272, the phase difference is determined between the phase value of the carrier frequency bin and the phase value of the bin from FT5 that has the same bin number as the carrier frequency bin. This phase difference corrects for the FT frequency which is not located at the exact center of the FT bin. At the step 274, the frequency offset is determined by dividing the phase difference in the step 272 by the time duration of one of the FTs. At the step 276, the carrier frequency is determined by adding the frequency offset from the step 274 to the carrier frequency estimate from the step 270. With reference now to FIG. 2E, the phase and timing of the incoming signal will now be determined from the steps 278 through 286. At the step 278, the phase of the bins containing the ∀1 MSK tones for the carrier frequency bin is determined. One method of determining the phase is by taking arctan(Q/I) for these bins, although any method for determining phase can be used. At the step 280, a delay offset is determined by calculating the difference in phase between the ∀1 MSK bins and dividing the difference in phase by the bin distance between the ∀1 MSK tones, which may be the same as twice the TBD. At the step 281, the baud rate offset is determined, by any method known in the art. The baud rate offset is a timing offset which is used to determine where the FT started with respect to the start of the preamble. At the step 282, the phase of the carrier frequency bin is determined by known methods and at the step 284, a modified phase for the carrier frequency bin is determined by subtracting the delay offset determined in the step 280 from the phase of the carrier frequency bin determined in the step 282. At the step 285, the frequency offset calculated in the step 274 is subtracted from the modified phase for the carrier frequency bin determined in the step 284 to thereby determine the phase of the incoming signal at the step 286. With reference now to the step 288, the signal level of the incoming signal is determined from the gain values from one of the FT bin normalizing steps. Accordingly, the frequency, phase, and signal level characteristics of the incoming signal are determined. With attention now drawn to FIG. 3, the present inventive system and method can also be used to determine, by the use of contiguous Fourier Transforms, the frequency of a received and detected continuous wave (“CW”) waveform. At the step 302, the CW waveform is received and detected by any means known in the art. At the step 304, a first FT is performed on a first set of samples. At the step 306, a second FT is performed on a second set of samples where the first and second set of samples are contiguous, as are the first and second FTs. At the step 308, the bin number of the bin with the largest amplitude, from both the first and second FTs, is determined. At the step 310, a phase difference is determined between the phase values for the bin with the largest amplitude, as determined in the step 308, and the bin with the same bin number as the largest amplitude bin but is from the FT that does not contain the bin with the largest amplitude. At the step 312, a frequency offset is determined by dividing the phase difference determined in the step 310 by the time duration of one FT. At the step 314, the frequency of the CW waveform is determined by adding the frequency offset determined in the step 312 to the center frequency for the bin with the largest amplitude determined in the step 308. It can be seen by those of skill in the art that the carrier frequency of a CPM waveform with a MSK preamble can be determined by information derived from contiguous FT blocks. While preferred embodiments of the present invention have been described, it is to be understood that the embodiments described are illustrative only and that the scope of the invention is to be defined solely by the appended claims when accorded a full range of equivalence, many variations and modifications naturally occurring to those of skill in the art from a perusal hereof.

An imaging design approach for optical systems consisting of two aspheres which is free of astigmatism is presented in this paper. A set of implicit differential equations is derived from generalized ray tracing. The solution of the derived equations provides the profiles of the two aspheres as well as the object to image mapping. The obtained design can be used as a good starting point for optimization. Particular examples are given. © 2015 Optical Society of America Aspheric surfaces can in general provide more compact imaging designs with better performance than their spherical counterparts. We call aspheric surface or asphere to a rotational symmetric surface which is not a portion of a sphere. Schmidt corrector plate is an early example of a single aspheric design to correct spherical aberration in a telescope , and Schwarzschild pioneered the design of two aspheres for an aplanatic system . Later, Wassermann and Wolf generalized Schwarzschild approach presenting a method for the design of two aspheres as solution of two simultaneous first-order ordinary differential equations. Standard integrating methods have been applied to obtain the aspheric profiles numerically. More recently Willstrop and Lynden-Bell gave general analytic solutions of the two-mirror aplanatic design problem [4, 5]. The 2D-SMS (Two-dimensional Simultaneous Multiple Surfaces) method, which can be seen as a system of functional differential equations, is another design procedure that has been used to design directly two or more aspheres for imaging . It has been used, for instance, to design a telephoto for the SWIR band . In this design the 6 SMS aspheres fitted with Forbes polynomials [8, 9] were used as the starting point of an optimization procedure. As yet there are 2D-SMS designs perfectly imaging up to 8 fields. This method can also be used to image several fields using less aspheres, only by controlling the size of the pupil aperture . Novel direct methods for designing an anastigmatic (free of astigmatism) single-asphere and an anastigmatic single-freeform-surface optical systems have been recently presented [11, 12]. These methods are based on finding a differential equation on the optical surface deduced by equating the principal curvatures of the output wavefronts. In this design procedure, the mapping between object and image points cannot be prescribed. This mapping and the shape of the image surface are obtained as a result of the design process. Here, we have extended this strategy to the design of a double asphere anastigmatic optical system. The second surface provides additional freedom and, unlike the preceding single surface method, now both the object and the image surface shapes can be prescribed. The derivation of the differential equations is done in section 3. In section 4, an anastigmatic design is presented and compared with an aplanatic design as a starting point for multi-parameter optimization. 2. Statement of the problem The design method introduced here adopts the definition of “supporting” and “supported” wavefronts from . The input supporting wavefront is normal to a set of rays in the object space, which once refracted or reflected by the two unknown aspheres, define the output supporting wavefront in the image space. There is a supported wavefront per each ray of the supporting wavefront. This ray is also normal to its corresponding supported wavefront. Then, the supported wavefronts are tangent to the supporting wavefront, but with different principal curvatures at the points of tangency. In our approach, we will consider a second order approximation of the supported wavefronts in the neighborhood of the point tangent to the supporting wavefront, therefore, each supported wavefront surface will be fully characterized by the tangent point, the ray direction and the value of the principal curvatures relative to those of the supporting wavefront, all these values taken at the tangent point. The design problem for double surface system is to find the optical surfaces (refractive or reflective) such that the second order approximations of the output supported wavefronts when they get the image are spheres centered at the image points, so the design is free from astigmatism. The image surface, where the centers of curvature of the spherical output supported wavefronts lay on, will be prescribed, as well as the input supporting and supported wavefronts. This description is adequate to model an optical system with a small aperture stop. A schematic drawing of the system is shown in Fig. 1. These design conditions lead to a group of three implicit ordinary differential equations in the general case as shown in the next section. 3. Differential equations for double optical surface profiles We consider a general double surface optical system with refractive index n2 between two surfaces and n1 elsewhere shown in Fig. 2. We will restrict our analysis henceforth to the particular case in which the input supporting wavefront is a sphere centered at the origin O. (rf (α), α) and (rs(β), β) are points on the first and the second surface respectively, in two polar coordinates with the same origin O. θ 1, θ 2, θ 3 and θ 4 are angles between the ray vector and the surface normal. The prescribed image surface is given in implicit form by S(x, y) = 0. The two surface profiles are defined by the three unknown functions rf (α), rs(β) and β(α). According to the generalized ray tracing equations , the change of the curvatures of the supported wavefronts due to the refractions are expressed by the following equations:14]. The radii of curvature of the wavefronts between refractions are expressed by: Anastigmatic designs require ρq4 = ρp4. Then the supported wavefronts are locally spherical after the second refraction. Henceforth we shall restrict the analysis to the case where the second order approximation of the input supported wavefronts are spheres with radius R(α) at the point of the first refraction. SoEqs. (1)-(3), (5) and the anastigmatic condition (ρq4 = ρp4) we can eliminate the 8 variables ρqi, ρpi (i = 1, 2, 3, 4) to get a single equation: The loci of the curvature centers of the output supported wavefronts are calculated from the radii of curvature and the direction of the output supported wavefronts. Since the curvature centers have to be on the prescribed image surface, the output supported wavefronts and the prescribed image surface are connected by: Substituting these equations into S(x, y) = 0, we get Now get the expression of ρp4 from the second equations in Eqs. (1)-(3) as Finally, let’s derive the following equation from the sine law applied to the triangle formed by the origin and the two points on the two surfaces where the refractions occur: Consider now Eq. (6) after substituting D (Eq. (4)) into it; Eq. (8) after substituting ρp4 (Eq. (9)) and D (Eq. (4)) into it; and Eq. (10). These three equations relate the variables n1, n2, R, α, β, rf, rs, ρpf, ρqf, ρps, ρqs, θ 1, θ 2, θ 3, θ 4 and the function S(x, y) = 0. ρpf, ρqf, ρps and ρqs can be expressed as functions of α, β, and rf´, rf´´, rs´, β´ (where ′ = d/dα and ″ = d2/dα 2) using the general formulas for rotational surfaces as: Since n1, n2, R(α) and the function S(x, y) = 0 is known, the three equations Eqs. (6), (8) and (10) are in fact three implicit ordinary differential equations with three unknown functions rf(α), rs(β) and β(α). The numerical solutions of these functions represent aspheric profiles. 4. Optical design example and evaluation A design example has been derived using the approach presented in section 3. For this example R(α) = ∞. The vertexes of the first and the second optical surface are on the axis of the rotational symmetry, 31.5 mm and 41.5 mm away from the origin respectively; the image surface is a flat plane perpendicular to the optical axis and is 85 mm away from the origin (S(x, y)≡x-85mm = 0), so that Eq. (8) becomesEq. (10). Then rs´/β´ can be obtained by solving Eq. (8). Finally rf´´, β and (rs´/β´)´ can be obtained by solving Eq. (6), differentials of Eqs. (8) and (10) simultaneously. As we can see, the initial value of rf´ will affect the boundary condition which leads to different numerical solutions. A series value of rf´ has been tested and a design in which the two surfaces converge to the same edge without total internal reflection is obtained and shown in Fig. 3(a). Once the design has been done, the two image surfaces where the centers of curvature of the tangential and sagittal output supported wavefronts lay on respectively are calculated separately from the surface profiles to check that the design is free from astigmatism. The difference between the two image surfaces and the image plane x = 85 mm is less than 1 μm. An aplanatic design with the same positions of lens surface vertexes and image plane (Fig. 3(b)) has been used as comparison. Since this design is not anastigmatic there will be in general two surfaces where the centers of curvature of output wavefronts lie and these two surfaces are in general curved unlike the flat single image plane of the anastigmatic design. All the optical surfaces are fitted and represented by Qcon asphere polynomials in CodeV. A subsequent optimization has been done with a 2mm diameter round pupil centered at the origin in CodeV for 25 field points uniformly distributed in the interval 0° ≤ α ≤ 29.6°. The layouts of the both designs after optimization are plotted in Fig. 3(c) and (d) respectively. The optimization and the following evaluation have been done for a monochromatic light of wavelength 580 nm. The RMS spot diameter distribution is shown in Fig. 4. The result optimized from anastigmatic design yields an average RMS spot diameter of 9.9 μm and a maximum value of 12.9 μm across 25 field points, while the result optimized from aplanatic design yields an average value of 13 μm and a maximum value of 19 μm. The Modulation Transfer Function has been evaluated across all the 25 field points and the result is shown in Fig. 4 for both optimized designs. The result optimized from anastigmatic design has achieved an average MTF of 46.9% and a minimum of 42.8% at the frequency of 30 cycles/mm across 25 field points, while the result optimized from aplanatic design achieves an average MTF of 44.5% and a minimum of 32.3% at the same frequency. 5. Discussion and conclusion The differential equation design approach shown hereinabove can control the tangential and sagittal ray propagation simultaneously, thus anastigmatic design can be readily obtained. We have extended this approach from single optical surface designs to double aspheric surface designs. With more freedoms provided by the second optical surface, both the object and the image surfaces can be prescribed. Nevertheless, in no case, nor single nor double surface design, the mapping from the object to image can be prescribed. This mapping is obtained after the design of the optical surfaces. Based on this approach, a double aspheric surface anastigmatic design example has been developed. As a good initial design for optimization, a result from the final design with an average RMS spot diameter of 9.9 μm and an average MTF value of 46.9% at the frequency of 30 cycs/mm has been achieved. Authors thank the European Commission (SMETHODS: FP7-ICT-2009-7 Grant Agreement No. 288526, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (ENGINEERING METAMATERIALS: CSD2008-00066, SEM: TSI-020302-2010-65 SUPERRESOLUCION: TEC2011-24019, SIGMAMODULOS: IPT-2011-1441-920000, PMEL: IPT-2011-1212-920000), UPM (Q090935C59) and the academic licence for CodeV from Synopsys for the support given to the research activity of the UPM-Optics Engineering Group, making the present work possible. References and links 1. B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931). 2. K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993). 3. G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949). [CrossRef] 4. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002). [CrossRef] 5. R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003). [CrossRef] 6. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005), Chap. 9. 7. J. M. Infante, “Optical Systems Design using SMS method and optimizations,” Ph.D. Thesis, Universidad Politécnica de Madrid (2013). 10. Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015). [CrossRef] 12. J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012). [CrossRef] 13. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications (Wiley-VCH, 2006). 14. W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 3. 15. E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Textbooks in Mathematics) (Chapman and Hall/CRC, 2006)

About GMC Network The research on mechanical and dynamical systems has had a deep impact in other research areas and in the development of several technologies. A big part of its advances has been based on numerical and analytical techniques. In the sixties, the most sophisticated and powerful techniques coming from Geometry and Topology were used in the study of dynamical systems. Those techniques led, for instance, to the beginning of the modern Hamiltonian Mechanics. Geometric techniques have been also applied to a wide range of control problems such as locomotion systems, robotics, etc. Most of these ideas have been developed in the last 30 years by mathematicians of a high scientific level such as J. Marsden, A. Weinstein, R. Abraham, V. Arnold or R. Brockett among others. The emphasis on geometry means an attempt of understanding the structure of the equations of motion of the system in order to analyze them and study their design. The symplectic structures play a fundamental role in the differential-geometric description of the Lagrangian and Hamiltonian Mechanics on the tangent and cotangent bundle of the configuration space. In particular, it is important in the study of “regular systems” (a notion linked to the non-degeneracy of the symplectic form). The symplectic geometry allows us to obtain, starting from the Hamiltonian energy and in a simple way the dynamics (the Hamiltonian vector field), whose integral curves satisfy the Hamilton’s equations. When the system is singular (the Lagrangian function is not regular), that is, when there exist internal constraints, we must replace the symplectic structure for a more general geometric object: Poisson structures. These objects also play an essential role in the geometric description of the reduction of mechanical systems which admit a Lie group of symmetries. It is particularly interesting the case of linear Poisson structures on vector bundles. There are other singular systems in which the constraints are external: the non-holonomic mechanical systems. The dynamics for this kind of systems is controlled by a “quasi-Poisson” structure, a geometric object with the same properties as a Poisson bivector, but the integrability condition is not satisfied. For all these topics, the members of the network have made relevant contributions along the last 25 years. A category which is closely related with Poisson Geometry is that of Lie algebroids. A Lie algebroid is a natural generalization of the tangent bundle and of the Lie algebras. On the other hand, there exists a one-to-one correspondence between Lie algebroid structures on a vector bundle and linear Poisson bivectors on the dual bundle. As a consequence, it is not surprising the usefulness of Lie algebroids in the geometric description of Mechanics. This fact was first noted by A. Weinstein and, after him, it has been recently studied by several authors, some of them belonging to this network. During the last decade, the construction of geometric integrators for Lagrangian systems using discrete variational principles, has been a topic of growing interest. Of a particular interest we find systems with discrete Lagrangian over the cartesian product of the configuration manifold with itself. This cartesian product is the discrete version of the phase space of velocities. When the discrete Lagrangian is an approximation of a continuous Lagrangian function, it is obtained a geometric integrator which inherits some properties of the continuous Lagrangian. Some extensions of these results with the purpose of construct geometric integrators for more general mechanical systems, such as non-holonomic or time-dependent systems, have been obtained by several members of this network. On the other hand, Moser and Veselov have recently considered discrete Lagrangian systems on Lie groups, what later motivated A. Weinstein to start the study of discrete Mechanics on Lie groupoids. Lie groupoids are a natural generalization of Lie groups and cartesian products of a manifold by itself and, moreover, Lie algebroids can be considered as the infinitesimal version of Lie groupoids. The work started by A. Weinstein, as well as some other problems arisen by him, are nowadays a research topic to some members of this network. Classical Mechanics can be considered as a Classical Field Theory of first order in which space-time has dimension one. As it is well-known, the space-time of a Classical Field Theory of first order is a smooth manifold and the dynamics is given as the solutions of a partial differential equation. One of the geometric formalisms, broadly accepted, which is used in the description of Classical Field Theories since the seventies, is the multisymplectic approach. Pioneer works in this sense were made by some of the members of this network. The notion of a multisymplectic structure became the key notion to definitely clarify the Hamiltonian Field Theory, which had been thoroughly treated in previously decades. This doctrine is an important interdisciplinary area whose many aspects of research are far away from being finished. In fact, nowadays it is one of the research lines of an extensive group of members of the network. To this fact we have to add a renovated interest in Multisymplectic Geometry partly due to the discovery of numerical integrators which preserve the multisymplectic form. In fact, some members of this network are starting some investigations in this line of research. All the lines of research previously mentioned are part of a wider research topic generically called Geometric Mechanics, which is a meeting point for several disciplines such as Geometry, Analysis, Algebra, or Partial Diferential Equations. Geometric Mechanics is a growing research area with fruitful connections with other disciplines such as Nonlinear Control Theory or Numerical Analysis. Starting in the seventies, Diferential Geometry has played an important role in the development of Control Theory. Since then, several questions such as controlability, observability, dynamical reaction, linearization and stabilization of the reaction or optimal control, are linked to geometric concepts such as Lie group theory, differential forms, distributions, homogeneous spaces, symplectic structures, Sub-Riemannian geometry, etc. In fact, some of the developments of Diferential Geometry are a consequence of the geometrization of control problems, creating a mutual benefit between both disciplines. Recently, several members of this network project has been interested by the application of geometric techniques to optimal control theory, in particular to obtain numerical integrators for optimal control problems. The organization of this network is based in on-going collaborations that have already been materialized in publications and joint activities between the different groups being part of this network. This fact guarantees the feasibility of the project, which incorporates at the same time some young researchers, who have received an excellent preparation in well-known research centers and who proportionate the network a guaranty of continuity for future activities. The members of this Thematic Network project center their work around these axis: Geometry, Mechanics and Control, all clearly interconnected. - Symplectic, Poisson and Jacobi manifolds - Lie group geometry in classical and quantum mechanics - Physics in spaces with constant curvature - Cayley-Klein geometries: models and applications in Physics - Geometry of gauge theories - Non-holonomic mechanics - Vakonomic dynamics - Calculus of variations - Systems and conservation laws. Noether theorems - Classical field theories and multisymplectic geometry - Numeric-geometrical integration of mechanical systems - Lie algebroids and geometric mechanics - Lie groupoids and discrete mechanics - Integrable and super-integrable systems - Geometric quantization - Symmetries. Lie group actions. Reduction of order - Relative equilibria and relative periodic orbits in symmetric Hamiltonian systems. Stability and bifurcations - Optimal control theory - Control of mechanical systems - Distributed control - Geometric control - Facilitate the exchange and transference of knowledge between the Spanish groups which work in different aspects related with Geometric Mechanics, Field Theory and Control Theory. - Define research lines as a result of the group activities among the different members of the network and organize the research in every group more precisely and in collaboration with the other groups. - Promote the collaboration among different reseach groups both in European and international networks. There already exist collaborations with several groups of these networks, what will definitely help to improve this cooperation. It is specially interesting the identification of a group in a topic which could play a Spanish reference role in the European Framework of Research. - Give to the new generations of researchers a common framework for their apprenticeship in aspects related with the research in Geometric Mechanics, Field theory and Control theory.

What is the Spanish 'Necesito tomar agua' in English? I need to drink water is an English equivalent of 'Necesito tomar agua'. The verb 'necesito' means '[I] am needing, do need, need'. The infinitive 'tomar' means 'to take, drink'. The feminine noun 'agua' means 'water'. All together, they're pronounced 'neh-seh-SEE-toh toh-MAH-RAH-gwah'. 61 people found this useful "Jacob" and "James" are standard English equivalents of the Spanish name Jaime . The pronunciation of the masculine proper name will be "HEYE-mey"* in Spanish. *The sound is similar to that in the English noun "eye." English wins by about 300 years. The history of English is dated from around 450, while the history of Spanish as distinct from Latin does not begin until the 8th Century, and the National standard, Castillian, did not exist until much later. estoy en colombia y quiero ir a costarrica unos 5 dias que requisitos nesecito para entrar a ese pais " Water " is an English equivalent of " agua ." Specifically, the Spanish word is a feminine noun. But it takes the masculine singular definite article " el " ("the") in the singular. One reason lies in the reluctance to follow the ending vowel " a " of the feminine singular definite article " la… " with the beginning vowel " a " of " agua ." The feminine definite plural article " las " is used with the plural " aguas " ("waters"). The pronunciation is "AH-gwah." ( Full Answer ) Indicative, of the portuguese verb 'tomar' Eu tomo Tu tomas Ele toma Nós tomamos Vós tomais Eles tomam. Past Eu tomei Tu tomaste Ele tomou Nós tomamos Vós tomastes Eles tomaram. Future Eu tomarei Tu tomarás Ele tomará Nós tomaremos Vós… tomareis Eles tomarão. ( Full Answer ) Agua Mala is the slang term used for Portuguese Man o' War, a sea creature. Agua Mala would translate to (if not refering to the slang term) bad water. Aguas Freacas is a Nortcarachinean ritual in which all the people of the tribe come down the the nearest fish well and bring up ten fish to swim in a large washbucket with all of the villagers white clothes. this is belived to purify and bring good luck to the wearers of the clothes. the villagers t…hen release the fish into a lake and wear the wet clothes home...usually about ten miles.this is done every new moon, because they all belive that they need good luck and purity in the darkness. The Moon is their Goddess and fish with their shimmering scales are belived tohave come from the moon. ( Full Answer ) Green water is an English equivalent of 'verde agua'. The adjective 'verde' means 'green'. The feminine noun 'agua' means 'water'. Together, they're pronounced 'behr-theh AH-gwah'. But it isn't the usual order in which the phrase is said in Spanish. Typically, 'green water' is said 'agua ve…rde' . It's pronounced 'AH-gwah-VEHR-theh'. ( Full Answer ) It is Hydrogen Peroxide H 2 O 2 , usually it is used to remove dirt and further infection to cuts. To drink water is an English equivalent of 'tomar agua'. The infinitive 'tomar' means 'to take, to drink'. The feminine noun 'agua' means 'water'. Together, they're pronounced 'toh-MAH-RAH-gwah'. Son éstas las conjugaciones del verbo estar: Infinitivo: Estar (to be) Infinitivo compuesto: Haber estado (to have been) Gerundio simple: Estando (being) Gerundio compuesto: Habiendo estado (having been) PArticipio: Estado (been) Presente de indicativo: Yo estoy (I am) Tú e…stás (you are) Ãl/Ella/Usted está (he / she / is; you are, polite singular) Nosotros estamos (we are) Vosotros estáis (you are, familiar plural) Ellos/Ellas/Ustedes están (they are; you are polite plural) Pretérito imperfecto de indicativo: Estaba, estabas, estaba, estábamos, estabais, estaban (I was, you were...) Pretérito de indicativo: Estuve, estuviste, estuvo, estuvimos, estuvisteis, estuvieron (I was, you were...) Futuro de indicativo: Estaré, estarás, estará, estaremos, estaréis, estarán (I will be...) Pretérito perfecto de indicativo: He, has, ha, hemos, habéis, han + estado (I, you, he, she, it we, they have been) Pretérito pluscuamperfecto de indicativo: HabÃa, habÃas, habÃa, habÃamos, habÃais, habÃan + estado (I had been, you had been...) Pretérito anterior: Hube, hubiste, hubo, hubimos, hubisteis, hubieron + estado (This tense does not exist in English) Futuro perfecto de indicativo: Habré, habrás, habrá, habremos, habréis, habrán + estado (I will have been...) Potencial simple o imperfecto / condicional simple o imperfecto: EstarÃa, estarÃas, estarÃa, estarÃamos, estarÃais, estarÃan (I would be...) Potencial perfecto o compuesto / condicional perfecto o compuesto: HabrÃa, habrÃas, habrÃa, habrÃamos, habrÃais, habrÃan + estado (You would have been...) Presente de subjuntivo: Esté, estés, esté, estemos, estéis, estén (This tense does not exist in English) Pretérito imperfecto de subjuntivo: Estuviera, estuvieras, estuviera, estuviéramos, estuvierais, estuvieran ( también: estuviese, stuvieses, estuviese, estuviésemos, estuvieseis, estuviesen) --- This tense does not exist in English Futuro imperfecto de subjuntivo: Estuviere, estuvieres, estuviere, estuviéremos, estuviereis, estuvieren --- (This tense does not exist in English) Pretérito perfecto de subjuntivo: Haya, hayas, haya, hayamos, hayáis, hayan + estado (This tense does not exist in English) Pretérito pluscuamperfecto de subjuntivo: Hubiera, hubieras, hubiera, hubiéramos, hubierais, hubieran + estado ( también: hubiese, hubieses, hubiese, hubiésemos, hubieseis, hubiesen + estado) --- This tense does not exist in English. Futuro perfecto de subjuntivo: Hubiere, hubieres, hubiere, hubiéremos, hubiereis, hubieren + estado (This tense does not exist in English) Imperativo directo \nEstate / está (tú), Estemos / estémonos (nosotros), Estese / esté \n(usted), Estense / estén (ustedes), Estaos / estad (vosotros) Imperativo indirecto \nQue se esté (usted / él / ella); que se estén (ustedes, ellos, ellas); \nque me esté (yo); que nos estemos (nosotros); que os estéis (vosotros), \nque te estés (tú) Que esté (usted / él / ella); que estén \n(ustedes, ellos, ellas); que esté (yo); que estemos (nosotros); que \nestéis (vosotros), que estés (tú) . ( Full Answer ) Water or The water may be English equivalents of 'el agua'. The feminine noun 'agua' means 'water'. In the singular, it takes the masculine definite article 'el' . In the plural, it takes the feminine definite article 'las' , to be written as 'las aguas' . The singular is pronounced 'eh-LA…H-gwah'. The plural is pronounced 'lah-SWAH-gwahs'. ( Full Answer ) Translate mi dia fue bueno excepto fot el hecho de que tuve que tomar una prueba de hoy from spanish to English? my day was good except for the fact that I had to take a test Water is what 'agua' means in Nicaragua. The Central American country is a Latin American nation that includes Spanish among its spoken languages. The Spanish word is a feminine noun whose definite article in the singular is the masculine form 'el' , but the feminine 'las' in the plural. It's …pronounced 'AH-gwah'. ( Full Answer ) "Necesito mas de ti, quiero que sepas mas, te amo" Translated directly into English "I need more from you, I want you to know more, I love you." It means, I need more of you, I hope that you know that I love you. en el trabajo = at work y = and necesito cafe = i need coffee It means "You (or he or she) can drink alcohol with Albendazole". Albendazole is a medicine for treating worms. The possessive "his" is "su" if it's singular (his book = su libro) and "sus" if it's plural (his books= sus libros) Because su/sus also means her, your, and their, many times the phrase will have a a qualifier "de él" tacked on. "English", as in the English language, is "inglés". "The English" as a nationality, is "los ingleses". "Soy de Inglaterra" means "I am from England". Three oaks is an English equivalent of 'Tres robles'. The number 'tres' means 'three'. The masculine noun 'robles' means 'oak trees'. Together, they're pronounced 'treh-SROH-blehs'. The ball point pen is an English equivalent of 'el bolÃgrafo' . The masculine definite article 'el' means 'the'. The masculine noun 'bolÃgrafo' means 'ball point pen'. Together, they're pronounced 'ehl boh-LEE-grah-FOH'. It is a bad word it means a male reproductive organ. I will not go into more specific description. Go to goggle translator your self if you want to actually know and you are not some mindless hooligan. ***************** Well, I disagree with this answer. "Pene" is not a bad word. It just means "pe…nis". ( Full Answer ) Good north [ ern exposure ] is an English equivalent of 'Buen norte'. The masculine adjective 'bueno' * means 'good, fortunate'. The masculine noun 'norte' means 'north'. Together, they're pronounced 'bwehn NOHR-teh'. *The vowel 'o' of 'bueno' may drop before a noun that begins with a co…nsonant. ( Full Answer ) The translation depends on how the word "but" is being used. If you wanted to say, "I like green, but my favorite color is blue," you would use the word "pero." If you wanted to say, "She is anything but pretty," you could use "salvo." When you're using "but" as an adverb, though, you don't use e…ither. The sentences are constructed differently. ( Full Answer ) A great internet resource is behindthename.com. This will give you a comprehensive list of every form of almost any name, current and past, male and female; you can also specify a language and gender you want a translation for. Need a quick translation at a time when you don't have internet access? …Biblical names are very commonly used in Spanish cultures, and these can be easily translated by comparing their usages in the English and Spanish bible translations. ( Full Answer ) Either "how I need you" or "how do I need you", depending on whether it's a statement or a question. gandules means beans. gandules are a specific type of beans put into rice together to make arroz con gandules, meaning rice with beans. unlike standard rice and beans which beans can either be served next to, on top of, or in the rice, arroz con gandules comes made with the beans specifically in the… rice. ( Full Answer ) well they are 1,2,3,4 but the writing is different like one = uno two = dos three = tres four = cuatro twenty = veinte additionally, some numbers ARE Different. in English a Billion = 1000,000,000 in Spanish a Billón = 1000,000,000,000 Speaking of Billones in Spanish is therefore… rare. ( Full Answer ) no beber el agua - Or...the proposal you made is correct if you are giving a command to someone. No beba el agua. "Beba" is the formal command form of beber. It depends on how you want to use it and who is doing it. However the verbs for "to boil" are hervir, which is to boil water, and cocer, which is to boil food. Let me know how you want to say it on my message board. As a question it means "What are you taking/carrying" As part of a statement it could mean "that they are taking/carrying" Voy a trabajar. Literally "I go to work", but normally translated as "I am going to work". If you are in the actual process of going to work, you could say "Estoy yendo a trabajar." back-espalda head-cabeza eyes-ojos leg-pierna elbow-coda heel-muy hair-pro chin-barbilla stomach-estomago lips-labios forehead-frente nose-nariz mouth-boca toe-dedo del pie dedo-finger tounge-lengua ear-oreja teeth-diebtes foot-pie hand-mano no translates to Spanish is simply "no." it also may take the place of words such as don't, can't, and won't. Ahijado is a Spanish equivalent of the English word"godson." The pronunciation of the masculine singular noun will be"eye-KHA-tho" in Spanish. many Spanish people have learned to speak English, the same as English speaking people have learned to speak Spanish, this goes for almost any one who thinks it is great to learn different languages and it serves them well when travelling to different parts of the world. 'Oil spill' is an English equivalent of 'marea negra'. The feminine noun 'marea' literally means 'tide'. The feminine adjective 'negra' means 'black'. Together, they're pronounced 'mah-REH-ah NEH-grah'. yo bebo agua = i drink water tu bebes = you drink el/ella bebe = he/she drink nosotros bebemos = we drink ellos/ellas beben = they drink at least thats what is says in my spanish book. "I need a laxative" is an English equivalent of the Spanish phrase Necesito laxante . The pronunciation of the declarative statement -- which typically includes no word for "a" ( un or uno in the masculine, una in the feminine) -- will be "NEY-they-SEE-to lak-SAN-tey" in Spain and "NEY-sey-SEE…-to lak-SAN-tey" throughout Latin America. ( Full Answer ) Yes, But most likely you will get The Dictionary At A Local Library or Order One online! The water Agua is water. El agua is "the water" or just water. It is not a water hole. A water hole is abrevadero, aguada, or bebedero You could say "inglés al español". Note that the names of languages in Spanish are not capitalized. You can say "Mi idioma es inglés, no español". Note that languages are not capitalized in Spanish. (el) inglés (language) inglés, inglesa (nationality) *NEVER WRITE WITH CAPITAL LETTERS: el Inglés (even the language); la Inglesa It is a chemical composed of Nitric and Hydrochloric acid that has the property to dissolve gold, platinum and palladium. Que quizás algo no le parezca a uno, pero debe de vérsele lo mejor del asunto. Translation: In fact, no, but I have to be pleased. This is a case where somebody translated word by word and ended up writing something that makes little sense as written. Literal Translation: You want my and I want your it allows to take slow. Probable intended Spanish: Me quieres y te quiero, permÃtenos tomarla lentamente. Intended Spa…nish Translation: You like me and I like you, let's take it slow. (presumably a relationship). ( Full Answer ) \n \n \n \n \n \n\n Translation: One cup of\nwater, One hour before study \n\n \n \n Normal \n 0 \n \n \n \n \n false \n false \n false \n \n EN-US \n X-NONE \n X-NONE \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n …\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n ( Full Answer ) The sentence is incomprehensible. Necesito que papa... > there is no sense... 'Necesito que....' = 'I need....'(followed by a verb). 'Papa''s position is ambiguous. 'Papa por que estas actuando asi' > 'Papa, why are you behaving like this?' nash aguas is an actor in luv u and i am a really big fan of himhis so cute and hot - iamroseannp

NCERT Solutions for class 7 Science Chapter-13 This page consist of details solution of chapter Motion and Time class 7 science . You can check NCERT Solutions for class 7 Science for other chapters of class 7 science prepared by Physics Wallah. do the questions by yourself and check your solution with Physics Wallah NCERT solution for the chapter Motion and Time. to have more depth in chapter Motion and Time class 7 science you must read the theory of Motion and Time and do the additional question of Physics Wallah click this link to do the questions and read theory of Motion and Time. along with NCERT solution of class 7 science NCERT Solutions for class 7 Science Chapter 13 Motion and Time Solutions of Chapter 13 Class 7 Science NCERT SOLUTION OF NCERT CLASS VII SCIENCE CHAPTER 13 EXERCISE-1 Question 1: Classify the following as motion along a straight line, circular or oscillatory motion: (i) Motion of your hands while running. (ii) Motion of a horse pulling a cart on a straight road. (iii) Motion of a child in a merry-go-round. (iv) Motion of a child on a see-saw. (v) Motion of the hammer of an electric bell. (vi) Motion of a train on a straight bridge. Answer: (i) Oscillatory motion While running, the hands move to and fro and repeat their motion after a given interval of time. Hence, it is an oscillatory motion. The horse is pulling a cart on a straight road. Therefore, it has a motion along a straight line. Merry-go-round has a circular motion. Therefore, a child sitting inside it will also have a circular motion. (iv) Oscillatory motion The child on a see-saw goes up and down continuously. It oscillates up-down. Therefore, it is an oscillatory motion. The hammer hits the electric bell and vibrates rapidly. Therefore, it is an oscillatory motion. (vi) Straight line The train is moving on a straight bridge. Therefore, it has a motion along a straight line. Question 2: Which of the following are not correct? (i) The basic unit of time is second. (ii) Every object moves with a constant speed. (iii) Distances between two cities are measured in kilometres. (iv) The time period of a given pendulum is not constant. (v) The speed of a train is expressed in m/h. Second is the SI unit of time. (ii) Not correct An object can move with constant or variable speed. The distance between two cities can be very large. Since kilometre is a bigger unit of distance, the distance between two cities is measured in kilometres. (iv) Not correct Time period of a pendulum depends on the length of the thread. Hence, it is constant for a particular pendulum. (v) Not correct The speed of a train is measured either in km/h or in m/s. Question 3: A simple pendulum takes 32 s to complete 20 oscillations. What is the time period of the pendulum? Answer: Number of oscillations = 20 Total time taken to complete 20 oscillations = 32 s Question 4: The distance between two stations is 240 km. A train takes 4 hours to cover this distance. Calculate the speed of the train. Answer: Distance between the two stations = 240 km Time taken = 4 h Question 5: The odometer of a car reads 57321.0 km when the clock shows the time 08:30 AM. What is the distance moved by the car, if at 08:50 AM, the odometer reading has changed to 57336.0 km? Calculate the speed of the car in km/min during this time. Express the speed in km/h also. Answer: Initial reading of the odometer of the car = 57321.0 km Final reading of the odometer of the car = 57336.0 km Distance covered by the car = Final reading of the odometer of the car − Initial reading of the odometer of the car = 57336.0 − 57321.0 = 15 km The given car starts at 8:30 a.m. and stops at 8:50 a.m. Therefore, time taken by the car to cover the distance is (8:50 − 8:30) min = 20 min Distance covered by the car = 15 km Time taken by the car = 20 min 60 min = 1 h Time taken by the car = 1/3 h Question 6: Salma takes 15 minutes from her house to reach her school on a bicycle. If the bicycle has a speed of 2 m/s, calculate the distance between her house and the school. Time taken by Salma to reach her school from her home = 15 min = 15 × 60 = 900 s Speed of her bicycle = 2 m/s Distance covered = Speed × Time taken = 2 × 900 = 1800 m 1000 m = 1 km 1800m = 1/1000 x 1800 = 1.8 km Question 7: Show the shape of the distance-time graph for the motion in the following cases: (i) A car moving with a constant speed. (ii) A car parked on a side road. Answer: (i)A car moving with a constant speed covers equal distance in equal intervals of time. Such motion of car is represented in the given distance-time graph. (ii)The distance-time graph of a car parked on a road side is such that with the increase in time, there is no change in distance, as shown in the given figure. Question 8: Which of the following relations is correct? (i) Speed = Distance × Time (ii) Speed of an object is given by the relation Question 9: The basic unit of speed is: Answer: (iv) m/s The basic unit of distance is metre (m). The basic unit of time is second (s). Therefore, the basic unit of speed is m/s. Question 10: A car moves with a speed of 40 km/h for 15 minutes and then with a speed of 60 km/h for the next 15 minutes. The total distance covered by the car is: (i) 100 km (ii) 25 km (iii) 15 km (iv) 10 km Answer: (ii) 25 km Speed of the car = 40 km/h Time taken = 15 min = 15/60 = 0.25 h Distance covered, d1 = Speed × Time taken = 40 × 0.25 = 10 km Speed of the car = 60 km/h Time taken = 15 min = 15/60 = 0.25 h Distance covered, d2 = Speed × Time taken = 60 × 0.25 = 15 km Total distance covered by the car, d = d1 + d2 = 10 + 15 = 25 km Therefore, the total distance covered by the car is 25 km. Question 11: Suppose the two photographs, shown in Figure 1 and Figure 2, had been taken at an interval of 10 seconds. If a distance of 100 metres is shown by 1 cm in these photographs, calculate the speed of the blue car. Figure 1 Vehicles moving in the same direction of on a road Figure 2 Position of vehicles shown in Figure 1 after some time Answer: The distance covered by the blue car (as evident from the photograph) from one white strip to another, which is measured by scale is 1.4 cm. It is given that 1 cm is equivalent to 100 m. Therefore, 1.4 cm is equivalent to 140 m. Distance travelled by the car = 140 m Time interval between the two photographs = 10 s = 140/10 = 14 m/s Question 12: Figure shows the distance-time graph for the motion of two vehicles A and B. Which one of them is moving faster? Distance-time graph for the motion of two cars Answer: Vehicle A is moving faster than vehicle B. Speed is given by the relation This relation shows that speed of a vehicle is greater if it covers maximum distance in a given interval of time. To compare the distance, draw a line perpendicular to the time-axis, as shown in the following distance-time graph. From the graph, it is evident that for a given time t, the distance covered by vehicle A is more than vehicle B. Hence, vehicle A is moving faster than vehicle B. Question 13: Which of the following distance-time graphs shows a truck moving with speed which is not constant? Answer: Graph (iii) In a distance-time graph, the constant speed of a truck will be represented by a straight line. In a distance-time graph, a straight line parallel to the time axis indicates that the truck is not moving. A curved line on the distance-time graph indicates that the truck is moving with a speed which is not constant. Extend Learning — Activities and Projects Question 1:You can make your own sundial and use it to mark the time of the day at your place. First of all find the latitude of your city with the help of an atlas. Cut out a triangular piece of a cardboard such that its one angle is equal to the latitude of your place and the angle opposite to it is a right angle. Fix this piece, called gnomon, vertically along a diameter of a circular board a shown in Fig. 13.16. One way to fix the gnomon could be to make a groove along a diameter on the circular board. Next, select an open space, which receives sunlight for most of the day. Mark a line on the ground along the North-South direction. Place the sundial in the sun as shown in Fig. 13.16. Mark the position of the tip of the shadow of the gnomon on the circular board as early in the day as possible, say 8:00 AM. Mark the position of the tip of the shadow every hour throughout the day. Draw lines to connect each point marked by you with the centre of the base of the gnomon as shown in Fig. 13.16. Extend the lines on the circular board up to its periphery. You can use this sundial to read the time of the day at your place. Remember that the gnomon should always be placed in the North-South direction as shown in Fig. 13.16. The activity can be performed as follows: 1. Find the latitude of the city with the help of Atlas. 2. Cut out a triangular piece of a cardboard such that its one angle is equal to the latitude of the location and the angle opposite to it is a right angle. 3. Fix this piece, called gnomon, vertically along a diameter of a circular board by making a groove along a diameter on the circular board. 4. Choose a space, which receives sunlight for most of the day. Mark a line on the ground along the North-South direction. Place the sundial in the sun. 5.Mark the position of the tip of the shadow of the gnomon on the circular board as early in the day as possible. Mark the position of the tip of the shadow every hour throughout the day. Draw lines to connect each point marked by you with the centre of the base of the gnomon. 6.Extend the lines on the circular board up to its periphery. 7. This Sundial can be used for reading the time at chosen place. Question 2: Collect information about time measuring devices that were used in the ancient times in different parts of the world. Prepare a brief write up on each one of them. The write up may include the name of the device, the place of its origin, the period when it was used, the unit in which the time was measured by it and a drawing or a photograph of the device, if available. The time measuring devices that were used in the ancient times in different parts of the world are: Sun Dial: This time measuring device uses a spot of light or shadow by the sun’s position on a reference scale. The prototypes of Sundials are seems to be originated in Russia and they were used in 1830s. This time measuring device uses regulated water flow of liquid i.e. inflow or outflow of liquid to calculate the time. They are supposed to be existed in 16th Century in Egypt. Sand Clocks: The sand clocks are supposed to be invented at Alexandria about 150 B.C. Question 3:Make a model of a sand clock which can measure a time interval of 2 minutes (Fig. 13.17). 1.Take one plastic cup and place it in upside down on a table. Put the other cup on the top of the first one so that the bottom parts of both cups touch each other. 2.Place two lids on the cups and make holes into them. And then invert one cup one another. 3.Measure the sand and put on top of one cup timer. Allow the sand to pass through the holes. 4.Place the cup timer on top of plate and note down the time taken by the sand to pass through the bottom of the timer. 5.Adjust the amount of sand until the time taken by it is 2 minutes for all the sand to pass through the bottom of the timer. Question 4:You can perform an interesting activity when you visit a park to ride a swing. You will require a watch. Make the swing oscillate without anyone sitting on it. Find its time period in the same way as you did for the pendulum. Make sure that there are no jerks in the motion of the swing. Ask one of your friends to sit on the swing. Push it once and let it swing naturally. Again, measure its time period. Repeat the activity with different persons sitting on the swing. Compare the time period of the swing measured in different cases. What conclusions do you draw from this activity? 1. Oscillate the swing without any person sitting on it and note down the time period for 5 oscillations. 2. Ask a person to sit on the swing and note down the time for 5 oscillations. 3. Repeat the same for different persons. 4. Calculate the average time taken by dividing to no. of oscillations. The time period of the swing will be different for different cases. The time period of swing when no one is sitting on it will be least and will depend on the weight of the object sitting on the swing. The higher is the mass of the object on swing, the more will be the time taken by the swing to complete one oscillation because speed becomes less.

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Inverses of the Trigonometric Functions. TEST. Proof of Sum and Difference Identities. Using trigonometric identities and fundamental trigonometric function values, find each of the. SECTION 5. PART A: EXAMPLE STRATEGIES AND SHOWING WORK. Verify the identity. Proof that normal distribution is a distribution. Substitute x x - μ superscript x normal- x μ xprimex-mu and y y - μ superscript y normal- y μ. Home Lesson 16: Normal Distributions. Also recall that in order to show that the normal p. is a valid p. f, we need to show that 1 fx is always. Lets start with the easy part first, namely, showing that fx is always positive. We know that for fx to be a probability distribution. These values for a and b are valid for both R1 and R2. PDF for the normal distribution is fx 1. By symmetry, we can negate the inequality. article, we will give a derivation of the normal probability density function suitable for. The broad applicability of the normal distribution can be seen from. Proof valid pdf part 1. A little change of variable, nothing fancy.Canadian style guide abbreviations for example

- software - Simplified Music Chord Theory Scale in key of C Understanding chords really begins with understanding a scale. Let us use the key of C for our scale since that will yield simplest results. A scale is generally taken as 8 notes, which is also considered one “octave”. The scale begins on a note and ends when it reaches that same type of note. Let us look at a simple C scale looking at it three different ways: Scale, Key of C to a Singer Do, re, mi, fa, so, la, ti, do! (note that it has 8 notes, and begins and ends on “do”) Scale, Key of C by name C, D, E, F, G, A, B, C (8 notes again, and begins and ends on C) Scale, Key of C by number 1, 2, 3, 4, 5, 6, 7, 8 (8 notes again, but this time we called the 8th note 8) The above three representations are all really the same. How do we use this for chords? Well let us start with a simple major chord. A major chord is always made up of the first, 3rd, and 5th notes of its scale. So C Major is simply C, E, and G. Not so hard, eh? What would a sixth be? Can you guess? Well we begin with a major chord, and then add the sixth note. So a major sixth (normal sixth) would be: C, E, G, A. Now the C scale above is not all of the possible notes. It is merely the 8 notes making up the C scale. If we think in terms of a piano, the C scale is all of the white keys. It does not use any of the shorter black keys, which are sharps and flats. When we look at all of the possible notes, we call that a chromatic scale. All of the possible notes in a chromatic scale, not counting any repeats, not even one, are 12 total. They are: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, (and then C would repeat next) Of course, that shows the chromatic scale using all sharps. A sharp sign # means to raise a tone one half step. So C# is one half step higher than C. For every sharp representation, there is also a flat representation possible. Instead of raising C one half step to get C#, we could have also lowered D one half step to get a Db. The “b” sign is for “flat” just as the “#” sign is for sharp. C# is the same note as Db. There are just two ways of showing it. Sometimes it s more convenient to show it as a sharp, and sometimes as a flat. Minor chords, Dominant 7ths It was necessary to talk about chromatic scales before talking about some chords, such as minors and dominant 7ths, since they go off of the normal C scale and involve half steps. We had said that a major chord was made up of the first, third and fifth notes of the scale, C, E and G. A minor is close, but it “flats” the third. So a C minor chord would be: C, Eb, G. Note that we could have shown this also as: C, D#,G since D# and Eb again are the same note. When we talk about 7th chords, one would think that we take a major chord such as C,E, and G, and add the seventh note of the scale, which would be “B”. That in fact is exactly what we do for a C major 7th chord. However, when we just write 7th, it is taken to mean a “dominant 7th” which flats the 7th note. So, C 7th is: C, E, G, and Bb. This again can also be written as C, E, G, A#. So, the above covers the 7th chord. The C major 7th chord would in fact be what you would have guessed: C, E, G, B - where B is clearly the 7th note of the scale. Now, how about a minor 7th chord? Let us begin with the minor, which flats the third, and then add the 7th. So, let us begin with C minor which is C, Eb, G and we will add the 7th which again is Bb and we get: C, Eb, G, Bb. Recall that when we just say “7th” it refers to the dominant 7th, which is a flatted 7th of the scale. There is a version of a minor chord that uses the major 7th, it is called mmaj7th. It is a somewhat confusing name, being a minor, major7th. I would agree that minor-major in the same sentence seems like a contradiction of terms, but they refer to two different parts of the chord. The minor refers to the first three notes, C Eb, G. The major 7th means that we do not flat the 7th note of the scale. So, therefore a C mmaj7th would be: C, Eb, G, B. Well, if you follow all of that, many of the rest of the chords should make sense. An augmented chord is also called a + chord, or a +5 chord. You can probably guess what that chord does. It raises the fifth note of the scale one half step. So a C major chord again is C, E, and G, which are the first, third and fifth notes of the scale. So, a +5 would cause the chord to become C, E, G#. On the other hand there is a -5 chord. So, a C-5 would be C, E, Gb or C, E, F#. One strange “chord” is simply called a “5”. It really is only two notes, the first and the the fifth of the scale. So, a C5 is merely C and G. Note that this chord is also sometimes called C major no 3rd. Can you see that? If C major is C, E, G, then C major with no 3rd is simply C and G. 9ths, 11ths and 13ths are somewhat understandable, but they do add some confusion. The understandable part is that they do in fact add the note that one would think. They also add a few more however. But let us start with the reasonable part. A C 9th would add a “D” as one might expect - the ninth note in the scale. A C 11th does add an “F”, the 11th note in the scale, and a C 13th does add an “A” which is the 13th note in the scale. However, C9th also adds a dominant 7th and therefore is: C, E, G, Bb, D. The 11th adds not only the dominant 7th but also the 9th and therefore is: C, E, G, Bb, D, F. The 13th is similar to the 11th and becomes: C, E, G, Bb, D, A. The minor versions of 9th, 11th and 13th? They really follow the same formula but begin with the minor rather than the major chord. So C minor 9th is: C, Eb, G, Bb, D. Scales / Chords other than C If the above makes sense to you, you are getting close to understanding the basic make up of chords. Some is simple mathematics, and some is convention, established years ago. Of course you could say that all this is in the scale and key of C. Other keys are harder. Well, yes and no. The other keys may have odder looking sequencing, but they follow all of the same rules. This is where the mathematics come in. To look at a D scale, for example, note that a D is two half steps in the chromatic scale higher than C. Use that then to calculate all of the notes of the D scale. Doing that a D scale is: D, E, F#, G, A, B, C#, D A D major chord is still the first, third and fifth of the scale, and is: D, F#, A. D minor is similar with the third “flatted” and would be: D, F, A. D 7th would be D, F#, A, C since flatting C#, the 7th note in the D scale is really just C. D major 7th would be D, F#, A, C# (the 1st, 3rd, 5th and 7th notes of the scale) If you understand the above, you may in fact be able to even figure out some of the unique chords that guitar players continue to come up with. I just noted a new one in one of my song books. It was a D major add 2. What? we might say? But then we would get a grip on ourselves. We would put together the D major chord of D, F# and A, and then merely add the 2nd note of the scale, an E. Then we would get D, E, F#, A. Voila ! See? not so bad. If you are like me and appreciate the mathematics, but then have a hard time memorizing 28 chords versus 12 notes of the chromatic scale, 336 possibilities. And even less likely to memorize the guitar fingerings for those 336 down 12 frets making 4032 possibilities, then you might need a simple aid. The software program being marketed by rpsoft 2000, called musicord, is reasonably priced and serves as a simple reference for most of those common chords,

By Weimin Han This ebook makes a speciality of the theoretical elements of small pressure concept of elastoplasticity with hardening assumptions. It presents a finished and unified therapy of the mathematical conception and numerical research. it truly is divided into 3 elements, with the 1st half supplying a close creation to plasticity, the second one half overlaying the mathematical research of the pliability challenge, and the 3rd half dedicated to blunders research of assorted semi-discrete and entirely discrete approximations for variational formulations of the elastoplasticity. This revised and accelerated version comprises fabric on single-crystal and strain-gradient plasticity. moreover, the complete publication has been revised to make it extra obtainable to readers who're actively all for computations yet much less so in numerical research. reports of previous version: “The authors have written a good publication that are steered for experts in plasticity who desire to be aware of extra in regards to the mathematical idea, in addition to people with a heritage within the mathematical sciences who search a self-contained account of the mechanics and arithmetic of plasticity theory.” (ZAMM, 2002) “In precis, the booklet represents a powerful accomplished review of the mathematical method of the speculation and numerics of plasticity. Scientists in addition to teachers and graduate scholars will locate the publication very beneficial as a reference for examine or for getting ready classes during this field.” (Technische Mechanik) "The booklet is professionally written and may be an invaluable connection with researchers and scholars drawn to mathematical and numerical difficulties of plasticity. It represents an incredible contribution within the zone of continuum mechanics and numerical analysis." (Math Reviews) Read or Download Plasticity: Mathematical Theory and Numerical Analysis PDF Similar number systems books The time period differential-algebraic equation was once coined to include differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such difficulties come up in quite a few functions, e. g. restricted mechanical structures, fluid dynamics, chemical response kinetics, simulation of electric networks, and keep watch over engineering. This monograph examines and develops the worldwide Smoothness protection estate (GSPP) and the form upkeep estate (SPP) within the box of interpolation of features. The learn is constructed for the univariate and bivariate instances utilizing famous classical interpolation operators of Lagrange, Grünwald, Hermite-Fejér and Shepard sort. Coupled with its sequel, this e-book offers a hooked up, unified exposition of Approximation idea for services of 1 actual variable. It describes areas of features similar to Sobolev, Lipschitz, Besov rearrangement-invariant functionality areas and interpolation of operators. different subject matters comprise Weierstrauss and top approximation theorems, homes of polynomials and splines. Specific numerical recommendations are already had to take care of nxn matrices for big n. Tensor facts are of measurement nxnx. .. xn=n^d, the place n^d exceeds the pc reminiscence via some distance. they seem for difficulties of excessive spatial dimensions. for the reason that average equipment fail, a specific tensor calculus is required to regard such difficulties. - Multiscale Finite Element Methods: Theory and Applications (Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 4) - Time Dependent Problems and Difference Methods - Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms (Computational Mathematics) - A Theoretical Introduction to Numerical Analysis - The Green Computing Book: Tackling Energy Efficiency at Large Scale (Chapman & Hall/CRC Computational Science) - Topics in Mathematical Analysis and Applications (Springer Optimization and Its Applications) Extra resources for Plasticity: Mathematical Theory and Numerical Analysis We define an elastic material to be one for which the constitutive equations take the form ψ = ψ( ), σ = σ( ). 48) That is, the free energy and stress depend only on the current strain; there is no dependence on the history of behavior, for example. It should be remarked that the more general point of departure is to take the free energy and stress to be functions of the displacement gradient ∇u rather than the strain. That these variables in fact depend on ∇u through its symmetric part, the strain , is a consequence of the principle of material frame indifference (see ). This slope will continue to decrease, and eventually a variety of phenomena may take place. For example, the material may rupture, at which point the experiment will necessarily be regarded as concluded. 1(c)). 1(d)). All of these features, and others yet, are important, the importance of any particular feature depending on the application in question and on the range of stress that is expected to be experienced. 1(b), in which the curve continues to rise, albeit at a slope less than that when σ < σ0 , is known as hardening behavior. We are using here the result from calculus that dy = Jdx, where dx and dy denote the volume elements in Ω and Ωt . A sufficient condition for the motion y to be invertible is that there exist a constant c(Ω) > 0, depending only on Ω, such that sup |∇y − I| < c(Ω). Ω This result, as well as others on the invertibility of the motion, may be found in . Instead of adopting the function y as the primary unknown variable, it is more convenient to introduce the displacement vector u by u(x, t) = y(x, t) − x and to replace the motion by the displacement as the primary unknown.

