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https://stats.stackexchange.com/questions/11473/uncorrected-pairwise-p-values-for-one-way-anova/167406	[ "Uncorrected pairwise p-values for one-way ANOVA?\n\nI'm performing a relatively simple one-way ANOVA as part of a class exercise. I'd like to get pairwise uncorrected p-values out of R, so I can do a sequential FDR test in another package (I realize there are FDR packages in R as well). I've set up my ANOVA as shown below, and it works fine, produces results, but I can't seem to figure out how to get the raw, uncorrected p-values back. Ultimately, I'd like to do the pairwise tests for both FDR and sequential Bonferroni in R, but this seems like a first step.\n\nIt looks like pairwiseCI or multcomp might get me where I'm trying to go, but I'm having a difficult time figuring out which will do what I'm looking for.\n\ntime breed\n1 27.4 type.A\n2 18.3 type.A\n3 24.3 type.B\n4 19.6 type.B\n5 21.6 type.C\n6 30.3 type.D\n\na10 <- aov(time~breed,data=d10)\n\n# reports the overall significance, but nothing on the pairs\nsummary(a10)\n\n# reports corrected values only\nTukeyHSD(a10)\n\nFor the multcomp package, see the help page for glht; you want to use the \"Tukey\" option; this does not actually use the Tukey correction, it just sets up all pairwise comparisons. In the example section there's an example that does exactly what you want.\n\nThis calculates the estimates and se's for each comparison but doesn't do p-values; for that you need summary.glht; on the help page for that, notice in particular the test parameter, that allows you to set the actual test that is run. For doing multiply-adjusted p-values, you use the adjusted function for this parameter, and to not multiply-adjust, you use test=adjusted(\"none\") (which isn't mention specifically in the help page, though it does say it takes any of the methods in p.adjust, which is where you'd find none.)\n\nYou can also compute the estimates and se's by hand using matrix multiplication, and then get the p-values however you want; this is what the glht function is doing behind the scenes. To get the matrices you need to start you'd use coef and vcov.\n\nI didn't put complete code as you say it's for a class project (thanks for being honest, by the way!) and the policy here is to provide helpful hints but not solutions.\n\n• thanks for the detailed answer! Re: homework; this is a graduate course and this work isn't being turned in, I know how to perform the same in the class-used JMP but would like to stick to R as much as possible.\n– scw\nJun 2 '11 at 18:42\n• Thanks for the update about the homework status. If you need more specific help, let us know. Jun 2 '11 at 19:04\n\nYou can use pairwise.t.test() with one of the available options for multiple comparison correction in the p.adjust.method= argument; see help(p.adjust) for more information on the available option for single-step and step-down methods (e.g., BH for FDR or bonf for Bonferroni). Of note, you can directly give p.adjust() a vector of raw p-values and it will give you the corrected p-values.\n\nSo, I would suggest to run something like\n\npairwise.t.test(time, breed, p.adjust.method =\"none\") # uncorrected p-value\npairwise.t.test(time, breed, p.adjust.method =\"bonf\") # Bonferroni p-value\n\nThe first command gives you t-test-based p-values without controlling for FWER or FDR. You can then use whatever command you like to get corrected p-values.\n\n• By the name, one might think it computes each pairwise t test separately (I did, anyway), but it actually uses the pooled SD by default, so the result is the same as using the anova result. Jun 2 '11 at 15:12\n• @Aaron You're right, it uses pooled SD by default, but it can be disabled. We could also construct every contrast we want with simple algebra. I like your response btw (+1).\n– chl\nJun 2 '11 at 15:38\nlibrary(multcomp)\ndf = mtcars\ndf$am = as.factor(df$am)\nm1 <- aov(mpg ~ am, data= df)\nht = glht(m1, linfct = mcp(am = \"Tukey\"))" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9185118,"math_prob":0.54167086,"size":927,"snap":"2022-05-2022-21","text_gpt3_token_len":267,"char_repetition_ratio":0.100758396,"word_repetition_ratio":0.0,"special_character_ratio":0.27508092,"punctuation_ratio":0.12676056,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.981482,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-22T12:58:15Z\",\"WARC-Record-ID\":\"<urn:uuid:de9dd4ec-a6ed-4632-af61-32cb3ee49537>\",\"Content-Length\":\"157532\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:36f4a54a-cd2e-4673-bb60-ca754a6fc02c>\",\"WARC-Concurrent-To\":\"<urn:uuid:9bc95552-35ba-4706-b51e-c296c2ffc47c>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/11473/uncorrected-pairwise-p-values-for-one-way-anova/167406\",\"WARC-Payload-Digest\":\"sha1:GOLLIMFCHK2PPOCIQG5P2E2GNSDLFZM5\",\"WARC-Block-Digest\":\"sha1:XQNG72IN5PDEPSI3CBFEMHH6JOMQENXY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320303845.33_warc_CC-MAIN-20220122103819-20220122133819-00512.warc.gz\"}"}
https://breathmath.com/2017/07/01/surface-area-and-volumes-exercise-13-5-class-ix/	[ "# Surface Area and Volumes – Exercise 13.5 – Class IX\n\n1. A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes?\n\nSolution:\n\nVolume of 1 match box = lbh [here, l = 4cm , b = 2.5cm, h = 1.5cm]\n\n= 4 x 2.5 x 1.5\n\n= 15 cm3\n\nVolume of 12 such match boxes = 15 x 12 = 180 cm3\n\n1. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1 m3 = 1000 l)\n\nSolution:\n\nVolume of water tank = lbh [here, l = 6m , b = 5m, h = 4.5m]\n\n= 6 x 5 x 4.5\n\n= 135 m3\n\nCapacity of water tank for 1m3 = 1000litres\n\nThen, capacity of water tank for 135m3 = 135 x 1000 = 135000 litres\n\n1. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?\n\nSolution:\n\nVolume of cuboidal vessel = lbh [here, l = 10m , b = 8m, volume = 380m3]\n\n380 = 10 x 8 x h\n\nh = 380/10×8\n\nh = 4.75 m\n\n1. Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of Rs 30 per m3\n\nSolution:\n\nVolume of cuboidal pit = lbh\n\n= 8 x 6 x 3\n\n= 144 m3\n\nThe cost of digging cuboidal pit for 1m3 = Rs. 30\n\nThe cost of digging cuboidal pit for 144m3 = 144 x 30 = Rs. 4320\n\n1. The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.\n\nSolution:\n\nGiven length of tank = 2.5m and height 10 m\n\ncapacity of water tank = 50000 litres = 50000/1000 m3 = 50m3\n\nCapacity of water tank = volume of water tank = lbh\n\n50 = 2.5 x b x 10\n\nb = 50/2.5×10 = 2 m\n\ni.e., breadth of the tank = 2m\n\n1. A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. For how many days will the water of this tank last?\n\nSolution:\n\nGiven, a village, having a population of 4000, requires 150 litres of water per head per day.\n\nWater required for 1 person for one day = 150 litres\n\nTherefore, water required for 4000 peoples for one day = 150 x 4000 = 6,00,000 litres\n\nVolume of the water tank = lbh\n\nl = 20m , b = 15 m and h = 6m\n\nVolume of watertank = 20 x 15 x 6\n\n= 1800m3\n\nWater holding capacity of the tank = 1800 x 1000 = 18,00,000 litres.\n\nThus, the number of days will the water of this tank last = water holding capacity of the tank/water required for 4000 peoples for one day\n\n= 18,00,000/6,00,000 = 3\n\nHence, one tank of water will last for 3 days.\n\n1. A godown measures 40 m × 25 m × 10 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.\n\nSolution:\n\nVolume of godown = lbh [here l = 40m, b = 25m , h = 10m]\n\n= 40 x 25 x 10\n\n= 10000 m3\n\nVolume of the wooden crates = lbh [l = 1.5m , b = 1 .25m , h = 0.5 m]\n\n= 1.5 x 1.25 x 0.5\n\n= 0.9375 m3\n\nNo. of wooden crates stored in godown = Volume of godown/ Volume of the wooden crates\n\n= 10000/0.9375\n\n= 10666. 67\n\n1. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.\n\nSolution:\n\nVolume of cube = a3\n\n= 123\n\n= 1728 cm3\n\nGiven, solid cube of side 12 cm is cut into eight cubes of equal volume. i.e., = 1728/8 = 216 cm3\n\nTherefore, volume of the each new cube = 216 cm3\n\nWe know, volume of each new cube = (s)3\n\n216 = s3\n\ns = 6 cm\n\nThe ratio between volume of a soild cube and volume of new cube = 12 : 6 = 2 : 1\n\n1. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?\n\nsolution:\n\nh = 3 m ; b = 40 m ;\n\nWater flow for 1 hour, l = 2 km = 2000m\n\nVolume of water flowing through the river in 1 hour = lbh\n\n= 2000 x 3 x 40\n\n= 240000 m3\n\nVolume of water flowing in 1 minute = 240000/60 = 4000 m3" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8546184,"math_prob":0.999887,"size":3667,"snap":"2021-43-2021-49","text_gpt3_token_len":1325,"char_repetition_ratio":0.15315315,"word_repetition_ratio":0.088757396,"special_character_ratio":0.41287157,"punctuation_ratio":0.12528217,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995415,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T00:23:18Z\",\"WARC-Record-ID\":\"<urn:uuid:d1bed45e-4b21-4892-a8c9-81d3fc1560bc>\",\"Content-Length\":\"94816\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e35f8f7d-40c8-4150-80cb-575685a2af1a>\",\"WARC-Concurrent-To\":\"<urn:uuid:e7fa27af-1643-4955-a349-332cba2d8889>\",\"WARC-IP-Address\":\"192.0.78.25\",\"WARC-Target-URI\":\"https://breathmath.com/2017/07/01/surface-area-and-volumes-exercise-13-5-class-ix/\",\"WARC-Payload-Digest\":\"sha1:VMX5BWR776FCDFWF262D4NQ424PMTPNN\",\"WARC-Block-Digest\":\"sha1:4LEH4DKYQI5AI46XM6LO5AYJHYLPLHFV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585290.83_warc_CC-MAIN-20211019233130-20211020023130-00688.warc.gz\"}"}
http://vreitmann.de/?site=topic2	[ "", null, "Home Research interests Publications Conferences/Talks Supported projects since 2007 Supported projects (1994 - 2006) Biographical Data\n\n The Kalman-Yakubovich-Popov Theorem in thermo-elastic problems\n The Kalman-Yakubovich-Popov Theorem for infinite-dimensional systems: Some new results a) Bounded control operator, C0-semigroups, Pritchard-Salamon systems Louis, J. Cl. and D. Wexler: The Hilbert space regulator problem and operator Riccati equation under stabilizability. Ann. Soc. Sci. Bruxelles, Ser. 1, 105, 1991, 137 -- 165. Van Keulen, B.: Equivalent conditions for the solvability of the nonstandard LQ-problem for Pritchard-Salamon systems. J. Control Optim., 33, 1995, 1326 -- 1356. Curtain, R. F.: The Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems. System & Control Lett., 27, 1996, 67 -- 72. Curtain, R. F. and J. C. Oostveen: The Popov criterion for strongly stable distributed parameter systems. Int. J. Control, 74 (3), 2001, 265 -- 280. b) Parabolic systems, holomorphic semigroups, heat equation Pandolfi, L.: Dissipativity and Lur'e problem for parabolic boundary control systems. SIAM J. Control Optimiz., 36, 1998, 2061 -- 2081. Bucci, F.: Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynamics of Continuous, Discrete and Impulsive Systems, 7, 2000, 351 -- 368.c) First order hyperbolic equations Triggiani, R.: An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation. Appl. Math. Lett., 10 (2), 1997, 95 -- 102. d) Second order hyperbolic systems, Euler-Bernoulli equation, Kirchhoff equation, Schrödinger equation, string and membrane equation Lasiecka, I. and R. Triggiani: Algebraic Riccati equations arising in boundary / point control: A review of theoretical and numerical results. In: Perspective in control theory, Jacubczyk, B., Malanowski, K., Raspondek, W. eds., Birkhäuser, Boston, 1990. Part 1: Continuous case, 175 -- 210; Part 2: Approximation theory, 211 -- 235. Bucci, F.: The non-standard LQR problem for boundary control systems. Rend. Sem. Mat. Univ. Pol. Torino, 56 (4), 1998, 105 -- 114. Pandolfi, L.: The Kalman-Yakubovich-Popov Theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory, 34, 1999, 478 -- 493. Barbu, V., Lasiecka, I. and R. Triggiani: Extended algebraic Riccati equations in the abstract hyperbolic case. Nonlinear. Anal. 40, 2000, 105 -- 129. e) Nonstandard Riccati equations arising in boundary control problems governed by damped wave and plate equations; Hammerstein integral equations with weekly singular kernels Flandoli, F.: Riccati equations arising in a boundary control problem with distributed parameters. SIAM Journal on Control and Optimization, 22, 1984, 76 -- 86. Lasiecka, I., D. Lukas and L. Pandolfi: Input dynamics and nonstandard Riccati equations with applications to boundary control of damped wave and plate equations. Journal of Optimization Theory and Applications, 84 (3), 1995, 549 -- 574. f) Discrete time distributed systems, approximation theory for boundary control systems Helton, J. W.: A spectral factorization approach to the distributed stable regulator problem; the algebraic Riccati equation. SIAM Journal on Control and Optimization, 14, 1976, 639 -- 661. Malinen, J.: Discrete time H^\\infty algebraic Riccati equations. Techn. Report A 428, Institute of Mathematics, Helsinki University, Finland, 2000. Arov, D. Z., Kaashoek, M. A., and D. R. Pik: The Kalman-Yakubovich-Popov inequality and infinite dimensional discrete time dissipative systems. J. Operator Theory, 2005, to appear. g) Generalized (possibly unbounded) solutions of the KYP inequality Arov, D. Z. and O. J. Staffans: The infinite-dimensional continuous time Kalman-Yakubovich-Popov inequality. Operator Theory: Advances and Applications, 1, 2005, Birkhäuser Verlag Basel, 1 -- 28. h) Overview articles Pandolfi, L.: The Kalman-Popov-Yakubovich Theorem: an overview and new results for hyperbolic control systems. Nonlinear Analysis, Methods & Applications, 30 (2), 1997, 735 -- 745. Pandolfi, L.: Recent results on the Kalman-Popov-Yakubovich problem. Proc. Int. Conf. on Mathematics and its Applications, Yagyarta, 1999.", null, "" ]	[ null, "http://vreitmann.de/Bilder/papahp_01.gif", null, "http://vreitmann.de/Bilder/papahp_05.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62640435,"math_prob":0.89119184,"size":4207,"snap":"2022-05-2022-21","text_gpt3_token_len":1249,"char_repetition_ratio":0.13680704,"word_repetition_ratio":0.013266998,"special_character_ratio":0.2776325,"punctuation_ratio":0.28674698,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9812739,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,8,null,8,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-18T01:45:29Z\",\"WARC-Record-ID\":\"<urn:uuid:79e5f848-4886-4c5c-aba8-c7a488a92a5f>\",\"Content-Length\":\"7849\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:96577d45-63f1-46d6-ba48-6e4b69f98a3f>\",\"WARC-Concurrent-To\":\"<urn:uuid:311c0372-9e60-468b-a94a-a347433e38ea>\",\"WARC-IP-Address\":\"81.169.145.143\",\"WARC-Target-URI\":\"http://vreitmann.de/?site=topic2\",\"WARC-Payload-Digest\":\"sha1:UURF34L6PJUO7D2F44SNJEGLQS4VBTE3\",\"WARC-Block-Digest\":\"sha1:YRFKW6ATC4ZXA4C7LBEGIG2V7UL4R35O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320300658.84_warc_CC-MAIN-20220118002226-20220118032226-00577.warc.gz\"}"}
https://metanumbers.com/231755	[ "# 231755 (number)\n\n231,755 (two hundred thirty-one thousand seven hundred fifty-five) is an odd six-digits composite number following 231754 and preceding 231756. In scientific notation, it is written as 2.31755 × 105. The sum of its digits is 23. It has a total of 2 prime factors and 4 positive divisors. There are 185,400 positive integers (up to 231755) that are relatively prime to 231755.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Odd\n• Number length 6\n• Sum of Digits 23\n• Digital Root 5\n\n## Name\n\nShort name 231 thousand 755 two hundred thirty-one thousand seven hundred fifty-five\n\n## Notation\n\nScientific notation 2.31755 × 105 231.755 × 103\n\n## Prime Factorization of 231755\n\nPrime Factorization 5 × 46351\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 2 Total number of prime factors rad(n) 231755 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 231,755 is 5 × 46351. Since it has a total of 2 prime factors, 231,755 is a composite number.\n\n## Divisors of 231755\n\n4 divisors\n\n Even divisors 0 4 2 2\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 4 Total number of the positive divisors of n σ(n) 278112 Sum of all the positive divisors of n s(n) 46357 Sum of the proper positive divisors of n A(n) 69528 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 481.409 Returns the nth root of the product of n divisors H(n) 3.33326 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 231,755 can be divided by 4 positive divisors (out of which 0 are even, and 4 are odd). The sum of these divisors (counting 231,755) is 278,112, the average is 69,528.\n\n## Other Arithmetic Functions (n = 231755)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 185400 Total number of positive integers not greater than n that are coprime to n λ(n) 92700 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 20512 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares\n\nThere are 185,400 positive integers (less than 231,755) that are coprime with 231,755. And there are approximately 20,512 prime numbers less than or equal to 231,755.\n\n## Divisibility of 231755\n\n m n mod m 2 3 4 5 6 7 8 9 1 2 3 0 5 6 3 5\n\nThe number 231,755 is divisible by 5.\n\n## Classification of 231755\n\n• Arithmetic\n• Semiprime\n• Deficient\n\n• Polite\n\n• Square Free\n\n### Other numbers\n\n• LucasCarmichael\n\n## Base conversion (231755)\n\nBase System Value\n2 Binary 111000100101001011\n3 Ternary 102202220112\n4 Quaternary 320211023\n5 Quinary 24404010\n6 Senary 4544535\n8 Octal 704513\n10 Decimal 231755\n12 Duodecimal b214b\n20 Vigesimal 18j7f\n36 Base36 4ytn\n\n## Basic calculations (n = 231755)\n\n### Multiplication\n\nn×y\n n×2 463510 695265 927020 1158775\n\n### Division\n\nn÷y\n n÷2 115878 77251.7 57938.8 46351\n\n### Exponentiation\n\nny\n n2 53710380025 12447649122693875 2884804922429919000625 668567964797745877989846875\n\n### Nth Root\n\ny√n\n 2√n 481.409 61.4247 21.941 11.8306\n\n## 231755 as geometric shapes\n\n### Circle\n\n Diameter 463510 1.45616e+06 1.68736e+11\n\n### Sphere\n\n Volume 5.21406e+16 6.74945e+11 1.45616e+06\n\n### Square\n\nLength = n\n Perimeter 927020 5.37104e+10 327751\n\n### Cube\n\nLength = n\n Surface area 3.22262e+11 1.24476e+16 401411\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 695265 2.32573e+10 200706\n\n### Triangular Pyramid\n\nLength = n\n Surface area 9.30291e+10 1.46697e+15 189227\n\n## Cryptographic Hash Functions\n\nmd5 098aa99957b3a6e7ce4dedf26e5d52b2 167a0e0da5b22796e6015ca6ff67dfc95a248786 832c1bcb29128105090d51540d4c1585564f8da3eb0e0bac38663a6aae37b112 4a62e146367e1f84ab49d8775c76de1de5a9b5d75547771a81802966888648dcf9f5aabd5318e097c8f9ab563564aca05f041423fe441928a210fbdf8fdf64f3 c3bde1d34c510093d9a9372b068d362184345626" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6190928,"math_prob":0.9775618,"size":4649,"snap":"2021-43-2021-49","text_gpt3_token_len":1626,"char_repetition_ratio":0.12034446,"word_repetition_ratio":0.031019202,"special_character_ratio":0.45816305,"punctuation_ratio":0.075351216,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9962485,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T18:36:13Z\",\"WARC-Record-ID\":\"<urn:uuid:48c25fcb-8ce8-480f-b6ed-1a8f9720eadb>\",\"Content-Length\":\"40034\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2d967113-ba34-41e7-8764-5c1136b22f39>\",\"WARC-Concurrent-To\":\"<urn:uuid:4cc66f1d-90d1-4e8b-9635-404f7e2b1564>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/231755\",\"WARC-Payload-Digest\":\"sha1:JT36E46QLBJDWP3ET6JK7AHSFQDHDVFQ\",\"WARC-Block-Digest\":\"sha1:Q7O5FALCUCJXCBK5YCHPYVMYG2HIY2MS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585348.66_warc_CC-MAIN-20211020183354-20211020213354-00558.warc.gz\"}"}
https://api.openml.org/d/34581	[ "Data\nQSAR-DATASET-FOR-DRUG-TARGET-CHEMBL4980\n\n# QSAR-DATASET-FOR-DRUG-TARGET-CHEMBL4980\n\nIssue #Downvotes for this reason By\n\nThis dataset contains QSAR data (from ChEMBL version 17) showing activity values (unit is pseudo-pCI50) of several compounds on drug target ChEMBL_ID: CHEMBL4980 (TID: 12167), and it has 1045 rows and 22 features (not including molecule IDs and class feature: molecule_id and pXC50). The features represent Basic Molecular Descriptors which were generated from SMILES strings. Missing value imputation was applied to this dataset (By choosing the Median). Feature selection was also applied.\n\n### 24 features\n\n pXC50 (target) numeric 539 unique values 0 missing molecule_id (row identifier) nominal 1045 unique values 0 missing Me numeric 62 unique values 0 missing RBF numeric 145 unique values 0 missing Mp numeric 123 unique values 0 missing Mv numeric 127 unique values 0 missing H. numeric 142 unique values 0 missing AMW numeric 521 unique values 0 missing nN numeric 13 unique values 0 missing nHet numeric 19 unique values 0 missing C. numeric 132 unique values 0 missing N. numeric 100 unique values 0 missing nBM numeric 25 unique values 0 missing MW numeric 625 unique values 0 missing RBN numeric 18 unique values 0 missing nAB numeric 12 unique values 0 missing nCsp2 numeric 23 unique values 0 missing nSK numeric 44 unique values 0 missing nBO numeric 46 unique values 0 missing SCBO numeric 83 unique values 0 missing nHM numeric 4 unique values 0 missing nDB numeric 12 unique values 0 missing nBT numeric 77 unique values 0 missing Se numeric 614 unique values 0 missing\n\n### 62 properties\n\n1045\nNumber of instances (rows) of the dataset.\n24\nNumber of attributes (columns) of the dataset.\n0\nNumber of distinct values of the target attribute (if it is nominal).\n0\nNumber of missing values in the dataset.\n0\nNumber of instances with at least one value missing.\n23\nNumber of numeric attributes.\n1\nNumber of nominal attributes.\n2.41\nSecond quartile (Median) of kurtosis among attributes of the numeric type.\n0.02\nNumber of attributes divided by the number of instances.\nAverage number of distinct values among the attributes of the nominal type.\n7.13\nSecond quartile (Median) of means among attributes of the numeric type.\nNumber of attributes needed to optimally describe the class (under the assumption of independence among attributes). Equals ClassEntropy divided by MeanMutualInformation.\n2.42\nMean skewness among attributes of the numeric type.\nSecond quartile (Median) of mutual information between the nominal attributes and the target attribute.\nPercentage of instances belonging to the most frequent class.\n10.26\nMean standard deviation of attributes of the numeric type.\n1.22\nSecond quartile (Median) of skewness among attributes of the numeric type.\nNumber of instances belonging to the most frequent class.\nMinimal entropy among attributes.\n0\nPercentage of binary attributes.\n3.53\nSecond quartile (Median) of standard deviation of attributes of the numeric type.\nMaximum entropy among attributes.\n-0.72\nMinimum kurtosis among attributes of the numeric type.\n0.07\nMinimum of means among attributes of the numeric type.\n0\nPercentage of instances having missing values.\nThird quartile of entropy among attributes.\n88.02\nMaximum kurtosis among attributes of the numeric type.\nMinimal mutual information between the nominal attributes and the target attribute.\n0\nPercentage of missing values.\n37.85\nThird quartile of kurtosis among attributes of the numeric type.\n320.23\nMaximum of means among attributes of the numeric type.\nThe minimal number of distinct values among attributes of the nominal type.\n95.83\nPercentage of numeric attributes.\n31.45\nThird quartile of means among attributes of the numeric type.\nMaximum mutual information between the nominal attributes and the target attribute.\n-0.56\nMinimum skewness among attributes of the numeric type.\n4.17\nPercentage of nominal attributes.\nThird quartile of mutual information between the nominal attributes and the target attribute.\nThe maximum number of distinct values among attributes of the nominal type.\n0.01\nMinimum standard deviation of attributes of the numeric type.\nFirst quartile of entropy among attributes.\n4.64\nThird quartile of skewness among attributes of the numeric type.\n8.19\nMaximum skewness among attributes of the numeric type.\nPercentage of instances belonging to the least frequent class.\n0.37\nFirst quartile of kurtosis among attributes of the numeric type.\n9.09\nThird quartile of standard deviation of attributes of the numeric type.\n131.94\nMaximum standard deviation of attributes of the numeric type.\nNumber of instances belonging to the least frequent class.\n1.01\nFirst quartile of means among attributes of the numeric type.\nStandard deviation of the number of distinct values among attributes of the nominal type.\nAverage entropy of the attributes.\n0\nNumber of binary attributes.\nFirst quartile of mutual information between the nominal attributes and the target attribute.\n20.77\nMean kurtosis among attributes of the numeric type.\n0.01\nFirst quartile of skewness among attributes of the numeric type.\n28.14\nMean of means among attributes of the numeric type.\n0.89\nFirst quartile of standard deviation of attributes of the numeric type.\n-0.18\nAverage class difference between consecutive instances.\nAverage mutual information between the nominal attributes and the target attribute.\nSecond quartile (Median) of entropy among attributes.\nEntropy of the target attribute values.\nAn estimate of the amount of irrelevant information in the attributes regarding the class. Equals (MeanAttributeEntropy - MeanMutualInformation) divided by MeanMutualInformation." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75127804,"math_prob":0.9224849,"size":4054,"snap":"2023-40-2023-50","text_gpt3_token_len":830,"char_repetition_ratio":0.28098765,"word_repetition_ratio":0.33333334,"special_character_ratio":0.20374939,"punctuation_ratio":0.12831241,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9840798,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T17:32:58Z\",\"WARC-Record-ID\":\"<urn:uuid:c0e88e99-963b-4f68-b1ef-fe348acd7988>\",\"Content-Length\":\"54748\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aa44e0e9-53f0-4931-a3ed-6d268356df56>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a7a9ce0-219d-4aa9-9550-a9d6d30867fe>\",\"WARC-IP-Address\":\"131.155.11.11\",\"WARC-Target-URI\":\"https://api.openml.org/d/34581\",\"WARC-Payload-Digest\":\"sha1:BEVL2OUPPPVMNBTXK6C2I6LPUTDELCHF\",\"WARC-Block-Digest\":\"sha1:GOR5OILLVLI7QRSEZI5B3YN5YCJUNEQJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510520.98_warc_CC-MAIN-20230929154432-20230929184432-00734.warc.gz\"}"}
http://slideplayer.com/slide/8382718/	[ "Presentation is loading. Please wait.", null, "# Section 7.1 Solving Linear Systems by Graphing. A System is two linear equations: Ax + By = C Dx + Ey = F A Solution of a system of linear equations in.\n\n## Presentation on theme: \"Section 7.1 Solving Linear Systems by Graphing. A System is two linear equations: Ax + By = C Dx + Ey = F A Solution of a system of linear equations in.\"— Presentation transcript:\n\nSection 7.1 Solving Linear Systems by Graphing\n\nA System is two linear equations: Ax + By = C Dx + Ey = F A Solution of a system of linear equations in two variables is and ordered pair (x, y) that ___________ each equation in the system. When graphing both equations for the linear system, you are looking for the ___________ of the linear equations. Where A, B, C, and D are integers and x & y are variables satisfy Solution\n\nChecking the intersection point Use the graph to solve the system of linear equations. Then check your solution algebraically.\n\nRules for Solving a linear system using graphing and check 1)Write each equation in slope intercept form (y=mx + b) 2)Graph both equations thoroughly (neatness counts) 3)Estimate the coordinates of the point of intersection 4)Check the coordinates algebraically by substituting into each equation of the original linear system.\n\nNO SOLUTION!!!!!!!!!!!!!!!!\n\nDownload ppt \"Section 7.1 Solving Linear Systems by Graphing. A System is two linear equations: Ax + By = C Dx + Ey = F A Solution of a system of linear equations in.\"\n\nSimilar presentations\n\nAds by Google" ]	[ null, "http://slideplayer.com/static/blue_design/img/slide-loader4.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8496492,"math_prob":0.9995566,"size":1099,"snap":"2021-04-2021-17","text_gpt3_token_len":252,"char_repetition_ratio":0.17260274,"word_repetition_ratio":0.0,"special_character_ratio":0.24294813,"punctuation_ratio":0.15566038,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999206,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-14T08:44:14Z\",\"WARC-Record-ID\":\"<urn:uuid:3c01960a-4d53-49e0-b871-2e31b0bd6207>\",\"Content-Length\":\"140535\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f6847902-790b-4eab-8b8c-7002ee756439>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c293e40-a686-4bfc-83e2-e2e179210923>\",\"WARC-IP-Address\":\"138.201.58.10\",\"WARC-Target-URI\":\"http://slideplayer.com/slide/8382718/\",\"WARC-Payload-Digest\":\"sha1:E2ASGAY5FVHPJPJWVBP7NOOLBZWUHFXD\",\"WARC-Block-Digest\":\"sha1:DJHCY63YGVUWKBZKXY5T3LSBH33EH5FP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038077336.28_warc_CC-MAIN-20210414064832-20210414094832-00063.warc.gz\"}"}
https://argoprep.com/math/1st-grade/money/parts-of-a-whole/	[ "# Parts of a Whole\n\n## Overview\n\nWe can divide shapes into parts. When we divide shapes up, we can find how many parts make up the whole. When we divide things fairly, the parts are EQUAL.\n\nFor example, look below. The rectangle is divided into four EQUAL parts. Four PARTS make up the WHOLE rectangle.\n\nThree PARTS are shaded. There are FOUR parts that make up ONE whole rectangle.\n\nTake a look at the examples below. Some of the shapes are EQUAL. Some of the shapes are UNEQUAL. In order to be equal, the pieces must be the same size.\n\nThe first column above shows shapes with equal parts. We could share those pieces fairly.\n\nNotice that the rectangle and the square are divided into TWO PARTS. TWO parts make up the WHOLE shape.\n\nThe circle is divided into EIGHT PARTS. EIGHT parts make up the whole circle.\n\nThe other side shows unequal parts. These pieces would not be shared in a fair way.\n\nLet’s go ahead and try some practice examples.\nWe will decide how many parts make up the whole shape.\n\nThis square is divided into equal parts.\nHow many parts make up the whole square?\nFour parts make up the whole square.\nThere are FOUR parts.\n\nLet’s take a look at this next example.\nWe will decide again how many parts make up the whole shape.\n\nThis circle is divided into equal parts.\nHow many parts make up the whole circle?\nTwelve parts make up the whole circle.\nThere are TWELVE parts.\n\nLet’s take it one step further!\n\nWe will decide how many parts make up the whole shape first.\n\nThis diamond is divided into parts.\nHow many parts make up the whole diamond?\nFour parts make up the whole diamond.\nThere are FOUR parts.\n\nHow many parts of the diamond are shaded in?\n\nTHREE of the parts are shaded in.\nThe diamond is made up of four parts with three of those parts shaded in.\n\nLet’s try another one!\n\nWe will decide how many parts make up the whole shape first.\n\nThis heart is divided into parts.\nHow many parts make up the whole heart?\nTwo parts make up the whole heart.\nThere are TWO parts.\n\nHow many parts of the heart are shaded in?\n\nONE of the parts is shaded in.\nThe heart is made up of two parts with one of those parts shaded in.\n\n### Practice Question 1\n\nHow many PARTS make up the WHOLE shape?\n\nSee Related Worksheets:\nDoggone Great Geometric Division Practice\nWorksheets\n(0)\nKids (and dogs) will love this practical \"equal parts\" practice! To complete each of the four sets o...\nHalves, Fourths, or Neither, Remix!\nWorksheets\n(0)\nThis is a second five-question quiz in which students must decide whether the parts are equal. If th...\nHalves, Fourths, or Neither?\nWorksheets\n(0)\nStudents decide whether each of these five divided shapes is divided into halves, fourths, or neithe...\nDixie Dino Identifies Shapes\nWorksheets\n(0)\nIt's time to \"dinosoar\" into identifying shapes! Dixie Dino is determined to find all the matching s...\nRelevant Topics:\n\nShare good content with friends and get 15% discount for 12-month subscription\n\nCount the pieces. How many make up the WHOLE shape?\nTIP:\n\n### Practice Question 2\n\nLet’s try another! How many PARTS make up the WHOLE shape?\n\n### Practice Question 3\n\nHow many PARTS make up the WHOLE shape?\n\n### Practice Question 4\n\nLook at the picture below.\nHow many PARTS make up the WHOLE shape?\n\n### Practice Question 5\n\nLook at the picture below.\nHow many PARTS make up the WHOLE shape?\n\nSee Related Worksheets:\nDoggone Great Geometric Division Practice\nWorksheets\n(0)\nKids (and dogs) will love this practical \"equal parts\" practice! To complete each of the four sets o...\nHalves, Fourths, or Neither, Remix!\nWorksheets\n(0)\nThis is a second five-question quiz in which students must decide whether the parts are equal. If th...\nHalves, Fourths, or Neither?\nWorksheets\n(0)\nStudents decide whether each of these five divided shapes is divided into halves, fourths, or neithe...\nDixie Dino Identifies Shapes\nWorksheets\n(0)\nIt's time to \"dinosoar\" into identifying shapes! Dixie Dino is determined to find all the matching s...\nRelevant Topics:\n\nShare good content with friends and get 15% discount for 12-month subscription\n\nYou should now easily be able to identify how many parts create a WHOLE. To figure that out, simply count all the parts.\nTIP:\n\nChallenge yourself and try the quiz below on what you have just learned!\n\nSee Related Worksheets:\nDoggone Great Geometric Division Practice\nWorksheets\n(0)\nKids (and dogs) will love this practical \"equal parts\" practice! To complete each of the four sets o...\nHalves, Fourths, or Neither, Remix!\nWorksheets\n(0)\nThis is a second five-question quiz in which students must decide whether the parts are equal. If th...\nHalves, Fourths, or Neither?\nWorksheets\n(0)\nStudents decide whether each of these five divided shapes is divided into halves, fourths, or neithe...\nDixie Dino Identifies Shapes\nWorksheets\n(0)\nIt's time to \"dinosoar\" into identifying shapes! Dixie Dino is determined to find all the matching s...\nRelevant Topics:" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9170443,"math_prob":0.8818538,"size":3058,"snap":"2021-31-2021-39","text_gpt3_token_len":706,"char_repetition_ratio":0.22396857,"word_repetition_ratio":0.22084805,"special_character_ratio":0.22302158,"punctuation_ratio":0.1105919,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9793685,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-20T07:15:40Z\",\"WARC-Record-ID\":\"<urn:uuid:ea8053c9-390c-41e1-9ac1-daaacdd5b4a5>\",\"Content-Length\":\"280430\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2177536b-5d00-449e-95c5-6221ce372a87>\",\"WARC-Concurrent-To\":\"<urn:uuid:5aa5862c-df61-4b04-a58f-faf811c8699e>\",\"WARC-IP-Address\":\"35.245.210.119\",\"WARC-Target-URI\":\"https://argoprep.com/math/1st-grade/money/parts-of-a-whole/\",\"WARC-Payload-Digest\":\"sha1:5K6XHX2PMOJFD6BVRZP3AH3NXP5ZQAUI\",\"WARC-Block-Digest\":\"sha1:NJNRWWSVV3UCA5W2D67EGIPF5UOPHYSH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057033.33_warc_CC-MAIN-20210920070754-20210920100754-00522.warc.gz\"}"}