Srinivasa Ramanujan is a legend of the mathematics world. The son of a shop clerk in rural India, he taught himself mathematics, primarily out of a book he borrowed from the library. The math that he did started out as rediscovering old results, and then became novel, and ultimately became revolutionary; he is considered to be one of the great minds of mathematical history, someone routinely mentioned in comparison with names like Gauss, Euler, or Einstein. Ramanujan's work became known beyond his village starting in 1913, when he sent a letter to the British mathematician G. H. Hardy. Ramanujan had been spamming mathematicians with his ideas for a few years, but his early writing in particular tended to be rather impenetrable, of the sort that today I would describe as "proof by proctological extraction:" he would present a result which was definitely true, and you could check that it was true, but it was completely incomprehensible how he got it. But by the time he wrote to Hardy, both his clarity and the strength of his results had improved, and Hardy was simply stunned by what he saw. He immediately invited Ramanujan to come visit him in Cambridge, and the two became lifelong friends. Alas, his life was very short: Ramanujan died at age 32 of tuberculosis (or possibly of a liver parasite; recent research suggests this may have been his underlying condition), less than six years after his letter to Hardy. When we talk about people whose early death was a tremendous loss to humanity, there are few people for whom it's as true as Ramanujan, and a recent discovery in his papers has just underlined why. This discovery ties together two stories separated by centuries: The "1729" story, and the great mystery of Pierre Fermat's last theorem. The 1729 story comes from a time that Hardy came to visit Ramanujan when he was ill. In Hardy's words: "I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" This has become the famous Ramanujan story (and in fact, 1729 is known to this day as the Hardy-Ramanujan Number), because it's just so ludicrously Ramanujan: he did have the reputation of being the sort of guy to whom you could mention an arbitrary four-digit number, and he would just happen to know (or maybe figure out on the spot) some profound fact about it, because he was just that much of a badass. The other story is that of Fermat's Last Theorem. Pierre de Fermat was a 17th-century French mathematician, most famous for a theorem he didn't prove. In 1637, he jotted down a note in the margins of a book he had, about a generalization of the Pythagorean Theorem. From Pythagoras, we know that the legs and hypotenuse of a right triangle are related by a²+b²=c². We also know that there are plenty of sets of integers that satisfy this relationship -- say, 3, 4, and 5. Fermat asked if this was true for higher powers as well: that is, when n>2, are there any integers a, b, and c such that aⁿ+bⁿ=cⁿ? He claimed that the answer was no, and that "he had a truly marvelous proof of this statement which was, unfortunately, too large to fit in this margin." The consensus of mathematicians ever since is that Pierre de Fermat was full of shit: he had no such proof, and was bluffing. In fact, this statement -- known as Fermat's Last Theorem, as his notes were only discovered after his death -- wasn't proven until 1995, when Andrew Wiles finally cracked it. Wiles' success was stunning because he didn't use any of the traditional approaches: instead, he took (and significantly extended) a completely unrelated-seeming branch of mathematics, the theory of elliptic curves, and figured out how to solve this. That theory is also at the heart of much of modern cryptography, not to mention several rather unusual bits of physics. (Including my own former field, string theory) And so these two stories bring us to what just happened. A few months ago, two historians digging through Ramanujan's papers were amused to find the number 1729 on a sheet of paper: not written out as such, but hidden in the very formula which expresses that special property of the number, 9³+10³=12³+1. What turned this from a curiosity into a holy-crap moment was when the rest of the page, and the pages that went with it, suddenly made it clear that Ramanujan hadn't come up with 1729 at random: that property was a side effect of him making an attempt at Fermat's Last Theorem. What Ramanujan was doing was looking at "almost-solutions" of Fermat's equation: equations of the form aⁿ+bⁿ=cⁿ±1. He had developed an entire mechanism of generating triples like these, and was clearly trying to use this to home in on a way to solve the theorem itself. In fact, the method he was using was precisely the method of elliptic curves which Wiles ended up using to successfully crack the theorem most of a century later. What makes this completely insane is this: Wiles was taking a previously-separate branch of mathematics and applying it to a new problem. But the theory of elliptic curves wasn't even invented until the 1940's. Ramanujan was making significant progress towards solving Fermat's Last Theorem, using the mathematical theory which would in fact prove to be the key to solving it, while making up that entire branch of mathematics sort of in passing. This is why Ramanujan was considered one of the greatest badasses in the history of mathematics. He didn't know about 1729 because his head was full of random facts; he knew about it because, oh yes, he was in the middle of doing yet another thing that might restructure math, but it didn't really solve the big problem he was aiming at so he just forgot about it in his stack of papers. I shudder to imagine what our world would be like if Ramanujan had lived a longer life. He alone would probably have pushed much of mathematics ahead by 30 or 40 years. If you want to know more about elliptic curves, Fermat, and how they're related, the linked article tells more, and links to more still. You can also read an outline of Ramanujan's life at https://en.wikipedia.org/wiki/Srinivasa_Ramanujan , and about Fermat's Last Theorem (and why it's so important) at https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem . Also, There are 1 too many flaws in Sujoy's film for it to be getting this much attention. "Prashant appeared to be a good boy, but it seems that he has no remorse about Akhlaq's death. Instead, he asked us that after the partition, when it had been decided that Hindus will stay here and Muslims will go to Pakistan, why did Gandhi and Nehru ask Muslims stay back in India? I couldn't help but feel dismayed. These are the typical beliefs that keep the pot of communalism boiling. Prashant and I had a heated argument, but I lost. People like us are losing arguments every day. All I could do was ask Prashant to reconsider his views, read a few more books, but he looked self-assured that whatever he knows is true. It is final. I wonder who would have taught Prashant all this? Did someone come amongst these young men well before they coagulated into the mob of that Monday night? Who are those people who have left young men like Prashant to be misled by the purveyors of false histories? Who are those scholars who have left the Prashants of our villages behind to submit their own useless PHDs to earn accolades in foreign universities? We are not understanding what is happening around us. We are not being able to make others understand. The sparks have been spread across our villages. Young men with their half-baked sense of history want me to pose with them for selfies, but are not willing to even consider my appeal that they give up their violent ideals. Our politics has become a collective of opportunists and cowards." - 2008 - present - Oracle Corporation2004 - 2008 - Microsoft2003 - 2004 Good Old Dravid… - The India Uncut Blog - India Uncut My first book, My Friend Sancho, was published in May 2009, and went on to become the biggest selling debut novel released that year in Indi How Much is a Homemaker Worth? | MintLife Blog | Personal Finance News &... The life of a homemaker includes endless demands and lists of to-dos, not to mention the long work hours that often go way beyond the standa MintLife Blog | Personal Finance News & Advice | The Time Value of M... The argument over the value of time vs. money is an age-old debate. Read how one father learned a valuable lesson while organizing his daugh Moving at the Speed of Creativity - So many CMS options: Why I mainly ... I do not consider myself a web design guru, but I've created a fair number of websites in the past fifteen years or so and continue to d Chart: The Great Indian Stock Market Story was only 5 good years The Great Indian Stock Market Story of the last twenty years is only about four years, 2003-2007. If you’re finicky, we could add the stella » A Powerful Three-Step Algorithm for Happiness :zenhabits Post written by Leo Babauta. Today I'm going to share a really simple secret that can make your day instantly better. If you're feel

One of the key characteristics of a transistor is that it can be used as an amplifier. Transistors can act as amplifiers while they are functioning in the active region or when it is correctly biased. The need for a transistor as an amplifier arises when we want to increase or amplify the input signal. A transistor can take in a very small weak signal through the base junction and release the amplified signal through the collector. Transistor amplifiers are used frequently in RF (radio frequency), OFC (optic fibre communication), audio amplification, etc. In this lesson, we will discuss how a transistor works as an amplifier. For a transistor to work as an amplifier, we usually use the common-emitter configuration. The figure below shows how the transistor is set up when it is connected to a circuit as an amplifier. In the figure given above, the input is connected in forward-biased, and the output is connected in reverse-biased. The input signal is applied on the base-emitter junction, and the output is taken through the load in the emitter-collector junction. There is also an application of DC voltage in the input circuit for amplification. Besides, a small change in signal voltage results in the change of emitter current, which is mainly due to the low resistance in the input circuit. Also Read: Forward Bias The output is taken across the load connected to the output side. The load can be in any combination of R, L or C. The load resistance is of high value, which causes a large voltage drop. Overall, the weak signal is thus amplified in the collector circuit. However, to work as an amplifier, the transistor has to work in the active region of the output voltage versus the input voltage curve, as seen in the figure below. In the above graph, we have AB as the cut-off region, BC as the active region, and from C, we have a line parallel to X-axis, which is the saturation region. Gain in Power, Current and Voltage From the figure that is given above, it is clear that on the output side, V0 = VCC– ICRC, where V0 is the output voltage, Ic is the collector current, Rc is the load resistance, and Vcc is the fixed voltage on the output side. If we consider ∆V0 and ∆Vi as small changes in output and input voltages, respectively, then ∆V0 / ∆Vi is called the small-signal voltage gain, Av of the amplifier. Therefore, ∆V0= 0 – Rc ∆IC The gain in terms of voltage when the changes in input and output currents are observed is called voltage gain. Similarly, in the input side, Vin = IB RB + VBE ∆Vin = ∆IBRB + ∆VBE ~ ∆Vin = ∆IBRB (∆VBE <<∆IBRB) Av = ∆V0 / ∆Vin = – Rc ∆IC / RB ∆IB = -βac Rc / RB where βac = ∆IC / ∆IB is the AC current gain. When there is a gain in terms of current due to the changes in input and output currents, it is called current gain. β value can range between 20 and 500. The power gain of an amplifier is defined as the product of the current gain and voltage gain of the amplifier. It is also defined as the rate of change in output power to the change in input power. Mathematically, Ap = βac x Av. Use of Capacitors We have a coupling capacitor that is used to couple the AC component and decouples the DC component contained in the input signal of the amplifier. At the output of the amplifier, the DC component will be invariantly present due to the amplification process. This is being removed by the coupling capacitor at the output, and hence we will have a pure AC signal being supplied to the load connected at the output. Also Read: Capacitor Types and Capacitance Similarly, we do have an Emitter-bypass Capacitor CE. When an AC signal is applied to an amplifier circuit, the variable current will flow through resistors connected at the collector and emitter, i.e., RC and RE. This current in RE will develop a variable voltage drop across RE and provides additional negative feedback to the emitter junction. This will result in an overall reduction of voltage gain associated with the amplifier. The capacitor CE connected across RE provides a short circuit path for the AC signal and reduces the effect of additional negative feedback due to the AC signal and a corresponding reduction in voltage gain. Role of Resistance The resistors R1 and R2 form the voltage division circuit to supply the DC voltage to the base of the transistor. The resistors RC and RE control the collector and emitter currents, respectively. Proper selection of these resistors helps us to control the value of the limiting currents. These resistors provide the required junction voltages between E-B, C-B, C-E and currents IE, IB and IC to work the transistor in the active region of the output characteristics. The emitter resistor RE produces the following changes in the performance of the CE amplifier: - It causes bias stabilization. - It causes the current gain to remain essentially unaltered. - Increases the input and output impedances. - It stabilizes the voltage gain. Also Read: Transformer The Need for CE Configuration We usually employ CE configuration for transistors as amplifiers because it provides large values of current gain, voltage gain and power gain. Moreover, there is a phase shift of 180 degrees between input and output. It implies the output signal will be an inverted amplified version of the signal given in the input. As we come to the end of the lesson, we should remember that a transistor amplifier, in order to function properly, must have the following things: - High input impedance - High gain - High slew rate - High bandwidth - High efficiency - High stability - High linearity 1. Consider a CE- transistor made to work as an amplifier. The audio signal voltage across the collector resistance of 2 KΩ is 2 volts. Suppose the current amplification factor of the transistor is 100, and the base resistance is 1 KΩ. Determine the input signal voltage and base current. Given RC = 2 KΩ = 2000 Ω; VC= 2V; βac = 100; RB = 1KΩ = 1000 Ω Collector current, Ic= Vc/Rc = 2/2000 = 1 mA IB = VB/RB = VB/1000 = VB mA βac = 100 = IC/IB = 1/VB VB = 1/100 = 0.01 V Therefore, IB= VB m A = 0.01 / 1000 = 10 * 10-6 A = 10 µA. 2. 2 amplifiers are connected in a series (cascaded). The voltage gain of the first amplifier is 10, and the second is 20. The input signal is given as 0.01 V. Calculate the output of the AC signal. Total voltage gain is AV = AV1 * AV2 = ∆Vo / ∆Vi ∆Vo = ∆Vi * AV1 * AV2 = 0.01 * 10 * 20 = 2V Frequently Asked Questions on Transistor as Amplifier How does a transistor work as an amplifier? A transistor works as an amplifier by taking in a very small weak signal through the base junction and raising the strength of the weak signal. This amplified signal is released through the collector. What are the functions of a transistor? A transistor is a device used for amplifying, controlling and generating electrical signals. What are the uses of transistors? Transistors are used in cameras, hearing aids, watches, cell phones, power regulators, etc. How many terminals does a transistor have? A transistor has three terminals.

Neutral Higgs-Boson Pair Production [0.5cm] at Hadron Colliders: QCD Corrections S. Dawson, S. Dittmaier and M. Spira***Heisenberg Fellow. Physics Department††† Supported by U.S. Department of Energy contract number DE-AC02-98CH10886., Brookhaven National Laboratory, Upton, NY 11973, USA Theory Division, CERN, CH–1211 Geneva 23, Switzerland II. Institut für Theoretische Physik‡‡‡Supported by Bundesministerium für Bildung und Forschung (BMBF), Bonn, Germany, under Contract 05 7 HH 92P (5), and by EU Program Human Capital and Mobility through Network Physics at High Energy Colliders under Contract CHRX–CT93–0357 (DG12 COMA)., Universität Hamburg, Luruper Chaussee 149, D–22761 Hamburg, Germany Neutral Higgs-boson pair production provides the possibility of studying the trilinear Higgs couplings at future high-energy colliders. We present the QCD corrections to the gluon-initiated processes in the limit of a heavy top quark in the loops and the Drell–Yan-like pair production of scalar and pseudoscalar Higgs particles. The cross sections are discussed for LHC energies within the Standard Model and its minimal supersymmetric extension. The QCD corrections are large, enhancing the total cross sections significantly. The Higgs mechanism is a cornerstone of the Standard Model (SM) and its supersymmetric extensions. Thus, the search for Higgs bosons is one of the most important endeavours at future high-energy experiments. In the SM one Higgs doublet, , has to be introduced in order to break the electroweak symmetry, leading to the existence of one elementary Higgs boson, . The scalar sector of the SM is uniquely fixed by the vacuum expectation value of the Higgs doublet and the mass of the physical Higgs boson . Once a Higgs particle is found, it is necessary to investigate its properties in order to reconstruct the Higgs potential and to verify that it is indeed the SM Higgs boson. A first step in this direction is the measurement of the trilinear self-couplings, which are uniquely specified by the scalar potential The parameter defines the strength of the Higgs self-interactions. In the SM it is given by . At tree level, can only be probed through multiple Higgs-boson interactions and there are, at present, no direct experimental limits on . In extensions of the SM, such as models with an extended scalar sector, with composite particles or with supersymmetric partners, the self-couplings of the Higgs boson may be significantly different from the SM predictions. The limits that may be obtained on the trilinear self-coupling of the Higgs boson at the LHC and the impact of QCD corrections represent a particular topic of this paper. Since the minimal supersymmetric extension of the Standard Model (MSSM) requires the introduction of two Higgs doublets in order to preserve supersymmetry, there are five elementary Higgs particles, two -even ()111We have taken care that no confusion can arise from using the same symbol for the SM and the heavy -even MSSM Higgs particle., one -odd () and two charged ones (). This leads to a large variety of self-interactions among them. At lowest order all couplings and masses of the MSSM Higgs sector are fixed by two independent input parameters, which are generally chosen as , the ratio of the two vacuum expectation values , and the pseudoscalar Higgs-boson mass . The self-interactions among the Higgs bosons (at lowest order) are given in terms of the electroweak gauge couplings, tg and , and may be quite different from the ones of the SM, which are governed by the parameter . Higher-order corrections to the MSSM Higgs sector turn out to be important owing to the large top-quark mass [?–?]. They increase the upper bound on the light scalar Higgs mass from the -boson mass to about 130 GeV, along with altering the Higgs-boson self-couplings with contributions proportional to . Higgs-boson pairs can be produced by several mechanisms at hadron colliders: vector-boson fusion , associated production , Higgs radiation off top and bottom quarks , At the LHC, gluon fusion is the dominant source of Higgs-boson pairs, although in some regions of the MSSM parameter space, vector-boson fusion can be important. Note, however, that represents an exceptional case, since this channel is suppressed with respect to the Drell–Yan-like process , so that it will be very difficult to separate the gluon-fusion process in this case experimentally. The gluon-fusion process , on the other hand, is competitive with the Drell–Yan-like process . In this paper we present the QCD corrections to the Drell–Yan-like production and the gluon–gluon collision processes. The gluon-fusion processes are, in the SM, mediated by triangle and box loops of top and bottom quarks; in the SM, the contributions of the bottom quark can always be neglected. In the MSSM, the squark contributions will be suppressed if the squarks are heavier than GeV, and, for small tg, the top-quark loops dominate the cross sections. The QCD corrections to the gluon-fusion processes have been obtained in the limit of a heavy top quark by means of low-energy theorems and also by explicitly expanding all relevant one- and two-loop diagrams. The results are expected to be valid for small tg in the MSSM and below the threshold of the top-quark loops in both the MSSM and the SM, since in this regime effects of a finite top-quark mass are expected to be small. In the case of single-Higgs production the same procedure reproduces the known exact result for the NLO cross section within 5%, for Higgs-boson masses below . The considered QCD corrections are important in the process of extracting limits on the Higgs-boson self-couplings reliably. The paper is organized as follows. In the next section the low-energy theorems for the interactions of gluons with light Higgs bosons will be reviewed, and the relevant interactions in the heavy-quark limit will be constructed. In Section 3 the details of the calculation will be described, and in Section 4 we present the results for the SM and MSSM Higgs bosons. In Section 5 we give some conclusions. 2 Low-Energy Theorems In the low-energy limit of vanishing Higgs four-momentum, the Higgs-field operator acts as a constant field. In this limit it is possible to derive an effective Lagrangian for the interactions of the Higgs bosons with gauge bosons, which is valid for light Higgs bosons. This effective Lagrangian has been successfully used to compute the QCD corrections to a number of processes, in particular to single-Higgs production from gluon fusion at the LHC [13, 14]. In this case, the result of using the low-energy theorems has been shown to agree with the exact two-loop calculation to better than even for as large as TeV. This lends legitimacy to our use of the low-energy theorems to compute QCD corrections to multiple Higgs-boson production via gluon fusion. In the limit of vanishing Higgs four-momentum the entire interaction of the scalar Higgs particles with a heavy quark can be generated by the substitution in the Lagrangian of a heavy quark of bare mass , where denotes the bare SM Yukawa coupling, which must not be included in the substitution. The symbol is the relative strength of the heavy-quark Yukawa coupling, In the SM, we have . The expressions for in the MSSM are given in Ref. . At higher orders this substitution has to be performed for the unrenormalized parameters [14, 16]. In the following we restrict our analysis to the top-quark contributions. At next-to-leading order (NLO) the effective interaction between several scalar Higgs fields and gluons can be obtained from the radiatively corrected effective Lagrangian of gluon fields, with denoting the top-quark contribution to the unrenormalized gluon vacuum polarization at zero-momentum transfer. At two-loop order, we have where the strong coupling constant includes five light flavours. This means that the top-quark contribution to the running of has been subtracted at vanishing momentum transfer. Hereafter, we drop the superscript on . Performing the substitution Eq. (2) and renormalizing the bare top mass via where denotes the pole mass, we end up with the NLO result The case of odd numbers of pseudoscalar Higgs bosons can be derived from the ABJ anomaly in the divergence of the axial vector current [14, 16, 18]. The interactions that are relevant in our case are222Note that in the earlier Refs. [14, 16, 18] a factor of 1/2 is missing in the effective Lagrangians for the single pseudoscalar Higgs-boson coupling to gluons. where denotes the dual gluon field-strength tensor. Owing to the Adler–Bardeen theorem there are no higher-order corrections to the effective Lagrangians involving odd numbers of pseudoscalar Higgs bosons. These Feynman rules can now be used to compute Higgs interactions beyond the lowest order. We recall that there is no contribution of light quarks (which are considered to be massless) to the effective couplings, but note that light-quark loops have to be included when the Higgs bosons do not directly couple to the quark loops. Such contributions arise, in particular, in and cannot be obtained from the low-energy theorems. 3 QCD Corrections 3.1 Gluon fusion: basic definitions At leading order (LO) neutral-Higgs pair production via gluon fusion is mediated by triangle and box diagrams of heavy quarks, as exemplified in Fig. 2. In the heavy-quark limit, the fermion triangles and boxes can be replaced by the effective vertices of Fig. 1. Throughout this analysis, we choose the squark masses to be 1 TeV so that squark-loop contributions can be neglected in the MSSM case. Generically the partonic LO cross section can be expressed as The Mandelstam variables for the parton process are given by where is the scattering angle in the partonic c.m. system with invariant mass , and The integration limits in Eq. (11) correspond to . The scale parameter is the renormalization scale. The complete dependence on the fermion masses is contained in the functions , , and . The full expressions of the form factors , , , including the exact dependence on the fermion masses, can be found in Ref. . The couplings and and the form factors in the heavy-quark limit are given by: with the trilinear coupling . The couplings for the processes are generically defined as () where denotes the Higgs particles of the -channel contributions. The trilinear couplings and the normalized Yukawa couplings can be found in Ref. . The individual expressions in the heavy-quark limit can be summarized as: where denotes the axial charge of the top quark. It should be noted that owing to the Ward identities for the vertex only the pseudoscalar Goldstone component of the bosons contributes to in the case of and production in the heavy-quark limit. The anomaly contributions of the top and bottom quarks cancel. The QCD corrections consist of two-loop virtual corrections, generated by gluon exchange between the quark lines and/or external gluons, and one-loop real corrections with an additional gluon or light quark in the final state. We have evaluated the QCD corrections in the heavy-quark limit by means of two different methods: (i) using the effective couplings based on the low-energy theorems, as presented in the previous section, and (ii) explicitly expanding all relevant one- and two-loop diagrams in the inverse heavy-quark mass. In the following we shall describe the details of both approaches. 3.2 Low-energy theorems We are now in a position to compute the NLO corrections to Higgs-boson pair production. Typical effective diagrams contributing to the virtual and real corrections are presented in Fig. 3. Adopting the Feynman rules of Fig. 1 for the effective interactions, the calculation has been carried out in dimensional regularization with dimensions. The strong coupling has been renormalized in the scheme including five light-quark flavours, i.e. decoupling the top quark in the running of . After summing the virtual and real corrections the infrared singularities cancel. However, collinear initial-state singularities are left over in the partonic cross sections. Those divergences have been absorbed into the NLO parton densities, defined in the scheme with five light-quark flavours. We end up with finite results, which can be cast into the form with the individual contributions The coefficient for the virtual corrections reads The objects denote the Altarelli–Parisi splitting functions : where in our case. The factorization scale of the parton–parton luminosities is denoted by . The coefficients for the individual final-state Higgs bosons are given by In order to improve the validity of our results, we have inserted the full expressions for the form factors and in Eqs. (11) and (22), i.e. including the exact dependence on the fermion masses. This procedure is reasonable since the QCD corrections are dominated by soft and collinear gluon radiation, which do not resolve the Higgs couplings to gluons, analogously to single-Higgs production via gluon fusion . A few remarks on the -channel -boson exchange in production are in order. For the virtual corrections, the factorization of the NLO corrections into the LO form factors and a universal correction factor is exact for and . This is due to the fact that only the pseudoscalar Goldstone component of the boson contributes as in LO, i.e. the QCD corrections coincide with the one to -channel pseudoscalar Higgs-boson exchange. For the real corrections, the factorization is not exact, but the applied correction factor correctly includes the dominant contributions, which are caused by soft and collinear gluon radiation. Additional infrared- and collinear-finite contributions, e.g. originating from box corrections in processes, are expected to be small, since they do not exhibit large contributions from soft and collinear gluon radiation. They are neglected in our analysis. 3.3 Explicit expansion of the gluon-fusion diagrams We have derived the above results also by explicitly performing the heavy-mass expansion of the corresponding one- and two-loop Feynman diagrams. The amplitudes for the individual diagrams have been automatically generated using the package FeynArts . The asymptotic expansion of the individual amplitudes in the heavy top-quark mass is carried out directly in the integrand, i.e. before the integration over the momentum space. We employ the general algorithm of Ref. (see also Ref. and references therein) for the asymptotic expansion of Feynman diagrams in dimensional regularization. This method expresses the coefficients of the expansion in terms of simpler diagrams. At the one-loop level, this procedure leads to simple one-loop vacuum integrals only. At the two-loop level, we get two-loop vacuum integrals and products of one-loop vacuum integrals and massless one-loop integrals with non-vanishing external momenta. The analytical calculation of all those integrals is straightforward when using the Feynman-parameter technique. Since the employed strategy leads to a very large number of terms in intermediate steps, and since each step is algorithmic, we have fully automatized the calculation in Mathematica . In the following we sketch the single steps of the calculation and give the results for the basic integrals. The general algorithm for the asymptotic expansion of any given Feynman graph in the limit for some internal masses can easily be summarized. Denoting the corresponding Feynman amplitude by and the corresponding integrand by , the large-mass expansion reads where are the integration momenta. The sum on the r.h.s. runs over all subgraphs of which contain all propagators with the heavy masses and which are irreducible with respect to those lines of that carry light masses . The integrand of the subgraph is denoted by . The reduced graph results from upon shrinking to a point, and the integrand is defined such that . The symbol represents an operator that replaces the integrand by its Taylor series in the expansion parameters and , where are the external momenta of the subgraph . Therefore, Eq. (26) expresses the original integral in terms of an infinite sum over simpler integrals. For any given power , this sum contains only a finite number of terms that are non-vanishing in after the scaling limit is taken. These terms can easily be determined by power counting. This general strategy for the heavy-mass expansion will become more transparent when we inspect in more detail the different types of graphs that are relevant in our case. We start by considering the relevant one-loop integrals in the limit . They contain only top-quark propagators in the loop, both for the LO calculation and for the real NLO corrections. According to the algorithm (26), there is only one relevant subgraph , namely the subgraph containing only the propagators of lines inside the loop. If is irreducible, we have . The Taylor-expansion operator replaces each propagator by where denotes the integration momentum, and is any combination of external momenta. These replacements express each one-loop diagram by a sum of terms containing one-loop vacuum integrals The terms that are non-vanishing in the heavy-mass limit can be determined by simple power counting, since an explicit factor of and the integration momentum contribute to the scaling factor in exactly in the same way. All non-vanishing vacuum tensor integrals (28) can be decomposed into terms that are products of metric tensors and coefficient factors. The coefficients for the different covariants, which span the whole tensor, can be algebraically expressed in terms of scalar vacuum integrals. This algebraic reduction, which proceeds recursively in the tensor rank, is standard. The trick is to contract the equation that expresses the integral in terms of covariants with a set of some suitably chosen covariants. On the side of the integral, this leads to integrals that are already known; on the other side of this equation, one gets linear combinations for the tensor coefficients. The coefficients are obtained by inverting a system of such linear equations for the coefficients. For the case of one-loop tensor vacuum integrals, at most a single covariant structure contributes, namely the totally symmetric tensor built of metric tensors . Only scalar one-loop integrals, i.e. the ones of (28) with , have to be computed explicitly. A simple calculation yields At the two-loop level, there are two basically different types of diagrams. The first type contains two independent top-quark loops. Such diagrams do not lead to genuine two-loop integrals and can be treated like the one-loop diagrams above. Topologically those diagrams are represented by the third graph of Fig. 3. The second type of graphs is formed by the genuine two-loop diagrams. Each of those diagrams contains a closed top-quark loop and one, two, or three internal gluon lines. Typical box diagrams are shown in Fig. 4. For all such genuine two-loop diagrams there are two subgraphs that are relevant in the expansion (26) for . The first subgraph is the diagram built of all lines inside the loops, the second is given by the closed top-quark loop. First we consider the case . The Taylor expansion of the integrand involves the consistent expansion of each propagator about the external momenta of the process. This means that each propagator is replaced by where is a linear combination of the two integration momenta and , and consists of external momenta . This replacement leads to two-loop vacuum integrals of the general form with the actual mass insertions and . In the power counting, which determines the non-vanishing terms for , factors of the integration momenta and contribute in the same way as explicit factors of . The algebraic reduction of two-loop tensor vacuum integrals proceeds along the same lines as described for the one-loop case above. The only difference is that tensors need not be totally symmetric beyond one loop. Finally, we are left with the scalar integral , which can be easily calculated, Now we identify the subgraph with the top-quark loop. In this case the Taylor expansion in the integrand concerns the momenta that are external with respect to the top-quark loop. Thus, all top-quark propagators are replaced by where is the loop momentum running through the top-quark loop. Note that includes all external momenta of the process as well as the loop momentum running through the internal gluon lines. The integration over leads to one-loop vacuum integrals of the form (28), the calculation of which is described above. The integration over involves only massless propagators to the first power. Since the integration does not involve any -terms, it does not affect the power counting in at all. For one-gluon exchange the integral over is a massless tadpole, which vanishes in dimensional regularization. For two-gluon and three-gluon exchange the integration leads to the one-loop tensor integrals The tensor integrals can again be recursively reduced to the corresponding scalar integrals in a fully algebraic manner. The relevant scalar integrals can be easily calculated and are given by For the integrals are zero in dimensional regularization. The case with occurs, for instance, in graphs like Fig. 4b. Therefore, we find that the contribution in the expansion (26) for which is the top-quark loop is only non-vanishing in diagrams like Figs. 4c and d, where both external gluons are attached to the internal gluon lines. A few more “physical” remarks on the formally described algorithm for the asymptotic expansion seem to be in order. The different contributions to a given Feynman graph that are associated with the subgraphs in the expansion (26) are directly related to the effective diagrams in the approach of the low-energy theorem described above. Shrinking the subgraph to a point leads to the corresponding effective diagram, where the point arising from is the point-like interaction of the effective Lagrangian. A non-vanishing contribution of a subgraph requires that at least one external momentum passes through a massless propagator; otherwise the loop integral involving the massless propagators is zero. In other words, it is necessary that there exists a cut through the diagram that passes exclusively massless lines. Therefore, only diagrams with such a “massless cut” can lead to contributions to effective diagrams in which an effective coupling appears in loops. But all diagrams in general contribute to tree-like effective diagrams, which result from shrinking the complete loop part to a point. Finally, we mention that , which appears in the case of pseudoscalar Higgs bosons, is treated according to the prescription of ’t Hooft and Veltman . Technically we substitute by before the evaluation of the Dirac trace so that the actual trace calculation can be carried out for usual -dimensional Dirac matrices. The correct projection of the trace result on the physical four-dimensional space is achieved upon the contraction with the four-dimensional -tensor. In this approach, all loop integrations can be carried out before the contraction with , i.e. -dimensional momenta can be used. Note, however, that the contraction with necessarily occurs before the integration over the phase space of the radiated parton in the real corrections, i.e. one has to take care of the four-dimensionality of there. Moreover, an additional spurious counter term has to be added to the vertex (see also [14, 18]). 3.4 Drell–Yan-like processes Pairs of scalar and pseudoscalar Higgs bosons can also be produced in collisons via -channel -boson exchange, see Fig. 5. At LO the partonic cross sections are given by where denotes the partonic c.m. energy squared, and , are the vectorial and axial charges of the initial-state quarks. The QCD corrections coincide analytically with the QCD corrections to the Drell–Yan process , if squarks and gluinos are heavy, so that their contributions can be neglected. Thus the NLO cross section can be expressed as:

At times, music notation can be confusing. Even for highly trained musicians, certain aspects of music notation requires them to pause and investigate how a particular rhythmic grouping should be performed. This is particularly true with rhythmic groupings known as tuplets. All tuplets are mathematical ratios. The first number in the ratio is the number of notes in a bracketed group. That number also often (but not always) appears over the bracket. The second number in the ratio is the number of the same kind of notes which represents the time span over which the tuplet should be completed. Triplets (usually just referred to as triplets, not tuplets), provide an easy example. An eighth note triplet consists of three eight notes bracketed together. Those three eighth notes in the triplet are to be played in the normal time it takes to play two eighth notes that are not part of a triplet. In this case, the ratio would be 3:2. Three eighth notes played in the time span usually taken up by two, that is played in one beat of common time. All triplets have the same ratio—only the note values change. For a quarter note triplet, the three quarter notes in the triplet are played over the time span otherwise occupied by two quarter notes, that is, played in two beats of common time. The kind of note (quarter, half, etc.) stays the same on both sides of the ratio. Quintuplets (five notes bracketed together), sextuplets (six notes) and septuplets (seven notes) are all played in the time span of four like notes in common time. The ratios are 5:4 for the quintuplet, 6:4 for the sextuplet and 7:4 for the septuplet. This is the conventional notation for tuples in modern music notation. These ratios can be seen in the example below. Unfortunately, composers do not always use conventional notation. In these cases, performers must interpret the notation that encounter and adjust their performance to the closest match. For example, an eighth note septuplet should be played over two beats according to its ratio (7:4) because four eighth notes occupy a time span of two beats. But if that eighth note septuplet occupies an entire measure in common time, clearly it must be played over four beats, not two. According to conventional notation, this would be incorrectly notated, but when it appears, the composer leaves no choice but to play it over four beats. A composer might also choose to notate a septuplet to be played over one beat in thirty-second notes because of the rapid speed at which such a septuplet must be played. Thirty-second notes are an accurate representation of how the music sounds, though it is not accurate according to conventional notation, in which it would be notated in sixteenth notes in accordance with its ration of 7:4. (Blatter, 2007) In practice, the musician must read what the allotted time-span for the tuple is in the notation, and evenly space the number of notes in the tuples across that time span. So far, I have only discussed tuplets in common time. There are also tuplets in triple meter, and they have their own ratios. For example, in three-four meter, two quarter notes bracketed together and filling out the entire measure have a ratio of 2:3—that is, the two quarter notes of the duplet must be played over three beats. A quarter note quintuplet in three-four meter and filling out the entire measure would require five notes to be played over three quarter note beats, a ratio of 5:3. It is even possible to have an eighth note septuplet filling out an entire measure in three-four meter, requiring that seven eighth notes be played in the time span normally assigned to six, that is, in three beats. That ratio would be 7:6. With tuplets becoming more frequently used by recently active composers, they have often clarified their intent by including the ratio above the music notation. This leaves no doubt how the rhythmic figure is to be interpreted. In the case of duplets in duple compound meter, two eighth notes might be bracketed together with a “2” over the bracket, or two dotted eighth notes might be notated with no bracket or number above. Both are ways of notating the same thing. The occurrence of three durations over the time span normally occupied by two of the same durations is most often called a triplet. A triplet is sometimes called a hemiola usually when its effect is to disrupt the established metric feel. Hemiola originally described the ratio of one and a half to one. We can see how this applies to triplets if we take an eighth note triplet. Three eight notes is one and a half beats in common time, and the three eighth notes of the triplet are performed over one beat. Brahms was fond of using them, and Bernstein famously used hemiolas in his song “America” from West Side Story. “I like to be in A”- is sung with six notes, two beats in six-eight meter, then “-mer-i-ca!” Is also two beats, but with three notes. That’s the hemiola—three notes during two beats, 3:2. Regardless of which tuplet occurs, and in which meter, the function of a tuplet is to borrow a rhythm that does not naturally occur in the current meter and impose it on that meter, changing the grouping structure of the musical phrase. Triplets occur in duple meter, and duplets occur in triple meter. Because rhythms are already notated in groups of three in triple meter, triplets are not necessary there, and because rhythms are already notated in groups of two in duple meter, duplets are not necessary there. Tuplets temporarily disrupt the metric structure, and derive their effectiveness from that disruption. If a particular tuplet is repeated at length, it no longer sounds like a tuplet, but instead like a change in meter. Conventional notation helps us understand tuplets in many but not all cases. Extrapolating the correct interpretation of a tuplet where conventional notation is not used is done by fitting it into the allotted time span according to the metric context given in the notation. Blatter A. (2007). Revisiting music theory : a guide to the practice. Routledge.

In the century, the universe seemed a simpler place. It appeared to run like clockwork according to Newton's laws. All that began to change early in the century, in a golden age for physics that saw the way we view the universe repeatedly turned inside out. Among the greatest of these changes was Einstein's theory of special relativity, which this course explores. The "relativity" in the theory's name refers to the relative nature of time, length, energy, and momentum, which Einstein declared to be no longer absolute concepts, but different based on the relative motion of an observer or detector. In this chapter, we will explore the motivation for Einstein's theory, but we will begin by recalling how relative motion is woven into Newtonian mechanics. We will begin our venture into Einstein's theory of special relativity on the high seas. You are exploring a desolate part of the ocean on a raft, and your friend Optimus is above you on a hot air balloon. Your raft moves with the current, while Optimus's balloon moves with the wind. While exploring Einstein's theory, we will be concerned about how someone who is making physical measurements—an observer—describes the events and objects around them. You will frequently come across the term reference frame to describe the coordinate system that an observer drags along with them. Because different observers, such as you and Optimus, can have their own reference frames, the measurements you make of the world will be different. Optimus is directly above you at You are both moving East, which you define as the -direction of your coordinate system. He is going faster than you, and his speed is greater than yours by an amount This is Optimus's speed relative to you—or more simply, his relative speed. Your position in your reference frame is always What is Optimus's position in your reference frame at time You spot a seagull at time at a distance of ahead of your current position. You call to Optimus and ask him the position of the seagull in his reference frame. He tells you he sees it at Using the answer to the last question, find the relationship between your two measurements. This relationship is called the Galilean transformation and provides a way to convert between your reference frame and Optimus's "primed" reference frame. Note: at Optimus is right above you, so at this moment, you both are the same distance from the gull. Although you and Optimus are measuring the same seagull, you report different measurements. This is what is meant by relativity: observers in different reference frames make different (but related) measurements of the same events. In classical mechanics, the Galilean transformation is the tool that translates between spatial positions in different frames. We will soon see in Einstein's relativity that observers won't agree on spatial positions, or even how much time passes between events, so the transformation between frames will not be so simple. We will need to employ a new tool to relate objects in different reference frames: the spacetime diagram. Much like position versus time graphs in kinematics, spacetime diagrams depict the set of positions occupied by an object over a range of times. The difference is that on a spacetime diagram, time is on the vertical axis, and a spatial dimension is on the horizontal axis. For example, on a spacetime diagram above, the seagull's motion is plotted in your frame, together with the point when you first observed it. The trajectories represented on a spacetime diagram are called worldlines. Since observers in different reference frames do not agree on position and speed measurements, they will draw different worldlines on a diagram. For example, in Optimus's frame, the gull follows the worldline on the diagram below. Relative to you, which is moving faster, Optimus or the gull? Note: in Galilean relativity, observers agree on the time that all events take place, so the time axis of the spacetime diagram in Optimus's frame remains labeled by instead of Optimus' speed in the air relative to your boat on the water is and consequently, you and Optimus make different measurements of the gull's speed. Speed along the -axis in your reference frame is which we abbreviate as On the other hand, Optimus measures speed in his frame. The Galilean transformation relates position measurements in your reference frames. You carefully observe the seagull, and in your reference frame, it has velocity Assuming Optimus' relative speed is constant, what is the seagull's speed in Optimus's primed coordinates? Suddenly, Optimus pulls a slingshot from his balloon basket. He calls back to you, "I'm low on food, looks like we're having gull tonight!" You know Optimus's slingshot releases with a speed of (in Optimus's reference frame). If you measured the shot's speed in your reference frame, what would you measure? Recall: is Optimus's speed relative to your reference frame. The idea of snacking on gull does not whet your appetite, and you hope Optimus misses his shot. In order for it to definitely get away, how should the gull's speed compare to the shot? Assume gravity is insignificant in this situation. Because it dovetails with our everyday intuition, Galilean relativity was the tool relating different reference frames until the beginning of the century, i.e. until a young patent clerk named Einstein demonstrated that the Galilean viewpoint was inconsistent with the properties of light and matter. Einstein's revisions of relativity were inspired by how observers in different reference frames measure, not gulls or slingshot stones, but the speed of light and high-energy particles. If Optimus had a rifle with a laser scope, Einstein would have asked, "In a different frame, with what speed does laser light leave the scope?" This is the central question that Einstein set out to answer when he constructed the theory of relativity. Let's set off in his footsteps to see how he answered this question and why his answer was ultimately correct.

1 chapter 8 Behind the Supply Curve: >> Inputs and Costs Section 2: Two Key Concepts: Marginal Cost and Average Cost We ve just seen how to derive a firm s total cost curve from its production function. Our next step is to take a deeper look at total cost by deriving two extremely useful measures: marginal cost and average cost. As we ll see, these two measures of the cost of production have a somewhat surprising relationship to each other. Moreover, they will prove to be vitally important in Chapter 9, where we will use them to analyze the firm s output decision and the market supply curve. Marginal Cost We defined marginal cost in Chapter 7: it is the change in total cost generated by producing one more unit of output. We ve already seen that marginal product is easiest to calculate if data on output are available in increments of one unit of input. Similarly, marginal cost is easiest to calculate if data on total cost are available in increments of one unit of output. When the data come in less convenient increments, it s still possible to calculate marginal cost over each interval. But for the sake of simplicity, let s work with an example in which the data come in convenient increments. 2 2 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST Ben s Boots produces leather footwear; Table 8-1 shows how its costs per day depend on the number of boots it produces per day. The firm has fixed cost of $ per day, shown in the second column, which represents the daily cost of its bootmaking machine. The third column shows the variable cost, and the fourth column shows the total cost. Panel (a) of Figure 8-5 plots the total cost curve. Like the total cost curve for George and Martha s farm in Figure 8-4 in Section 1: The Production Function, this curve is upward sloping, getting steeper as you move up it to the right. The significance of the slope of the total cost curve is shown by the fifth column of Table 8-1, which calculates marginal cost: the cost of each additional unit. The general formula for marginal cost is Change in total cost Change in total cost generated by one (8-3) Marginal cost = = Change in quantity of output additional unit of output or MC = TC/ Q As in the case of marginal product, marginal cost is equal to rise (the increase in total cost) divided by run (the increase in the quantity of output). So just as marginal product is equal to the slope of the total product curve, marginal cost is equal to the slope of the total cost curve. Now we can understand why the total cost curve gets steeper as we move up it to the right: as you can see in Table 8-1, the marginal cost at Ben s Boots rises as output increases. Panel (b) of Figure 8-5 shows the marginal cost curve corresponding to the data in Table 8-1. Notice that, as in Figure 8-2 in Section 1: The Production Function, we plot the marginal cost for increasing output from 0 to 1 pair of boots halfway between 0 and 1, the marginal cost for increasing output from 1 to 2 pairs of boots halfway between 1 and 2, and so on. 3 3 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST Why is the marginal cost curve upward sloping? Because there are diminishing returns to inputs in this example. As output increases, the marginal product of the variable input declines. This implies that more and more of the variable input must be used to produce each additional unit of output as the amount of output already produced rises. And since each unit of the variable input must be paid for, the cost per additional unit of output also rises. TABLE 8-1 Costs at Ben s Boots Quantity of boots Variable Q Fixed cost cost Total cost (pairs) FC VC TC = FC + VC $ $ ,200 $ ,080 $1,308 Marginal cost of pair MC = TC/ Q $ 4 4 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST In addition, recall that the flattening of the total product curve is also due to diminishing returns to inputs in production: the marginal product of an input falls as more of that input is used if the quantities of other inputs are fixed. The flattening of the total product curve as output increases and the steepening of the total cost Figure 8-5 Total Cost and Marginal Cost Curves for Ben s Boots Total cost $1,400 1,200 1, nd pair of boots increases total cost by $36. (a) Total Cost 8th pair of boots increases total cost by $180. TC Marginal cost of pair $ (b) Marginal Cost MC Quantity of boots (pairs) Quantity of boots (pairs) Panel (a) shows the total cost curve from Table 8-1. Like the total cost curve in Figure 8-4 in Section 1: The Production Function, it slopes upward and gets steeper as we move up it to the right. Panel (b) shows the marginal cost curve. It also slopes upward, reflecting diminishing returns to the variable input. >web... 5 5 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST curve as output increases are just flip-sides of the same phenomenon. That is, as output increases, the marginal cost of output also increases because the marginal product of the variable input is falling. We will return to marginal cost in Chapter 9, when we consider the firm s profitmaximizing output decision. But our next step is to introduce another measure of cost: average cost. Average total cost, often referred to simply as average cost, is total cost divided by quantity of output produced. Average Cost In addition to total cost and marginal cost, it s useful to calculate one more measure, average total cost, often simply called average cost. The average total cost is total cost divided by the quantity of output produced; that is, it is equal to total cost per unit of output. If we let ATC denote average total cost, the equation looks like this: Total cost (8-4) ATC = = TC/Q Quantity of output Average total cost is important because it tells the producer how much the average or typical unit of output costs to produce. Marginal cost, meanwhile, tells the producer how much the last unit of output costs to produce. Although they may look very similar, these two measures of cost typically differ. And confusion between them is a major source of error in economics, both in the classroom and in real life. Table 8-2 uses the data from Ben s Boots to calculate average total cost. For example, the total cost of producing 4 pairs of boots is $300, consisting of $ in fixed cost and $192 in variable cost (see Table 8-1). You can see from Table 8-2 that as quantity of output increases, average total cost first falls, then rises. Figure 8-6 plots that data to yield the average total cost curve, which shows how average total cost depends on output. As before, cost in dollars is measured on the vertical axis and quantity of output is measured on the horizontal axis. The average total 6 6 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST A U-shaped average total cost curve falls at low levels of output, then rises at higher levels. Average fixed cost is the fixed cost per unit of output. Average variable cost is the variable cost per unit of output. cost curve has a distinctive U shape that corresponds to how average total cost first falls and then rises as output increases. Economists believe that such U-shaped average total cost curves are the norm for producers in many industries. To help our understanding of why the average total cost curve is U-shaped, Table 8-2 breaks average total cost into its two underlying components, average fixed cost and average variable cost. Average fixed cost, or AFC, is fixed cost divided by the quantity of output, also known as the fixed cost per unit of output. For example, if Ben s Boots produces 4 pairs of boots, average fixed cost is $/4 = $27 per pair of boots. Average variable cost, or AVC, is variable cost divided by the quantity of out- TABLE 8-2 Average Costs for Ben s Boots Quantity of boots Total Average total Average fixed Average variable Q cost cost of pair cost of pair cost of pair (pairs) TC ATC = TC/Q AFC = FC/Q AVC = VC/Q 1 $120 $ $.00 $ , , 7 7 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST put, also known as variable cost per unit of output. At an output of 4 pairs of boots, average variable cost is $192/4 = $48 per pair. Writing these in the form of equations, Fixed cost (8-5) AFC = = FC/Q Quantity of output Figure 8-6 Average Total Cost Curve for Ben s Boots Average total cost of pair The average total cost curve at Ben s Boots is U-shaped. At low levels of output, average total cost falls because the spreading effect of falling average fixed cost dominates the diminishing returns effect of rising average variable cost. At higher levels of output, the opposite is true and average total cost rises. At point M, corresponding to an output of three pairs of boots per day, average total cost is at its minimum level. >web... $ M Minimum average total cost Average total cost, ATC Minimum-cost output Quantity of boots (pairs) 8 8 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST AVC = Variable cost Quantity of output = VC/Q Average total cost is the sum of average fixed cost and average variable cost; it has a U shape because these components move in opposite directions as output rises. Average fixed cost falls as more output is produced because the numerator (the fixed cost) is a fixed number but the denominator (the quantity of output) increases as more is produced. Another way to think about this relationship is that, as more output is produced, the fixed cost is spread over more units of output; the end result is that the fixed cost per unit of output the average fixed cost falls. You can see this effect in the fourth column of Table 8-2: average fixed cost drops continuously as output increases. Average variable cost, however, rises as output increases. As we ve seen, this reflects diminishing returns to inputs in production: each additional unit of output incurs more variable cost to produce than the previous unit. So variable cost rises at a faster rate than the quantity of output increases. Increasing output, therefore, has two opposing effects on average total cost the spreading effect and the diminishing returns effect : The spreading effect: the larger the output, the more production that can share the fixed cost, and therefore the lower the average fixed cost. The diminishing returns effect: the more output produced, the more variable inputs it requires to produce additional units, and therefore the higher the average variable cost. At low levels of output, the spreading effect is very powerful because even small increases in output cause large reductions in average fixed cost. So at low levels of output, the spreading effect dominates the diminishing returns effect and causes the average total cost curve to slope downward. But when output is large, average fixed cost is 9 9 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST already quite small, so increasing output further has only a very small spreading effect. Diminishing returns, on the other hand, usually grow increasingly important as output rises. As a result, when output is large, the diminishing returns effect dominates the spreading effect, causing the average total cost curve to slope upward. At the bottom of the U-shaped average total cost curve, point M in Figure 8-6, the two effects exactly balance each other. At this point average total cost is at its minimum level. Figure 8-7 brings together in a single picture four members of the family of cost curves that we have derived from the total cost curve: the marginal cost curve (MC), the average total cost curve (ATC), the average variable cost curve (AVC), and the average fixed cost curve (AFC). All are based on the information in Tables 8-1 and 8-2. As before, cost is measured on the vertical axis and quantity is measured on the horizontal axis. Let s take a moment to note some features of the various cost curves. First of all, marginal cost is upward sloping the result of diminishing returns that make an additional unit of output more costly to produce than the one before. Average variable cost also is upward sloping again, due to diminishing returns but is flatter than the marginal cost curve. This is because the higher cost of an additional unit of output is averaged across all units, not just the additional units, in the average variable cost measure. Meanwhile, average fixed cost is downward sloping because of the spreading effect. Finally, notice that the marginal cost curve intersects the average total cost curve from below, crossing it at its lowest point, point M in Figure 8-7. This last feature is our next subject of study. Minimum Average Total Cost For a U-shaped average total cost curve, average total cost is at its minimum level at the bottom of the U. Economists call the quantity that corresponds to the minimum 10 10 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST The minimum-cost output is the quantity of output at which average total cost is lowest the bottom of the U-shaped average total cost curve. average total cost the minimum-cost output. In the case of Ben s Boots, the minimum-cost output is three pairs of boots per day. In Figure 8-7, the bottom of the U is at the level of output at which the marginal cost curve crosses the average total cost curve from below. Is this an accident? No it reflects general principles that are always true about a firm s marginal cost and average cost curves: Figure 8-7 Marginal Cost and Average Cost Curves for Ben s Boots Marginal, average costs of pair Here we have the family of cost curves for Ben s Boots: the marginal cost curve (MC), the average total cost curve (ATC), the average variable cost curve (AVC), and the average fixed cost curve (AFC). Note that the average total cost curve is U-shaped and the marginal cost curve crosses the average total cost curve at the bottom of the U, point M, corresponding to the minimum average total cost in Table 8-2 and Figure 8-6. >web... $ M MC ATC AVC AFC Minimum-cost output Quantity of boots (pairs) 11 11 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST At the minimum-cost output, average total cost is equal to marginal cost. At output less than the minimum-cost output, marginal cost is less than average total cost and average total cost is falling. And at output greater than minimum-cost output, marginal cost is greater than average total cost and average total cost is rising. To understand this principle, think about how your grade in one course say, a 3.0 in physics affects your overall grade point average. If your GPA before receiving that grade was more than 3.0, the new grade lowers your average. Similarly, if marginal cost the cost of producing one more unit is less than average total cost, producing that extra unit lowers average total cost. This is shown in Figure 8-8 by the movement from A 1 to A 2. In this case, the marginal cost of producing an additional unit of output is low, as indicated by the point MC L, on the marginal cost curve. And when the cost of producing the next unit of output is less than average total cost, increasing production reduces average total cost. So any level of output at which marginal cost is less than average total cost must be on the downward-sloping segment of the U. But if your grade in physics is more than the average of your previous grades, this new grade raises your GPA. Similarly, if marginal cost is greater than average total cost, producing that extra unit raises average total cost. This is illustrated by the move from B 1 to B 2 in Figure 8-8, where the marginal cost, MC H, is higher than average total cost. So any level of output at which marginal cost is greater than average total cost must be on the upward-sloping segment of the U. Finally, if a new grade is exactly equal to your previous GPA, the additional grade neither raises nor lowers that average it stays the same. This corresponds to point M in Figure 8-8: when marginal cost equals average total cost, we must be at the bottom of the U, because only at that point is average total cost neither falling nor rising. 12 12 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST Figure 8-8 Does the Marginal Cost Curve Always Slope Upward? Up to this point, we have emphasized the importance of diminishing returns, which lead to a marginal product curve that is always downward sloping and a marginal cost curve that is always upward sloping. In practice, however, economists believe that marginal cost curves often slope downward as a firm increases its production from zero up to some low level, sloping upward only at higher levels of production: they look like the curve MC in Figure 8-9. The Relationship Between the Average Total Cost and the Marginal Cost Curves To see why the marginal cost curve (MC) must cut through the average total cost curve at the minimum average total cost (point M), corresponding to the minimum-cost output, we look at what happens if marginal cost is different from average total cost. If marginal cost is less than average total cost, an increase in output must reduce average total cost, as in the movement from A 1 to A 2. If marginal cost is greater than average total cost, an increase in output must increase average total cost, as in the movement from B 1 to B 2. Marginal, average costs of unit MC L MC H A 1 M B 1 A 2 If marginal cost is above average total cost, average total cost is increasing. If marginal cost is below average total cost, average total cost is decreasing. MC B 2 ATC Quantity 13 13 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST This initial downward slope occurs because a firm that employs only a few workers often cannot reap the benefits of specialization of labor. For example, one individual producing boots would have to perform all the tasks involved: making soles, shaping the upper part, sewing the pieces together, and so on. As more workers are employed, they can divide the tasks, with each worker specializing in one or a few aspects of boot-making. This specialization can lead to increasing returns at first, and so to a downward-sloping marginal cost curve. Once there are enough workers to per- Figure 8-9 More Realistic Cost Curves In practice, the marginal cost curve often begins with a section that slopes downward. As output rises from a low level, a firm is capable of engaging in specialization and division of labor, which leads to increasing returns. At higher levels of output, however, diminishing returns lead to upward-sloping marginal cost. When marginal cost has a downwardsloping section, average variable cost is U-shaped. However, the basic results U-shaped average total cost, and marginal cost that cuts through the minimum average total cost remain the same. >web... Marginal, average costs of unit MC AVC ATC Quantity 14 14 CHAPTER 8 SECTION 2: TWO KEY CONCEPTS: MARGINAL COST AND AVERAGE COST mit specialization, however, diminishing returns set in. So typical marginal cost curves actually have the swoosh shape shown by MC in Figure 8-9. For the same reason, average variable cost curves typically look like AVC in Figure 8-9: they are U- shaped rather than strictly upward sloping. However, as Figure 8-9 also shows, the key features we saw from the example of Ben s Boots remain true: the average total cost curve is U-shaped, and the marginal cost curve passes through the point of minimum average total cost as well as through the point of minimum average variable cost. Microeconomics Topic 6: Be able to explain and calculate average and marginal cost to make production decisions. Reference: Gregory Mankiw s Principles of Microeconomics, 2 nd edition, Chapter 13. Long-Run Chapter 13 1. Joe runs a small boat factory. He can make ten boats per year and sell them for 25,000 each. It costs Joe 150,000 for the raw materials (fibreglass, wood, paint, and so on) to build the ten S171-S184_Krugman2e_PS_Ch12.qxp 9/16/08 9:22 PM Page S-171 Behind the Supply Curve: Inputs and Costs chapter: 12 1. Changes in the prices of key commodities can have a significant impact on a company s Chapter 10 Technology, Production, and Costs 10.1 Technology: An Economic Definition 10.1 LEARNING OBJECTIVE Learning Objective 1 Define technology and give examples of technological change. A firm s technology Chapter 11 PERFECT COMPETITION Competition Topic: Perfect Competition 1) Perfect competition is an industry with A) a few firms producing identical goods B) a few firms producing goods that differ somewhat 1. Which of the following would shift the demand curve for new textbooks to the right? a. A fall in the price of paper used in publishing texts. b. A fall in the price of equivalent used text books. c. Econ 101: Principles of Microeconomics Chapter 12 - Behind the Supply Curve - Inputs and Costs Fall 2010 Herriges (ISU) Ch. 12 Behind the Supply Curve Fall 2010 1 / 30 Outline 1 The Production Function chapter 7 >> Making Decisions Section : Making How Much Decisions: The Role of Marginal Analysis As the story of the two wars at the beginning of this chapter demonstrated, there are two types of decisions: Chapter 12 The Costs of Produc4on Copyright 214 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. What will you learn MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Firms that survive in the long run are usually those that A) remain small. B) strive for the largest Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools The Cost of Production 1. Opportunity Costs 2. Economic Costs versus Accounting Costs 3. All Sorts of Different Kinds of Costs 4. Cost in the Short Run 5. Cost in the Long Run 6. Cost Minimization 7. The Chapter 22 The Cost of Production Extra Multiple Choice Questions for Review 1. Implicit costs are: A) equal to total fixed costs. B) comprised entirely of variable costs. C) "payments" for self-employed chapter 9 The industry supply curve shows the relationship between the price of a good and the total output of the industry as a whole. Perfect Competition and the >> Supply Curve Section 3: The Industry Production and Cost Analysis The entire production process begins with the supply of factors of production or inputs used towards the production of a final good we all consume in the final good market. Name: Solutions Cosumnes River College Principles of Microeconomics Problem Set 6 Due Tuesday, March 24, 2015 Spring 2015 Prof. Dowell Instructions: Write the answers clearly and concisely on these sheets Chapter 8. Competitive Firms and Markets We have learned the production function and cost function, the question now is: how much to produce such that firm can maximize his profit? To solve this question, ECON 103, 2008-2 ANSWERS TO HOME WORK ASSIGNMENTS Due the Week of June 9 Chapter 6 WRITE Gomez runs a small pottery firm. He hires one helper at $12,000 per year, pays annual rent of $5,000 for his MERSİN UNIVERSITY FACULTY OF ECONOMICS AND ADMINISTRATIVE SCİENCES DEPARTMENT OF ECONOMICS MICROECONOMICS MIDTERM EXAM DATE 18.11.2011 TİIE 12:30 STUDENT NAME AND NUMBER MULTIPLE CHOICE. Choose the one N. Gregory Mankiw Principles of Economics Chapter 13. THE COSTS OF PRODUCTION Solutions to Problems and Applications 1. a. opportunity cost; b. average total cost; c. fixed cost; d. variable cost; e. total MATH MODULE Total, Average, and Marginal Functions 1. Discussion A very important skill for economists is the ability to relate total, average, and marginal curves. Much of standard microeconomics involves Perfect Competition Chapter 10 CHAPTER IN PERSPECTIVE In Chapter 10 we study perfect competition, the market that arises when the demand for a product is large relative to the output of a single producer. Pre-Test Chapter 20 ed17 Multiple Choice Questions 1. In the above diagram it is assumed that: A. some costs are fixed and other costs are variable. B. all costs are variable. C. the law of diminishing WSG8 7/7/03 4:34 PM Page 113 8 Market Structure: Perfect Competition and Monopoly OVERVIEW One of the most important decisions made by a manager is how to price the firm s product. If the firm is a profit Chapter 04 Firm Production, Cost, and Revenue Multiple Choice Questions 1. A key assumption about the way firms behave is that they a. Minimize costs B. Maximize profit c. Maximize market share d. Maximize CHAPTER 8 PROFIT MAXIMIZATION AND COMPETITIVE SUPPLY TEACHING NOTES This chapter begins by explaining what we mean by a competitive market and why it makes sense to assume that firms try to maximize profit. ECON 103, 2008-2 ANSWERS TO HOME WORK ASSIGNMENTS Due the Week of June 23 Chapter 8 WRITE Use the demand schedule that follows to calculate total revenue and marginal revenue at each quantity. Plot Practice Questions Week 6 Day 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Economists assume that the goal of the firm is to a. maximize total revenue Economics 101 Fall 2011 Homework #6 Due: 12/13/2010 in lecture Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework Practice Questions Week 8 Day 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The characteristics of a market that influence the behavior of market participants Microeconomics Topic 7: Contrast market outcomes under monopoly and competition. Reference: N. Gregory Mankiw s rinciples of Microeconomics, 2 nd edition, Chapter 14 (p. 291-314) and Chapter 15 (p. 315-347). Chapter 6 Cost-Volume-Profit Relationships Solutions to Questions 6-1 The contribution margin (CM) ratio is the ratio of the total contribution margin to total sales revenue. It can be used in a variety HW #7: Solutions QUESTIONS FOR REVIEW 8. Assume the marginal cost of production is greater than the average variable cost. Can you determine whether the average variable cost is increasing or decreasing? Name: Date: 1. Most electric, gas, and water companies are examples of: A) unregulated monopolies. B) natural monopolies. C) restricted-input monopolies. D) sunk-cost monopolies. Use the following to answer Chapter 9 Lecture Notes 1 Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter 9: Profit Maximization Profit Maximization The basic assumption here is that firms are profit maximizing. Managerial Economics & Business Strategy Chapter 5 The Production Process and Costs McGraw-Hill/Irwin Copyright 2010 by the McGraw-Hill Companies, Inc. All rights reserved. Overview I. Production Analysis Factor Markets Problem 1 (APT 93, P2) Two goods, coffee and cream, are complements. Due to a natural disaster in Brazil that drastically reduces the supply of coffee in the world market the price of coffee Chapter 12 Production and Cost 12.1 Economic Cost and Profit 1) The primary goal of a business firm is to A) promote fairness. B) make a quality product. C) promote workforce job satisfaction. D) maximize Chapter 13 Real Business Cycle Theory Real Business Cycle (RBC) Theory is the other dominant strand of thought in modern macroeconomics. For the most part, RBC theory has held much less sway amongst policy-makers CHAPTER 8 COSTS OF PRODUCTION Chapter in a Nutshell This chapter gives an in-depth look at the costs of production for firms, both in the short run and in the long run. Although production techniques may Instructions: The following instructions accompany the excel file 806-Final_project_data-09.xlsx. This project requires you to combine some basics of Economics, Math and Excel skills. Marks will be for Chapter 8 Production Technology and Costs 8.1 Economic Costs and Economic Profit 1) Accountants include costs as part of a firm's costs, while economists include costs. A) explicit; no explicit B) implicit; CE2451 Engineering Economics & Cost Analysis Dr. M. Selvakumar Associate Professor Department of Civil Engineering Sri Venkateswara College of Engineering Objectives of this course The main objective of Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in 385 356 PART FOUR Capital Budgeting a large number of NPV estimates that we summarize by calculating the average value and some measure of how spread out the different possibilities are. For example, it Pre-Test Chapter 21 ed17 Multiple Choice Questions 1. Which of the following is not a basic characteristic of pure competition? A. considerable nonprice competition B. no barriers to the entry or exodus Answers to Text Questions and Problems Chapter 22 Answers to Review Questions 3. In general, producers of durable goods are affected most by recessions while producers of nondurables (like food) and services Understanding Economics 2nd edition by Mark Lovewell and Khoa Nguyen Chapter 5 Perfect Competition Chapter Objectives! In this chapter you will: " Consider the four market structures, and the main differences Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9 print name on the line above as your signature INSTRUCTIONS: 1. This Exam #2 must be completed within the allocated time (i.e., between Costs of Production and Profit Maximizing Production: 3 examples. In this handout, we analyze costs and profit maximizing output decisions by looking at three different possible costs structures. Three Equilibrium in Perfectly Competitive Markets (Assume for simplicity that all firms have access to the same technology and input markets, so they all have the same cost curves.) Market Supply in the Short Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on Dear Profit Master, Congratulations for taking the next step in improving the profitability and efficiency of your company! Profit Guard will provide you with comparative statistical and graphical measurements 9 COST OVERVIEW 1. Total fixed cost is the cost which does not vary with output. Total variable cost changes as output changes. Total cost is the sum of total fixed cost and total variable cost. 2. Explicit TRUE OR FALSE STATEMENTS SHORT-RUN PRODUCTION 1. According to the law of diminishing returns, additional units of the labour input increase the total output at a constantly slower rate. 2. In the short-run Chapter 3 EXPENDITURE MULTIPLIERS: THE KEYNESIAN MODEL* Key Concepts Fixed Prices and Expenditure Plans In the very short run, firms do not change their prices and they sell the amount that is demanded. Multiple choice review questions for Midterm 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A consumption point inside the budget line A) is I. Learning Objectives In this chapter students will learn: A. The significance of resource pricing. B. How the marginal revenue productivity of a resource relates to a firm s demand for that resource. COST THEORY Cost theory is related to production theory, they are often used together. However, the question is how much to produce, as opposed to which inputs to use. That is, assume that we use production Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued MPP 801 Monopoly Kevin Wainwright Study Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The marginal revenue facing a monopolist A) is Yield to Maturity Outline Outline and Suggested Reading Yield to maturity on bonds Coupon effects Par rates Buzzwords Internal rate of return, Yield curve Term structure of interest rates Suggested reading Chapter 15: While a competitive firm is a taker, a monopoly firm is a maker. A firm is considered a monopoly if... it is the sole seller of its product. its product does not have close substitutes. The ECON 103, 2008-2 ANSWERS TO HOME WORK ASSIGNMENTS Due the Week of July 14 Chapter 11 WRITE: Complete the following labour demand table for a firm that is hiring labour competitively and selling its Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram AP Microeconomics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be Lecture Notes on MONEY, BANKING, AND FINANCIAL MARKETS Peter N. Ireland Department of Economics Boston College email@example.com http://www.bc.edu/~irelandp/ec61.html Chapter 6: The Risk and Term Structure EXAM TWO REVIEW: A. Explicit Cost vs. Implicit Cost and Accounting Costs vs. Economic Costs: Economic Cost: the monetary value of all inputs used in a particular activity or enterprise over a given period. Break-even analysis On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart. In order to survive businesses must at least break even, which Prof. Gustavo Indart Department of Economics University of Toronto ECO209 MACROECONOMIC THEORY Chapter 11 MONEY, INTEREST, AND INCOME Discussion Questions: 1. The model in Chapter 9 assumed that both the uestion 3: What is an average cost function? When a business produces goods or services, it incurs costs associated with the production of these goods and services. Some of these costs are variable and Business Economics Theory of the Firm II Production and Cost in the ong Run Two or more variable input factors Thomas & Maurice, Chapter 9 Herbert Stocker firstname.lastname@example.org Institute of International Dr. McGahagan Graphs and microeconomics You will see a remarkable number of graphs on the blackboard and in the text in this course. You will see a fair number on examinations as well, and many exam questions, Econ 202 Exam 3 Practice Problems Principles of Microeconomics Dr. Phillip Miller Multiple Choice Identify the choice that best completes the statement or answers the question. Chapter 13 Production and Test 2 Review Econ 201, V. Tremblay MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Barbara left a $25,000 job as an architect to run a catering Problems: Table 1: Labor Hours needed to make one Amount produced in 90 hours: Quilt Dress Quilts Dresses Helen 50 10 1.8 9 Carolyn 90 45 1 2 1. Refer to Table 1. For Carolyn, the opportunity cost of 1 Chap 13 Monopolistic Competition and Oligopoly These questions may include topics that were not covered in class and may not be on the exam. MULTIPLE CHOICE. Choose the one alternative that best completes Agenda Productivity, Output, and Employment, Part 1 3-1 3-2 A production function shows how businesses transform factors of production into output of goods and services through the applications of technology. CHAPTER 10 Bond Prices and Yields Interest rates go up and bond prices go down. But which bonds go up the most and which go up the least? Interest rates go down and bond prices go up. But which bonds go Break-even Analysis An enterprise, whether or not a profit maximizer, often finds it useful to know what price (or output level) must be for total revenue just equal total cost. This can be done with a CHAPTER 9: PURE COMPETITION Introduction In Chapters 9-11, we reach the heart of microeconomics, the concepts which comprise more than a quarter of the AP microeconomics exam. With a fuller understanding Chapter 3 The Concept of Elasticity and Consumer and roducer Surplus Chapter Objectives After reading this chapter you should be able to Understand that elasticity, the responsiveness of quantity to changes

Learn How Far Your Savings May Last In Retirement With This Free Calculator. Have A $500K Portfolio And Nearing Or Already In Retirement? This Is For You Monte Carlo Retirement Calculator. Confused? Try the simple retirement calculator. About Your Retirement ? Current Age. Retirement Age. Current Savings $ Annual Deposits $ Annual Withdrawals $ Stock market crash. Portfolio ? In Stocks % In Bonds % In Cash % Modify Stock Returns. 0%. This calculator uses a logic known as a Monte Carlo simulation to illustrate how long your retirement portfolio might last, on average, given input information. Under a Monte Carlo simulation, probabilities are calculated for different scenarios, based on random samplings of past performance The IFA Retirement Income Calculator uses 10,000 Monte Carlo Simulations based on normal distribtuions of mean and standard deviations from back-tested 93 year data of IFA's Index Portfolios (see ifabt.com for important disclosure information). Initial inflation rate of 3% is based on 93 years of Consumer Price Index data A Monte Carlo simulation is like a stress test for your financial future. Using financial planning software and retirement calculators, you can leverage these powerful forecasting models in your retirement planning. Learn to understand the terms, use the simulators, and interpret their results Monte Carlo simulations are what most online financial calculators use in order to determine the probability that you will not outlive your savings for retirement. The biggest problem with retirement savings is the sequence of return risks - the risk that your withdrawals will be greater than your returns for the remaining years of your life Monte Carlo Powered Retirement Planning Made Easy! Build and run a sophisticated retirement planning simulation in just a few minutes. Quickly create 'what-if' scenarios to explore the impact of unlikely or unexpected events. Capture extra financial details with year-by-year control of all input parameters. Perform sensitivity analysis to. A Monte Carlo simulation can be used to test if one will have enough income throughout retirement. Unlike a traditional retirement calculator, the Monte Carlo method incorporates many variables to. In a Monte Carlo Retirement calculator, the inflation and returns (pre- and post-retirement) are varied randomly (hence the gambling analogy). We could then determine. how long a corpus is likely to last. how much corpus we are likely to need. how much monthly investment is needed for maximising the probability of success Whether you are looking for a retirement score or a retirement income calculator, Fidelity's retirement tools & calculators can help you plan for your retirement. Discover the financial guidance to help with your retirement goals An example of what we show for Monte Carlo simulations is below: Our Monte Carlo calculator runs 1,000 scenarios where the rates of return for every investment changes in each year. We take the number of scenarios where money never runs out in retirement and divide it by 1,000 to find the probability of success (never running out of money) T. Rowe Price's Retirement Income Calculator also uses a Monte Carlo simulation to estimate the likelihood your retirement savings will be enough. The unique feature of this calculator, though. Download Sample Report This report is offered as a tool for helping investors understand key factors in retirement investing. The charts and graphs produced from the information provided by you in the form below will be based on a Monte Carlo simulation method The three primary modes that are available in the early retirement calculator are: (1) constant, single fixed-percentage real return rates, (2) historical series of real returns are applied to account for likely variability in future returns and (3) monte carlo simulation of the variable returns based upon user-specified input parameters Monte Carlo simulation using the nest egg calculator; Cons: Lacks ability to add spousal information; Lacks ability to adjust 4% withdrawal rate assumption #6: Bankrate. Bankrate is well known for its many different financial calculators from retirement calculators to mortgage calculators Overview: The Flexible Retirement Planner may be the most powerful free retirement calculator on the web. The user interface is rich and well executed, and the modeling capabilities are extensive, without being overwhelming at the start. The model is a Monte Carlo simulation relying on a Gaussian distribution computed using an average-return plus standard deviation How it works - the philosophy: FIRECalc makes a single fundamental assumption: If your retirement strategy would have withstood the worst ravages of inflation, the Great Depression, and every other financial calamity the US has seen since 1871, then it is likely to withstand whatever might happen between now and the day you no longer have any need for your retirement funds The user-friendly calculator uses 100,000 of what are called Monte Carlo simulations of potential future returns to the financial markets to arrive at the probability that a household's invested savings will last through the end of retirement. To get to this number, older workers enter their information into the calculator - 401 (k) account. My financial adviser ran Monte Carlo simulations that say I have an 85% chance my retirement savings will last into my 90s. Is an 85% probability safe enough? -- Tracey G Uses Monte Carlo simulations for the rate of return and chances you would run out of money. The Cons . Must estimate your allocation between stocks, bonds, and short-term investments this is not easy to do if you have a lot of balanced funds and/or multiple accounts. This information is used for the Monte Carlo simulations Monte Carlo Retirement Simulators Usable by Everyone (Preliminary Version) Floyd Vest, Sept. 2015 For an example, go to the internet and Search: monte carlo simulation in finance. Go down to Vanguard - Retirement nest egg calculator which addresses the question, How long will retirement funds last This is no ordinary retirement calculator. Run Monte Carlo simulations, what-if scenarios on market downturns, life insurance needs, retirement income analysis, safe withdrawal rates, and Roth IRA transfers. Use the solution finder to overcome planning roadblocks and for your retirement modeling MC = Monte Carlo. The calculator uses a Monte Carlo model of stock and bond investment returns as a guide. Initial Retirement Spending Abbreviations. These are the potential types of spending models that could be used to describe retiree spending at the start of retirement. RepR = Replacement Rate; STot = Single Budget, total spending onl However, this calculator does adjust the withdraw amount by the CPI each year of the simulation. For example, given a 30 year retirement and an initial withdraw amount of $50,000, the simulation starting in 1975 would increase the withdraw amount all the way to $181,440 in 2005 (in the final year of that simulation run) based on the change in CPI While its supporters are quick to point out that Monte Carlo simulations generally provide much more realistic scenarios than simple projections that The Retirement Calculator from Hell. Posted on May 23, 2016 by firstname.lastname@example.org. Retirement calculators are a popular topic, and for good reason. They're one of the best ways to monitor whether you're on track for retirement, and many of them are free. Even if you're already retired, they can be a useful tool to insure your money lasts longer than you do The NewRetirement Planner has been named a best retirement calculator by the American Association of Individual Investors (AAII), Forbes Magazine, The Center for Retirement Research at Boston College, MoneyBoss, CanIRetireyet and many more When it comes to projecting income in retirement, the best financial advisors for retirement often use a retirement calculator called Monte Carlo Simulation. If you're like many of our clients , the term Monte Carlo may take your mind to a seaside town in France as you enter one of the most famous casinos in the world Monte Carlo Models In the final analysis, most Monte Carlo simulations forecast outcomes that are too optimistic. That is the reason why I designed and developed the Otar Retirement Calculator which is based on actual market history. But that is another story. A Better Mousetrap Monte Carlo Retirement Calculator. We're now ready for an improved calculator that lets you include volatility in your plans. The Monte Carlo in the title doesn't mean that the odds are against you and you're going to crap out in the big casino of life; it's the generic term used for a computer model that runs many random iterations and lets you see your chance of meeting your goal The retirement calculator runs 5,000 Monte Carlo simulations to deliver a robust, personalized retirement projection. The simulations incorporate expected return and volatility, annual savings, income, spending goals, retirement spending, social security, and tax rules for taxable, tax-deferred, and tax-free investment accounts A Monte Carlo analysis, using an average return plus a standard deviation, takes volatility into account, but requires expertise (or trust) for choosing the necessary mathematical parameters. And there are arguments that the artificial randomness introduced by a Monte Carlo simulation doesn't mimic the real world accurately . Spend your retirement in luxury. This is a user-friendly model. Spend a minute reading the description below and start investing. This model assumes 3 timelines based on an individual's age and income capacity and suggests 3 different portfolios, each contributing proportionately to the retirement funds. The first is a high period growth when a. The Flexible Retirement Planner is a Monte Carlo Simulation calculator. It allows you a substantial amount of input into the calculator. This lets you find ways to improve your retirement plan. The calculator uses your inputs to run 10,000 market simulations to determine your retirement's probability of success It is recommended that investors use a retirement-plan calculator with Monte Carlo simulations for a more refined, personalized estimate. What You Can Do Next Explore additional steps for retirement planning in these related articles about choosing the right mix of investments to help you reach your goals and withdrawing from your savings in. . When using one of the many online retirement calculators based on the Monte Carlo method, or if your financial advisor uses a Monte Carlo simulation in shaping your retirement plan, it can't hurt. A typical Monte Carlo simulation runs thousands of scenarios to come up with a probability of success. Vanguard Retirement Nest Egg Calculator is one of the simplest online calculators to determine the probability of success for your retirement portfolio How an Investment Simulation Can Be Used to Help Build Your Portfolio - Monte Carlo Simulation William Noel Oct 07, 2016 comments off Tweet on Twitter Share on Facebook Pinteres The Personal Retirement Calculator is provided by one or more third party service providers. However, the information generated by the calculator is developed by Merrill Lynch to estimate how current savings and estimated future contributions may help to meet estimated income in retirement The Optimal Retirement Planner (ORP)is a retirement planning calculator that computes the maximum amount of money available for retirement spending based on age, spouse's age current saving balances, savings plans, taxes, and inflation. Tax-deferred and after-tax savings are treated separately. ORP is available at this web site at no charge Free Business Owner Calculators: What Monte Carlo software is good for, what it's not good for, and why you've been hearing so much about It. Monte Carlo simulations are also known as stochastic modeling, stress testing, and worst-to-best case scenario analysis. We're not a fan of Monte Carlo simulators in financial planning software How Monte Carlo Analysis Can Calm Your Fears About Running Out of Money in Retirement Running different scenarios through this forecasting model can help you make smarter decisions both before and. To see what a Monte Carlo simulation looks like, check out Vanguard Group's retirement nest egg calculator. While Monte Carlo analysis is widely used in financial planning, I would advise caution, for two reasons: 1. Subjective inputs. Monte Carlo simulation works well when forecasting physical or mechanical processes—things that act in. Monte-Carlo Simulation Monte-Carlo Simualtion is a technique in which the statistics of possible future outcomes are investigated by creating and averaging many different realizations, subject to some assumed probabality distribution. Monte-Carlo simulation is used in our Monte-Carlo retirement simulator MCRetire The Wade Pfau Updated Trinity Study. Wade Pfau is a professor and PhD in Financial Planning. He's written excellent pieces on the Trinity Study, including an updated Trinity Study using data through the year 2014.. Along with the extended data set, Pfau also changed the type of bond that the study assumed This calculator will tell you. Start. Retirement Nest Egg. See a long-term projection of how big your retirement nest egg will be with this calculator. Start. Retirement Plan Loan. Before you take a loan from your retirement plan, use this calculator to determine its true cost. Start. More tools and calculators. College savings option tool It allows you to get much more detailed in modeling potential future expenses and they just recently added Monte Carlo simulation. I'm not sure I would call it a retirement planner, so much as a lifetime financial modeling tool to help you explore different choices. But you can use it for retirement planning too OnTrajectory Features. Create multiple scenarios. Monitor investments & expenses. Calculate retirement success rate. Run Monte Carlo simulations. Track actual progress against your plan. Add unlimited income, expense, and account entries. Your plan will never increase in price One nice feature is they allow for you to adjust the Monte Carlo Simulation's sensitivity. Pros: Flexible Retirement Planner gives you a very detailed picture of your retirement plan. Cons: If you want a quick snapshot of your retirement savings, Flexible Retirement Planner isn't the right retirement calculator. There are plenty of other. Overall, the calculators in the guide are standard but the layout and organization are great to help walk you through the steps of financial independence. Vanguard Retirement Next Egg. Vanguard offers a very easy to use Nest Egg Calculator that takes a handful of inputs and gives you an answer based on a Monte Carlo simulation Calculators are also ideal for retirement planning. A good online retirement tool can make crunching decades of numbers and assumptions a snap. You can even configure the Monte Carlo simulator. Going to Monte Carlo. Online retirement calculators may also rely on what's known as Monte Carlo simulations. Rather than choose one rate of return to base calculations on, Monte Carlo. I've played around with some other Monte Carlo simulation calculators and found that if you go with 3%, you get close to 90% probability of not depleting your assets assuming you live longer than roughly 90% of the people in the US population. And in retirement, I'm going to assume I don't work at all for pay so that's my target. Thanks The likelihood of success is determined by a statistical methodology know as Monte Carlo Simulation. Retirement Advisor runs your current strategy through a minimum of 500 investment simulations and determines how many of the simulations provide you with assets until the end of your estimated retirement need, and how many simulations indicate a. Retirement planning software using Monte Carlo simulation. Funds or plans which follow a glidepath strategy (eg lifecycle or target-date funds) where the strategic asset allocation is varied over time -- usually to reduce the proportion of higher risk asset classes as the retirement date approaches -- are easily handled by the Retirement Planner It's a calculator to determine the age at which you can comfortably retire. It's a spreadsheet, which means you can easily customize it for YOU. It's NOT a Monte Carlo simulation for expected returns TIPSTER Monte Carlo Retirement Calculator. T IP$TER is a free patent-pending financial planner, retirement calculator, and portfolio simulator. It is designed to. The calculator uses this age to figure out how many years your retirement plan needs to generate income. You can choose an age from 85 to 100 as the end of your retirement. This will impact how much you will have, and consequently change your RPM. The default planning age used in the calculator is 93 To run a stress test on your own retirement plan, head to Vanguard, which hosts a free Monte Carlo retirement calculator that's very simple to use. The calculator wants to know: Your total pension pot - I used the figure projected by Hargreaves Lansdown's calculator based on my existing salary and contributions (as we've previously. Using one Monte Carlo calculator, an analysis showed that, if someone retires with $1.5 million in retirement assets and withdraws 4% ($60,000) during the first year of retirement from a portfolio consisting of 50% stocks, 30% bonds, and 20% cash, savings is projected to last 34.12 years, on average, with a 95% probability of lasting between 28. Run a Monte Carlo Simulation. Monte Carlo simulations are a popular test simulation for retirement planning. It involves using actuarial insights to extrapolate your financial data through your expected end of life. By adjusting variables like age, portfolio worth, inflation, annual withdrawals and more, the simulation can predict your. This calculator is what they call a 'disciplined investor' in that it doesn't panic during the bad times, nor does it go crazy and buy a diamond studded phone case in the good times. How the simulations work: This calculator is NOT a Monte-Carlo simulator in that it does not generate any fake or random data 4) Real-Time Data & Accuracy: By using Monte Carlo simulations, account aggregation and real-time data, Retirement Planner's level of accuracy is unlike many others in the market. The Retirement Planner literally calculates thousands of different scenarios to come up with their output We can do this with a Monte Carlo Simulation. I used Vanguard's retirement nest egg calculator. We'll start with a safe withdrawal rate of 4 percent per year: The results: There's a 93 percent probability that this portfolio lasts 30 years. When I re-run the simulation using a withdrawal rate of 5.3 percent (spending $40,000 per year. Monte Carlo Simulation, also known as the Monte Carlo Method or a multiple probability simulation, is a mathematical technique, which is used to estimate the possible outcomes of an uncertain event. The Monte Carlo Method was invented by John von Neumann and Stanislaw Ulam during World War II to improve decision making under uncertain conditions Our retirement calculator and planner estimates monthly retirement income and efficient retirement savings spending, providing useful financial insights investor's retirement calculator are obtained from Monte Carlo simulations and are hypothetical. A Monte Carlo simulation is a mathematical technique for simulating real-world situations that involve a high degree of uncertainty. The assets needed for income are based on an average for the type of income annuit However, retirement calculators are a good way to get a ballpark figure of how your plan is doing. One very good type of calculator is a Monte Carlo Simulator. This calculator simulates how your portfolio will do under a range of real world market conditions. Most retirement calculators calculate based on a set rate of return The key to using Monte Carlo simulation is to take many random values, recalculating the model each time, and then analyze the results. Step 2: Running a Monte Carlo Simulation. A Monte Carlo simulation calculates the same model many many times, and tries to generate useful information from the results Retirement income need of $85k with a real return of 5% before and during retirement and a 50% cut to Social Security income. Using Monte Carlo Simulation to run thousands of trials, each time using a different sequence of returns, we get the following results: Elaine has a 93% probability of success if she spends $75k/year in retirement and. You can use the Monte Carlo Simulation to improve your retirement planning. In his paper, The Retirement Calculator from Hell, William Bernstein clearly illustrates this shortcoming. He uses. T. Rowe Price Monte Carlo Simulation Calculator; Visual Retirement Calculator; Your Retirement Calculator; Top of page. Reality Check. Find helpful financial information based on where you are in life, explore financial information from trusted resources, and use tools to make more informed financial decisions Retirement Nest Egg calculator (using Monte Carlo simulations) Retirement Plan Loan calculator The first two calculators listed above are similar to many of the others on this list

Lucid Electronics Workbench: A Handy Tool for Electronic Circuit Design If you are a hobbyist or a student who wants to design simple electronic circuits, you may find Lucid Electronics Workbench very useful. This software program can help you calculate resistor, capacitor, regulator and power values for various components and circuits. It can also display integrated circuit (IC) and semiconductor drawing outlines, and show you the closest component match based on color-coded values. Lucid Electronics Workbench has several features that make it easy and convenient to use. You can select from a variety of power supply types and enter the transformer voltage and power supply load. The program will then calculate the values for diodes, capacitors, output voltage and load current. You can also use the Ohm's Law calculator to find the values of resistance, power, current and voltage when any two of them are given. Another feature of Lucid Electronics Workbench is the regulator circuit component values calculator. You can choose from a number of adjustable and fixed regulator types, and the program will display the case styles, pin outs, capacitor values and resistor values for the desired output voltage and supply voltage. You can also use the web links to regulator components provided by the program when available. Lucid Electronics Workbench can also help you with transistor circuit component values. It can calculate the bias resistors, emitter and collector resistors, and the capacitor values for input, output and emitter bypass. It can also display actual component values as well as standard component values. You can view or print the output along with an applicable schematic. One more feature of Lucid Electronics Workbench is the 555 timer component values calculator. You can use this feature to calculate the frequency and duty cycle based on C1, R1 and R2 values; or calculate C1 and R2 values based on frequency and R2 values; or calculate R1 and R2 values based on frequency, duty cycle and frequency tolerance values. The program can also show you actual component values as well as standard component values. Lucid Electronics Workbench is a user-friendly and affordable software program that can save you time and money when designing electronic circuits. You can download a free evaluation copy from their website https://www.lucidcc.com/LEWB.php or purchase a full version for $14.95. The program runs on Windows XP/Vista/7/8/10 operating systems. In this article, we will show you some examples of how to use Lucid Electronics Workbench to design and calculate some common electronic circuits. We will also provide some screenshots of the program interface and the output results. Example 1: LED Load Resistor Calculator One of the simplest and most common electronic circuits is the LED circuit. An LED (light emitting diode) is a device that emits light when a current passes through it. However, an LED cannot be directly connected to a power supply, because it will draw too much current and burn out. Therefore, a load resistor is needed to limit the current and protect the LED. To calculate the value of the load resistor for an LED circuit, you can use the LED Load Resistor Calculator feature of Lucid Electronics Workbench. You just need to enter the supply voltage and select an LED product from the list. The program will then calculate the value of the load resistor and display it in ohms. It will also show you a table of standard resistor values so that you can choose the closest one for your circuit. For example, suppose you want to connect a red LED to a 9V battery. You can enter 9 in the supply voltage box and select Red LED from the list. The program will then calculate that the load resistor value should be 470 ohms. It will also show you that the closest standard resistor value is 470 ohms as well. You can then view or print the output report, which includes a schematic diagram of the circuit. Example 2: Power Supply Component Values Calculator Another common electronic circuit is the power supply circuit. A power supply is a device that converts an alternating current (AC) input voltage into a direct current (DC) output voltage. A power supply usually consists of four main components: a transformer, a rectifier, a filter capacitor and a regulator. To calculate the values of these components for a power supply circuit, you can use the Power Supply Component Values Calculator feature of Lucid Electronics Workbench. You just need to select a power supply type and enter the transformer voltage and power supply load. The program will then calculate the values for diodes, capacitors, output voltage and load current. For example, suppose you want to design a full-wave rectifier power supply with a center-tapped transformer that has an input voltage of 120V AC and an output voltage of 12V AC. You also want to have an output voltage of 5V DC and a load current of 1A. You can select Full Wave Rectifier from the list and enter 12 in the transformer voltage box and 1 in the power supply load box. The program will then calculate that you need four diodes with a peak inverse voltage (PIV) of at least 16.8V, a filter capacitor with a capacitance of at least 6600 microfarads and a voltage rating of at least 10V, and a fixed regulator with an output voltage of 5V and a current rating of at least 1A. You can then view or print the output report, which includes a schematic diagram of the circuit.

2 Chapter Objectives Be able to: Explain what capacity is, how firms measure capacity, and the difference between theoretical and rated capacity.Describe the pros and cons associated with three different capacity strategies: lead, lag, and match.Apply a wide variety of analytical tools to capacity decisions, including expected value and break-even analysis, decision trees, learning curves, the Theory of Constraints, waiting line theory, and Little’s Law. 6 Indifference Point Examples Capacity for a PC Assembly Plant(800 units per line per shift)×(# of lines)×(# of shifts)Controllable FactorsUncontrollable Factors1 or 2 shifts?2 or 3 lines? Employee training?Supplier problems?98% or 100% good?Late or on time? 7 Three Common Capacity Strategies Lead capacity strategy – A capacity strategy in which capacity is added in anticipation of demand.Lag capacity strategy – A capacity strategy in which capacity is added only after demand has materialized.Match capacity strategy – A capacity strategy that strikes a balance between the lead and lag capacity strategies by avoiding period of high under or overutilization. 10 Cost ComparisonFixed costs – The expenses an organization incurs regardless of the level of business activity.Variable costs – Expenses directly tied to the level of business activity. 11 Cost Comparison TC = FC + VC * X TC = Total Cost FC = Fixed Cost VC = Variable cost per unit of business activityX = amount of business activity 12 Cost Comparison - Example 6.1 Table 6.2Figure 6.2 13 Cost Comparison - Example 6.1 Total cost of common carrier option = Total cost of contract carrier option$0 + $750X = $5,000 + $300XX = or 11 shipmentsFind the indifference point – the output level at which the two alternatives generate equal costs.Total cost of contract carrier option = Total cost of leasing$5,000 + $300X = $21,000 + $50XX = 64 shipments 14 Expected ValueExpected value – A calculation that summarizes the expected costs, revenues, or profits of a capacity alternative, based on several demand levels with different probabilities. 17 Decision TreesDecision tree – A visual tool that decision makers use to evaluate capacity decisions to enable users to see the interrelationships between decisions and possible outcomes. 18 Decision Tree RulesDraw the tree from left to right starting with a decision point or an outcome point and develop branches from there.Represent decision points with squares.Represent outcome points with circles.For expected value problems, calculate the financial results for each of the smaller branches and move backward by calculating weighted averages for the branches based on their probabilities. 19 Decision Trees – Example 6.3 Original Expected Value ExampleFigure 6.4 20 Break-Even AnalysisBreak-even point – The volume level for a business at which total revenues cover total costs.Where:BEP = break-even pointFC = fixed costsVC = variable cost per unit of business activityR = revenue per unit of business activity 21 Break-Even Analysis – Example 6.4 Suppose the firm makes $1,000 profit on each shipment before transportation costs are considered. What is the break-even point for each shipping option?Contracting: BEP = $5,000 / $700 = 7.1 or 8 shipmentsCommon: BEP = $0 / $250 = 0 shipmentsLeasing: BEP = $21,000 / $950 = 22.1 or 23 shipments 22 Learning CurvesLearning curve theory – A theory that suggests that productivity levels can improve at a predictable rate as people and even systems “learn” to do tasks more efficiently.For every doubling of cumulative output, thereis a set percentage reduction in the amountof inputs required. 24 Learning Curve – Example 6.5 What is the learning percentage?4/5 = 80% or .80 25 Learning Curve – Example 6.5 How long will it take to answer the 25th call?Figure 6.6 26 Other Capacity Considerations The strategic importance of an activity to a firm.The desired degree of managerial control.The need for flexibility. 27 The Theory of Constraints Theory of Constraints – An approach to visualizing and managing capacity which recognizes that nearly all products and services are created through a series of linked processes, and in every case, there is at least one process step that limits throughput for the entire chain.Figure 6.7 28 The Theory of Constraints Identify the constraintExploit the constraintKeep it busy!Subordinate everything to the constraintMake supporting it the overall priorityElevate the constraintTry to increase its capacity — more hours, screen out defective parts from previous step.Find the new constraint and repeatAs one step is removed as a constraint, a new one will emerge. 29 Theory of Constraints – Example 6.6 Table 6.5Where is the Bottleneck? Cut and Style 30 Theory of Constraints – Example 6.6 Current ProcessFigure 6.9 31 Theory of Constraints – Example 6.6 Adding a Second StylistFigure 6.10 32 Theory of Constraints – Example 6.6 Adding One Shampooerand Two StylistsFigure 6.11 33 Waiting Line TheoryWaiting Line Theory – A theory that helps managers evaluate the relationship between capacity decisions and important performance issues such as waiting times and line lengths.Figure 6.12 34 Waiting Line Theory Waiting Line Concerns: What percentage of the time will the server be busy?On average, how long will a customer have to wait in line? How long will the customer be in the system?On average, how may customers will be in line?How will those averages be affected by the arrival rate of customers and the service rate of the workers? 35 Waiting Lines – Example 6.7 The probability of arrivals in a time period =Example: Customers arrive at a drive-up window at a rate of 3 per minute. If the number of arrivals follows a Poisson distribution, what is the probability that two or fewer customers would arrive in a minute?P(< 2) = P(0) + P(1) + P(2) = = .423 or 42.3% 36 Waiting Lines – Example 6.7 The average utilization of the system:Example: Suppose that customers arrive at a rate of four per minute and that the worker at the window is able to handle on average 5 customers per minute. The average utilization of the system is: 37 Waiting Lines – Example 6.8 The average number of customers waiting in the system (CW) =The average number of customers in the system (CS) =Example: Given an arrival rate of four customers per minute and a service rate of five customers per minute:Average number of customers waiting:Average number in the system: 38 Waiting Lines – Example 6.9 The average time spent waiting (TW) =The average time spent in the system (TS) =Example: Given an arrival rate of four customers per minute and a service rate of five customers per minute:Average time spent waiting:Average time spent in the system: 39 Little’s LawLittle’s Law is a law that holds for any system that has reached a steady state that enables us to understand the relationship between inventory, arrival time, and throughput time.I = RT 41 Little’s Law - Example 6.11 Average Throughput Time = T = I/R = (25 orders) / (100 orders per day)= .25 days in order processingAverage time an order spends in workcenter A =T = I/R = (14 orders)/(70 orders per day)= .2 days in workcenter AAmount of time the average A order spends in the plant =Order processing time + workcenter A time= .25 days + .2 days = .45 daysAmount of time the average B order spends in the plant =Order processing time + workcenter B time= .25 days days = .30 days 42 Little’s Law - Example 6.11Average time an order spends in the plant =70% x .45 days + 30% *.30 days= .405 daysEstimate average throughout time for the entire system =T = I/R = (40.5 orders)/(100 orders per day)= .405 days for the average order 44 Printed in the United States of America. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.Printed in the United States of America.