https://www.codevscolor.com/c-find-volume-cube	[ "# 5 different ways to find the volume of a cube in C", null, "## Introduction :\n\nIn this tutorial, we will learn how to find the volume of a cube in C programming language. The program will take the side of the cube as input and prints the volume. The formula used to calculate the volume of a cube is as below :\n\n``Volume = side ^ 3``\n\nIf the side is 2 cm, its volume will be 8 cm^3.\n\nI will show you different ways to solve this problem. Go through the examples and drop one comment below if you have any questions.\n\n### Method 1: Standard way :\n\nLet’s take a look at the below program :\n\n``````#include <stdio.h>\n\nint main()\n{\nfloat side, volume;\n\nside = 4;\nvolume = side * side * side;\n\nprintf(\"The volume is : %.02f\\n\", volume);\nreturn 0;\n}``````\n\nHere, we are using two floating-point variables side and volume. This program calculates the volume for side = 4.\n\nWe are multiplying the value of side three times to get the cube of it.\n\nIt will print the below output :\n\n``The volume is : 64.00``\n\n### Method 2: Using math.pow() function :\n\nmath.h header library defines one function called pow() to find out the power of a number. We can use it to find the cube of a number as like below :\n\n``````#include <stdio.h>\n#include <math.h>\n\nint main()\n{\nfloat side, volume;\n\nside = 4;\nvolume = pow(side, 3);\n\nprintf(\"The volume is : %.02f\\n\", volume);\nreturn 0;\n}``````\n\nIt will print :\n\n``The volume is : 64.00``\n\n### Method 3: By taking the side as user input :\n\nWe can use scanf to read the side as user input and calculate the volume as like below :\n\n``````#include <stdio.h>\n#include <math.h>\n\nint main()\n{\nfloat side, volume;\n\nprintf(\"Enter the side of the cube : \");\nscanf(\"%f\",&side);\n\nvolume = pow(side, 3);\n\nprintf(\"The volume is : %.02f\\n\", volume);\nreturn 0;\n}``````\n\nIt reads the side into the variable side and calculates the volume.\n\nIt can take any value as side and calculate the value of volume. One sample output :\n\n``````Enter the side of the cube : 6\nThe volume is : 216.00\n\nEnter the side of the cube : 4\nThe volume is : 64.00``````\n\n### Method 4: Using a different function :\n\nWe can use one different function to do the calculation and the main function can call this function :\n\n``````#include <stdio.h>\n#include <math.h>\n\nfloat findCube(float side)\n{\nreturn pow(side, 3);\n}\nint main()\n{\nfloat side, volume;\n\nprintf(\"Enter the side of the cube : \");\nscanf(\"%f\", &side);\n\nvolume = findCube(side);\n\nprintf(\"The volume is : %.02f\\n\", volume);\nreturn 0;\n}``````\n\nUsing a different function is useful if you want to reuse that function. We can put the implementation in one place and call that method from other places.\n\n### Method 5: Using pointers :\n\nWe can also use pointers to do the same thing as below :\n\n``````#include <stdio.h>\n#include <math.h>\n\nfloat findCube(float side)\n{\nreturn pow(side, 3);\n}\nint main()\n{\nfloat side, volume;\n\nprintf(\"Enter the side of the cube : \");\nscanf(\"%f\", &side);\n\nvolume = findCube(&side);\n\nprintf(\"The volume is : %.02f\\n\", volume);\nreturn 0;\n}``````\n\nHere, we are passing the address of side to findCube. *side gets the value of side." ]	[ null, "data:image/png;base64,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", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7545416,"math_prob":0.9920456,"size":3240,"snap":"2021-43-2021-49","text_gpt3_token_len":822,"char_repetition_ratio":0.16100124,"word_repetition_ratio":0.24172185,"special_character_ratio":0.2882716,"punctuation_ratio":0.17114569,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982834,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-28T08:57:51Z\",\"WARC-Record-ID\":\"<urn:uuid:0e8ab176-c5d9-4864-b2c1-b023cfc674ae>\",\"Content-Length\":\"88709\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2f6de79c-d955-4a96-9404-d60cf36ee7f2>\",\"WARC-Concurrent-To\":\"<urn:uuid:ad0e280c-bbd0-4351-a622-1fdd4c0a1837>\",\"WARC-IP-Address\":\"35.175.60.16\",\"WARC-Target-URI\":\"https://www.codevscolor.com/c-find-volume-cube\",\"WARC-Payload-Digest\":\"sha1:MQHDQRZT56V256PEWC3JJ3NGTLZFG2KE\",\"WARC-Block-Digest\":\"sha1:WFKGXMPEY7BV2PHCRW56GD4TAAHUZZJ6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588282.80_warc_CC-MAIN-20211028065732-20211028095732-00392.warc.gz\"}"}
https://electronoobs.com/eng_arduino_tut83.php	[ "", null, "Help me by sharing this post\n\n←PREVIOUS TUTORIAL NEXT TUTORIAL→", null, "PART 1 - Schematic\n\nThe connection is simple. On the one hand, we feed the module from Arduino through GND and 5V and connect the SDA and SCL pin of Arduino with the corresponding pins of the sensor. The ADDR pin determines the I2C address of the device, according to the following table. The default mode is to connect the ADDR pin to GND, which results in the address 0x48.\n\nWe need:\n1 x Arduino NANO/UNO: LINK eBay\n1 x i2c LCD: LINK eBay\n1 x Jump wires: LINK eBay", null, "The following code shows the use of single end mode. We read the four channels and show the value per serial port. To convert the value to voltage it is necessary to use a multiplier, which at the same time depends on the gain of the PGA that we have set. We print the values on the serial monitor or LCD. See codes below.\n\n``````\n#include <Wire.h>\n\nconst float multiplier = 0.1875F;\n\nvoid setup(void)\n{\nSerial.begin(9600);\n\n// Descomentar el que interese\n// ads.setGain(GAIN_TWOTHIRDS); +/- 6.144V 1 bit = 0.1875mV (default)\n// ads.setGain(GAIN_ONE); +/- 4.096V 1 bit = 0.125mV\n// ads.setGain(GAIN_TWO); +/- 2.048V 1 bit = 0.0625mV\n// ads.setGain(GAIN_FOUR); +/- 1.024V 1 bit = 0.03125mV\n// ads.setGain(GAIN_EIGHT); +/- 0.512V 1 bit = 0.015625mV\n// ads.setGain(GAIN_SIXTEEN); +/- 0.256V 1 bit = 0.0078125mV\n}\n\nvoid loop(void)\n{\n\nSerial.println(\" \");\n\ndelay(1000);\n}\n``````\n\nUpload the code and make the connections. Then open the serial monitor at 9600 bauds and you will have the results printed to the serial monitor. Or go below and download the code with the i2c LCD.\n\n``````\n#include <Wire.h>\n#include <LiquidCrystal_I2C.h>\n// Set the LCD address to 0x27 or 0x3f for a 16 chars and 2 line display\nLiquidCrystal_I2C lcd(0x27, 20, 4);\nfloat Voltage = 0.0;\n\nvoid setup(void)\n{\nlcd.init();\nlcd.backlight();\nSerial.begin(9600);\n}\n\nvoid loop(void)\n{\n\nSerial.print(\"AIN0: \");\nSerial.print(\"\\tVoltage: \");\nSerial.println(Voltage, 7);\nSerial.println();\nlcd.clear();\nlcd.setCursor(0,0);\nlcd.setCursor(3,1);\nlcd.print(Voltage,7);\ndelay(100);\n}\n``````\n\nUpload the code and make the connections. Then open the serial monitor at 9600 bauds or just see the results on the LCD screen.\n\n←PREVIOUS TUTORIAL NEXT TUTORIAL→\n\nHelp me by sharing this post" ]	[ null, "https://electronoobs.com/images/BG_GIF4.gif", null, "https://electronoobs.com/images/Arduino/tut_83/connection_1.jpg", null, "https://electronoobs.com/images/Arduino/tut_83/sch_1.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6735674,"math_prob":0.9363059,"size":3401,"snap":"2023-40-2023-50","text_gpt3_token_len":983,"char_repetition_ratio":0.1262879,"word_repetition_ratio":0.05511811,"special_character_ratio":0.31931785,"punctuation_ratio":0.21348314,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9720579,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-26T03:52:49Z\",\"WARC-Record-ID\":\"<urn:uuid:5a4fd944-b79e-4926-b511-e088a55bcd25>\",\"Content-Length\":\"42646\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a8b966d8-9832-4b71-a334-57154d877e2e>\",\"WARC-Concurrent-To\":\"<urn:uuid:dcf0b1b1-5cfb-4e30-8fa6-d0f07e6c4691>\",\"WARC-IP-Address\":\"217.160.0.92\",\"WARC-Target-URI\":\"https://electronoobs.com/eng_arduino_tut83.php\",\"WARC-Payload-Digest\":\"sha1:IJTHAXXLZ3BFNAAX2LFVXIX6MJDHVVTE\",\"WARC-Block-Digest\":\"sha1:ZS6AOAOT5JROC7TXJCNDSBMH7EXNDVLT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510130.53_warc_CC-MAIN-20230926011608-20230926041608-00127.warc.gz\"}"}
https://www.numberempire.com/49559	[ "Home \| Menu \| Get Involved \| Contact webmaster", null, "", null, "", null, "", null, "", null, "# Number 49559\n\nforty nine thousand five hundred fifty nine\n\n### Properties of the number 49559\n\n Factorization 49559 Divisors 1, 49559 Count of divisors 2 Sum of divisors 49560 Previous integer 49558 Next integer 49560 Is prime? YES (5093rd prime) Previous prime 49549 Next prime 49597 49559th prime 606049 Is a Fibonacci number? NO Is a Bell number? NO Is a Catalan number? NO Is a factorial? NO Is a regular number? NO Is a perfect number? NO Polygonal number (s < 11)? NO Binary 1100000110010111 Octal 140627 Duodecimal 2481b Hexadecimal c197 Square 2456094481 Square root 222.61850776609 Natural logarithm 10.810919157977 Decimal logarithm 4.6951225346115 Sine -0.36710018317012 Cosine -0.9301814100037 Tangent 0.39465439668232\nNumber 49559 is pronounced forty nine thousand five hundred fifty nine. Number 49559 is a prime number. The prime number before 49559 is 49549. The prime number after 49559 is 49597. Number 49559 has 2 divisors: 1, 49559. Sum of the divisors is 49560. Number 49559 is not a Fibonacci number. It is not a Bell number. Number 49559 is not a Catalan number. Number 49559 is not a regular number (Hamming number). It is a not factorial of any number. Number 49559 is a deficient number and therefore is not a perfect number. Binary numeral for number 49559 is 1100000110010111. Octal numeral is 140627. Duodecimal value is 2481b. Hexadecimal representation is c197. Square of the number 49559 is 2456094481. Square root of the number 49559 is 222.61850776609. Natural logarithm of 49559 is 10.810919157977 Decimal logarithm of the number 49559 is 4.6951225346115 Sine of 49559 is -0.36710018317012. Cosine of the number 49559 is -0.9301814100037. Tangent of the number 49559 is 0.39465439668232\n\n### Number properties\n\nExamples: 3628800, 9876543211, 12586269025" ]	[ null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6307376,"math_prob":0.9817644,"size":2157,"snap":"2020-45-2020-50","text_gpt3_token_len":687,"char_repetition_ratio":0.19136089,"word_repetition_ratio":0.052173913,"special_character_ratio":0.42002782,"punctuation_ratio":0.12533334,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99641734,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-22T01:08:58Z\",\"WARC-Record-ID\":\"<urn:uuid:40164e9b-bd07-4905-832a-cf1004df7649>\",\"Content-Length\":\"21551\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:955cc954-6f0d-40fc-9842-4d01aa71e4d7>\",\"WARC-Concurrent-To\":\"<urn:uuid:46b22083-19cb-4006-b55e-bd59af08b553>\",\"WARC-IP-Address\":\"104.24.112.69\",\"WARC-Target-URI\":\"https://www.numberempire.com/49559\",\"WARC-Payload-Digest\":\"sha1:HPSZHY3SAITSX3CFPXVR6FPF7V34NFL7\",\"WARC-Block-Digest\":\"sha1:LHPHMIPQAAZDKFRX3Y73RXZ5MSSFVF5X\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107878662.15_warc_CC-MAIN-20201021235030-20201022025030-00236.warc.gz\"}"}
https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/	[ "This is a file in the archives of the Stanford Encyclopedia of Philosophy.\n\nFirst published Wed 10 Jan, 2001\n\nConsider the following list of sentences, named ‘The List’:\n\n1. Tasmanian devils have strong jaws.\n2. The second sentence on The List is circular.\n3. If the third sentence on The List is true, then every sentence is true.\n4. The List comprises exactly four sentences.\nAlthough The List itself is not paradoxical, the third sentence (a conditional) is. Is it true? Well, suppose, for conditional proof, that its antecedent is true. Then\nthe third sentence of The List is true\nis true. By substitution, it follows that\nIf the third sentence of The List is true, then every sentence is true\nis true. But, then, Modus Ponens on the above two sentences yields that\nevery sentence is true\nis true. So, by conditional proof, we conclude that\nIf the third sentence of The List is true, then every sentence is true\nis true. By substitution, it follows that\nthe third sentence of The List is true\nis true. But, now, by Modus Ponens on the above two sentences we get that\nevery sentence is true\nis true. By naive truth theory we disquote (or, in this case, dis-display, as it were) to conclude: Every sentence is true! So goes (one version of) Curry's paradox.\n\n## 1. Brief History and Some Caveats\n\nIn 1942 Haskell B. Curry presented what is now called Curry's paradox. Perhaps the most intuitive version of the paradox is due to Arthur N. Prior (1955), who recast Curry's paradox as a \"proof\" of God's existence. (Let C = ‘If C is true then God exists’.) The version presented above is (in effect) Prior's version.\n\nThere are basically two different versions of Curry's paradox, a truth-theoretic (or proof-theoretic) and a set-theoretic version; these versions will be presented below. For now, however, there are a few caveats that need to be issued.\n\nCaveat 1. Loeb's Paradox. Prior's version is (in effect) rehearsed by Boolos and Jeffrey (1989), where neither Prior nor Curry is given credit; rather, Boolos and Jeffrey point out the similarity of the paradox to reasoning used within the proof of Loeb's Theorem; and subsequent authors, notably Barwise & Etchemendy (1984), have called the paradox Loeb's paradox. While there is no doubt strong justification for the alternative name (given the similarity of Curry's paradox to the reasoning involved in proving Loeb's Theorem) the paradox does appear to have been first discovered by Curry.\n\nCaveat 2. Geometrical Curry Paradox (Jigsaw Paradox). This is not the same Curry paradox under discussion; it is a well-known paradox, due to Paul Curry, having to do with so-called geometrical dissection. (The so-called Banach-Tarski geometrical paradox is related to Paul Curry's geometrical paradox.) See Gardner 1956 and Fredrickson 1997 for full discussion of this (geometrical) Curry paradox.\n\n## 2. Curry's Paradox: Truth- and Set-Theoretic Versions\n\n### Truth-Theoretic Version\n\nAssume that our truth predicate satisfies the following T-schema:\nT-Schema: T[A]", null, "A,\nwhere ‘[ ]’ is a name-forming device. Assume, too, that we have the principle called Assertion (also known as pseudo modus ponens):\nAssertion: (A & (A", null, "B))", null, "B\n(NB: We could also use the principle called Contraction: ((A", null, "(A", null, "B))", null, "(A", null, "B).) Curry's paradox quickly generates triviality, the case in which everything is true.\n\nBy diagonalization, self-reference or the like we can get an arbitrary sentence, C, such that:\n\nC = T[C]", null, "F,\nwhere F is anything you like. (For effect, though, make F something obviously false.) By an instance of the T-schema (‘T[C]", null, "C’) we immediately get:\nT[C]", null, "(T[C]", null, "F),\nAgain, using the same instance of the T-Schema, we can substitute C for T[C] in the above to get (1):\n 1 C", null, "(C", null, "F) [by T-schema and Substitution] 2 (C & (C", null, "F))", null, "F [by Assertion] 3 (C & C)", null, "F [by Substitution, from 2] 4 C", null, "F [by Equivalence of C and C&C, from 3] 5 C [by Modus Ponens, from 1 and 4] 6 F [by Modus Ponens, from 4 and 5]\nLetting F be anything entailing triviality Curry's paradox quickly \"shows\" that the world is trivial!\n\n### Set-Theoretic Version\n\nThe same result ensues within naive set theory. Assume, in particular, the (unrestricted) axiom of abstraction (or comprehension):\nUnrestricted Abstraction: x", null, "{y \| A(y)}", null, "A(x).\nMoreover, assume that our conditional,", null, ", satisfies Contraction (as above), which permits the deduction of\n(s", null, "s", null, "A)\nfrom\ns", null, "s", null, "(s", null, "s", null, "A).\nIn the set-theoretic case, let C =df {x \| x", null, "x", null, "F}, where F remains as you please (but something obviously false, for effect). From here we reason thus:\n 1 x", null, "C", null, "(x", null, "x", null, "F) [by Naive Abstraction] 2 C", null, "C", null, "(C", null, "C", null, "F) [by Universal Specification, from 1] 3 C", null, "C", null, "(C", null, "C", null, "F) [by Simplification, from 2] 4 C", null, "C", null, "F [by Contraction, from 3] 5 C", null, "C [by Modus Ponens, from 2 and 4] 6 F [by Modus Ponens, from 4 and 5]\nSo, coupling Contraction with the naive abstraction schema yields, via Curry's paradox, triviality.\n\n## Significance, Solutions, and Open Problems\n\n### Significance\n\nWhat is the significance of Curry's paradox? The answer depends on one's approach to paradox in general. Any comprehensive theory of language has to give some sort account of the paradoxes (e.g., the liar, or Russell's, or etc.). Classical approaches tend to fiddle with the T-schema (or naive abstraction) or reject the existence of certain (paradoxical) sentences. Such classical approaches tend to respond to Curry's paradox in the same fashion — by rejecting the existence of Curry sentences or fiddling with the unrestricted T-schema (or naive abstraction). Some popular variations of these two options include Gupta-Belnap revision theory (1993), Tarski's familiar hierarchical theory (or Russellian type theory), Simmons's singularity theory (1993), Burge's indexical theory (1979), Kripke's fixed point semantics (1975), Gaifman's pointer semantics (1988), Barwise-Etchemendy situation-cum-Aczel-set-theory (1984), and others. (NB: These theories are quite different from each other; however, each of them fits under one of the two so-called classical options mentioned above; they either modify the naive T-schema or reject the existence of so-called strengthened liar sentences.) Where Curry's paradox becomes especially significant is not with classical approaches but rather with certain non-classical approaches; specifically, Curry's paradox is a direct challenge to any non-classical approach that attempts to preserve naive truth (or set) theory in full. Such approaches attempt to preserve naive truth (or set) theory, preserve the apparent existence of Curry sentences, and avoid the apparent non-triviality of the world. Satisfying these desiderata requires a paraconsistent logic, one that affords inconsistent but non-trivial theories. What Curry's paradox shows is that not just any old paraconsistent logic will do; in particular, on pain of triviality, no connective in the language can satisfy contraction or absorption and support the T-scheme or Naive comprehension scheme. Among other things, this constraint rules out quite a few popular candidates for implicative conditionals -- including, for example, various popular relevant conditionals, including those of E and R.\n\n### A Solution\n\nThere is great interest in resolving the paradoxes in the sort of non-classical fashion suggested above. Such interest, coupled with Curry's paradox, has fostered ongoing interest in non-classical (paraconsistent) semantics for entailment. One area in which such research is growing is substructural logic. While there is no generally accepted (non-classical) solution to Curry's paradox one approach is particularly promising, an approach due to Graham Priest (1992) and based upon Kripke's invocation of non-normal worlds. (Kripke invoked such worlds for purposes of modeling Lewis systems weaker than S4, not for purposes of solving Curry's paradox.) The idea may be seen easily through its semantics, as follows.\n\nSetting negation aside (for purposes of Curry), we assume a propositional language with the following connectives: conjunction (&), disjunction (", null, "), and entailment (", null, "). (For purposes of resolving Curry's paradox, negation may be set aside; however, the current semantics allow for a variety of approaches to negation, as well as quantifiers.) An interpretation is a 4-tuple, (W,N,[ ], f), where W is a non-empty set of worlds (index points), N is a non-empty subset of W, [ ] is a function from propositional parameters to the powerset of W; we may, for convenience, see the range of [ ] as comprising propositions (sets of worlds at which various sentences are true), and so call the values of [ ] propositions. We let NN be the set of so-called non-normal worlds, namely NN = W", null, "N. In turn, f is a function from (ordered) pairs of propositions to NN. Now, [ ] is extended to all sentences (A, B, ...) via the following clauses:\n\n[A&B] = [A]", null, "[B]\n\n[A", null, "B] = [A]", null, "[B]\n\nThe value of an entailment is the union of two sets: N, the class of normal worlds where the entailment is true, and NN, the of non-normal worlds where the entailment is true. Assuming the usual S5 truth conditions, N and NN are specified thus:\nN = W, if [A]", null, "[B]; otherwise, N=", null, ".\n\nNN = f([A],[B]).\n\nWith all this in hand, validity is defined in the usual way: namely, as truth-preservation at all normal worlds of all interpretations.\n\nWhy restrict the definition merely to normal worlds? The explanation goes hand-in-hand with the informal interpretation of non-normal worlds; according to Priest's suggestion, non-normal worlds should be understood to be worlds where the laws of logic are different — different from the actual laws, where such laws are expressed by (true) entailment claims. Accordingly, since our definition of validity is an attempt to capture our (actual) logical laws, we need not, and should not, worry about worlds where the logical laws are different, at least not in our definition of validity. Such worlds, however, are otherwise very important; as one can easily verify, such worlds afford the usual logical laws (within the positive fragment at issue) but do not sanction the unwanted \"laws\" — e.g., Assertion and the like. In this way, one can enjoy naive truth theory (or naive set theory) without tripping into triviality as a result of Curry sentences.\n\nPriest (1992) gives a sound and complete proof theory for the given semantics, but this is left for the reader to consult.\n\n### Open Issues and Problems\n\nWith the foregoing semantics one sneed not reject the existence of Curry sentences (which are difficult to reject when one's language is a natural language) or naive truth theory; however, there are various philosophical issues that need to be addressed, a few of which are canvassed below.\n\nOne philosophical issue confronting the given semantics is the very nature of such non-normal worlds. What are they? As intimated, Priest's suggestion is that they are simply (impossible) worlds where the laws of logic are different. But is there any reason, independent of Curry's paradox, to admit such worlds? Fortunately, the answer seems to be ‘yes’. One reason has to do with the common (natural language) reasoning involving counter-logicals, including, for example, sentences such as ‘If intuitionistic logic is correct, then double negation elimination is invalid’. Invoking non-normal worlds provides a simple way of modelling such sentences and the reasoning involving them.\n\nAnother objection also arises. Notice that, on the foregoing semantics, there are (non-normal) worlds where the law of simplification, i.e., A&B", null, "B, is false; however, there is no world (normal or otherwise) at which we have a false B but true A&B. Likewise for all other worlds where the logical laws differ; the worlds themselves, as it were, do not break the laws, even though the laws are false at such worlds. What explains this \"lack of supervenience\" at non-normal worlds? Priest himself offers no explanation, and the problem remains an open one. None the less, here is a suggestion (which has yet to be explored in print): What would it take for logical laws to fail? Most philosophers will agree that it is hard to imagine worlds in which there are events that contravene logical laws. My suggestion is that the only way for logical laws to fail is via arbitrary \"fiat\", as it were. No world (possible or otherwise) comprises events that refute, contravene, or otherwise show the actual logical laws to be false; what is required to falsify logical laws is mere arbitrariness; and such arbitrariness is precisely what one gets from the function, f. The suggestion, then, is simply this: For logical laws to fail at any world (and, hence, at non-normal worlds) one requires arbitrariness and thereby a lack of the supervenience at issue. Whether this suggestion solves the (philosophical) problem at hand is an (other) open problem.\n\nThere are other philosophical (and logical) problems that remain open. One of the most important recent papers discussing such problems is Restall's \"Costing Non-Classical Solutions to Paradoxes of Self-Reference\" (see Other Internet Resources). Restall shows that the sorts of non-classical approach discussed above must give up either transitivity of entailment, infinitary disjunction or distributive lattice logic (i.e., an infinitary disjunction operator distributing over finite conjunction); otherwise, as Restall shows, Curry's paradox arises immediately and triviality ensues. The importance of Restall's point lies not only in the formal constraints imposed on suitable non-classical approaches to Curry; its importance lies especially in the philosophical awkwardness imposed by such constraints. For example, one (formal) upshot of Restall's point is that, on a natural way of modelling propositions (e.g., in familiar world-semantics), some classes of propositions will not have disjunctions on the (given sort of) non-classical approach; the philosophical upshot (and important open problem) is that there is no known explanation for why such classes lack such a disjunction. (Needless to say, it is not a sufficient explanation to note that the presence of such a disjunction would otherwise generate triviality via Curry's paradox.)\n\nThe foregoing issues and open problems confront various non-classical approaches to paradox, problems that arise particularly sharply in the face of Curry's paradox. It should be understood, however, that such problems may remain pressing even for those who are firmly committed to classical approaches to paradox; for one might be interested not so much in accepting or believing such non-classical proposals but, rather, merely in using such proposals to model various naive but non-trivial theories — naive truth theory, naive set theory, naive denotation theory, etc.. One need not believe or accept such theories to have an interest in modeling them accurately. If one has such an interest, then the foregoing problems arising from Curry's paradox must be addressed. (See Slaney 1989, and the classic Meyer, Dunn, and Routley 1979, and also Restall 2000 for further discussion.)\n\n## Bibliography\n\n### Works Cited or Further Reading\n\n• Barwise, J., and Etchemendy, J., 1984. The Liar, New York: Oxford University Press.\n• Boolos, G., and Jeffrey, R., 1989. Computability and Logic, 3rd edition, New York: Cambridge University Press.\n• Burge, T., 1979. \"Semantical Paradox\", Journal of Philosophy 76:169-198.\n• Curry, H., 1942, \"The inconsistency of certain formal logics\", Journal of Symbolic Logic 7, pp. 115-117.\n• Frederickson, G., 1997, Dissections: Plane and Fancy, Cambridge: CUP.\n• Gardner, M., 1956. Mathematics, Magic and Mystery, New York, Dover Publ.\n• Gaifman, H., 1988. \"Operational pointer semantics: Solution to self-referential puzzles I\", in Vardi, M., ed, Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge, Morgan Kaufmann.\n• Goldstein, L. 1986. \"Epimenides and Curry\". Analysis 463:117-121.\n• Gupta and Belnap, 1993. The Revision Theory of Truth, Cambridge, MA: MIT Press.\n• Kripke, S., 1975. \"Outline of a theory of truth\", Journal of Philosophy 72:690-716.\n• Meyer, R. K., Routley, R. and Dunn, J.M., 1979, \"Curry's paradox\", Analysis 39, pp. 124- 128.\n• Myhill, J., 1975. \"Levels of Implication\". In A. R. Anderson, R. C. Barcan-Marcus, annd R. M. Martin, editors, The Logical Enterprise, pp. 179-185. Yale Univ. Press.\n• Myhill, J., 1984. \"Paradoxes\". Synthese, 60:129-143.\n• Priest, G., 1987. In Contradiction, Martinus Nijhoff.\n• Priest, G., 1992. \"What is a non-normal world?\", Logique & Analyse 139-40:291-302.\n• Prior, A. N., 1955. \"Curry's Paradox and 3-Valued Logic\", Australasian Journal of Philosophy 33:177-82.\n• Read, S., 2001, \"Self-Reference and Validity Revisited\", in Medieval Formal Logic, M. Yrjonsuuri (ed.), Kluwer 2001, pp. 183-96.\n• Restall, G., 2000, An Introduction to Substructural Logics, Routledge. ( online précis)\n• Simmons, K., 1993. Universaliity and the Liar, New York: Cambridge University Press.\n• Moh Skaw-Kwei. \"Logical Paradoxes for Many-Valued Systems\". Journal of Symbolic Logic, 19 (1954), pp. 37-39.\n• Slaney, John. 1989. \"RWX is not Curry Paraconsistent\", in G. Priest, R. Routley, and J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 472--480.\n\n## Related Entries\n\nlogic: paraconsistent \| logic: relevance \| logic: substructural \| Russell's paradox" ]	[ null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/lra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, "https://stanford.library.sydney.edu.au/archives/sum2007/entries/curry-paradox/ra.gif", null, 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https://assignmentpoint.com/electromagnetic-induction/	[ "Physics\n\n# Electromagnetic Induction", null, "Electromagnetic Induction\n\nDefinition\n\nElectromagnetic Induction is the production of voltage or electromotive force due to a change in the magnetic field. Electromagnetic induction was discovered by Michael Faraday in the 1830s. Many electrical components and types of equipment work based on the principle of electromagnetic induction. It has found many applications in technology, including electrical components such as inductors and transformers, and devices such as electric motors and generators.", null, "Electromagnetic Induction method is based on the measurement of the change in mutual impedance between a pair of coils on or above the earth’s surface. Most Electromagnetic instruments are comprised of two or more sets of coils. These coils are electrically connected and are separated by a fixed distance. The transmitter coil is used to generate an electromagnetic field at a specific frequency. This is known as the primary field. The primary field causes electrical currents to flow in conductive materials in the subsurface. The flow of currents in the subsurface, called eddy currents, generate a secondary magnetic field, which is sensed by the receiver coil. The magnitude of the secondary field sensed by the receiver depends upon the type and distribution of conductive material in the subsurface. Both the induced secondary field, along with the primary field, is detected at the receiver coil. The number of turns in the coils/wire is directly proportional to induced voltage. The induced voltage in an electromagnetic induction is described by the following equation as:\n\ne = N × dΦdt\n\nWhere,\n\ne = voltage induced (measured in volts)\n\nt = time (measured in seconds)\n\nN = number of turns found in the coil\n\nΦ = magnetic flux (measured in Webers)\n\nMany types of electrical equipment such as motors, generators and transformers function based on the principle of the electromagnetic induction.", null, "Applications of Electromagnetic Induction\n\nElectromagnetic Induction is the production of an electromotive force (i.e., voltage) across an electrical conductor due to its dynamic interaction with a magnetic field. Induction is used in power generation and power transmission, and it’s worth taking a look at how that’s done. There are other effects with some interesting applications to consider, too, such as eddy currents.\n\nAn Electric Generator – An electric motor is a device for transforming electrical energy into mechanical energy; an electric generator does the reverse, using mechanical energy to generate electricity. At the heart of both motors and generators is a wire coil in a magnetic field. In fact, the same device can be used as a motor or a generator.\n\nWhen the device is used as a motor, a current is passed through the coil. The interaction of the magnetic field with the current causes the coil to spin. To use the device as a generator, the coil can be spun, inducing a current in the coil.\n\nEddy Currents – An eddy current is a swirling current set up in a conductor in response to a changing magnetic field. By Lenzıs law, the current swirls in such a way as to create a magnetic field opposing the change; to do this in a conductor, electrons swirl in a plane perpendicular to the magnetic field. More accurately, eddy currents transform more useful forms of energy, such as kinetic energy, into heat, which is generally much less useful. In many applications the loss of useful energy is not particularly desirable, but there are some practical applications.\n\nTransformers – Electricity is often generated a long way from where it is used, and is transmitted long distances through power lines. Although the resistance of a short length of power line is relatively low, over a long distance the resistance can become substantial. A power line of resistance R causes a power loss of I2R; this is wasted as heat. By reducing the current, therefore, the I2R losses can be minimized. A transformer is made up of two coils, each with a different number of loops, linked by an iron core so the magnetic flux from one passes through the other. When the flux generated by one coil changes, as it does continually if the coil is connected to an AC power source, the flux passing through the other will change, inducing a voltage in the second coil. With AC power, the voltage induced in the second coil will also be AC.", null, "" ]	[ null, "https://assignmentpoint.com/wp-content/uploads/2017/05/Electromagnetic-Induction.jpg", null, "https://assignmentpoint.com/wp-content/uploads/2017/05/Electromagnetic-Induction0.jpg", null, "https://assignmentpoint.com/wp-content/uploads/2017/05/Electromagnetic-Induction-0.jpg", null, "https://assignmentpoint.com/wp-content/uploads/2017/05/Electromagnetic-Induction-01.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.953327,"math_prob":0.96799475,"size":5090,"snap":"2023-14-2023-23","text_gpt3_token_len":1011,"char_repetition_ratio":0.15454188,"word_repetition_ratio":0.007202881,"special_character_ratio":0.18762279,"punctuation_ratio":0.09130435,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.973829,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-05T09:47:39Z\",\"WARC-Record-ID\":\"<urn:uuid:d010031f-06ad-4f05-beda-e3955fddb1fe>\",\"Content-Length\":\"30307\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4258fd51-912f-4f6d-9985-de79c34abad3>\",\"WARC-Concurrent-To\":\"<urn:uuid:81a97c91-ede6-4523-adab-5805b3af5848>\",\"WARC-IP-Address\":\"162.55.81.222\",\"WARC-Target-URI\":\"https://assignmentpoint.com/electromagnetic-induction/\",\"WARC-Payload-Digest\":\"sha1:R56YOLOEJDLREO7EHZ2UEUFC7TRAECLM\",\"WARC-Block-Digest\":\"sha1:6TSSPTLDYZFWSX4M2OQ6WIY77NBKNUIH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224651815.80_warc_CC-MAIN-20230605085657-20230605115657-00052.warc.gz\"}"}
https://theincidentaleconomist.com/wordpress/why-consumer-surplus-negative-the-integral-of-the-demand-function/	[ "# Why Consumer Surplus is Negative the Integral of the Demand Function\n\nHaving just spent too many hours on this, I’m posting it so I won’t forget it. Maybe it’ll save some other folks some time too.\n\nI’ve learned that, up to a constant, consumer surplus is the integral of the demand function. That’s wrong. It’s negative the integral of the demand function. The reason has to do with the fact that what we call the demand function–that which translates price into quantity–is not what we draw on a graph. Thanks to Alfred Marshall, the custom is to draw the inverse demand function, with price on the vertical axis and quantity on the horizontal axis.", null, "To show formally that consumer surplus is the negative of the integral of demand (or that demand is negative the derivative of consumer surplus), one can use the rule for integrating inverses. To do so, let\n\n• Q denote quantity and P denote price,\n• Q = D(P) be the demand function,\n• and the equilibrium quantity and price be Q0 and P0, respectively.\n\nThen, consumer surplus (CS) is", null, "where the first line is the definition of consumer surplus, the second follows from the rule of integrating inverses and the facts that Q=D(P) and P=D-1(Q), and the third holds if PD(P) is zero when P is infinity.\n\nThis is important for the very reason I spent hours trying to find or work out the above explanation. If one is working with discrete choice models (logit, nested logit, conditional logit, etc.) and wishes to derive the expression for consumer surplus, one might do so by recalling two things:\n\n1. The functional form of consumer surplus is the log of the denominator of the discrete choice probability model, which is also the demand function. That is, consumer surplus has a log-sum of exponentials form.\n2. The derivative of consumer surplus is negative the demand function, i.e. the discrete choice model.\n\nRecalling point 1 and not 2, which is what I did, one is tempted to put the wrong sign on consumer surplus. That leads to very incorrect (exactly backwards) results. One is tempted to scrutinize one’s code for bugs. But the problem isn’t the code, it’s Alfred Marshall’s graphing convention.\n\n## Subscribe\n\n* indicates required\nEmail Format" ]	[ null, "https://theincidentaleconomist.com/wordpress/wp-content/uploads/2010/05/350px-Economic-surpluses.svg.png", null, "https://theincidentaleconomist.com/wordpress/wp-content/uploads/2010/05/CS.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90450895,"math_prob":0.962219,"size":2075,"snap":"2022-40-2023-06","text_gpt3_token_len":454,"char_repetition_ratio":0.16513762,"word_repetition_ratio":0.011204482,"special_character_ratio":0.21590361,"punctuation_ratio":0.10688836,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9956475,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,10,null,10,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-05T11:37:20Z\",\"WARC-Record-ID\":\"<urn:uuid:0cda6f59-6838-4362-99f6-3fe9c33455a4>\",\"Content-Length\":\"69281\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:094de04b-6880-4cc7-a848-655e33428aa8>\",\"WARC-Concurrent-To\":\"<urn:uuid:d9e966de-f928-4a0d-ab3c-1be221b36cbc>\",\"WARC-IP-Address\":\"108.160.150.161\",\"WARC-Target-URI\":\"https://theincidentaleconomist.com/wordpress/why-consumer-surplus-negative-the-integral-of-the-demand-function/\",\"WARC-Payload-Digest\":\"sha1:7ZLQJ72KLTT5JBULMJ2YS2T6KMUK4CQZ\",\"WARC-Block-Digest\":\"sha1:5WNIEZQAP2M56EMUXDRZKPQD3NXUY7HY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500251.38_warc_CC-MAIN-20230205094841-20230205124841-00726.warc.gz\"}"}
https://answers.everydaycalculation.com/multiply-fractions/14-70-times-4-8	[ "# Answers\n\nSolutions by everydaycalculation.com\n\n## Multiply 14/70 with 4/8\n\nThis multiplication involving fractions can also be rephrased as \"What is 14/70 of 4/8?\"\n\n14/70 × 4/8 is 1/10.\n\n#### Steps for multiplying fractions\n\n1. Simply multiply the numerators and denominators separately:\n2. 14/70 × 4/8 = 14 × 4/70 × 8 = 56/560\n3. After reducing the fraction, the answer is 1/10\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:\nAndroid and iPhone/ iPad\n\n#### Multiply Fractions Calculator\n\n×\n\n© everydaycalculation.com" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79912704,"math_prob":0.9498388,"size":410,"snap":"2021-21-2021-25","text_gpt3_token_len":157,"char_repetition_ratio":0.18965517,"word_repetition_ratio":0.0,"special_character_ratio":0.44390243,"punctuation_ratio":0.06382979,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9857381,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-20T00:46:27Z\",\"WARC-Record-ID\":\"<urn:uuid:873423ab-3111-43dd-bdc7-251229781481>\",\"Content-Length\":\"7234\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4173b298-4c47-45a7-8d86-fdd22376d258>\",\"WARC-Concurrent-To\":\"<urn:uuid:86257ad0-7196-41c1-b43e-4663972c4de3>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/multiply-fractions/14-70-times-4-8\",\"WARC-Payload-Digest\":\"sha1:DJH34O4K3PAIG626SR3KTSAK6SD27ITI\",\"WARC-Block-Digest\":\"sha1:YIP6KK5D6HIWEPZURDSL2PK2EMYKAXVV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487653461.74_warc_CC-MAIN-20210619233720-20210620023720-00228.warc.gz\"}"}
https://runestone.academy/ns/books/published/fcla/section-SD.html	[ "## SectionSDSimilarity and Diagonalization\n\nThis section's topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.\n\n### SubsectionSMSimilar Matrices\n\nThe notion of matrices being “similar” is a lot like saying two matrices are row-equivalent. Two similar matrices are not equal, but they share many important properties. This section, and later sections in Chapter R will be devoted in part to discovering just what these common properties are.\n\nFirst, the main definition for this section.\n\n#### DefinitionSIM.Similar Matrices.\n\nSuppose $$A$$ and $$B$$ are two square matrices of size $$n\\text{.}$$ Then $$A$$ and $$B$$ are similar if there exists a nonsingular matrix of size $$n\\text{,}$$ $$S\\text{,}$$ such that $$A=\\similar{B}{S}\\text{.}$$\n\nWe will say “$$A$$ is similar to $$B$$ via $$S$$” when we want to emphasize the role of $$S$$ in the relationship between $$A$$ and $$B\\text{.}$$ Also, it does not matter if we say $$A$$ is similar to $$B\\text{,}$$ or $$B$$ is similar to $$A\\text{.}$$ If one statement is true then so is the other, as can be seen by using $$\\inverse{S}$$ in place of $$S$$ (see Theorem SER for the careful proof). Finally, we will refer to $$\\similar{B}{S}$$ as a similarity transformation when we want to emphasize the way $$S$$ changes $$B\\text{.}$$ OK, enough about language, let us build a few examples.\n\nIf you wondered if there are examples of similar matrices, then it will not be hard to convince you they exist. Define\n\n\\begin{align} B=\\begin{bmatrix} -4 & 1 & -3 & -2 & 2 \\\\ 1 & 2 & -1 & 3 & -2 \\\\ -4 & 1 & 3 & 2 & 2 \\\\ -3 & 4 & -2 & -1 & -3 \\\\ 3 & 1 & -1 & 1 & -4 \\end{bmatrix}&& S=\\begin{bmatrix} 1 & 2 & -1 & 1 & 1 \\\\ 0 & 1 & -1 & -2 & -1 \\\\ 1 & 3 & -1 & 1 & 1 \\\\ -2 & -3 & 3 & 1 & -2 \\\\ 1 & 3 & -1 & 2 & 1\\\\ \\end{bmatrix}\\text{.} \\end{align}\n\nCheck that $$S$$ is nonsingular and then compute\n\n\\begin{align} &A=\\similar{B}{S}\\\\ &= \\begin{bmatrix} 10 & 1 & 0 & 2 & -5 \\\\ -1 & 0 & 1 & 0 & 0 \\\\ 3 & 0 & 2 & 1 & -3 \\\\ 0 & 0 & -1 & 0 & 1 \\\\ -4 & -1 & 1 & -1 & 1 \\end{bmatrix} \\begin{bmatrix} -4 & 1 & -3 & -2 & 2 \\\\ 1 & 2 & -1 & 3 & -2 \\\\ -4 & 1 & 3 & 2 & 2 \\\\ -3 & 4 & -2 & -1 & -3 \\\\ 3 & 1 & -1 & 1 & -4 \\end{bmatrix} \\begin{bmatrix} 1 & 2 & -1 & 1 & 1 \\\\ 0 & 1 & -1 & -2 & -1 \\\\ 1 & 3 & -1 & 1 & 1 \\\\ -2 & -3 & 3 & 1 & -2 \\\\ 1 & 3 & -1 & 2 & 1 \\end{bmatrix}\\\\ &= \\begin{bmatrix} -10 & -27 & -29 & -80 & -25 \\\\ -2 & 6 & 6 & 10 & -2 \\\\ -3 & 11 & -9 & -14 & -9 \\\\ -1 & -13 & 0 & -10 & -1 \\\\ 11 & 35 & 6 & 49 & 19 \\end{bmatrix}\\text{.} \\end{align}\n\nSo by this construction, we know that $$A$$ and $$B$$ are similar.\n\nLet us do that again.\n\nDefine\n\n\\begin{align} B=\\begin{bmatrix} -13 & -8 & -4 \\\\ 12 & 7 & 4 \\\\ 24 & 16 & 7 \\end{bmatrix}&& S=\\begin{bmatrix} 1 & 1 & 2 \\\\ -2 & -1 & -3 \\\\ 1 & -2 & 0 \\end{bmatrix}\\text{.} \\end{align}\n\nCheck that $$S$$ is nonsingular and then compute\n\n\\begin{align} A&=\\similar{B}{S}\\\\ &= \\begin{bmatrix} -6 & -4 & -1 \\\\ -3 & -2 & -1 \\\\ 5 & 3 & 1 \\end{bmatrix} \\begin{bmatrix} -13 & -8 & -4 \\\\ 12 & 7 & 4 \\\\ 24 & 16 & 7 \\end{bmatrix} \\begin{bmatrix} 1 & 1 & 2 \\\\ -2 & -1 & -3 \\\\ 1 & -2 & 0 \\end{bmatrix}\\\\ &= \\begin{bmatrix} -1 & 0 & 0 \\\\ 0 & 3 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix}\\text{.} \\end{align}\n\nSo by this construction, we know that $$A$$ and $$B$$ are similar. But before we move on, look at how pleasing the form of $$A$$ is. Not convinced? Then consider that several computations related to $$A$$ are especially easy. For example, in the spirit of Example DUTM, $$\\detname{A}=(-1)(3)(-1)=3\\text{.}$$ Similarly, the characteristic polynomial is straightforward to compute by hand, $$\\charpoly{A}{x}=(-1-x)(3-x)(-1-x)=-(x-3)(x+1)^2$$ and since the result is already factored, the eigenvalues are transparently $$\\lambda=3,\\,-1\\text{.}$$ Finally, the eigenvectors of $$A$$ are just the standard unit vectors (Definition SUV).\n\n### SubsectionPSMProperties of Similar Matrices\n\nSimilar matrices share many properties and it is these theorems that justify the choice of the word “similar.” First we will show that similarity is an equivalence relation. Equivalence relations are important in the study of various algebras and can always be regarded as a kind of weak version of equality. Sort of alike, but not quite equal. The notion of two matrices being row-equivalent is an example of an equivalence relation we have been working with since the beginning of the course (see Exercise RREF.T11). Row-equivalent matrices are not equal, but they are a lot alike. For example, row-equivalent matrices have the same rank. Formally, an equivalence relation requires three conditions hold: reflexive, symmetric and transitive. We will illustrate these as we prove that similarity is an equivalence relation.\n\nTo see that $$A$$ is similar to $$A\\text{,}$$ we need only demonstrate a nonsingular matrix that effects a similarity transformation of $$A$$ to $$A\\text{.}$$ $$I_n$$ is nonsingular (since it row-reduces to the identity matrix, Theorem NMRRI), and\n\n\\begin{equation} \\similar{A}{I_n}=I_nAI_n=A\\text{.} \\end{equation}\n\nIf we assume that $$A$$ is similar to $$B\\text{,}$$ then we know there is a nonsingular matrix $$S$$ so that $$A=\\similar{B}{S}$$ by Definition SIM. By Theorem MIMI, $$\\inverse{S}$$ is invertible, and by Theorem NI is therefore nonsingular. So\n\n\\begin{align} \\similar{A}{(\\inverse{S})}&=SA\\inverse{S}&& \\knowl{./knowl/theorem-MIMI.html}{\\text{Theorem MIMI}}\\\\ &=S\\similar{B}{S}\\inverse{S}&& \\knowl{./knowl/definition-SIM.html}{\\text{Definition SIM}}\\\\ &=\\left(S\\inverse{S}\\right)B\\left(S\\inverse{S}\\right)&& \\knowl{./knowl/theorem-MMA.html}{\\text{Theorem MMA}}\\\\ &=I_nBI_n&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=B&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}} \\end{align}\n\nand we see that $$B$$ is similar to $$A\\text{.}$$\n\nAssume that $$A$$ is similar to $$B\\text{,}$$ and $$B$$ is similar to $$C\\text{.}$$ This gives us the existence of two nonsingular matrices, $$S$$ and $$R\\text{,}$$ such that $$A=\\similar{B}{S}$$ and $$B=\\similar{C}{R}\\text{,}$$ by Definition SIM. (Notice how we have to assume $$S\\neq R\\text{,}$$ as will usually be the case.) Since $$S$$ and $$R$$ are invertible, so too $$RS$$ is invertible by Theorem SS and then nonsingular by Theorem NI. Now\n\n\\begin{align} \\similar{C}{(RS)}&=\\similar{\\similar{C}{R}}{S}&& \\knowl{./knowl/theorem-SS.html}{\\text{Theorem SS}}\\\\ &=\\similar{\\left(\\similar{C}{R}\\right)}{S}&& \\knowl{./knowl/theorem-MMA.html}{\\text{Theorem MMA}}\\\\ &=\\similar{B}{S}&& \\knowl{./knowl/definition-SIM.html}{\\text{Definition SIM}}\\\\ &=A \\end{align}\n\nso $$A$$ is similar to $$C$$ via the nonsingular matrix $$RS\\text{.}$$\n\nHere is another theorem that tells us exactly what sorts of properties similar matrices share.\n\nLet $$n$$ denote the size of $$A$$ and $$B\\text{.}$$ Since $$A$$ and $$B$$ are similar, there exists a nonsingular matrix $$S\\text{,}$$ such that $$A=\\similar{B}{S}$$ (Definition SIM). Then\n\n\\begin{align} \\charpoly{A}{x}&=\\detname{A-xI_n}&& \\knowl{./knowl/definition-CP.html}{\\text{Definition CP}}\\\\ &=\\detname{\\similar{B}{S}-xI_n}&& \\knowl{./knowl/definition-SIM.html}{\\text{Definition SIM}}\\\\ &=\\detname{\\similar{B}{S}-x\\similar{I_n}{S}}&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}}\\\\ &=\\detname{\\similar{B}{S}-\\inverse{S}xI_nS}&& \\knowl{./knowl/theorem-MMSMM.html}{\\text{Theorem MMSMM}}\\\\ &=\\detname{\\similar{\\left(B-xI_n\\right)}{S}}&& \\knowl{./knowl/theorem-MMDAA.html}{\\text{Theorem MMDAA}}\\\\ &=\\detname{\\inverse{S}}\\detname{B-xI_n}\\detname{S}&& \\knowl{./knowl/theorem-DRMM.html}{\\text{Theorem DRMM}}\\\\ &=\\detname{\\inverse{S}}\\detname{S}\\detname{B-xI_n}&& \\knowl{./knowl/property-CMCN.html}{\\text{Property CMCN}}\\\\ &=\\detname{\\inverse{S}S}\\detname{B-xI_n}&& \\knowl{./knowl/theorem-DRMM.html}{\\text{Theorem DRMM}}\\\\ &=\\detname{I_n}\\detname{B-xI_n}&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=1\\detname{B-xI_n}&& \\knowl{./knowl/definition-DM.html}{\\text{Definition DM}}\\\\ &=\\charpoly{B}{x}&& \\knowl{./knowl/definition-CP.html}{\\text{Definition CP}}\\text{.} \\end{align}\n\nSo similar matrices not only have the same set of eigenvalues, the algebraic multiplicities of these eigenvalues will also be the same. However, be careful with this theorem. It is tempting to think the converse is true, and argue that if two matrices have the same eigenvalues, then they are similar. Not so, as the following example illustrates.\n\nDefine\n\n\\begin{align} A&=\\begin{bmatrix}1&1\\\\0&1\\end{bmatrix} & B&=\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix} \\end{align}\n\nand check that\n\n\\begin{equation} \\charpoly{A}{x}=\\charpoly{B}{x}=1-2x+x^2=(x-1)^2 \\end{equation}\n\nand so $$A$$ and $$B$$ have equal characteristic polynomials. If the converse of Theorem SMEE were true, then $$A$$ and $$B$$ would be similar. Suppose this is the case. More precisely, suppose there is a nonsingular matrix $$S$$ so that $$A=\\similar{B}{S}\\text{.}$$\n\nThen\n\n\\begin{equation} A=\\similar{B}{S}=\\similar{I_2}{S}=\\inverse{S}S=I_2\\text{.} \\end{equation}\n\nClearly $$A\\neq I_2$$ and this contradiction tells us that the converse of Theorem SMEE is false.\n\n#### SageSM.Similar Matrices.\n\nIt is quite easy to determine if two matrices are similar, using the matrix method .is_similar(). However, computationally this can be a very difficult proposition, so support in Sage is incomplete now, though it will always return a result for matrices with rational entries. Here are examples where the two matrices are, and are not, similar. Notice that the keyword option transformation=Truewill cause a pair to be returned, such that if the matrices are indeed similar, the matrix effecting the similarity transformation will be in the second slot of the pair.\n\nA = matrix(QQ, [[ 25, 2, -8, -1, 11, 26, 35],\n[ 28, 2, -15, 2, 6, 34, 31],\n[ 1, -17, -25, 28, -44, 26, -23],\n[ 36, -2, -24, 10, -1, 50, 39],\n[ 0, -7, -13, 14, -21, 14, -11],\n[-22, -17, -16, 27, -51, 1, -53],\n[ -1, 10, 17, -18, 28, -18, 15]])\nB = matrix(QQ, [[-89, -16, -55, -23, -104, -48, -67],\n[-15, 1, -20, -21, -20, -60, -26],\n[ 48, 6, 37, 25, 59, 64, 46],\n[-96, -16, -49, -16, -114, -23, -67],\n[ 56, 10, 33, 13, 67, 29, 37],\n[ 10, 2, 2, -2, 12, -9, 4],\n[ 28, 6, 13, 1, 32, -4, 16]])\nis_sim, trans = A.is_similar(B, transformation=True)\nis_sim\n\n\nSince we knew in advance these two matrices are similar, we requested the transformation matrix, so the output is a pair. The similarity matrix is a bit of a mess, so we will use three Sage routines to clean up trans. We convert the entries to numerical approximations, clip very small values (less than $$10^{-5}$$) to zero and then round to three decimal places. You can experiment printing just transall by itself.\n\ntrans.change_ring(RDF).zero_at(10^-5).round(3)\n\n\nThe matrix Cis not similar to A(and hence not similar to Bby Theorem SER), so we illustrate the return value when we do not request the similarity matrix (since it does not even exist).\n\nC = matrix(QQ, [[ 16, -26, 19, -56, 26, 8, -49],\n[ 20, -43, 36, -102, 52, 23, -65],\n[-18, 29, -21, 62, -30, -9, 56],\n[-17, 31, -27, 73, -37, -16, 49],\n[ 18, -36, 30, -82, 43, 18, -54],\n[-32, 62, -56, 146, -77, -35, 88],\n[ 11, -19, 17, -44, 23, 10, -29]])\nC.is_similar(A)\n\n\n### SubsectionDDiagonalization\n\nGood things happen when a matrix is similar to a diagonal matrix. For example, the eigenvalues of the matrix are the entries on the diagonal of the diagonal matrix. And it can be a much simpler matter to compute high powers of the matrix. Diagonalizable matrices are also of interest in more abstract settings. Here are the relevant definitions, then our main theorem for this section.\n\n#### DefinitionDIM.Diagonal Matrix.\n\nSuppose that $$A$$ is a square matrix. Then $$A$$ is a diagonal matrix if $$\\matrixentry{A}{ij}=0$$ whenever $$i\\neq j\\text{.}$$\n\n#### DefinitionDZM.Diagonalizable Matrix.\n\nSuppose $$A$$ is a square matrix. Then $$A$$ is diagonalizable if $$A$$ is similar to a diagonal matrix.\n\nArchetype B has a $$3\\times 3$$ coefficient matrix\n\n\\begin{equation} B=\\begin{bmatrix} -7&-6&-12\\\\ 5&5&7\\\\ 1&0&4 \\end{bmatrix} \\end{equation}\n\nand is similar to a diagonal matrix, as can be seen by the following computation with the nonsingular matrix $$S\\text{,}$$\n\n\\begin{align} \\similar{B}{S}&= \\inverse{\\begin{bmatrix}-5&-3&-2\\\\3&2&1\\\\1&1&1\\end{bmatrix}}\\begin{bmatrix} -7&-6&-12\\\\ 5&5&7\\\\ 1&0&4 \\end{bmatrix} \\begin{bmatrix}-5&-3&-2\\\\3&2&1\\\\1&1&1\\end{bmatrix}\\\\ &=\\begin{bmatrix}-1&-1&-1\\\\2&3&1\\\\-1&-2&1\\end{bmatrix} \\begin{bmatrix} -7&-6&-12\\\\ 5&5&7\\\\ 1&0&4 \\end{bmatrix} \\begin{bmatrix}-5&-3&-2\\\\3&2&1\\\\1&1&1\\end{bmatrix}\\\\ &= \\begin{bmatrix}-1&0&0\\\\0&1&0\\\\0&0&2\\end{bmatrix}\\text{.} \\end{align}\n\nExample SMS3 provides yet another example of a matrix that is subjected to a similarity transformation and the result is a diagonal matrix. Alright, just how would we find the magic matrix $$S$$ that can be used in a similarity transformation to produce a diagonal matrix? Before you read the statement of the next theorem, you might study the eigenvalues and eigenvectors of Archetype B and compute the eigenvalues and eigenvectors of the matrix in Example SMS3.\n\n##### (⇐)\n\nLet $$S=\\set{\\vectorlist{x}{n}}$$ be a linearly independent set of eigenvectors of $$A$$ for the eigenvalues $$\\scalarlist{\\lambda}{n}\\text{.}$$ Recall Definition SUV and define\n\n\\begin{align} R&=\\matrixcolumns{x}{n}\\\\ D&= \\begin{bmatrix} \\lambda_1 & 0 & 0 &\\cdots & 0\\\\ 0 &\\lambda_2 & 0 &\\cdots & 0\\\\ 0 & 0 &\\lambda_3 &\\cdots & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots\\\\ 0 & 0 & 0 &\\cdots & \\lambda_n \\end{bmatrix} =[\\lambda_1\\vect{e}_1\|\\lambda_2\\vect{e}_2\|\\lambda_3\\vect{e}_3\|\\ldots\|\\lambda_n\\vect{e}_n]\\text{.} \\end{align}\n\nThe columns of $$R$$ are the vectors of the linearly independent set $$S$$ and so by Theorem NMLIC the matrix $$R$$ is nonsingular. By Theorem NI we know $$\\inverse{R}$$ exists. We have\n\n\\begin{align} \\inverse{R}AR&= \\inverse{R}A\\matrixcolumns{x}{n}\\\\ &=\\inverse{R}[A\\vect{x}_1\|A\\vect{x}_2\|A\\vect{x}_3\|\\ldots\|A\\vect{x}_n]&& \\knowl{./knowl/definition-MM.html}{\\text{Definition MM}}\\\\ &=\\inverse{R}[\\lambda_1\\vect{x}_1\|\\lambda_2\\vect{x}_2\|\\lambda_3\\vect{x}_3\|\\ldots\|\\lambda_n\\vect{x}_n]&& \\knowl{./knowl/definition-EEM.html}{\\text{Definition EEM}}\\\\ &=\\inverse{R}[\\lambda_1R\\vect{e}_1\|\\lambda_2R\\vect{e}_2\|\\lambda_3R\\vect{e}_3\|\\ldots\|\\lambda_nR\\vect{e}_n]&& \\knowl{./knowl/definition-MVP.html}{\\text{Definition MVP}}\\\\ &=\\inverse{R}[R(\\lambda_1\\vect{e}_1)\|R(\\lambda_2\\vect{e}_2)\|R(\\lambda_3\\vect{e}_3)\|\\ldots\|R(\\lambda_n\\vect{e}_n)]&& \\knowl{./knowl/theorem-MMSMM.html}{\\text{Theorem MMSMM}}\\\\ &=\\inverse{R}R[\\lambda_1\\vect{e}_1\|\\lambda_2\\vect{e}_2\|\\lambda_3\\vect{e}_3\|\\ldots\|\\lambda_n\\vect{e}_n]&& \\knowl{./knowl/definition-MM.html}{\\text{Definition MM}}\\\\ &=I_nD&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=D&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}}\\text{.} \\end{align}\n\nThis says that $$A$$ is similar to the diagonal matrix $$D$$ via the nonsingular matrix $$R\\text{.}$$ Thus $$A$$ is diagonalizable (Definition DZM).\n\n##### (⇒)\n\nSuppose that $$A$$ is diagonalizable, so there is a nonsingular matrix of size $$n$$\n\n\\begin{align} T&=\\matrixcolumns{y}{n}\\\\ \\end{align}\n\nand a diagonal matrix (recall Definition SUV)\n\n\\begin{align} E&=\\begin{bmatrix} d_1 & 0 & 0 &\\cdots & 0\\\\ 0 &d_2 & 0 &\\cdots & 0\\\\ 0 & 0 &d_3 &\\cdots & 0\\\\ \\vdots & \\vdots & \\vdots & & \\vdots\\\\ 0 & 0 & 0 &\\cdots & d_n \\end{bmatrix} =[d_1\\vect{e}_1\|d_2\\vect{e}_2\|d_3\\vect{e}_3\|\\ldots\|d_n\\vect{e}_n]&&\\text{} \\end{align}\n\nsuch that $$\\inverse{T}AT=E\\text{.}$$\n\nThen consider,\n\n\\begin{align} [A\\vect{y}_1\|A\\vect{y}_2\|A\\vect{y}_3&\|\\ldots\|A\\vect{y}_n]\\\\ &=A\\matrixcolumns{y}{n}&& \\knowl{./knowl/definition-MM.html}{\\text{Definition MM}}\\\\ &=AT\\\\ &=I_nAT&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}}\\\\ &=T\\inverse{T}AT&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=TE\\\\ &=T[d_1\\vect{e}_1\|d_2\\vect{e}_2\|d_3\\vect{e}_3\|\\ldots\|d_n\\vect{e}_n]\\\\ &=[T(d_1\\vect{e}_1)\|T(d_2\\vect{e}_2)\|T(d_3\\vect{e}_3)\|\\ldots\|T(d_n\\vect{e}_n)]&& \\knowl{./knowl/definition-MM.html}{\\text{Definition MM}}\\\\ &=[d_1T\\vect{e}_1\|d_2T\\vect{e}_2\|d_3T\\vect{e}_3\|\\ldots\|d_nT\\vect{e}_n]&& \\knowl{./knowl/definition-MM.html}{\\text{Definition MM}}\\\\ &=[d_1\\vect{y}_1\|d_2\\vect{y}_2\|d_3\\vect{y}_3\|\\ldots\|d_n\\vect{y}_n]&& \\knowl{./knowl/definition-MVP.html}{\\text{Definition MVP}}\\text{.} \\end{align}\n\nThis equality of matrices (Definition ME) allows us to conclude that the individual columns are equal vectors (Definition CVE). That is, $$A\\vect{y}_i=d_i\\vect{y}_i$$ for $$1\\leq i\\leq n\\text{.}$$ In other words, $$\\vect{y}_i$$ is an eigenvector of $$A$$ for the eigenvalue $$d_i\\text{,}$$ $$1\\leq i\\leq n\\text{.}$$ (Why does $$\\vect{y}_i\\neq\\zerovector\\text{?}$$). Because $$T$$ is nonsingular, the set containing $$T$$'s columns, $$S=\\set{\\vectorlist{y}{n}}\\text{,}$$ is a linearly independent set (Theorem NMLIC). So the set $$S$$ has all the required properties.\n\nNotice that the proof of Theorem DC is constructive. To diagonalize a matrix, we need only locate $$n$$ linearly independent eigenvectors. Then we can construct a nonsingular matrix using the eigenvectors as columns ($$R$$) so that $$\\inverse{R}AR$$ is a diagonal matrix ($$D$$). The entries on the diagonal of $$D$$ will be the eigenvalues of the eigenvectors used to create $$R\\text{,}$$ in the same order as the eigenvectors appear in $$R\\text{.}$$ We illustrate this by diagonalizing some matrices.\n\nConsider the matrix\n\n\\begin{equation} F= \\begin{bmatrix} -13 & -8 & -4\\\\ 12 & 7 & 4\\\\ 24 & 16 & 7 \\end{bmatrix} \\end{equation}\n\nof Example CPMS3, Example EMS3 and Example ESMS3. $$F$$'s eigenvalues and eigenspaces are\n\n\\begin{align} \\lambda&=3&\\eigenspace{F}{3}&=\\spn{\\set{\\colvector{-\\frac{1}{2}\\\\\\frac{1}{2}\\\\1}}}\\\\ \\lambda&=-1&\\eigenspace{F}{-1}&=\\spn{\\set{\\colvector{-\\frac{2}{3}\\\\1\\\\0},\\,\\colvector{-\\frac{1}{3}\\\\0\\\\1}}}\\text{.} \\end{align}\n\nDefine the matrix $$S$$ to be the $$3\\times 3$$ matrix whose columns are the three basis vectors in the eigenspaces for $$F\\text{,}$$\n\n\\begin{equation} S= \\begin{bmatrix} -\\frac{1}{2} & -\\frac{2}{3} & -\\frac{1}{3}\\\\ \\frac{1}{2} & 1 & 0\\\\ 1 & 0 & 1 \\end{bmatrix}\\text{.} \\end{equation}\n\nCheck that $$S$$ is nonsingular (row-reduces to the identity matrix, Theorem NMRRI or has a nonzero determinant, Theorem SMZD). Then the three columns of $$S$$ are a linearly independent set (Theorem NMLIC). By Theorem DC we now know that $$F$$ is diagonalizable. Furthermore, the construction in the proof of Theorem DC tells us that if we apply the matrix $$S$$ to $$F$$ in a similarity transformation, the result will be a diagonal matrix with the eigenvalues of $$F$$ on the diagonal. The eigenvalues appear on the diagonal of the matrix in the same order as the eigenvectors appear in $$S\\text{.}$$ So,\n\n\\begin{align} \\similar{F}{S}&= \\inverse{ \\begin{bmatrix} -\\frac{1}{2} & -\\frac{2}{3} & -\\frac{1}{3}\\\\ \\frac{1}{2} & 1 & 0\\\\ 1 & 0 & 1 \\end{bmatrix} } \\begin{bmatrix} -13 & -8 & -4\\\\ 12 & 7 & 4\\\\ 24 & 16 & 7 \\end{bmatrix} \\begin{bmatrix} -\\frac{1}{2} & -\\frac{2}{3} & -\\frac{1}{3}\\\\ \\frac{1}{2} & 1 & 0\\\\ 1 & 0 & 1 \\end{bmatrix}\\\\ &= \\begin{bmatrix} 6 & 4 & 2\\\\ -3 & -1 & -1\\\\ -6 & -4 & -1 \\end{bmatrix} \\begin{bmatrix} -13 & -8 & -4\\\\ 12 & 7 & 4\\\\ 24 & 16 & 7 \\end{bmatrix} \\begin{bmatrix} -\\frac{1}{2} & -\\frac{2}{3} & -\\frac{1}{3}\\\\ \\frac{1}{2} & 1 & 0\\\\ 1 & 0 & 1 \\end{bmatrix}\\\\ &= \\begin{bmatrix} 3 & 0 & 0\\\\ 0 & -1 & 0\\\\ 0 & 0 & -1 \\end{bmatrix}\\text{.} \\end{align}\n\nNote that the above computations can be viewed two ways. The proof of Theorem DC tells us that the four matrices ($$F\\text{,}$$ $$S\\text{,}$$ $$\\inverse{F}$$ and the diagonal matrix) will interact the way we have written the equation. Or as an example, we can actually perform the computations to verify what the theorem predicts.\n\nThe dimension of an eigenspace can be no larger than the algebraic multiplicity of the eigenvalue by Theorem ME. When every eigenvalue's eigenspace is this large, then we can diagonalize the matrix, and only then. Three examples we have seen so far in this section, Example SMS5, Example DAB and Example DMS3, illustrate the diagonalization of a matrix, with varying degrees of detail about just how the diagonalization is achieved. However, in each case, you can verify that the geometric and algebraic multiplicities are equal for every eigenvalue. This is the substance of the next theorem.\n\nSuppose $$A$$ has size $$n$$ and $$k$$ distinct eigenvalues, $$\\scalarlist{\\lambda}{k}\\text{.}$$ Let $$S_i=\\set{\\vect{x}_{i1},\\,\\vect{x}_{i2},\\,\\vect{x}_{i3},\\,\\ldots,\\,\\vect{x}_{i\\geomult{A}{\\lambda_i}}}\\text{,}$$ denote a basis for the eigenspace of $$\\lambda_i\\text{,}$$ $$\\eigenspace{A}{\\lambda_i}\\text{,}$$ for $$1\\leq i\\leq k\\text{.}$$ Then\n\n\\begin{equation} S=S_1\\cup S_2\\cup S_3\\cup\\cdots\\cup S_k \\end{equation}\n\nis a set of eigenvectors for $$A\\text{.}$$ A vector cannot be an eigenvector for two different eigenvalues (see Exercise EE.T20) so $$S_i\\cap S_j=\\emptyset$$ whenever $$i\\neq j\\text{.}$$ In other words, $$S$$ is a disjoint union of $$S_i\\text{,}$$ $$1\\leq i\\leq k\\text{.}$$\n\n##### (⇐)\n\nThe size of $$S$$ is\n\n\\begin{align} \\card{S} &=\\sum_{i=1}^k\\geomult{A}{\\lambda_i}&& S\\text{ disjoint union of }S_i\\\\ &=\\sum_{i=1}^k\\algmult{A}{\\lambda_i}&& \\text{Hypothesis}\\\\ &=n&& \\knowl{./knowl/theorem-NEM.html}{\\text{Theorem NEM}}\\text{.} \\end{align}\n\nWe next show that $$S$$ is a linearly independent set. So we will begin with a relation of linear dependence on $$S\\text{,}$$ using doubly-subscripted scalars and eigenvectors,\n\nDefine the vectors $$\\vect{y}_i\\text{,}$$ $$1\\leq i\\leq k$$ by\n\nThen the relation of linear dependence becomes\n\n\\begin{align} \\zerovector&=\\vect{y}_1+\\vect{y}_2+\\vect{y}_3+\\cdots+\\vect{y}_k\\text{.} \\end{align}\n\nSince the eigenspace $$\\eigenspace{A}{\\lambda_i}$$ is closed under vector addition and scalar multiplication, $$\\vect{y}_i\\in\\eigenspace{A}{\\lambda_i}\\text{,}$$ $$1\\leq i\\leq k\\text{.}$$ Thus, for each $$i\\text{,}$$ the vector $$\\vect{y}_i$$ is an eigenvector of $$A$$ for $$\\lambda_i\\text{,}$$ or is the zero vector. Recall that sets of eigenvectors whose eigenvalues are distinct form a linearly independent set by Theorem EDELI. Should any (or some) $$\\vect{y}_i$$ be nonzero, the previous equation would provide a nontrivial relation of linear dependence on a set of eigenvectors with distinct eigenvalues, contradicting Theorem EDELI. Thus $$\\vect{y}_i=\\zerovector\\text{,}$$ $$1\\leq i\\leq k\\text{.}$$\n\nEach of the $$k$$ equations, $$\\vect{y}_i=\\zerovector\\text{,}$$ is a relation of linear dependence on the corresponding set $$S_i\\text{,}$$ a set of basis vectors for the eigenspace $$\\eigenspace{A}{\\lambda_i}\\text{,}$$ which is therefore linearly independent. From these relations of linear dependence on linearly independent sets we conclude that the scalars are all zero, more precisely, $$a_{ij}=0\\text{,}$$ $$1\\leq j\\leq\\geomult{A}{\\lambda_i}$$ for $$1\\leq i\\leq k\\text{.}$$ This establishes that our original relation of linear dependence on $$S$$ has only the trivial relation of linear dependence, and hence $$S$$ is a linearly independent set.\n\nWe have determined that $$S$$ is a set of $$n$$ linearly independent eigenvectors for $$A\\text{,}$$ and so by Theorem DC is diagonalizable.\n\n##### (⇒)\n\nNow we assume that $$A$$ is diagonalizable. Aiming for a contradiction (Proof Technique CD), suppose that there is at least one eigenvalue, say $$\\lambda_t\\text{,}$$ such that $$\\geomult{A}{\\lambda_t}\\neq\\algmult{A}{\\lambda_t}\\text{.}$$ By Theorem ME we must have $$\\geomult{A}{\\lambda_t}\\lt\\algmult{A}{\\lambda_t}\\text{,}$$ and $$\\geomult{A}{\\lambda_i}\\leq\\algmult{A}{\\lambda_i}$$ for $$1\\leq i\\leq k\\text{,}$$ $$i\\neq t\\text{.}$$\n\nSince $$A$$ is diagonalizable, Theorem DC guarantees a set of $$n$$ linearly independent vectors, all of which are eigenvectors of $$A\\text{.}$$ Let $$n_i$$ denote the number of eigenvectors in $$S$$ that are eigenvectors for $$\\lambda_i\\text{,}$$ and recall that a vector cannot be an eigenvector for two different eigenvalues (Exercise EE.T20). $$S$$ is a linearly independent set, so the subset $$S_i$$ containing the $$n_i$$ eigenvectors for $$\\lambda_i$$ must also be linearly independent. Because the eigenspace $$\\eigenspace{A}{\\lambda_i}$$ has dimension $$\\geomult{A}{\\lambda_i}$$ and $$S_i$$ is a linearly independent subset in $$\\eigenspace{A}{\\lambda_i}\\text{,}$$ Theorem G tells us that $$n_i\\leq\\geomult{A}{\\lambda_i}\\text{,}$$ for $$1\\leq i\\leq k\\text{.}$$\n\nPutting all these facts together gives,\n\n\\begin{align} n &=n_1+n_2+n_3+\\cdots+n_t+\\cdots+n_k&& \\knowl{./knowl/definition-SU.html}{\\text{Definition SU}}\\\\ &\\leq\\geomult{A}{\\lambda_1}+\\geomult{A}{\\lambda_2}+\\geomult{A}{\\lambda_3}+\\cdots+\\geomult{A}{\\lambda_t}+\\cdots+\\geomult{A}{\\lambda_k}&& \\knowl{./knowl/theorem-G.html}{\\text{Theorem G}}\\\\ &\\lt \\algmult{A}{\\lambda_1}+\\algmult{A}{\\lambda_2}+\\algmult{A}{\\lambda_3}+\\cdots+\\algmult{A}{\\lambda_t}+\\cdots+\\algmult{A}{\\lambda_k}&& \\knowl{./knowl/theorem-ME.html}{\\text{Theorem ME}}\\\\ &=n&& \\knowl{./knowl/theorem-NEM.html}{\\text{Theorem NEM}}\\text{.} \\end{align}\n\nThis is a contradiction (we cannot have $$n\\lt n\\text{!}$$) and so our assumption that some eigenspace had less than full dimension was false.\n\nExample SEE, Example CAEHW, Example ESMS3, Example ESMS4, Example DEMS5, Archetype B, Archetype F, Archetype K and Archetype L are all examples of matrices that are diagonalizable and that illustrate Theorem DMFE. While we have provided many examples of matrices that are diagonalizable, especially among the archetypes, there are many matrices that are not diagonalizable. Here is one now.\n\nIn Example EMMS4 the matrix\n\n\\begin{equation} B= \\begin{bmatrix} -2 & 1 & -2 & -4\\\\ 12 & 1 & 4 & 9\\\\ 6 & 5 & -2 & -4\\\\ 3 & -4 & 5 & 10 \\end{bmatrix} \\end{equation}\n\nwas determined to have characteristic polynomial\n\n\\begin{equation} \\charpoly{B}{x}=(x-1)(x-2)^3 \\end{equation}\n\nand an eigenspace for $$\\lambda=2$$ of\n\n\\begin{equation} \\eigenspace{B}{2}=\\spn{\\set{\\colvector{-\\frac{1}{2}\\\\1\\\\-\\frac{1}{2}\\\\1}}}\\text{.} \\end{equation}\n\nSo the geometric multiplicity of $$\\lambda=2$$ is $$\\geomult{B}{2}=1\\text{,}$$ while the algebraic multiplicity is $$\\algmult{B}{2}=3\\text{.}$$ By Theorem DMFE, the matrix $$B$$ is not diagonalizable.\n\nArchetype A is the lone archetype with a square matrix that is not diagonalizable, as the algebraic and geometric multiplicities of the eigenvalue $$\\lambda=0$$ differ. Example HMEM5 is another example of a matrix that cannot be diagonalized due to the difference between the geometric and algebraic multiplicities of $$\\lambda=2\\text{,}$$ as is Example CEMS6 which has two complex eigenvalues, each with differing multiplicities. Likewise, Example EMMS4 has an eigenvalue with different algebraic and geometric multiplicities and so cannot be diagonalized.\n\n#### SageMD.Matrix Diagonalization.\n\nThe third way to get eigenvectors is the matrix method .eigenmatrix_right()(and the analogous .eigenmatrix_left()). It always returns two square matrices of the same size as the original matrix. The first matrix of the output is a diagonal matrix with the eigenvalues of the matrix filling the diagonal entries of the matrix. The second matrix has eigenvectors in the columns, in the same order as the corresponding eigenvalues. For a single eigenvalue, these columns/eigenvectors form a linearly independent set.\n\nA careful reading of the previous paragraph suggests the question: what if we do not have enough eigenvectors to fill the columns of the second square matrix? When the geometric multiplicity does not equal the algebraic multiplicity, the deficit is met by inserting zero columns in the matrix of eigenvectors. Conversely, when the matrix is diagonalizable, by Theorem DMFE the geometric and algebraic multiplicities of each eigenvalue are equal, and the union of the bases of the eigenspaces provides a complete set of linearly independent vectors. So for a matrix $$A\\text{,}$$ Sage will output two matrices, $$D$$ and $$S$$ such that $$\\inverse{S}AS=D\\text{.}$$\n\nWe can rewrite the relation above as $$AS=SD\\text{.}$$ In the case of a non-diagonalizable matrix, the matrix of eigenvectors is singular (it has zero columns), but the relationship $$AS=SD$$ still holds. Here are examples of the two scenarios, along with demonstrations of the matrix method is_diagonalizable().\n\nA = matrix(QQ, [[ 2, -18, -68, 64, -99, -87, 83],\n[ 4, -10, -41, 34, -58, -57, 46],\n[ 4, 16, 59, -60, 86, 70, -72],\n[ 2, -15, -65, 57, -92, -81, 78],\n[-4, -7, -32, 31, -45, -31, 41],\n[ 2, -6, -22, 20, -32, -31, 26],\n[ 0, 7, 30, -27, 42, 37, -36]])\nD, S = A.eigenmatrix_right()\nD\n\nS\n\nS.inverse()AS == D\n\nA.is_diagonalizable()\n\n\nNow for a matrix that is far from diagonalizable.\n\nB = matrix(QQ, [[ 37, -13, 30, 81, -74, -13, 18],\n[ 6, 26, 21, -11, -46, -48, 19],\n[ 16, 10, 29, 16, -42, -39, 26],\n[-24, 8, -24, -53, 54, 13, -15],\n[ -8, 3, -8, -20, 24, 4, -5],\n[ 31, 12, 46, 48, -97, -56, 35],\n[ 8, 5, 16, 12, -34, -20, 11]])\nD, S = B.eigenmatrix_right()\nD\n\nS\n\nBS == SD\n\nB.is_diagonalizable()\n\n\nLet $$\\scalarlist{\\lambda}{n}$$ denote the $$n$$ distinct eigenvalues of $$A\\text{.}$$ Then by Theorem NEM we have $$n=\\sum_{i=1}^n\\algmult{A}{\\lambda_i}\\text{,}$$ which implies that $$\\algmult{A}{\\lambda_i}=1\\text{,}$$ $$1\\leq i\\leq n\\text{.}$$ From Theorem ME it follows that $$\\geomult{A}{\\lambda_i}=1\\text{,}$$ $$1\\leq i\\leq n\\text{.}$$ So $$\\geomult{A}{\\lambda_i}=\\algmult{A}{\\lambda_i}\\text{,}$$ $$1\\leq i\\leq n$$ and Theorem DMFE says $$A$$ is diagonalizable.\n\nIn Example DEMS5 the matrix\n\n\\begin{equation} H= \\begin{bmatrix} 15 & 18 & -8 & 6 & -5\\\\ 5 & 3 & 1 & -1 & -3\\\\ 0 & -4 & 5 & -4 & -2\\\\ -43 & -46 & 17 & -14 & 15\\\\ 26 & 30 & -12 & 8 & -10 \\end{bmatrix} \\end{equation}\n\nhas characteristic polynomial\n\n\\begin{equation} \\charpoly{H}{x}=x(x-2)(x-1)(x+1)(x+3) \\end{equation}\n\nand so is a $$5\\times 5$$ matrix with 5 distinct eigenvalues.\n\nBy Theorem DED we know $$H$$ must be diagonalizable. But just for practice, we exhibit a diagonalization. The matrix $$S$$ contains eigenvectors of $$H$$ as columns, one from each eigenspace, guaranteeing linear independent columns and thus the nonsingularity of $$S\\text{.}$$ Notice that we are using the versions of the eigenvectors from Example DEMS5 that have integer entries. The diagonal matrix has the eigenvalues of $$H$$ in the same order that their respective eigenvectors appear as the columns of $$S\\text{.}$$ With these matrices, verify computationally that $$\\similar{H}{S}=D\\text{.}$$\n\n\\begin{align} S&= \\begin{bmatrix} 2 & 1 & -1 & 1 & 1\\\\ -1 & 0 & 2 & 0 & -1\\\\ -2 & 0 & 2 & -1 & -2\\\\ -4 & -1 & 0 & -2 & -1\\\\ 2 & 2 & 1 & 2 & 1 \\end{bmatrix} &D&= \\begin{bmatrix} -3 & 0 & 0 & 0 & 0\\\\ 0 & -1 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 1 & 0\\\\ 0 & 0 & 0 & 0 & 2 \\end{bmatrix}\\text{.} \\end{align}\n\nNote that there are many different ways to diagonalize $$H\\text{.}$$ We could replace eigenvectors by nonzero scalar multiples, or we could rearrange the order of the eigenvectors as the columns of $$S$$ (which would subsequently reorder the eigenvalues along the diagonal of $$D$$).\n\nArchetype B is another example of a matrix that has as many distinct eigenvalues as its size, and is hence diagonalizable by Theorem DED.\n\nPowers of a diagonal matrix are easy to compute, and when a matrix is diagonalizable, it is almost as easy. We could state a theorem here perhaps, but we will settle instead for an example that makes the point just as well.\n\nSuppose that\n\n\\begin{equation} A=\\begin{bmatrix} 19 & 0 & 6 & 13 \\\\ -33 & -1 & -9 & -21 \\\\ 21 & -4 & 12 & 21 \\\\ -36 & 2 & -14 & -28 \\end{bmatrix} \\end{equation}\n\nand we wish to compute $$A^{20}\\text{.}$$ Normally this would require 19 matrix multiplications, but since $$A$$ is diagonalizable, we can simplify the computations substantially.\n\nFirst, we diagonalize $$A\\text{.}$$ With\n\n\\begin{equation} S=\\begin{bmatrix} 1 & -1 & 2 & -1 \\\\ -2 & 3 & -3 & 3 \\\\ 1 & 1 & 3 & 3 \\\\ -2 & 1 & -4 & 0 \\end{bmatrix} \\end{equation}\n\nwe find\n\n\\begin{align} D&=\\similar{A}{S}\\\\ &= \\begin{bmatrix} -6 & 1 & -3 & -6 \\\\ 0 & 2 & -2 & -3 \\\\ 3 & 0 & 1 & 2 \\\\ -1 & -1 & 1 & 1 \\end{bmatrix} \\begin{bmatrix} 19 & 0 & 6 & 13 \\\\ -33 & -1 & -9 & -21 \\\\ 21 & -4 & 12 & 21 \\\\ -36 & 2 & -14 & -28 \\end{bmatrix} \\begin{bmatrix} 1 & -1 & 2 & -1 \\\\ -2 & 3 & -3 & 3 \\\\ 1 & 1 & 3 & 3 \\\\ -2 & 1 & -4 & 0 \\end{bmatrix}\\\\ &= \\begin{bmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 2 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix}\\text{.} \\end{align}\n\nNow we find an alternate expression for $$A^{20}\\text{,}$$\n\n\\begin{align} A^{20} &=AAA\\ldots A\\\\ &=I_nAI_nAI_nAI_n\\ldots I_nAI_n\\\\ &=\\left(S\\inverse{S}\\right)A\\left(S\\inverse{S}\\right)A\\left(S\\inverse{S}\\right)A\\left(S\\inverse{S}\\right)\\ldots \\left(S\\inverse{S}\\right)A\\left(S\\inverse{S}\\right)\\\\ &=S\\left(\\inverse{S}AS\\right)\\left(\\inverse{S}AS\\right)\\left(\\inverse{S}AS\\right)\\ldots \\left(\\inverse{S}AS\\right)\\inverse{S}\\\\ &=SDDD\\ldots D\\inverse{S}\\\\ &=SD^{20}\\inverse{S}\\\\ \\end{align}\n\nand since $$D$$ is a diagonal matrix, powers are much easier to compute,\n\n\\begin{align} &= S \\begin{bmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 2 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix}^{20} \\inverse{S}\\\\ &= S \\begin{bmatrix} (-1)^{20} & 0 & 0 & 0 \\\\ 0 & (0)^{20} & 0 & 0 \\\\ 0 & 0 & (2)^{20} & 0 \\\\ 0 & 0 & 0 & (1)^{20} \\end{bmatrix} \\inverse{S}\\\\ &= \\begin{bmatrix} 1 & -1 & 2 & -1 \\\\ -2 & 3 & -3 & 3 \\\\ 1 & 1 & 3 & 3 \\\\ -2 & 1 & -4 & 0 \\end{bmatrix} \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1048576 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} -6 & 1 & -3 & -6 \\\\ 0 & 2 & -2 & -3 \\\\ 3 & 0 & 1 & 2 \\\\ -1 & -1 & 1 & 1 \\end{bmatrix}\\\\ &= \\begin{bmatrix} 6291451 & 2 & 2097148 & 4194297 \\\\ -9437175 & -5 & -3145719 & -6291441 \\\\ 9437175 & -2 & 3145728 & 6291453 \\\\ -12582900 & -2 & -4194298 & -8388596 \\end{bmatrix}\\text{.