Finding Your Way In the Sky Term: sky map The sky map shows the entire sky as viewed from a given location at a specified time and date. A stereographic projection is used, as is the convention for printed star maps. Coordinates and Time Astronomical coordinate systems are virtually all spherical coordinate systems; defined by a great circle and its poles. A latitude coordinate measures the angle above or below the circle, a longitude coordinate measures the angle along the circle from some arbitrarily defined point. Terrestrial (latitude/longitude) The horizon system is a "local" system centered at the individual observer. The great circle is the observer's horizon. The poles of that circle are the zenith (directly overhead) and the nadir. The reference point on the horizon is the north point, defined by a great circle (the meridian) from the zenith through the north celestial pole to the horizon. Extending the meridian in the oposite direction establishes the south point. The east and west points are defined by the intersections of the celestial equator with the horizon. The latitude coordinate is the altitude (or elevation), measured from the horizon along a great circle running through the zenith. The longitude coordinate is azimuth measured either from the north point increasing towards the east (geographic definition) or from the south point increasing towards the west (astronomical definition). Because of this ambiguity in definition, it is important to determine which form is in use. The geographic definition is most common, even in use by astronomers. Equatorial (declination/hour angle or declination/right ascension) The Equatorial system is defined by the celestial equator, the projection of the Earth's equator onto to celestial sphere. The poles ore the north celestial pole and the south celestial pole. The reference point is the vernal equinox which is the point where the ecliptic (the apparent path of the sun) crosses the celestial equator with the sun moving towards the summer solstice. The latitude coordinate is declination measured from the celestial equator. The longitude coordinate is right ascension (RA) - measured from the vernal equinox increasing in the direction of the sun's motion; ( 0 Hour Angle and Right Ascension The introduction of hour angle and right ascension brings us to a transition between coordinates and timekeeping. An observer has a meridian running overhead from north to south. Objects rise in the east, transit the meridian, and set in the west. An object on the celestial equator (declination = 0) takes 12 hours to cross the sky from east to west. Let's actually build a celestial sphere. First, consider the sky in relation to the earth. Take the north and south poles and extend them into the sky; these become the north and south celestial poles. The earth's equator can be projected outward to form the celestial equator. Now consider yourself as standing on top of the earth. Your zenith is the point directly above your head; your horizon is a circle perpindicular to the line between you and the zenith, and marks the boundary of the hemisphere of the sky which you can actually see. Now, to position the celestial poles with respect to your location on the earth, you have to place them at an angle that will match your lattitude. When you're done, you should have a celestial sphere very like the one at the top of this section. Celestial Coordinates The earth is spinning on its axis once every 24 hours. The axis of the earth is also moving in a circular motion but maintains a constant 23.4 tilt. This in effect makes the earth like a wobbling gyroscope. Compounding things, the earth is also moving around the Sun once a year and the Sun is moving around our galaxy every 220 million years. One more problem, the galaxy is moving through space. In essence everything in the universe is moving constantly in different direction and at different speeds Because of the slow movement of just about everything in the universe, we can catalog objects in the universe by devising a system that allows us to locate and establish the objects position.Declination Right Ascension Alt-Azimuth Angular Measure AltitudeZenithLongitude and LatitudeMeridianNadirCelestial Equator Angular Measure You will need to know how to specify a direction by angular measure Angles Astronomy involves thinking about angles--often very tiny, tiny angles. A circle is divided into 360 degrees (or 360). Each degree is divided into 60 arcminutes (or 60'). Each arcminute can be divided into 60 arcseconds (or 60"). Therefore there are 360x60x60 = 1,296,000 ~ 1.3 x 106 arcminutes in a full circle. Yes, this seems a crazy system and not very `metric'--it goes back to the Babylonians who were hung up on the number 60. No one has been bothered to convince everyone else to make a better system Since we know the Earth has to rotate completely in 24 hours, and one entire rotation means circling 360, the rate in degrees that the Earth rotates per hour is: Rotation rate = ------------- = ------------ 360 degrees 15 degrees 24 hour hour Or 15 per hour There are angles like longitude and latitude on the celestial sphere that describe the location of a star, such as Betelguese. These angles are called Right Ascension and Declination. 1) Go outside and find Polaris and the celestial pole in the night sky. The easiest way is to look roughly north and use the 2 "pointer stars" of the Big Dipper. 2) Point with your left arm to Polaris. Make a right angle to your left arm with your right arm. Your right arm is now pointing to somewhere on the celestial equator. Keeping your left arm pointing to Polaris and your right arm at a rightangle, rotate about your left arm--your right arm will be sweeping out the celestial equator. (3) Point to the south celestial pole. (4) Look for a bright star (preferable 2 or 3 different ones) that is fairly close to the horizon or an object on the skyline--such as the roof of a house or a tree or a lampost. Note where you are standing, where the star is located and the time. Come back in about 1 hour and note how the stars have moved.

World Esports Association Adds Members Merch Sales Up Big For Final Four Teams Herschel Supply Launches MLB-Licensed Backpacks Spalding Renews NCAA Partnership Johnson Controls To Sponsor Bucks' Arena Topps Chooses Cubs' Bryant As Cover Athlete StubHub, Blue Jays Sign Deal Former Big Ten Commissioner Wayne Duke Dies SI Media Podcast Talks All Things ESPN NCAA Women's Tournament Strugges With Attendance SBD/May 15, 2012/CollegesPrint All Florida State Univ. President Eric Barron has written a memo that “includes four key points that would support a move from the ACC to the Big 12 and seven longer key points that argue against it,” according to Joe Schad of ESPN.com. The memo “seems to be an argument by Barron, who has stated the school is ‘committed’ to the ACC.” In support of changing conferences, Barron said, "The ACC is too North Carolina centric," adding, "The Big 12 contract (which actually isn't signed yet) is rumored to be 2.9 M more per year than the ACC contract." In support of remaining in the ACC, he said, "The ACC is an equal share conference. ... So when fans realize that Texas would get more dollars than FSU, always having a competitive advantage, it would be interesting to see the fan reaction." Barron also said that FSU "would lose the rivalry with University of Miami that does fill our stadium." He added, "It will cost between $20M and $25M to leave the ACC and we have no idea where that money would come from." Barron: "The faculty are adamantly opposed to joining a league that is academically weaker" (ESPN.com, 5/14). Barron also wrote that negotiations between the school and the Big 12 “are not taking place” (AP, 5/15). Read Barron's memo. HEARSAY: Univ. of Texas men's AD DeLoss Dodds yesterday said there was "no traction" to the story of FSU jumping to the Big 12. Dodds: "There've been no conversations between Florida State and the Big 12.” He added that he “preferred to keep the league at its current 10 teams” (AUSTIN AMERICAN-STATESMAN, 5/15). FSU football coach Jimbo Fisher said, “We’re in the ACC. We’re happy to be here. … We have a good football conference” (ORLANDO SENTINEL, 5/15). In Tallahassee, Jim Lamar notes Barron spent “at least part of his day trying to deliver a message to the university’s alumni, fans and other stakeholders.” Barron said, “I tried to let them know that this is a complicated issue.” He added, "I have no idea what the next few weeks or even days will bring. I know I have alumni irritated because they think I made up my mind. What I am trying to say is this is not some simple thing.” Barron continued, "It's amazing how many people will say to me, 'You don't understand. If you go to the Big 12, Clemson will go with you and so will Miami and you can play them.'” Barron asked, “How do you know that? How can you say that?” (TALLAHASSEE DEMOCRAT, 5/15). UNPRECEDENTED & UNWARRANTED: WARCHANT.com’s Williams & Reeves wrote “It's unprecedented for a university president to write a detailed response to fans on a subject like conference expansion when reportedly the two sides haven't even touched base.” Barron's memo “was so unorthodox that many fans originally thought it was a hoax” (WARCHANT.com, 5/14). In Virginia, David Teel wrote Barron’s statement was “beyond unusual for a university president -- silence may have been the more prudent course” (DAILYPRESS.com, 5/14). SI.com’s Andy Staples wrote Barron “seems intent on pumping the brakes on this process before it reaches move-or-else territory.” Staples: “No one has to make any decisions this week or even this month. But before the year is out, FSU's leaders will have to answer the questions of an increasingly aggravated fan base by either exploring the idea of another conference or by committing completely to the ACC” (SI.com, 5/14). NBCSPORTS.com’s John Taylor wrote Barron’s “commitment” to the ACC or reasons for remaining in their current conference “doesn’t ensure in any way, shape or form that FSU will or won’t remain in the ACC.” What it does, though, “is continue to highlight the divide between the upper levels of the university’s administration” (NBCSPORTS.com, 5/14). SEMINOLES ARE NOT READY: ESPN.com’s Heather Dinich asked, “If Florida State is so desperate for more cash, then, how on earth is it going to afford the ACC's $20 million exit fee?” The conversation “should stop right there.” Barron “makes some very good points and helps clarify some misinformation that's been out there.” But based on what he wrote, “it looks like FSU's options are limited” (ESPN.com, 5/14). ESPN.com’s Andrea Adelson wrote the damage “has been done and the expansion rumors have begun to swirl once again, leading to renewed questions about the future of Florida State and the ACC” (ESPN.com, 5/14). YAHOO SPORTS’ Graham Watson wrote FSU is “not ready for the Big 12,” and the university is "not ready to leave the comfy confines of the ACC” (SPORTS.YAHOO.com, 5/14). CBSSPORTS.com’s Brett McMurphy wrote the Big 12 “may -- or may not -- be interested in expanding.” Big 12 Commissioner Bob Bowlsby “doesn't even officially start until next month and the reality is any candidate the Big 12 would pursue (Florida State, Louisville, etc.) has no other options.” So the Big 12 “can sit back and wait to see how the new 2014 playoff revenue is divvied up among the conferences before deciding who, if anyone, they want to add.” McMurphy noted the Big 12 “is very satisfied with its current 10-member makeup” (CBSSPORTS.com, 5/14). An annual analysis of college athletics finances shows that over the past three years “no college athletics program has out-earned or outspent the colossus that is Texas,” according to a front-page piece by Wieberg, Upton & Berkowitz of the USA TODAY. The Univ. of Texas took in “a little more" than $150M in '10-11, which "outdistanced second-place Ohio State by $18.5M. UT's "outlay for football and 19 other varsity sports" was $133.7M, almost $11.5M more than Ohio State put into its 36 teams. The UT athletic department is “one of only 22 across Division I that operate in the black,” and a year ago "kicked" $6M back to the school’s academic side. UT's “unabashed athletics growth comes, however, as the NCAA continues to preach fiscal temperance, particularly to schools spending beyond their means in the chase for athletics success.” The study shows that “ten programs, all anchored by football," made or spent more than $100M a year ago. Nearly two dozen topped $80M "on one side of the ledger or both.” The Longhorn Network has “just started paying off" for UT, including a first installment of about $8M, with a total of $247.5M due over a 20-year contract with ESPN. Overall athletics revenue for ‘11-12 is “projected to approach $160 million, expenditures to rise by almost $20 million to $153.5 million.” Former Texas A&M AD Bill Byrne, who retired last week after 10 years at the school, said of UT, "They do everything they can to set themselves apart, that they are the very best, they are the elite." He added, "They're absolutely committed to having the best athletics program in the country. You've got to tip your hat to them." UT men's AD DeLoss Dodds said, "Whatever we do, we want to do it well. Whatever sport we have, we want it totally funded -- I mean totally funded.” Former Univ. of Arizona President Peter Likins “doesn't object to Texas' approach specifically.” However, he objects “to what it represents: unbridled escalation that dwarfs the growth of universities in general and all but mocks the financial straits that have led many schools to pull back on the academic side” (USA TODAY, 5/15). THE HAVES AND THE HAVE NOTS: USA TODAY’s Steve Wieberg writes UT “poured as much money into its athletics programs a year ago as eight of the nine schools in the more modestly resourced Sun Belt Conference did combined.” UT President Bill Powers said, “We may get to a point -- I want to underline the word may -- where many schools are really not in a position to compete at the level of the Floridas and the Notre Dames and the Texases and the USCs.” He added, “I am not a fan of some national league, but we may end up with 50 schools in (the upper football division of the NCAA's) Division I." In the study of 99 public-school programs in ’10-11, the top 50 revenue producers -- led by UT -- generated an average of nearly $81.5M and the bottom 49 an average of a little more than $28M. Below is a chart listing the schools in order with the largest difference in total revenue gained and the athletic department’s operating expenses (USA TODAY, 5/15). SCHOOLTOTAL REVENUEGENERATED REVENUEALLOCATED REVENUEOPERATING REVENUEDIFFERENCE Kansas State$69,947,834$66,656,183$3,291,651$46,549,248$20,106,935 Texas$150,295,926$150,295,926$0$133,686,815$16,609,111 LSU$107,259,352$107,259,352$0$91,796,925$15,462,427 Penn State$116,118,025$116,118,025$0$101,336,483$14,781,542 Alabama$124,498,616$119,263,316$5,235,300$105,068,152$14,195,164 Florida$123,514,257$119,147,186$4,367,071$107,157,831$11,989,355 Michigan$122,739,052$122,466,368$272,684$111,844,553$10,621,815 Arkansas$91,768,112$89,917,612$1,850,500$79,392,988$10,524,624 Oklahoma$104,338,844$104,338,844$0$94,363,928$9,974,916 Ohio State$131,815,821$131,815,821$0$122,286,869$9,528,952 Oklahoma State$82,631,915$76,444,582$6,187,333$66,937,207$9,507,375 Texas A&M$87,296,532$87,286,676$9,856$78,310,805$8,975,871 Georgia$92,341,067$89,143,680$3,197,387$80,759,498$8,384,182 Oregon$85,819,699$83,399,844$2,419,855$76,274,142$7,125,702 Purdue$66,202,493$66,202,493$0$59,429,383$6,773,110 Tennessee$104,368,922$103,368,992$1,000,000$97,580,406$5,788,586 Iowa$93,353,561$92,788,881$564,680$88,057,486$4,731,395 Mississippi State$58,981,769$54,162,116$4,819,653$51,588,743$2,573,373 Nebraska$83,679,756$83,679,756$0$81,916,484$1,763,272 Kentucky$84,878,311$84,059,187$819,124$82,840,006$1,219,181 South Carolina$83,813,226$81,564,951$2,248,275$80,525,711$1,039,240 Illinois$77,863,883$73,880,243$3,983,640$73,476,818$403,425

Many factors affect the flow of blood in the arteries and other small vessels among which are the diameters of the various vessels, the viscosity of the blood and various constituents of the blood. People with the HbS type of red blood cell have their cell sickle. The present study used a one-dimension model of flow in the arteries to investigate the effect of increase viscosity on small pressure disturbance and on the arterial compliance and the sequence of shock. We deduced that increase viscosity which all sickle cell patient are prone to leads to reduced flow and hampers the distribution of oxygen to essential areas of the body. The higher the viscosity the more like the crisis will occur. PDF Abstract XML References Citation How to cite this article The desire to quantitatively understand both the microscopic and macroscopic behaviour of blood when it flows through the vessels has supplied the motivation for appropriate investigations. Young summarized the essence of this desire as understanding the physical events that take place in normal animal and thereby make contribution to physiology. A secondary aim is to make contribution to medicine by analyzing particular abnormal or diseased states in the hope of improving diagnosis or treatment (Pedley, 1980) . Within the blood vessels are red blood cells, which contain a protein called haemoglobin whose function is to carry oxygen from the lungs to all parts of the body. Most people have normal haemoglobin or HbA; their blood cells are doughnut shaped and flexible, allowing them to pass easily through even the smallest vessels. Those with sickle cell anaemia have a different type of haemoglobin, HbS. Their blood cells are sickle shaped and rigid and therefore cannot pass through small blood vessels. This results in blockage and a lack of oxygen reaching parts of the body causing pain or damage to vital organs. There are three primary factors that determine resistance to blood flow within a single vessel: diameter R (radius), length of vessel L and viscosity of blood η (Klahunde, 2005). The relationship is given by: The relationship between flow F, the Pressure gradient ΔP and Resistance R is given by the well-known Poiseuille equation as This implies that flow is inversely proportional to blood viscosity. The viscosity (thickness) of blood or any liquid, for that matter, is controlled by both the chemical components and temperature (Bedham, 2004). Temperature induced changes are attributable to the way molecules in the liquid react to a gain or loss of thermal energy. Just as the Poiseuille equation suggests viscous liquids resist internal flow; viscous blood would resist flow through blood vessels and would consequently not be as effective at distributing heat to either areas of the body that are in need of it or away from places that have excess of it. It is well known that the viscosity of a sickle cell patient is greater than that of a normal patient. Mazzoni et al. (2002) found that as blood viscosity decreases, cardiac output increases. This implies also that increase in blood viscosity decreases cardiac output. Chiens measurement (Chien, 1970) of effective viscosity as a function of particle shape in dilute suspension found that thin discs have higher viscosity. Chiens results also suggest that continuous surface deformations in response to flow conditions may further reduce their contribution to blood viscosity. This implies that in a sickle cell patient where the cell are sickle and resist deformation the viscosity will be higher than normal. In this study we investigate the effect of increase viscosity on small pressure disturbance and the arterial compliance. We also determine the effect of this increase on the time of shock formation i.e., the onset of sickle cell anaemia patient crisis. Due to the large wavelength of the arterial pulse waves compared to the vessel diameters, one-dimensional models permits an efficient simulation of the wave propagation in the large arteries which allows us to study the effects of local changes on the global system. The governing equations of the one dimensional model are obtained by applying conservation of mass and momentum to a one dimensional impermeable tubular control volume of inviscid, incompressible and Newtonian fluid and by considering a tube law that relates changes in pressure to changes in cross-sectional area. The continuity equation is one of the equations governing the fluid dynamics of the blood and states the conservation of matter i.e., what goes into a section must either come out or be stored. Under the assumption of one dimensionality the continuity equation is where A denotes the cross sectional area of the artery, u the axial blood velocity (average over the cross-section), x the axial coordinate and t is the time. The first term is the rate of storage of blood within the arterial segment; the second is the difference between the blood flowing into and out of the segment in the axial direction. The second fluid dynamics relation is the momentum equation, which balances inertia, friction and pressure forces acting on the blood. In one dimension it is given as where P is the pressure difference across the tube and p is the fluid density. The first term is the acceleration of the blood; the second is the convective acceleration due to axial rate of change of velocity, the third the pressure gradient and the fourth term represents the friction at walls of the arteries. It is assumed that the lumen area is related instantaneously to the pressure (Vander Werff, 1974) and there is no phase lag; thus the one dimension state equation is given as Although the shear stress in pulsating flows is not directly related to the instantaneous mean velocity, it is assumed to be given by the laminar Poiseuille relation where μ is the blood viscosity. Equation 4 then becomes METHOD OF SOLUTION If we assume that the wave amplitude is smaller than the wavelength, then the cross-sectional area can be assumed to be constant along the elastic vessel. and Eq. 3 reduces to A special example of a pressure area relationship which has been empirically formulated to model the pulmonary artery is where R0 is the vessel radius at zero pressure and α is the compliance constant of the vessel wall. We can thus use Eq. 7, 9 and 10 to determine the speed of propagation of a small disturbance in an elastic vessel. This may allow us quantify the compliance and the effect of increase in viscosity on it. If we substitute Eq. 10 into 9 we have If the initial fluid velocity is zero, then again because the wave amplitude is small the convective acceleration term in Eq. 7 can be neglected. Thus we have Equation 12 is the wave equation where the quantity c is the wave speed and is given as We can the determine the vessel compliance α as Therefore we can determine the vessel compliance if we know the other measurable parameters. Let us introduce the wave speed in the form Note that this form is conforms with Eq. 14 Using (5) and (16) Eq. 3 becomes if we consider a uniform arterial segment as in Eq. 8, we have Now let us consider the flow as a perturbation of the undisturbed state u = 0, P = P0= constant, A = A0 = constant (Hoogstraten and Smit, 1978; Akinrelere and Ayeni, 1983; Ayeni and Akinrelere, 1984). We also assume that for x≥0 i.e., the tube wall react instantaneously on pressure change in the blood. Thus if we use the subscript 0 to denote the initial value then we have Let also the wave expansion for P, u, T be of the form. This leads us to have If we let then equation can be written in the form We consider first the case when μ = 0 i.e., b = 0 and we have This gives us P1 becomes infinite when We now consider when the viscosity μ ≠ 0 by solving the equation This gives us We plot below the graph of P1(t) against t for various values of μ P1(t) and invariably v1(t) become infinite at time This is the time when shock occurs. The circulation time of blood is given by the ratio of volume to flow. A minimum volume will result in a reduced transformation time of information through the circulation delivered by hormones, oxygen carbon dioxide and other blood solute. Chien (1970) has shown that continuous surface deformation in response to flow conditions reduces their contribution to blood viscosity. For a sickle cell anaemia patient the sickle nature which causes rigidity prevents this thus making the blood density higher than would have been reduced. From Eq. (14) we see that the wave speed is inversely proportional to the to the density of the fluid. Therefore as the density and invariably the viscosity increases, the wave-speed decreases. The wall compliance is also inversely proportional to the density and as it increases the compliance decreases and the vessel response is decreased. Therefore there is a blockage and a lack of oxygen reaching parts of the body causing pain or damage to vital organs. The occurrence of shock is synonymous with the breakdown of flow or what is termed crisis in the patient. Equation (35) clearly shows when this is likely to occur and we can also see that as μ increases t decreases i.e., the existence of shock becomes prominent as the viscosity increases. - Mazzoni, M.C., T.G. Amy and I. Marcos, 2002. Blood and plasma viscosity and microvascular function in hemodilution. Eur. Sur. Res., 34: 101-105.

The CV of a variable or the CV of a prediction model for a variable can be considered as a reasonable measure if the variable contains only positive values. In actuarial sciencethe CV is known as unitized risk. The model RMSE and mean of the predicted variable are expressed in the same units, so taking the ratio of these two allows the units to cancel. Distribution[ edit ] Provided that negative and small positive values of the sample mean occur with negligible frequency, the probability distribution of the coefficient of variation for a sample of size n has been shown by Hendricks and Robey to be d. Even if the mean of a variable is not zero, but the variable contains both positive and negative values and the mean is close to zero, then the CV can be misleading. The higher the CV, the greater the dispersion in the variable. If all values are negative, then we can regard the sign as just a convention that can be ignored. When an intercept is included, then r2 is simply the square of the sample correlation coefficient i. There is occasionally a tendency to regard it as a kind of magic summary measure that encapsulates both mean and standard deviation. Similarly, the RMSE of two models both measure the magnitude of the residuals, but they cannot be compared to each other in a meaningful way to determine which model provides better predictions of an outcome. For comparison between data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation. In this case standard error in percent is suggested to be superior. In both settings, the CV is often presented as the given ratio multiplied by In statisticsthe coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable s. Unlike the standard deviation, it cannot be used directly to construct confidence intervals for the mean. This is naturally primitive thinking, as even when the ratio makes sense, the mean and standard deviation cannot be recovered from it. The standard deviation of an exponential distribution is equal to its mean, so its coefficient of variation is equal to 1. There are cases Coefficient of variation the computational definition of R2 can yield negative values, depending on the definition used. While intra-assay and inter-assay CVs might be assumed to be calculated by simply averaging CV values across CV values for multiple samples within one assay or by averaging multiple inter-assay CV estimates, it has been suggested that these practices are incorrect and that a more complex computational process is required. Both are unitless measures that are indicative of model fit, but they define model fit in two different ways: As in the case of the bizarre examples from climatology, which I leave unreferenced as the authors deserve neither the credit nor the shame, the coefficient of variation has been over-used in some fields. Calculate the mean of the data set. The CV for a variable can easily be calculated using the information from a typical variable summary and sometimes the CV will be returned by default in the variable summary. This follows from the fact that the variance and mean both obey this principle. Formulas to calculate coefficient of variation: Disadvantages[ edit ] When the mean value is close to zero, the coefficient of variation will approach infinity and is therefore sensitive to small changes in the mean. The method of measuring the ratio of standard deviation to mean is also known as relative standard deviation often abbreviated as RSD. As it happens, the coefficient of variation is not especially useful even for temperatures measured in kelvin, but for physical reasons rather than mathematical or statistical. Even more bizarrely, I have seen suggestions that the problem is solved by using Fahrenheit instead. The lower the CV, the smaller the residuals relative to the predicted value. This is a definite disadvantage of CVs.coefficient of variation calculator - to find the ratio of standard deviation (σ) to mean (μ); along with formula, example & complete step by step relative variability calculation. In statistics the coefficient of variation is a fairly natural parameter if variation follows either the gamma or the lognormal, as may be seen by looking at the form of the coefficient of variation for those distributions. The coefficient of partial determination can be defined as the proportion of variation that cannot be explained in a reduced model, but can be explained by the predictors specified in a full(er) model. A coefficient of variation (CV) can be calculated and interpreted in two different settings: analyzing a single variable and interpreting a model. The standard formulation of the CV, the ratio of the standard deviation to the mean, applies in the single variable setting. The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree. The coefficient of variation (CV), also known as the relative standard deviation (RSD) is commonly used in probability. Enter the values separated by a comma in this coefficient of variation calculator to know the relative standard deviation.Download

3. Is the origin stabile in the following cases: (i)x000+ 6x00+ 11x0+ 6x= 0, (ii)x000−6x00+ 11x0−6x= 0, (iii)x000+ax00+bx0+cx= 0, for all possible values ofa, b andc. 4. Consider the system x1 x2 x3 0 2 0 −2 0 0 0 0 0 x1 x2 x3 Show that no non-trivial solution of this system tends to zero as t → ∞. Is every solution bounded ? Is every solution periodic ? 5. Prove that for 1< α <√ 2, x0= (sin logt+ cos logt−α)x is asymptotically stable. 6. Consider the equation Show that the origin is asymptotically stable if and only if Z ∞ Under what condition the zero solution is stable ? These conditions guarantee the existence of local solutions of (5.35) on some interval. The solutions may not be unique. However, for stability we assume that solutions of (5.35) uniquely exist on I. Let Φ(t) denote a fundamental matrix of (5.36) such that Φ(t0) = E, where E is the n×n identity matrix. As a first step, we obtain necessary and sufficient conditions for the stability of the linear system (5.36). Note thatx≡0,on I satisfies (5.36) or in other wordsx≡0 or the zero solution or or the null the origin is an equilibrium state of (5.36). Theorem 5.5.1. The zero solution of equation (5.36) is stable if and only if a positive constant k exists such that |Φ(t)| ≤k, t≥t0. (5.37) Proof. The solution y of (5.36) which takes the valuec att0 ∈I (or y(t0) =c) is given by y(t) = Φ(t)c (Φ(t0) =E). Suppose that the inequality (5.37) hold. Then, fort∈I |y(t)|=|Φ(t)c| ≤k|c|< ², if|c|< ²/k. The origin is thus stable. |y(t)|=|Φ(t)c|< ², t≥t0, for all csuch that |c|< δ. Then,|Φ(t)|< ²/δ.By Choosing k=²/δ the inequality (5.37) follows and hence the proof. The result stated below concerns about the asymptotic stability of the zero (or null) solution of the system (5.36). Theorem 5.5.2. The null solution of the system(5.36)is asymptotically stable if and only if |Φ(t)| →0 as t→ ∞. (5.38) Proof. Firstly we note that (5.37) is a consequence of (5.38) and so the origin is obviously stable. Since |Φ(t)| →0as t→ ∞ in view of (5.38) we have |y(t)| → 0 as t → ∞ or in other words the zero solution is asymptotically stabile. The stability of (5.36) has already been considered when A(t) =A is a constant matrix. We have seen earlier that if the characteristic roots of the matrixAhave negative real parts then every solution of (5.36) tends to zero as t → ∞. In fact, this is asymptotic stability. We already are familiar with the fundamental matrix Φ(t) which is given by Φ(t) =e(t−t0)A, t0, t∈I. (5.39) When the characteristic roots of the matrix Ahave negative real parts then,there exist two positive constantsM and ρ such that |e(t−t0)A| ≤M e−ρ(t−t0), t0, t∈I. (5.40) Let the functionf satisfy the condition |f(t, x)|=o(|x|) (5.41) uniformly int fort∈I. This implies that forx in a sufficiently small neighborhood of the origin, |f(t, x)| |x| can be made arbitrarily small. The proof of the following result depends on the Gronwall’s inequality. Theorem 5.5.3. In equation (5.35), let A(t) be a constant matrix A and let all the char- acteristic roots of A have negative real parts. Assume further that f satisfies the condition (5.41). Then, the origin for the system (5.35) is asymptotically stable. Proof. By the variation of parameters formula, the solutiony of the equation (5.35) passing through (t0, y0) satisfies the integral equation y(t) =e(t−t0)Ay0+ Z t e(t−s)Af(s, y(s))ds. (5.42) The inequality (5.40) together with (5.42) yields |y(t)| ≤M|y0|e−ρ(t−t0)+M Z t e−ρ(t−s)|f(s, y(s))|ds. (5.43) which takes the form |y(t)|eρt≤M|y0|eρt0 +M Z t Let|y0|< α. Then, the relation (5.42) is true in any interval [t0, t1) for which|y(t)|< α. In view of the condition (5.41), for a given² >0 we can find a positive numberδ such that |f(t, x)| ≤²|x|, t∈I, f or|x|< δ. (5.44) Let us assume that|y0|< δ. Then, there exists a numberT such that|y(t)|< δfort∈[t0, T]. Using (5.44) in (5.43), we obtain eρt|y(t)| ≤M|y0|eρt0+M ² Z t eρs|y(s)|ds, (5.45) fort0≤t < T. An application of Gronwall’s inequality to (5.45), yields eρt|y(t)| ≤M|y0|eρt0.eM ²(t−t0) (5.46) or fort0≤t < T, we obtain |y(t)| ≤M|y0|e(M ²−ρ)(t−t0). (5.47) Choose M ² < ρand y(t0) =y0. If|y0|< δ/M, then, (5.47) yields |y(t)|< δ, t0≤t < T. The solutiony of the equation (5.35) exists locally at each point (t, y), t≥t0, |y|< α. Since the function f is defined on I×Sα, we extend the solutiony interval by interval by preserving its bound byδ. So given any solutiony(t) =y(t;t0, y0) with|y0|< δ/M, y exists on t0 ≤t < ∞ and satisfies |y(t)|< δ. In the above discussion, δ can be made arbitrarily small. Hence,y≡0 is asymptotically stable whenM ² < ρ. When the matrix A is a function oft (ie A is not a constant matrix), still the stability properties solutions of (5.35) and (5.36) are shared but now the fundamental matrix needs to satisfy some stronger conditions. Let r :I → R+ be a non-negative continuous function such that Z ∞ Letf be continuous and satisfy the inequality |f(t, x)| ≤r(t)|x|,(t, x)∈I×Sα, (5.48) The condition (5.48) guarantees the existence of a null solution of (5.35). Now the following is a result on asymptotic stability of the zero solution of (5.35). Theorem 5.5.4. Let the fundamental matrix Φ(t) satisfy the condition |Φ(t)Φ−1(s)| ≤K, (5.49) where K is a positive constant and t0 ≤s ≤t < ∞. Let f satisfy the hypotheses given by (5.48). Then, there exists a positive constantM such that ift1≥t0, any solutiony of (5.35) is defined and satisfies |y(t)| ≤M|y(t1)|, t≥t1 whenever |y(t1)|< α/M. Moreover, if |Φ(t)| →0 as t→ ∞, then |y(t)| →0 as t→ ∞. Proof. Let t1 ≥t0 and y be any solution of (5.35) such that |y(t1)|< α. We know thar y satisfies the integral equation y(t) = Φ(t)Φ−1(t1)y(t1) + Z t Φ(t)Φ−1(s)f(s, y(s))ds. (5.50) fort1≤t < T, where |y(t)|< αfort1≤t < T. By hypotheses (5.48) and (5.49) we obtain |y(t)| ≤K|y(t1)|+K Z t r(s)|y(s)|ds The Gronwall’s inequality now yields By the condition (5.48) the integral on the right side is bounded. With M =Kexp |y(t)| ≤M|y(t1)|. (5.52) Clearly this inequality holds if |y(t1)|< α/M. Following the lines of proof of in Theorem 5.5.3, we extend the solution for all t≥t1. Hence, the inequality (5.52) holds for t≥t1. The general solutiony of (5.35) also satisfies the integral equation y(t) = Φ(t)Φ−1(t0)y(t0) + = Φ(t)y(t0) + Z t1 Φ(t)Φ−1(s)f(s, y(s))ds+ Z t Note that Φ(t0) =E. By using the conditions (5.48), (5.49) and (5.52), we obtain |y(t)| ≤ |Φ(t)||y(t0)|+|Φ(t)| |Φ−1(s)||f(s, y(s))|ds+K Z ∞ r(s)ds. (5.53) The last term of the right side of the inequality (5.53) can be made less than (arbitrary)²/2 by choosingt1 sufficiently large. By hypotheses Φ(t)→0 ast→ ∞. The first two terms on the right side contain the term |Φ(t)|. Hence, their sum together can be made arbitrarily small say less than ²/2 and by choosing tlarge enough, . Thus, |y(t)|< ² for larget. This proves that|y(t)| →0 ast→ ∞. The inequality (5.52) shows that the origin is stable for t ≥ t1. But note that t1 ≥ t0 is any arbitrary number. Here, condition (5.52) holds for any t1 ≥ t0. Thus, we have es- tablished a stronger than the stability of the origin .In literature such a property is called uniform stability. We do not propose to go into the detailed study of such types of stability properties. 1. Prove that all solutions of the system (5.36) are stable if and only if they are bounded. 2. Letb:I →Rn be a continuous function. Prove that a solution x of linear nonhomo- geneous system is stable, asymptotically stable, unstable, if the same holds for the null solution of the corresponding homogeneous system (5.36). 3. Prove that if the characteristic polynomial of the matrixA is stable, the matrix C(t) is continuous on 0≤t <∞ and R∞ 0 |C(t)|dt <∞, then all solutions of x0= (A+C(t))x are asymptotically stable. 4. Prove that the system (5.36) is unstable if Re ³ Z t t0 → ∞, as t→ ∞. 5. Define the norm of a matrix A(t) by µ(A(t)) = lim h , where E is the n×n identity matrix. (i) Prove thatµ is a continuous function oft. (ii) For any solutiony of (5.36) prove that − Z t ≤ |y(t)| ≤ |y(t0)|exp Z t £Hint : Letr(t) =|y(t)|. Then r+0 (t) = lim |y(t) +hy0(t)| − |y(t)| (iii) When A(t) =A a constant matrix, show that|exp(tA)| ≤exp[tµ(A)]. (iv) Prove that the trivial solution is stable if lim sup (v) Show that the trivial solution is asymptotically stable if Z t µ(A(s))ds→ −∞ as t→ ∞. (vi) Establish that the solution is unstable if lim inf

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field . The unit for energy in the International System of Units (SI) is the joule , which has the symbol J. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine , although it has links to Greek philosopher Aristotle 's concept of potentiality Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called ''conservative forces'', can be represented at every point in space by vectors expressed as gradients of a certain scalar function called ''potential''. Since the work of potential forces acting on a body that moves from a start to an end position is determined only by these two positions, and does not depend on the trajectory of the body, there is a function known as ''potential'' that can be evaluated at the two positions to determine this work. There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy ; work of the strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels , is the work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their configuration. Forces derivable from a potential are also called conservative force s. The work done by a conservative force is is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are ''PE'', ''U'', ''V'', and ''Ep Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring force s such as a spring or the force of gravity . The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. Consider a ball whose mass is m and whose height is h. The acceleration g of free fall is approximately constant, so the weight force of the ball mg is constant. Force × displacement gives the work done, which is equal to the gravitational potential energy, thus The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position. Work and potential energy Potential energy is closely linked with forces . If the work done by a force on a body that moves from ''A'' to ''B'' does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from ''A'' assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application. This means that there is a function ''U''(x), called a "potential," that can be evaluated at the two points xA to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is where ''C'' is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, ''C'', from A to B. The function ''U''(x) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces. Derivable from a potential In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve ''C'' takes a special form if the force F is related to a scalar field Φ(x) so that In this case, work along the curve is given by which can be evaluated using the gradient theorem This shows that when forces are derivable from a scalar field, the work of those forces along a curve ''C'' is computed by evaluating the scalar field at the start point ''A'' and the end point ''B'' of the curve. This means the work integral does not depend on the path between ''A'' and ''B'' and is said to be independent of the path. Potential energy ''U''=-Φ(x) is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is In this case, the application of the del operator to the work function yields, and the force F is said to be "derivable from a potential." This also necessarily implies that F must be a conservative vector field . The potential ''U'' defines a force F at every point x in space, so the set of forces is called a force field Computing potential energy Given a force field F(x), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve γ(t)=r(t) from γ(a)=A to γ(b)=B, and computing, For the force field F, let v= dr/dt, then the gradient theorem The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is Examples of work that can be computed from potential functions are gravity and spring forces. Potential energy for near Earth gravity For small height changes, gravitational potential energy can be computed using where m is the mass in kg, g is the local gravitational field (9.8 metres per second squared on earth), h is the height above a reference level in metres, and U is the energy in joules. In classical physics, gravity exerts a constant downward force ''F''=(0, 0, ''Fz '') on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory ''r''(t) = (''x''(t), ''y''(t), ''z''(t)), such as the track of a roller coaster is calculated using its velocity, ''v''=(''v''x ), to obtain where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve ''r''(t). Potential energy for a linear spring A horizontal spring exerts a force F = (−''kx'', 0, 0) that is proportional to its deformation in the axial or ''x'' direction. The work of this spring on a body moving along the space curve s(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t'')), is calculated using its velocity, v = (''v''x ), to obtain For convenience, consider contact with the spring occurs at ''t'' = 0, then the integral of the product of the distance ''x'' and the ''x''-velocity, ''xvx '', is ''x''2 is called the potential energy of a linear spring. Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy Potential energy for gravitational forces between two bodies The gravitational potential function, also known as gravitational potential energy The negative sign follows the convention that work is gained from a loss of potential energy. The gravitational force between two bodies of mass ''M'' and ''m'' separated by a distance ''r'' is given by Newton's law is a vector of length 1 pointing from ''M'' to ''m'' and ''G'' is the gravitational constant Let the mass ''m'' move at the velocity v then the work of gravity on this mass as it moves from position r(t1 ) to r(t2 ) is given by The position and velocity of the mass ''m'' are given by are the radial and tangential unit vectors directed relative to the vector from ''M'' to ''m''. Use this to simplify the formula for work of gravity to, This calculation uses the fact that Potential energy for electrostatic forces between two bodies The electrostatic force exerted by a charge ''Q'' on another charge ''q'' separated by a distance ''r'' is given by Coulomb's Law is a vector of length 1 pointing from ''Q'' to ''q'' and ''ε''0 is the vacuum permittivity . This may also be written using Coulomb constant The work ''W'' required to move ''q'' from ''A'' to any point ''B'' in the electrostatic force field is given by the potential function The potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state; it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inverse-square law forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience. Typically the potential energy of a system depends on the ''relative'' positions of its components only, so the reference state can also be expressed in terms of relative positions. Gravitational potential energy Gravitational energy is the potential energy associated with gravitational force , as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount. Consider a book placed on top of a table. As the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy . When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact. The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. "Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail. The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant ("standard gravity "). In this case, a simple expression for gravitational potential energy can be derived using the ''W'' = ''Fd'' equation for work , and the equation The amount of gravitational potential energy held by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember ''W = Fd''). The upward force required while moving at a constant velocity is equal to the weight, ''mg'', of an object, so the work done in lifting it through a height ''h'' is the product ''mgh''. Thus, when accounting only for mass , and altitude , the equation is: where ''U'' is the potential energy of the object relative to its being on the Earth's surface, ''m'' is the mass of the object, ''g'' is the acceleration due to gravity, and ''h'' is the altitude of the object. If ''m'' is expressed in kilogram s, ''g'' in m/s2 and ''h'' in metre s then ''U'' will be calculated in joule Hence, the potential difference is However, over large variations in distance, the approximation that ''g'' is constant is no longer valid, and we have to use calculus and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation , with respect to the distance ''r'' between the two bodies. Using that definition, the gravitational potential energy of a system of masses ''m''1 at a distance ''r'' using gravitational constant where ''K'' is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that ''K''=0 (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making ''U'' negative; for why this is physically reasonable, see below. Given this formula for ''U'', the total potential energy of a system of ''n'' bodies is found by summing, for all pairs of two bodies, the potential energy of the system of those two bodies. Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy . This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity. Negative gravitational energy As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and the choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite ''r'' over another, there seem to be only two reasonable choices for the distance at which ''U'' becomes zero: . The choice of at infinity may seem peculiar, and the consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative. in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with , would result in potential energy being positive, but infinitely large for all nonzero values of ''r'', and would make calculations involving sums or differences of potential energies beyond what is possible with the real number system. Since physicists abhor infinities in their calculations, and ''r'' is always non-zero in practice, the choice of at infinity is by far the more preferable choice, even if the idea of negative energy in a gravity well appears to be peculiar at first. The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the total energy of the universe can meaningfully be considered; see inflation theory for more on this. Gravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity . For example, in Dinorwig , Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction. Pumped storage in Switzerland – an outlook beyond 2000 ''Stucky''. Accessed: 13 February 2012. [Levine, Jonah G] Pumped Hydroelectric Energy Storage and Spatial Diversity of Wind Resources as Methods of Improving Utilization of Renewable Energy Sources page 6, ''University of Colorado'', December 2007. Accessed: 12 February 2012. Pumped Hydroelectric Storage ''Duke University''. Accessed: 12 February 2012. ''Hawaiian Electric Company''. Accessed: 13 February 2012. Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism. It's also used by counterweight s for lifting up an elevator , crane, or sash window are an entertaining way to utilize potential energy – chains are used to move a car up an incline (building up gravitational potential energy), to then have that energy converted into kinetic energy as it falls. Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the kinetic energy obtained from the potential energy of descent may be used to start ascending the next grade such as what happens when a road is undulating and has frequent dips. The commercialization of stored energy (in the form of rail cars raised to higher elevations) that is then converted to electrical energy when needed by an electrical grid, is being undertaken in the United States in a system called Advanced Rail Energy Storage [Packing Some Power: Energy Technology: Better ways of storing energy are needed if electricity systems are to become cleaner and more efficient](_blank) ''The Economist'', 3 March 2012 Ski Lifts Help Open $25 Billion Market for Storing Power Bloomberg News online, 6 September 2012 Chemical potential energy Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bond s within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction . As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis , and electrical energy can be converted to chemical energy through electrochemical The similar term chemical potential is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc. Electric potential energy An object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy). Electrostatic potential energy Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge ''Q'' on another charge ''q'' which is given by is a vector of length 1 pointing from ''Q'' to ''q'' and ''ε''0 is the vacuum permittivity . This may also be written using Coulomb's constant If the electric charge of an object can be assumed to be at rest, then it has potential energy due to its position relative to other charged objects. The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object. This energy will generally be non-zero if there is another electrically charged object nearby. The work ''W'' required to move ''q'' from ''A'' to any point ''B'' in the electrostatic force field is given by typically given in ''J'' for Joules. A related quantity called ''electric potential '' (commonly denoted with a ''V'' for voltage) is equal to the electric potential energy per unit charge. Magnetic potential energy The energy of a magnetic moment in an externally produced magnetic B-field has potential energy in a field is where the integral can be over all space or, equivalently, where is nonzero. Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned with the north and south poles of the Earth's magnetic field. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart. Nuclear potential energy Nuclear potential energy is the potential energy of the particles inside an atomic nucleus . The nuclear particles are bound together by the strong nuclear force . Weak nuclear force s provide the potential energy for certain kinds of radioactive decay, such as beta decay Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them can have less mass than if they were individually free, in which case this mass difference can be liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy , which is radiated into space. Forces and potential energy Potential energy is closely linked with forces . If the work done by a force on a body that moves from ''A'' to ''B'' does not depend on the path between these points, then the work of this force measured from ''A'' assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. For example, gravity is a conservative force . The associated potential is the gravitational potential , often denoted by , corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass ''M'' and ''m'' separated by a distance ''r'' is The gravitational potential (specific energy ) of the two bodies is is the reduced mass The work done against gravity by moving an infinitesimal mass from point A with to point B with and the work done going back the other way is so that the total work done in moving from A to B and returning to A is If the potential is redefined at A to be and the potential at B to be is a constant (i.e. can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is In practical terms, this means that one can set the zero of anywhere one likes. One may set it to be zero at the surface of the Earth , or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section). A conservative force can be expressed in the language of differential geometry as a closed form . As Euclidean space , its de Rham cohomology vanishes, so every closed form is also an exact form , and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field. External linksWhat is potential energy? Category:Forms of energy