} \\end{align}\n\nNotice how we effectively replaced the twentieth power of $$A$$ by the twentieth power of $$D\\text{,}$$ and how a high power of a diagonal matrix is just a collection of powers of scalars on the diagonal. The price we pay for this simplification is the need to diagonalize the matrix (by computing eigenvalues and eigenvectors) and finding the inverse of the matrix of eigenvectors. And we still need to do two matrix products. But the higher the power, the greater the savings.\n\n### SubsectionFSFibonacci Sequences\n\nThe Fibonacci sequence is a sequence of integers defined recursively by\n\n\\begin{align} a_0&=0 & a_1&=1 & a_{n+1}&=a_n+a_{n-1},\\quad n\\geq 1\\text{.} \\end{align}\n\nSo the initial portion of the sequence is $$0,\\,1,\\,1,\\,2,\\,3,\\,5,\\,8,\\,13,\\,21,\\,\\ldots\\text{.}$$ In this subsection we will illustrate an application of eigenvalues and diagonalization through the determination of a closed-form expression for an arbitrary term of this sequence.\n\nTo begin, verify that for any $$n\\geq 1$$ the recursive statement above establishes the truth of the statement\n\n\\begin{align} \\colvector{a_n\\\\a_{n+1}} &= \\begin{bmatrix}0&1\\\\1&1\\end{bmatrix} \\colvector{a_{n-1}\\\\a_n}\\text{.} \\end{align}\n\nLet $$A$$ denote this $$2\\times 2$$ matrix. Through repeated applications of the statement above we have\n\n\\begin{align} \\colvector{a_n\\\\a_{n+1}} & =A\\colvector{a_{n-1}\\\\a_n} =A^2\\colvector{a_{n-2}\\\\a_{n-1}} =A^3\\colvector{a_{n-3}\\\\a_{n-2}} =\\cdots =A^n\\colvector{a_{0}\\\\a_{1}}\\text{.} \\end{align}\n\nIn preparation for working with this high power of $$A\\text{,}$$ not unlike in Example HPDM, we will diagonalize $$A\\text{.}$$ The characteristic polynomial of $$A$$ is $$\\charpoly{A}{x}=x^2-x-1\\text{,}$$ with roots (the eigenvalues of $$A$$ by Theorem EMRCP)\n\n\\begin{align} \\rho&=\\frac{1+\\sqrt{5}}{2} & \\delta&=\\frac{1-\\sqrt{5}}{2}\\text{.} \\end{align}\n\nWith two distinct eigenvalues, Theorem DED implies that $$A$$ is diagonalizable. It will be easier to compute with these eigenvalues once you confirm the following properties (all but the last can be derived from the fact that $$\\rho$$ and $$\\delta$$ are roots of the characteristic polynomial, in a factored or unfactored form)\n\n\\begin{align} \\rho+\\delta&=1 & \\rho\\delta&=-1 & 1+\\rho&=\\rho^2 & 1+\\delta&=\\delta^2 & \\rho-\\delta&=\\sqrt{5}\\text{.} \\end{align}\n\nThen eigenvectors of $$A$$ (for $$\\rho$$ and $$\\delta\\text{,}$$ respectively) are\n\n\\begin{align} &\\colvector{1\\\\\\rho} & &\\colvector{1\\\\\\delta} \\end{align}\n\nwhich can be easily confirmed, as we demonstrate for the eigenvector for $$\\rho\\text{,}$$\n\n\\begin{align} \\begin{bmatrix}0&1\\\\1&1\\end{bmatrix}\\colvector{1\\\\\\rho} & =\\colvector{\\rho\\\\1+\\rho} =\\colvector{\\rho\\\\\\rho^2} =\\rho\\colvector{1\\\\\\rho}\\text{.} \\end{align}\n\nFrom the proof of Theorem DC we know $$A$$ can be diagonalized by a matrix $$S$$ with these eigenvectors as columns, giving $$D=\\inverse{S}AS\\text{.}$$ We list $$S\\text{,}$$ $$\\inverse{S}$$ and the diagonal matrix $$D\\text{,}$$\n\n\\begin{align} S&=\\begin{bmatrix}1&1\\\\\\rho&\\delta\\end{bmatrix} & \\inverse{S}&=\\frac{1}{\\rho-\\delta}\\begin{bmatrix}-\\delta&1\\\\\\rho&-1\\end{bmatrix} & D&=\\begin{bmatrix}\\rho&0\\\\0&\\delta\\end{bmatrix}\\text{.} \\end{align}\n\nOK, we have everything in place now. The main step in the following is to replace $$A$$ by $$SD\\inverse{S}\\text{.}$$ Here we go,\n\n\\begin{align} \\colvector{a_n\\\\a_{n+1}} &=A^n\\colvector{a_{0}\\\\a_{1}}\\\\ &=\\left(SD\\inverse{S}\\right)^n\\colvector{a_{0}\\\\a_{1}}\\\\ &=SD\\inverse{S}SD\\inverse{S}SD\\inverse{S}\\cdots SD\\inverse{S}\\colvector{a_{0}\\\\a_{1}}\\\\ &=SDDD\\cdots D\\inverse{S}\\colvector{a_{0}\\\\a_{1}}\\\\ &=SD^n\\inverse{S}\\colvector{a_{0}\\\\a_{1}}\\\\ &= \\begin{bmatrix}1&1\\\\\\rho&\\delta\\end{bmatrix} \\begin{bmatrix}\\rho&0\\\\0&\\delta\\end{bmatrix}^n \\frac{1}{\\rho-\\delta}\\begin{bmatrix}-\\delta&1\\\\\\rho&-1\\end{bmatrix} \\colvector{a_{0}\\\\a_{1}}\\\\ &= \\frac{1}{\\rho-\\delta} \\begin{bmatrix}1&1\\\\\\rho&\\delta\\end{bmatrix} \\begin{bmatrix}\\rho^n&0\\\\0&\\delta^n\\end{bmatrix} \\begin{bmatrix}-\\delta&1\\\\\\rho&-1\\end{bmatrix} \\colvector{0\\\\1}\\\\ &= \\frac{1}{\\rho-\\delta} \\begin{bmatrix}1&1\\\\\\rho&\\delta\\end{bmatrix} \\begin{bmatrix}\\rho^n&0\\\\0&\\delta^n\\end{bmatrix} \\colvector{1\\\\-1}\\\\ &= \\frac{1}{\\rho-\\delta} \\begin{bmatrix}1&1\\\\\\rho&\\delta\\end{bmatrix} \\colvector{\\rho^n\\\\-\\delta^n}\\\\ &= \\frac{1}{\\rho-\\delta} \\colvector{\\rho^n-\\delta^n\\\\\\rho^{n+1}-\\delta^{n+1}}\\text{.} \\end{align}\n\nPerforming the scalar multiplication and equating the first entries of the two vectors, we arrive at the closed form expression\n\n\\begin{align} a_n&=\\frac{1}{\\rho-\\delta}\\left(\\rho^n-\\delta^n\\right)\\\\ &=\\frac{1}{\\sqrt{5}} \\left(\\left(\\frac{1+\\sqrt{5}}{2}\\right)^n-\\left(\\frac{1-\\sqrt{5}}{2}\\right)^n\\right)\\\\ &=\\frac{1}{2^n\\sqrt{5}} \\left(\\left(1+\\sqrt{5}\\right)^n-\\left(1-\\sqrt{5}\\right)^n\\right)\\text{.} \\end{align}\n\nNotice that it does not matter whether we use the equality of the first or second entries of the vectors, we will arrive at the same formula, once in terms of $$n$$ and again in terms of $$n+1\\text{.}$$ Also, our definition clearly describes a sequence that will only contain integers, yet the presence of the irrational number $$\\sqrt{5}$$ might make us suspicious. But no, our expression for $$a^n$$ will always yield an integer!\n\nThe Fibonacci sequence, and generalizations of it, have been extensively studied (Fibonacci lived in the 12th and 13th centuries). There are many ways to derive the closed-form expression we just found, and our approach may not be the most efficient route. But it is a nice demonstration of how diagonalization can be used to solve a problem outside the field of linear algebra.\n\nWe close this section with a comment about an important upcoming theorem that we prove in Chapter R. A consequence of Theorem OD is that every Hermitian matrix (Definition HM) is diagonalizable (Definition DZM), and the similarity transformation that accomplishes the diagonalization uses a unitary matrix (Definition UM). This means that for every Hermitian matrix of size $$n$$ there is a basis of $$\\complex{n}$$ that is composed entirely of eigenvectors for the matrix and also forms an orthonormal set (Definition ONS). Notice that for matrices with only real entries, we only need the hypothesis that the matrix is symmetric (Definition SYM) to reach this conclusion (Example ESMS4). Can you imagine a prettier basis for use with a matrix? I cannot.\n\nThese results in Section OD explain much of our recurring interest in orthogonality, and make the section a high point in your study of linear algebra. A precise statement of this diagonalization result applies to a slightly broader class of matrices, known as “normal” matrices (Definition NRML), which are matrices that commute with their adjoints. With this expanded category of matrices, the result becomes an equivalence (Proof Technique E). See Theorem OD and Theorem OBNM in Section OD for all the details.\n\n#### 1.\n\nWhat is an equivalence relation?\n\n#### 2.\n\nState a condition that is equivalent to a matrix being diagonalizable, but is not the definition.\n\n#### 3.\n\nFind a diagonal matrix similar to\n\n\\begin{equation} A=\\begin{bmatrix} -5 & 8\\\\-4 & 7 \\end{bmatrix}\\text{.} \\end{equation}\n\n### ExercisesSDExercises\n\n#### C20.\n\nConsider the matrix $$A$$ below. First, show that $$A$$ is diagonalizable by computing the geometric multiplicities of the eigenvalues and quoting the relevant theorem. Second, find a diagonal matrix $$D$$ and a nonsingular matrix $$S$$ so that $$\\similar{A}{S}=D\\text{.}$$ (See Exercise EE.C20 for some of the necessary computations.)\n\n\\begin{equation} A= \\begin{bmatrix} 18 & -15 & 33 & -15\\\\ -4 & 8 & -6 & 6\\\\ -9 & 9 & -16 & 9\\\\ 5 & -6 & 9 & -4 \\end{bmatrix} \\end{equation}\nSolution.\n\nUsing a calculator, we find that $$A$$ has three distinct eigenvalues, $$\\lambda=3,\\,2,\\,-1\\text{,}$$ with $$\\lambda=2$$ having algebraic multiplicity two, $$\\algmult{A}{2}=2\\text{.}$$ The eigenvalues $$\\lambda=3,\\,-1$$ have algebraic multiplicity one, and so by Theorem ME we can conclude that their geometric multiplicities are one as well. Together with the computation of the geometric multiplicity of $$\\lambda=2$$ from Exercise EE.C20, we know\n\n\\begin{align} \\geomult{A}{3}&=\\algmult{A}{3}=1& \\geomult{A}{2}&=\\algmult{A}{2}=2& \\geomult{A}{-1}&=\\algmult{A}{-1}=1\\text{.} \\end{align}\n\nThis satisfies the hypotheses of Theorem DMFE, and so we can conclude that $$A$$ is diagonalizable.\n\nA calculator will give us four eigenvectors of $$A\\text{,}$$ the two for $$\\lambda=2$$ being linearly independent presumably. Or, by hand, we could find basis vectors for the three eigenspaces. For $$\\lambda=3,\\,-1$$ the eigenspaces have dimension one, and so any eigenvector for these eigenvalues will be multiples of the ones we use below. For $$\\lambda=2$$ there are many different bases for the eigenspace, so your answer could vary. Our eigenvectors are the basis vectors we would have obtained if we had actually constructed a basis in Exercise EE.C20 rather than just computing the dimension.\n\nBy the construction in the proof of Theorem DC, the required matrix $$S$$ has columns that are four linearly independent eigenvectors of $$A$$ and the diagonal matrix has the eigenvalues on the diagonal (in the same order as the eigenvectors in $$S$$). Here are the pieces, “doing” the diagonalization,\n\n\\begin{equation} \\inverse{ \\begin{bmatrix} -1 & 0 & -3 & 6\\\\ -2 & -1 & -1 & 0\\\\ 0 & 0 & 1 & -3\\\\ 1 & 1 & 0 & 1 \\end{bmatrix} } \\begin{bmatrix} 18 & -15 & 33 & -15\\\\ -4 & 8 & -6 & 6\\\\ -9 & 9 & -16 & 9\\\\ 5 & -6 & 9 & -4 \\end{bmatrix} \\begin{bmatrix} -1 & 0 & -3 & 6\\\\ -2 & -1 & -1 & 0\\\\ 0 & 0 & 1 & -3\\\\ 1 & 1 & 0 & 1 \\end{bmatrix} = \\begin{bmatrix} 3 & 0 & 0 & 0\\\\ 0 & 2 & 0 & 0\\\\ 0 & 0 & 2 & 0\\\\ 0 & 0 & 0 & -1 \\end{bmatrix}\\text{.} \\end{equation}\n\n#### C21.\n\nDetermine if the matrix $$A$$ below is diagonalizable. If the matrix is diagonalizable, then find a diagonal matrix $$D$$ that is similar to $$A\\text{,}$$ and provide the invertible matrix $$S$$ that performs the similarity transformation. You should use your calculator to find the eigenvalues of the matrix, but try only using the row-reducing function of your calculator to assist with finding eigenvectors.\n\n\\begin{equation} A= \\begin{bmatrix} 1 & 9 & 9 & 24 \\\\ -3 & -27 & -29 & -68 \\\\ 1 & 11 & 13 & 26 \\\\ 1 & 7 & 7 & 18 \\end{bmatrix} \\end{equation}\nSolution.\n\nA calculator will provide the eigenvalues $$\\lambda=2,\\,2,\\,1,\\,0\\text{,}$$ so we can reconstruct the characteristic polynomial as\n\n\\begin{equation} \\charpoly{A}{x}=(x-2)^2(x-1)x \\end{equation}\n\nso the algebraic multiplicities of the eigenvalues are\n\n\\begin{align} \\algmult{A}{2}&=2& \\algmult{A}{1}&=1& \\algmult{A}{0}&=1\\text{.} \\end{align}\n\nNow compute eigenspaces by hand, obtaining null spaces for each of the three eigenvalues by constructing the correct singular matrix (Theorem EMNS),\n\n\\begin{align} A-2I_4&= \\begin{bmatrix} -1 & 9 & 9 & 24 \\\\ -3 & -29 & -29 & -68 \\\\ 1 & 11 & 11 & 26 \\\\ 1 & 7 & 7 & 16 \\end{bmatrix} \\rref \\begin{bmatrix} 1 & 0 & 0 & -\\frac{3}{2} \\\\ 0 & 1 & 1 & \\frac{5}{2} \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\\\ \\eigenspace{A}{2}&=\\nsp{A-2I_4} =\\spn{\\set{\\colvector{\\frac{3}{2}\\\\-\\frac{5}{2}\\\\0\\\\1},\\,\\colvector{0\\\\-1\\\\1\\\\0}}} =\\spn{\\set{\\colvector{3\\\\-5\\\\0\\\\2},\\,\\colvector{0\\\\-1\\\\1\\\\0}}}\\\\ A-1I_4&= \\begin{bmatrix} 0 & 9 & 9 & 24 \\\\ -3 & -28 & -29 & -68 \\\\ 1 & 11 & 12 & 26 \\\\ 1 & 7 & 7 & 17 \\end{bmatrix} \\rref \\begin{bmatrix} 1 & 0 & 0 & -\\frac{5}{3} \\\\ 0 & 1 & 0 & \\frac{13}{3} \\\\ 0 & 0 & 1 & -\\frac{5}{3} \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\\\ \\eigenspace{A}{1}&=\\nsp{A-I_4} =\\spn{\\set{\\colvector{\\frac{5}{3}\\\\-\\frac{13}{3}\\\\\\frac{5}{3}\\\\1}}} =\\spn{\\set{\\colvector{5\\\\-13\\\\5\\\\3}}}\\\\ A-0I_4&= \\begin{bmatrix} 1 & 9 & 9 & 24 \\\\ -3 & -27 & -29 & -68 \\\\ 1 & 11 & 13 & 26 \\\\ 1 & 7 & 7 & 18 \\end{bmatrix} \\rref \\begin{bmatrix} 1 & 0 & 0 & -3 \\\\ 0 & 1 & 0 & 5 \\\\ 0 & 0 & 1 & -2 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\\\ \\eigenspace{A}{0}&=\\nsp{A}=\\spn{\\set{\\colvector{3\\\\-5\\\\2\\\\1}}}\\text{.} \\end{align}\n\nFrom this we can compute the dimensions of the eigenspaces to obtain the geometric multiplicities,\n\n\\begin{align} \\geomult{A}{2}&=2& \\geomult{A}{1}&=1& \\geomult{A}{0}&=1\\text{.} \\end{align}\n\nFor each eigenvalue, the algebraic and geometric multiplicities are equal and so by Theorem DMFE we now know that $$A$$ is diagonalizable. The construction in Theorem DC suggests we form a matrix whose columns are eigenvectors of $$A$$\n\n\\begin{equation} S= \\begin{bmatrix} 3 & 0 & 5 & 3 \\\\ -5 & -1 & -13 & -5 \\\\ 0 & 1 & 5 & 2 \\\\ 2 & 0 & 3 & 1 \\end{bmatrix}\\text{.} \\end{equation}\n\nSince $$\\detname{S}=-1\\neq 0\\text{,}$$ we know that $$S$$ is nonsingular (Theorem SMZD), so the columns of $$S$$ are a set of 4 linearly independent eigenvectors of $$A\\text{.}$$ By the proof of Theorem SMZD we know\n\n\\begin{equation} \\similar{A}{S}= \\begin{bmatrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} \\end{equation}\n\nis a diagonal matrix with the eigenvalues of $$A$$ along the diagonal, in the same order as the associated eigenvectors appear as columns of $$S\\text{.}$$\n\n#### C22.\n\nConsider the matrix $$A$$ below. Find the eigenvalues of $$A$$ using a calculator and use these to construct the characteristic polynomial of $$A\\text{,}$$ $$\\charpoly{A}{x}\\text{.}$$ State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for $$A$$ by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is $$A$$ diagonalizable? If not, explain why. If so, find a diagonal matrix $$D$$ that is similar to $$A\\text{.}$$\n\n\\begin{equation} A= \\begin{bmatrix} 19 & 25 & 30 & 5 \\\\ -23 & -30 & -35 & -5 \\\\ 7 & 9 & 10 & 1 \\\\ -3 & -4 & -5 & -1 \\end{bmatrix} \\end{equation}\nSolution.\n\nA calculator will report $$\\lambda=0$$ as an eigenvalue of algebraic multiplicity of 2, and $$\\lambda=-1$$ as an eigenvalue of algebraic multiplicity 2 as well. Since eigenvalues are roots of the characteristic polynomial (Theorem EMRCP) we have the factored version\n\n\\begin{equation} \\charpoly{A}{x}=(x-0)^2(x-(-1))^2=x^2(x^2+2x+1)=x^4+2x^3+x^2\\text{.} \\end{equation}\n\nThe eigenspaces are then\n\n\\begin{align} \\lambda&=0\\\\ A-(0)I_4&= \\begin{bmatrix} 19 & 25 & 30 & 5 \\\\ -23 & -30 & -35 & -5 \\\\ 7 & 9 & 10 & 1 \\\\ -3 & -4 & -5 & -1 \\end{bmatrix} \\rref \\begin{bmatrix} \\leading{1} & 0 & -5 & -5 \\\\ 0 & \\leading{1} & 5 & 4 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\\\ \\eigenspace{A}{0}&=\\nsp{C-(0)I_4}= \\spn{\\set{\\colvector{5\\\\-5\\\\1\\\\0},\\,\\colvector{5\\\\-4\\\\0\\\\1}}}\\\\ \\lambda&=-1\\\\ A-(-1)I_4&= \\begin{bmatrix} 20 & 25 & 30 & 5 \\\\ -23 & -29 & -35 & -5 \\\\ 7 & 9 & 11 & 1 \\\\ -3 & -4 & -5 & 0 \\end{bmatrix} \\rref \\begin{bmatrix} \\leading{1} & 0 & -1 & 4 \\\\ 0 & \\leading{1} & 2 & -3 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\\\ \\eigenspace{A}{-1}&=\\nsp{C-(-1)I_4}= \\spn{\\set{\\colvector{1\\\\-2\\\\1\\\\0},\\,\\colvector{-4\\\\3\\\\0\\\\1}}}\\text{.} \\end{align}\n\nEach eigenspace above is described by a spanning set obtained through an application of Theorem BNS and so is a basis for the eigenspace. In each case the dimension, and therefore the geometric multiplicity, is 2.\n\nFor each of the two eigenvalues, the algebraic and geometric multiplicities are equal. Theorem DMFE says that in this situation the matrix is diagonalizable. We know from Theorem DC that when we diagonalize $$A$$ the diagonal matrix will have the eigenvalues of $$A$$ on the diagonal (in some order). So we can claim that\n\n\\begin{equation} D= \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 \\end{bmatrix}\\text{.} \\end{equation}\n\n#### T15.\n\nSuppose that $$A$$ and $$B$$ are similar matrices of size $$n\\text{.}$$ Prove that $$A^3$$ and $$B^3$$ are similar matrices. Generalize.\n\nSolution.\n\nBy Definition SIM we know that there is a nonsingular matrix $$S$$ so that $$A=\\similar{B}{S}\\text{.}$$ Then\n\n\\begin{align} A^3&=(\\similar{B}{S})^3\\\\ &=(\\similar{B}{S})(\\similar{B}{S})(\\similar{B}{S})\\\\ &=\\inverse{S}B(S\\inverse{S})B(S\\inverse{S})BS&& \\knowl{./knowl/theorem-MMA.html}{\\text{Theorem MMA}}\\\\ &=\\inverse{S}B(I_n)B(I_n)BS&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=\\inverse{S}BBBS&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}}\\\\ &=\\inverse{S}B^3S\\text{.} \\end{align}\n\nThis equation says that $$A^3$$ is similar to $$B^3$$ (via the matrix $$S$$).\n\nMore generally, if $$A$$ is similar to $$B\\text{,}$$ and $$m$$ is a non-negative integer, then $$A^m$$ is similar to $$B^m\\text{.}$$ This can be proved using induction (Proof Technique I).\n\n#### T16.\n\nSuppose that $$A$$ and $$B$$ are similar matrices, with $$A$$ nonsingular. Prove that $$B$$ is nonsingular, and that $$\\inverse{A}$$ is similar to $$\\inverse{B}\\text{.}$$\n\nSolution.\n\n$$A$$ being similar to $$B$$ means that there exists an $$S$$ such that $$A=\\inverse{S}BS\\text{.}$$ So, $$B=SA\\inverse{S}$$ and because $$S\\text{,}$$ $$A\\text{,}$$ and $$\\inverse{S}$$ are nonsingular, by Theorem NPNT, $$B$$ is nonsingular.\n\n\\begin{align} \\inverse{A} &= \\inverse{\\left(\\inverse{S}BS\\right)}&& \\knowl{./knowl/definition-SIM.html}{\\text{Definition SIM}}\\\\ &= \\inverse{S}\\inverse{B}\\inverse{\\left(\\inverse{S}\\right)}&& \\knowl{./knowl/theorem-SS.html}{\\text{Theorem SS}}\\\\ &= \\inverse{S}\\inverse{B}S&& \\knowl{./knowl/theorem-MIMI.html}{\\text{Theorem MIMI}}\\text{.} \\end{align}\n\nThen by Definition SIM, $$\\inverse{A}$$ is similar to $$\\inverse{B}\\text{.}$$\n\n#### T17.\n\nSuppose that $$B$$ is a nonsingular matrix. Prove that $$AB$$ is similar to $$BA\\text{.}$$\n\nSolution.\n\nThe nonsingular (invertible) matrix $$B$$ will provide the desired similarity transformation,\n\n\\begin{align} \\inverse{B}\\left(BA\\right)B &=\\left(\\inverse{B}B\\right)\\left(AB\\right)&& \\knowl{./knowl/theorem-MMA.html}{\\text{Theorem MMA}}\\\\ &=I_nAB&& \\knowl{./knowl/definition-MI.html}{\\text{Definition MI}}\\\\ &=AB&& \\knowl{./knowl/theorem-MMIM.html}{\\text{Theorem MMIM}}\\text{.} \\end{align}" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8641342,"math_prob":0.9999801,"size":33634,"snap":"2023-14-2023-23","text_gpt3_token_len":9142,"char_repetition_ratio":0.20758252,"word_repetition_ratio":0.069671325,"special_character_ratio":0.2724029,"punctuation_ratio":0.10321213,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999547,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-01T09:59:07Z\",\"WARC-Record-ID\":\"<urn:uuid:ed274529-8b9a-4f04-8719-540426541a3b>\",\"Content-Length\":\"135269\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6a6bf255-0c01-4405-9cb3-ce9f7f293b1d>\",\"WARC-Concurrent-To\":\"<urn:uuid:ce7b1604-a6ef-4daf-ab3a-e92d4ddb0c3e>\",\"WARC-IP-Address\":\"134.122.31.165\",\"WARC-Target-URI\":\"https://runestone.academy/ns/books/published/fcla/section-SD.html\",\"WARC-Payload-Digest\":\"sha1:IVL6HLGOO3NUE322HDEA36QELKGVATMU\",\"WARC-Block-Digest\":\"sha1:XTZBBSWWKFIFCPFBZPWG2WLZBNU2ME7W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224647639.37_warc_CC-MAIN-20230601074606-20230601104606-00029.warc.gz\"}"}
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Kids will be inspired to stay active this summer, plus they can sneak in some math, logic and fine motor practice too!\nKindergarten\nMath\nWorkbook\nToy Time Counting\nWorksheet\nToy Time Counting\nPractice counting and writing numbers with this simple worksheet, which challenges your child to look at each group of toys and count how many.\nKindergarten\nMath\nWorksheet\nKindergarten Fall Review Packet - Week 3\nWorkbook\nKindergarten Fall Review Packet - Week 3\nWeek 3 of our Kindergarten Fall Review Packet offers five more days of learning activities for new kindergarteners in the subject areas of reading, writing, and math.\nKindergarten\nScience\nWorkbook\nDot-to-Dot Alphabet: A\nWorksheet\nDot-to-Dot Alphabet: A\nCount 1-2-3 to learn A-B-C with this fun dot-to-dot for the letter A.\nKindergarten\nWorksheet\nSubtraction for Kids\nWorksheet\nSubtraction for Kids\nReady, set... go! Start off subtraction with some picture equations, perfect for helping your child understand a new math concept.\nKindergarten\nMath\nWorksheet\nWorksheet\nHow many zebras do you see? What about tigers? Practice your math skills with animals. Count and add up the drawings to practice addition.\nKindergarten\nMath\nWorksheet\nKindergarten Fall Review Packet - Week 4\nWorkbook\nKindergarten Fall Review Packet - Week 4\nIn this final installment of our Kindergarten Fall Review Packet, learners are provided five more days of engaging activities in math, literacy, and other fun topics.\nKindergarten\nScience\nWorkbook\nCounting Practice Worksheet\nWorksheet\nCounting Practice Worksheet\nThrow counting practive into reverse with this counting-backwards worksheet.\nKindergarten\nMath\nWorksheet" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75434077,"math_prob":0.45603263,"size":7876,"snap":"2021-31-2021-39","text_gpt3_token_len":1894,"char_repetition_ratio":0.22802338,"word_repetition_ratio":0.15566836,"special_character_ratio":0.1829609,"punctuation_ratio":0.060653187,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95721704,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-04T05:17:53Z\",\"WARC-Record-ID\":\"<urn:uuid:a05d5bda-bdae-4ee5-bb56-ff66bcf21b3e>\",\"Content-Length\":\"151312\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6548f2ae-24ea-47eb-a733-3552721a1025>\",\"WARC-Concurrent-To\":\"<urn:uuid:e2968d24-b831-4ba5-a059-34e6ac01e0f6>\",\"WARC-IP-Address\":\"199.232.65.185\",\"WARC-Target-URI\":\"https://www.education.com/resources/kindergarten/writing-numbers-0-10/\",\"WARC-Payload-Digest\":\"sha1:VZLG3XQRAYCLV4TC3OXXHKT5GDTTBIZO\",\"WARC-Block-Digest\":\"sha1:U2FKT6HUTNORVB3HJRO5575AZWWLQXCW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154796.71_warc_CC-MAIN-20210804045226-20210804075226-00059.warc.gz\"}"}
https://sikademy.com/answer/computer-science/discrete-mathematics/survey-on-subjects-being-taken-by-250-college-students-i-i601/	[ "is below this banner.\n\nCan't find a solution anywhere?\n\nNEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?\n\nYou will get a detailed answer to your question or assignment in the shortest time possible.\n\n## Here's the Solution to this Question", null, "$N(M\\cup F\\cup E)=N(M)+N(F)+N(E)$\n\n$-N(M\\cap F)-N(M\\cap E)-N(F\\cap E)$\n\n$+N(M\\cap F\\cap E)=145+90+88$\n\n$-25-38-59+15=216$\n\na)\n\n$N(not\\ any)=N(U)-N(M\\cup F\\cup E)$\n\n$=250-216=34$\n\nb)\n\n$N(only\\ one)=N(only\\ M)$\n\n$+N(only\\ F)+N(only\\ E)$\n\n$N(only\\ M)=N(M)-N(M\\cap F)-N(M\\cap E)$\n\n$+N(M\\cap F\\cap E)=145-25-59+15=76$\n\n$N(only\\ F)=N(F)-N(M\\cap F)-N(F\\cap E)$\n\n$+N(M\\cap F\\cap E)=90-25-38+15=42$\n\n$N(only\\ E)=N(E)-N(M\\cap E)-N(F\\cap E)$\n\n$+N(M\\cap F\\cap E)=88-38-59+15=6$\n\n$N(only\\ one)=76+42+6=124$\n\nc)\n\n$N((M\\cap F)-E)=N(M\\cap F)-N(M\\cap F\\cap E)$\n\n$=25-15=10$\n\nd)\n\n$N((F\\cap E)-M)=N(F\\cap E)-N(M\\cap F\\cap E)$\n\n$=38-15=23$" ]	[ null, "https://www.assignmentexpert.com/image", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86541635,"math_prob":0.9999683,"size":14713,"snap":"2022-27-2022-33","text_gpt3_token_len":4852,"char_repetition_ratio":0.12835678,"word_repetition_ratio":0.24625044,"special_character_ratio":0.32651398,"punctuation_ratio":0.106710024,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000094,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-06T10:45:12Z\",\"WARC-Record-ID\":\"<urn:uuid:2a25b69f-22bf-4db5-ac67-c4fc5642aabb>\",\"Content-Length\":\"164105\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cce377da-95a1-4053-a262-4388f1797eaa>\",\"WARC-Concurrent-To\":\"<urn:uuid:a5c43386-73af-4e9d-a401-2de34a5b7c2c>\",\"WARC-IP-Address\":\"172.67.147.141\",\"WARC-Target-URI\":\"https://sikademy.com/answer/computer-science/discrete-mathematics/survey-on-subjects-being-taken-by-250-college-students-i-i601/\",\"WARC-Payload-Digest\":\"sha1:CVWDHGXZDBVVCXOPVLB6NYVFELZJTXXI\",\"WARC-Block-Digest\":\"sha1:YIGOXLJEKVUMC5ES6DL6R5YTK6YGZWRB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104669950.91_warc_CC-MAIN-20220706090857-20220706120857-00387.warc.gz\"}"}
http://methods.sagepub.com/Reference/encyc-of-research-design/n483.xml	[ "# Unbiased Estimator\n\nEncyclopedia\nEdited by: Published: 2010\n\n• ## Subject Index\n\nOne of the important objectives of scientific studies is to estimate quantities of interest from a population of subjects. These quantities are called parameters, and the actual nature of the parameter varies from population to population and from study to study. For example, in a study designed to estimate the proportion of U.S. citizens who approve the President's economic stimulus plan, the parameter of interest is a proportion. In other cases, parameters may be the population mean, variance, median, and so on. In cases where the population values are represented by a parametric model, the parameters represent the whole population instead of a particular characteristic of it.\n\nParameters are estimated based on the sample values. Estimators are functions of sample observations used to estimate the parameter. ...\n\n• All\n• A\n• B\n• C\n• D\n• E\n• F\n• G\n• H\n• I\n• J\n• K\n• L\n• M\n• N\n• O\n• P\n• Q\n• R\n• S\n• T\n• U\n• V\n• W\n• X\n• Y\n• Z\n\n## Methods Map", null, "Research Methods\n\nCopy and paste the following HTML into your website" ]	[ null, "http://methods.sagepub.com/images/img-bg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7325397,"math_prob":0.692758,"size":1574,"snap":"2020-34-2020-40","text_gpt3_token_len":360,"char_repetition_ratio":0.14076433,"word_repetition_ratio":0.0,"special_character_ratio":0.19504447,"punctuation_ratio":0.09126984,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9865989,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T15:54:01Z\",\"WARC-Record-ID\":\"<urn:uuid:4c0daeed-7f78-4125-acc4-7501c32912c4>\",\"Content-Length\":\"250144\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0fff8106-b767-4187-8e4b-61ef17c9b866>\",\"WARC-Concurrent-To\":\"<urn:uuid:63f35449-b76a-49d0-a0f4-f831941d4df1>\",\"WARC-IP-Address\":\"128.121.3.195\",\"WARC-Target-URI\":\"http://methods.sagepub.com/Reference/encyc-of-research-design/n483.xml\",\"WARC-Payload-Digest\":\"sha1:YN4BJH33VI5O3E5VFXFWWC2DVZQZZWGT\",\"WARC-Block-Digest\":\"sha1:52YVUUVBJXGXJNAW334A6OGES6WCU75T\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402131777.95_warc_CC-MAIN-20201001143636-20201001173636-00058.warc.gz\"}"}
http://ipine.me/2019-05-14/	[ "### 简介\n\n#### 数据集的简介\n\n• PatientId：病人ID，\n• AppointmentID：预约流水号ID，\n• Gender：预约者的性别，\n• ScheduledDay：作出预约的具体时间，\n• AppointmentDay：预约的就诊日期，\n• Age：病人年龄，\n• Neighbourhood：医院所在位置，\n• Scholarship：是否参加巴西福利项目 Bolsa Família\n• Hipertension：是否是高血压，\n• Diabetes：是否是糖尿病，\n• Alcoholism：是否是酗酒，\n• Handcap：是否是残障，\n• No-show：no表示病人如约就诊，yes表示病人没有前往就诊。\n\n### 提出问题\n\n• 病人的年龄是如何分布的？哪个年龄段的病人更多？\n• 未按约去就诊的病人有多少？占多大的比例？\n• 人们一般会预约在一周的哪一天就诊？\n• 一天中的哪个时段，预约的人最多？\n• 一般会提前几天预约挂号，即病人等待就诊的时长是多久？\n• 哪些重要因素会影响病人是否如约去就诊？例如，年龄，性别，是否参加福利项目，一周中的哪天就诊等。\n\n### 理解数据\n\n#### 导入必要的包和数据：\n\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport seaborn as sns\n%matplotlib inline\n\n\n\n#### 查看基本信息\n\ndf.head()\n\n\ndf.info()\ndf.duplicated().sum()\n\n\ndf.describe()\ndf.Age.describe()\n\n\n count 110527.000000\nmean 37.088874\nstd 23.110205\nmin -1.000000\n25% 18.000000\n50% 37.000000\n75% 55.000000\nmax 115.000000\nName: Age, dtype: float64\n\n\n### 数据清洗\n\n#### 1、将所有列名统一修改成小写并以下划线连接，便于操作。\n\ndf.rename(columns=lambda x: x.strip().lower().replace('-','_'), inplace=True )\n\n\n#### 2、删除与问题无关的列\n\n• PatientId：病人ID，\n\n• neighbourhood: 医院位置\n\n• Hipertension：是否是高血压，\n\n• Diabetes：是否是糖尿病，\n\n• Alcoholism：是否是酗酒，\n\n• Handcap：是否是残障，\n\ndf.tail(2)\n\n#### 3、处理异常值。\n\ndf = df[df['age'] > 0]\ndf.age.describe()\n\n\n#### 4、对age列进行分段，并创建新的列 age_group\n\nbin_labels = ['Teenager','Young','Middle','Old']\nbin_edges = [0,19,38,56,116]\ndf['age_group'] = pd.cut(df['age'], bin_edges, labels=bin_labels)\ndf['age_group'].isnull().sum()\n\n\n### 探索性数据分析（建立指标模型与可视化）\n\n#### 问题 1：病人的年龄是如何分布的？哪个年龄段的病人更多？\n\nplt.subplots(figsize=(6,4))\ndf['age_group'].value_counts().plot(kind='bar')\nplt.title('Age Distribution')\nplt.xlabel('Age')\nplt.ylabel('Frequence');", null, "#### 问题2：未按约去就诊的病人有多少？占多大的比例？\n\nx_vaule = df['no_show'].value_counts()\nprint(x_vaule)\nlabels = 'Show up','No show'\nexplode = [0, 0.1]\nplt.subplots(figsize=(5,5))\nplt.axes(aspect=1)\n# 设置饼图样式，标签，突出显示，圆上文本格式，显示阴影，文本位置离圆心距离，起始角度，百分比的文本离圆心距离\nplt.pie(x=x_vaule, labels=labels,explode=explode,autopct='%.1f %%', shadow=True, labeldistance=1.1, startangle = 90,pctdistance = 0.6 )\nplt.title('The proportion of patients who did not go to the hospital as appointment ');", null, "#### 问题3：人们一般会预约在一周的哪一天就诊？\n\n# 先将预约日期转换成日期类型，然后换成一周表示\ndf['appointmentday'] = pd.to_datetime(df['appointmentday'], errors='coerce')\ndf['appointment_weekday'] =df['appointmentday'].dt.weekday_name\n\nplt.subplots(figsize=(8,5))\nplt.plot(df['appointment_weekday'].value_counts().keys(),df['appointment_weekday'].value_counts().values)\nplt.title('Appointment Day Distribution')\nplt.xlabel('Appointment Day')\nplt.ylabel('Frequence');", null, "#### 问题4：一天中的哪个时段，预约的人最多？\n\n#先将scheduleday处理成日期类型，然后提取小时\ndf['scheduledday'] = pd.to_datetime(df['scheduledday'], errors='coerce')\ndf['scheduledhour'] = df['scheduledday'].dt.hour\n# df['scheduledhour'].describe()\n\n#将一天的时间分成三个时段，上午，下午，晚上\nbin_labels = ['Morning','Afternoon','Night']\nbin_edges = [5,12,18,22]\ndf['scheduledhour_cut'] = pd.cut(df['scheduledhour'], bin_edges, labels=bin_labels)\n# df['scheduledhour_cut'].isnull().sum()\ndf['scheduledhour_cut'].value_counts()\n\nplt.subplots(figsize=(6,4))\ndf['scheduledhour_cut'].value_counts().plot(kind='bar')\nplt.title('Scheduled Time Distribution')\nplt.xlabel('Scheduled Time')\nplt.ylabel('Frequence');", null, "#### 问题5：一般会提前几天预约挂号，即病人等待就诊的时长是多久？\n\ndf['waittime'] = df['appointmentday'].dt.date - df['scheduledday'].dt.date\ndf['waittime'].describe()\n\n\n等待时长的统计信息如下：\n\ncount 106987\nmean 10 days 04:00:04.710852\nstd 15 days 06:19:27.050399\nmin -6 days +00:00:00\n25% 0 days 00:00:00\n50% 4 days 00:00:00\n75% 14 days 00:00:00\nmax 179 days 00:00:00\nName: waittime, dtype: object\n\n\ndf = df[df['appointmentday'].dt.date >= df['scheduledday'].dt.date]\ndf['waittime'] = (df['appointmentday'].dt.date - df['scheduledday'].dt.date)\ndf['waittime'].describe()\n\n\ncount 106982\nmean 10 days 04:00:53.840833\nstd 15 days 06:19:37.756884\nmin 0 days 00:00:00\n25% 0 days 00:00:00\n50% 4 days 00:00:00\n75% 14 days 00:00:00\nmax 179 days 00:00:00\nName: waittime, dtype: object\n\n\n#### 问题6：哪些重要因素会影响病人是否如约去就诊？\n\ndef cal_proportion(col_name):\n\"\"\"按传入的列与no-show分组，统计不同类的数量，并计算各类的比例\n\"\"\"\ncounts = df.groupby(['no_show',col_name]).count()['appointmentid']\ntotal = df.groupby(['no_show']).count()['appointmentid']\nno_show_proportions = counts['Yes'] / total['Yes']\nshow_proportions = counts['No']/ total['No']\nreturn no_show_proportions,show_proportions\n\n\ndef double_bar(bar_data1,bar_data2,xlabels,xtick_label):\n# 绘制条柱,先设置每个等级组的x坐标位置和每个条柱的宽度\nind = np.arange(len(bar_data1))\nwidth = 0.35\nred_bar = plt.bar(ind,bar_data1, width, color='r', alpha=0.7, label='No show')\nblue_bar = plt.bar(ind+width, bar_data2, width, color='b',alpha=0.7, label='Show up')\n\n# 标题和标签\nplt.ylabel('Proportion')\nplt.xlabel(xlabels)\nplt.title('Proportion by '+ xlabels + ' and No-Show')\nlocations = ind + width / 2 # x 坐标刻度位置\nplt.xticks(locations, xtick_label)\n\n# 图例\nplt.legend()\n\n##### 6.1 不同年龄段的病人，如约去就诊的情况怎么样？\n\nno_show = cal_proportion('age_group')\nshow_up = cal_proportion('age_group')\nxtick_label = ['Teenager', 'Young', 'Middle', 'Old']\ndouble_bar(no_show,show_up,'Age Group',xtick_label)", null, "##### 6.2 不同性别的病人，如约去就诊的情况怎么样？\nno_show = cal_proportion('gender')\nshow_up = cal_proportion('gender')\nxtick_label = ['Female', 'Male']\ndouble_bar(no_show,show_up,'Gender',xtick_label)", null, "##### 6.3 是否参加福利项目，影响患者如约去就诊吗？\nno_show = cal_proportion('scholarship')\nshow_up = cal_proportion('scholarship')\nxtick_label = ['False', 'True']\ndouble_bar(no_show,show_up,'Scholarship',xtick_label)", null, "##### 6.4 在一周的哪天就诊，是否会影响病人如约去就诊？\nno_show = cal_proportion('appointment_weekday')\nshow_up = cal_proportion('appointment_weekday')\nxtick_label = ['Friday', 'Monday', 'Saturday', 'Thursday', 'Tuesday', 'Wednesday']\ndouble_bar(no_show,show_up,'Appointment Weekday',xtick_label)", null, "### 结论/交流\n\n• 交流结果的方法有多种：报告、幻灯片、博客帖子、电子邮件、演示文稿，甚至对话。\n• 数据可视化在这个过程中可以发挥很大的价值。\n\n-------------完-------------" ]	[ null, "http://ipineimg.lijundong.com/output_19_0.png", null, "http://ipineimg.lijundong.com/output_22_1.png", null, "http://ipineimg.lijundong.com/output_26_0.png", null, "http://ipineimg.lijundong.com/output_30_0.png", null, "http://ipineimg.lijundong.com/output_45_0.png", null, "http://ipineimg.lijundong.com/output_48_0.png", null, "http://ipineimg.lijundong.com/output_51_0.png", null, "http://ipineimg.lijundong.com/output_54_0.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.5557848,"math_prob":0.8912937,"size":7682,"snap":"2020-24-2020-29","text_gpt3_token_len":4447,"char_repetition_ratio":0.11969263,"word_repetition_ratio":0.088974856,"special_character_ratio":0.28065607,"punctuation_ratio":0.19949067,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97244686,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-02T23:35:41Z\",\"WARC-Record-ID\":\"<urn:uuid:3dfadb3a-94b5-499a-b6ee-916b1c309b6d>\",\"Content-Length\":\"56600\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b4c7eca3-a753-4257-b902-22c65e0cd161>\",\"WARC-Concurrent-To\":\"<urn:uuid:161af1fa-8466-4525-9250-e6d87877b9b9>\",\"WARC-IP-Address\":\"104.31.73.5\",\"WARC-Target-URI\":\"http://ipine.me/2019-05-14/\",\"WARC-Payload-Digest\":\"sha1:7GMUMZKMY27XAO3X5SC3FRR3GER6EC2M\",\"WARC-Block-Digest\":\"sha1:SXTG2W35AADHUX6MMFLFVYKUCY47UTXN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655880243.25_warc_CC-MAIN-20200702205206-20200702235206-00394.warc.gz\"}"}
https://www.numberempire.com/535848	[ "Home \| Menu \| Get Involved \| Contact webmaster", null, "", null, "", null, "", null, "", null, "# Number 535848\n\nfive hundred thirty five thousand eight hundred forty eight\n\n### Properties of the number 535848\n\n Factorization 2 * 2 * 2 * 3 * 83 * 269 Divisors 1, 2, 3, 4, 6, 8, 12, 24, 83, 166, 249, 269, 332, 498, 538, 664, 807, 996, 1076, 1614, 1992, 2152, 3228, 6456, 22327, 44654, 66981, 89308, 133962, 178616, 267924, 535848 Count of divisors 32 Sum of divisors 1360800 Previous integer 535847 Next integer 535849 Is prime? NO Previous prime 535811 Next prime 535849 535848th prime 7937431 Is a Fibonacci number? NO Is a Bell number? NO Is a Catalan number? NO Is a factorial? NO Is a regular number? NO Is a perfect number? NO Polygonal number (s < 11)? NO Binary 10000010110100101000 Octal 2026450 Duodecimal 21a120 Hexadecimal 82d28 Square 287133079104 Square root 732.01639325906 Natural logarithm 13.191605817746 Decimal logarithm 5.7290416140901 Sine -0.77867559825559 Cosine 0.6274267388957 Tangent -1.2410621830145\nNumber 535848 is pronounced five hundred thirty five thousand eight hundred forty eight. Number 535848 is a composite number. Factors of 535848 are 2 * 2 * 2 * 3 * 83 * 269. Number 535848 has 32 divisors: 1, 2, 3, 4, 6, 8, 12, 24, 83, 166, 249, 269, 332, 498, 538, 664, 807, 996, 1076, 1614, 1992, 2152, 3228, 6456, 22327, 44654, 66981, 89308, 133962, 178616, 267924, 535848. Sum of the divisors is 1360800. Number 535848 is not a Fibonacci number. It is not a Bell number. Number 535848 is not a Catalan number. Number 535848 is not a regular number (Hamming number). It is a not factorial of any number. Number 535848 is an abundant number and therefore is not a perfect number. Binary numeral for number 535848 is 10000010110100101000. Octal numeral is 2026450. Duodecimal value is 21a120. Hexadecimal representation is 82d28. Square of the number 535848 is 287133079104. Square root of the number 535848 is 732.01639325906. Natural logarithm of 535848 is 13.191605817746 Decimal logarithm of the number 535848 is 5.7290416140901 Sine of 535848 is -0.77867559825559. Cosine of the number 535848 is 0.6274267388957. Tangent of the number 535848 is -1.2410621830145\n\n### Number properties\n\nExamples: 3628800, 9876543211, 12586269025" ]	[ null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6106264,"math_prob":0.9885105,"size":2532,"snap":"2020-45-2020-50","text_gpt3_token_len":914,"char_repetition_ratio":0.16772152,"word_repetition_ratio":0.21052632,"special_character_ratio":0.5086888,"punctuation_ratio":0.21327968,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9849794,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-25T19:12:23Z\",\"WARC-Record-ID\":\"<urn:uuid:0e7d876d-6842-438e-8d9e-f399dc77af72>\",\"Content-Length\":\"23347\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d3e958c2-5f90-4b6d-9b4c-2a28a92e312b>\",\"WARC-Concurrent-To\":\"<urn:uuid:be0fb032-bb76-4774-bdab-42ca18c1eff3>\",\"WARC-IP-Address\":\"104.24.112.69\",\"WARC-Target-URI\":\"https://www.numberempire.com/535848\",\"WARC-Payload-Digest\":\"sha1:UTVERRLYWE3QLTSM3J3PB6DUSCKO36NE\",\"WARC-Block-Digest\":\"sha1:X3BLH4QYFWTRZZULV6GWIRP57ZVDHRMB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141184123.9_warc_CC-MAIN-20201125183823-20201125213823-00461.warc.gz\"}"}