Financial Modeling Tutorials - Excel Modeling - Financial Functions in Excel - Sensitivity Analysis in Excel - Sensitivity Analysis - Capital Budgeting Techniques - Time Value of Money - Future Value Formula - Present Value Factor - Perpetuity Formula - Present Value vs Future Value - Annuity vs Pension - Present Value of an Annuity - Doubling Time Formula - Annuity Formula - Present Value of an Annuity Formula - Future Value of Annuity Due Formula - Maturity Value - Annuity vs Perpetuity - Annuity vs Lump Sum - Deferred Annuity Formula - Internal Rate of Return (IRR) - IRR Examples (Internal Rate of Return) - NPV vs XNPV - NPV vs IRR - NPV Formula - NPV Profile - NPV Examples - Advantages and Disadvantages of NPV - Mutually Exclusive Projects - PV vs NPV - IRR vs ROI - Break Even Point - Break Even Analysis - Breakeven Analysis Examples - Break Even Chart - Benefit Cost Ratio - Payback Period & Discounted Payback Period - Payback period Formula - Discounted Payback Period Formula - Payback Period Advantages and Disadvantages - Profitability Index - Feasibility Study Examples - Cash Burn Rate - Interest Formula - Simple Interest - Simple Interest vs Compound Interest - Simple Interest Formula - CAGR Formula (Compounded Annual Growth Rate) - Growth Rate Formula - Effective Interest Rate - Loan Amortization Schedule - Mortgage Formula - Loan Principal Amount - Interest Rate Formula - Rate of Return Formula - Effective Annual Rate - Effective Annual Rate Formula (EAR) - Compounding Formula - Compound Interest - Compound Interest Examples - Daily Compound Interest - Monthly Compound Interest Formula - Discount Rate vs Interest Rate - Discounting Formula - Rule of 72 - Geometric Mean Return - Geometric Mean vs Arithmetic Mean - Real Rate of Return Formula - Continuous compounding Formula - Weighted average Formula - Average Formula - EWMA (Exponentially Weighted Moving Average) - Average Rate of Return Formula - Mean Formula - Mean Examples - Population Mean Formula - Weighted Mean Formula - Harmonic Mean Formula - Median Formula in Statistics - Range Formula - Outlier Formula - Decile Formula - Midrange Formula - Quartile Deviation - Expected Value Formula - Exponential Growth Formula - Margin of Error Formula - Decrease Percentage Formula - Relative Change - Percent Error Formula - Holding Period Return Formula - Cost Benefit Analysis - Cost Benefit Analysis Examples - Cost Volume Profit Analysis - Opportunity Cost Formula - Opportunity Cost Examples - APR vs APY - Mortgage APR vs Interest Rate - Normal Distribution Formula - Standard Normal Distribution Formula - Normalization Formula - Bell Curve - T Distribution Formula - Regression Formula - Regression Analysis Formula - Multiple Regression Formula - Correlation Coefficient Formula - Correlation Formula - Correlation Examples - Coefficient of Determination - Population Variance Formula - Covariance Formula - Coefficient of Variation Formula - Sample Standard Deviation Formula - Relative Standard Deviation Formula - Standard Deviation Formula - Standard Deviation Examples - Effect Size - Sample Size Formula - Volatility Formula - Binomial Distribution Formula - Hypergeometric Distribution - Exponential Distribution - Central Limit Theorem - Poisson Distribution - Central Tendency - Hypothesis Testing - Gini Coefficient - Quartile Formula - P Value Formula - Skewness Formula - R Squared Formula - Adjusted R Squared - Regression vs ANOVA - Z Test Formula - Z Score Formula - Z Test vs T Test - F-Test Formula - Quantitative Research - Histogram Examples Capital Budgeting Techniques Capital budgeting techniques help to determine whether the long term investments like the new plants, new machinery, and research development products are worth the cash investment by the firm. The top 5 capital budgeting techniques are as below:- - Profitability index - Payback period - Net present value - Internal rate of return - Modified rate of return Let us discuss these capital budgeting techniques one by one in detail – Top 5 Capital Budgeting Techniques (with examples) Now we will discuss the Capital budgeting methods one by one with examples. #1 – Profitability Index Profitability Index is one of the most important techniques of capital budgeting and it signifies a relationship between the investment of the project and the payoff of the project. The formula of profitability index given by:- Where PV is the present value. It is mainly used for ranking projects. According to the rank of the project, a suitable project is chosen for investment. #2 – Payback Period This method of capital budgeting helps to find a profitable project. The payback period is calculated by dividing the initial investment by the annual cash flows. But the main drawback is it ignores the time value of money. By time value of money, we mean that money is more today than the same value in the future. So if we payback to an investor tomorrow, it includes an opportunity cost. As already mentioned, the payback period disregards the time value of money. It is calculated by how many years it is required to recover the amount of investment done. Shorter paybacks are more attractive than longer payback periods. Let’s calculate the payback period for the below investment:- For example, there is an initial investment of ₹1000 in a project and it generates a cash flow of ₹ 300 for the next 5 years. Therefore the payback period is calculated as below: - Payback period = no. of years – (cumulative cash flow/cash flow) - Payback period = 5- (500/300) - = 3.33 years Therefore it will take 3.33 years to recover the investment. #3 – Net Present Value Net Present value is the difference between the present value of incoming cash flow and the outgoing cash flow over a certain period of time. It is used to analyze the profitability of a project. The formula for the calculation of NPV is as below:- 4.9 (1,067 ratings) Here i is the discount rate and n is the number of years. Let us see an example to discuss it. Let us assume the discount rate is 10% - NPV = -1000 + 200/(1+0.1)^1 + 300/(1+0.1)^2+400/(1+0.1)^3+600/(1+0.1)^4+ 700/(1+0.1)^5 - = 574.731 We can also calculate it by excel formula. There is in-built excel formula of “NPV” which can be used. The discounting rate and the series of cash flows from the 1st year to the last year is taken as arguments. We should not include the year zero cash flow in the formula. We should later subtract it. - = NPV (Discount rate, cash flow of 1st year: cash flow of 5th year) + (-Initial investment) - = NPV (10%, 200:700) – 1000 - = 574.731 As NPV is positive, it is recommended to go ahead with the project. But not only NPV, but IRR is also used for determining the profitability of the project. #4 – Internal rate of return The Internal rate of return is also among the top capital budgeting techniques that are used to determine whether the firm should take up the investment or not. It is used together with NPV to determine the profitability of the project. IRR is the discount rate when all the NPV of all the cash flows is equal to zero. Here we need to find “i” which is the discount rate. Now we shall discuss an example to understand the internal rate of return in a better way. While calculating, we need to find out the rate at which NPV is zero. This is usually done by error and trial method else we can use excel for the same. Let us assume the discount rate to be 10%. NPV at a 10 % discount is ₹ 574.730. So we need to increase the discount percentage to make NPV as 0. So if we increase the discount rate to 26.22 %, the NPV is 0.5 that is almost zero. There is in-built excel formula of “IRR” which can be used. The series of cash flows is taken as arguments. - =IRR (Cash flow from 0 to 5th year) - = 26 % Therefore in both the ways, we get 26 % as the internal rate of return. #5 – Modified Internal Rate of return The main drawback of internal rate of return that it assumes that the amount will be reinvested at the IRR itself which is not the case. MIRR solves this problem and reflects the profitability in a more accurate manner. The formula is as below:- - N = the number of periods - FVCF = the future value of positive cash flow at the cost of capital - PVCF = the present value of negative cash flows at the financing cost of the company. We can calculate MIRR for the below example: Let us assume the cost of capital at 12%. In MIRR we need to take into account the reinvested rate which we assume as 14%. In Excel, we can calculate as the below formulae - MIRR= (cash flows from year 0 to 4th year, cost of capital rate, reinvestment rate) - MIRR= (-1000: 600, 12%, 14%) - MIRR= 22% Modified rate of return is a better estimation than an internal rate of return. Therefore capital budgeting methods help us to decide the profitability of investments which needs to be done in a firm. There are different techniques to decide the return of investment. This has been a guide to Capital Budgeting Techniques. Here we will discuss the Top 5 methods of Capital Budgeting along with formula, explanation & examples. You can learn more about accounting from following articles –

The story of mathematical genius S. Ramanujan His father was K. Srinivasa Iyengar, an accounting clerk for clothing merchant. His mother was Komalatammal, who earned a small amount of money each month as a singer at the local temple of town Kumbakonam. Ramanujan mother thought him Brahmin traditions and culture. His family was high caste Brahmin, but they were very poor. Ramanujan’s parents moved around a lot, and he attended a variety of different elementary schools. By the age of 10, Ramanujan topped in his district Kumbakonam. In the story of mathematical genius S.Ramanujan, an incident happen when he was 15 years old, he finds a book in pure mathematics by George S. Carr from his school library. Srinivasa Ramanujan education Ramanujan verified the results in Carr’s book and went beyond it with developing his own theorems and ideas. When he was 12 years old, he had begun the study of calculus. During school days he worked on arithmetic and geometric series and cubic equations. He discovered his own method of solving quartic equations. In 1903 he secured a scholarship from the University of Madras but Ramanujan could not qualify his non-mathematical exams and lost his scholarship. During the year 1905, he traveled to the city of Madras (presently known as Chennai) and enrolled himself at Pachaiyappa’s College. But he again could not qualify his non-mathematical exams. Better we should say education system remains to fail to judge the knowledge of Ramanujan. The mathematics he was doing was original and highly advanced. Even though he had the very little formal higher mathematical education he was able to discover new theorems. He independently finds out the results originally discovered by some of the great mathematicians in history, such as Gauss, Euler Riemann. Ramanujan continued his work, in the poorest circumstances. Ramanujan married S. Janaki Ammal in1909. After marrying he started to search for a job. He gave an interview with a government official, Ramachandra Rao. Ramachandra Rao was very impressed by Ramanujan and he wants support to his research for a time, but Ramanujan, unwilling to exist on charity. So he helped him obtained a clerical post with the Madras Port Trust. In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. Ramanujan’s knowledge of mathematics (most of which he had worked out for himself) was astonished. He was almost completely unaware of modern developments in mathematics, but his mastery of continued fractions was unequaled by any living mathematician. Srinivasa Ramanujan contribution to mathematics He discovers a new theory of divergent series and he also worked out the Riemann series, the elliptic integrals, hypergeometric series and the functional equations of the zeta function. He amazed classmates with his ability to recite the values of irrational numbers like π, e, and √2 to as many decimal places as they asked for. His memory for memorizing mathematical formulas and constants was very good. Ramanujan wrote letters to a number of British professors, but only one of them respond – he was G. H. Hardy pure mathematician at the University of Cambridge, who received a letter from S.Ramanujan in January 1913. By this time, Ramanujan was 25 years old. Professor Hardy puzzled over the nine pages of mathematical notes Ramanujan had sent. They seemed rather incredible. Could it be that one of his colleagues was playing a trick on him? Hardy reviewed the letters with J. E. Littlewood, another Cambridge mathematician. He was telling to Littlewood they had been written by either a crank or a genius, but he wasn’t quite sure which. After spending two and a half hours poring over the outlandishly original work, the mathematicians came to a conclusion. They were looking at the papers of some mathematical genius. The story of mathematical genius S. Ramanujan: Some of the Ramanujan theorems have no proof that’s why they are known as “Ramanujan Conjectures”. Hardy was eager for Ramanujan to move to Cambridge, but in accordance with his Brahmin beliefs, Ramanujan refused to travel overseas. Instead, an arrangement was made to fund two years of work at the University of Madras. During this time, Ramanujan’s mother had a dream in which the goddess Namagiri told her she should give her son permission to go to Cambridge, and this she did. Ramanujan began work with Hardy and Littlewood during world war first because Ramanujan arrived in Cambridge in April 1914, three months before the outbreak of World War 1. Two years later, he was awarded the equivalent of a Ph.D. in Cambridge University for his work. When Ramanujan left to study at the University of Cambridge, his wife moved in with Ramanujan’s parents. In England Ramanujan research very advanced. His papers were published in English and European journals, and in 1918 he was elected to the Royal Society of London. In 1917 he was diagnosed with tuberculosis and worryingly low weight. Srinivasa Ramanujan death In February 1919 he returned to India and his health seemed to recover sufficiently. But sadly he died within a year on his return. The reason behind the illness of Ramanujan was a deficiency of vitamin levels due to the irregular diet. He spent several months in nursing homes in London. He recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (1707–83) and Carl Jacobi (1804–51). Like Riemann, Jacobi, Gauss, Euler, Srinivasa Ramanujan also died in early aged 32 in Madras on April 26, 1920. Ramanujan was very shy, quiet and a dignified man with pleasant manners. He lived for a very short interval of life at Cambridge. Ramanujan was rigorously orthodox Hindu Brahmin. He credited his acumen to his family goddess, Mahalakshmi of Namakkal. Goddess Namakkal is the wife of Lord Narashimha (Incarnation of Lord Vishnu). Ramanujan looked to Goddess for inspiration in his work. The story of mathematical genius S. Ramanujan: Actually, Ramanujan and his family members were Vaishnav and his mother gave the name to Ramanujan from a Vaishnav saint Ramanujacharya. He often said, “An equation for me has no meaning unless it represents a thought of God.” Hardy cites Ramanujan as remarking that all religions seemed equally true to him. he remarked that Ramanujan was strict pure vegetarian. Ramanujan left behind three notebooks and a sheaf of pages (also called the “lost notebook”) containing many unpublished results that mathematicians continued to verify long after his death. Ramanujan’s prodigious mathematical output amazed Hardy and Littlewood and many more other mathematicians. The lost notebooks of Ramanujan which he had brought from India were filled with thousands of mathematical identities, equations, and theorems which he had discovered for himself in the school days and in years 1903 – 1914.

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On the non-minimal gravitational coupling to matter The connection between theories of gravity and scalar-tensor models with a “physical” metric coupled to the scalar field is well known. In this work, we pursue the equivalence between a suitable scalar theory and a model that generalises the scenario, encompassing both a non-minimal scalar curvature term and a non-minimum coupling of the scalar curvature and matter. This equivalence allows for the calculation of the PPN parameters and and, eventually, a solution to the debate concerning the weak-field limit of theories. pacs:04.20.Fy, 04.80.Cc, 11.10.Ef Preprint DF/IST-2.2008 Contemporary cosmology is faced with the outstanding challenge of understanding the existence and nature of the so-called dark components of the Universe: dark energy and dark matter. The former is required to explain the accelerated expansion of the Universe, and accounts for about of the energy content of the Universe; the latter is hinted, for instance, by the flattening of galactic rotation curves and cluster dynamics bullet , and constitutes about of the Universe’s energy budget. Several theories have been put forward to address these issues, usually resorting to the introduction of new fields; for dark energy, the so-called “quintessence” models consider the slow-roll down of a scalar field, thus inducing the observed accelerated expansion scalar ; Amendola . For dark matter, several weak-interacting particles (WIMPs) have been suggested, many arising from extensions to the Standard Model (e.g. axions, neutralinos). A scalar field can also account for an unified model of dark energy and dark matter Rosenfeld . Alternatively, one can implement this unification through an exotic equation of state, such as the generalized Chaplygin gas Chaplygin . Other approaches consider that these observational challenges do not demand the inclusion of extra energy content in the Universe but, instead, they hint at an incompleteness of the fundamental laws and tenets of General Relativity (GR). Following this line of reasoning, one may resort e.g. to extensions of the Friedmann equation to include higher order terms in the energy density have been suggested (see e.g. Maartens and references therein). Another approach considers changes on the fundamental action functional: a rather straight forward approach lies in replacing the linear scalar curvature term in the Einstein-Hilbert action by a function of the scalar curvature, ; alternatively, one could resort to other scalar invariants of the theory f(R) (see paramos and references therein for a discussion). As with several other theories solar1 ; solar2 , solar system tests could shed some light onto the possible form and behaviour of these theories; amongst other considerations, this approach is based either in the more usual metric affine connection, or in the so-called Palatini approach Palatini , where both the metric and the affine connection are taken as independent variables. As an example of a phenomenological consequence of this extension of GR, it has been shown that theories yield a gravitational potential which displays an increasing, repulsive contribution, besides the Newtonian term flat . Another line of action lies in the comparison between present and future observational signatures and the parameterized post-Newtonian (PPN) metric coefficients arising from this extension of GR, taken in the weak field limit and when the added degree of freedom may be characterized by a light scalar field solar2 . Regarding this, some disagreement exists in the community, some arguing that no significant changes are predicted at a post-Newtonian level (see e.g. PPN and references therein); others defending that theories yield the PPN parameter , which is clearly disallowed by the current experimental constraint Cassini . This result first arose from the equivalence of the theory with a scalar field model analogy , which led to criticism from several fronts PPN2 ; however, a later study implied that the result could be obtained directly from the original theory chiba (see Ref. Faraoni for a follow-up and criticism). Despite the significant literature on these models, another interesting possibility has been neglected until recent times: including not only a non-minimal scalar curvature term in the Einstein-Hilbert Lagrangian density, but also a non-minimum coupling between the scalar curvature and the matter Lagrangian density; indeed, these are only implicitly related in the action functional, since one expects that covariantly invariant terms in should be constructed by contraction with the metric (e.g. the kinetic term of a real scalar field, ). In regions where the curvature is high (which, in GR, are related to regions of high energy density or pressure), the implications of such theory could deviate considerably from those predicted by Einstein’s theory Lobo . Related proposals have been put forward previously to address the problem of the accelerated expansion of the Universe expansion and the existence of a cosmological constant cosmological . Other studies have studied the behaviour of matter, namely changes to geodetic behaviour Lobo , the possibility of modelling dark matter dark matter , the violation of the highly constrained equivalence principle equivalence 222The connection between the equivalence principle and the interaction between dark energy and dark matter has been discussed in Ref. equivalence2 . and the effect on the hydrostatic equilibrium of spherical bodies such as the Sun; also, a viability criterion for these generalized theories has been obtained viability . In this study, we focus on the equivalence of a theory displaying a non-minimal coupling of the scalar curvature with matter with a scalar-tensor theory; through a conformal transformation of the metric, this yields a purely scalar theory, that is, one in which the curvature term appears isolated from any scalar field contribution. For definitiveness, we recast the theory in a form that is as consistent as possible with the work of Ref. damour , and in close analogy with the available equivalence with models analogy . This work is divided into the following sections: first, we introduce the gravity model and discuss some of its features; then, we recast it as a scalar-tensor theory with a suitable dynamical identification of the scalar fields, and then as a scalar theory with a conformally related metric and redefined scalar fields. The later prompts for a computation of the PPN parameters and , which is followed by a discussion of our results. Ii The model Following the discussion of the previous section, one considers the following action Lobo , where , (with ) are arbitrary functions of the scalar curvature , is the Lagrangian density of matter and is the metric determinant; the metric signature is . The standard Einstein-Hilbert action is recovered by taking and , and is the cosmological constant (from now on, one works in a unit system where ). Variation with respect to the metric yields the modified Einstein equations of motion, here arranged as where one defines for convenience, as well as , omitting the argument. The matter energy-momentum tensor is, as usually, defined by The Bianchi identities, together with the identity , imply the non-(covariant) conservation law and, as expected, in the limit , one recovers the conservation law . Since the energy-momentum tensor is not covariantly conserved, one concludes that the motion of matter distribution characterized by a Lagrangian density is non-geodesical. This, of course, may yield a violation of the Equivalence principle, if the right-hand side of Eq. (4) differs between distinguishable matter distributions, which could be used to experimentally test the model and obtain constraints on . ii.1 Equivalence with scalar-tensor theory As in usual models, one may rewrite the considered mixed curvature model as a scalar-tensor theory. One wishes to establish an equivalent action in the Jordan frame (where the scalar curvature appears coupled linearly to a function of scalar fields); this may be obtained in a similar fashion to usual models (with ), so that the equations of motion derived from the action functional coincide with those derived directly from the action Eq. (1). However, one cannot simply consider an action of the form The equivalence with the action (1) must stem from the equations of motion obtained through the variation of with respect to the non-dynamical scalar field ; through a suitable definition of the functions , and the potential , the variation of the action (5) should yield the dynamic identification , so that both actions are equal, that is From action (5), the equation of motion for reads where the prime denotes differentiation with respect to . Integrating, one gets which then yields an action without matter content, indicating that one can resort to just one scalar field only when dealing with pure models (that is, with ). This pathology has already been pointed out in Ref. viability , and in a following study Sotiriou it is shown that one must resort to a second auxiliary real scalar field . In this approach, the authors chose an action of the form so that the second scalar field acts as a Lagrange multiplier, enforcing the identification ; however, variation of the above with respect fo yields the relation This differs from the unobtainable form of Eq. (5) in the dependence of on both the scalar field and the Lagrangian matter density. As in usual models, a suitable conformal transformation to the metric could be used to transform the above action of Eq. (13) into a functional where the scalar curvature appears decoupled from other fields. However, the presence of the Lagrangian density in this transformation would render any comparison with standard scalar theories too complicated (including the extraction of the PPN parameters and ); for this reason, the second scalar field will be kept for the remainder of this study. Notice also that one also could opt for the equivalence with a theory with just one scalar field, by taking the action However, the latter is not written in the Jordan frame, as the scalar curvature appears in a non-linear fashion. ii.1.2 Two-field scalar-tensor model where and are scalar fields and one defines As discussed in the previous section, variation of the metric with respect to both scalar fields yields the dynamical equivalence and . Substituting into Eq. (15), one recovers the action for the mixed curvature model, Eq. (1). With the above taken into consideration, one should notice that the second scalar field is not a function of the curvature (or ) alone, but also of the matter Lagrangian (which is itself a scalar). The degrees of freedom of the matter fields which appear in the Lagrangian (including any kinetic terms) are displayed in the Einstein field equations, through the corresponding energy-momentum tensor : which, introducing and , recovers Eq. (2). Using the Bianchi identities and the previous relation, one obtains which, upon the substitution, and , collapses back to Eq. (4). Iii Equivalence with a scalar theory One may now perform a conformal transformation (see e.g. Ref. conformal ), so that the curvature appears decoupled from the scalar fields , (yielding the action in the so-called Einstein frame), by writing , with , one obtains where denotes the D’Alembertian operator, defined from the metric . From the definition of the energy-momentum tensor, this implies that . Introducing the above into the action (15) yields where one defines Notice that there are two couplings between the scalar fields , and matter: the explicit coupling given by the factor , and an additional coupling due to the rewriting of the metric (in the Jordan frame) in terms of the new metric . One now attempts to recast the action in terms of two other scalar fields, endowed with a canonical kinetic term. For this, one first integrates the covariant derivative term by parts and uses the metric compatibility relations, obtaining By resorting to the divergence theorem, the first integral may be dropped, yielding One obtains the action One may redefine the two scalar fields, so that their kinetic terms may be recast in the canonical way. Specifically, one aims at writing (see Ref. damour ): with ; is the metric of the two-dimensional space of scalar fields (field-space metric, for short), and , the two new scalar fields. Clearly, this kinetic term prompts for the identification and the field metric which indicates that only has a kinetic term. Despite this, it is clear that is a distinct degree of freedom, since one cannot rewrite the potential in terms of one scalar field alone: In the trivial case or , one gets , that is, , so that one degree of freedom is lost and this potential may be written as a function of just one the fields. Since the inverse field metric will be required to raise latin indexes throughout the text, one still has do deal with the particular form of , which is non-invertible. In order to cope with this caveat, one is free to add an antisimmetric part, rewriting it as Clearly, no physical results can depend on the value that shows in the off-diagonal part of – in particular, the values of the PPN parameters and , as shall be shown. With this choice, the action now reads where one defines , with . By varying the action with respect to the metric , one obtains Defining and , so that one obtains, after some algebra so that the Euler-Lagrange equation for each field read where one defines . Alternatively, one may use the field-space metric to raise and lower latin indexes, so that and , rewriting Eq. (36) as and Eq. (III) becomes This form enables a prompt comparison with the limit, yielding the usual expression PPN for just one scalar field, (with and ). Using the Bianchi identities, the expression for the non-covariant conservation of the energy-momentum tensor in the Einstein frame is also attained: If, for consistency, one rewrites this in the Jordan frame, the conformal transformation properties of the contravariant derivative eliminate the first term in the r.h.s., and the substitution yields Eq. (4). Again, taking one recovers (or, in the Jordan frame, the covariant conservation law ). Iv Parameterized Post-Newtonian formalism Assuming that the effect of the non-minimum coupling of curvature to matter is perturbative, one may write , with , so that the current bounds on the equivalence principle are respected. Substituting into Eq. (40) one gets, at zeroth-order in , which amounts to ignoring the factor in the action (III), so that manifests itself only through the coupling present in , and the derivative of (since and ). In this case, the matter action reads If there is no characteristic length ruling the added gravitational interaction (that is, both scalar fields are light, leading to long range interactions), this manifestation of a “physical metric” in the matter action allows one to resort to calculate the PPN parameters and damour : where and ; the subscript indicates that the quantities should be evaluated at their asymptotic values , related with the cosmological values of the curvature and matter Lagrangian density (as shall be further discussed in the following section). Since the ’s are constant for , one gets that ; hence, the PPN parameter is unitary. Moreover, since is orthogonal to , one gets . Hence, the PPN parameter is also unitary. As argued previously, these results do not reflect the particular value of chosen in the antisymmetric part of . In light of the overall discussion presented, it is clear that the two degrees of freedom embodied in the two independent scalar fields and stem not only from the non-minimal curvature term in action (1), but also of the non-minimal coupling of to the matter Lagrangian density. Given this, it is clear that a “natural” choice for the two scalar fields (in the Jordan frame) would be and : this more physical interpretation ab initio comes at the cost of more evolved calculations, since both redefined scalar fields in the Einstein frame will depend on and . This less pedagogical approach is deferred to Appendix A. V Discussion and Conclusions The result is key to our study: it is clear that, in the standard theories, does not vanish (it is a purely algebraic, not matricial result and ), and the resulting PPN parameter , which violates well-known observational bounds! A more thorough discussion on the ongoing debate concerning the value of the PPN parameter for theories is deferred to Appendix B – with special focus to what is believed to be a misconception in the identification of the equivalence with a scalar-tensor theory. In the case, the added degree of freedom that a non-minimal coupling of curvature to matter implies yields not one, but two scalar fields: as a result, a two-dimensional field-space metric arises; from the redefinition of the fields necessary to absorb non-canonical kinetic terms after the conformal transformation to the Einstein frame, it follows that this enables a vanishing term, yielding no post-Newtonian observational signature that discriminates these models from General Relativity. However, this conclusion is valid only in zeroth-order in : if more terms are allowed, the non-covariant conservation law for the energy-momentum tensor is no longer of the form treated in Ref. damour , and more elaborate calculations would have to be performed in order to extract the PPN parameters and . It is important to highlight that a naive analysis of the model under scrutiny might predict no difference between the PPN parameters arising from the trivial and the non-minimal cases: indeed, since this function is coupled to the matter Lagrangian density one might expect that, outside of the matter distribution (), the theory would collapse back to the usual scenario. However, this misinterpretation is resolved by Eq. (43), which is evaluated at the cosmological values : for this reason, the relevant value for is given by the Lagrangian density of the overall cosmological fluid, not of the local environment. The issue of suitably identificating this contribution is discussed in Ref. BLP . Finally, note that the result is approximate, since it corresponds to dropping the term in action (III): a future work should consider a perturbative approach, which could perhaps yield a small, -dependent deviation from unity, thus marking a clear (even if small) departure between the model studied here and General Relativity. Appendix A Alternative formulation of the two-field equivalent scalar-tensor theory We present here an alternative formulation where one chooses instead to express action (1) through the equivalent expression where one defines One obtains the equivalence with the model under scrutiny by writing the equations of motion for the scalar fields, It should be stressed that the GR limit and disables the identification : one may write and , so that Eqs. (46) become and taking the limit gives a trivial identity. For this reason, it is misleading to insert the above approximations in results that stem from the scalar field approach to theories, and only then consider the GR limit: the formalism itself breaks down at its inception. For this reason, one concludes that one cannot simply take the limit and argue that, as a theory collapses back to GR, so should the PPN parameter approach unity, which does not happen if and does not show a dependence on . Variation of the action (44) with respect to the metric yield the Einstein equations, which, introducing and , recovers Eq. (2). Using the Bianchi identities and the previous relation, one obtains which, upon the substitution and , collapses back into Eq. (4). a.1 Equivalence with a scalar theory In this alternative formulation, the adequate conformal transformation to decouple the scalar curvature from the scalar fields , is given by , with . One obtains Introducing the above into the action (44) yields where one defines . One now attempts to recast the action in terms of two other scalar fields, endowed with canonical kinetic term. For this, one repeats the integration of the covariant derivative term by parts and uses the metric compatibility relations, finally obtaining the action As one aims to allow for an immediate comparison with the scenario, it is interesting to isolate the contributions arising from the non-minimal scalar curvature term and from its coupling with matter as clearly as possible, namely assuming that . If one defines then the comparison is transparent: when , the second scalar field vanishes, and the first scalar field coincides with the usual redefined scalar field arising in models (although in many studies some terms are overlooked, see Appendix B). Also, when , the first scalar field vanishes. For the particular choice of fields Eqs. (57), one obtains cross-products between the derivatives of and , so that the field-space metric displays non-vanishing off-diagonal elements; since one has absorved the numerical factors in the redifinition, this field-space metric is trivially given by for . However, this field-space metric is not invertible, a problem already found in the main body of this work; as before, this issue may be surpassed by adding an antisymmetric part to this field metric, adopting instead the form As already discussed, the physical results will not depend on the value . With this choice, the action now reads where one defines as before, but using the new redefined fields ; notice that the non-minimal coupling is written in terms of alone. The Einstein field equations are and variation of the matter action with respect to each scalar field yields, after some algebra where, as before, one defines and, given definitions Eqs. (57), The Euler-Lagrange equation for each field reads In the limit, one recovers , as before. Using the Bianchi identities, the expression for the non-covariant conservation of the energy-momentum tensor is

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By calculating the derivative of your position vector within B, you are determining how quickly this vector changes direction and=or magnitude as seen from the B system. You would find the time variation of your position to be zero when viewed from the Earth-fixed frame. This should be no big surprise; after all, you are standing still and not walking around on Earth. Analytical Mechanics of Space Systems by H. Schaub, J. Junkins

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Q=mass × specific heat capacity x temperature is the formula, temperature cannot be removed from the equation to calculate specific heat, start by reading the. Part of ncssm core collection: this video shows the collection of data to determine the specific heat of a metal please attribute t. The metal cooling off could change the heat capacity because the temperature wasn't kept constant and would most likely lower the calculated value of the specific heat capacity the calorimeter losing heat as a result of stirring would most likely raise the calculated value of the specific heat capacity. A heat exchange experiment is required after which you plug your data into q = mcdeltat you can not calculate specific heat capacity from such data as atomic number, atomic weight, density, magnetic susceptibility, vicker's hardness etc. Specific heat capacity table substance: specific heat capacity at 25 o c in j/g o c: h 2 gas: 14267: he gas: 5300: h 2 o (l): 4184: lithium: 356. Specific heat of metal alloys like brass, bronze and more sponsored links the specific heat capacity of some common metal alloys are indicated in the table below. Is there a correlation between material density and specific heat why low density materials have a higher specific heat capacities, and vice-versa the specific heat of 152sm metal between 0. The ability of a substance to store heat is one way of defining the heat capacity or specific heat of a substance] table 1 lists the specific heats of six metals. Calorimetry lab - specific heat capacity introduction experience tells us that if a hot piece of metal is added to water, the temperature of the water will rise. The specific heat capacity of a metal (or any other substance) is the amount of heat energy required to increase the temperature of 1 kilogram of the metal by 1 degree celsius this typically is measured in joules per kilogram per degree celsius depending on the context, either degrees celsius or. Assuming all the heat lost by the water is gained by the metal and that the cup is perfectly insulated, determine the specific heat capacity of the unknown metal the specific heat capacity of water is 418 j/g/°c. Engineering thermal properties of metals, conductivity, thermal expansion, specific heat data - metals in general have high electrical conductivity, high thermal conductivity, and high density. Heat capacity and specific heat capacity if the final temperature of the mixture is 25 o c, calculate the specific heat capacity (c) of the metal of the block. High precision heat capacity measurements of metals by modulated accurate measurement of specific heat capacity (cp) by differential scanning it has been. Specific heat capacity questions and equation all about specific heat capacity questions and equation physics notes measuring the specific heat capacity of a metal. It is there so you notice the difference between heat capacity and specific heat capacity problem #3: a 432 g block of an unknown metal at 890 °c was dropped into an insulated vesssel containing 4300 g of ice and 2600 g of water at 0 °c. Specific heats of metals page content we know that different materials have the ability to store different amounts of heat, given their temperature changes, masses and material. The experimental determination of the specific heat of a metal by the method of mixtures consists in dropping a known mass of the metal at a known high temperature into a known mass of water at a known low temperature. The formula for specific heat capacity is q=mcδt q stands for heat, usually given in joules m is the mass of the given substance c is the specific heat capacity of that substance, and δt is the change in temperature (initial temperature minus final temperature) in degrees celsius. For solids this gives a value of joules/mole/degree in fact, at room temperature most solids (in particular, metals) have heat capacities which lie remarkably close to this value. Materials specific heat capacity of metals table chart engineering materials specific heat capacity of metals table chart the specific heat is the amount of heat enregy per unit mass required to raise the temperature by one degree celsius. Heat capacity of minerals: a hands-on introduction to chemical thermodynamics multiplying the specific heat capacities ofeach metal by their densities you should. This specific heat calculator is a tool that determines the heat capacity of a heated or a cooled sample specific heat is just the amount of thermal energy you need to supply to a sample weighing 1 kg to increase its temperature by 1 k read on to learn how to apply the heat capacity formula. Purpose the purpose of this lab experiment is to measure the specific heat capacity of unknown metal samples and also to determine the latent heat of fusion of water in addition, we will study the effectiveness of different calorimet. - investigating the heat capacity of metals aim: my aim is to measure the specific heat capacity of 4 metals and find out if they all have the same specific heat capacity or different prediction: the specific heat capacity is the amount of heat energy needed to raise 1kg by 1 c and is measured in joules. Specific heat capacity j/(mol ) j/(g ) 1 h table 63, enthalpies and gibbs energies of formation, entropies, and heat capacities of the elements and. To measure the specific heat capacity of a metal (solid block method) this method is suitable for a metal which is a good thermal conductor, eg, copper or aluminium. The molar heat capacity is the heat capacity per unit amount (si unit: mole) of a pure substance and the specific heat capacity, often simply called specific heat, is the heat capacity per unit mass of a material occasionally, in engineering contexts, the volumetric heat capacity is used.

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As it is already known, the pattern of SBI PO has undergone a change this year. The prelims exam is going to be followed by a mains exam. The mains exam will have an objective test worth 200 marks. In addition to the objective test, a descriptive test of 50 marks will also be there. The objective test will be on the same lines as the last year. You will be tested on four sections of 50 marks each: (I) English (II) Data Analysis & Interpretation (III) Reasoning (High Level) (IV) General Awareness, Marketing & Computers. However, over here, it is important to note that SBI clearly mentions in its recruitment notification that the Reasoning section will have a high level of difficulty. Quite often students who are going to attempt SBI PO are in dark regarding how they should go about preparing for this section. In order to guide them in the right direction, Vidya Guru, the best Bank Coaching Center in Delhi, has come up with an analysis of the reasoning section as seen in 2014 exam. SBI PO Reasoning 2014 Analysis - The section can be divided into around 10 topics - Non-verbal reasoning, also known as visual reasoning, was there in 2013 but not in 2014. - Questions on Direction test were not there in 2013, but they were included in 2014. - Input-Output, Seating Arrangement and Mathematical Inequations (Inequalities) are the 3 most scoring areas. - Questions on Data Sufficiency are also present in the Reasoning section. Topic Wise Analysis: 1. Puzzle Test – A tabular / grid approach is recommended to solve these questions. The information given in the question generally has 3 categories / variables which have to be arranged systematically to prepare the table. Out of the 3 categories / variables, one each is shown on X axis and Y axis of the table respectively. The third variable (the person) is then fitted into the table after carefully looking at the conditions provided in the question. 2. Direction Test – At the start of any direction test question, you must make sure that you get a sense of where you will point the North, East, West and South directions. Once it is done, you need to understand how directions are changing as the question progresses. This should be done by assuming that you yourself are following the path mentioned in the question and are covering the distance given from initial to the final point. Quite often questions are also based on calculating the distance travelled between these two points. In such a case, remember to calculate the shortest distance (straight line distance) between the points. Pythagoras theorem can be used to calculate this distance. 3. Blood Relations – These questions can be best attempted by preparing a family tree. Family tree is a diagrammatic structure demonstrating the relationships within the family. The relationships are shown by drawing arrow like lines to connect the members of the family with one another. Over here, it is also critical to point out the male and female members of the family. Finally when it comes to answering the question, make sure you clearly understand from whose perspective the relationship needs to be determined. The answer may turn out to be different, if the reference point is changed. Let’s say we have the following information: A is the father of B who is a male member of the family. So, if you have the question: How is B related to A? The answer will be: SON. 4. Input – Output – From this topic, generally 3 types of questions are framed in the exam: (I) Questions based on numerical operations (II) Questions based on alphabetical / verbal operations (III) Questions based on alpha-numeric operations. One aspect is common in all these questions: the processing, which the input goes through to give the final output, has to be identified. Out here, the first thing you need to do is compare the input with the final output (last step). A number of intermediate steps are given that show the sequence in which processing happens from the start to the end. Once the step-wise processing has been understood the same can be applied on the given input to obtain the desired output. 5. Seating Arrangement – Questions based on this topic can be arranged into 3 categories: Circular (facing the centre, facing outside of the centre & a combination of both), Linear (single row arrangement & multiple row arrangement) and Others (square, hexagon and octagon etc.). CIRCULAR: At first, assume yourself to be a part of this seating arrangement so that you can easily decide where you are seated in respect to others and vice versa. Since, it is a circular pattern, you can begin allotting positions from the first statement itself. For example, the first statement is: A and B are opposite to each other. So, you can start like this: Now, you can move ahead and keep allotting further positions as per the given information. At times, the seating arrangement may not even get completed. You need not worry about it as the question may have been designed that way. LINEAR: Over here, you need to start from the person whose position has been clearly defined. Once it is done, the information from subsequent statements can be used in reference to the position of this person. The information which is not clear has to be withheld till you are sure about how it will figure out in the arrangement. MIXED: Since these are closed figures and follow a cyclical arrangement, questions based on them can be solved in a manner similar to that of a circular arrangement. 6. Mathematical Inequations (Inequalities) – There are five inequality symbols that we deal with in these questions: Greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤) and equal to (=). These inequalities are, at times, encoded using symbols such as @, #, $, * etc. The inequality symbols given should be arranged in a sequential manner to as much extent as possible. Towards the end, a conclusion will be drawn on the basis of this arrangement. 7. Critical Reasoning – Critical reasoning questions require the application of verbal logic. These questions can be based on any of the following areas (I) Assumptions (II) Inferences or Conclusions (III) Cause & Effect (IV) Course of Action (V) Strengthening the Argument & Weakening the Argument (VI) Degree of Truth or Falsity (Probably True / Definitely True / Probably False / Definitely False). To solve such questions, you have to read between the lines and logically analyze the meaning of the information provided in the statements. 8. Syllogisms / Venn Diagrams – Inferential reasoning is an integral part of any bank exam. Syllogisms or venn diagrams is the most common type of questions based on it. In such questions, certain statements are there and you have to decide which conclusions can be drawn on the basis of these statements; you must be able to find out which conclusions are valid and which are not. The statements can primarily be of 4 types: (I) Some As are Bs (II) Some As are not Bs (III) All As are Bs (IV) No A is B. The first thing you need to do is go through the statement and draw the relevant venn diagrams. Once it has been done, the possibilities should be neglected as they can’t be a part of the conclusion. Only certainties can be there in the conclusion. 9. Data Sufficiency – In these questions information is given in the form of separate statements (two statements or three statements). Your job is to analyze how much information is needed to answer the question. As the name suggests, it is about finding out whether the data given is sufficient or not. At times, one statement alone is sufficient to answer the question. However, sometimes it is a combination of two or more statements which is required to answer the question. All such possibilities are given in the form of options and you have to choose the most appropriate option as your answer. 10. Coding – Decoding – There are primarily 4 types of questions which are there from this topic: (I) Alphabet to Alphabet coding (II) Alphabet to Number coding (III) Alphabet to Symbol coding (IV) Statement Coding. The logic which has been used to encode the information has to be decoded. Once you have understood the logic behind the encoding, it can be used to encode or decode the given problem. Finally, the methodology of approaching the above mentioned questions can be better understood by enrolling yourself at an institute for Bank PO Coaching or Bank clerk coaching. Doing so will give you much greater clarity as to how you can handle even the trickiest aspects of the reasoning section. This article presents a detailed analysis of the Reasoning section which you can expect to see in SBI Probationary Officer exam. To learn further by getting in touch with the experts who have done this analysis, you can write to email@example.com.

Questions 1. Pat purchased a used five-year class asset on March 15, 2009, for $60,000. He did not elect § 179 expensing. Determine the cost recovery deduction for 2009 for earnings and profits purposes. A) $2,000. B) $3,000. C) $6,000. D) $12,000. E) None of the above. 2. Bill is the regional manager for a national chain of auto-parts stores and is based in Salt Lake City. When the company opens new stores in Boise, Bill is given the task of supervising their initial operation. For three months, he works weekdays in Boise and returns home on weekends. He spends $410 returning to Salt Lake City but would have spent $390 had he stayed in Boise for the weekend. As to the weekend trips, how much, if any, qualifies as a deduction? A) $0, since the trips are personal and not work related. B) $0, since Bill’s tax home has changed from Salt Lake City to Boise. C) $390. D) $410. E) None of the above. 3. If a taxpayer has tentative AMT of $60,000 and AMT of $15,000, what is the regular income tax liability? A) $0. B) $15,000. C) $45,000. D) $75,000. E) None of the above. 4. Phil is advised by his family physician that his dependent son, Tony, needs surgery for a benign tumor in his leg. Phil and his son travel to Rochester, Minnesota, for in-patient treatment at the Mayo Clinic, which specializes in this type of surgery. Phil incurred the following costs: Round-trip airfare ($375 each) $ 750 Phil’s hotel in Rochester for four nights ($105 per night) 420 Phil’s meals while in Rochester 150 Tony’s medical treatment 1,500 Tony’s prescription medicine 300 Compute Phil’s medical expenses for the trip (before the 7.5% floor). A) $2,550. B) $2,750. C) $2,970. D) $3,120. E) None of the above. 5. Amy works as an auditor for a large major CPA firm. During the months of September thru and November of each year, she is permanently assigned to the team auditing Garnet Corporation. As a result, every day she drives from her home to Garnet and returns home after work. Mileage is as follows: Miles Home to office 10 Home to Garnet 30 Office to Garnet 35 For these three months, Amy’s deductible mileage for each workday is: A) 0. B) 30. C) 35. D) 60. E) None of the above. 6. Erin owns a mineral property that had a basis of $10,000 at the beginning of the year. The property qualifies for a 15% depletion rate. Gross income from the property was $120,000 and net income before the percentage depletion deduction was $50,000. What is Erin’s tax preference for excess depletion? A) $8,000 B) $10,000. C) $18,000. D) $0. E) None of the above. 7. Byron owned stock in Blossom Corporation that he donated to a museum (a qualified charitable organization) on June 8 this year. What is the amount of Byron’s deduction assuming that he had purchased the stock for $10,500 last year on August 7, and the stock had a fair market value of $13,800 when he made the donation? A) $3,300. B) $10,500. C) $12,150. D) $13,800. E) None of the above. 8. Michael is the city sales manager for “Chick-Stick,” a national fast food franchise. Every working day, Michael drives his car as follows: Miles Home to office 20 Office to Chick-Stick No. 1 15 Chick-Stick No. 1 to No. 2 18 Chick-Stick No. 2 to No. 3 14 Chick-Stick No. 3 to home 30 Michael’s deductible mileage is: A) 0 miles. B) 30 miles. C) 47 miles. D) 77 miles. E) None of the above. 9. For regular income tax purposes, Yolanda, who is single, is in the 35% tax bracket. Her AMT base is $220,000. Her tentative AMT is: A) $57,200. B) $58,100. C) $61,600. D) $77,000. E) None of the above. 10. During its first year of operations, Sherry’s business incurred circulation expenditures of $90,000. Since the income of the business is small, Sherry decides to capitalize the expenditures and to amortize them over 3 years for regular income tax purposes. The AMT adjustment for circulation expenditures for the first year of operations is: A) $0. B) Negative adjustment of $30,000. C) Positive adjustment of $30,000. D) Positive adjustment of $60,000. E) None of the above. 11. On September 3, 2008, Able purchased § 1244 stock in Red Corporation for $6,000. On December 31, 2008, the stock was worth $8,500. On August 15, 2009, Able was notified that the stock was worthless. How should Able report this item on his 2008 and 2009 tax returns? A) 2008—$0; 2009—$6,000 ordinary loss. B) 2008—$0; 2009—$6,000 long-term capital loss. C) 2008—$2,500 short-term capital loss; 2009—$8,500 short-term capital loss. D) 2008—$2,500 short-term capital gain; 2009—$3,800 ordinary loss. E) None of the above. 12. Leigh, who owns a 50% interest in a sporting goods store, was a material participant in the activity for the last fifteen years. She retired from the sporting goods store at the end of last year and will not participate in the activity in the future. However, she continues to be a material participant in an office supply store in which she is a 50% partner. The operations of the sporting goods store resulted in a loss for the current year and Leigh’s share of the loss is $40,000. Leigh’s share of the income from the office supply store is $75,000. She does not own interests in any other activities. A) Leigh cannot deduct the $40,000 loss from the sporting goods store because she is not a material participant. B) Leigh can offset the $40,000 loss from the sporting goods store against the $75,000 of income from the office supply store. C) Leigh will not be able to deduct any losses from the sporting goods store until future years. D) Leigh will not be able to deduct any losses from the sporting goods store until she has been retired for at least four years. E) None of the above. 13. Diane purchased a factory building on November 15, 1993, for $5,000,000. She sells the factory building on February 2, 2009. Determine the cost recovery deduction for the year of the sale. A) $16,025. B) $19,844. C) $26,458. D) $158,750. E) None of the above. 14. In 2009, Ray incurs $60,000 of mining exploration expenditures, and deducts the entire amount for regular income tax purposes. Which of the following statements is incorrect? A) For AMT purposes, Ray will have a positive adjustment of $54,000 in 2009. B) Ray will have a negative AMT adjustment of $6,000 in 2014. C) Over a 10-year period, positive and negative adjustments will net to zero. D) The AMT adjustment for mining exploration expenditures cannot be avoided if the taxpayer elects to write the expenditures off over a 10-year period. E) All of the above are correct. 15. On May 15, 2009, Brent purchased new farm equipment for $40,000. Brent used the equipment in connection with his farming business. Brent does not elect to expense assets under § 179. Brent does elect not to take additional first-year depreciation. Determine the cost recovery deduction for 2009. A) $4,000. B) $6,000. C) $8,000. D) $10,000. E) None of the above. 16. On June 1, 2009, Irene places in service a new automobile that cost $21,000. The car is used 70% for business and 30% for personal use. (Assume this percentage is maintained for the life of the car.) She does elect not to take additional first-year depreciation. Determine the cost recovery deduction for 2010. A) $3,060. B) $3,290. C) $3,360. D) $6,720. E) None of the above. 17. In Shelby County, the real property tax year is the calendar year. The real property tax becomes a personal liability of the owner of real property on January 1 in the current real property tax year, 2009. The tax is payable on June 1, 2009. On April 30, 2009, Julio sells his house to Anita for $230,000. On June 1, 2009, Anita pays the entire real estate tax of $7,300 for the year ending December 31, 2009. How much of the property taxes may Julio deduct? A) $0. B) $2,380. C) $2,400. D) $4,920 E) None of the above. 18. Kim made a gift to Sam of a passive activity (adjusted basis of $50,000, suspended losses of $20,000, and a fair market value of $80,000). No gift tax resulted from the transfer. A) Sam’s adjusted basis is $80,000. B) Sam’s adjusted basis is $50,000, and Sam can deduct the $20,000 of suspended losses in the future. C) Sam’s adjusted basis is $80,000, and Sam can deduct the $20,000 of suspended losses in the future. D) Sam’s adjusted basis is $50,000, and the suspended losses are lost. E) None of the above. 19. Elizabeth has the following items for the current year: Nonbusiness capital gains $ 5,000 Nonbusiness capital losses (3,000) Interest income 3,000 Itemized deductions (including a $20,000 casualty loss) (27,000) In calculating Elizabeth’s net operating loss, and with respect to the above amounts only, what amount must be added back to taxable income (loss)? A) $0. B) $1,000. C) $2,000. D) $20,000. E) None of the above. 20. Janet purchased a new car on June 5, 2009, at a cost of $18,000. She used the car 80% for business and 20% for personal use in 2009. She used the automobile 40% for business and 60% for personal use in 2010. Determine Janet’s cost recovery recapture for 2010. A) $0. B) $928. C) $1,008. D) $1,440. E) None of the above. 21. Sandra is single and does a lot of business entertaining at home. Because Arthur, Sandra’s 80-year old dependent grandfather who lived with Sandra, needs medical and nursing care, he moved to Twilight Nursing Home. During the year, Sandra made the following payments on behalf of Arthur: Room at Twilight $4,500 Meals for Arthur at Twilight 850 Doctor and nurse fees 700 Cable TV service for Arthur’s room 107 Total $6,157 Twilight has medical staff in residence. Disregarding the 7.5% floor, how much, if any, of these expenses qualify for a medical deduction by Sandra? A) $6,157. B) $6,050. C) $5,200. D) $1,550. E) None of the above. 22. James purchased a new business asset (three-year property) on July 23, 2009, at a cost of $50,000. He did not elect to expense any of the asset under § 179, nor did he elect straight-line cost recovery. Determine the cost recovery deduction for 2009. A) $8,333. B) $16,665. C) $33,333. D) $41,665. E) None of the above. 23. Jim had a car accident in 2009 in which his car was completely destroyed. At the time of the accident, the car had a fair market value of $30,000 and an adjusted basis of $40,000. Jim used the car 100% of the time for personal use. Jim received an insurance recovery of 80% of the value of the car at the time of the accident. If Jim’s AGI for the year is $50,000, determine his deductible loss on the car. A) $500. B) $6,000. C) $10,500. D) $30,000. E) None of the above. 24. On June 1, 2009, Sam purchased new farm machinery for $50,000. Sam used the machinery in connection with his farming business. Sam does not elect to expense assets under § 179. Sam has, however, made an election to not have the uniform capitalization rules apply to the farming business. Sam does elect not to take additional first-year depreciation. Determine the cost recovery deduction for 2009. A) $5,000. B) $7,500. C) $10,000. D) $12,500. E) None of the above. 25. Agnes is able to reduce her regular income tax liability from $40,000 to $36,000 as the result of the alternative tax on net capital gain. Agnes’ tentative AMT is $50,000. A) Agnes’ tax liability is reduced by $4,000 as the result of the alternative tax calculation on net capital gain. B) Agnes’ AMT is increased by $4,000 as the result of the alternative tax calculation on net capital gain. C) Agnes’ tax liability is $46,000. D) Agnes’ tax liability is $54,000. E) None of the above. 26. On March 1, 2009, Lana leases and places in service a passenger automobile. The lease will run for five years and the payments are $900 per month. During 2009, she uses her car 20% for business and 80% for personal activities. Assuming the dollar amount from the IRS table is $233, determine Lana’s deduction for the lease payments. A) $0. B) $1,800. C) $2,000. D) $2,330. E) None of the above. 27. Doug purchased a new factory building on January 15, 1987, for $4,000,000. On March 1, 2009, the building was sold. Determine the cost recovery deduction for the year of the sale assuming he did not use the MACRS straight-line method. A) $0. B) $15,870. C) $26,450. D) $126,960. E) None of the above. 28. If a taxpayer has an NOL of $20,000, of which $8,000 is attributable to a theft of personal use property, the taxpayer may: A) Carry all of the NOL of $20,000 back 5 years. B) Carry all of the NOL of $20,000 back 3 years. C) Carry $8,000 of the NOL back 3 years and the remainder of the NOL of $12,000 back 2 years. D) All of the above. E) None of the above. 29. Hans purchased a new passenger automobile on August 17, 2009, for $40,000. During the year the car was used 40% for business and 60% for personal use. Determine his cost recovery deduction for the car for 2009. A) $500. B) $1,000. C) $1,184. D) $1,500. E) None of the above. 30. Norm’s car, which he uses 100% for personal purposes, was completely destroyed in an accident in 2009. The car’s adjusted basis at the time of the accident was $13,000. Its fair market value was $11,500. The car was covered by a $2,000 deductible insurance policy. Norm did not file a claim against the insurance policy because of a fear that reporting the accident would result in a substantial increase in his insurance rates. His adjusted gross income was $14,000 (before considering the loss). What is Norm’s deductible loss? A) $0. B) $100. C) $500. D) $9,500. E) None of the above. 31. White Corporation, a closely held personal service corporation, has $150,000 of passive losses, $120,000 of active business income, and $30,000 of portfolio income. How much of the passive loss may White Corporation deduct? A) $0. B) $30,000. C) $120,000. D) $150,000. E) None of the above. 32. On June 1 of the current year, Tab converted a machine to rental property. At the time of the conversion, the machine was worth $90,000. Five years ago Tab purchased the machine for $120,000. The machine is still encumbered by a $50,000 mortgage. What is the basis of the machine for cost recovery? A) $70,000. B) $90,000. C) $120,000. D) $140,000. E) None of the above. 33. George purchases used office furniture (seven-year class property) at a cost of $50,000 on April 20, 2009. Determine George’s cost recovery deduction for 2009 for alternative minimum tax purposes, assuming George does not elect § 179 and the maximum cost recovery deduction is taken for regular income tax purposes. A) $2,500. B) $3,750. C) $5,355. D) $6,212. E) None of the above. 34. Three years ago, Sharon loaned her sister $30,000 to buy a car. A note was issued for the loan with the provision for monthly payments of principal and interest. Last year, Sharon purchased a car from the same dealer, Hank’s Auto. As partial payment for the car, the dealer accepted the note from Sharon’s sister. At the time Sharon purchased the car, the note had a balance of $18,000. During the current year, Sharon’s sister died. Hank’s Auto was notified that no further payments on the note would be received. At the time of the notification, the note had a balance due of $15,500. What is the amount of loss, with respect to the note, that Hank’s Auto may claim on the current year tax return? A) $0. B) $3,000. C) $15,500. D) $18,000. E) None of the above. 35. Terry pays $8,000 this year to become a charter member of Mammoth University’s Athletic Council. The membership ensures that Terry will receive choice seating at all of Mammoth’s home basketball games. In addition, Terry pays $2,200 (the regular retail price) for season tickets for himself and his wife. For these items, how much qualifies as a charitable contribution? A) $6,200. B) $6,400. C) $8,000. D) $10,200. E) None of the above. 36. Pedro’s child attends a school operated by the church the family attends. Pedro made a donation of $1,000 to the church in lieu of the normal registration fee of $200. In addition, Pedro paid the regular tuition of $6,000 to the school. Based on this information, what is Pedro’s charitable contribution? A) $0. B) $800. C) $1,000. D) $6,800. E) $7,000. 37. Maria, who owns a 50% interest in a restaurant, has been a material participant in the restaurant activity for the last 20 years. She retired from the restaurant at the end of last year and will not participate in the restaurant activity in the future. However, she continues to be a material participant in a retail store in which she is a 50% partner. The restaurant operations produce a loss for the current year, and Maria’s share of the loss is $80,000. Her share of the income from the retail store is $150,000. She does not own interests in any other activities. A) Maria cannot deduct the $80,000 loss from the restaurant because she is not a material participant. B) Maria can offset the $80,000 loss against the $150,000 of income from the retail store. C) Maria will not be able to deduct any losses from the restaurant until she has been retired for at least three years. D) Assuming Maria continues to hold the interest in the restaurant, she will always treat the losses as active. E) None of the above. 38. Last year, Lucy purchased a $100,000 account receivable for $80,000. During the current year, Lucy collected $85,000 on the account. What are the tax consequences to Lucy associated with the collection of the account receivable? A) $0. B) $5,000 gain. C) $10,000 loss. D) $15,000 loss. E) None of the above. 39. Wes’s at-risk amount in a passive activity is $25,000 at the beginning of the current year. His current loss from the activity is $35,000 and he has no passive activity income. At the end of the current year, which of the following statements is incorrect? A) Wes has a loss of $25,000 suspended under the passive loss rules. B) Wes has an at-risk amount in the activity of $0. C) Wes has a loss of $10,000 suspended under the at-risk rules. D) Wes has a loss of $35,000 suspended under the passive loss rules. E) None of the above is incorrect. 40. Omar acquires used 7-year personal property for $100,000 to use in his business in February 2009. Omar does not elect § 179 expensing or additional first-year depreciation, but does take the maximum regular cost recovery deduction. As a result, Omar will have a positive AMT adjustment in 2009 of what amount? A) $0. B) $3,580. C) $10,710. D) $14,290. E) None of the above.

S/N : Signal-to-Noise ratio the difference between the nominal or maximum operating level and the noise floor in dB Sampling Frequency the rate at which a continuous waveform is digitised, given in Hz. Sampling Interval how long the data is measured at each sample point. Sampling Theorem says that ideally a signal should be sampled at a rate twice its highest frequency component. Scalar Quantity a quantity fully described by a magnitude or numerical value, for example density, mass and speed. As opposed to a vector quantity which has both magnitude and direction, for example acceleration, force and velocity Scalar Quantities may be added, subtracted or multiplied like ordinary numbers, Vector Quantities can not. Second : s the second, symbol s, is the name of the SI base unit of time. 0.001 s = 1 ms millisecond : 0.000001 s = 1 μs microsecond Seismic relating to earthquakes or other vibration in the earth Seismic Reflection the reflection of waves at boundaries between different rock formations Seismic Refraction the refraction of waves passing through formations of 'earth' having different seismic velocities Seismic Velocity the velocity of wave propagation in particular ground or rock formation Seismogram a record produced by a seismograph Seismograph A measuring instrument for detecting and measuring the intensity and direction and duration of movements of the ground (as in ground-borne vibration) - Certified Seismographs Serial Frequency Analysis the measurement of octave or third octave bands of noise where a single filter is stepped across the different bands one at a time, suitable mainly for steady noise signals only. Superseded, in the main, by real time analysis. Shock rapid transient transmission of mechanical energy. Shock Pulse IEC Definition, excitation of a system characterized by rise and fall in a time interval short in comparison with the half-period of any mode of oscillation of the system SI Units is the world's most widely system of units devised around the convenience of the number 10. There are 7 base units from which other units are derived and known as SI derived units.SI Unit prefixes : used together with a SI unit to form decimal multiples or submultiples of the unit Example 10-6 g = 1 μg = 1 microgram or one millionth of a gram. Signal-to-Noise ratio the difference between the nominal or maximum operating level and the noise floor in dB Single Event Noise Exposure Level : SENEL there are two variations of this term:- Both are similar on the sound exposure level : SEL but we believe the second was developed to take account of single events like aircraft noise where the Lmax is important but the duration should also be taken factored in. An event with a higher Lmax can have a lower SEL than a longer event. Single Number Rating : SNR a single number rating system for hearing protectors - BS EN ISO 4869 See also : Noise Reduction Rating - NNR used in the USA. Single Number Rating : SNR a rating system for duct silencers. Slow Time Weighting - also known as Slow Response and Slow Time Constant. SNR : see Single Number Rating above Sones a unit to compare the loudness of two sounds. Sound IEC Definition, movement of particles in an elastic medium about an equilibrium position Sound any pressure variation that the human ear can detect. Depending on the medium, sound extends and affects a greater area (propagates) at different speeds. In air, sound propagates at a speed of approximately 343 m/s. In liquids and solids, the propagation velocity is greater - 1480 m/s in water and 5120 m/s in steel, for example. Sound Analyser under Spectrum Analyser. Sound Flux under Sound Energy Flux Sound Propagation Coefficient IEC Definition, with respect to a uniform system, natural logarithm of the complex ratio of particle velocities (or pressures) measured at two successive points separated by unit distance, when this system is assumed to be of infinite length. Also known as Linear Exponent of Sound Propagation Sound Source IEC Definition, a simple sound source (monopole) radiates sound equally in all directions. A complex sound source is composed of various sources, multiple frequencies and directivity patterns. Point Sound Source IEC Definition, source that radiates sound as if from a single point. Sound Speed Gradient The Speed of Sound decreases with decreasing temperature and creates a negative sound speed gradient. An increase in temperature results in a positive sound speed gradient Sound Transmission passage of a sound wave through a medium or series of media. Sound Velocity Level under Particle Velocity Level Spatial Averaging taking measurements at various positions and averaging the results. Mandatory in Sound Insulation measurements and recommended anywhere multiple reflections are present. See also other types of Averaging Specific Acoustic Admittance IEC Definition, reciprocal of the Specific Acoustic Impedance Specific Acoustic Impedance : z the ratio of the effective sound pressure at a point in an acoustic medium to the effective particle velocity at that point. z = p/v the SI units are Pa·s/m3 or rayls in MKS units. Specific Noise noise from the source under investigation as defined in BS 4142 Method for rating industrial noise affecting mixed residential and industrial areas. The specific noise is compared to the Background Noise to assess the likelihood of complaints. Specific Volume : v the number of cubic metres occupied by one kilogram of the substance : m3/kg. Spectra is the plural of spectrum Spectral Density IEC Definition, limit as the bandwidth approaches zero, of the mean square value of a field quantity divided by bandwidth. The kind of field quantity must be specified, such as sound pressure, particle velocity, particle acceleration The Spectral Density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, per unit frequency, known as the power spectral density (PSD) of the signal and is commonly expressed in watts per Hertz (W/Hz). Spectral Density Limit as the bandwidth approaches zero, of the mean square value of a field quantity divided by bandwidth. See also : Energy Spectral Density. Spectral Lines the number of constant bandwidth lines used in the measurement of spectra. Spectrum the description of a sound wave's resolution into its components of frequency and amplitude. Spectrum Analyser an instrument to analyse a sound or vibration wave into it's frequency components. A spectrum analyser converts a signal from the time domain into the frequency domain,. The FFT, Octave and 1/3-octave analysers are the most common type today, but there are many other types. Spectrum Averaging a short term spectrum analysis may include information due to external sources, for example background noise. Repeating the measurements over a longer period and averaging the spectra will cause any random signals to be 'discarded' and your confidence in the measurement will improve. See also other types of Averaging Spectrum Density Level IEC Definition, level of the limit, as the width of the band approaches zero, of the ratio of a specified quantity distributed within a frequency band to the width of the band. ● Note 1 : the kind of quantity must be specified, such as by (squared) sound pressure spectrum level. ● Note 2 : in view of the fact that filters have finite bandwidths, practically the sound pressure spectrum level Lps is obtained for the centre frequency of the band by the formula: Lps = 10 log10 (p2/B) ÷ (po2/Bo) dB, where p and po are respectively the given field quantity and the reference quantity; B and Bo are respectively the effective bandwidth of the filter and the reference bandwidth of 1 Hz. When Lp is the band pressure level observed through the filter, the above relation reduces to Lps = Lp - Log10 (B/Bo) dB Speech - Articulation Index : AI Speech - Articulation Intelligibility Speech - Intelligibility Speech Interference Level : SIL Speech Interference Level : SIL3 Speech Transmission Index : STI and STIPA Speed of Sound : c ≈ 331.5 + 0.60 T(°C), at 20 °C, the speed of sound in air is approximately 343 m/s and the decrease of speed with temperature is referred to as a negative sound speed gradient. The speed of sound is also dependent, to a minor extent, on atmospheric pressure and relative humidity. Sound travels faster in liquids and solids. For example the speed of sound in water is 1,480 m/s and for iron 5,120 m/s, these values are also temperature dependent, also giving rise to Sound Speed Gradients. The Speed of Sound should not to be confused with the Sound Particle Velocity. Spreading Loss IEC Definition, that part of the transmission loss due to the divergence, i.e. spreading, of sound waves in accordance with the configuration of the system, also known as Divergence Loss. ● Note : Spreading Loss exists, for example, for spherical waves emitted by a point source. square metre : m2 : Area square root : √ the square root of a number is a smaller number that, when multiplied by itself equals the original number. Standard Atmospheric Pressure : atm atmospheric pressure is the force per unit area exerted on a surface by the weight of air above that surface. Standard Atmospheric Pressure is equal to 101.325 kPa the preferred SI units or 8760 mmHg and 1013.25 millibars. Standardized measurement in accordance with a Standard or 'Norm'. Stationary Signal a stationary signal is a signal whose average statistical properties over a time interval of interest are constant. In general, the vibration signatures of rotating machines are stationary. Statistical Analysis a calculation performed by a Sound Level Meter on the noise levels measured during the measurement period to describe the Statistical Levels Ln of the noise. Steady-state Oscillation IEC Definition, oscillation that continues without change Stimulus IEC Definition, external force, or other input, applied to a system Stochastic details of individual events may be unpredictable but the overall character of the sound is. For example rain falling, sound of insects, birds, etc. Strength of a Monopole and Simple Sound SourceIEC Definition, maximum instantaneous volume velocity produced by a simple source small compared with wavelength, emitting a wave with sinusoidal variation with time Structure-borne Noise a significant portion of the transmission path from source to receiver takes place in a solid structure rather than through the air. Subharmonic Response IEC Definition, periodic response of a system at a frequency that is a submultiple of the excitation frequency Superposition principle under Sound Waves. SVL : Sound Velocity Level under Particle Velocity Level Quality Symbols IEC Definition, character or combination of characters denoting a quantity. A simple quantity symbol is preferably one, or in some cases two, letters of the Latin or Greek alphabets and may include subscripts, superscripts, or other modifying signs. The letters are in italic (sloping) type, using preferably a font with serifs. The subscripts and superscripts are printed either in roman (upright) type, or, when they denote quantities, variables, or running numbers, in italic (sloping) type. See ISO 80000-1 and IEC 60027-1 for more details and for the combination of symbols. Synchronous Averaging see Time Domain Averaging. System of Units set of base units and derived units, together with their multiples and submultiples, defined in accordance with given rules, for a given system of quantities - see the SI units above

US 3986924 A A nuclear reactor of the type in which fissionable material is placed as fuel in the reactor core in such a manner as to provide the most effective utilization of the material, wherein a fuel assembly is divided into three concentric regions around the core center and is loaded with fuel in such a way that if, for convenience, the regions are named the first region, second region and third region, respectively, in the order of distance from the core center, the infinite multiplication factors in these regions will be such that the factor in the third region is smaller than that in the first region and that in the first region is smaller than that in the second region. 1. In a nuclear reactor having a core containing fissionable materials as fuel, the improvement which comprises a reactor core arrangement that is divided into three concentric regions of fuel assemblies including a first region, a second region, and a third region from the core center, said first region being positioned in the center of the core, said third region being positioned at the outer periphery of the core and said second region being positioned between said first and third regions, and the loading of fuel assemblies being such that said third region is smaller in its infinite multiplication factor than that of said first region and said first region is smaller in its infinite multiplication factor that that of said second region, and said three regions being of almost equal volume. 2. The reactor of claim 1 in which the second region contacts the first and third regions. 3. In a nuclear reactor having a core containing fissionable materials as fuel, the improvement which comprises a reactor core arrangement made up of three concentric regions of fuel assemblies, including a first region, a second region, and a third region from the core center, the loading of fuel assemblies being such that said first region, which is positioned in the center of the core, will be smaller in its infinite multiplication factor than that of said second region, such that said third region, which is positioned at the outer periphery of the core, will be smaller in its infinite multiplication factor than that of said first region, and such that said second region, which is positioned between said first and third regions, will be higher in its infinite multiplication factor to any of said first and third regions, the three regions being of almost equal volume. This invention relates to a nuclear reactor of the type in which fissionable material is placed in the reactor core as fuel in such a way as to provide most effective and economical utilization of the material. In general, operation of a nuclear reactor is carried out by controlling the excess reactivity of the fuel which is composed of fissionable materials loaded in the reactor core, by a neutron absorber mounted in the reactor. When the excess reactivity is reduced to zero, the reactor is temporarily shut down to perform refueling, that is, the used up fuel in the reactor core is removed and fresh fuel is charged. If need be, shuffling or rearranging of the used and fresh fuels may be performed in the reactor core to effectuate flattening of the power distribution. Generally, it is of essential requirements for the most economical operation of a nuclear reactor, supposing a certain required fuel enrichment was given, to realize, firstly, optimization of the refueling schedule and, secondarily, optimization of controlling by use of control material, so as to minimize the amount of fuel required. Needless to say, the schemes for these two types of optimization are closely related to each other and cannot be treated separately from each other. So far, however, there is known no theoretically established method of realizing such optimizations to any satisfactory extent. Heretofore, there has been popularly employed a uniform scatter refueling method which places emphasis on flattening of power distribution and in which, for achieving optimization of the refueling schedule, a fixed quantity of fuel (for example, 1/4 of the entire amount of fuel) in the reactor core is withdrawn successively, beginning with the portion with higher extent of burn-up or use, and a corresponding quantity of fresh fuel is charged and scattered uniformly in the reactor core. In a simplest example of such system, the reactor core is divided into many units so that each unit consists of four fuel assemblies, and refueling is performed by discharging the fuel in each unit successively, beginning with the oldest fuel. This refueling method features a simple refueling principle and minimized tendency of causing local power peaking as the fresh and old fuels are arranged alternately. If refueling operation is continued according to such method, there is produced a situation in which burn-up or use of fuel is lowered proportionally to the distance from the center and hence the infinite multiplication factor becomes higher as the distance from the center increases. This is because the power distribution in the reactor core generally has a tendency to decline as the distance from the center increases. This promotes flattening of power as well as temperature distributions of fuel, and consequently, certain thermal allowance is provided. However, such determined intra-reactor nuclear properties (such as infinite multiplication factor, material buckling, etc., which define the fission productivity) are not necessarily advantageous from the viewpoint of economical utilization (combustion or useful life) of the nuclear fuel. Also, for performing initial fuel loading, there is employed either a method in which fuel of uniform enrichment is loaded evenly or a method in which, for ensuring flattening of power distribution, a fuel mixture of two to three different degrees of enrichment is loaded such that the fuel of higher enrichment is positioned on the outside. But, either of such methods proves unsatisfactory in always providing effective utilization of nuclear fuel. On the other hand, for achieving optimization of control by the control material, there is generally employed a method in which a control rod pattern that will maintain power distribution constant during the operation is decided through a trial and error policy by controlling the reactor core uniformly with a liquid poison (such as aqueous solution of boron) or controlling the reactivity by a control rod. In the study of the problems concerning the above-mentioned two types of optimization, controlling by control material may be regarded as a sub-problem concomitant to the refueling schedule, and it is hence considered that the solution of the latter problem will provide several times as greater effect of optimization than the solution of the former problem. It is therefore an object of the present invention to provide a nuclear reactor core constitution specifically arranged to realize most effective utilization of fissionable materials as fuel. It is another object of the present invention to provide a nuclear reactor core constitution of the type described, by theoretically elucidating the optimum nuclear property distribution, which has been equivocal in the past, from the standpoint of economical utilization of nuclear fuel, so as to determine a general principle for such optimization and adapt such principle in arrangement of fuel in the reactor core. According to the present invention, there is provided a nuclear reactor in which the fuel assembly using fissionable material as fuel is divided into three concentric regions about the core center and is loaded with fuel such that, if the regions are named the inner (first) region, middle (second) region and outer (third) region in the order of distance from the core center, the infinite multiplication factor in each of said regions will be such that it is smaller in the outer region than in the inner region and smaller in the inner region than in the middle region, thereby realizing a fuel arrangement that provides the optimal nuclear property distribution. FIG. 1 is a characteristic diagram showing the optimal space distribution of infinite multiplication factor in the optimal nuclear fuel loading according to the present invention; FIG. 2 is a planar development showing flow distribution in a 1/4 reactor core according to an embodiment of the present invention; FIG. 3 is a chart showing a pattern of fuel loading according to an embodiment of the present invention; FIG. 4 is a graphical representation of FIG. 3; FIG. 5 is a characteristic diagram showing the average power distribution and the maximum power distribution, as observed in the axial direction, according to an embodiment of the present invention; and FIG. 6 is a characteristic diagram showing the power distributions (in the a- and b-directions) as observed radially, further embodying the present invention. Before proceeding to the discussion on practical embodiments of the present invention, it will be of importance to describe here the basic principles of the invention. Firstly, in order to accomplish the aforementioned objects of the invention, the problem of optimal fuel loading pattern is theoretically dealt with as follows. Assuming that the one-dimensional one-group neutron diffusion equation can apply with neutron, the following formula is given: M2 V2 φ(x, t) + (ko (x)-a.sup.. e (x, t) - 1 - u(x, t))φ (x, t) = 0 0≦x≦H, 0=t≦tf (1) ______________________________________M2 neutron migration area (cm2)Φ(x, t) neutron flux (per cm2. sec.)e(x, t) burn-up (MWD/T)u(x, t) control material enrichment (reactivity unit)ko (t) initial nuclear property (infinite multiplication factor)a depletion coefficient of property by fuel burn-up (reactivity/MWD/T)x spatial coordinate (cm)H core dimension (cm)t time (year)tf operation period (year)______________________________________ The burn-up or use increase in proportion to the neutron fluxes, hence: ##EQU1## The boundary condition of the formula (1) is given as follows: ##EQU2## where stands for extrapolated distance provided by the reflector. The initial condition of the formula (2) is: e(x, o) = 0 0≦x≦H (4) the constraints include: power definite condition given by: ##EQU3## restriction of power peaking factor given by: 0≦ φ (x, t) ≦fp, 0≦t≦tf, 0≦x≦H (6) restriction of control material enrichment given by: 0≦u(x, t) ≦U, 0≦t≦tf, o≦x≦H (7) and restriction of fuel property given by: k min≦ko (x)≦k max, 0≦x≦H (8) the performance index is: ##EQU4## The problem is therefore how to determine ko (x) and u(x, t) which satisfy the formulae (1) to (8) and minimize the formula (9). In order to solve this problem, the formulae (1) to (9) are discretized concerning time and space and converted into a non-linear programming problem. If the space is equally divided into N - 1 number of sections and the time into M - 1 number of sections with the variables at the n-minute point and at the m-hour point being expressed by letters n and m, respectively, the formulae (1) to (9) may be unified into a non-linear constraint equation of the following formula (10): g(ko.n, φn. m, en.m, un.m)≦0 n= 1 ˜ N, m= 1 ˜ M (10) likewise, the formula (9) may be defined as follows: J = f(ko.n) n= 1 ˜ N (11) the non-linear programming problem in the formulae (10) and (11) can be solved by using an approximation programming method. If the feasible solution that satisfies the formula (10) is expressed by letter o and the formulae (10) and (11) are linearized around such feasible solution, with the variables being expressed by x collectively, then the following equations are given: ##EQU5## where δx stands for small variation of x. It will be noted that the formula (12) presents a linear programming problem concerning the variable δx. There is need of imposing a certain restriction to the range of variation of the variable δx for bettering accuracy of linearization, but even if such matter is considered, no change is made in that a linear programming problem is presented. A δ x ≦ B δJ = C δ x (13) After all, solution to the optimization problems in the formulae (1) to (9) is obtained by repetitively solving the linear programming problem in the form of formula (13). FIG. 1 shows, in graphical representation, the results obtained from application of the above-described method to a light water moderated power reactor under the following conditions: l = 0.0 cm, H = 200 cm, M2 = 80 cm2, P = H, fp = 1.4, tf = 1 year, V = 0.5, k min = 0.9, and k max = 1.2. In FIG. 1, the solid line represents the optimal loading. What is most typical of these results is that the reactor core is divided into three regions of almost equal volume, with the following relations being established: ko (x) = k max in the second region and ko (x) = k min in the third region. In the first region, ko (x) is of a constant value which is greater than k min but smaller than k max (k min < ko (x ) < k max), and its value is determined by the power peaking factor. In Table 1 below, three examples of combined methods for fuel loading and control rod programming are compared. The first example is based on the combination of optimal loading and optimal control rod programming which were obtained in the above-described method. The second example uses optimal solution only for loading and a conventional constant power shape method for control rod programming. The third example employs the optimal loading of the first example, which was uniformly homogenized in the core as shown by dotted lines in FIG. 1, while employing a constant power shape method for control rod programming. Table 1______________________________________ Δk: Δt: difference of difference of loaded fuel infi- operation feasible nite multiplica- period tion factor______________________________________Optimal loading andoptimal control rod 0.00 0.00programmingOptimal loading andconstant power shape -0.01 0.0013controllingUniform loading andconstant power shape -0.45 --controlling______________________________________ The numerical values in the left column in the above table compare the operation feasible time (burn-up time), showing that if the loading is optimal, the effect of optimization of control rod programming is only about 1%, while the difference of loading method can produce as high as 45% optimization effect. The numeral figures in the right column of Table 1 show the difference of loaded fuel infinite multiplication factor in case the operating time is one year. These results are indicative of extremely high effect of optimal loading. The general characteristics concomitant to optimal loading have been described above, but when including approximate optimal loading in consideration of its prominent effect, the general characteristics may be educed in the following way. Namely, when the reactor core is tentatively divided into equal-volume sections, it is essential that the fuel is loaded in such a manner that the average nuclear property in the regions will be arranged in the order of 2- 1- 3 from the core center. Now, the present invention is described in further detail by way of its preferred embodiments. FIG. 2 is a plane view of the core portion of a nuclear reactor, depicting the rate of distribution of the coolant flow through an orifice. It shows the flow rate of each fuel assembly as measured with the rated flow being given as 1.0. It will be understood that the Regions 5, 6 and 7 have flow rates of 1.127, 0.62 and 0.55, respectively. Flow distribution is so made that the coolant flow will be reduced at the core end where the power density is low because the power distribution will not be uniform in the reactor core. One section corresponds to one fuel assembly. The drawing shows only a 1/4 type core. There are total 400 pieces of fuel assemblies in the entire core. FIG. 3 shows the fuel loading method according to the present invention, where Region 8 (first region counted from the core center), Region 9 (second region) and Region 10 (third region) are loaded with 164 (41 × 4) fuel assemblies, 120 (30 × 4) fuel assemblies and 116 (29 × 4) fuel assemblies, respectively. That is, they are loaded with fuel of different enrichments --2.09 w/o weight per cent and 1.83 w/o, respectively. The corresponding infinite multiplication factors are 1.119, 1.200 and 1.035, respectively. FIG. 4 is a diagrammatic demonstration of the above infinite multiplication factors as considered in relation to the respective fuel assemblies in the a- and b-directions on the plane (longitudinal) of FIG. 3. The operative performance obtained from this embodiment of the invention is shown comparatively with that from a conventional example in Table 2 below. Table 2______________________________________ Conventional Present example embodiment______________________________________Burn-up GWD/T 5.575 6.785Difference % -- 21.70Power peaking factor* 2.523 2.398Location of peak (1,1,3) (4,7,2)Minimum criticalheat flux ratio 3.3096 3.4820Location of minimumcritical heat flux (1,1,3) (4,7,2)Maximum linear powergeneration kw/ft 13.211 12.557Effective multiplica-tion factor 0.99972 1.00037Average infinite multi-plication factor 1.0448 1.0348______________________________________ *Local power peaking factor of 1.257 and over-power (10%) peaking factor of 1.100 are included. Here the constant power shape method is used for control rod programming. The conventional example in the above table was given to clarify the actual effects of the embodiment of the present invention, and where the same fuel is used in the first region was loaded uniformly in the reactor core, and also the average value of infinite multiplication factor in the reactor core was selected same as in the instant embodiment of the present invention so as to ensure a valid comparison. The above results dictate that the instant embodiment of the present invention provides surprising 21.7% higher burn-up than the conventional example and also produces more flattened power distribution and better thermal characteristics. FIGS. 5 and 6 show power distributions, as viewed in the axial and radial directions of the core, according to the present invention and a conventional example. The difference between the two is most apparent at the core periphery, where the present invention develops more flattened power distribution than the conventional example. As reviewed above, the present invention provides far better thermal characteristics and as much as about 20% higher burn-up than the conventional example, in spite of the fact that the same amount of fissionable material is loaded in the reactor core. This allows appreciable economization of the fuel cost, rendering a great contribution and benefit to the industries concerned. While the novel embodiments of the invention have been described, it will be understood that various omissions, modifications and changes in these embodiments may be made by one skilled in the art without departing from the spirit and scope of the invention.

Fresnel diffraction accounts for the curvature of the wavefront. Difference between fresnel diffraction and fraunhofer. Similarly the wave fronts leaving the obstacle are not plane. As in fresnel diffraction, well typically assume a plane wave incident field, well neglect the phase factors, and well explicitly write the aperture function in the integral. Calculation of fresnel diffraction is based on an approximation. Now consider a circular aperture of radius, illuminated by a point source r at a finite distance. A fresnel diffraction pattern, which is a convolution ot the normalized aperture 2function with expi. Note the concentric diffraction rings and the fresnel bright spot at the center of the pattern. Comparison of normalized intensity distribution on the fresnel diffraction pattern between two singlemode optical fibers. If the observation plane is rather close to the transmittance function under free propagation, then fresnel diffraction ap pears, and as it occurs in most cases, it. Fresnel diffraction fraunhofer diffraction the source and the screen are at finite distance from the diffracting aperture the source and the screen are at infinite distance from the diffracting aperture for obtaining fresnel diffraction, zone plates are used for this single double ae plane diffraction grating are used. A free powerpoint ppt presentation displayed as a flash slide show on id. When a transparent planeparallel plate is illuminated at the edge region by a quasimonochromatic parallel beam of light, diffraction fringes appear on a plane perpendicular to the transmitted beam direction. How to remember fraunhofer and fresnel diffraction. Between the wave source v and the observation point p, a thin planar object, which neither reflects nor transmits electromagnetic waves, so called absorbing object is placed. This represents a bessel beam a socalled diffractionfree beam, which is. In optics, the fresnel diffraction equation for nearfield diffraction is an approximation of the kirchhofffresnel diffraction that can be applied to the propagation of waves in the near field. Fresnel developed an equation using the huygens wavelets together with the principle of. Computer simulation of fresnel diffraction from rectangular. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close. Fraunhofer diffraction from a single slit consider the geometry shown below. Fresnel diffraction must be used in all other cases the fresnel and fraunhofer regions are used as synonyms for near field and far field, respectively in fresnel region, geometric optics can be used for the most part. Fresnel diffraction article about fresnel diffraction by. Fraunhofer diffraction vs fresnel diffraction difference. Diffraction of light, fraunhofer and fresnel diffraction. Photographs of transition for square aperture are shown and discussed. Analytical solutions are not possible for most configurations, but the fresnel diffraction equation and fraunhofer diffraction equation, which are approximations of kirchhoffs formula for the near field and far fieldcan be applied to a very wide range of optical systems. Numerical calculation of near field scalar diffraction using. What are the differences between fresnel and fraunhofer. He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source. Eindhoven university of technology bachelor fresnel diffraction. Objectoriented crystallographic library and program objectoriented crystallographic library and program, for the analysis of crystal structures from scattering experiments. In the study of fresnel diffraction it is convenient to divide the aperture into regions called fresnel zones. Mar 02, 2008 fresnel and fraunhofer diffraction patterns and dynamic transition between them can be easily obtained on a distance of few meters, what gives an opportunity to use our setup as a lecture experiment. According to the superposition principle, the net displacement is simply given by the. He introduce a quantity of the obliquity factor, but he did little more than. The diffraction of light by a rectangular aperture is discussed and analyzed in many textbooks. The size of the pattern depends on the distance between a. Theory and applications 3 what is a diffraction grating. Fresnel diffraction fraunhofer diffraction f 1 f 1 f. We are going to study the wave diffraction in the following situation. I wrote a short matlab script file that is suppose to run fresnels propagation diffraction, such that given a certain input field u0, it will tell you how the field looks after distance z0. More accurately, it is the diffraction case when the fresnel number is large and thus the fraunhofer approximation diffraction of. Find materials for this course in the pages linked along the left. Although a relic of the early 19th century, the fresnel diffraction integral 2 is still used today in its original form and has remained a most useful, reliable tool for diffraction calculations that have consistently yielded results which agree with experience and are well documented 2. Full text of computer simulation of fresnel diffraction from. Oct 22, 2018 when waves come from a finite distance, they can be treated as divergent rays, and they gives fresnel diffraction. In optics, the fresnel diffraction equation for nearfield diffraction is an approximation of the. Diffraction theory has been further developed by huygens, fresnel. In the order of increasing distance from the aperture, diffraction pattern is a shadow of the aperture. Fresnel diffraction, or nearfield diffraction, occurs when a wave passes via a small hole and diffracts, creating a diffraction pattern. Fraunhofer diffraction the area a 1 above is replaced by a wavefront from p 0which almost fills the aperture, and a portion of a cone with a vertex at p 0which is labeled a 4 in the diagram. Pdf spectral transfer from phase to intensity in fresnel. The different patterns can also be registered using a pattern scanner as described in. In optics, fresnel diffraction or nearfield diffraction is a process of diffraction which occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to. Closer to the aperture the diffraction pattern does change with distance. But realize that turning a vernier is less visible to an audience than shifting a lens across the table. Users may download and print one copy of any publication from the public portal for the purpose of private. Use lenses to expand a 632 nm laser beam and form a line focus source. Note the lines of alternating maximum and minimum intensity. Use ocw to guide your own lifelong learning, or to teach others. More accurately, it is the diffraction case when the fresnel number is large and thus the fraunhofer approximation diffraction of parallel beams can not be used. The sharp change in the refractive index at the plate boundary imposes an abrupt change on the phase of the illuminating beam that leads to the fresnel diffraction. Mar 08, 2016 fresnel diffraction fraunhofer diffraction the source and the screen are at finite distance from the diffracting aperture the source and the screen are at infinite distance from the diffracting aperture for obtaining fresnel diffraction, zone plates are used for this single double ae plane diffraction grating are used. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction. An optical component that is free of aberrations is called diffraction. Scribd is the worlds largest social reading and publishing site. The different patterns can also be registered using a pattern scanner as described in diffraction1, introduction in this database. To understand, briefly, the basis of calculation of fresnel diffraction from a. In optics, the fraunhofer diffraction equation is used to model the diffraction of waves when the. Ppt fresnel diffraction powerpoint presentation free to. Difference between fresnel diffraction and fraunhofer diffraction. The concepts of fresnel and fraunhofer diffraction. Electromagnetic fields in free space are governed in general by the scalar wave equation 11. Diffraction diffraction describes the tendency for light to bend around corners. Mujibur rahman 2, kazi monowar abedin 3 physics department, college of science, sultan. Determination of refractive indices of liquids by fresnel. One dimensional surface profilometry by analyzing the fresnel. And, of course, if you want to make your own fresnel diffraction patterns, download my program. Dynamic transition between fresnel and fraunhofer diffraction. Diffraction, free diffraction software downloads, page 3. A b p manifestation of the wave nature of light light bends around corners. Fresnel diffraction on a semiinfinite opaque screen. Fresnel diffraction or nearfield diffraction is the diffraction pattern of an electromagnetic wave obtained close to the diffracting object often a source or aperture. Fraunhofer diffraction last lecture numerical aperture of optical fiber allowed modes in fibers attenuation modal distortion, material dispersion, waveguide dispersion this lecture diffraction from a single slit diffraction from apertures. I compared the result to textbook results, and it seems like my program works fine. Fresnelkirchhoff diffraction formula is simplified for the limiting case of. Nov 23, 2015 in optics, the fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is. Freely browse and use ocw materials at your own pace. A nondestructive technique, based on fresnel diffraction from phase objects, is proposed for specifying the refractive indices of optical fibers without requiring index matching liquid. Pdf applications of fresnel diffraction from phase objects. Huygens principle all points on a wavefront can be considered as point sources for the production of secondary wavelets, and at a later time the new wavefront position is the envelope or surface of tangency to these secondary wavelets. For fresnel diffraction, the observing screen is a finite distance from the slit or edge and the light rays are not rendered paral. That book has many diagrams, examples, explanations for both types of diffraction. Fraunhofer and fresnel diffraction in one dimension revised november 15, 2017 introduction the simplest and most commonly described examples of di. Difference between fraunhofer and fresnel diffraction. In this situation the wave fronts falling on the obstacle are not plane. It does not aim to be a formal course or a tutorial in optics, and remains in the theme of the school, for. A fraunhofer diffraction pattern, which is the squaredabsolute value of the fourier transform of the aperture. When a coherent quasimonochromatic light is reflected from a step, a diffraction pattern is formed that can be described by fresnelkirchhoff integral. Diffraction 6 fresnel and fraunhofer diffraction fresnel diffraction near field from phys 1114 at the hong kong university of science and technology. Diffraction software free download diffraction page 3. Light fresnel diffraction experimental study set up an arrangement in the lab to observe light fresnel diffraction from a slit. Osa applications of fresnel diffraction from the edge of a. Jun 18, 2017 fresnel diffraction of circular aperture. Let the array of length a be parallel to the y axis with its center at the origin as indicated in the figure to the right. Every unobstructed point of a wavefront acts as a source of spherical secondary wavelets. On the basis of the following assumptions, fresnel developed the theory of diffraction which explained the various diffraction effects.1535 298 377 988 158 547 1235 172 599 727 1104 1386 1287 184 1395 1286 390 1564 1256 1224 350 340 1331 81 38 848 473 1167 595 824 1253 635 318 1135

Spring 2014 Dr. S. Print Your Name:_ Score:_ Print your name above and also somewhere on the back of this quiz. Write your answers directly on this quiz, not on a separate sheet of paper. You must show adequate amount of work to get STAT 3401 SOLUTIONS FOR SECTIONS 2.10 AND 2.11 2.124 Let F be the event that a randomly selected voter favors the election issue. Let R be the event that the voter is a Republican and let D be the event that the voter is a Democrat. The probl STAT 3401 SOLUTIONS FOR SECTION 3.11 3.167 Y has mean = 11 and standard deviation = 9 = 3 . a) P ( 6 < Y < 16) = P ( 6 < Y < 16 ) = P ( 6 11 < Y < 16 11) = P ( 5 < Y < 5) = P ( Y < 5) 1 k12 , where 5 = k . So k = and 1 k12 = 1 (5 /13)2 = 1 ( 3 )2 STAT 3401 SOLUTIONS FOR SECTION 3.8 3.121 Y is Poisson with = 2 . e 2 2 4 a) P (Y = 4) = b) P (Y 4) = 1 P (Y 3) . But P (Y 3) = p(0) + p(1) + p(2) + p(3) = e 2 20 e 2 21 e 2 22 e 2 23 = e 2 (1 + 2 + 2 + 8 ) = e 2 ( 19 ) . So STAT 3401 SOLUTIONS FOR SECTION 3.9 3.149 Suppose that the random variable Y has the mgf m(t ) = (.6et + .4)3 = ( pet + q )n . This is the mgf of the binomial distribution with n = 3 and p = .6 (and q = .4 ). By the uniqueness property, Y must have this d STAT 3401 SOLUTIONS FOR SECTION 3.7 3.103 Let X be the number of non-defective machines out of the five selected machines. Then X is hypergeometric with N = 10, r = 10 4 = 6 (non-defectives), n = 5 . So P ( X = 5) = = STAT 3401 SOLUTIONS FOR SECTION 3.6 3.90 Let Y be the number of employees that must be tested in order to find three that test positive for asbestos in their lungs. Then Y is negative binomial with r = 3 , p = .40 , and q = 1 p = .60 . So P (Y = 10) = STAT 3401 SOLUTIONS FOR SECTION 3.5 3.67 Let Y be the number of applicants that are interviewed until the first one is found who has advanced training in programming. Then Y is geometric with p = .30 and q = 1 p = .7 . So P (Y = 5) = q 51p = (.7)4 (.3) = STAT 3401 SOLUTIONS FOR SECTION 3.4 3.39 Let Y be the number of components out of the four components that operate longer than 1000 hours. Then Y is binomial with n = 4 and p = 1 .2 = .8 . a) P (Y = 2) = (.8)2 (.2)2 = 6(.8)2 (.2)2 = .1536 . b) The s STAT 3401 SOLUTIONS FOR SECTION 3.2 3.1 Let A be the event that impurity A is found in the well and B, the event that impurity B is found. The problem gives P ( A) = .40, P (B) = .50, P ( A B ) = .20 . By DeMorgans rule and the complement rule, P ( A B) = STAT 3401 SOLUTIONS FOR SECTION 2.9 2.110 Let D be the event that a randomly selected item is defective. Let I and II be the events that correspond to which line the item came from. The problem gives: P ( I) = .40, P ( II) = .60, P (D | I) = .08, and STAT 3401 SOLUTIONS FOR SECTION 2.8 2.85 Since, A and B are independent, P ( A B) = P ( A) P (B) and so it follows that P ( A B) = P ( A) P ( A B) = P ( A) P ( A) P (B) = P ( A) (1 P (B) = P ( A) P (B) . Therefore, A and B are independent. Another way: By STAT 3401 SOLUTIONS FOR SECTION 2.6 2.35 Consider the 2-stage job: In stage 1, you select a flight from New York to California; in stage 2, you select a flight from California to Hawaii. The job can be done in 6 7 = 42 ways. 2.41 Consider the 7-stage job: STAT 3401 SOLUTIONS FOR SECTION 2.7 P ( AB) .1 1 P ( B) P (BA) .1 1 P (B | A) = P ( A) First note that P ( A B) = P ( A) + P (B) P ( AB) = .5 + .3 .1 = .7 . Also note that A ( A B) = A since P ( A ( A B) P ( A) A ( A B) . T STAT 3401 SOLUTIONS FOR SECTION 2.5 2.26 Label the four cans as: 1, 2, 3, 4. Let the labels 1 and 2 represent the cans with water. = cfw_ cfw_1,2,cfw_1,3,cfw_1, 4,cfw_2,3,cfw_2, 4,cfw_3, 4 . Each outcome represents the two cans that the expert says STAT 3401 SOLUTIONS FOR SECTION 2.4 2.12 Let L, R, and S be the events that the vehicle turns left, turns right, and goes straight, respectively. = cfw_L, R, S . b) P ( It turns ) = P ( cfw_L, R ) = a) P (E1 ) + P (E3 ) + P (E4 ) = .01 + .09 STAT 3401 SOLUTIONS FOR SUPPLEMENTARY EXERCISES The sample space is the set of all combinations of five cards chosen from the 52 cards and the number of such combinations is = 52 5 = 2,598,960 . There are four suits: spades, clubs, diamonds, STAT 3401 SOLUTIONS FOR SECTION 2.3 A = cfw_FF . B = cfw_MM . C = cfw_FM , MF, MM . A B = . A B = cfw_FF, MM . A C = . A C = S. B C = B. B C = C. C B = cfw_FM , MF . a) AB = A B . b) A B . c) AB = A B . d) ( AB ) (BA) . 2.7 First note that the sam GEOMETRIC AND NEGATIVE BINOMIAL STATISTICS 3401 Consider a sequence of independent Bernoulli trials with constant probability of success p, 0 < p < 1 . Let X denote the number of trials until the first success. Then X is said to ha HYPERGEOMETRIC DISTRIBUTION STATISTICS 3401 Suppose that a population consists of N objects each of which is categorized as success (S) or failure (F). Assume that r objects are successes and the other N r objects are failures. Consider the random BERNOULLI AND BINOMIAL STATISTICS 3401 The binomial, geometric, and negative binomial distributions are defined in terms of a sequence of independent Bernoulli trials. Definition: A Bernoulli trial is a trial of a random experiment that has only two possi CONVERGENCE IN DISTRIBUTION AND THE CENTRAL LIMIT THEOREM STAT 3401/6204 Definition: Suppose that X1 , X 2 , X 3 , is an infinite sequence of random variables and FXn ( x ) is the cdf of X n for each n = 1, 2,3, . Further suppose that X is a random variab MGFS FOR CONTINUOUS DISTRIBUTIONS STATISTICS 3401 Recall that the moment-generating function (mgf) of a random variable X is the function m(t ) = E (e tX ) . For a continuous X having density function f(x), the LOTUS implies that m (t ) = etx f ( x ) dx . RANDOM VARIABLES AND THEIR DISTRIBUTIONS STATISTICS 3401 A random variable is a variable that represents a numerical characteristic of the outcomes of a random experiment. In mathematical terms, a random variable is a rule that assigns a real number to ea THE POISSON DISTRIBUTION STATISTICS 3401 The Poisson distribution is named after Simeon Poisson who, in 1837, used the distribution as an approximation to the binomial distribution. The Poisson random variable has the countably infinite range cfw_ 0, 1, 2 THE NORMAL DISTRIBUTION STATISTICS 3401 The normal distribution is the most important distribution in all of statistics and probability. The normal density function describes the classic symmetric, bell-shaped curve. One of the first applications of the n MOMENT-GENERATING FUNCTION STATISTICS 3401/4412 Definition: The moments (about the origin) of a random variable X are the following expected values: E ( X ), E ( X 2 ), E ( X 3 ), E ( X 4 ), . . For each positive integer k, let = E ( X k ) , the kth momen INDEPENDENCE OF RANDOM VARIABLES Recall that events A and B are independent if and only if P ( A B) = P ( A)P (B) . This is the basis for the definition of independence of random variables. Definition: Random variables X and Y, defined on the same sample