https://scholarworks.wmich.edu/dissertations/3309/	[ "## Dissertations\n\n#### Title\n\nInduced Graph Colorings\n\n6-2018\n\n#### Degree Name\n\nDoctor of Philosophy\n\n#### Department\n\nMathematics\n\nDr. Ping Zhang\n\nDr. Gary Chartrand\n\nDr. Allen Schwenk\n\nDr. Ebrahim Salehi\n\n#### Keywords\n\nGraph theory, vertex colorings, edge colorings, majestic coloring, majestic t-tone coloring\n\n#### Abstract\n\nAn edge coloring of a nonempty graph G is an assignment of colors to the edges of G. In an unrestricted edge coloring, adjacent edges of G may be colored the same. If every two adjacent edges of G are colored differently, then this edge coloring is proper and the minimum number of colors in a proper edge coloring of G is the chromatic index χ/(G) of G. A proper vertex coloring of a nontrivial graph G is an assignment of colors to the vertices of G such that every two adjacent vertices of G are colored differently. The minimum number of colors in a proper vertex coloring of G is the chromatic number χ(G) of G. In this work, we study a proper vertex coloring of a graph that is induced by an unrestricted edge coloring of the graph.\n\nFor a connected graph G of order at least 3, let c : E(G) → {1, 2, . . . , k} be an\n\nunrestricted edge coloring of G where adjacent edges may be colored the same. Then c induces a vertex coloring c/ of G obtained by assigning to each vertex v of G the set of colors of the edges incident with v. The edge coloring c is called a majestic k-edge coloring of G if the induced vertex coloring c/ is a proper vertex coloring of G. The minimum positive integer k for which a graph G has a majestic k-edge coloring is the majestic chromatic index of G and and denoted by maj(G). For a graph G with majestic chromatic index k, the minimum number of distinct vertex colors induced by a majestic k-edge coloring is the majestic chromatic number of G and denoted by ψ(G). Thus, ψ(G) is at least as large as the chromatic number χ(G) of a graph G.\n\nMajestic chromatic indexes and numbers are determined for several well-known classes of graphs, including complete graphs, complete multipartite graphs, paths, cycles and connected bipartite graphs and the Cartesian products G K2 when G is a bipartite graph as well as two classes of nonbipartite graphs G, namely odd cycles and complete multipartite graph. Sharp bounds have been established for these two parameters and relationship between the majestic chromatic number and the chromatic number of a graph has been studied.\n\nThe idea of assigning a color a to an edge of a graph can be looking as assigning the color {a} to the edge, which gives rise to a natural generalization of majestic colorings. For a positive integer k, let P([k]) be the power set of the set [k] = {1, 2, . . . , k} and let P∗([k]) = P([k]) − {∅} be the set of nonempty subsets of [k]. For each integer t with 1 ≤ t < k, let Pt([k]) be the set of t-element subsets of P([k]). For an edge coloring c : E(G) → Pt([k]) of a graph G, where adjacent edges may be colored the same, the vertex coloring c/ : V (G) → P∗([k]) is defined by c/(v) to be the union of the colors of edges incident with the vertex v in G. If c/ is a proper vertex coloring of G, then c is a majestic t-tone k-coloring of G. For a fixed positive integer t, the minimum positive integer k for which a graph G has a majestic t-tone k-coloring is the majestic t-tone index majt(G) of G. In particular, a majestic 1-tone k-coloring is a majestic k-coloring and the majestic 1-tone index of a graph G is the majestic index of G.\n\nFirst, our emphasis is on the majestic 2-tone colorings in graphs. The values of maj2(G) are determined for several well-known classes of graphs G. We study the re- lationship between maj2(G K1) and maj2(G) where G K1 is the join of a graph G and K1. We then investigate the majestic t-tone indexes of connected graphs for t ≥ 2 in general with main emphasis on connected bipartite graphs. It is shown that if G is a connected bipartite graph of order at least 3, then majt(G) = t + 1 or majt(G) = t + 2 for each positive integer t. All trees, unicyclic bipartite graphs and connected bipartite graphs of cycle rank 2 having majestic t-tone index t + 1 have been characterized.\n\nWe also investigate the majestic t-tone indices of connected bipartite graphs having large cycles. In particular, we consider 2-connected bipartite graphs with large cycles. It is shown that (i) if G is a 2-connected bipartite graph of sufficiently large order n whose longest cycles have length £ where n − 5 ≤ £ n and t ≥ 2 is an integer, then majt(G) = t + 1 and (ii) there is a 2-connected bipartite graph F of sufficiently large order n whose longest cycles have length n − 6 and maj2(F ) = 4. Furthermore, it is shown for integers k, t ≥ 2 that there exists a k-connected bipartite graph G such that majt(G) = t + 2. Other results and open questions are also presented.\n\n#### Access Setting\n\nDissertation-Open Access\n\nCOinS" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88703805,"math_prob":0.9768092,"size":5004,"snap":"2020-10-2020-16","text_gpt3_token_len":1260,"char_repetition_ratio":0.1774,"word_repetition_ratio":0.1075741,"special_character_ratio":0.24040768,"punctuation_ratio":0.08751229,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99523073,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-30T17:25:21Z\",\"WARC-Record-ID\":\"<urn:uuid:ffb13745-605f-4f2c-a991-4eb08b706eab>\",\"Content-Length\":\"43934\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ee38919d-1306-4abe-b4a7-3f19345cc2ae>\",\"WARC-Concurrent-To\":\"<urn:uuid:1bbccb28-be14-455d-92bb-a2c3bb784fae>\",\"WARC-IP-Address\":\"50.18.241.247\",\"WARC-Target-URI\":\"https://scholarworks.wmich.edu/dissertations/3309/\",\"WARC-Payload-Digest\":\"sha1:ZF32PCA7ZVOONIAN6U334KI2HQWAZEGP\",\"WARC-Block-Digest\":\"sha1:AAZL3D5CDLTROCBCEN6UDDDFO6RHNYNJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370497171.9_warc_CC-MAIN-20200330150913-20200330180913-00150.warc.gz\"}"}
https://physics.stackexchange.com/questions/470640/what-fraction-of-events-goes-through-a-small-area-da-a-distance-r-from-a-sph	[ "# What fraction of events goes through a small area $dA$ a distance $r$ from a spherically symmetric point source as $r\\to 0$?\n\nI'm not sure how to use limiting principle to estimate the N fraction when r -> 0 and small area dA stays fixed\n\n$$\\lim N_{r\\to\\infty} = \\lim\\delta A/(4\\pi r^2)_{r\\to\\infty}$$ = 1/2 N ? How?\n\n• Since $\\delta A$ is fixed, then it can't vary with $r$, hence the $\\lim_{r->\\infty} \\delta A/4\\pi r^2=(\\delta A/4 \\pi )\\lim_{r->\\infty} \\frac {1}{r^2}=0$. And the mystery variable $N$ must have a value of $N=0$. – Cinaed Simson Apr 5 at 3:09\n• Answer is 1/2 number of events N. how do you interpret the final explanation for flashcard 16 under \"mathematics and statistics\"? -> great.cwru.edu – Quesop Apr 5 at 3:25\n• To send me email, you need @cinaed in the comment. Also, I don't do flash cards - that's your homework. – Cinaed Simson Apr 5 at 18:38\n• Since $4\\pi r^2$ is the surface of sphere, and $\\delta A$ is a small area of the sphere, you probably want the $lim_{r \\rightarrow 0} \\delta A/4\\pi r^2$. Think of the surface as a balloon and $\\delta A$ is small fixed patch on the ballon. Let the air out of the balloon until the small patch prevents the balloon from from shrinking further. The patch should now shield roughly half of the balloon. So you probably don't want to take $r \\rightarrow 0$. Maybe $r \\rightarrow \\delta r$ such that $lim_{r \\rightarrow \\delta r} \\delta A/4\\pi r^2 =1/2$. It's hard to be precise. – Cinaed Simson Apr 5 at 18:43\n• Sorry, I did get an email - I was just looking for different one. – Cinaed Simson Apr 5 at 19:15\n\nYour problem is a mismatch between the expression of which you are finding the limit and the answer you are expecting. They are answers to different questions.\n\nYour expression 𝛿𝐴/(4𝜋𝑟2) is the fraction of a sphere at radius r covered by dA if the area were spread onto the surface of the sphere. This will actually diverge as r goes to 0 because dA will be many times the area of the sphere for sufficiently small r.\n\nYour expected answer, 1/2, is the limit as r goes to 0 of the fraction of events passing through a FLAT dA which is tangent to the sphere of radius r.\n\nI suggest taking the limit of a new expression derived from the geometry I describe in my sentence about your expected answer.\n\n• so dA = r^2 dsolidangle = r^2 theta^2 phi for small theta and phi angles. so working backwards theta^2 phi must be 2pi. I don't see how setting theta^2 phi to 2pi would make dA flat? please clarify? – Quesop Apr 5 at 2:08\n• No, flat is not the only possibility. There are infinitely many expressions which have a limit of 1/2 as r goes to 0. I don’t know what your goal is with this limit, so I just chose the first physically reasonable possibility that occurred to me which would yield your expected result of 1/2. Maybe the quantity you were trying to find was something else entirely. – Duncan Harris Apr 5 at 2:19\n• The argument I am reading is \"There are dA/(4pir^2) events with dA much smaller than 4*pir^2 (total surface area). As r goes to 0 (dA fixed) we get half the total events since dA now gets hit by emissions into half the spherical volume\" My question is about the limit in second sentence. I think it is a non sequitur. – Quesop Apr 5 at 2:31\n• You can’t assume both that dA is constant and that it is much smaller than r^2 as r goes to zero. They are contradictory assumptions so you can only derive nonsense. – Duncan Harris Apr 5 at 2:34\n• I find the last question on that flash card silly if a quantitative answer was expected. How do you put the finite area dA an infinitesimal distance from the origin? There isn’t some obviously best choice of how to do so, and your choice dramatically affects the answer. I think that if I had been answering the question on a test I would have written the same expression as you, and then said that it was meaningless for spheres smaller than dA. – Duncan Harris Apr 5 at 3:17" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92672294,"math_prob":0.9653468,"size":1072,"snap":"2019-43-2019-47","text_gpt3_token_len":303,"char_repetition_ratio":0.111423224,"word_repetition_ratio":0.3265306,"special_character_ratio":0.26399255,"punctuation_ratio":0.05263158,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99841875,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-18T14:09:21Z\",\"WARC-Record-ID\":\"<urn:uuid:1855118d-69c1-4532-8613-9e1dc1aa8126>\",\"Content-Length\":\"148852\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2786fc69-d310-4f19-9515-9b1c89157af8>\",\"WARC-Concurrent-To\":\"<urn:uuid:9a6b4fb0-16c1-42db-a7a2-3db1acea06c5>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/470640/what-fraction-of-events-goes-through-a-small-area-da-a-distance-r-from-a-sph\",\"WARC-Payload-Digest\":\"sha1:4ETMFYR3A2SQCBJRXJCZWDT3G64YK4BP\",\"WARC-Block-Digest\":\"sha1:VH3ZUPKVGXW4GESMFLBQ7ZJC7R3F5EHQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669795.59_warc_CC-MAIN-20191118131311-20191118155311-00162.warc.gz\"}"}
https://blogs.sap.com/2015/07/14/from-r-to-custom-pa-component-part-2/	[ "# From R to Custom PA Component Part 2\n\nIntro\n\nThis is part 2 of a series of blogs.\n\nPlease refer to part 1 that can be found here From R to Custom PA Component Part 1\n\nIn this blog I will be focusing on beginner syntax that is required to create a Predictive Analytics component, will cover the following\n\n• Basic syntax\n• Functions\n• Variables\n• Help\n\nWe will then have a look at RStudio again to understand the difference between RStudio and RGui.\n\nBasic Syntax\n\nSo open the RGui as shown in part 1 of my this blog series. You can immediately use it as a calculator by typing in 4+4 and then pressing enter. Then 8 will then be shown as the answer. As seen below I have done +, / and *. Give it a try and see the same results as shown below", null, "You can then also do Boolean expressions by typing 5<7, then pressing enter. Will respond with TRUE or FALSE. In this case TRUE. Note to test if 8=7+1, in R = is shown as ==.", null, "In R, the value T is short syntax for TRUE. Same for the value F being short for FALSE.", null, "When working with text you should enter the text in double quotes. As shown below I just typed the text and got an error back. The second time I enterd the text correctly. With text it just responds with what you have entered.", null, "Functions\n\nR comes with built in functions. So lets go through some of the functions to see how they work.\n\nTo use the function sum, type sum(value1,value2,value3,etc.). Below is some examples", null, "To use the square root function, type sqrt(value1)", null, "There is a repeat function, below is an example.", null, "More examples of functions. Give them a try. Feel free to google some more functions.", null, "Variables\n\nWhen working with variables, most programming languages you need to declare the variable, say whether it is string or integer. In R you don’t need to do that. You can just type x<-10. This syntax is saying the variable is X, we assigning the value 10 to X.\n\nSo below you can se I assigned X to have the value 10, after pressing enter the console is waiting for another command. If you just type X again it will tell you the value in X.", null, "You can then start working with x. So if x has a value of 10, then if I want to divide by 2 you just simply type X/2, then press enter. Below you can see I have divided by 2, later on I add 10. Bear in mind I’m not changing X value, I’m just calculating what X would be should I divide by 2 or add 10 to it. X still has the value 10.", null, "To change the value you must use <- to assign the value. So below I’m saying X has a new value that is X/2 which is 5.", null, "We can do the same with text. Here I have a variable called Y, I then assign the value Hello World to it.", null, "You can check the data types R has assigned to x and y by using str function. Will show the type and the value.", null, "Help\n\nYou can also use the help function to get more info. So to get more help just type help(function), so below I have typed help(sum). This will then launch a browser that shows help for this function.", null, "RStudio Differences\n\nSo as mentioned in part 1 of my blog series we can code in the RGui or RStudio. So lets see what RStudio offers.\n\nSo in the R Console we can type the commands the same as we did in the RGui console. By looking at the top right window we can then see that it shows the X and Y variables and values. So in the RGui we have to call x and y to see the value where in the RStudio we can see it immediately in the top right hand window. The bottom right window you can access the help.", null, "When working in the console you can press ctrl+space bar and then RStudio will assist with available functions or syntax with regard to the text you typed. Will also provide info on how to use the function. So for example below we can see for substr it requires x which is the string, then a start value, and a stop value.", null, "Hope this helps, part 3 can be found here From R to Custom PA Component Part 3.\n\n/", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "•", null, "•", null, "" ]	[ null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui1_747102.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui2_747103.png", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui3_747110.png", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui4_747111.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui5_747112.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui6_747126.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui7_747127.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui8_747136.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui9_747146.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui10_747148.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui11_747161.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui12_747162.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui13_747163.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui14_747168.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rconsole1_747191.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rstudio2_747181.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui1_747102.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui2_747103.png", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui3_747110.png", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui4_747111.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui5_747112.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui6_747126.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui7_747127.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui8_747136.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui9_747146.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui10_747148.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui11_747161.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui12_747162.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui13_747163.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rgui14_747168.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rconsole1_747191.jpg", null, "https://blogs.sap.com/wp-content/uploads/2015/07/rstudio2_747181.jpg", null, "https://avatars.services.sap.com/images/mister.makmerphy_small.png", null, "https://avatars.services.sap.com/images/former.member_small.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91335803,"math_prob":0.87591475,"size":3957,"snap":"2020-45-2020-50","text_gpt3_token_len":959,"char_repetition_ratio":0.13736403,"word_repetition_ratio":0.020278834,"special_character_ratio":0.2405863,"punctuation_ratio":0.09775281,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9709604,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68],"im_url_duplicate_count":[null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,7,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-25T22:43:27Z\",\"WARC-Record-ID\":\"<urn:uuid:c9670470-0ae1-4bcc-8b27-9ad7e0079c80>\",\"Content-Length\":\"54678\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c3f4f620-d002-4207-bc48-732ca14f0aa9>\",\"WARC-Concurrent-To\":\"<urn:uuid:82d2aa56-b136-4ef4-9f63-47875886fcb2>\",\"WARC-IP-Address\":\"104.122.70.223\",\"WARC-Target-URI\":\"https://blogs.sap.com/2015/07/14/from-r-to-custom-pa-component-part-2/\",\"WARC-Payload-Digest\":\"sha1:BOBBPZJYGE2LM6YSCC4I2IMKWD4EJPFP\",\"WARC-Block-Digest\":\"sha1:KMC4JMFINUEZWUZLVQPIN5UVX4VG6Q7Y\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107890028.58_warc_CC-MAIN-20201025212948-20201026002948-00057.warc.gz\"}"}
https://dekegisapiwe.anvgames.com/algebraic-systems-and-computational-complexity-theory-book-16901on.php	[ "Last edited by Mezikazahn\nMonday, May 18, 2020 \| History\n\n2 edition of Algebraic Systems and Computational Complexity Theory (Mathematics and Its Applications) found in the catalog.\n\nAlgebraic Systems and Computational Complexity Theory (Mathematics and Its Applications)\n\nZ. Wang\n\n# Algebraic Systems and Computational Complexity Theory (Mathematics and Its Applications)\n\n## by Z. Wang\n\nWritten in English\n\nSubjects:\n• Algebra,\n• Numerical analysis,\n• Mathematics,\n• Computer Science,\n• Science/Mathematics,\n• Programming - Algorithms,\n• Number Systems,\n• Mathematics / Number Systems,\n• Computational complexity,\n• Equations,\n• Homotopy Theory,\n• Numerical solutions\n\n• The Physical Object\nFormatHardcover\nNumber of Pages256\nID Numbers\nOpen LibraryOL7807327M\nISBN 100792325338\nISBN 109780792325338\n\ne-books in Theory of Computation category Models of Computation: Exploring the Power of Computing by John E. Savage - Addison-Wesley, The book re-examines computer science, giving priority to resource tradeoffs and complexity classifications over the . anvgames.com: Algebraic Complexity Theory (Grundlehren der mathematischen Wissenschaften) () by Bürgisser, Peter; Clausen, Michael; Shokrollahi, Mohammad A. and a great selection of similar New, Used and Collectible Books available now at great anvgames.com Range: \\$ - \\$\n\nEngineering Computational Intelligence and Complexity. Parametric Interval Algebraic Systems. Authors: Skalna, Iwona Free Preview. Provides a comprehensive overview of the theory related to the problem of solving parametric interval linear systems and investigates the basic properties of parametric interval matrices. Presents a framework Brand: Springer International Publishing. Get this from a library! Computational complexity of bilinear forms: algebraic coding theory and applications of digital communication systems. [Hari Krishna].\n\nThis beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and 5/5. This chapter discusses algebraic complexity theory. Complexity theory, as a project of lower bounds and optimality, unites two quite different traditions. The first comes from mathematical logic and the theory of recursive functions. In this, the basic computational model is the Turing machine.\n\nYou might also like\nLegal essentials of health care administration\n\nLegal essentials of health care administration\n\nChurch statistics\n\nChurch statistics\n\nAmenity handbook\n\nAmenity handbook\n\nThe Prokaryotes: Vol. 7: Proteobacteria\n\nThe Prokaryotes: Vol. 7: Proteobacteria\n\nSociology of religion\n\nSociology of religion\n\nThe science of air\n\nThe science of air\n\nHeterogeneous engine in-cylinder mixing measurements using a tracer technique\n\nHeterogeneous engine in-cylinder mixing measurements using a tracer technique\n\nmurdered queen! or, Caroline of Brunswick\n\nmurdered queen! or, Caroline of Brunswick\n\nThe military balance.\n\nThe military balance.\n\nCAPITAL ASSET PRICING MODEL : EQUITY RISK PREMIUMS AND THE PRIVATELY-HELD BUSINESS\n\nCAPITAL ASSET PRICING MODEL : EQUITY RISK PREMIUMS AND THE PRIVATELY-HELD BUSINESS\n\nA King and his minister instructed in divers important points of present concern\n\nA King and his minister instructed in divers important points of present concern\n\nhistory of labor in modern Japan\n\nhistory of labor in modern Japan\n\npersecution of the Jews in Russia.\n\npersecution of the Jews in Russia.\n\nGiles Sunday Express & Daily Express cartoons.\n\nGiles Sunday Express & Daily Express cartoons.\n\nHearts ease in heart-trouble, or, A soverign remedy against all trouble of heart that Christs disciples are subject to ...\n\nHearts ease in heart-trouble, or, A soverign remedy against all trouble of heart that Christs disciples are subject to ...\n\nElementary phonetics\n\nElementary phonetics\n\nExtreme Ultraviolet Explorer (EUVE) Guest Observer Program\n\nExtreme Ultraviolet Explorer (EUVE) Guest Observer Program\n\n### Algebraic Systems and Computational Complexity Theory (Mathematics and Its Applications) by Z. Wang Download PDF EPUB FB2\n\nFoundations, Applications, Systems ; [with CD-ROM] Author: Johannes Grabmeier This beginning graduate textbook describes both recent achievements and classical results of computational complexity. Algebraic systems of equations and computational complexity theory.\n\n[Zeke Wang; Sen-lin Hsü; Tang'an Gao] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library.\n\nCreate Book, Internet Resource. Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate anvgames.com algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.\n\nThe fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of. \"This book is certainly the most complete reference on algebraic complexity theory that is available hitherto.\n\nsuperb bibliographical and historical notes are given at the end of each chapter. this book would most certainly make a great textbook for a graduate course on algebraic complexity theory.\n\nBuy Algebraic Systems of Equations and Computational Complexity Theory (Mathematics and Its Applications (closed)) on anvgames.com FREE SHIPPING on qualified ordersAuthor: Wang Zeke, Xu Senlin, Gao Tangan.\n\nGeometric complexity theory (GCT), is a research program in computational complexity theory proposed by Ketan Mulmuley and Milind Sohoni. The goal of the program is to answer the most famous open problem in computer science – whether P = NP – by showing that the complexity class P is not equal to the complexity class NP.\n\nis is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified.\n\nJan 01, · Algebraic Complexity Theory by Peter Bürgisser, Algebraic Complexity Theory \"The book contains interesting exercises and useful bibliographical notes.\n\nIn short, this is a nice book.\"- He is associate editor of the journal Computational Complexity and he was invited speaker at the International Congress anvgames.com: This is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified.\n\nin approximative complexity Proceedings of the 33rd Computational Complexity Conference, () decomposition of semi-algebraic systems Proceedings of the International Symposium on.\n\nCompleteness and Reduction in Algebraic Complexity Theory (Algorithms and Computation in Mathematics Book 7) - Kindle edition by Peter Bürgisser. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Completeness and Reduction in Algebraic Complexity Theory (Algorithms and Computation in Manufacturer: Springer.\n\nTheory of Computation at Princeton Theoretical computer science (TCS) studies efficient algorithms and protocols, which ultimately enable much of modern computing. But even more than that, the very concept of computation gives a fundamental new lens for examining the world around us. Algebraic complexity theory, the study of the minimum number of operations suficient to perform algebraic computations, is surveyed with emphasis on the general theory of bilinear forms and two of its applications: polynomial multiplication and matrix multiplication.\n\nThough by no means exhausting algebraic complexity theory, these topics. Each book emphasizes a different area, corresponding to the author's tastes and interests.\n\nThe most famous, but unfortunately the oldest, is Knuth's Art of Computer Programming, especially Chapter 4. The present book has two goals. First, to give a reasonably comprehensive introductory course in computational number theory. Another phrasing of the question would be \"What papers should I read to create a connection from computational complexity to algebraic geometry / topology?\" I have looked at Geometric Complexity Theory already.\n\nAlso papers in Topological Quantum Computation which I have read enough papers that I am already familiar with the field. Avi Wigderson Mathematics and Computation Draft: March 27, Dedicated to the memory of my father, Pinchas Wigderson ({), who loved people, loved puzzles, and inspired me.\n\nComputability and computational complexity were first considered over the fields of real and complex numbers and generalized to arbitrary algebraic systems. This article approaches the theory of computational complexity over an arbitrary algebraic system by taking computability over the list extension for a computational model of anvgames.com: A.\n\nRybalov. geometric invariant theory, computational complexity, algebraic geometry, representation theory, stability. AMS Subject Headings 20G05, Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma.\n\nSIAM Journal on ComputingCited by: A GENTLE INTRODUCTION TO COMPUTATIONAL COMPLEXITY THEORY, AND A LITTLE BIT MORE SEAN HOGAN Abstract. We give the interested reader a gentle introduction to computa-tional complexity theory, by providing and looking at the background leading up to a discussion of the complexity classes P and NP.\n\nWe also introduce. Mar 14, · The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm.\n\nTheory of Computational Complexity, Second Edition, is an excellent textbook for courses on computational theory and complexity at the graduate level. The book is also a useful reference for practitioners in the fields of computer science, engineering, and mathematics who utilize state-of-the-art software and computational methods to conduct.\n\nThank you for downloading the computational complexity of algebraic and numeric problems elsevier computer science library theory of computation series 1. As you may know, people have search numerous times for their chosen readings like this the computational complexity of algebraic and numeric problems elsevier computer science library theory of computation series 1, but end up in harmful.Algebraic number theory involves using techniques from (mostly commutative) algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects (e.g., functions elds, elliptic curves, etc.).\n\nThe main objects that we study in this book .We finally note that the relationship between multi-linear functions (polynomials in general) and arithmetic circuits is a classical subject of algebraic complexity theory . In this field of Author: Joachim Von Zur Gathen." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8766595,"math_prob":0.6734383,"size":10104,"snap":"2022-05-2022-21","text_gpt3_token_len":1991,"char_repetition_ratio":0.18009901,"word_repetition_ratio":0.07317073,"special_character_ratio":0.17666271,"punctuation_ratio":0.11376147,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9596159,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-25T16:02:51Z\",\"WARC-Record-ID\":\"<urn:uuid:fe06c74d-bd4f-4c74-a463-82aeaeb6005e>\",\"Content-Length\":\"25066\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:396b33cf-5db6-4db5-af08-f5619aeb72fe>\",\"WARC-Concurrent-To\":\"<urn:uuid:f25eca9a-f4f6-4897-a16c-a7ad97d69f48>\",\"WARC-IP-Address\":\"172.67.173.116\",\"WARC-Target-URI\":\"https://dekegisapiwe.anvgames.com/algebraic-systems-and-computational-complexity-theory-book-16901on.php\",\"WARC-Payload-Digest\":\"sha1:V44OQKHLMN3K2ZSJBSDYUJGVA4VEI46K\",\"WARC-Block-Digest\":\"sha1:OY5EYVDNRZX6IPS7H24VG2VIE3PLXHRS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662588661.65_warc_CC-MAIN-20220525151311-20220525181311-00368.warc.gz\"}"}