Cosmic Structure Formation with Topological Defects - 1 Introduction - 2 Symmetry Breaking Phase Transitions and the Formation of Topological Defects 3 Theoretical Framework - 3.1 Linear cosmological perturbations with seeds - 3.2 The seed energy momentum tensor - 3.3 Einstein’s equations and the fluid equations - 3.4 Boltzmann equation, polarization and CMB power spectra - 3.5 Neutrinos - 3.6 Computing power spectra in seed models - 4 Numerical Implementation - 5 Result - 6 Generalizations - 7 Conclusion - A Definitions of all gauge-invariant perturbation variables - B Boltzmann equation and polarization Topological defects are ubiquitous in physics. Whenever a symmetry breaking phase transition occurs, topological defects may form. The best known examples are vortex lines in type II super conductors or in liquid Helium, and declination lines in liquid crystals [109, 24]. In an adiabatically expanding universe which cools down from a very hot initial state, it is quite natural to postulate that topological defects may have emerged during a phase transition in the early universe and that they may have played the role of initial inhomogeneities seeding the formation of cosmic structure. This basic idea goes back to Kibble (1976) . In this report we summarize the progress made in the investigation of Kibble’s idea during the last 25 years. Our understanding of the formation and evolution of topological defects is reported almost completely in the beautiful book by Vilenkin & Shellard or the excellent Review by Hindmarsh & Kibble , and we shall hence be rather short on that topic. Nevertheless, in order to be self contained, we have included a short chapter on spontaneous symmetry breaking and defect formation. Our main topic is however the calculation of structure formation with defects, results which are not included in and . Besides the formation of structure in the universe, topological defects may be relevant for the baryon number asymmetry of the universe . Superconducting cosmic strings or vortons might produce the high energy cosmic rays , or even gamma ray bursts . The brane worlds which have have focussed a lot of attention recently, may actually just represent topological defects in a higher dimensional space [65, 61, 62]. There have also been interesting results on chiral strings and their observational signatures [21, 138]. GUT scale cosmic strings could be detected by their very peculiar lensing properties. For a straight cosmic string lensing is very simple . For a more realistic network of strings, characteristic caustics and cusps in the lensing signal are very generically expected [96, 148, 14]. The relevant energy scale for a topological defect is , the phase transition temperature. Hence a good estimate for the amplitude of the dimensionless gravitational potential induced by topological defects is where denotes the Planck mass. The measurements of cosmic microwave background anisotropies on large scales by the cosmic background explorer (COBE) satellite have found that this potential, which is of the same order as the temperature fluctuations on large scales, is about . Hence, for cosmic structure formation, we are interested in phase transitions at GeV. Interestingly, this is just the scale of the grand unification phase transition (GUT scale) of supersymmetric GUT’s (grand unified theories). Topological defects represent regions in space-time where the corresponding field (order parameter in condensed matter physics or Higgs field in particle physics) is frustrated. It cannot relax into the vacuum state, the lowest energy state, by topological obstructions. They represent positions of higher energy density and are thus inherently inhomogeneous distributions of energy and momentum. We shall discuss concrete examples later. In the remainder of this introduction we give a brief overview of the problem of structure formation and we present the main results worked out in this report. In Chapter 2 we introduce the concept of topological defect formation during symmetry breaking phase transitions, we classify the defects and illustrate them with examples. In Chapter 3 we present in detail the theoretical framework used to investigate structure formation with topological defects. This chapter together with two appendices is self contained and should enable a non-specialist in the field to fully understand the often somewhat sketchy literature. In Chapter 4 we discuss numerical simulation of topological defects. We distinguish global and local defects which have to be treated in a very different way. We specify the approximations made in different numerical implementations and discuss their validity and drawbacks. In Chapter 5 we present the results of simulations of structure formation with topological defects and compare them with present observations. In Chapter 6 we investigate the question in whether the results discussed in Chapter 5 are generic or whether they are just special cases. We derive properties of the unequal time correlators of generic causal scaling seeds. Since these are the sole ingredients in the calculation of the fluctuation power spectra, they determine the ’phase space’ of defect models of structure formation. We discuss a model of causal scaling seeds which mimics the cosmic microwave background (CMB) anisotropy spectrum of inflation. We also consider the possibility that large scale structure may be due to a mixture of adiabatic inflationary initial perturbations and topological defects. We study especially the fluctuations in the CMB. We investigate to which extent CMB parameter estimations are degraded if we allow for an admixture of defects. We end with a brief summary of the main results. Throughout this work we use units with . The basic unit is thus either an energy (we usually take MeV’s) or a length, for which we take cm or Mpc depending on the situation. We choose the metric signature . Three-dimensional vectors are denoted in boldface. The variables and are comoving position and comoving wave vector in Fourier space. Greek indices, denote spacetime components of vectors and tensors while Latin indices denote three dimensional spatial components. We mostly use conformal time with , where is cosmic time and is the scale factor. Derivatives with respect to conformal time are indicated by an over-dot, . 1.1 Main results Before we start to discuss models of structure formation with topological defects in any detail, let us present the main results discussed in this review. We concentrate primarily on CMB anisotropies. Since these anisotropies are small, they can be calculated (almost fully) within linear cosmological perturbation theory. To compare models with other data of cosmic structure, like the galaxy distribution or velocities, one has to make some assumptions concerning the not well understood relation between the distribution of galaxies and of the dark matter, the problem of biasing. Furthermore, on small scales one has to study non-linear Newtonian clustering which is usually done by -body simulations. But to lay down the initial conditions for -body simulations, one does not only need to know the linear power spectrum, but also the statistical distribution of the fluctuations which is largely unknown for topological defects. Fluctuations induced by topological defects are generically non-Gaussian, but to which extent and how to characterize their non-Gaussianity is still an open question. In this report, we therefore concentrate on CMB anisotropies and their polarization and shall only mention on the side the induced matter fluctuations and bulk velocities on large scales. Like inflationary perturbations, topological defects predict a Harrison-Zel’dovich spectrum of perturbations [69, 163]. Therefore, the fluctuation spectrum is in good agreement with the COBE DMR experiment , which has measured CMB anisotropies on large angular scales and found that they are approximately constant as expected from a Harrison-Zel’dovich spectrum of initial fluctuations (see e.g. ). Since quite some time it is known, however, that topological defect differ from adiabatic inflationary models in the acoustic peaks of the CMB anisotropy spectrum . Due to the isocurvature nature of defects, the position of the first acoustic peak is shifted from an angular harmonic of about to (or up to in the case of cosmic strings) for a spatially flat universe. More important, the peaks are much lower in defect models and they are smeared out into one broad hump with small wiggles at best. Even by changing cosmological parameters at will, this second characteristics cannot be brought in agreement with present CMB anisotropy data like [110, 66, 97]. Also the large scales bulk velocities, which measure fluctuation amplitudes on similar scales turn out to be too small . As the CMB anisotropy signals from cosmic strings and from global defects are quite different, it is natural to wonder how generic these results may be. Interestingly enough, as we shall see in Chapter 6, one can define ’scaling causal seeds’, i.e. initial perturbations which obey the basic constraints for topological defects and which show a CMB anisotropy spectrum resembling adiabatic inflationary predictions very closely . This ’Turok model’ can nevertheless be distinguished from adiabatic perturbations by the CMB polarization spectrum. Also mixed models with a relatively high degree of defect admixture, up to more than 50%, are in good agreement with the data. 1.2 Cosmic structure formation The geometry of our universe is to a very good approximation isotropic and therefore (if we assume that we are not situated in a special position) also homogeneous. The best observational evidence for this fact is the isotropy of the cosmic microwave background which is (apart from the dipole anisotropy) on the level of about – . Nevertheless, on galaxy and cluster scales, the matter distribution is very inhomogeneous. If these inhomogeneities have grown by gravitational instability from small initial fluctuations, the amplitude of the initial density fluctuations have to be about to become of order unity or larger today. Radiation pressure inhibits growth of perturbations as long as the universe is radiation dominated, and even in a matter dominated universe small perturbations grow only like the scale factor. The discovery of large angular scale fluctuations in the CMB by the DMR experiment aboard the COBE satellite , which are just about of the needed amplitude, is an important support of the gravitational instability picture. The DMR experiment also revealed that the spectrum of fluctuations is ’flat’, which means that the fluctuations have a fixed amplitude when entering the Hubble horizon. This implies that temperature fluctuations on large scales are constant, as measured by COBE: independent of the angle , for . These fluctuations are of the same order of magnitude as the gravitational potential. Their smallness therefore justifies the use of linear perturbation theory. Within linear perturbation theory, the gravitational potential does not grow. This observation originally led Lifshitz to abandon gravitational instability as the origin for cosmic structure . But density fluctuations of dust can become large and can collapse to form galaxies and even black holes. At late times and on scales which are small compared to the horizon, linear perturbation theory is no longer valid and numerical N-body simulations have to be performed to follow the formation of structure. But also with N-body simulations one cannot compute the details of galaxy formation which strongly depend on thermal processes like cooling and formation of heavy elements in stars. Therefore, the relation of the power spectrum obtained by N-body simulations to the observed galaxy power spectrum may not be straight forward. This problem is known under the name of ’biasing’. Within linear perturbation analysis, structure formation is described by an equation of the form where is a time dependent linear differential operator, is the wave vector and is a long vector describing all the cosmic perturbation variables for a given -mode, like the dark matter density and velocity fluctuations, the CMB anisotropy multipole amplitudes and so on. is a source term. In Chapter 3 we will write down the system (3) explicitly. There are two basically different classes of models for structure formation. In the first class, the linear perturbation equations are homogeneous, i.e. , and the resulting structure is determined by the initial conditions and by the cosmological parameters of the background universe alone. Inflationary initial perturbations are of this type. In most models of inflation, the initial fluctuations are even Gaussian and the entire statistical information is contained in the initial power spectrum. The evolution is given by the differential operator which depends on the cosmological parameters. In the second class, the linear perturbation equations are inhomogeneous, having a so called source term or ’seed’, , on the right hand side which evolves non-linearly in time. Topological defects are of this second class. The difficulty of such models lies in the non-linear evolution of the seeds, which in most cases has to be determined by numerical simulations. Without additional symmetries, like e.g. scaling in the case of topological defects, it is in general not possible to simulate the seed evolution over the long time-scale and for the considerable dynamical range needed in cosmology. We shall see in Chapter 4 how this problem is overcome in the case of topological defects. An additional difficulty of models with seeds is their non-Gaussian nature. Due to non-linear evolution, even if the initial conditions are Gaussian, the seeds are in general not Gaussian at late times. The fluctuation power spectra therefore do not contain the full information. All the reduced higher moments can be relevant. Unfortunately only very little work on these non-Gaussian aspects of seed models has been published and the results are highly incomplete [60, 43, 59]. 2 Symmetry Breaking Phase Transitions and the Formation of Topological Defects 2.1 Spontaneous symmetry breaking Spontaneous symmetry breaking is a concept which originated in condensed matter physics. As an example consider the isotropic model of a ferro-magnet: although the Hamiltonian is rotationally invariant, the ground state is not. The magnetic moments point all in the same direction. In models of elementary particle physics, symmetry breaking is most often described in terms of a scalar field, the Higgs field. In condensed matter physics this field is called the order parameter. It can also be a vector or tensor field. A symmetry is called spontaneously broken if the ground state is not invariant under the full symmetry of the Lagrangian (or Hamiltonian) density. Since the symmetry group can be represented as a group of linear transformations, this implies that the vacuum expectation value of the Higgs field is non-zero. The essential features of a spontaneously broken symmetry can be illustrated with a simple model which was first studied by Goldstone (1961) . This model has the classical Lagrangian density Here is a complex scalar field and and are real positive constants. The potential in (4) is called the ’Mexican hat potential’ as it looks somewhat like a Mexican sombrero. This Lagrangian density is invariant under the group of global phase transformations, The minima of the potential lie on the circle which is called the ’vacuum manifold’, (here and in what follows denotes an -sphere of radius and denotes an -sphere of radius ). The notion ’global’ indicates that the symmetry transformation is global, i.e., is independent of the spacetime position . The quantum ground states (vacuum states) of the model are characterized by A phase transformation changes into , hence a ground state is not invariant under the symmetry transformation given in Eq. (5). (Clearly, the full vacuum manifold is invariant under symmetry transformations and thus a mixed state which represents a homogeneous mixture of all vacuum states is still invariant even though no pure state is.) The only state invariant under the symmetry (5), characterized by , corresponds to a local maximum of the potential. Small perturbations around this ’false vacuum’ have ’negative mass’ which indicates the instability of this state: The vacuum states of the broken symmetry are all equivalent and we can thus reveal their properties by studying one of them. For convenience we discuss the vacuum state with vanishing phase, . Expanding the field around this state yields where and are real fields. The Lagrangian density in terms of and becomes The interaction Lagrangian is easily determined from the original Lagrangian, (4). This form of the Lagrangian shows that the degree of freedom is massive with mass while describes a massless particle (circular excitations), a Goldstone boson. This simple model is very generic: whenever a continuous global symmetry is spontaneously broken, massless Goldstone bosons emerge. Their number equals the dimension of the vacuum manifold, i.e., the dimension of a group orbit (in the space of field values). In our case the space of field values is . A group orbit is a circle of dimension leading to one massless boson, the excitations tangential to the circle which cost no potential energy. The general result can be formulated in the following theorem: (Goldstone, 1961) If a continuous global symmetry, described by a symmetry group is spontaneously broken to a sub-group , massless particles emerge. Their number is equal to the dimension of the vacuum manifold (the “number of broken symmetries”). Generically, where here means topological equivalence. In our example , and . Another well-known example are the three pions, , , which are the Goldstone bosons of isospin (proton/neutron) symmetry. There the original symmetry, is completely broken leading to Goldstone bosons (see. e.g. ). Very often, symmetries in particle physics are gauged (local). The simplest gauge theory is the Abelian Higgs model (sometimes also called scalar electrodynamics). It is described by the Lagrangian density where is again a complex scalar field and is the covariant derivative w.r.t. the gauge field . is the gauge field-strength, the gauge coupling constant and is the potential given in Eq. (4). This Lagrangian is invariant under the group of local transformations, The minima of the potential, , are not invariant, the symmetry is spontaneously broken. Expanding as before around the vacuum expectation value , we find where, as in the global case, . Here is no longer a physical degree of freedom. It can be absorbed by a gauge transformation. After the gauge transformation the Lagrangian given in Eq. (11) becomes with and . The gauge boson “absorbs” the massless Goldstone boson and becomes massive. It has now three independent polarizations (degrees of freedom) instead of the original two. The phenomenon described here is called the ’Higgs mechanism’. It works in the same way also for more complicated non-Abelian gauge theories (Yang Mills theories). On the classical level, what we have done here is just rewriting the Lagrangian density in terms of different variables. However, on a quantum level, particles are excitations of a vacuum state, a state of lowest energy, and these are clearly not described by the original field but by the fields and in the global case and by and in the local case. The two models presented here have very close analogies in condensed a) The non-relativistic version of Eq. (4) is used to describe super fluids where is the Bose condensate (the best known example being super fluid He). b) The Abelian Higgs model, Eq. (10) is the Landau Ginzburg model of super-conductivity, where represents the Cooper pair wave function. A very physical and thorough account of the problem of spontaneous symmetry breaking can be found in Weinberg . It is possible that also the scalar fields in particle physics (e.g. the Higgs of the standard model which is supposed to provide the masses of the and ) are not fundamental but “condensates” as in condensed matter physics. Maybe the fact that no fundamental scalar particle has been discovered so far has some deeper significance. 2.2 Symmetry restoration at high temperature In particle physics like in condensed matter systems, symmetries which are spontaneously broken can be restored at high temperatures. The basic reason for this is that a system at finite temperature is not in the vacuum state which minimizes energy, but in a thermal state which tends to maximize entropy. We thus have to expand excitations of the system about a different state. More precisely, it is not the potential energy, but the free energy which has to be minimized. The equilibrium value of at temperature , , is in general temperature dependent . At low temperature, the entropy term is unimportant. But as the temperature raises, low entropy becomes more and more costly and the state tends to raise its entropy. The field becomes less and less ordered and thus its expectation value becomes smaller. Above a certain critical temperature, , the expectation value vanishes. If the coupling constants are not extremely small, the critical temperature is of order . To calculate the free energy of quantum fields at finite temperature, one has to develop a perturbation theory similar to the Feynman diagrams, where ordinary Greens functions are replaced by thermal Greens functions. The inverse temperature, plays the role of an imaginary time component. It would lead us too far from the main topic of this review to introduce thermal perturbation theory, and there are excellent reviews on the subject available, see, e.g. [11, 157, 86, 156, 39, 90]. Here we give a much simplified derivation of the lowest order (tree level) thermal correction to the effective potential . In lowest order the particles are non-interacting and their contributions to the free energy can be summed (each degree of freedom describes one particle), Here is the zero temperature effective potential and is the free energy of each degree of freedom, as known from statistical mechanics. The upper sign is valid for bosons and the lower one for fermions. . For the free energy is exponentially small. But it can become considerable at high temperature, , where we obtain If symmetry restoration occurs at a temperature well above all the mass thresholds, we can approximate by Here, is the formal mass given by . If the potential contains a -term, the mass includes a term , which leads to a positive quadratic term, . If the temperature is sufficiently high, this term overcomes the negative quadratic term in the Mexican hat potential and becomes a global minimum of the potential. The temperature at which this happens is called the critical temperature. In the Abelian Higgs model, the critical temperature , becomes For non-Abelian models one finds analogously The critical temperature for global symmetry breaking, i.e. without gauge field, is obtained in the limit . As expected, for one finds Like in condensed matter systems, a phase transition is second order if is a local maximum and first order if is a local minimum. In the example of the Abelian Higgs model, the order depends on the parameters and of the model. In models, or any other model where the vacuum manifold (i.e. the space of minima of the effective potential) of the broken symmetry phase is non-trivial, minimization of the effective potential fixes the absolute value of but the direction, , is arbitrary. The field can vary in the vacuum manifold, given by the sphere for models. At low temperature, the free energy is minimized if the phase is constant (no gradient energy) but after the phase transition will vary in space. The size of the patches with roughly constant direction is given by the correlation length which is a function of time. In the early universe is bounded by the size of the causal horizon, Formally diverges at the phase transition, but also our perturbative treatment is no longer valid in the vicinity of the phase transition since fluctuations become big. A thorough treatment of the physics at the phase transition is the subject of modern theory of critical phenomena and goes beyond the scope of this review. Very often, the relevant correlation length is the correlation length at the Ginsburg temperature, , the temperature at which thermal fluctuations are comparable to the mass term. However, in the cosmological context there is also another scale, the expansion time. As the system approaches the phase transition, it takes longer and longer to reach thermal equilibrium, and at some temperature, expansion is faster than the speed at which the system equilibrates and it falls out of thermal equilibrium. It has been argued that it is the correlation length at this moment, somewhat before the phase transition, which is relevant. If the phase transition is second order, the order parameter changes continuously with time. In a first order transition, the state is meta-stable (false vacuum) and the phase transition takes place spontaneously at different positions in space and different temperatures via bubble nucleation (super cooling). Thermal fluctuations and/or tunneling take the field over the potential barrier to the true vacuum. The bubbles of true vacuum grow and eventually coalesce thereby completing the phase transition. It is interesting to note that the order of the phase transition is not very important in the context of defects and structure formation. Even though the number of defects per horizon volume formed at the transition does depend on the order and, especially on the relevant correlation length , this can be compensated by a slight change of the phase transition temperature to obtain the required density of defects. As we have seen, a non-trivial vacuum manifold, , in general implies that shortly after a phase transition the order parameter has different values at different positions in space. Such non-trivial configurations are generically unstable and will eventually relax to the configuration constant, which has the lowest energy. Naturally, we would expect this process to happen with the speed of light. However, it can be slowed significantly for topological reasons and intermediate long lived configurations with well confined energy may form, these are topological defects. Such defects can have important consequences in cosmology. Several exact solutions of topological defects can be found in the literature, see e.g. . In the case of global defects, i.e. defects due to global symmetry breaking, the energy density of the defect is mainly due to gradient energy in the scalar field and is therefore not well localized in space. The scalar field gradient of local defects (defects due to the braking of a local, or gauge symmetry) is compensated by the gauge field and the energy is well confined to the location of the defect. To exemplify this, we present the solutions for a global and a local straight cosmic string. 2.3 Exact solutions for strings 2.3.1 Global strings We consider a complex scalar field, , with Lagrangian at low temperature. The vacuum manifold is a circle of radius , . At high temperature, , the effective potential has a single minimum at . As the temperature drops below the critical value , a phase transition occurs and assumes a finite vacuum expectation value which is uncorrelated at sufficiently distant points in physical space. If we now consider a closed curve in space it may happen that winds around in the circle . We then have with with . Since the integer (the winding number of the map ) cannot change if we shrink the curve continuously, the function must be ill defined somewhere in the interior of , i.e. must assume the value and thus have higher potential energy somewhere in the interior of . If we continue this argument into the third dimension, a string of higher potential energy must form. The size of the region within which leaves the vacuum manifold, the diameter of the string, is of the order . For topological reasons, the string cannot end. It is either infinite or closed.111 The only exception may occur if other defects are present. Then a string can end on a monopole. We now look for an exact solution of a static, infinite straight string along the -direction. We make the ansatz with and , is the usual polar angle. The field equation of motion then reduces to an ordinary differential equation for , where and . A solution of this differential equation which satisfies the boundary conditions and can be found numerically. It is a function of . and behaves like The energy momentum tensor of the string is given by The energy per unit length of a cross-section of string out to radius is The log divergence for large results from the angular dependence of , the gradient energy, the last term in Eq. (26), which decays only like . In realistic configurations an upper cutoff is provided by the curvature radius of the string or by its distance to the next string. Also for a single, spherically symmetric global monopoles solution the total energy divergies (linearly). A non-trivial results shows, however, that the energy needed to deform the monopole into a topologically trivial configuration is finite . 2.3.2 Local strings We also describe a string solution of the Abelian Higgs model, the Nielson-Oleson or Abrikosov vortex . The Lagrangian density is the one of scalar electrodynamics, Eq. (10), We are looking for a cylindrically symmetric, static solution of the field equations. For we want the solution to approach a vacuum state, i.e. and so that (the gauge field ‘screens’ the gradient energy). We insert the following ansatz into the field equations which leads to two coupled ordinary differential equations for and Solutions which describe a string along the -axis satisfy the asymptotics above which require The solution to this system of two coupled ordinary differential equations is easily obtained numerically. Asymptotically, for , the -equation reduces to the differential equation for a modified Bessel function and we have For large values of , the falloff of is controlled by the gauge field coupling, . For the gauge field coupling can be neglected at large radii and we obtain for This field configuration leads to , hence ; and The energy per unit length of the string is with . All the terms in the integral are regular and decay exponentially for large . The energy per unit length of gauge string is finite. The gradient energy which leads to the divergence for the global string is ‘screened’ by the gauge field. The integral is of the order . This value is exact in the case , where it can be computed analytically. In the general case with , we have where is a slowly varying function of order unity. The thickness of a Nielson Oleson string is about and on length scales much larger than we can approximate its energy momentum tensor by 2.4 General remarks on topological defects In three spatial dimensions four different types of defects can form. The question whether and what kind of topological defects form during a symmetry breaking phase transition is determined by the topology of the vacuum manifold : If is disconnected, domain walls from. Example: if the symmetry for a real scalar field is spontaneously broken, . Domain walls form when discrete symmetries are broken. Discrete symmetries are not continuous and therefore cannot be gauged. Hence domain walls are always global defects. If there exist loops in which cannot be continuously shrunk into a point, is not simply connected, strings form. Example: if is completely broken by a complex scalar field, , see previous subsection. If contains non-contractible spheres, monopoles form. Example: if is broken to by a three component scalar field, . If contains non-contractible 3-spheres, textures form. Example: if is broken to by a four component scalar field, . These topological properties of are best described by the homotopy groups, . The group is relevant for the existence of textures, decides about monopoles, is relevant for strings and for domain walls . If a symmetry group is spontaneously broken to a subgroup , the vacuum manifold is in general equivalent to the quotient space, . In the monopole example above, we have and . The vacuum manifold is . 2.5 Defect formation and evolution in the expanding universe Our universe which is to a good approximation an expanding Friedman universe was much denser and hotter in the past. During expansion the universe may cool through a certain critical temperature at which a symmetry is spontaneously broken down to . If is topologically non-trivial, topological defects can form during the phase transition. This scenario is called the Kibble mechanism . We apply the Kibble mechanism to estimate the energy density in defects from phase transitions with different vacuum manifolds at a given temperature . Consider a field theory with symmetry group and Higgs-field with a self-interaction potential . For illustration we use , and At finite temperature, the free energy is of the form where is a real constant given by combinations of and other coupling constants (e.g. gauge couplings, Yukawa couplings). The sign of depends on the number of fermions. We assume , i.e. that there are only few fermions and sufficiently small Yukawa-couplings. Then, from Eqs. (39) and (40), we see that the effective masses of the field at temperature and are At , this field theory undergoes a second order phase transition: the equilibrium point becomes unstable for ( becomes negative) and assumes a non-vanishing vacuum expectation value. For another form of , the equilibrium at can be meta-stable so that the phase transition is of first order. Hence, to decide whether the transition is of first or second order, it is important that we can rely on the form of the effective potential which is obtained by perturbation theory or by numerical lattice calculations. This is in general a difficult problem. For the electro-weak theory, e.g. , it was discovered only recently that the electroweak transition is probably not a real phase transition but just a continuous cross-over . The correlations of the field are described by the thermal Greens functions: For massive particles at where one can write where are the zero temperature contributions. For such that we have, with : For , and therefore, at for large . The correlation length for the phase transition (of order) is defined as . This definition is different from the definition used in solid state physics. There one defines the correlation length as the length above which the correlation decreases exponentially. In this sense, the correlation length would be infinite at (). In cosmology the correlation length cannot diverge because of causality. It is bounded from above by the distance a photon can travel during the age of the universe until . This distance is (for non-inflationary expansion) given by Hence another meaningful definition of the correlation length would be . Often also the correlation length at the Ginzburg temperature or at the temperature (before the phase transition) at which the system drops out of thermal equilibrium is chosen. For the following it is not important which of the above definitions we use. We only require that . We now suppose that directly after the phase transition, the vacuum expectation value takes arbitrary uncorrelated values in points with distance , but stays continuous (finite gradient energy!). If is non-trivial for , the map for a large enough -sphere in physical space, may represent a non-trivial element of . Then cannot be contracted continuously to a point on and, somewhere inside , has to leave the vacuum manifold, . These positions of higher potential energy are topological defects. The type of defect formed depends on the order of the non-trivial homotopy group: : 2-dimensional defects, domain walls, : line-like defects, cosmic strings, : point-like defects, monopoles, : event-like defects, texture, . Here is the spacetime dimension of the defect, . If a vacuum manifold has several nontrivial homotopy groups with , generically only the lowest defects survive and the higher order defects are instable. As an example, in the isotropic to nematic phase transition of liquid crystals is broken to leading to . This allows for texture, monopoles and strings, but textures decay into monopole anti-monopole pairs and monopole/anti-monopole pairs are connected by strings and attract each other until they annihilate. Only strings scale . The case of texture, can be described in this context only if either the universe is closed and physical space is a three sphere of if is asymptotically parallel, i.e. . Then the points can be identified in all directions and we can regard as a map from to and ask whether this map is topologically trivial or not. In the cosmological context this concept violates causality. However, the texture case allows for a texture winding number density whose integral over all of space only takes integer values if is asymptotically constant (or space is a three sphere). The integral of the winding number density over a region of space tells us whether the field configuration inside contains textures. According to the above description of the process of defect formation after a cosmological phase transitions, called the Kibble mechanism, we typically expect on the order of one defect per horizon volume. Simulations and analytical arguments show that the actual number is somewhat larger for cosmic strings and significantly smaller for texture. If defects are local, the scalar field gradiants are compensated by the gauge field and they do not interact at large distance other than gravitationally in the simplest model, where no massless charged particles ’live’ inside the defect. An exeption to this are superconducting cosmic string . For example local monopoles do not annihilate once they are formed and their density just scales with the expansion of the Universe, like . Since they are non-relativistic, , their energy density scales the same way and they soon dominate the total energy density of the universe. Every simple GUT group produces monopoles when it breaks down to the standard model symmetry group, . The observed absence of monopoles therefore represents a serious problem for the unification of standard cosmology with grand unified theories . Local texture, on the contrary, soon thin out and do not induce sufficiently strong perturbations to generate structure in the universe. Only local strings scale, i.e. contribute a constant small fraction to the energy density of the universe, and are therefore possible candidates of topological defects for structure formation. If the group is non-Abelian ( is the only homotopy group which can be non-Abelian), the cosmic string network becomes ’frustrated’ and does not scale. Such a low energy frustrated string network has been proposed as candidate for the cosmic dark energy . The situation is different for global defects. There, the main contribution to the energy density comes from the Higgs field and scales as , like the background energy density in the universe (up to logarithmic corrections for global strings). The only exception are domain walls which are forbidden, since they soon come to dominate, leading to a very inhomogeneous universe. Recently, however ’soft domain walls’ , i.e. domain walls forming at a late time phase transition, have been studied. 3 Theoretical Framework 3.1 Linear cosmological perturbations with seeds A basic tool for cosmic structure formation is linear cosmological perturbation theory. The fact that CMB anisotropies are small shows that at least initially also perturbations in the matter density have been much smaller than unity and therefore they may be treated within linear perturbation theory. It is generally assumed (an assumption which is supported by several observational facts, see e.g. ) that perturbations are still linear on scales above about Mpc. On smaller scales non-linear N-body simulations are needed to compute the evolution of density fluctuations. The principal difference in perturbation theory in models with topological defects as compared to the more familiar inflationary models, is the fact that here cosmic perturbation equations are not homogeneous. The perturbations are induced by ’seeds’ which are not present in the background energy momentum tensor.. The defect energy momentum tensor enters in the perturbation equation as ’source’ or ’seed’ term on the right hand side, but the defects themselves evolve according to the background space-time. Perturbations in the defect evolution are of second order. (This procedure has sometimes also been termed the ’stiff approximation’ , but it is actually nothing else than consistent linear perturbation theory.). Gauge-invariant perturbation equations for cosmological models with seeds have been derived in Refs. [41, 42]. Here we follow the notation and use the results presented in Ref. . Definitions of all the gauge-invariant perturbation variables used in this Review in terms of perturbations of the metric, the energy momentum tensor and the photon and neutrino brightness are given in Appendix A for completeness. We consider a background universe with density parameter , consisting of photons, cold dark matter (CDM), baryons and neutrinos. At very early times , photons and baryons form a perfectly coupled ideal fluid. As time evolves, and as the electron density drops due to recombination of primordial helium and hydrogen, Compton scattering becomes less frequent and higher moments in the photon distribution develop. This process has to be described by a Boltzmann equation. Long after recombination, free electrons are so sparse that the collision term can be neglected, and photons evolve according to the collisionless Boltzmann or Liouville equation. During the epoch of interest here, neutrinos are always collisionless and thus obey the Liouville equation. In the next section, we parameterize in a completely general way the degrees of freedom of the seed energy momentum tensor. Section 3.3 is devoted to the perturbation of Einstein’s equations and the fluid equations of motion. Next we treat the evolution of CMB photons by the Boltzmann perturbation equation, including polarization. The detailed derivations as well as the expressions for the CMB anisotropy and polarization power spectra are given in Appendix B. We finally explain how to compute the power spectra of density fluctuations, CMB anisotropies and peculiar velocities by means of the derived perturbation equations and the unequal time correlators of the seed energy momentum tensor which are obtained by numerical simulations. 3.2 The seed energy momentum tensor Since the energy momentum tensor of the seeds, , does not contribute to the background Friedman universe, it is gauge invariant by itself according to the Stewart-Walker Lemma . can be calculated by solving the matter equations for the seeds in the Friedman background geometry. Since has no background component it satisfies the unperturbed “conservation” equations. We decompose into scalar, vector and tensor contributions. They decouple within linear perturbation theory and it is thus possible to write the equations for each of these contributions separately. As always (unless noted otherwise), we work in Fourier space, is the comoving wave number and . We parameterize the scalar vector and tensor contributions to in the form Here denotes a typical mass scale of the seeds. In the case of topological defects we set , where is the symmetry breaking scale . The vectors and are transverse and is a transverse traceless tensor, From the full energy momentum tensor which contains scalar, vector and tensor contributions, the scalar parts and of a Fourier mode are given by On the other hand and are also determined in terms of and by energy and momentum conservation, Once is known it is easy to extract For we use Again, can also be obtained in terms of by means of momentum conservation, The geometry perturbations induced by the seeds are characterized by the Bardeen potentials, and , for scalar perturbations, by the potential for the shear of the extrinsic curvature, , for vector perturbations and by the gravitational wave amplitude, , for tensor perturbations. Detailed definitions of these variables and their geometrical interpretation are given in Ref. (see also Appendix A). Einstein’s equations for an unperturbed cosmic background fluid with seeds relate the seed perturbations of the geometry to the energy momentum tensor of the seeds. Defining the dimensionless small parameter we obtain to first order in Eqs. (56) to (59) would determine the geometric perturbations if the cosmic fluid were perfectly unperturbed. In a realistic situation, however, we have to add the fluid perturbations which are defined in the next subsection. Only the total geometrical perturbations are determined via Einstein’s equations. In this sense, Eqs. (56) to (59) should be regarded as definitions for and . A description of the numerical calculation of the energy momentum tensor of the seeds for global defects and cosmic strings is given in Chapter 4. 3.3 Einstein’s equations and the fluid equations 3.3.1 Scalar perturbations Scalar perturbations of the geometry have two degrees of freedom which can be cast in terms of the gauge-invariant Bardeen potentials, and [8, 91]. For Newtonian forms of matter, is nothing else than the Newtonian gravitational potential. For matter with significant anisotropic stresses, and differ. In geometrical terms, the former represents the lapse function of the zero-shear hyper-surfaces while the latter is a measure of their 3-curvature . In the presence of seeds, the Bardeen potentials are given by To describe the scalar perturbations of the energy momentum tensor of a given matter component, we use the gauge invariant variables for density fluctuations, corresponding to the usual density fluctuation in the ’flat gauge’, , for the potential of peculiar velocity fluctuations, corresponding to the usual velocity potential in the longitudinal gauge and , a potential for anisotropic stresses (which vanishes for CDM and baryons). A definition of these variables in terms of the components of the energy momentum tensor of the fluids and the metric perturbations can be found in Refs. or and in Appendix A. Subscripts and superscripts , , or denote the photons, CDM, baryons and neutrinos respectively. Einstein’s equations yield the following relation for the matter part of the Bardeen potentials Note the appearance of on the r.h.s. of Eq. (62). Using the decompositions (60,61) we can solve for and in terms of the fluid variables and the seeds. With the help of Friedman’s equation, Eqs. (62) and (63) can then be written in the form

Assigned: Nov. 10/11 Due: Nov. 24/27 assignment is due (in hardcopy form) by the beginning of your class on the date shown above. Assignments up to 24 hours late will receive a 25% penalty. Assignments more than 24 hours late will not be accepted. Late assignments can be given to Helen Ho in HA 266 or to Ines Bilec in the Undergraduate office or can be given to your instructor. are possible only for genuine emergencies. Permission for the extension must be in advance (i.e. at least 24 hours before the assignment is due) and must be accompanied by appropriate documentation. See Ines Belic in the Undergraduate 3. Show your working for all questions. Please be concise with your answers, and clearly identify the answer to the specific question you are being asked. Please carry 2 non-zero digits to the right of the decimal for answers that do not work out to whole numbers. Explain your reasoning where asked to do so. 4. Please use the remaining pages (not this page) as a template.You may type in your answers before printing out the document or you may print out the document and print your answers neatly by Your answers must be neat and easily readable. If not, marks will be deducted or the answer will be ignored entirely. If in doubt, type your answer. Please confine your answer to the allowed space. Do question 7 before printing out the template as question 7 requires you to cut and paste a spreadsheet into the template. 5. Some of the questions have lines on which you are to write the answer. Do not exceed this space. Part of your task is to choose the right thing to say in a small space. 6. If you do not follow these instructions you will not get full credit for your work. often like to work together in doing assignments. However, this is NOT a group assignment. It is primarily an individual effort. The basic rule about joint work is that it is acceptable to discuss questions with classmates but you must do the actual write-up of the assignment on your own. You must not copy someone else’s answer. Copied or plagiarized answers will be subject to appropriate 8. You may use pen or pencil but if you use pencil we cannot review situations where you think there is a marking mistake (except addition errors). assignment is challenging. Do not leave it to the last minute. When doing each question it is a good idea to find and read the relevant sections of the textbook first. Good Luck. UBC Sauder School of Business Commerce 295 / FRE 295 7 _ /10 8 _______ /10 TOTAL _________ / 100 question is worth 10 pts. Each part of each question is worth 5 pts. Show your 1. (Game Theory I)Suppose Air Canada and West Jet are deciding whether to offer a frequent flyer program (FFP). The annual profits of the two firms from each combination of strategies they choose is given in the following payoff matrix. The firms choose simultaneously. a)Identify any dominant strategies. Determine the Nash equilibrium in this game.(State the strategies and the payoffs.) Determine the maximin solution. Is this an example of a Prisoner’s Dilemma? In answering this question provide the relevant definitions and explain briefly how they b) Here is an entry game. Do Not Enter Do Not Enter any Nash equilibria in pure strategies. If firm 1 plays a mixed strategy of entering with 90% probability, what is the best response of firm 2? Is this combination of strategies a Nash equilibrium? Explain briefly. (Hint: when considering whether a proposed solution is a Nash equilibrium ask whether either firm regrets its choice – whether its best response is different from what it actually chose.) 2. (Coordination Games). Consider the following network scheduling payoff matrix for Shaw Cable and Telus. any Nash equilibria in this game. Explain briefly. b)Would pre-play communication or application of the Pareto criterion be helpful in this game? Explain briefly. Games)Apple and Microsoft must decide on a pricing strategy for their tablet devices, the iPad and the Surface. Their respective marketing departments have determined two potential prices, one high and one low. The payoffs (in millions of dollars) are summarized in the table below. a) What is the Nash equilibrium in the one-shot (static) game? Now suppose that the companies repeat this game every year with new versions of the tablets. Describe two different Nash equilibria of this infinitely repeated game and explain why each is an equilibrium. Your description of each equilibrium should be a pair of strategies (one for each firm), and the equilibrium payoffs for each firm per round. b) A start-up competitor has developed a new technology that will make current tablet computers obsolete. There is no uncertainty about the success of the new venture, but it will take an additional 3 years to manufacture the required components. The founder of the start-up makes a public announcement that the new device will go on sale in three years. Do we expect Apple and Microsoft to make the most of the next three years by colluding on high prices? Explain your answer using appropriate Oligopoly)Consider the competition between Honda and Toyota in the compact car market. Each firm must commit to a production level at the beginning of the year. The payoffs for three possible outputs of each firm (in millions of dollars) are given in the table below. Suppose Honda preempts Toyota by finishing plans on its latest upgrade early, and therefore makes a public commitment on quantity before Toyota. a) Draw the relevant extensive form representation (game tree) for this strategic interaction. What is the sub-game perfect Nash Equilibrium? b) Suppose Honda must spend an additional 15 million dollars to be in the position to announce first, and otherwise the two firms announce simultaneously. Is it profitable for Honda to expedite planning? Now suppose that Toyota goes first unless Honda spends the $15 million to expedite planning, in which case Honda goes first. Would the 15 million dollar investment be worthwhile in this case? Explain. 5. Sequential Games a) Consider the following game tree describing strategic interaction between an incumbent monopolist and a For what range of values of X and Y will the incumbent deter entry by investing? Show your working. b) A mining company is considering starting operations in a foreign country known to be rich in mineral deposits but with an unreliable government. Extracting the minerals will be very profitable, but requires extensive and costly digging before anything can be extracted from the ground. Explain briefly using appropriate terminology why the company might be hesitant to start work on this potentially profitable project. (No diagram is needed.) 6. Behavioral Economics a. Consider a game with 10 players. Each person submits an integer between 0 and 100. The winner of the game is the person who submits a number closest to 60% of the average number submitted. The winner gets a prize of $1000 and no one else gets anything. If two or more people tie for the win they share the prize equally. Assuming that all players are fully rational and that rationality is common knowledge, what is the Nash equilibrium? If you were playing this game with randomly chosen first year UBC students, would you bid the Nash equilibrium The Nash equilibrium is We know this is a Nash equilibrium because If I were trying to win this game I would submit the value _________________________. I think this offers the best chance of success because _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. b. State whether prospect theory, the reciprocity norm, or neither can explain the following phenomena. Circle the correct response and explain briefly. Framing effects. Prospect Theory or Reciprocity Norm or Neither. The Certainty Effect. Prospect Theory or Reciprocity Norm or Neither. Overconfidence.Prospect Theory or Reciprocity Norm or Neither. Behavior in Ultimatum Games. Prospect Theory or Reciprocity Norm or Neither. Bounded Rationality. Prospect Theory or Reciprocity Norm or Neither. 7. (Uncertainty)A manager must choose one of four advertising strategies, with the following characteristics. The manager cares only about the expected value and variance of the projects. a. Put this data into a spreadsheet, add three columns with headings EV, VAR, and SD, and use the spreadsheet to calculate the expected value, variance, and standard deviation. Paste the resulting spreadsheet in the space below. Which strategy would be chosen by a risk preferring manager? a risk neutral manager? a risk averse manager? A risk preferring manager would choose _____ because _________________________________ A risk neutral manager would choose ________ because _________________________________ For a risk averse manager we can say ________________________________________________ b. Suppose utility is given by U = ?Y where Y is the return. For Strategy B what is the expected utility and what is the risk premium? Draw the associated diagram showing the risk premium. (Do not worry about making the diagram precise. Just illustrate the general idea, but include the relevant numbers on the axes.) Selection)Consider the used car example described in Q&A 15.1, except the numbers are different. Buyers value lemons at $5000 and good used cars at $10,000. The reservation price of lemon owners is $4000 and the reservation price of owners of good used cars is $7,500. Owners know the quality of the cars but buyers do not. (Show your working.) a. Suppose that 40% of the used cars are good used cars and 60% are lemons. Describe the equilibrium. b. Now suppose that 60% of the cars are good used cars and 40% are lemons. Describe the equilibrium now. 9. (Agency) Priscilla hires Arnie to manage her business; both of them are risk neutral. The following table shows Priscilla’s profits (before paying Arnie) for two different levels of Arnie’s effort and two different market situations. If Arnie provides low effort, his cost of effort is zero but if he provides high effort his cost is 10. Weak demand and strong demand are equally likely. (Show your working.) Weak Demand Strong Demand Low Effort 20 40 High Effort 40 80 If Priscilla offers a profit sharing contract in which Arnie gets 30% of the profits, what effort level would Arnie provide? Calculate Priscilla’s expected net profits (after paying to Arnie) under this contract. If Priscilla offers a bonus contract in which Arnie gets a base salary of 4 (no matter whether demand is weak or strong) plus 80% of all profits exceeding 40, what effort level would Arnie choose now? Which contract (profit sharing or bonus) would Priscilla prefer? Which contract would Arnie prefer? Holloway, Lee and Shen (2014) study the determinants of cross-border leveraged buyouts. a) The authors present a stylized model of the global competition among buyout firms. i) What factors in the model determine the number of acquisitions by buyout firm in each country? ii) Can you think of any other factors that likely play a role in the real world? iii) Where do these other factors implicitly enter the equation for the number of completed buyout deals by each investor in each country? b) The authors test the predictions of the model using a sample of worldwide buyout deals. i) Which two countries dominate as home countries for buyout firms in the sample? ii) How do the authors test the prediction that transaction costs associated with remote ownership impact the number of deals? How do they measure the theoretical cost mitigation ability? iii) Are results consistent with the predictions of the model? How can you tell?

General Astronomy/The Distance Ladder The Cosmic Distance ladder—each step is less accurate that the one before - Radar is the most accurate method, but it works only on solid bodies in the solar system. The Sun and gaseous planets such as Jupiter do not reflect radio waves and hence cannot be ranged this way. - Stellar parallax, or distance based on how a star's apparent location varies with the observer's location, is the original method used to determine stellar distances. This is the most accurate method for estimating distances to stars, but is limited to only the nearest stars. Parallax is smaller the farther away a star is, and can be detected for only several thousand stars as viewed from Earth. However, parallax is detectable for many more by spacecraft above the Earth's obscuring atmosphere. - Cepheid variables, is a method of determining cosmic distances based on the "period luminosity" characteristics of certain stars. Cepheids are stars whose total brightness (luminosity) varies with specific periods. They vary in brightness from brightest to dimmest to brightest in certain time periods. It turns out that the period is related to the star's absolute magnitude or luminosity. Luminosity is a measure of how bright the star truly is at all wavelengths.[By comparison, light bulbs come in various luminosities, such as 60 watts or 100 watts. A 60 watt light bulb will appear much brighter at 5 feet than at 50 feet, but it will still have a 60-watt luminosity. If you know what the luminosity is, and can measure the apparent brightness, you can determine the distance by a simple mathematical relation.] By observing the star's period, we know its luminosity (which is directly related to another quantity called "absolute magnitude". By then comparing the known brightness [absolute magnitude] to the apparent brightness [apparent magnitude], the distance can be found by applying the inverse square law [just as you could determine the distance to a 60-watt light bulb if you could measure how bright it appears at your distance]. The specific equation is known as the "distance modulus." This is most useful for finding distances to stars clusters of which the Cepheid is a part. (Other stars with a similar relationship are the RR Lyra variables.) - Spectroscopic parallax and related techniques based on spectral classification and the HR diagram (below). That is, if we know the spectral type of a "normal" star, we know its luminosity (actual total energy output at all wavelengths). This is because stars follow a pattern. That is, all G-type "normal" (non-Giant) stars are like all other G-type stars. All B-type normal stars are like all other B-type normal stars. More specifically, all G2 normal stars (such as our Sun) have approximately the same luminosity. Thus, if we can identify a star as being a G2 from its spectrum, then we know its luminosity, because we know that it is essentially the same as the Sun's. If we know the luminosity or absolute magnitude, we can compare that to the observed brightness [apparent magnitude] to deduce mathematically the actual distance by the distance modulus. - Main Sequence fitting is a process related to "spectroscopic parallax" that compares the Hertzsprung -Russell (HR) diagram of the stars in a star cluster, with a calibrated HR diagram to determine absolute magnitude. This again is compared to the apparent magnitude of the stars and the distance determined by the distance modulus relation. The processes above are useful for determining distances within the Milky Way Galaxy. Several other techniques for determining distances beyond our Milky Way Galaxy: - Cepheid variables. The technique can be used for a few nearby galaxies, but is limited because outside of a few nearby galaxies, individual stars cannot be seen. - Supernova magnitudes. Type Ia supernovae tend to be caused by the same series of events and all tend to be the same absolute magnitude. Since they are so bright (sometimes exceeding the brilliance of an entire galaxy) they can be seen much farther than individual stars and distance can be obtained through the distance modulus by comparing the apparent magnitude to the known absolute magnitude. - The Tully-Fisher relationship. Astronomers have noted that the mass of spiral galaxy is related to its rotation rate. Mass is related to the number of stars, and the number of stars is related to the absolute magnitude of the galaxy (the greater the mass, the more stars and the brighter the galaxy). Thus by measuring rotation rate of the galaxy, and estimation of its true brightness can be made, and distance follows through the distance modulus. - Red-Shift. Galaxies show shifts in the position of spectral lines in their light, dependent on how far the galaxy is. The farther the galaxy is, the more its light (and the spectral lines contained in that light) is shifted toward the red end of the spectrum. Thus by simply measuring the amount of shift, astronomers obtain an idea of how quickly the galaxy is moving away from us; given an assumed cosmological model this then allows the distance to the galaxy to be calculated.

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals? Can you arrange 5 different digits (from 0 - 9) in the cross in the way described? You have 5 darts and your target score is 44. How many different ways could you score 44? Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information? In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins? Using the statements, can you work out how many of each type of rabbit there are in these pens? In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case? Use these head, body and leg pieces to make Robot Monsters which are different heights. This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15! This challenge is about finding the difference between numbers which have the same tens digit. This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards. You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how? There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it? Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own? In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible. Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions. This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! Try this matching game which will help you recognise different ways of saying the same time interval. Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make? There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs. This challenge focuses on finding the sum and difference of pairs of two-digit numbers. Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs? Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make? There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places. Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it? In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make? Can you find all the ways to get 15 at the top of this triangle of numbers? This task follows on from Build it Up and takes the ideas into three dimensions! Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens? There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find? Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families? Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it. Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest? Can you put the numbers 1 to 8 into the circles so that the four calculations are correct? Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it. Ben has five coins in his pocket. How much money might he have? What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way? El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps? Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number. Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be? Find all the numbers that can be made by adding the dots on two dice. Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon? On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.? Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores. Moira is late for school. What is the shortest route she can take from the school gates to the entrance? Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag? What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters. Can you substitute numbers for the letters in these sums?

Pi, denoted by the Greek letter π, is one of the most famous and intriguing constants in mathematics. Its value, approximately 3.14159, is etched into the minds of countless students as they learn about geometry and trigonometry. But beyond its simple numerical representation, pi hides a wealth of mysteries and mathematical wonders that have captivated scholars for centuries. In this article, we will explore the history, properties, and significance of pi123, delving deep into the enigmatic world of this irrational number. A Brief History of Pi The earliest known approximation of pi123 dates back to ancient Egypt around 1900 BC. The Rhind Papyrus, an ancient Egyptian mathematical document, contains a rough estimation of pi as 3.125. Ancient Babylonians and Chinese mathematicians also had their own approximations of pi. However, it was the ancient Greeks who made significant contributions to the understanding of pi. The Greek mathematician Archimedes is often credited with one of the earliest rigorous methods for approximating pi. He used the method of exhaustion, which involved inscribing and circumscribing polygons around a circle, to determine that pi was between 3 1/7 and 3 10/71. pi123 continued to be a subject of fascination throughout history. In the 18th century, the Swiss mathematician Leonhard Euler introduced the symbol π to represent this constant, making it more convenient for mathematicians to work with. The continued development of calculus in the 17th and 18th centuries allowed for more precise calculations and approximations of pi. The 19th and 20th centuries saw remarkable advancements in pi’s calculation, with mathematicians employing various mathematical techniques and computer algorithms to compute its decimal expansion to millions, and even billions, of digits. Defining the Irrationality of Pi One of the most fascinating aspects of pi123 is its irrationality. An irrational number is a number that cannot be expressed as a simple fraction of two integers. Pi cannot be precisely represented as a fraction, making it an infinitely non-repeating, non-terminating decimal. The proof of pi’s irrationality was established in the 18th century by Johann Lambert. His proof demonstrated that pi cannot be expressed as a fraction p/q, where p and q are integers. This discovery added a new layer of intrigue to pi, as it highlighted the complexity and richness of this seemingly simple number. Moreover, pi123 is not just irrational; it is also transcendental. A transcendental number is a number that is not the root of any non-zero polynomial equation with integer coefficients. The proof of pi’s transcendence was achieved in the 19th century by Charles Hermite. This means that pi cannot be the solution to any polynomial equation with whole number coefficients, making it an even more elusive and mysterious mathematical constant. Calculating pi123 to high precision has been a mathematical challenge for centuries. Before the advent of computers, mathematicians used various formulas and methods to calculate pi to more and more decimal places. Archimedes’ method of exhaustion was one such approach, and it provided a reasonably accurate approximation for its time. In the 17th century, the English mathematician John Wallis introduced a formula that involved an infinite product of fractions to approximate pi123. Wallis’s formula was one of the earliest examples of a series expansion for pi and helped improve its accuracy. The 18th century saw the emergence of more sophisticated mathematical tools, such as calculus, which allowed mathematicians to develop series expansions and infinite products that converged more rapidly to the value of pi123. One of the most famous of these series is the Leibniz formula for pi, discovered by the German mathematician Gottfried Wilhelm Leibniz. However, the real breakthroughs in the calculation of pi came with the advent of computers in the 20th century. With the help of algorithms like the Gauss-Legendre algorithm and the Bailey-Borwein-Plouffe (BBP) formula, mathematicians and computer scientists were able to calculate pi to billions of decimal places. In fact, as of my last knowledge update in September 2021, pi had been calculated to trillions of digits using powerful supercomputers. The Ubiquity of Pi pi123 significance goes far beyond the realm of pure mathematics. It appears in various branches of science and engineering, making it one of the most widely used constants in the world. Here are a few examples of its ubiquity: In geometry, pi123 plays a fundamental role in calculations related to circles, spheres, and other curved shapes. The formula for the circumference of a circle, C = 2πr, and the formula for the area of a circle, A = πr², are well-known examples of its importance. Pi appears in trigonometric functions such as sine and cosine. For instance, the period of the sine and cosine functions is 2π, which means that these functions repeat themselves every 2π radians. In physics, pi123 appears in equations describing waveforms, oscillations, and electromagnetic phenomena. It is also crucial in calculations related to the principles of fluid dynamics and electromagnetism. Engineers use pi123 extensively in various calculations, from designing gears and pulleys to analyzing structural stability in architecture. 5. Statistics and Probability In statistics and probability theory, pi is often used to calculate probabilities and areas under curves in various distributions. In the world of cryptography, pi123 is used in algorithms to generate random numbers, which are essential for securing data and communications. Pi in Popular Culture pi123 mystique has not been limited to the realm of academia. It has captured the imagination of artists, writers, and filmmakers. In Darren Aronofsky’s cult film “Pi,” the protagonist becomes obsessed with finding patterns in the decimal expansion of pi, believing that it holds the key to unlocking the secrets of the universe. This film, while highly stylized and fictional, reflects the enduring fascination with pi’s enigmatic nature. Pi has also made appearances in literature, music, and art. Authors like Yann Martel have used pi as a symbol of the infinite and the unknowable in their works. Musicians have composed pieces inspired by pi, and artists have created visual representations of its digits, turning the numerical constant into a source of artistic inspiration. The Quest for More Digits The calculation of pi123 to ever-increasing numbers of decimal places has become a competitive pursuit among mathematicians and computer enthusiasts. This quest for more digits serves both practical and intellectual purposes. On the practical side, more digits of pi are needed for various scientific and engineering calculations, such as those used in GPS technology and the modeling of physical systems. On the intellectual side, the search for more digits is a testament to human curiosity and the desire to explore the limits of mathematical knowledge. The record for calculating the most digits of pi has been repeatedly broken over the years, thanks to the ever-increasing power of supercomputers. As of my last knowledge update in September 2021, the record stood at trillions of digits, a mind-boggling achievement that highlights the remarkable capabilities of modern computing. pi123, the irrational and transcendental number, has fascinated mathematicians, scientists, and enthusiasts for centuries. Its history is a journey through the annals of mathematics, with each era contributing to our understanding of this mysterious constant. Pi’s ubiquity in the sciences and its presence in popular culture make it a symbol of mathematical beauty and the infinite complexity of the universe. As technology continues to advance, the quest to calculate more digits of pi will undoubtedly continue. Yet, pi will remain a symbol of the endless mysteries of mathematics, reminding us that there are always deeper layers of knowledge to uncover and explore. In conclusion, pi123 is not just a number; it’s a symbol of human curiosity, ingenuity, and the enduring quest to understand the mathematical underpinnings of our world. Its allure will continue to inspire generations of mathematicians and scientists, ensuring that the mysteries of pi are explored and celebrated for years to come.

By Sergei Abramovich This publication addresses middle innovations via the convention Board of the Mathematical Sciences - an umbrella company together with 16 expert societies within the usa - concerning the mathematical education of the academics. based on the Board, the concept that of a capstone direction in a arithmetic schooling software contains academics' studying to exploit usually to be had academic software program with the objective to arrive a definite intensity of the maths curriculum via competently designed computational experiments. In flip, the suggestion of test within the educating of arithmetic units up a course towards bettering the 'E' section of the lecturers' literacy within the STEM disciplines. This publication discusses studies in instructing a computer-enhanced capstone direction for potential lecturers of highschool arithmetic. Read or Download Computer-Enabled Mathematics: Integrating Experiment and Theory in Teacher Education (Education in a Competitive and Globalizing World) PDF Similar study & teaching books If freshmen within the school room are to be all for arithmetic, lecturers must be either good trained approximately present projects and ready to see how what's anticipated of them should be translated into wealthy and stimulating school room recommendations. The publication examines present projects that have an effect on instructing arithmetic and identifies guidelines for motion within the lecture room. New 4th version thoroughly revised and up to date with new DVD now on hand; ISBN 1-56368-283-4 Offers an exam of the historical past to checking out vocabulary wisdom in a moment language and specifically considers the impact that note frequency and lexical assurance have on studying and communique in a international language. content material: factors and definitions -- notice hassle, note frequency and acquisition : lexical profiles -- Frequency and insurance -- Measuring vocabulary breadth : passive attractiveness vocabulary -- Measuring different features of vocabulary breadth -- Measuring effective vocabulary wisdom -- Measuring vocabulary intensity -- Vocabulary acquisition and checks of language point -- Vocabulary acquisition and lecture room enter -- Vocabulary acquisition and casual language enter -- Implications for studying and instructing : concept and perform. Deutsch als Fremdsprache im Klinikalltag meistern! Der berufliche Erfolg eines jeden Arztes basiert auf der gekonnten Kommunikation mit Patienten und Kollegen. Sei es im Anamnesegespräch, bei der körperlichen Untersuchung oder der Diskussion von Differentialdiagnosen. Internationale Ärzte stellt die Arbeit in deutschen Kliniken und Praxen vor eine besonders große Aufgabe, da sie sich sowohl in der fachspezifischen Ausdrucksweise als auch mit kulturellen Besonderheiten der Sprache zurechtfinden müssen. - Teaching and Learning History: Understanding the Past 11-18 - A Teacher's Guide to The Bill of Rights: A History in Documents - Critical Mathematics Education: Theory, Praxis and Reality - Second Language Acquisition in a Study Abroad Context - Boya Chinese: Elementary Starter I Extra info for Computer-Enabled Mathematics: Integrating Experiment and Theory in Teacher Education (Education in a Competitive and Globalizing World) 9(n 1) 9n 8(n 1) 8n Therefore, inequalities (27) can be replaced by inequalities (26). This completes the proof. Proposition 21. The series v(n) diverges. n 1 2 2 1 . 9 n 1 n n 1 9n Proof. It follows from Proposition 17 that v(n) n 1 7 In the words of Langtangen and Tveito (2001): ―Much of the current focus on algebraically challenging, lengthy, error-prone paper and pencil work can be significantly reduced. The reason for such an evolution is that the computer is simply much better than humans on any theoretically phrased well-defined repetitive operation‖ (pp. 78... , demonstrating an extremely slow divergence which can even be taken for n 1 convergence if approached from a computational perspective alone. More information on the divergence of the harmonic series can be found in Chapter 6. Proposition 15. The series q ( n) converges to the number 2. n 1 Proof. Due to (21), q(n) n 1 n 1 1 q ( n ) 2 n 2 2 n2 and, thereby, due to Proposition 14, we have . However, noting that q(n) 1 2 1 1 2( ) tn n(n 1) n n 1 1 2 . 14. For what values of parameter c does equation (1) have two real roots? Reflections. 25. However, the last inequality can be directly derived from Figure 1 if one interprets the roots of equation (1) as the x-intercepts of the locus and the horizontal line c = constant. 25. 5. Exploration 2. For what values of parameter c does equation (1) have two positive roots? Reflections. Note that in order to answer this question through the locus approach, one does not need to construct a series of graphs y = x2 + x + c for different values of parameter c (using graphing technology) or to carry out transformation of inequalities involving radicals Algebraic Equations with Parameters 39 that result from the use of the quadratic formula. Computer-Enabled Mathematics: Integrating Experiment and Theory in Teacher Education (Education in a Competitive and Globalizing World) by Sergei Abramovich

The concept of dimension and shape fractal geometry Fractal geometry a fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale if the replication is exactly the same at every scale, it is called a self-similar pattern. Analytic and numerical calculations of fractal dimensions arman shamsgovara [email protected] under the direction of gaultier lambert. Request pdf on researchgate | fractal methods and results in cellular morphology - dimensions, lacunarity and multifractals | this paper discusses the concepts of fractal geometry in a cellular biological context it defines the concept of the fractal dimension d, as a measure of complexity and illustrates the two different general. Fractal geometry almost all geometric forms used for building man made objects belong to euclidean geometry, they are comprised of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc these elements can be classified as belonging to an integer dimension, either 1, 2, or 3 this concept of dimension can be described both. The koch snowflake (also known as the koch curve, koch star, or koch island) is a mathematical curve and one of the earliest fractal curves to have been described it is based on the koch curve, which appeared in a 1904 paper titled on a continuous curve without tangents, constructible from elementary geometry (original french title: sur. This paper has applied the concept of fractal geometry in designing a grid-shell-like complex spatial structure the property of the fractal dimension which characterizes the level of roughness of a shape has been particularly explored in this study for designing a complex-shaped spatial structure by taking a paraboloid as a basic shape of reference. Question sent by jose galán (alacant) eduardo ros answers: «clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel a straight line» are benoît mandelbrot’s genius opening lines for his book the fractal geometry of naturemandelbrot enunciated the concept of fractal. Concept of fractals introduced by mandelbrot , which has the shape formed in nature, has been usually analyzed using euclidian geometry the key parameter for fractal the key parameter for fractal. In mathematics , more specifically in fractal geometry , a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured it has also been characterized as a measure of the space-filling capacity. In fractal geometry, on the other hand, dimension is treated as a continuum a a curve’s dimension, for example, can take on any non-integer value between 1 and 2. A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers fractals are typically self-similar patterns, where self-similar means they are the same from near as from far [3. Any mathematical concept now well-known to school children has gone through decades, if not centuries of refinement a typical student will, at various points in her mathematical career -- however long or brief that may be -- encounter the concepts of dimension, complex numbers, and geometry if. The key concept in chaos theory is that an order is laid in any disorder this means that discipline should not be sought only in a scale chaos is a theory that considers various issues, but requires a language for its concepts visualization and this was an area that created geometry of chaos or fractals fractal geometry provides us a tool for. Fractal geometry math carla moreno 6th period the concept fractal geometry , also called the “true geometry of nature” could be explained as: non-triangular or squared shaped figures, but as more complex ones fractals are used to explain concrete objects slideshow 4940703. The concept of fractal dimension was defined for true fractal objects and is formally defined in ref 3 3 b b mandelbrot, the fractal geometry of nature b b mandelbrot, the fractal geometry of nature. He stated that fractals (through fractal geometry) are more useful to describe natural shapes than the classic euclidian geometry describes financial volatility measurement using fractal dimension keywords: discontinuous functions interpolation approximation functional spaces fractals. The relationship between the fractal dimension and shape properties of particles the shape of particles is not accurately described by euclidian geometry however, fractal geometry uses the concept of fractal dimension, dr, as a way to describe the shape of particles in this study, the fractal dimensions and shape properties of particles were. The shape in the antenna that can have non-integer values, called a fractal dimension this means that, shapes in the antenna can have irregular shape than regular shape. Dimension is at the heart of all fractal geometry, which provides a reasonable basis for an invariant between different fractal objects there are also experimental techniques. Geometry ( geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space geometry is one of the oldest sciences initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century bc geometry was put into an axiomatic form by euclid, whose treatment—geometry. The fractal geometry of nature dimension and fractal dimension • the discovery of space-filling curves, such as the peano curve, had a major impact in the development of the concept of dimension: questioned the intuitive perception of curves as one-dimensional objects • points are zero-dimensional, lines and curves are one. 15 jagadeesha, ambresh, raj and madhusudhana ii design consideration fractal geometry with base shape of plus shaped with ebg. Is made by using concept of fractal geometry vertical length of 27 mm is divided into 3 parts, each of length 9 mm vertical length of 27 mm is divided into 3 parts, each of length 9 mm these two cuts are made in vertical direction to form h shape patch h shaped patch vs having five squares this h.

Dynamics is the study of moving objects. In previous chapters we have considered the sum total of forces on a body (resultant) and our next consideration is the movement that these forces bring about. Forces of friction, thrust, gravity and tension will be considered and the subsequent motion will be measured by the application of constant acceleration equations and the equation of motion. As in most mechanics questions a certain amount of modeling will have to be used in our working. Newton’s First Law A body will remain at rest, or will continue to move with constant velocity, unless external forces force it to do otherwise. A change in state of motion of a body is caused by a force. The unit of force is the Newton, (N). A body of mass 3500Kg moves horizontally at a constant speed of 5ms-1 subject to the forces shown. Find P and S. There is no vertical motion therefore: 2S = 3500g S = 1750g The horizontal acceleration is zero, therefore: Newton’s Second Law The force F applied to a particle is proportional to the product of mass of the particle and the acceleration produced. A force of 1N produces an acceleration of 1ms-2 in a body of mass 1kg. Newton’s Second Law is summarized by the equation: F = ma this is often termed the equation of motion. It is vitally important to realise that F is the overall resultant and not Friction (FR). If the object in Example 1 is slightly modified to take account of the fact that there is a pushing force 600N, calculate the acceleration. The resultant of the two horizontal forces is 250N pushing the object to the right. So by setting up an equation of motion: If a particle is falling in the earth’s atmosphere then it will accelerate at 9.8ms-2. By Newton’s first law the particle must be experiencing a force and the only force present is the weight, so by considering the equation of motion: F = ma Weight = mg A man of mass 95kg is traveling up in a lift. Given that the acceleration of the lift is 0.8ms-2, find the force exerted on the man by the floor of the lift. Calculate the same force when the lift is descending with the same acceleration. Taking up to be positive and setting up an equation of motion: F = ma R – 95g = 95a R – 931 = 95 × 0.8 R = 1007N When the lift is traveling in the opposite direction let down be positive so the equation of motion becomes: 95g – R = 95a 931 –R = 95 × 0.8 R = 855N It is worth pointing out that the reaction from the floor is less than the weight of the man when the lift is descending (and vice versa). The next example introduces constant acceleration equations into the problem. A ball of mass 0.9 kg falls from a height of 22m above horizontal ground. The ball reaches the ground after t seconds. The ball sinks into the ground a distance of 1.9cm before coming to rest. The ground is assumed to exert a constant resistive force of magnitude F newtons. Find: - the value of t to 3 sig fig; - the value of F to 3 sig Using constant acceleration equations to find t: The ball is brought to rest in 1.9cm so by using constant acceleration equations we can find the deceleration and hence the force: Firstly we need to find the speed with which the ball hits the surface And now for the deceleration: And finally using an equation of motion for the particle: F = ma F = 0.9 × 11359 F = 10.2KN Note that we were asked for the magnitude hence the positive answer. How realistic is this answer? Ask a physics teacher for some other examples. A ball of mass 5kg falls from a height of 6m into a jar containing a viscous liquid. The upward force exerted by the liquid is of magnitude 75N. How far will the ball sink into the liquid? Calculate the total time that the ball is in motion. Firstly we need to calculate the speed with which the ball hits the liquid. Secondly we can set up an equation of motion for the ball to work out the deceleration. Thirdly we need to calculate the distance that the ball travels through the liquid before coming to rest. Finally the total time taken by the ball in motion must be done in two parts seeing as before the ball hits the liquid it has an acceleration of whereas, whilst falling through the liquid, it has an acceleration of Time to meet the surface of the liquid: Time to come to rest: Therefore the total time in motion is 3.3 sec. The liquid is rather viscous, how realistic is the resistance force? What about a ball falling into a jar of syrup? The next example introduces friction on a slope. The questions are beginning to get more challenging but a good diagram is always the best place to start. Examiners regularly report that the most successful candidates in mechanics M1 and M2 always draw diagrams. A ball of mass 2kg is projected up a line of greatest slope inclined at an angle of 30º to the horizontal. The coefficient of friction between the plane and the ball is 0.4. The initial speed of the ball is Find: - the frictional force acting whilst the ball moves up the plane. - the distance moved up the plane by the ball before it comes to instantaneous - Since the ball is moving then Friction must be at its maximum value. Resolving perpendicular to the plane gives: - The frictional force and the weight component of the ball are trying to slow the ball down. By setting up an equation of motion for the ball we can calculate the Assuming that uphill is positive: F = ma – FR – weight comp down the plane = ma -6.79 – 2g cos 30º = 2a a = -8.295ms-2 At the point the ball comes to instantaneous rest it will have zero velocity and we can use constant acceleration equations to calculate the distance traveled. Motion of Two Connected Particles Newton’s Third Law Before we can consider the motion of two connected particles we need to discuss Newton’s Third Law. This law states that action and reaction are equal and opposite. If two bodies A and B are in contact and exert forces on each other, then the force exerted by A on B is equal in magnitude and opposite in direction to the force exerted by B on A. This principle will be applied to tow truck problems and pulleys to name but two. Consider the situation below where the cartoon car is towing a racing car. The racing car is pulled forward by tension in the tow bar. The racing car will exert an equal but opposite force on the car. If the car is slowing down and there are no breaks on the racing car then some force must be acting in the opposite direction to the direction of motion of the two cars. In this case the tow bar will exert a thrust on both cars (arrows change directions). The AA man is towing a car along a straight horizontal road. The truck has a mass of 1500kg and the car has a mass of 850kg. The truck is connected to the car by a bar which is to be modelled as a light inextensible string. The truck’s engine produces a constant driving force of 2500N. The resistance to motion of the truck and the car are constant and of magnitude 750N and 400N respectively. Find: the acceleration of the truck and the car; the tension in the When the truck and the car are traveling at the tow bar breaks. If the magnitude of the resistance to motion of the truck remains at 750N calculate: the time difference in achieving a speed of with and without the car in Setting up equations of motion for the car and truck separately gives: Finding the tension: Substituting the value into the car’s equation of motion gives: T – 400 = 850 × 0.574 T = 888N If we assume that the bar doesn’t break then the time required to reach 30ms-1 is calculated by using the constant acceleration At the point that the tow bar breaks, the tension in the bar is no longer acting against the truck. Therefore the equation of motion of the truck becomes: Therefore there is a time difference of 7 seconds. In all questions in M1 the pulley system will be smooth. This implies that the motion of the particles at the end of the string are unaffected by the string passing over the pulley. A further assumption is that the string is light and inextensible. These modelling assumptions make the problem simpler but we can still get pretty realistic answers. If two particles are connected by a string where the string passes over a smooth pulley, then we can assume that the particles will have equal accelerations but in opposite directions. Two particles P and Q are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest. Find: the magnitude of the acceleration; find the tension in the To start the problem set up an equation of motion for particle P. The particle will accelerate upwards hence: F = ma T – 8g = 8a (1) When considering particle Q, its weight will cause it to accelerate down hence the equation of motion is: F = ma 20g – T = 20a (2) By adding the two equations the tension will be eliminated: 12g = 28a a = 4.2ms-2 Substituting the value of the acceleration into equation (1) will give the tension. T – 8g = 8 × 4.2 T = 112N The following example is more algebraic but this does not mean that it is more complicated. Two particles P and Q of masses have masses 8m and Km, where K > 8. They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown below. Initially P has an acceleration of magnitude of - Find, in terms of m and g, the tension, T, in the string - Find the value of K. - Adding forces to the system: Setting up an equation of motion for particle P: The next problem involves a pulley system where one of the particles is being dragged across a horizontal table as the other particle is falling. The problem will be made more complex when the horizontal table is considered to be rough. The questions are increasing in complexity but the same basic principles apply. A particle A, of mass 0.9kg, rests on smooth horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley P fixed at the edge of the table. The other end of the string is attached to a particle B of mass 1.8kg which hangs freely below the pulley. The system is released from rest with the string taut and B at a height of 3.2m above the ground. In the subsequent motion A does not reach the pulley before B reaches the ground. Find: - the tension in the string before B reaches the ground - the time taken by B to reach the ground Then, to make the model more realistic, assume that the coefficient of friction between the particle and the table is 0.3. Using this modification find the time taken by B to reach the ground. Setting up equations of motion for the two particles gives: Adding the two equations gives: The particle B is falling with acceleration 6.53ms-2. So by using constant acceleration equation we can find the time it takes to reach the floor. Seeing as the particle is moving friction must be at its maximum value and hence Resolving vertically for A: The new value of acceleration can now be used to calculate the new time: In the next problem a particle is being pulled up a rough inclined plane by the motion of another particle falling towards a floor. This is very similar to an exam question and would be worth in excess of 10 marks on an M1 paper. A particle, A of mass 6kg, rests on a rough plane inclined at an angle of 35º to the horizontal. The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a light smooth pulley P fixed at the top of the plane. The other end of the string is attached to a particle B of mass 16kg. The particles are released from rest with the string taut. The particle B moves down with an acceleration of (a) the tension, T, in the string (b) the coefficient of friction between the plane and A a) Setting up equations of motion for A and B gives: Using (2) to find the tension: Resolving vertically for A: Rearranging for μ: Assume that in the example above the particle is 5m above a level surface and that after 0.5sec the string breaks. Calculate the total time that the particle B is in flight and the distance that A moves up the plane before it comes to instantaneous rest (assume that it does not reach the pulley). This is a rather large value for μ and this obviously accounts for the low value for the acceleration. What is the resultant force on the pulley? Momentum and Impulse The momentum of a body of mass m, having a velocity v is mv. The units of momentum are Newton seconds (Ns). Momentum = mv The momentum of a body is dependent upon its velocity therefore momentum is a vector quantity. This implies that direction is very important and great care must be taken with signs. Find the momentum of a hockey ball of mass 0.9kg hit at Momentum = mass × velocity Momentum = 0.9 × 18 = 16.2Ns Change in Momentum If a particle experiences a change in velocity then, by definition, its momentum must change. Let the initial velocity be u and the final velocity be v then the change in momentum is given by: Change in Momentum = mv – mu = m( v – u ) If the hockey ball from example 12 hits a wall directly and returns with a velocity of Find the change in momentum. Always draw a diagram. Taking left to right as positive, therefore u = 18, v = -12 Change in Momentum = m( v – u ) = 0.9 × (-12 – 18 ) The wall in the above example has experienced a force as the hockey ball hits it. This action takes place in a very short time period and is called the impulse. When a force F, is applied to a particle for a period of time t, then this quantity is defined as the impulse of the force. Obviously an impulse will bring about a change in velocity and therefore momentum will change. Impulse = F × t = m(v- u ) The derivation of the formula comes from the equation of motion and constant acceleration equations: A particle of mass 7.5kg is acted on by a force for 6 seconds and in the process its velocity increases from Find the magnitude of the force. Impulse = change in momentum F × t = m(v- u ) F × 6 = 7.5 (15 – 6) F = 11.25N A ball of mass 1.2kg is moving vertically with a speed of when it hits a smooth horizontal floor. It rebounds with a speed Find the magnitude of the impulse exerted by the floor on the ball. Take care with signs Assuming down to be positive Impulse = Change in Momentum = m( v – u ) = 1.2( -8 – 14 ) = 1.2( – 22 ) = -26.4 Ns The Principle of the Conservation of Momentum When a collision occurs between two bodies, A and B, then the force exerted on A by B will be equal and opposite to the force exerted on B by A (by application of Newton’s third law). If no other forces are present then the change in momentum in one particle will equate to the loss of momentum in the other particle. Momentum is conserved and therefore the sum of the momentum of the particles before collision must equal the sum of momentum after the collision. This is referred to as the Principle of Conservation of Momentum. If two particles of masses, m1 and m2, with initial velocities u1 and u2, collide then, given that their final velocities are v1 and v2 we can say that: The following examples illustrate the principle and once again it is always best to draw a diagram as this will help to avoid mistakes with signs and direction. Two particles A and B have masses of 1.2kg and 1kg respectively. Particle A is moving towards a stationary particle B with a velocity of Immediately after the collision the speed of A is and its direction is unchanged. Find: the speed of B after the collision; the magnitude of the impulse exerted on A in the collision. By conservation of momentum: Two small balls A and B have masses 1.2kg and 2.5kg respectively. They are moving in opposite directions on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of A is and the speed of B is The speed of A immediately after the collision is The direction of A remains unchanged after the collision. Find: the speed of B immediately after the collision; the magnitude of the impulse exerted on B in the collision. By conservation of momentum: 1.2 × 4.5 – 2.5 × 0.9 = 1.2 × 1.3 + 2.5v v = 0.636ms-1 Impulse = Change in Momentum = m( v – u ) = 2.5 ( 0.636 – – 0.9 ) A locomotive A, of mass 1800kg is moving along a straight horizontal track with a speed of It collides directly with a stationary coal truck, B, of mass 1000kg. In the collision, A and B are coupled and move off together. Find the speed of the combined train. After collision a constant braking force of magnitude R. Newtons is applied. The train comes to rest after 15 seconds. Find the value of R. The combined train comes to rest in 15 seconds therefore we need to calculate the acceleration for use in an equation of motion. Assuming that the train acts as one body: Equation of motion: F = ma R = -2800 × 0.386 R = 1080.8N = 1.10KN Jerk in a String If two particles are connected by a light inextensible string and one of the particles is projected away from the other then at some point there will be a jerk in the string. At the instant before the jerk one of the particles will have momentum. As soon as the string becomes taut the particles will move onwards with the same velocity. The overall momentum must be conserved so therefore the velocity after the jerk must be lower than the initial velocity. This idea is best explained through an example. Two particles P and Q of masses 4kg and 7.5kg respectively are connected by a light inextensible string which is initially slack. Q is projected away from P with velocity When the string becomes taught the two particles move on together with a common speed. Find the common speed and the impulse exerted on P by the string. Using the conservation of momentum where Impulse is the change in momentum so considering particle P: Impulse = m(v – u) = 4(3.26 – 0)

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle? Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have? Can you maximise the area available to a grazing goat? Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star. A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly? Can you work out the dimensions of the three cubes? In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . . It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into? 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? Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . . ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates? Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end. If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? 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? A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner. A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . . How efficiently can you pack together disks? You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area. A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels? A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be? 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? Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? Show that among the interior angles of a convex polygon there cannot be more than three acute angles. We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used? 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? Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. A huge wheel is rolling past your window. What do you see? In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened? At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again? Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? 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? If you move the tiles around, can you make squares with different coloured edges? Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . . Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. How many moves does it take to swap over some red and blue frogs? Do you have a method? 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. The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . . Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . . Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw? Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares? The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone? Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? How many different symmetrical shapes can you make by shading triangles or squares? Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

ROCK SLOPE STABILITY ANALYSIS OF QUARRY SITES AT WEST MALVERN UK. ABSTRACT: This paper analyzes the scan line survey data of the geological discontinuities collected from three quarry sites of West Malvern in Malvern Hills UK which remaine done of the major resource of geological materials for building stones and aggregates in the area. The kinematic analysisof the discontinuity data of the quarry sites was performed by using the DIPS software which showed the likelihood of wedge plane and toppling failures of the slopes. However no failure is likely even for the steepest slope faces if planned to cut in the dip direction range of 30o to 70o and 130o to 230o. Keywords: Kinematic Analysis Malvern Hills Slope Stability DIPS Discontinuities. Numerous methods are available for assessing the stability of slopes as reported by (Hoek and Bray 1981; Park and West 2001; Wyllie and Mah 2004).Kinematic analysis is one of the analytical tools used for the assessment of a slope's stability (Olaleye and Ajibade 2011).Normally kinematic analysis is performed prior to detailed investigations in almost all slope stability analysis(Kulatilake et al. 2011 and Aksoy and Ercanoglu 2007).Kinematic analysis of rock slopes is purely a geometric technique which utilizes angular relationships between slope surfaces and discontinuity planes through their stereographic projections. The outcome of this kind of analysis is used to determine the likelihood and the modes of failures (Yoon et al. 2002; Iqbalet al. 2013; Park et al. 2005).Generally four types of failures exist in rock slopes namely plane wedge toppling and circular. The conditions suitable for these failure types are elaborated in detail by (Hoek and Bray1981). The purpose of this study was to evaluate the geological discontinuities data gathered from old quarry sites near West Malvern in the Malvern Hills UK. The location map of the site is shown in Figure-1. The data of discontinuities collected from the site was then used to perform kinematic analysis using the DIPS software to propose the safe slope orientations with steepest possible face angles. Geology: The geology of the area is dominated by the metamorphosed granites of the Malvern Hills. The materials are described according to BS5930 (British Standard 1999) as pink fresh coarse grained crystalline granites extremely strong. Naturally some variations in the properties of these materials can be observed from location to location. Structurally and stratigraphically above the Malvern Hills Granites are sediments of Ordovician and Silurian age. These sedimentary rocks vary but are characterized by sandstones siltstones and mudstones in a deep water setting. Faults occur throughout the area and present varying levels of difficulty in construction (Murphy 2005). Geomorphology: The geomorphology of the study area is strongly controlled by the juxtaposition of rocks of different materials and mass properties. In addition to this the area has been affected by several periglacial periods during the quaternary and questions remain as to whether the Malvern Hills were a local ice center during the devencian. There is a fragmentary evidence of what has been interpreted as glacial tills throughout the area(Murphy 2005).Description of the rocks found in the quarries is given in Table 1 as per BS5930(British Standard 1999). Table 1 showing rock description of quarry sites as per BS5930 (British Standard 1999). Quarry 1###Pinkish gray coarse-grained fresh crystalline GRANITE extremely strong Quarry 2###Dark gray medium grained fresh crystalline GRANITE strong Quarry 3###Dark gray and brown coarse grained crystalline DOLERITE very strong MATERIALS AND METHODS The discontinuities data was collected by scan line survey along the exposed cut faces of the old quarries. Dip and dip directions of geological discontinuities were measured and analyzed through stereographic projections using DIPS software. The data set consisting of 62 discontinuities was then used for pole plotting (Figure 2a) which was subsequently utilized for contour plotting (Figure 2b). Plotting of the data identified three major pole concentrations. The pole concentrations helped in identifying the representative great circles of the discontinuities (Figure 2c). A friction angle of 40 was assessed for calculation and analysis based on the recommendations of Hoek and Bray (1981). Possible slope face orientations: The pole concentrations of the discontinuities data showed that the quarry sites had three major discontinuity planes (J1 J2 and J3); two of them were steeper while the other one was relatively flat (Table 2). Although all three major discontinuity planes were making wedges but two of them were lying outside the friction circle whereas the wedge formed by the intersection of J1 and J2 was inside the friction circle (Figure-3). Keeping in view the likely future orientations of the slope faces at the quarry sites (Table-3) different slope failure cases are discussed. Table 2: showing mean discontinuity planes based on major pole concentrations. Name of Discontinuity###Symbol###Trend###Plunge Joint Plane 1###J1###098###18 Joint Plane 2###J2###176###35 Joint Plane 3###J3###256###62 Table 3: showing possible face orientations: Face Slope Orientation Dip###Dip Direction CASE : Considering slope face orientation of 69/318 and stereographic projection of mean discontinuity planes drawn through DIPS showed a phenomena of kinematic instability (Figure-4). Sliding envelope shown by shaded area represented the conditions of Markland's wedge failure (Markland 1972) i.e. plunge of line of intersection of two joint planes must be less than the slope face angle and greater than friction angle. In the current situation the plunge of line of intersection of J1 and J2 measured was 54 greater than the 40 (the friction angle) and less than 69 (plunge of slope face). Therefore potential of sliding movement of block shown by dark shaded area existed at quarry site having a plunge of slope face greater than 60 and dip direction between 315 and 355. In this particular case the wedge block was likely moving in the direction of N19W along the line of intersection (Figure-4). Kinematic analysis was performed for another possible orientation of slope face of 71/004. In this case there was likelihood of plane and wedge failures (Figure-5). Another important aspect was the fulfillment of (Hocking's refinement 1976) which was introduced to differentiate between the sliding of a wedge along the line of intersection or along one of the planes forming the base of the wedge. According to Hocking if the dip direction of either of plane fell between the dip direction of the slope face the trend of the line of intersection sliding will occur on that plane instead of along the line of intersection. Therefore in this case the wedge block moved along J2 instead of line of intersection of J1 and J2. Figure-5 clearly fulfills the Hocking's refinement where: a1 = Trend of the line of intersection a2 =Dip direction of joint plane 2 af=Dip direction of slope face Moreover J2 satisfied the conditions of plane failure as described by (Hoek and Bray 1981). The dip of sliding plane should be less than the dip of slope face and greater than friction angle and the trend of sliding plane should be within 20 of the slope face trend. In this particular case dip of J2 was 55 which was less than that of slope face that was inclined at 69 and greater than the friction circle i.e. 40. Both wedge block and sliding mass by plane failure are shown by relatively dark shaded area (Figure-5). CASE : Case III was concerned with the stability of slopes having face orientation of 78/268. In this case the possibility of plane failure was apparent (Figure-6). The angle difference between the dip direction of J1 and the slope face was only 10. Sliding of the mass shown by the shaded area was along the joint plane J1. Along with other conditions role of the release surfaces in plane failure of the slopes was critical. Release surfaces may exist if some other discontinuities intersect the sliding plane. If other discontinuities intersected the sliding plane at two points in the sliding envelope bounded by the region between slope face and sliding plane (shown by shaded area) then plane failure was evident. In this case J2 intersected J1 only at one point in the sliding envelope which suggested a relatively remote chance of plane failure of the slope. However other factors such as strength of rock mass and ground water conditions should be considered to check the factor of safety. CASE V: If the slope face had a dip angle in the range of 80o to 90o and dip direction of 80o to 100o then there were fair chances of toppling failure (Figure-7). This was due to the reason that great circle of J1 had a steeper angle of 72 which dipped into the slope face. Table 4: showingsummary of results of kinematic analysis of quarry slope ###Orientation of###Dip of Slope face###Failure/Stable###Mode of###Recommended safe ###quarry slope###Failure###quarry slope ###000o to 020o###60o to 90o###Failure###Plane###fless than 50o ###030o to 070o###Any###Stable###-###- ###080o to 100o###80o to 90o###Failure###Toppling###fless than 45o ###130o to 230o###75 to 90###Stable###-###- ###260 to 275###75o to 90o###Failure###Plane###fless than 65o ###310o to 350o###60o to 90o###Failure###Wedge###fless than 50o Conclusions and Recommendations: The results of analysis performed on the discontinuities data collected from the old quarry sites of West Malvern along with the recommended dip angles for safe slope faces are summarized in Table 4. It was noted that slope faces were stable with the steepest slope angles within the range of orientation from 030 to 070. If there is a need of cutting slope faces for any purpose in future then these faces should not be cut with the resultant orientations within the range of 000- 020 080-100 260-275 and 310-350. If there is no choice other than these orientations then the slope angle should be less than 45. Among the most important parameters describing the discontinuities are orientation spacing persistence roughness aperture and infilling materials (Zhou and Maerz 2002). Usually the kinematic analysis suffers from the inability to consider most of these parameters except the orientations of the discontinuities. For a detailed stability analysis inclusion of all important discontinuity features into the analysis is warranted. Aksoy H. and M. Ercanoglu. Fuzzified Kinematic Analysis of Discontinuity-Controlled Rock Slope Instabilities. Engineering Geology 89 206219: (2007). British Standard (BS 5930).Code of practice for site investigations (1999). Hocking G.A Method for Distinguishing between Single and Double Plane Sliding of Tetrahedral Wedges. International Journal of Rock Mechanics and Mining Science 13 225226: (1976). Hoek E. and J. W. Bray. Rock Slope Engineering. The institute of Mining and Metallurgy London (1981). Iqbal M. M. M. Z. Abu Bakar M.Akram M. Shahzad and Y. Majeed.Slope Stability Analysis of Dandot Plateau Punjab Pakistan. Pakistan Journal of Science 65(4): 531-538 (2013). Kulatilake P.H.S.W. L. Wang H. Tang and Y. Liang. Evaluation of Rock Slope Stability for Yujian River Dam Site by Kinematic and Block Theory Analyse. Computers and Geotechnics 38: 846860 (2011). Markland J. T. A Useful Technique for Estimating the Stability of Rock Slopes when the Rigid Wedge Sliding Type of Failure is Expected. Imp. Coll. Rock Mech. Res. Rep. 19: 10 (1972). Murphy W.; Broadway and the Malverns Field Course The University of Leeds UK (2005). Olaleye B.M and Z.F Ajibade. Kinematic Analyses of Different Types of Rock Slope Failures in a Typical Limestone Quarry in Nigeria. Journal of Emerging Trends in Engineering and Applied Sciences 2(6): 914-920 (2011). Park H. and T. R. West. Development of a Probabilistic Approach for Rock Wedge Failure. Engineering Geology 59: 233-251 (2001). Park H. J. T. R. West and I. Woo. Probabilistic Analysis of Rock Slope Stability and Random Properties of Discontinuity Parameters Interstate Highway 40 Western North Carolina USA.Engineering Geology 79: 230-250 (2005). www.maps.google.com Wyllie D. C and C. W Mah. Rock Slope Engineering: Civil and Mining (4thed.). Spon Press 270 Madison Avenue New York NY 10016 USA (2004). Yoon W.S. U.J. Jeong and J.H. Kim.Kinematic Analysis for Sliding Failure of Multi-Faced Rock Slopes.Engineering Geology 67: 5161 (2002). Zhou W. and N.H. Maerz. Implementation of Multivariate Clustering Methods for Characterizing Discontinuities Data from Scanlines and Oriented Boreholes. Computers and Geosciences 28: 827-839 (2002). |Printer friendly Cite/link Email Feedback| |Publication:||Pakistan Journal of Science| |Date:||Mar 31, 2015| |Previous Article:||IMPACT ASSESSMENT OF SEWERAGE DRAINS ON GROUNDWATER QUALITY OF FAISALABAD PAKISTAN. A PHYSIO-CHEMICAL ANALYSIS.| |Next Article:||ROLE OF CASSIA OCCIDENTALIS IN THALASSEMIC PATIENTS.|

Each unit has its own set of interactive tutorials one for each section within that unit. Hold your mouse over the name of a skill to view a sample question. What are the best introductory books on mathematical. Books are a primary resource for the serious student of financial math. This means you obtained 55 100 ths of the marks available. Download free ebooks at financial mathematics and its applications 6 preface preface is pocket book is meant for anyone who is interested in the applications of nance, particularly business students. Bass department of mathematics university of connecticut these notes are c 2003 by richard bass. The basics of financial mathematics spring 2003 richard f. What it is about is figuring out the price of options and derivatives. Nov 25, 2011 imaths 7 shares the fundamental philosophy of the australian curriculum giving maths reallife relevance through investigations. Toland university of bath other books in this series a first course in discrete mathematics i. A pricing model on the competitive financial market 8. Using the imaths 7 student book together with free additional downloadable resources will ensure that teachers can cover. Hence, there are many books coming into pdf format. Multiplication of a twodigit number by an amount try worksheet for free. Ks3 maths year 7 workbook collins ks3 revision paperback 20 jun 2014. Along with the year 8 and 9 course books full coverage of the ks3 programme of study is provided. It uses quite advanced mathematics including random matrices, stable distributions and so on. Each student book gives access to nelsonnet and an interactive online ebook. Optimization of the portfolio with risk free investment possibility 8. Opitonal mathematics of investment and credit, 6 edition. Section b money and financial mathematics acmna2 investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies sequenced activities. Jun 20, 2014 not only does this maths book encourage students to achieve their best possible results, but it will build confidence by ensuring fluency in all year 7 maths skills taught at key stage 3. Nelson think maths for the australian curriculum 710 is a new series that has been developed to support teachers implementing the australian mathematics curriculum for years 710 students in victoria. Week 8 homework, exams be able to determine the value of a portfolio using swap rates and forward rates and determine if the investments are immunized. The obook is a cloudbased webbook available anywhere, anytime, on any device, navigated by topic or by page view. So if the test is actually marked out of 40, then you have 55 100 of 40 55 100. Grade 6 b money and financial mathematics backtofront. Here is a partial list of some of the books i used at the beginning of my phd 1. If you dont see any interesting for you, use our search form on bottom v. Mathematical finance is not about predicting the price of a stock. Anderson analytic methods for partial differential equations g. From statistical physics to risk management amazon. From popular books for dummies to classic investment books like benjamin grahams, security analysis. Collins ks3 ks3 maths year 7 workbook collins ks3 revision. Nelson think maths for the australian curriculum year 7. National council of educational research and training ncert book for class vii subject. The classic 1951 edition, and steven kellisons modern classic, theory of interest, this financial math bibliography covers the spectrum. Centre for innovation in mathematics teaching year 7. Click on the name of a skill to practise that skill. Not only does this maths book encourage students to achieve their best possible results, but it will build confidence by ensuring fluency in all year 7 maths skills taught at key stage 3. Until this book and series are required in all high schools, along with a full suite of economic education classes for all four years, enlightened parents will have to buy this book on their own but be assured, it is money well spent. Note that the following support material forms part of the year 7 interactive maths second edition homework licence software. Feb 02, 2016 there are several interesting introductory math finance books. The mathematics of investment this book provides an elementary course in the theory and the application of annuities certain and in the mathematical aspects of life insurance. Book 7, part 1 of 2 by john mighton author, jump math author 4. Class vii ncert maths text book chapter 2 fractions and decimals is given below. Math1510 financial mathematics i university of leeds. The book 2 describes the rst exam that you need to pass to become an accredited actuary in the uk. Year 6 and 7 maths booster booklets teaching resources. See more ideas about teaching money, money activities and australian money. Ks3 maths year 7 workbook by collins ks3 waterstones. This website is created solely for jee aspirants to download pdf, ebooks, study materials for free. The study guide is designed for students with the major 230 applied mathematics, 230700 application informatics, and masters program students with the major 140400. Stochastic calculus for finance 2 volumes by shreve and karatzas. Three have names on them but you can put a name sticker over them. Zastawniak, probability through problems, springerverlag, new york, 2001. Mathematics books for free math questions and answers. On this page you can read or download financial accounting n6 textbook pdf in pdf format. Some expert physicist, working partly in finance recommended me the book. Improve your skills with free problems in what decimal number is illustrated. Financial math books on financial and economic subjects. Nelson think maths for the australian curriculum year 7 buy. Present, forward and futures prices futures contract day 0 123 t. As a beginner in mathematical finance, what book should i. Written by experienced educators to support year 7 australian curriculum outcomes, and to improve high school numeracy skills through naplan test practice, this range includes excel revise in a month year 7 naplanstyle tests, excel essential skills maths workbooks, the popular. Signpost year7 maths sold a monster calls sold skate pearson science year 7 sold big rain coming islamic studies year 7. Remember to write all amounts as exact values and fractions in their simplest form. Until then, you can view a complete list of year 7 objectives below. Using the imaths 7 student book together with free additional downloadable resources will ensure that teachers can cover all the requirements of the australian curriculum. It uses quite advanced mathematics including random matrices. Be able to determine the price, book value and market value of a bond. The statistical analysis of the financial market questions for selftest bibliography. An introduction to financial engineering springer undergraduate mathematics series book online at best prices in india on. This folder contains 45 pdf booklets and are perfect for supporting intervention sessions in year 6 and 7. An introduction to financial engineering springer undergraduate mathematics series 2 by marek capinski, tomasz zastawniak isbn. Valuating security contribution to the total expected portfolio return 8. Delve into mathematical models and concepts, limit value or engineering mathematics and find the answers to all your questions. In many numerical examples and exercises it may be helpful to use a com. Basic ideas of financial mathematics 1 percentage the word \percent simply means \out of 100. The book is particularly adapted to the needs of students in colleges of business administration, but it is also fitted for study by college students of mathematics who. Grade 6 b money and financial mathematics backtofront maths. Book 5 contains many exercises, but does not go quite as deep. Accounting principles a business perspective, financial. Find great content to download for free, right here. Our year 7 maths range is designed to develop essential maths numeracy skills for high school students aged 12. Basics of financial mathematics a study guide 2012. The activities below are organised sequentially from the easiest to the hardest. Reading is beneficial, because we can get a lot of information from the reading materials. The signpost maths exercise book has been damaged by water but is still useable. Pearson mathematics 7 student book second edition created date. Bsc maths book downloded pdf in trichy 2019 fraud bible download link political lists jfk jr cs class 12 python preeti arora bsc maths book downloded pdf in. They may be used for personal use or class use, but not for commercial purposes. Objectives are in black and ixl maths skills are in dark green. For more ks3 maths study materials, try our ks3 maths standard allinone revision and practice 9780007562770 and advanced allinone revision and practice. We have taken every year 6 sats question from 2000 2007 and regrouped them into topic sets. We have also aligned them with the new test framework used to produce sats tests from 2016. This course book covers topics appropriate for ks3 year 7 maths and accurately reflects the language and content of the new programme of study.1564 206 1427 1018 1486 1504 845 1140 164 288 43 1185 1438 433 1296 317 397 485 255 849 1546 686 880 1229 956 1071 359 119 1231 1019 237

FLOW, EXCEDANCE and PERCENTILES Jubilee River Flow Schematic An attempt to estimate the whole river flow. Where Thames Smooth Waters Glide Thames flow is usually measured in cubic metres per second: m³/sec. Its easy to miss what this means. A cubic metre is 1000Kg of water - 1 tonne. The Thames flow at the Farmoor Gauge at the moment is online as m³/sec. That's tonnes per minute, or tonnes per hour, or tonnes per day, or Megalitres per day. Is that a lot or a little? London's water requirement, a good proportion of which comes out of the Thames is about 27m³/sec. So the flow at Farmoor, in the upper reaches above Oxford, at this moment, is about % of what Londoners use for one purpose, (or another). (To be fair to them they do use it several times!) But is that a flood or a drought or somewhere in between? To compare it with other places up and down the river, where the character of the river varies and various inflows from other watercourses add their contributions, and then various reservoirs take it away again , some way of standardising the assessment is required. The chosen method is Excedance. It is a percentage. (Percentile) The percentage of the time that the present flow is exceeded. That of course results in a somewhat counter intuitive figure in that it is low when the river is in flood and high when it is in drought. The NRFA give some standard figures to judge the flow against. These are for Farmoor: |This is a drought situation. (95% of the time the flow exceeds this).| |The summer norm will be around this figure.| |Only don't think this is the mean average because its not!| |That's the average, skewed from 50% by the fewer very high flows| |Not boating weather! We are getting towards flood!| |and very definitely flood conditions!| So how do the NRFA come up with these magic figures? Its not just "suck it and see"! They measure, and how they measure! Online at the moment are 27 numbers some or all of which are applicable to each and every significant place (and in my reckoning there are 101 such places from Ewen to Teddington plus the significant side streams. That makes a database of 2727 figures some of which will change every 15 minutes. So how are the Excedance percentages calculated? In the case of Farmoor, every day since 7th September 1992 the mean daily flow has been recorded: 6.31, 5.52, 6.81, 23.2, 32.3, 32.7, 36.9, 30.4, 20.3, .... and so on - lots and lots of apparently random numbers. So how to make sense of them? Well the Mean figure is easy enough: add them up and divide by the number of figures. NRFA answer m³/sec. But what about all those Excedance percentages? This is where computers come in. It simply wasn't feasible in the days of calculators - and one shudders to think of trying to do it by hand, but on a computer is actually quite easy once you understand what is going on. You simply get hold of all the figures and sort them into order from the largest to the smallest. Then the 50% figure will be exactly half way through your sorted list. That's the Q50% figure. And in that long list the figure halfway through is m³/sec. So what is the percentile of the present figure m³/sec? You can see by eye how it fits in to the percentiles given. And that is enough for most purposes maybe. But what is its actual percentile? Go through the sorted list and the fraction through the list that you find the nearest figure is, represents the percentile. All you need is a handy list of the 365 x 27 days worth. It can be done, but maybe life is a little short to do it too often. And you may have noticed that these figures are all comparative. A wet year will shift them. Climate change may alter them significantly. But this last year for which figures are available (the twelve months from September 2017) taken alone would reduce the figures significantly. The Farmoor mean for that twelve months was 13.674m³/sec. One of the issues for hydrologists and meteriorologists is how you cope with a shifting baseline in such data. The longer your period the more stable - but in a time of shifting climate maybe less relevant? Given the NRFA data this is my current (pun warning!) calculation. The slight discrepancies are not significant but probably caused by selecting very slightly different dates (or treating the blanks differently). Q0% is the largest daily mean flow in the period and for no day in the data period was that mean flow exceeded. Q100% is the smallest daily mean flow in the period and every other day in the data period the mean flow was greater. The NRFA quote their figures (mine in brackets) as: Q5%=52.1 (52.1) ; Q10%= 39.3 (39.4) ; Q50%=9.22 (9.24) ; Q70%=4.03(4.03) ; Q95%=0.96 (0.96) ; Meanflow=14.906 (14.92) (Qmeanflow%= 35%) Here are the Sutton Courtenay Gauge figures: Q5%=95.69 (95.69) ; Q10%= 68.4 (68.4) ; Q50%=16 (16) ; Q70%=7.89(7.89) ; Q95%=2.5 (2.5) ; Meanflow=27.495(27.5) (Qmeanflow%= 34%) And the Reading Gauge figures: Q5%=129(129) ; Q10%= 93.7 (93.5) ; Q50%=22.9 (22.9) ; Q70%=12.5(12.5) ; Q95%=5.26 (5.26) ; Meanflow=37.764(37.76) (Qmeanflow%= 33%) And, sorry but you're not going to believe this, the Taplow and Maidenhead figures are not available from the NRFA! The flow and heights are online but not from NRFA The Jubilee River and the arguments over its effectiveness in flood prevention, and the question of just how much of the flow should be taken for reservoirs, would seem to me worth the effort to make these available online. However I do have 24 months worth at Maidenhead. The percentiles would not make sense because they will be skewed by the flow in the Jubilee River. And I also have 24 months of Taplow and equally the percentiles would not be relevant But adding Maidenhead and Taplow daily flows, it then becomes possible to create a CLIVEDEN (ie whole channel flow including Taplow and Maidenhead) percentiles table: Then going downstream we can see what comes out after the Jubilee River reunites with the main channel: Q5%=173(173) ; Q10%= 128 (128) ; Q50%=40.74 (40.729) ; Q70%=26.65(26.64) ; Q95%=14.79 (14.78) ; Meanflow=59.192(59.19) (Qmeanflow%= 34%) Here are the Staines Gauge Figures: Q5%=191(191) ; Q10%= 137 (137) ; Q50%=33.1 (33.1) ; Q70%=17.81(17.81) ; Q95%=10.7 (10.7) ; Meanflow=55.745(55.74) (Qmeanflow%= 33%) Notice the high flow figures continue to rise as we go downstream, but Q50%, Q70% and Q95% figures are all significantly reduced, presumably by water abstraction. Londoners are drinking the Thames! Q5%=201 (201) ; Q10%= 143 (143) ; Q50%=33.23 (33.2) ; Q70%=17.3(17.3) ; Q95%=9.8 (9.8) ; Meanflow=58.249(58.25) (Qmeanflow%= 35%) For Kingston there is an embarrassment of riches, a complete data set since 1883. Only of course the methods have changed since then so here is the 1991-2018 set to match (more or less) the other gauges: Q5%=216 (227) ; Q10%= 161 (161) ; Q50%=40.2 (33.2) ; Q70%=21.7 (14.7) ; Q95%=7.56 (6.18) ; Meanflow=65.298 (61.83) (Qmeanflow%= 33%) Whoops, suddenly my figures do not match! The method is proved by the other Gauge figures which are essentially identical - so this must be down to using a different date range. Repeating the exercise with the full range 1883 to 2018 produced the following table: Q5%=216 (216) ; Q10%= 161 (161) ; Q50%=40.2 (40.2) ; Q70%=21.7 (21.7) ; Q95%=7.56 (7.56) ; Meanflow=65.298 (65.3) (Qmeanflow%= 31%) So my figures are vindicated! BUT my doubts are raised - should we really be using figures from 1883 for this one Gauge? Wouldn't it be better to use the more or less matching range to match the other gauges? Not only have the measuring methods and physical gauges changed but a whole new regime of pipeline extraction has been implemented.