https://pepada.com/classes/promises-in-vba/complex-aynchronous-web-services	[ "Here’s where it gets really interesting. Let’s say that we need to get some large number of spreadsheets from Google Docs. We want to do it asynchronously, and we also know that google docs will fail if it receives to many requests, or it’s too busy. So we have to orchestrate and deal with all of the following.\n\n• Getting the schema to know which sheets exists\n• Getting some unknown number of sheets and populating the respective Excel sheets when they return\n• Throttle failures on Google Docs. We’ll use exponential backoff and SetTimer and VBA to deal with that\n• Ensuring that things don’t slip out of memory when the calling procedure exits.\n• JSON and Google Wire deciphering to interpret the returned data\nAll of this will be promise based, which should make the whole thing digestible. As input we’ll use the university tables that were used in the javacript app, University Rankings visualized\n\n#### The calling app\n\nI’ve generalized this so it can be used to retrieve any Google Spreadsheet. Here’s the shell. You can substitute in the key for any Google Spreadsheet. The only thing of consequence here is that the import process is kicked off asynchronously, and control will immediately be returned to the caller, which will then exit.\n\nNOTE: with the new Google Sheets, this method has been largely deprecated. There is a complete revamp of this here Get Data From Google Docs. You can also check out the Sheets apps from Apps built with data abstraction, which can both read and write from Excel to Sheets, but this post may be useful since it is about asynchronicity more than the utility of the app itself.\n\n`Public Sub asynchDocs()`\n` Dim url As String, key As String`\n` key = \"0At2ExLh4POiZdFd0YUhpZVRPUGxFcW85X2xkMm1vY2c\"`\n` url = \"https://spreadsheets.google.com/feeds/worksheets/\" + key + \"/public/basic?alt=json\"`\n` `\n` ' kick this off in the background`\n` doDocsWithBackoff url`\n` '`\n` ' we can go and do something else now`\n` Debug.Print \"Im doing something else - have a nice day\"`\n` `\n`End Sub`\n\n#### The promise orchestrator\n\nHere’s the code that kicks everything off. Let’s walk through\n\n`Public Sub doDocsWithBackoff(url As String)`\n` Dim b As cBackoff, p As cPromise, prs As cPromise, _`\n` process As cDocsImporter, doneSchema As cDeferred, doneSheets As cDeferred`\n` `\n` ' always to this for a fresh start - it will clear up any memory/timers from previously`\n` clearRegister`\n` `\n` ' this class knows how to get google spreadsheet`\n` Set process = New cDocsImporter`\n` Set doneSchema = New cDeferred`\n` `\n` ' first phase, we get the schema, and release up memory`\n` Set b = New cBackoff`\n` b.execute(url) _`\n` .done(process, \"getSchema\", doneSchema) _`\n` .fail process, \"show\"`\n` `\n` ' although this will be done synchronously, using a promise means we can know its done`\n` Set prs = deleteAllTheSheets`\n` `\n` ' we'll have resolved doneSchema when the previous .done is executed`\n` ' we'll need another deferred to tell us when all this is over`\n` Set doneSheets = New cDeferred`\n` when(Array(prs, doneSchema.promise)) _`\n` .done(process, \"getSheets\", doneSheets, url) _`\n` .fail process, \"show\"`\n\n` ' now we can clear everything up`\n` doneSheets.promise _`\n` .done(register, \"teardown\") _`\n` .done(process, \"teardown\") _`\n` .fail process, \"show\"`\n\n`End Sub`\n\n#### clearRegister\n\nWhen executing this asynchronously, one of the issues is to stop things disappearing from memory when the module that initiates them goes out of scope. One way is create a tangle of public variables – but that will be hard to manage, even harder to clear out, and impossible when the task is data driven as it is in this case. The promise framework maintains a register of objects that are going to be needed later, and keeps them in memory. This simply clears out any hanging around from previous asynch sessions. It should be the first thing called.\n\n` ' always to this for a fresh start - it will clear up any memory/timers from previously`\n` clearRegister`\n\n#### callBack class\n\nOne of the limitations of VBA is that you can only callback class members. I recommend that you centralize all specific methods needed for a particular project in a class like this – in my case I’ve created a class called cDocsImporter that will know all about how to handle google Docs Data.\n` ' this class knows how to get google spreadsheet`\n\n` Set process = New cDocsImporter`\n\n#### getting the schema\n\nHere’s the first deferred class – doneSchema. It ‘s promise will be used as a signal that we’ve managed to get (or failed to get) schema data from Google Docs. This is the definition of all the sheets in the Spreadsheet.\n` Set doneSchema = New cDeferred`\n\n#### getting schema data using cBackoff\n\nThe cBackoff class knows how to get data asynchronously – it uses Promises in VBA too – and also how to deal with throttling using an exponential backoff algorithm. For that it needs to use SetTimer and VBA. The cBackoff will kick off retrieval data from the given url, and immediately give control back. It returns a promise.\n\nWhen that gets resolved (within the cBackoff class), it will execute b.done() by calling process.getSchema(). The data retrieved by the cBackoff class will get passed to process.getSchema() also. If the cBackoff class is rejected (meaning it failed to get data), it will execute b.fail() by calling process.show() to report whatever the cause of the failure was.\n\nFinally it will resolve (or reject) doneSchema. We can use doneSchema.promise() as a signal shortly to know that we’ve finished getting the schema\n\n` ' first phase, we get the schema, and release up memory`\n`set b = New cBackoff``b.execute(url) _`\n` .done(process, \"getSchema\", doneSchema) _``.fail process, \"show\"`\n\n#### using promises for synchronous activities\n\nThe use of promises is not limited to asynchronous activities. You can use it to detect whether something has happened successfully or failed. You can also start synchronously, but later convert to asynchronous without having to modify the code structure. In this case we need to execute something that Excel doesnt know how to do asynchronously – delete all the sheets in the workbook. Since we have a lull in activities whilst waiting for the schema to come back, this is the perfect time.\n` ' although this will be done synchronously, using a promise means we can know its done`` Set prs = deleteAllTheSheets`` `\n\nHere is the deleteallthesheets function. Note that it creates and resolves a promise in line, since the activity is synchronous, and returns a promise.\n\n`Private Function deleteAllTheSheets() As cPromise`\n` ' its possible to use promises for synchronous tasks too`\n` Dim d As cDeferred`\n` Set d = New cDeferred`\n` ' dont want complaints`\n` Application.DisplayAlerts = False`\n` `\n` ' delete all but one sheet`\n` While sheets.Count > 1`\n` sheets(1).Delete`\n` Wend`\n` Application.DisplayAlerts = True`\n` `\n` d.resolve (Array(\"sheets deleted\"))`\n` Set deleteAllTheSheets = d.promise`\n`End Function`\n\nThe reason we use a promise here is that next we’ll want to retrieve each of the spreadsheets from withing the Google Workbook. However we don’t want to start doing that until both the schema has finished being retrieved and the old sheets have been deleted. It is almost certain that the sheet deletion would be finished being executed before the Schema was returned anyway, but we can use both promises now before continuing.\n\n#### Using When\n\nYou can test that multiple things have completed using when(). In our case we want to check that both the sheets are finished being deleted and the schema has been retrieved before starting to get data back.\n\n` ' we'll have resolved doneSchema when the previous .done is executed`` ' we'll need another deferred to tell us when all this is over`` Set doneSheets = New cDeferred`\n` when(Array(prs, doneSchema.promise)) _`` .done(process, \"getSheets\", doneSheets, url) _``.fail process, \"show\"`\n\nWhen actually returns a promise. This means that you can use .done() and .fail(). All the promises need to be successful for when().done() to execute. If any one fails, then when().fail() will be executed instead. The promises to be evaluated are presented as an array(promise1,promise2,…promisen). When() will receive a signal when each of those promises is either resolved or rejected. In our case we want both of array(prs, doneSchema.promise()) to be successfully resolved before continuing. We’ll also need another promise. This time to signal that we’ve got all the data. process.getSheets() will resolve (or reject) doneSheets when all the data has been retrieved and the sheets populated. Aside from taking multiple promises, When() will pass on the data from the last promise in the array – in this case the data that was signalled by doneScheme.resolve(data).\n\n#### Clearing up\n\nSince the procedure will have long exited by now, its up to you to clean up the objects that will have been forced to remain in memory beyond their regular scope. We can detect when everything is completed either by using When() . done() or fail(). In our case, we know that doneSheets.promise() could only have been resolved if everything else had been done, so we can simply test for the doneSheets.promise().done() signal\n\n` ' now we can clear everything up`\n` doneSheets.promise _`\n` .done(register, \"teardown\") _`\n` .done(process, \"teardown\") _`\n` .fail process, \"show\"`\n` `\n\nIt is good practice to create a .tearDown method for classes in order to properly remove it from memory. In many cases, setting its reference to Nothing will not work (where classes have dual links to each other), leading to a memory leak. In our case, we’ve managed to retain them in memory by creating a link to them in a public register. Register.teardown() will not only remove the link, but also execute each objects teardown method (if it has one). Doing this at the end of everything will recover the memory used during all the asynchronous activity.\n\n#### The cDocsImporter class\n\nAll the hard work interpreting google docs formats is done by calling back various methods in this class. There’s nothing much new in here, json and google docs importing having been covered elsewhere in this site, but here’s how it all fits together, and how the promises are rejected or resolved.\n\n`Option Explicit`\n\n`Public Sub getSchema(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called when I get the schema.`\n` ' the schema will be the 2nd argument of a(), 1st will be the url`\n` Dim c As cJobject`\n` Set c = getJsonDataFromArray(a)`\n` ' register it so it gets cleaned up later`\n` keepInRegister c, \"schema\", \"cJobject\"`\n` ' we'll pass back the parsed schema`\n` defer.resolve (Array(c))`\n`End Sub`\n`Public Sub storeData(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called each time I have some data from the vizAPI`\n` ' the schema will be the 2nd argument of a(), 1st will be the url`\n` `\n` Dim c As cJobject, job As cJobject, jr As cJobject, jc As cJobject, _`\n` s As String, joc As cJobject, r As Range, w As Worksheet`\n` Set c = getJsonDataFromArray(a, eDeserializeGoogleWire)`\n` Application.ScreenUpdating = False`\n` Application.Calculation = xlCalculationManual`\n` ' add a worksheet`\n` Set w = Worksheets.add()`\n` w.name = CStr(args(LBound(args)))`\n` Set r = w.Cells(1, 1)`\n` '`\n` 'put the data`\n` '`\n` Set jr = c.find(\"rows\")`\n` Set jc = c.find(\"cols\")`\n` `\n` ' here's the column headings`\n` '`\n` If Not jc Is Nothing Then`\n` For Each job In jc.children`\n` s = job.child(\"label\").value`\n` If s = vbNullString Then`\n` s = job.child(\"id\").value`\n` End If`\n` r.Offset(, job.childIndex - 1).value = s`\n` Next job`\n` End If`\n` '`\n` ' and these are the rows`\n` '`\n` If Not jr Is Nothing Then`\n` For Each job In jr.children`\n` For Each joc In job.child(\"c\").children`\n` r.Offset(job.childIndex, joc.childIndex - 1).value = joc.child(\"v\").value`\n` Next joc`\n` Next job`\n` End If`\n` Application.Calculation = xlCalculationAutomatic`\n` Application.ScreenUpdating = True`\n` ' release the cJobject memory`\n` c.tearDown`\n` ' signal that all is good`\n` defer.resolve (a)`\n`End Sub`\n`Public Sub getSheets(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called when I want to get all the data`\n` ' the schema will be the 1st argument of a()`\n` '`\n` `\n` Dim c As cJobject, cj As cJobject, job As cJobject, _`\n` joc As cJobject, s As String, u() As String, url As String, i As Long, _`\n` d() As cPromise, b As cBackoff, def As cDeferred, testing As Boolean, maxU As Long`\n` `\n` ' limit processing for debugging by setting this true`\n` testing = False`\n` `\n` Set c = a(LBound(a))`\n` Set cj = c.find(\"feed.entry\")`\n` If cj Is Nothing Then`\n` defer.reject (Array(\"could not find feed urls in schema entries\"))`\n` Exit Sub`\n` End If`\n` ' these will be the urls for each sheet to get`\n` maxU = cj.children.Count`\n` If (testing) Then maxU = 1`\n` ReDim u(0 To maxU - 1)`\n` ReDim d(0 To maxU - 1)`\n` ReDim t(0 To maxU - 1)`\n` `\n` If (arrayLength(u) = 0) Then`\n` defer.reject (Array(\"there were no sheets in the schema\"))`\n` Exit Sub`\n` End If`\n` `\n` ' extract the vizapi links`\n` For Each job In cj.children`\n` If job.childIndex > maxU Then Exit For`\n` url = vbNullString`\n` For Each joc In job.child(\"link\").children`\n` s = joc.toString(\"rel\")`\n` ' we're going to use the link for viz api`\n` If InStr(1, s, \"visualizationApi\") Then`\n` url = joc.toString(\"href\")`\n` End If`\n` Next joc`\n` If (url = vbNullString) Then`\n` defer.reject (Array(\"couldnt find the vizapi link in the schema\"))`\n` Exit Sub`\n` End If`\n` ' store it`\n` u(job.childIndex - 1) = url`\n` If (job.child(\"title.\\$t\") Is Nothing) Then`\n` defer.reject (Array(\"couldnt find the tiles node in the schema\"))`\n` Exit Sub`\n` End If`\n` t(job.childIndex - 1) = job.toString(\"title.\\$t\")`\n` Next job`\n` ' when we get here, we have to set up a promise for each of the urls and get the data`\n` For i = LBound(u) To UBound(u)`\n` Set b = New cBackoff`\n` Set def = New cDeferred`\n` Set d(i) = def.promise`\n` b.execute(u(i)) _`\n` .done(Me, \"storeData\", def, Array(t(i))) _`\n` .fail Me, \"show\"`\n` `\n` Next i`\n` 'signal the urls have been processed`\n` when(d) _`\n` .done(Me, \"resolve\", defer) _`\n` .fail Me, \"reject\", defer`\n` `\n`End Sub`\n`Public Sub resolve(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` defer.resolve (Array(\"urls done\"))`\n`End Sub`\n`Public Sub reject(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` defer.resolve (Array(\"urls done\"))`\n`End Sub`\n\n`Public Sub tearDown(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n\n`End Sub`\n`Private Sub class_initialize()`\n` keepInRegister Me, , \"cProcessData\"`\n`End Sub`\n\n`Private Function getJsonDataFromArray(a As Variant, _`\n` Optional t As eDeserializeType = eDeserializeNormal) As cJobject`\n` ' this will organize the data returned from a standard httpdeferred`\n` Dim c As cJobject, s As String`\n` Set c = New cJobject`\n` If t = eDeserializeGoogleWire Then`\n` s = cleanGoogleWire(CStr(a(LBound(a) + 1)))`\n` Else`\n` s = CStr(a(LBound(a) + 1))`\n` End If`\n` `\n` With c.init(Nothing)`\n` .add \"url\", CStr(a(LBound(a)))`\n` .add(\"data\").append JSONParse(s, t)`\n` End With`\n` Set getJsonDataFromArray = c`\n`End Function`\n\n`Public Sub show(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` Dim i As Long`\n` Debug.Print \"show:\";`\n` For i = LBound(a) To UBound(a)`\n` Debug.Print a(i)`\n` Next i`\n` Debug.Print \"\"`\n`End Sub`\n\n#### getSchema method\n\nThis will be called when it’s time to extract the schema from the data returned by the cBackoff class. Every method that is likely to be called by a .done() or a .fail() should have the same argument list.\n\na As Variant – this is the data that will have been signalled in .resolve(array(data)) and will be an array of whatever\nOptional defer As cDeferred – this can be used to signal completion of this task through a promise\nOptional args As Variant – this is another array that this time can be passed via .done()\n\n` `\n`Public Sub getSchema(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called when I get the schema.`\n` ' the schema will be the 2nd argument of a(), 1st will be the url`\n` Dim c As cJobject`\n` Set c = getJsonDataFromArray(a)`\n` ' register it so it gets cleaned up later`\n` keepInRegister c, \"schema\", \"cJobject\"`\n` ' we'll pass back the parsed schema`\n` defer.resolve (Array(c))`\n`End Sub`\n` `\nThe json of the schema get parsed and creates a cJobject . We register it so that it doesnt slip out of memory, but also so that it can be easily cleaned up later (cjobject has a teardown class that will be executed by register.teardown()). Finally we resolve the work and pass on the cJobject containing the schema data.\n\n#### getSheets method\n\nOnce getSchema (and deleteallthesheets) is resolved, this will be called to go and get each of the sheets referenced in the schema. Much f the code here is simply about converting google wire protocol. The interesting part is this section….\n\n` ' when we get here, we have to set up a promise for each of the urls and get the data`\n` For i = LBound(u) To UBound(u)`\n` Set b = New cBackoff`\n` Set def = New cDeferred`\n` Set d(i) = def.promise`\n` b.execute(u(i)) _`\n` .done(Me, \"storeData\", def, Array(t(i))) _`\n` .fail Me, \"show\"`\n` `\n` Next i`\n` `\n\nEach sheet is retrieved asynchronously using a cBackoff, and its promise is stored in an array of promises. On each resolution, the storeData() method will be called to move the data into the Excel sheet. So there are a whole stream of data fetches and data stores all happening at the same time.\n\n`Public Sub getSheets(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called when I want to get all the data`\n` ' the schema will be the 1st argument of a()`\n` '`\n` `\n` Dim c As cJobject, cj As cJobject, job As cJobject, _`\n` joc As cJobject, s As String, u() As String, url As String, i As Long, _`\n` d() As cPromise, b As cBackoff, def As cDeferred, testing As Boolean, maxU As Long`\n` `\n` ' limit processing for debugging by setting this true`\n` testing = False`\n` `\n` Set c = a(LBound(a))`\n` Set cj = c.find(\"feed.entry\")`\n` If cj Is Nothing Then`\n` defer.reject (Array(\"could not find feed urls in schema entries\"))`\n` Exit Sub`\n` End If`\n` ' these will be the urls for each sheet to get`\n` maxU = cj.children.Count`\n` If (testing) Then maxU = 1`\n` ReDim u(0 To maxU - 1)`\n` ReDim d(0 To maxU - 1)`\n` ReDim t(0 To maxU - 1)`\n` `\n` If (arrayLength(u) = 0) Then`\n` defer.reject (Array(\"there were no sheets in the schema\"))`\n` Exit Sub`\n` End If`\n` `\n` ' extract the vizapi links`\n` For Each job In cj.children`\n` If job.childIndex > maxU Then Exit For`\n` url = vbNullString`\n` For Each joc In job.child(\"link\").children`\n` s = joc.toString(\"rel\")`\n` ' we're going to use the link for viz api`\n` If InStr(1, s, \"visualizationApi\") Then`\n` url = joc.toString(\"href\")`\n` End If`\n` Next joc`\n` If (url = vbNullString) Then`\n` defer.reject (Array(\"couldnt find the vizapi link in the schema\"))`\n` Exit Sub`\n` End If`\n` ' store it`\n` u(job.childIndex - 1) = url`\n` If (job.child(\"title.\\$t\") Is Nothing) Then`\n` defer.reject (Array(\"couldnt find the tiles node in the schema\"))`\n` Exit Sub`\n` End If`\n` t(job.childIndex - 1) = job.toString(\"title.\\$t\")`\n` Next job`\n` ' when we get here, we have to set up a promise for each of the urls and get the data`\n` For i = LBound(u) To UBound(u)`\n` Set b = New cBackoff`\n` Set def = New cDeferred`\n` Set d(i) = def.promise`\n` b.execute(u(i)) _`\n` .done(Me, \"storeData\", def, Array(t(i))) _`\n` .fail Me, \"show\"`\n` `\n` Next i`\n` 'signal the urls have been processed`\n` when(d) _`\n` .done(Me, \"resolve\", defer) _`\n` .fail Me, \"reject\", defer`\n` `\n`End Sub`\nFinally, we can use when() to resolve that we have completely finished all the retrievals, since we have an array of the all the promises that have been issued, each of which will have been resolved when the data from them has been transferred to excel by storeData.\n\n#### storeData method\n\nThis is the final step. Data from each sheet is passed here and converting to a cJobject. When done its promise will be resolved. These promises make up the list that getSheets() is waiting for before finally declaring victory.\n\n`Public Sub storeData(a As Variant, Optional defer As cDeferred, Optional args As Variant)`\n` ' this will be called each time I have some data from the vizAPI`\n` ' the schema will be the 2nd argument of a(), 1st will be the url`\n` `\n` Dim c As cJobject, job As cJobject, jr As cJobject, jc As cJobject, _`\n` s As String, joc As cJobject, r As Range, w As Worksheet`\n` Set c = getJsonDataFromArray(a, eDeserializeGoogleWire)`\n` Application.ScreenUpdating = False`\n` Application.Calculation = xlCalculationManual`\n` ' add a worksheet`\n` Set w = Worksheets.add()`\n` w.name = CStr(args(LBound(args)))`\n` Set r = w.Cells(1, 1)`\n` '`\n` 'put the data`\n` '`\n` Set jr = c.find(\"rows\")`\n` Set jc = c.find(\"cols\")`\n` `\n` ' here's the column headings`\n` '`\n` If Not jc Is Nothing Then`\n` For Each job In jc.children`\n` s = job.child(\"label\").value`\n` If s = vbNullString Then`\n` s = job.child(\"id\").value`\n` End If`\n` r.Offset(, job.childIndex - 1).value = s`\n` Next job`\n` End If`\n` '`\n` ' and these are the rows`\n` '`\n` If Not jr Is Nothing Then`\n` For Each job In jr.children`\n` For Each joc In job.child(\"c\").children`\n` r.Offset(job.childIndex, joc.childIndex - 1).value = joc.child(\"v\").value`\n` Next joc`\n` Next job`\n` End If`\n` Application.Calculation = xlCalculationAutomatic`\n` Application.ScreenUpdating = True`\n` ' release the cJobject memory`\n` c.tearDown`\n` ' signal that all is good`\n` defer.resolve (a)`\n`End Sub`\n\n#### Summary\n\nAlthough conceptually this is a little harder to deal with than if this had been done synchronously, the use of promises helps control the complex asynchronous orchestration. In fact I can’t imagine doing it without it now. See more about Promises in VBA" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78860503,"math_prob":0.7428204,"size":16511,"snap":"2019-51-2020-05","text_gpt3_token_len":4327,"char_repetition_ratio":0.12437148,"word_repetition_ratio":0.55563664,"special_character_ratio":0.24365574,"punctuation_ratio":0.11584997,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.962102,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-18T05:58:11Z\",\"WARC-Record-ID\":\"<urn:uuid:26849e56-a0b0-4569-81dc-6250bd74f181>\",\"Content-Length\":\"184157\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f4387b69-15fc-4d6d-80a4-74c6789cdcc6>\",\"WARC-Concurrent-To\":\"<urn:uuid:247aeebc-45c4-446e-8a86-257e8550203c>\",\"WARC-IP-Address\":\"192.241.148.82\",\"WARC-Target-URI\":\"https://pepada.com/classes/promises-in-vba/complex-aynchronous-web-services\",\"WARC-Payload-Digest\":\"sha1:YR4I57FZKN3Q2WIHHDZKH5NCWSAAOIEZ\",\"WARC-Block-Digest\":\"sha1:VYJOSSKLSSQSBZHDS5UHSNWHX2HTYBJS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250592261.1_warc_CC-MAIN-20200118052321-20200118080321-00441.warc.gz\"}"}
https://encyclopedia2.thefreedictionary.com/Poincar%C3%A9+recurrence+theorem	[ "Poincaré recurrence theorem\n\nAlso found in: Wikipedia.\n\nPoincaré recurrence theorem\n\n[‚pwän‚kä′rā ri′kə·rəns ‚thir·əm]\n(mathematics)\nA volume preserving homeomorphism T of a finite dimensional Euclidean space will have, for almost all points x, infinitely many points of the form T i (x), i = 1, 2, … within any open set containing x.\nA measure preserving transformation on a space with finite measure is recurrent.\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7427124,"math_prob":0.9663047,"size":412,"snap":"2019-43-2019-47","text_gpt3_token_len":113,"char_repetition_ratio":0.10539216,"word_repetition_ratio":0.0,"special_character_ratio":0.21601942,"punctuation_ratio":0.118421055,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9621117,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-19T12:38:44Z\",\"WARC-Record-ID\":\"<urn:uuid:159b55d6-c060-46fa-9617-dcbd25f06cb8>\",\"Content-Length\":\"39756\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:117d6a7a-3ca1-4f08-bfd6-671d9a4fcc14>\",\"WARC-Concurrent-To\":\"<urn:uuid:fd8047f2-8e78-4082-bd31-8da03a13e058>\",\"WARC-IP-Address\":\"91.204.210.230\",\"WARC-Target-URI\":\"https://encyclopedia2.thefreedictionary.com/Poincar%C3%A9+recurrence+theorem\",\"WARC-Payload-Digest\":\"sha1:3PEZFVFPVEIDXULH45BRJXD5KVBLEOSQ\",\"WARC-Block-Digest\":\"sha1:RWJNCHBWEZX6A632E3TVSCVP2UGNCXPF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986693979.65_warc_CC-MAIN-20191019114429-20191019141929-00105.warc.gz\"}"}
http://rossriveryukon.info/100-math-facts-worksheets-printable/division-worksheet-printable-math-rounding-coloring-worksheets-moreover-grade-3-grade-3-math-worksheets-table-worksheet-100-math-facts-worksheets-printable-worksheets-for-1st-grade-reading/	[ "# Division Worksheet Printable Math Rounding Coloring Worksheets Moreover Grade 3 Grade 3 Math Worksheets Table Worksheet 100 Math Facts Worksheets Printable Worksheets For 1st Grade Reading", null, "division worksheet printable math rounding coloring worksheets moreover grade 3 grade 3 math worksheets table worksheet 100 math facts worksheets printable worksheets for 1st grade reading.\n\nworksheets4kids perimeter worksheets for 4th grade geometry of math addition within teachers,worksheets for 1st grade reading comprehension 2nd math pdf facts printable addition and subtraction timed,worksheets for kindergarten science preschool math counting printable facts worksheet teachers,addition worksheets worksheetworks solving multi step equations for kindergarten pdf 4th grade science,worksheets for preschool numbers 2nd grade writing math facts printable main ideas 4th,addition worksheets for you to print right now 1st grade reading comprehension teachers english work handwriting,worksheets for 4th grade english 1st reading pre k 3 math facts fourth printable download,worksheetworks coordinate picture worksheets for pre k math 1st grade reading addition facts worksheet unique best 1 bis images on free,worksheets for preschool numbers multiplication math facts problems worksheets4kids area kindergarten reading,worksheets work times tables to free printable multiplication facts worksheet 2 for 1st graders reading grade." ]	[ null, "http://rossriveryukon.info/wp-content/uploads/2019/05/division-worksheet-printable-math-rounding-coloring-worksheets-moreover-grade-3-grade-3-math-worksheets-table-worksheet-100-math-facts-worksheets-printable-worksheets-for-1st-grade-reading.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80666375,"math_prob":0.68754387,"size":1234,"snap":"2019-13-2019-22","text_gpt3_token_len":231,"char_repetition_ratio":0.25609756,"word_repetition_ratio":0.0,"special_character_ratio":0.15640195,"punctuation_ratio":0.06077348,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97881746,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-20T16:30:48Z\",\"WARC-Record-ID\":\"<urn:uuid:9e030ec7-bfc8-45a6-890e-f77002b3befb>\",\"Content-Length\":\"52924\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:50457a3e-745c-4442-9b0d-1e8f36802835>\",\"WARC-Concurrent-To\":\"<urn:uuid:3894f75f-5d93-4a05-8d02-1cf25ceef293>\",\"WARC-IP-Address\":\"104.31.64.193\",\"WARC-Target-URI\":\"http://rossriveryukon.info/100-math-facts-worksheets-printable/division-worksheet-printable-math-rounding-coloring-worksheets-moreover-grade-3-grade-3-math-worksheets-table-worksheet-100-math-facts-worksheets-printable-worksheets-for-1st-grade-reading/\",\"WARC-Payload-Digest\":\"sha1:VBGCDMBA4NQXPGEZPW6QWEPW6QGNNUAI\",\"WARC-Block-Digest\":\"sha1:44AEE6CPY4ZMCOF42B6J5MV2OAXU3GK5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232256082.54_warc_CC-MAIN-20190520162024-20190520184024-00491.warc.gz\"}"}
https://arduino.stackexchange.com/questions/63495/issue-with-indenting-the-serial-print-data-in-the-serial-monitor	[ "# Issue with Indenting the Serial Print data in the Serial Monitor\n\nI have a question with Serial Print. I want the Sensor reading values to fall under 3 columns, and each column should have the readings from the same sensor.\n\nRaw Value \| Sensor 1 \| Sensor 2 \| Sensor 3\n\nhow can I modify this code do this?\n\nBelow I have posted my code and what I currently see using the Serial Monitor\n\n`````` void loop(){\n\nVoltage0 = (RawValue0 / 1023.0) * 5000; // 5000 to get millivots.\ntempC0 = (Voltage0-500) * 0.1; // 500 is the offset\ntempF0 = (tempC0 * 1.8) + 32; // conver to F\nSerial.print(\"Raw Value = \" ); // shows pre-scaled value\nSerial.print(RawValue0);\nSerial.print(\"\\t milli volts = \"); // shows the voltage measured\nSerial.print(Voltage0,0); //\nSerial.print(\"\\t Sensor 1: Temperature in C = \");\nSerial.print(tempC0,1);\nSerial.print(\"\\t Sensor 1: Temperature in F = \");\nSerial.println(tempF0,1);\ndelay(500);\n``````", null, "• you have no code for dealing with sensor 2 and sensor 3 ..... there is no way to modify non-existent code – jsotola Apr 14 '19 at 19:53\n• is this schoolwork? – jsotola Apr 14 '19 at 19:55\n• you already know how to print in column format .... it is entirely unclear what you are having a problem with – jsotola Apr 14 '19 at 19:56\n• think very carefully about this .... do you really want `Raw Value \| Sensor 1 \| Sensor 2 \| Sensor 3`? – jsotola Apr 14 '19 at 20:17\n\nColumns with lines?\n\nThe `sprintf` function can fill data with spaces to align them. However, the Arduino `sprintf` function does not support floating point variables. The `dtostrf` function has already a fixed output width by itself.\nThe sketch below is a combination of UTF-8 characters and the `dtostrf` and `sprintf` functions.\n\n``````void setup() {\nSerial.begin(9600);\n\nSerial.println();\nSerial.println(\"┏━━━━━━┳━━━━━━━━┳━━━━━━━━┓\");\nSerial.println(\"┃ raw ┃ °C ┃ °F ┃\");\nSerial.println(\"┣━━━━━━╋━━━━━━━━╋━━━━━━━━┫\");\n}\n\nvoid loop() {\nint raw = random(0, 1024);\nfloat c = random(-1000, 1000) / 10.0;\nfloat f = random(-500, 1500) / 10.0;\n\nchar buffer_c;\nchar buffer_f;\n\ndtostrf(c, 6, 1, buffer_c);\ndtostrf(f, 6, 1, buffer_f);\n\nchar buffer;\nsprintf(buffer, \"┃ %4d ┃ %s ┃ %s ┃\", raw, buffer_c, buffer_f);\nSerial.println(buffer);\n\ndelay(500);\n}\n``````\n\nThe strlen() function can not be used before printing a floating point number to the serial port, since its length is determined while printing characters to the serial port. The final number of bytes that is written is the return value.\nThere is a way to use that return value, by using a dummy write to a dummy serial port. After that the number of spaces to align the floating point number can be calculated.\n\nThat reduces the code size from 6160 to 4708 for an Arduino Uno.\n\n``````class DummyClass : public Stream\n{\nint available() {return 0;}\nint peek() {return 0;}\nvoid flush() {return;}\nsize_t write(uint8_t) {return 1;}\nusing Print::write;\n};\n\nDummyClass Dummy;\n\nconst int width_r = 4;\nconst int width_c = 6;\nconst int width_f = 6;\n\nvoid setup() {\nSerial.begin(9600);\n\nSerial.println();\nSerial.println(\"┏━━━━━━┳━━━━━━━━┳━━━━━━━━┓\");\nSerial.println(\"┃ raw ┃ °C ┃ °F ┃\");\nSerial.println(\"┣━━━━━━╋━━━━━━━━╋━━━━━━━━┫\");\n}\n\nvoid loop() {\nint raw = random(0, 1024);\nfloat c = random(-1000, 1000) / 10.0;\nfloat f = random(-500, 1500) / 10.0;\n\nint i;\nSerial.print(\"┃ \");\nSpaces(width_r - Dummy.print(raw));\nSerial.print(raw);\nSerial.print(\" ┃ \");\nSpaces(width_c - Dummy.print(c,1));\nSerial.print(c,1);\nSerial.print(\" ┃ \");\nSpaces(width_f - Dummy.print(f,1));\nSerial.print(f,1);\nSerial.print(\" ┃\");\nSerial.println();\n\ndelay(500);\n}\n\nvoid Spaces(int n)\n{\nfor(int i=0; i<n; i++) {\nSerial.print(' ');\n}\n}\n``````\n\nI advise not to use this \"Dummy\" trick. It is mere a coding exercise. I can not find a problem with this trick, but I'm not 100% sure that it is okay.\n\n• use dtostrf for `raw` too and and save 1kB flash on `sprintf` – Juraj Apr 15 '19 at 9:53\n• @Juraj I can do better than that. After reading your answer here: arduino.stackexchange.com/questions/62783/… I tried a dummy class. To my surprise it worked. – Jot Apr 15 '19 at 11:00\n\nHere is a sketch that uses a different approach than Jot uses in his answer. There is no question that Jot's answer is excellent and takes very few lines of code to accomplish.\n\nI thought it would look nice if the Raw Value, an integer, was centered under the \"Raw Value\" column when it was at it's maximum value of 1023. A value of 1 would be right-ish justified, but still aligned with the center in mind.\n\nThe floating point numbers, which could be positive or negative, should be right justified I thought, so I wrote a second function to deal with those numbers.\n\n``````int RawValue0;\nfloat Voltage0, tempC0, tempF0;\n\nvoid setup(){\nSerial.begin(9600);\n\nSerial.println();\nSerial.println(\"Raw Value \| Sensor 1 \| Sensor 2 \| Sensor 3\");\nSerial.println(\"------------------------------------------\");\n\nVoltage0 = 11.0; //(RawValue0 / 1023.0) * 5000; // 5000 to get millivots.\ntempC0 = -100.0; //(Voltage0-500) * 0.1; // 500 is the offset\ntempF0 = 1212.0; //(tempC0 * 1.8) + 32; // conver to F\n}\n\nvoid loop(){\n\n// Increment the variables to simulate test data.\nRawValue0 += 51;\nif(RawValue0 > 1023){RawValue0 = 0;}\nVoltage0 += 99.99;\nif(Voltage0 > 9999.99){Voltage0 = 0.0;}\ntempC0 -= 99.99;\nif(tempC0 < -9999.99){tempC0 = 0.0;}\n\n// Print out the data in columns.\nprintNumber(RawValue0);\nSerial.print(\" \| \");\nprintNumber(Voltage0);\nSerial.print(\" \| \");\nprintNumber(tempC0);\nSerial.print(\" \| \");\nprintNumber(tempF0);\nSerial.println();\ndelay(250);\n}\n\nlong printNumber(int number){\nSerial.print(\" \");\nint spaces = 0;\nif(number < 1000){spaces = 1;}\nif(number < 100){spaces = 2;}\nif(number < 10){spaces = 3;}\nfor(int i = 0; i < spaces; i++){\nSerial.print(\" \");\n}\nSerial.print(number);\n}\n\nfloat printNumber(float number){\nint spaces = 0;\nif(number > -1000.00){Serial.print(\" \");}\nif(number < 1000){spaces = 1;}\nif(number < 100){spaces = 2;}\nif(number < 10){spaces = 3;}\nif(number < 0){spaces = 0;}\nif(number == -99.99){spaces = 1;}\nfor(int i = 0; i < spaces; i++){\nSerial.print(\" \");\n}\nSerial.print(number);\n}\n``````\n\nWhen I compile Jot's sketch, it's 6640 bytes. If you remove his \"fancy\" header, it drops to 6414 bytes. My sketch is 5124 bytes. There is a certain amount of \"overhead\" involved when using `sprintf()`.\n\nThe question you have to ask yourself is, \"Does saving 1000 bytes matter?\"." ]	[ null, "https://i.stack.imgur.com/gKF4R.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5676142,"math_prob":0.98467034,"size":2466,"snap":"2020-24-2020-29","text_gpt3_token_len":762,"char_repetition_ratio":0.17912267,"word_repetition_ratio":0.1769437,"special_character_ratio":0.33090025,"punctuation_ratio":0.23483366,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9872849,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-15T03:00:48Z\",\"WARC-Record-ID\":\"<urn:uuid:fbdad8c7-6b61-44d2-84b5-122e7f2f551d>\",\"Content-Length\":\"160226\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8494db2b-67b9-49c1-8da6-5aa68b469769>\",\"WARC-Concurrent-To\":\"<urn:uuid:b15eccb7-d3a9-4f87-ab9b-ed495f3f31f2>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://arduino.stackexchange.com/questions/63495/issue-with-indenting-the-serial-print-data-in-the-serial-monitor\",\"WARC-Payload-Digest\":\"sha1:BN7BU26EX24RVIDK7ZAEM6Y5NBADQHZF\",\"WARC-Block-Digest\":\"sha1:3ZVXCEK6BOSTKWVRIJYQOXGIU2FCI6HZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657154789.95_warc_CC-MAIN-20200715003838-20200715033838-00218.warc.gz\"}"}
https://jonlabelle.com/snippets/view/sql/winning-streaks-in-mysql	[ "Given a table of IDs and won-lost results, how do we find the longest winning streak?\n\n``````-- ---------------------------------------------------------------------------\n-- Winning Streaks\n--\n-- from the Artful Common Queries page\n-- http://www.artfulsoftware.com/infotree/qrytip.php?id=803\n-- ---------------------------------------------------------------------------\n\n-- Given a table of IDs and won-lost results, how do we find the longest\n-- winning streak?\n\ndrop table if exists results;\ncreate table results(id int,result char(1));\n\ninsert into results values\n(1,'w'),(2,'l'),(3,'l'),(4,'w'),(5,'w'),(6,'w'),(7,'l'),(8,'w'),(9,'w');\n\nselect * from results;\n-- +------+--------+\n-- \| id \| result \|\n-- +------+--------+\n-- \| 1 \| w \|\n-- \| 2 \| l \|\n-- \| 3 \| l \|\n-- \| 4 \| w \|\n-- \| 5 \| w \|\n-- \| 6 \| w \|\n-- \| 7 \| l \|\n-- \| 8 \| w \|\n-- \| 9 \| w \|\n-- +------+--------+\n\n-- We can find streaks of two with a left join on a.id=b.id+1. To count streak\n-- lengths, initialise a counter to 0 then increment it for every hit:\n\nset @count=0;\nselect a.id, a.result, b.result, @count := IF(a.result = b.result, @count + 1, 1) as Streak\nfrom results a\nleft join results b on a.id = b.id + 1\nwhere a.result = 'w';\n\n-- The longest winning streak is the longest such streak found:\n\nset @count=0;\nselect MAX(@count:=IF(a.result = b.result, @count + 1, 1)) as LongestStreak\nfrom results a\nleft join results b on a.id = b.id + 1\nwhere a.result = 'w';\n-- +---------------+\n-- \| LongestStreak \|\n-- +---------------+\n-- \| 3 \|\n-- +---------------+\n\n-- That solution is from a response by Jon Roshko to a question by Ed Ball on\n-- the MySQL Newbie Forum. Scott Noyes points out that our query pattern for\n-- sequence starts and ends also works for winning streaks:\n\nSELECT MIN( c.id ) - a.id + 1 as LongestStreak\nFROM results AS a\nLEFT JOIN results AS b ON a.id = b.id + 1 AND b.result = 'w'\nLEFT JOIN results AS c ON a.id <= c.id AND c.result = 'w'\nLEFT JOIN results AS d ON c.id = d.id - 1 AND d.result = 'w'\nWHERE\na.result = 'w'\nAND b.id IS NULL\nAND c.id IS NOT NULL\nAND d.id IS NULL\nGROUP BY a.id\nORDER BY LongestStreak DESC LIMIT 1;``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6182502,"math_prob":0.9021894,"size":2095,"snap":"2023-40-2023-50","text_gpt3_token_len":613,"char_repetition_ratio":0.24390244,"word_repetition_ratio":0.1689008,"special_character_ratio":0.45202863,"punctuation_ratio":0.19069767,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9722706,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-30T04:38:28Z\",\"WARC-Record-ID\":\"<urn:uuid:6560cb62-73d4-4e75-9172-6a47d46022aa>\",\"Content-Length\":\"11773\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f92b79a2-1ec8-4eef-85a7-93859fca5032>\",\"WARC-Concurrent-To\":\"<urn:uuid:0fb41a2a-dc92-40d6-a121-73c19b976ed8>\",\"WARC-IP-Address\":\"45.32.199.83\",\"WARC-Target-URI\":\"https://jonlabelle.com/snippets/view/sql/winning-streaks-in-mysql\",\"WARC-Payload-Digest\":\"sha1:57QCRPC2FCYWF33WSFGJRGRERVKNBNJC\",\"WARC-Block-Digest\":\"sha1:DAXEFDYDB7UC3M67VH7K53FKL7LJ6QGT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100164.87_warc_CC-MAIN-20231130031610-20231130061610-00445.warc.gz\"}"}
https://smallbusiness.chron.com/mean-depreciation-computed-straightline-basis-20495.html	[ "What Does It Mean That Depreciation Is Computed on a Straight-Line Basis?\n\nDepreciation can refer to the decline in an asset's value due to its use in business activities, its gradual obsolescence, its depletion and other causes. It can also refer to the accounting procedure used to represent this change in value on the accounts, where the asset has a portion of its value deducted in each time period in which it continues to be used. Multiple methods exist for calculating depreciation, and the straight-line method is one of the most popular.\n\nDepreciation\n\nOnly long-term assets with physical existence and finite usefulness are depreciated to reflect their declining value. Depreciation is done in order to accurately portray the process through which depreciable assets become useless. This way, assets that become useless do not have their entire values recorded as a single enormous loss at the end of their usefulness and instead have those losses spread across the entire time span of their usefulness.\n\nStraight-Line Method\n\nMultiple methods exist to calculate an depreciation. The straight-line method is one of the simplest, and is the most popular. The straight-line method allocates an equal amount of depreciation during each period during its useful life.\n\nParameters Used in Straight-Line Method\n\nBook value, useful lifespan, and residual value upon disposal are parameters used in the straight-line depreciation method. Book value is the value of the asset at its time of acquisition, and is usually the price paid for it. Useful lifespan is the length of time that the asset is expected to remain useful. Residual value upon disposal is the estimated salvage value of the asset at the end of its usefulness, and is often zero. Book value minus residual value upon disposal is equal to the asset's depreciable value.\n\nImplementation of Straight-Line Method\n\nSuppose that a business purchased a \\$20,000 machine with an estimated lifespan of 10 years and a residual value upon disposal of \\$2,000. Its depreciable value is thus \\$18,000, to be allocated across 10 years. The machine is depreciated by \\$1,800 in each year of its useful lifespan." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95059186,"math_prob":0.9114664,"size":2481,"snap":"2022-05-2022-21","text_gpt3_token_len":486,"char_repetition_ratio":0.14210738,"word_repetition_ratio":0.015151516,"special_character_ratio":0.19145505,"punctuation_ratio":0.078125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9655782,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-20T22:49:49Z\",\"WARC-Record-ID\":\"<urn:uuid:f2a10d0c-2ce5-4793-b082-216cb2e0bc8c>\",\"Content-Length\":\"115801\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:54bf97ef-c933-44cc-adae-91c0552d369d>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a0b7d3f-2bb7-471c-a591-2c2dfef40c74>\",\"WARC-IP-Address\":\"13.32.204.39\",\"WARC-Target-URI\":\"https://smallbusiness.chron.com/mean-depreciation-computed-straightline-basis-20495.html\",\"WARC-Payload-Digest\":\"sha1:NBYSS2OJLOEZZNDP4WC4XIQSQ3ZXUKPY\",\"WARC-Block-Digest\":\"sha1:5JCASDF4MLCPWI2EI6DOROHX74G2AYZA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320302706.62_warc_CC-MAIN-20220120220649-20220121010649-00596.warc.gz\"}"}
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We NEVER charge anything unless you requested it.\nAll payment are encrypted, stored and securely processed by Stripe.\nEdit Payee Info\nCountry\nState / Province\nCity\nZIP Code / Postal Code\nLegal name\nDate of birth\nSave\nEdit Debit Card\nDebit Card Number\nExp Month\nExp Year\nCVV / CVC\nSave\nUpdating...\n64\n0\n0\n0\n1\nmannikinism1975\nEdit · Unsubscribe · Report · Sat Nov 14 2020 15:14:47 GMT-0500 (EST) chemistry\nLive Preview\nInsert symbol in textarea\nLogic\nGreek letters\nExponent\nTrigonometry\nCalculus\n$\\div$\n$\\times$\n$-$\n$+$\n$\\pm$\n$<$\n$>$\n$\\le$\n$\\ge$\n$=$\n$\\approx$\n$\\neq$\n$\\cup$\n$\\subset$\n$\\supset$\n$\\subseteq$\n$\\supseteq$\n$\\in$\n$\\ni$\n$\\emptyset$\n$\\notin$\n$\\forall$\n$\\exists$\n$\\exists!$\n$\\nexists$\n$\\neg$\n$\\lor$\n$\\land$\n$\\implies$\n$\\Longleftarrow$\n$\\iff$\n$\\equiv$\n$\\otimes$\n$\\oplus$\n$\\cdot$\n$\\cdots$\n$\\ddots$\n$\\alpha$\n$\\beta$\n$\\gamma$\n$\\delta$\n$\\epsilon$\n$\\zeta$\n$\\eta$\n$\\theta$\n$\\iota$\n$\\kappa$\n$\\lambda$\n$\\mu$\n$\\nu$\n$\\xi$\n$\\pi$\n$\\rho$\n$\\sigma$\n$\\tau$\n$\\upsilon$\n$\\phi$\n$\\chi$\n$\\psi$\n$\\omega$\n$\\Gamma$\n$\\Delta$\n$\\Theta$\n$\\Lambda$\n$\\Xi$\n$\\Pi$\n$\\Sigma$\n$\\Phi$\n$\\Psi$\n$\\Omega$\n$\\infty$\n$\\mathbb{R}$\n$\\mathbb{Z}$\n$\\mathbb{Q}$\n$\\mathbb{N}$\n$\\Re$\n$\\Im$\n$x^{a}$\n$x_{b}$\n$x_{a}^{b}$\n$\\sqrt{x}$\n$\\sqrt[n]{x}$\n$\\left\| x \\right\|$\n$\\frac{a}{b}$\n$\\binom{a}{b}$\n$\\ln$\n$\\log_{n}$\n$\\cos$\n$\\sin$\n$\\tan$\n$\\sec$\n$\\csc$\n$\\cot$\n$\\sin^{-1}$\n$\\cos^{-1}$\n$\\tan^{-1}$\n$\\exp$\n$\\sinh$\n$\\cosh$\n$\\tanh$\n$\\coth$\n$\\overline{\\rm AB}$\n$\\overrightarrow{\\rm AB}$\n$\\angle$\n$\\measuredangle$\n$\\triangle$\n$\\square$\n$\\cong$\n$\\ncong$\n$\\sim$\n$\\nsim$\n$\\\|$\n$\\nparallel$\n$\\perp$\n$\\not\\perp$\n$\\sum_{i=1}^{k}$\n$\\prod_{i=1}^{k}$\n$\\int_{a}^{b}$\n$\\int_{a}^\\infty$\n$\\int_\\infty^{b}$\n$\\oint_{a}^{b}$\n$\\iint_{a}^{b}$\n$\\iiint_{a}^{b}$\n$\\lim_{a \\rightarrow b}$\n$\\frac{dx}{dt}$\n$\\frac{\\partial u}{\\partial t}$\n$f^{\\prime}$\n$f^{\\prime \\prime}$\n$\\infty$\n$d\\theta$\n$dV$\n$dA$\n$dxdydz$\n$rdrd\\theta$\n$\\rho^2\\sin \\phi d \\rho d\\phi d \\theta$\nInsert drawing\nInsert\nRecord audio\n00:00\nChoose a subject\nChoose a subject\nbiology\nchemistry\ncomputer science\nenglish\nfrench\ngeography\nhealth\nhistory\nmathematics" ]	[ null, "https://www.peeranswer.com/images/newMessage.png", null, "https://www.peeranswer.com/images/newMessage.png", null, "https://www.peeranswer.com/images/paypal/blue.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9860834,"math_prob":0.999782,"size":1998,"snap":"2021-21-2021-25","text_gpt3_token_len":435,"char_repetition_ratio":0.09729187,"word_repetition_ratio":0.0,"special_character_ratio":0.1986987,"punctuation_ratio":0.1142132,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99943376,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-17T17:11:47Z\",\"WARC-Record-ID\":\"<urn:uuid:02bb3995-d4d6-4bcb-9027-2f5d17c8f29f>\",\"Content-Length\":\"144766\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a7476a3e-958d-448d-972a-1789079144b0>\",\"WARC-Concurrent-To\":\"<urn:uuid:a48d4760-9e05-4821-b955-9a379da2d9ed>\",\"WARC-IP-Address\":\"138.68.235.190\",\"WARC-Target-URI\":\"https://www.peeranswer.com/question/5fb03ab7f67cda5211f062fb\",\"WARC-Payload-Digest\":\"sha1:F5LZUSWC5OC4UXJXIPCTC35ZRQOCYDOE\",\"WARC-Block-Digest\":\"sha1:LGTVCLMX2RQZRSPA6VOMDWI22JSRUZDM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991258.68_warc_CC-MAIN-20210517150020-20210517180020-00590.warc.gz\"}"}
https://programmingwords.com/home/recursion-tail-call-optimization	[ "## Aug 31 Recursion & Tail Call Optimization\n\nRecursion is, simply put, a function calling itself.\n\n``````def someFunction(x):\nsomeFunction(x - 1)``````\n\nIf you try to run the above code, you'll notice it will run in an infinite loop until something on your machine crashes terribly. This is why every recursive function has two parts: a base case and recursive case. In the above example, we only have a recursive case: that's the part where you recursively call the function. For the function to ever complete, we need a way to break out of the loop.\n\n``````def someFunction(x):\nif (x == 0)\nreturn 0;\nsomeFunction(x - 1)``````\n\nA base case is usually a very simple return value, based on some sort of check. In the above example we are checking if we reduced `x` to 0, and in that case we simply return 0, breaking the recursion.\n\nRecursion lets you solve a bunch of different problems. The canonical example of recursion is a factorial function. The factorial of a natural number is the product of all natural numbers smaller or equal than that number. For instance, the factorial of 4 is `4 * 3 * 2 * 1`.\n\nTo implement this for an arbitrary number, we need some way to keep multiplying and decrementing the number, all the way until we reach 1. There's one thing that makes this easy: the factorial of 4, is the same as factorial of 3 (`3 * 2 * 1`) multiplied by 4. We can generally say that `fact(x) = x * fact(x - 1)`. Using this, we write our function.\n\n``````def fact(n):\nif (n <= 1)\nreturn 1;\n\nreturn n * fact(n - 1);``````\n\nWe can recursively call the `fact` function until we reach 1. When we call the function `fact(4)`, it will call `fact(3)`, and then that function will call `fact(2)`, and so on. When the base case is reached, the function returns `1` to `fact(2)`. `fact(2)` then finishes executing and returns `2 * 1`. `fact(3)` can then return `3 * 2 * 1`, and so on until the root function returns.\n\nWe defined the factorial function with just a few lines of code, and it looks similar to the mathematical definition from above.\n\nIf you don't like recursion, you can also write this as a regular old loop.\n\n``````def fact(n):\nreturnValue = 1;\nfor (i = 1; i < n; i++)\nreturnValue = i\n\nreturn returnValue``````\n\nIn fact, you can write all recursive programs as loops, and vice versa. This is because recursion is Turing complete. Every program in the world can be written using recursion. Since this is also true for loops, both of these tools can be used to solve the same problems. At least in theory.\n\n## The Trouble With Recursion\n\nOur two `fact` functions behave exactly the same as far as we can see, but behind the scenes there might be big and very important differences. It has to do with the way computers call functions.\n\nYour code does not get executed line-by-line, it has to keep jumping around. When you call a function, you are jumping to the code inside that function, and then returning back to the call site. The compiler needs to keep track of these jumps so it doesn't lose its place. It does this by saving the location it needs to return to in the call stack.\n\nA call stack is a stack that contains function calls, their parameters and local variables, as well as the return address. This allows the compiler to go back from the code inside the function to where the function was called. It's the compiler's version of leaving breadcrumbs so it knows how to find its way back.\n\nWhen you call a function, the compiler adds the function, its parameters, and the return location to the stack. This function will only get cleared off the stack once it's done running.\n\n``[(fact, n = 3, returnAddress = 1)]``\n\nIf `fact` calls another function, the compiler will add another item to the stack. In our case, `fact` is calling itself with `2`, so our stack would have two elements.\n\n``````[(fact, n = 3, returnAddress = 1),\n(fact, n = 2, returnAddress = 5)]``````\n\nThis other function calls `fact(1)`, incrementing the length of the stack to three.\n\n``````[(fact, n = 3, returnAddress = 1),\n(fact, n = 2, returnAddress = 5),\n(fact, n = 1, returnAddress = 5)]``````\n\n`fact(1)` will return `1`, and that's is when the compiler will pop `fact(1)` off the stack, and resume execution at the `returnAddress`. It will do this over and over until the root function returns.\n\nTo calculate the factorial of 3 recursively, we had to store three function calls at once. This is much worse than implementing the function with a for loop. With a for loop, you are not jumping to a new function, so there's no need to add items to the call stack. Our loop example takes up much less memory than the recursive one.\n\nFor small numbers this is not a problem, but for larger numbers and functions with a lot of data, memory consumption builds up. Recursive functions expect the compiler to store all of their calls at once. If the compiler runs out of memory, you will get a stack overflow exception. A problem so bad that they named a whole Q&A site after it.\n\n## The Fix\n\nThe above is only true for some recursive functions and in some languages. There is a way to go around the whole stack overflow issue, and it's called tail call optimization.\n\nA tail call is simply the last instruction in a function. In the following function, the tail call is adding the numbers 2 and 5:\n\n``````def f():\nprint(\"I am some instruction!\")\nreturn 2 + 5``````\n\nA tail call is the last instruction that will be executed, not the instruction that is last in the text. For instance, in our `fact` example, the tail call is performing multiplication, and not the recursive call, because the multiplication operands have to get evaluated before the multiplication can take place.\n\nFunctions that do call themselves as the last instruction are called tail recursive functions. Tail recursive functions don't have the problem of stack overflows, because we can do a little trick that avoids adding new items onto the call stack.\n\nThere is nothing left to do inside a function after the tail call. You won't return back to the function after the tail call. This means you don't need to pass a return address to the function called inside the tail call. Since the whole point of the call stack is knowing where you need to return, you don't need a new item in the stack if you're calling a function inside a tail call.\n\nTo fix our memory issue, we can rewrite our `fact` function to be tail recursive.\n\n``````def fact(n, i):\nif n < 0\nreturn 0\nif n == 0\nreturn 1\nif n == 1:\nreturn i\nelse\nreturn fact(n - 1, n i)``````\n\nIf we want to calculate the factorial of 3, we would call `fact(3, 1)`. This function behaves the same as our previous one, but you'll notice that the last instruction is a call to `fact`. This means that this function is tail recursive.\n\nWhy is this better than our previous example? Let's take a look at the call stack to see what's going on. When we call `fact(3, 1)` a new item is added to the call stack.\n\n``[(fact, n = 3, i = 1, returnAddress = 1)]``\n\nThe function will call `fact(2, 3)`. However, since we are at the end of the function, we no longer need the function's item in the call stack. It also means the return address for the next call will be the same as it is for the current call. So, we can make a shortcut: instead of allocating a new item on the stack, we simply use the one we already have, and just replace the parameters.\n\n``[(fact, n = 2, i = 3, returnAddress = 1)]``\n\nWe can keep doing this for every recursive call until we get a result. During the whole time of the factorial calculation we are only using a maximum of one item in the call stack. And the best part is this is true no matter how large the number is!\n\nTail call elimination makes tail recursive functions as efficient as loops. Some languages only have recursion, and no loops. If it weren't for this optimization, they would be constantly running our of memory.\n\nIt's important to note that not all languages do this. Even mainstream languages like C and Java don't have tail call optimization. Newer languages like Swift and Kotlin perform tail call optimization, as do most functional programming languages.\n\n## Recursion Is Cool*\n\nRecursion lets you write your code in a different, more expressive style. You can elegantly solve a lot of problems and write your code in a more mathematical style. Just keep in mind the memory and performance requirements of recursion in your language, and make sure you recurse in your tail call!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91700375,"math_prob":0.9682283,"size":8191,"snap":"2019-35-2019-39","text_gpt3_token_len":1928,"char_repetition_ratio":0.16941492,"word_repetition_ratio":0.020956123,"special_character_ratio":0.24038579,"punctuation_ratio":0.11570248,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99242556,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-19T09:41:14Z\",\"WARC-Record-ID\":\"<urn:uuid:a0742802-4885-41bb-99b1-bf6f0bff37ed>\",\"Content-Length\":\"65460\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3e6d0ac2-b383-49e5-a03f-6106f16f612d>\",\"WARC-Concurrent-To\":\"<urn:uuid:a360c18d-3ca9-4fa0-bc6a-4b38f548af23>\",\"WARC-IP-Address\":\"198.49.23.144\",\"WARC-Target-URI\":\"https://programmingwords.com/home/recursion-tail-call-optimization\",\"WARC-Payload-Digest\":\"sha1:N5LMBC2JHCJWJFPCD5KWKUENS26KTPPW\",\"WARC-Block-Digest\":\"sha1:GCLIEV64HKPX4UXDE2RQIUURDMTPZFEO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573465.18_warc_CC-MAIN-20190919081032-20190919103032-00366.warc.gz\"}"}
https://kupdf.net/download/acct-4400-ch3_5966242fdc0d60686fa88e76_pdf	[ "# ACCT 4400 CH3\n\nJuly 12, 2017 \| Author: Caroline Roberson \| Category: Forecasting, Seasonality, Regression Analysis, Moving Average, Errors And Residuals\n\n#### Short Description\n\nAudit chapter 3 test bank...\n\n#### Description\n\nChapter 03 Forecasting\n\nTrue / False Questions\n\n1. Forecasting techniques generally assume an existing causal system that will continue to exist in the future. True\n\nFalse\n\n2. For new products in a strong growth mode, a low alpha will minimize forecast errors when using exponential smoothing techniques. True\n\nFalse\n\n3. Once accepted by managers, forecasts should be held firm regardless of new input since many plans have been made using the original forecast. True\n\nFalse\n\n4. Forecasts for groups of items tend to be less accurate than forecasts for individual items because forecasts for individual items don't include as many influencing factors. True\n\nFalse\n\n5. Forecasts help managers both to plan the system itself and to provide valuable information for using the system. True\n\nFalse\n\n6. Organizations that are capable of responding quickly to changing requirements can use a shorter forecast horizon and therefore benefit from more accurate forecasts. True\n\nFalse\n\n7. When new products or services are introduced, focus forecasting models are an attractive option. True\n\nFalse\n\n8. The purpose of the forecast should be established first so that the level of detail, amount of resources, and accuracy level can be understood. True\n\nFalse\n\n9. Forecasts based on time-series (historical) data are referred to as associative forecasts. True\n\nFalse\n\n10. Time-series techniques involve the identification of explanatory variables that can be used to predict future demand. True\n\nFalse\n\n11. A consumer survey is an easy and sure way to obtain accurate input from future customers since most people enjoy participating in surveys. True\n\nFalse\n\n12. The Delphi approach involves the use of a series of questionnaires to achieve a consensus forecast. True\n\nFalse\n\n13. Exponential smoothing adds a percentage (called alpha) of the last period's forecast to estimate the next period's demand. True\n\nFalse\n\n14. The shorter the forecast period, the more accurately the forecasts tend to track what actually happens. True\n\nFalse\n\n15. Forecasting techniques that are based on time-series data assume that future values of the series will duplicate past values. True\n\nFalse\n\n16. Trend-adjusted exponential smoothing uses double smoothing to add twice the forecast error to last period's actual demand. True\n\nFalse\n\n17. Forecasts based on an average tend to exhibit less variability than the original data. True\n\nFalse\n\n18. The naive approach to forecasting requires a linear trend line. True\n\nFalse\n\n19. The naive forecast is limited in its application to series that reflect no trend or seasonality. True\n\nFalse\n\n20. The naive forecast can serve as a quick and easy standard of comparison against which to judge the cost and accuracy of other techniques. True\n\nFalse\n\n21. A moving average forecast tends to be more responsive to changes in the data series when more data points are included in the average. True\n\nFalse\n\n22. In order to update a moving average forecast, the values of each data point in the average must be known. True\n\nFalse\n\n23. Forecasts of future demand are used by operations people to plan capacity. True\n\nFalse\n\n24. An advantage of a weighted moving average is that recent actual results can be given more importance than what occurred a while ago. True\n\nFalse\n\n25. Exponential smoothing is a form of weighted averaging. True\n\nFalse\n\n26. A smoothing constant of .1 will cause an exponential smoothing forecast to react more quickly to a sudden change than a smoothing constant value of .3. True\n\nFalse\n\n27. The T in the model TAF = S + T represents the time dimension (which is usually expressed in weeks or months). True\n\nFalse\n\n28. Trend-adjusted exponential smoothing requires selection of two smoothing constants. True\n\nFalse\n\n29. An advantage of trend-adjusted exponential smoothing over the linear trend equation is its ability to adjust over time to changes in the trend. True\n\nFalse\n\n30. A seasonal relative (or seasonal indexes) is expressed as a percentage of average or trend. True\n\nFalse\n\n31. In order to compute seasonal relatives, the trend of past data must be computed or known, which means that for brand-new products this approach cannot be used. True\n\nFalse\n\n32. Removing the seasonal component from a data series (deseasonalizing) can be accomplished by dividing each data point by its appropriate seasonal relative. True\n\nFalse\n\n33. If a pattern appears when a dependent variable is plotted against time, one should use time series analysis instead of regression analysis. True\n\nFalse\n\n34. Curvilinear and multiple regression procedures permit us to extend associative models to relationships that are nonlinear or involve more than one predictor variable. True\n\nFalse\n\n35. The sample standard deviation of forecast error is equal to the square root of MSE. True\n\nFalse\n\n36. Correlation measures the strength and direction of a relationship between variables. True\n\nFalse\n\n37. MAD is equal to the square root of MSE, which is why we calculate the easier MSE and then calculate the more difficult MAD. True\n\nFalse\n\n38. In exponential smoothing, an alpha of 1.0 will generate the same forecast that a naive forecast would yield. True\n\nFalse\n\n39. A forecast method is generally deemed to perform adequately when the errors exhibit an identifiable pattern. True\n\nFalse\n\n40. A control chart involves setting action limits for cumulative forecast error. True\n\nFalse\n\n41. A tracking signal focuses on the ratio of cumulative forecast error to the corresponding value of MAD. True\n\nFalse\n\n42. The use of a control chart assumes that errors are normally distributed about a mean of zero. True\n\nFalse\n\n43. Bias exists when forecasts tend to be greater or less than the actual values of time series. True\n\nFalse\n\n44. Bias is measured by the cumulative sum of forecast errors. True\n\nFalse\n\n45. Seasonal relatives can be used to deseasonalize data or incorporate seasonality in a forecast. True\n\nFalse\n\n46. The best forecast is not necessarily the most accurate. True\n\nFalse\n\nMultiple Choice Questions\n\n47. Which of the following is a potential shortcoming of using sales force opinions in demand forecasting?\n\nA. Members of the sales force often have substantial histories of working with and understanding their customers. B. Members of the sales force often are well aware of customers' future plans. C. Members of the sales force have direct contact with consumers. D. Members of the sales force can have difficulty distinguishing between what customers would like to do and what they actually will do. E. Customers often are quite open with members of the sales force with regard to future plans. 48. Suppose a four-period weighted average is being used to forecast demand. Weights for the periods are as follows: wt-4 = 0.1, wt-3 = 0.2, wt-2 = 0.3 and wt-1 = 0.4. Demand observed in the previous four periods was as follows: A t-4 = 380, At-3 = 410, At-2 = 390, At-1 = 400. What will be the demand forecast for period t?\n\nA. B. C. D. E.\n\n402 397 399 393 403\n\n49. Suppose a three-period weighted average is being used to forecast demand. Weights for the periods are as follows: wt-3 = 0.2, wt-2 = 0.3 and wt-1 = 0.5. Demand observed in the previous three periods was as follows: At-3 = 2,200, At-2 = 1,950, At-1 = 2,050. What will be the demand forecast for period t?\n\nA. B. C. D. E.\n\n2,000 2,095 1,980 2,050 1,875\n\n50. When choosing a forecasting technique, a critical trade-off that must be considered is that between:\n\nA. B. C. D. E.\n\ntime series and associative. seasonality and cyclicality. length and duration. simplicity and complexity. cost and accuracy.\n\n51. The more novel a new product or service design is, the more forecasters have to rely on:\n\nA. B. C. D. E.\n\nsubjective estimates. seasonality. cyclicality. historical data. smoothed variation.\n\n52. Forecasts based on judgment and opinion do not include:\n\nA. B. C. D. E.\n\nexecutive opinion. salesperson opinion. second opinions. customer surveys. Delphi methods.\n\n53. Which of the following is/are a primary input into capacity, sales, and production planning?\n\nA. B. C. D. E.\n\nproduct design market share ethics globalization demand forecasts\n\n54. Which of the following features would not generally be considered common to all forecasts?\n\nA. Assumption of a stable underlying causal system. B. Actual results will differ somewhat from predicted values. C. Historical data is available on which to base the forecast. D. Forecasts for groups of items tend to be more accurate than forecasts for individual items. E. Accuracy decreases as the time horizon increases. 55. Which of the following is not a step in the forecasting process?\n\nA. B. C. D. E.\n\nDetermine the purpose and level of detail required. Eliminate all assumptions. Establish a time horizon. Select a forecasting model. Monitor the forecast.\n\n56. Minimizing the sum of the squared deviations around the line is called:\n\nA. B. C. D. E.\n\nmean squared error technique. mean absolute deviation. double smoothing. least squares estimation. predictor regression.\n\n57. The two general approaches to forecasting are:\n\nA. B. C. D. E.\n\nmathematical and statistical. qualitative and quantitative. judgmental and qualitative. historical and associative. precise and approximation.\n\n58. Which of the following is not a type of judgmental forecasting?\n\nA. B. C. D. E.\n\nexecutive opinions sales force opinions consumer surveys the Delphi method time series analysis\n\n59. Accuracy in forecasting can be measured by:\n\nA. B. C. D. E.\n\nMSE. MRP. MPS. MTM. MTE.\n\n60. Which of the following would be an advantage of using a sales force composite to develop a demand forecast?\n\nA. The sales staff is least affected by changing customer needs. B. The sales force can easily distinguish between customer desires and probable actions. C. The sales staff is often aware of customers' future plans. D. Salespeople are least likely to be influenced by recent events. E. Salespeople are least likely to be biased by sales quotas.\n\n61. Which phrase most closely describes the Delphi technique?\n\nA. B. C. D. E.\n\nassociative forecast consumer survey series of questionnaires developed in India historical data\n\n62. The forecasting method which uses anonymous questionnaires to achieve a consensus forecast is:\n\nA. B. C. D. E.\n\nsales force opinions. consumer surveys. the Delphi method. time series analysis. executive opinions.\n\n63. One reason for using the Delphi method in forecasting is to:\n\nA. B. C. D. E.\n\navoid premature consensus (bandwagon effect). achieve a high degree of accuracy. maintain accountability and responsibility. be able to replicate results. prevent hurt feelings.\n\n64. Detecting nonrandomness in errors can be done using:\n\nA. B. C. D. E.\n\nMSEs. MAPs. control charts. correlation coefficients. strategies.\n\n65. Gradual, long-term movement in time series data is called:\n\nA. B. C. D. E.\n\nseasonal variation. cycles. irregular variation. trend. random variation.\n\n66. The primary difference between seasonality and cycles is:\n\nA. B. C. D. E.\n\nthe duration of the repeating patterns. the magnitude of the variation. the ability to attribute the pattern to a cause. the direction of the movement. there are only four seasons but 30 cycles.\n\n67. Averaging techniques are useful for:\n\nA. distinguishing between random and nonrandom variations. B. smoothing out fluctuations in time series. C. eliminating historical data. D. providing accuracy in forecasts. E. average people. 68. Putting forecast errors into perspective is best done using\n\nA. B. C. D. E.\n\nexponential smoothing. MAPE. linear decision rules. MAD. hindsight.\n\n69. Using the latest observation in a sequence of data to forecast the next period is:\n\nA. B. C. D. E.\n\na moving average forecast. a naive forecast. an exponentially smoothed forecast. an associative forecast. regression analysis.\n\n70. For the data given below, what would the naive forecast be for period 5?\n\nA. B. C. D. E.\n\n58 62 59.5 61 cannot tell from the data given\n\n71. Moving average forecasting techniques do the following:\n\nA. B. C. D. E.\n\nImmediately reflect changing patterns in the data. Lead changes in the data. Smooth variations in the data. Operate independently of recent data. Assist when organizations are relocating.\n\n72. Which is not a characteristic of simple moving averages applied to time series data?\n\nA. B. C. D. E.\n\nsmoothes random variations in the data weights each historical value equally lags changes in the data requires only last period's forecast and actual data smoothes real variations in the data\n\n73. In order to increase the responsiveness of a forecast made using the moving average technique, the number of data points in the average should be:\n\nA. B. C. D. E.\n\ndecreased. increased. multiplied by a larger alpha. multiplied by a smaller alpha. eliminated if the MAD is greater than the MSE.\n\n74. A forecast based on the previous forecast plus a percentage of the forecast error is:\n\nA. B. C. D. E.\n\na naive forecast. a simple moving average forecast. a centered moving average forecast. an exponentially smoothed forecast. an associative forecast.\n\n75. Which is not a characteristic of exponential smoothing?\n\nA. B. C. D. E.\n\nsmoothes random variations in the data weights each historical value equally has an easily altered weighting scheme has minimal data storage requirements smoothes real variations in the data\n\n76. Which of the following smoothing constants would make an exponential smoothing forecast equivalent to a naive forecast?\n\nA. B. C. D. E.\n\n0 .01 .1 .5 1.0\n\n77. Simple exponential smoothing is being used to forecast demand. The previous forecast of 66 turned out to be four units less than actual demand. The next forecast is 66.6, implying a smoothing constant, alpha, equal to:\n\nA. B. C. D. E.\n\n.01. .10. .15. .20. .60.\n\n78. Given an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the forecast for the next period be using simple exponential smoothing?\n\nA. B. C. D. E.\n\n36.9 57.5 60.5 62.5 65.5\n\n79. Given an actual demand of 105, a forecasted value of 97, and an alpha of .4, the simple exponential smoothing forecast for the next period would be:\n\nA. B. C. D. E.\n\n80.8. 93.8. 100.2. 101.8. 108.2.\n\n80. Which of the following possible values of alpha would cause exponential smoothing to respond the most quickly to forecast errors?\n\nA. B. C. D. E.\n\n0 .01 .05 .10 .15\n\n81. A manager uses the following equation to predict monthly receipts: Y t = 40,000 + 150t. What is the forecast for July if t = 0 in April of this year?\n\nA. B. C. D. E.\n\n40,450 40,600 42,100 42,250 42,400\n\nA. an exponentially smoothed forecast and a smoothed trend factor. B. an exponentially smoothed forecast and an estimated trend value. C. the old forecast adjusted by a trend factor. D. the old forecast and a smoothed trend factor. E. a moving average and a trend factor.\n\n83. In the additive model for seasonality, seasonality is expressed as a ______________ adjustment to the average; in the multiplicative model, seasonality is expressed as a __________ adjustment to the average.\n\nA. B. C. D. E.\n\nquantity; percentage percentage; quantity quantity; quantity percentage; percentage qualitative; quantitative\n\n84. Which technique is used in computing seasonal relatives?\n\nA. B. C. D. E.\n\ndouble smoothing Delphi mean squared error centered moving average exponential smoothing\n\n85. A persistent tendency for forecasts to be greater than or less than the actual values is called:\n\nA. B. C. D. E.\n\nbias. tracking. control charting. positive correlation. linear regression.\n\n86. Which of the following might be used to indicate the cyclical component of a forecast?\n\nA. B. C. D. E.\n\nleading variable mean squared error Delphi technique exponential smoothing mean absolute deviation\n\n87. The primary method for associative forecasting is:\n\nA. B. C. D. E.\n\nsensitivity analysis. regression analysis. simple moving averages. centered moving averages. exponential smoothing.\n\n88. Which term most closely relates to associative forecasting techniques?\n\nA. B. C. D. E.\n\ntime series data expert opinions Delphi technique consumer survey predictor variables\n\n89. Which of the following corresponds to the predictor variable in simple linear regression?\n\nA. B. C. D. E.\n\nregression coefficient dependent variable independent variable predicted variable demand coefficient\n\n90. The mean absolute deviation is used to:\n\nA. B. C. D. E.\n\nestimate the trend line. eliminate forecast errors. measure forecast accuracy. seasonally adjust the forecast. compute periodic forecast errors.\n\n91. Given forecast errors of 4, 8, and -3, what is the mean absolute deviation?\n\nA. B. C. D. E.\n\n4 3 5 6 12\n\n92. Given forecast errors of 5, 0, -4, and 3, what is the mean absolute deviation?\n\nA. B. C. D. E.\n\n4 3 2.5 2 1\n\n93. Given forecast errors of 5, 0, -4, and 3, what is the bias?\n\nA. B. C. D. E.\n\n-4 4 5 12 6\n\n94. Which of the following is used for constructing a control chart?\n\nA. B. C. D.\n\nmean absolute deviation mean squared error tracking signal bias\n\n95. The two most important factors in choosing a forecasting technique are:\n\nA. B. C. D. E.\n\ncost and time horizon. accuracy and time horizon. cost and accuracy. quantity and quality. objective and subjective components.\n\n96. The degree of management involvement in short-range forecasts is:\n\nA. B. C. D. E.\n\nnone. low. moderate. high. total.\n\n97. Which of the following is not necessarily an element of a good forecast?\n\nA. B. C. D. E.\n\nestimate of accuracy timeliness meaningful units low cost written\n\n98. Forecasting techniques generally assume:\n\nA. the absence of randomness. B. continuity of some underlying causal system. C. a linear relationship between time and demand. D. accuracy that increases the farther out in time the forecast projects. E. accuracy that is better when individual items, rather than groups of items, are being considered. 99. A managerial approach toward forecasting which seeks to actively influence demand is:\n\nA. B. C. D. E.\n\nreactive. proactive. influential. protracted. retroactive.\n\n100 Customer service levels can be improved by better: . A. B. C. D. E.\n\nmission statements. control charting. short-term forecast accuracy. exponential smoothing. customer selection.\n\n101 Given the following historical data, what is the simple three-period moving average . forecast for period 6?\n\nA. B. C. D. E.\n\n67 115 69 68 68.67\n\n102 Given the following historical data and weights of .5, .3, and .2, what is the three. period moving average forecast for period 5?\n\nA. B. C. D. E.\n\n144.20 144.80 144.67 143.00 144.00\n\n103 Use of simple linear regression analysis assumes that: . A. variations around the line are nonrandom. B. deviations around the line are normally distributed. C. predictions can easily be made beyond the range of observed values of the predictor variable. D. all possible predictor variables are included in the model. E. the variance of error terms (deviations) varies directly with the predictor variable. 104 Given forecast errors of -5, -10, and +15, the MAD is: . A. B. C. D. E.\n\n0. 10. 30. 175. 225.\n\n105 The president of State University wants to forecast student enrollments for this . academic year based on the following historical data:\n\nWhat is the forecast for this year using the naive approach?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,000 22,800\n\n106 The president of State University wants to forecast student enrollments for this . academic year based on the following historical data:\n\nWhat is the forecast for this year using a four-year simple moving average?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\n107 The president of State University wants to forecast student enrollments for this . academic year based on the following historical data:\n\nWhat is the forecast for this year using exponential smoothing with alpha = .5, if the forecast for two years ago was 16,000?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\n108 The president of State University wants to forecast student enrollments for this . academic year based on the following historical data:\n\nWhat is the forecast for this year using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\n109 The president of State University wants to forecast student enrollments for this . academic year based on the following historical data:\n\nWhat is the forecast for this year using trend-adjusted (double) smoothing with alpha = .05 and beta = .3, if the forecast for last year was 21,000, the forecast for two years ago was 19,000, and the trend estimate for last year's forecast was 1,500?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\n110 The business analyst for Video Sales, Inc. wants to forecast this year's demand for . DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using the naive approach?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\n111 The business analyst for Video Sales, Inc. wants to forecast this year's demand for . DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using a three-year weighted moving average with weights of .5, .3, and .2?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\n112 The business analyst for Video Sales, Inc. wants to forecast this year's demand for . DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using exponential smoothing with alpha = .4, if the forecast for two years ago was 750?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\n113 The business analyst for Video Sales, Inc. wants to forecast this year's demand for . DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\n114 The business analyst for Video Sales, Inc. wants to forecast this year's demand for . DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using trend-adjusted (double) smoothing with alpha = .3 and beta = .2, if the forecast for last year was 310, the forecast for two years ago was 430, and the trend estimate for last year's forecast was -150?\n\nA. B. C. D. E.\n\n162.4 180.3 301.4 403.2 510.0\n\n115 Professor Very Busy needs to allocate time next week to include time for office hours. . He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using the naive approach?\n\nA. B. C. D. E.\n\n45 50 52 65 78\n\n116 Professor Very Busy needs to allocate time next week to include time for office hours. . He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using a three-week simple moving average?\n\nA. B. C. D. E.\n\n49 50 52 65 78\n\n117 Professor Very Busy needs to allocate time next week to include time for office hours. . He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using exponential smoothing with alpha = .2, if the forecast for two weeks ago was 90?\n\nA. B. C. D. E.\n\n49 50 52 65 77\n\n118 Professor Very Busy needs to allocate time next week to include time for office hours. . He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n49 50 52 65 78\n\n119 Professor Very Busy needs to allocate time next week to include time for office hours. . He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using trend-adjusted (double) smoothing with alpha = .5 and beta = .1, if the forecast for last week was 65, the forecast for two weeks ago was 75, and the trend estimate for last week's forecast was -5?\n\nA. B. C. D. E.\n\n49.3 50.6 51.3 65.4 78.7\n\n120 A concert promoter is forecasting this year's attendance for one of his concerts based . on the following historical data:\n\nWhat is this year's forecast using the naive approach?\n\nA. B. C. D. E.\n\n22,000 20,000 18,000 15,000 12,000\n\n121 A concert promoter is forecasting this year's attendance for one of his concerts based . on the following historical data:\n\nWhat is this year's forecast using a two-year weighted moving average with weights of .7 and .3?\n\nA. B. C. D. E.\n\n19,400 18,600 19,000 11,400 10,600\n\n122 A concert promoter is forecasting this year's attendance for one of his concerts based . on the following historical data:\n\nWhat is this year's forecast using exponential smoothing with alpha = .2, if last year's smoothed forecast was 15,000?\n\nA. B. C. D. E.\n\n20,000 19,000 17,500 16,000 15,000\n\n123 A concert promoter is forecasting this year's attendance for one of his concerts based . on the following historical data:\n\nWhat is this year's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n20,000 21,000 22,000 23,000 24,000\n\n124 A concert promoter is forecasting this year's attendance for one of his concerts based . on the following historical data:\n\nThe previous trend line had predicted 18,500 for two years ago, and 19,700 for last year. What was the mean absolute deviation for these forecasts?\n\nA. B. C. D. E.\n\n100 200 400 500 800\n\n125 The dean of a school of business is forecasting total student enrollment for this year's . summer session classes based on the following historical data:\n\nWhat is this year's forecast using the naive approach?\n\nA. B. C. D. E.\n\n2,000 2,200 2,800 3,000 4,300\n\n126 The dean of a school of business is forecasting total student enrollment for this year's . summer session classes based on the following historical data:\n\nWhat is this year's forecast using a three-year simple moving average?\n\nA. B. C. D. E.\n\n2,667 2,600 2,500 2,400 2,333\n\n127 The dean of a school of business is forecasting total student enrollment for this year's . summer session classes based on the following historical data:\n\nWhat is this year's forecast using exponential smoothing with alpha = .4, if last year's smoothed forecast was 2,600?\n\nA. B. C. D. E.\n\n2,600 2,760 2,800 3,840 3,000\n\n128 The dean of a school of business is forecasting total student enrollment for this year's . summer session classes based on the following historical data:\n\nWhat is the annual rate of change (slope) of the least squares trend line for these data?\n\nA. B. C. D. E.\n\n0 200 400 180 360\n\n129 The dean of a school of business is forecasting total student enrollment for this year's . summer session classes based on the following historical data:\n\nWhat is this year's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n3,600 3,500 3,400 3,300 3,200\n\n130 The owner of Darkest Tans Unlimited in a local mall is forecasting this month's . (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using the naive approach?\n\nA. B. C. D. E.\n\n100 160 130 140 120\n\n131 The owner of Darkest Tans Unlimited in a local mall is forecasting this month's . (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using a four-month weighted moving average with weights of .4, .3, .2, and .1?\n\nA. B. C. D. E.\n\n120 129 141 135 140\n\n132 The owner of Darkest Tans Unlimited in a local mall is forecasting this month's . (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using exponential smoothing with alpha = .2, if August's forecast was 145?\n\nA. B. C. D. E.\n\n144 140 142 148 163\n\n133 The owner of Darkest Tans Unlimited in a local mall is forecasting this month's . (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is the monthly rate of change (slope) of the least squares trend line for these data?\n\nA. B. C. D. E.\n\n320 102 8 -.4 -8\n\n134 The owner of Darkest Tans Unlimited in a local mall is forecasting this month's . (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n1,250 128.6 102 158 164\n\n135 Which of the following mechanisms for enhancing profitability is most likely to result . from improving short-term forecast performance?\n\nA. B. C. D. E.\n\nincreased inventory reduced flexibility higher-quality products greater customer satisfaction greater seasonality\n\n136 Which of the following changes would tend to shorten the time frame for short-term . forecasting?\n\nA. bringing greater variety into the product mix B. increasing the flexibility of the production system C. ordering fewer weather-sensitive items D. adding more special-purpose equipment E. investing in the production system to make it more task-specific 137 Which of the following helps improve supply chain forecasting performance? . A. B. C. D. E.\n\ncontracts that require supply chain members to formulate long-term forecasts penalties for supply chain members that adjust forecasts sharing forecasts or demand data across the supply chain increasing lead times for critical supply chain members increasing the number of suppliers at critical junctures in the supply chain\n\n138 Which of the following would tend to decrease forecast accuracy? . A. a reduction in demand variability B. a shortening of the forecast time horizon C. an attempt to forecast demand for a group of similar items rather than an individual item D. a change in the underlying causal system 139 Which of the following is the most valuable piece of information the sales force can . bring into forecasting situations?\n\nA. what customers are most likely to do in the future B. what customers most want to do in the future C. what customers' future plans are D. whether customers are satisfied or dissatisfied with their performance in the past E. what the salesperson's appropriate sales quota should be\n\nEssay Questions\n\n140 Develop a forecast for the next period, given the data below, using a three-period . moving average.\n\n141 Consider the data below: .\n\nUsing exponential smoothing with alpha = .2, and assuming the forecast for period 11 was 80, what would the forecast for period 14 be?\n\n142 A manager is using exponential smoothing to predict merchandise returns at a . suburban branch of a department store chain. Given a previous forecast of 140 items, an actual number of returns of 148 items, and a smoothing constant equal to .15, what is the forecast for the next period?\n\n143 A manager is using the equation below to forecast quarterly demand for a product: . Yt = 6,000 + 80t where t = 0 at Q2 of last year Quarter relatives are Q1 = .6, Q2 = .9, Q3 = 1.3, and Q4 = 1.2. What forecasts are appropriate for the last quarter of this year and the first quarter of next year?\n\n144 Over the past five years, a firm's sales have averaged 250 units in the first quarter of . each year, 100 units in the second quarter, 150 units in the third quarter, and 300 units in the fourth quarter. What are appropriate quarter relatives for this firm's sales? Hint: Only minimal computations are necessary.\n\n145 A manager has been using a certain technique to forecast demand for gallons of ice . cream for the past six periods. Actual and predicted amounts are shown below. Would a naive forecast have produced better results?\n\n146 A new car dealer has been using exponential smoothing with an alpha of .2 to . forecast weekly new car sales. Given the data below, would a naive forecast have provided greater accuracy? Explain. Assume an initial exponential forecast of 60 units in period 2 (i.e., no forecast for period 1).\n\n147 A CPA firm has been using the following equation to predict annual demand for tax . audits: Yt = 55 + 4t. Demand for the past few years is shown below. Is the forecast performing as well as it might? Explain.\n\n148 Given the data below, develop a forecast for period 6 using a four-period weighted . moving average and weights of .4, .3, .2 and .1.\n\n149 Use linear regression to develop a predictive model for demand for burial vaults . based on sales of caskets.\n\n150 Given the following data, develop a linear regression model for y as a function of x. .\n\n151 Given the following data, develop a linear regression model for y as a function of x. .\n\n152 Develop a linear trend equation for the data on bread deliveries shown below. . Forecast deliveries for period 11 through 14.\n\n153 Demand for the last four months was: .\n\nA) Predict demand for July using each of these methods: 1) a three-period moving average 2) exponential smoothing with alpha equal to .20 (use a naive forecast for April for your first forecast) B) If the naive approach had been used to predict demand for April through June, what would MAD have been for those months?\n\n154 A manager wants to choose one of two forecasting alternatives. Each alternative was . tested using historical data. The resulting forecast errors for the two are shown in the table. Analyze the data and recommend a course of action to the manager.\n\n155 A manager uses this equation to predict demand: Y t = 20 + 4t. Over the past eight . periods, demand has been as follows. Are the results acceptable? Explain.\n\n156 Data on demand over the last few years are available as follows: .\n\nWhat would this year's forecast be if we were using the naive approach?\n\n157 Data on demand over the last few years are available as follows: .\n\nWhat is this year's forecast using a four-year simple moving average?\n\n158 Data on demand over the last few years are available as follows: .\n\nWhat is this year's forecast using exponential smoothing with alpha = .25, if last year's smoothed forecast was 45?\n\n159 Data on demand over the last few years are available as follows: .\n\nWhat are this and next year's forecasts using the least squares trend line for these data?\n\n160 Data on demand over the last few years are available as follows: .\n\nWhat is this year's forecast using trend-adjusted (double) smoothing with alpha = .2 and beta = .1, if the forecast for last year was 56, the forecast for two years ago was 46, and the trend estimate for last year's forecast was 7?\n\n161 Data on the last three years of demand are available as follows: .\n\nWhat is the centered moving average for spring two years ago?\n\n162 Data on the last three years of demand are available as follows: .\n\nWhat is the spring's seasonal relative?\n\n163 Data on the last three years of demand are available as follows: .\n\nWhat is the linear regression trend line for these data (t = 0 for spring, three years ago)?\n\n164 Data on the last three years of demand are available as follows: .\n\nWhat is this year's seasonally adjusted forecast for each season?\n\nTrue / False Questions\n\n1.\n\nForecasting techniques generally assume an existing causal system that will continue to exist in the future. TRUE Forecasts depend on the rules of the game remaining reasonably constant.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 2 Medium Topic: Features Common to All Forecasts\n\n2.\n\nFor new products in a strong growth mode, a low alpha will minimize forecast errors when using exponential smoothing techniques. FALSE If growth is strong, alpha should be large so that the model will catch up more quickly.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n3.\n\nOnce accepted by managers, forecasts should be held firm regardless of new input since many plans have been made using the original forecast. FALSE Flexibility to accommodate major changes is important to good forecasting.\n\nBlooms: Understand Learning Objective: 03-04 Outline the steps in the forecasting process. Level of Difficulty: 1 Easy Topic: Steps in the Forecasting Process\n\n4.\n\nForecasts for groups of items tend to be less accurate than forecasts for individual items because forecasts for individual items don't include as many influencing factors. FALSE Forecasting for an individual item is more difficult than forecasting for a number of items.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 2 Medium Topic: Features Common to All Forecasts\n\n5.\n\nForecasts help managers both to plan the system itself and to provide valuable information for using the system. TRUE Both planning and use are shaped by forecasts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 1 Easy Topic: Forecasting\n\n6.\n\nOrganizations that are capable of responding quickly to changing requirements can use a shorter forecast horizon and therefore benefit from more accurate forecasts. TRUE If an organization can react more quickly, its forecasts need not be so long term.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-03 List the elements of a good forecast. Level of Difficulty: 2 Medium Topic: Elements of a Good Forecast\n\n7.\n\nWhen new products or services are introduced, focus forecasting models are an attractive option. FALSE Because focus forecasting models depend on historical data, they're not so attractive for newly introduced products or services.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n8.\n\nThe purpose of the forecast should be established first so that the level of detail, amount of resources, and accuracy level can be understood. TRUE All of these considerations are shaped by what the forecast will be used for.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 2 Medium Topic: Steps in the Forecasting Process\n\n9.\n\nForecasts based on time-series (historical) data are referred to as associative forecasts. FALSE Forecasts based on time-series data are referred to as time-series forecasts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 1 Easy Topic: Associative Forecasting Techniques\n\n10.\n\nTime-series techniques involve the identification of explanatory variables that can be used to predict future demand. FALSE Associative forecasts involve identifying explanatory variables.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n11.\n\nA consumer survey is an easy and sure way to obtain accurate input from future customers since most people enjoy participating in surveys. FALSE Most people do not enjoy participating in surveys.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n12.\n\nThe Delphi approach involves the use of a series of questionnaires to achieve a consensus forecast. TRUE A consensus among divergent perspectives is developed using questionnaires.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n13.\n\nExponential smoothing adds a percentage (called alpha) of the last period's forecast to estimate the next period's demand. FALSE Exponential smoothing adds a percentage to the last period's forecast error.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n14.\n\nThe shorter the forecast period, the more accurately the forecasts tend to track what actually happens. TRUE Long-term forecasting is much more difficult to do accurately.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 1 Easy Topic: Monitoring the Forecast Error\n\n15.\n\nForecasting techniques that are based on time-series data assume that future values of the series will duplicate past values. FALSE Time-series forecasts assume that future patterns in the series will mimic past patterns in the series.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n16.\n\nTrend-adjusted exponential smoothing uses double smoothing to add twice the forecast error to last period's actual demand. FALSE Trend-adjusted smoothing smoothes both random and trend-related variation.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n17.\n\nForecasts based on an average tend to exhibit less variability than the original data. TRUE Averaging is a way of smoothing out random variability.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n18.\n\nThe naive approach to forecasting requires a linear trend line. FALSE The naive approach is useful in a wider variety of settings.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n19.\n\nThe naive forecast is limited in its application to series that reflect no trend or seasonality. FALSE When a trend or seasonality is present, the naive forecast is more limited in its application.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n20.\n\nThe naive forecast can serve as a quick and easy standard of comparison against which to judge the cost and accuracy of other techniques. TRUE Often the naive forecast performs reasonably well when compared to more complex techniques.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n21.\n\nA moving average forecast tends to be more responsive to changes in the data series when more data points are included in the average. FALSE More data points reduce a moving average forecast's responsiveness.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n22.\n\nIn order to update a moving average forecast, the values of each data point in the average must be known. TRUE The moving average cannot be updated until the most recent value is known.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n23.\n\nForecasts of future demand are used by operations people to plan capacity. TRUE Capacity decisions are made for the future and therefore depend on forecasts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-04 Outline the steps in the forecasting process. Level of Difficulty: 1 Easy Topic: Forecasting\n\n24.\n\nAn advantage of a weighted moving average is that recent actual results can be given more importance than what occurred a while ago. TRUE Weighted moving averages can be adjusted to make more recent data more important in setting the forecast.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n25.\n\nExponential smoothing is a form of weighted averaging. TRUE The most recent period is given the most weight, but prior periods also factor in.\n\nAccessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n26.\n\nA smoothing constant of .1 will cause an exponential smoothing forecast to react more quickly to a sudden change than a smoothing constant value of .3. FALSE Smaller smoothing constants result in less reactive forecast models.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n27.\n\nThe T in the model TAF = S + T represents the time dimension (which is usually expressed in weeks or months). FALSE The T represents the trend dimension.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n28.\n\nTrend-adjusted exponential smoothing requires selection of two smoothing constants. TRUE One is for the trend and one is for the random error.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n29.\n\nAn advantage of trend-adjusted exponential smoothing over the linear trend equation is its ability to adjust over time to changes in the trend. TRUE A linear trend equation assumes a constant trend; trend-adjusted smoothing allows for changes in the underlying trend.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n30.\n\nA seasonal relative (or seasonal indexes) is expressed as a percentage of average or trend. TRUE Seasonal relatives are used when the seasonal effect is multiplicative rather than additive.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n31.\n\nIn order to compute seasonal relatives, the trend of past data must be computed or known, which means that for brand-new products this approach cannot be used. TRUE Computing seasonal relatives depends on past data being available.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n32.\n\nRemoving the seasonal component from a data series (deseasonalizing) can be accomplished by dividing each data point by its appropriate seasonal relative. TRUE Deseasonalized data points have been adjusted for seasonal influences.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n33.\n\nIf a pattern appears when a dependent variable is plotted against time, one should use time series analysis instead of regression analysis. TRUE Patterns reflect influences such as trends or seasonality that go against regression analysis assumptions.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 3 Hard Topic: Associative Forecasting Techniques\n\n34.\n\nCurvilinear and multiple regression procedures permit us to extend associative models to relationships that are nonlinear or involve more than one predictor variable. TRUE Regression analysis can be used in a variety of settings.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n35.\n\nThe sample standard deviation of forecast error is equal to the square root of MSE. TRUE The MSE is equal to the sample variance of the forecast error.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 1 Easy Topic: Forecast Accuracy\n\n36.\n\nCorrelation measures the strength and direction of a relationship between variables. TRUE The association between two variations is summarized in the correlation coefficient.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n37.\n\nMAD is equal to the square root of MSE, which is why we calculate the easier MSE and then calculate the more difficult MAD. FALSE MAD is the mean absolute deviation.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n38.\n\nIn exponential smoothing, an alpha of 1.0 will generate the same forecast that a naive forecast would yield. TRUE With alpha equal to 1 we are using a naive forecasting method.\n\nAccessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n39.\n\nA forecast method is generally deemed to perform adequately when the errors exhibit an identifiable pattern. FALSE Forecast methods are generally considered to be performing adequately when the errors appear to be randomly distributed.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n40.\n\nA control chart involves setting action limits for cumulative forecast error. FALSE Control charts set action limits for the tracking signal.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 2 Medium Topic: Monitoring the Forecast Error\n\n41.\n\nA tracking signal focuses on the ratio of cumulative forecast error to the corresponding value of MAD. TRUE Large absolute values of the tracking signal suggest a fundamental change in the forecast model's performance.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 2 Medium Topic: Monitoring the Forecast Error\n\n42.\n\nThe use of a control chart assumes that errors are normally distributed about a mean of zero. TRUE Over time, a forecast model's tracking signal should fluctuate randomly about a mean of zero.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 3 Hard Topic: Monitoring the Forecast Error\n\n43.\n\nBias exists when forecasts tend to be greater or less than the actual values of time series. TRUE A tendency in one direction is defined as bias.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 1 Easy Topic: Monitoring the Forecast Error\n\n44.\n\nBias is measured by the cumulative sum of forecast errors. TRUE Bias would result in the cumulative sum of forecast errors being large in absolute value.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Monitoring the Forecast Error\n\n45.\n\nSeasonal relatives can be used to deseasonalize data or incorporate seasonality in a forecast. TRUE Seasonal relatives are used to deseasonalize data to forecast future values of the underlying trend, and they are also used to reseasonalize deseasonalized forecasts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n46.\n\nThe best forecast is not necessarily the most accurate. TRUE More accuracy often comes at too high a cost to be worthwhile.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 2 Medium Topic: Elements of a Good Forecast\n\nMultiple Choice Questions\n\n47.\n\nWhich of the following is a potential shortcoming of using sales force opinions in demand forecasting?\n\nA. Members of the sales force often have substantial histories of working with and understanding their customers. B. Members of the sales force often are well aware of customers' future plans. C. Members of the sales force have direct contact with consumers. D. Members of the sales force can have difficulty distinguishing between what customers would like to do and what they actually will do. E. Customers often are quite open with members of the sales force with regard to future plans. Customers themselves may be unclear regarding what they'd like to do versus what they'll actually do.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n48.\n\nSuppose a four-period weighted average is being used to forecast demand. Weights for the periods are as follows: wt-4 = 0.1, wt-3 = 0.2, wt-2 = 0.3 and wt-1 = 0.4. Demand observed in the previous four periods was as follows: A t-4 = 380, At-3 = 410, At-2 = 390, At-1 = 400. What will be the demand forecast for period t?\n\nA. B. C. D. E.\n\n402 397 399 393 403\n\nThe forecast will be (.1 * 380) + (.2 * 410) + (.3 * 390) + (.4 * 400) = 397.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n49.\n\nSuppose a three-period weighted average is being used to forecast demand. Weights for the periods are as follows: wt-3 = 0.2, wt-2 = 0.3 and wt-1 = 0.5. Demand observed in the previous three periods was as follows: A t-3 = 2,200, At-2 = 1,950, At-1 = 2,050. What will be the demand forecast for period t?\n\nA. B. C. D. E.\n\n2,000 2,095 1,980 2,050 1,875\n\nThe forecast for will be (.2 * 2,200) + (.3 * 1,950) + (.5 * 2,050) = 2,050.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n50.\n\nWhen choosing a forecasting technique, a critical trade-off that must be considered is that between:\n\nA. B. C. D. E.\n\ntime series and associative. seasonality and cyclicality. length and duration. simplicity and complexity. cost and accuracy.\n\nThe trade-off between cost and accuracy is the critical consideration when choosing a forecasting technique.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 1 Easy Topic: Choosing a Forecasting Technique\n\n51.\n\nThe more novel a new product or service design is, the more forecasters have to rely on:\n\nA. B. C. D. E.\n\nsubjective estimates. seasonality. cyclicality. historical data. smoothed variation.\n\nNew products and services lack historical data, so forecasts for them must be based on subjective estimates.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 1 Easy Topic: Choosing a Forecasting Technique\n\n52.\n\nForecasts based on judgment and opinion do not include:\n\nA. B. C. D. E.\n\nexecutive opinion. salesperson opinion. second opinions. customer surveys. Delphi methods.\n\nSecond opinions generally refer to medical diagnoses, not demand forecasting.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 2 Medium Topic: Qualitative Forecasts\n\n53.\n\nWhich of the following is/are a primary input into capacity, sales, and production planning?\n\nA. B. C. D. E.\n\nproduct design market share ethics globalization demand forecasts\n\nDemand forecasts are direct inputs into capacity, sales, and production plans.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 1 Easy Topic: Features Common to All Forecasts\n\n54.\n\nWhich of the following features would not generally be considered common to all forecasts?\n\nA. Assumption of a stable underlying causal system. B. Actual results will differ somewhat from predicted values. C. Historical data is available on which to base the forecast. D. Forecasts for groups of items tend to be more accurate than forecasts for individual items. E. Accuracy decreases as the time horizon increases. In some forecasting situations historical data are not available.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 3 Hard Topic: Features Common to All Forecasts\n\n55.\n\nWhich of the following is not a step in the forecasting process?\n\nA. B. C. D. E.\n\nDetermine the purpose and level of detail required. Eliminate all assumptions. Establish a time horizon. Select a forecasting model. Monitor the forecast.\n\nWe cannot eliminate all assumptions.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-04 Outline the steps in the forecasting process. Level of Difficulty: 2 Medium Topic: Features Common to All Forecasts\n\n56.\n\nMinimizing the sum of the squared deviations around the line is called:\n\nA. B. C. D. E.\n\nmean squared error technique. mean absolute deviation. double smoothing. least squares estimation. predictor regression.\n\nLeast squares estimations minimize the sum of squared deviations around the estimated regression function.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n57.\n\nThe two general approaches to forecasting are:\n\nA. B. C. D. E.\n\nmathematical and statistical. qualitative and quantitative. judgmental and qualitative. historical and associative. precise and approximation.\n\nForecast approaches are either quantitative or qualitative.\n\nAccessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 1 Easy Topic: Approaches to Forecasting\n\n58.\n\nWhich of the following is not a type of judgmental forecasting?\n\nA. B. C. D. E.\n\nexecutive opinions sales force opinions consumer surveys the Delphi method time series analysis\n\nTime series analysis is a quantitative approach.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n59.\n\nAccuracy in forecasting can be measured by:\n\nA. B. C. D. E.\n\nMSE. MRP. MPS. MTM. MTE.\n\nMSE is mean squared error.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 3 Hard Topic: Forecast Accuracy\n\n60.\n\nWhich of the following would be an advantage of using a sales force composite to develop a demand forecast?\n\nA. The sales staff is least affected by changing customer needs. B. The sales force can easily distinguish between customer desires and probable actions. C. The sales staff is often aware of customers' future plans. D. Salespeople are least likely to be influenced by recent events. E. Salespeople are least likely to be biased by sales quotas. Members of the sales force should be the organization's tightest link with its customers.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 3 Hard Topic: Qualitative Forecasts\n\n61.\n\nWhich phrase most closely describes the Delphi technique?\n\nA. B. C. D. E.\n\nassociative forecast consumer survey series of questionnaires developed in India historical data\n\nThe questionnaires are a way of fostering a consensus among divergent perspectives.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n62.\n\nThe forecasting method which uses anonymous questionnaires to achieve a consensus forecast is:\n\nA. B. C. D. E.\n\nsales force opinions. consumer surveys. the Delphi method. time series analysis. executive opinions.\n\nAnonymity is important in Delphi efforts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 1 Easy Topic: Qualitative Forecasts\n\n63.\n\nOne reason for using the Delphi method in forecasting is to:\n\nA. B. C. D. E.\n\navoid premature consensus (bandwagon effect). achieve a high degree of accuracy. maintain accountability and responsibility. be able to replicate results. prevent hurt feelings.\n\nA bandwagon can lead to popular but potentially inaccurate viewpoints to drown out other important considerations.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 2 Medium Topic: Qualitative Forecasts\n\n64.\n\nDetecting nonrandomness in errors can be done using:\n\nA. B. C. D. E.\n\nMSEs. MAPs. control charts. correlation coefficients. strategies.\n\nControl charts graphically depict the statistical behavior of forecast errors.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 2 Medium Topic: Approaches to Forecasting\n\n65.\n\nGradual, long-term movement in time series data is called:\n\nA. B. C. D. E.\n\nseasonal variation. cycles. irregular variation. trend. random variation.\n\nTrends move the time series in a long-term direction.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n66.\n\nThe primary difference between seasonality and cycles is:\n\nA. B. C. D. E.\n\nthe duration of the repeating patterns. the magnitude of the variation. the ability to attribute the pattern to a cause. the direction of the movement. there are only four seasons but 30 cycles.\n\nSeasons happen within time periods; cycles happen across multiple time periods.\n\nBlooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n67.\n\nAveraging techniques are useful for:\n\nA. B. C. D. E.\n\ndistinguishing between random and nonrandom variations. smoothing out fluctuations in time series. eliminating historical data. providing accuracy in forecasts. average people.\n\nSmoothing helps forecasters see past random error.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n68.\n\nPutting forecast errors into perspective is best done using\n\nA. B. C. D. E.\n\nexponential smoothing. MAPE. linear decision rules. MAD. hindsight.\n\nMAPE depicts the forecast error relative to what was being forecast.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Monitoring the Forecast Error\n\n69.\n\nUsing the latest observation in a sequence of data to forecast the next period is:\n\nA. B. C. D. E.\n\na moving average forecast. a naive forecast. an exponentially smoothed forecast. an associative forecast. regression analysis.\n\nOnly one piece of information is needed for a naive forecast.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n70.\n\nFor the data given below, what would the naive forecast be for period 5?\n\nA. B. C. D. E.\n\n58 62 59.5 61 cannot tell from the data given\n\nPeriod 5's forecast would be period 4's demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n71.\n\nMoving average forecasting techniques do the following:\n\nA. B. C. D. E.\n\nImmediately reflect changing patterns in the data. Lead changes in the data. Smooth variations in the data. Operate independently of recent data. Assist when organizations are relocating.\n\nVariation is smoothed out in moving average forecasts.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n72.\n\nWhich is not a characteristic of simple moving averages applied to time series data?\n\nA. B. C. D. E.\n\nsmoothes random variations in the data weights each historical value equally lags changes in the data requires only last period's forecast and actual data smoothes real variations in the data\n\nSimple moving averages can require several periods of data.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n73.\n\nIn order to increase the responsiveness of a forecast made using the moving average technique, the number of data points in the average should be:\n\nA. B. C. D. E.\n\ndecreased. increased. multiplied by a larger alpha. multiplied by a smaller alpha. eliminated if the MAD is greater than the MSE.\n\nFewer data points result in more responsive moving averages.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n74.\n\nA forecast based on the previous forecast plus a percentage of the forecast error is:\n\nA. B. C. D. E.\n\na naive forecast. a simple moving average forecast. a centered moving average forecast. an exponentially smoothed forecast. an associative forecast.\n\nExponential smoothing uses the previous forecast error to shape the next forecast.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n75.\n\nWhich is not a characteristic of exponential smoothing?\n\nA. B. C. D. E.\n\nsmoothes random variations in the data weights each historical value equally has an easily altered weighting scheme has minimal data storage requirements smoothes real variations in the data\n\nThe most recent period of demand is given the most weight in exponential smoothing.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n76.\n\nWhich of the following smoothing constants would make an exponential smoothing forecast equivalent to a naive forecast?\n\nA. B. C. D. E.\n\n0 .01 .1 .5 1.0\n\nAn alpha of 1.0 leads to a naive forecast.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n77.\n\nSimple exponential smoothing is being used to forecast demand. The previous forecast of 66 turned out to be four units less than actual demand. The next forecast is 66.6, implying a smoothing constant, alpha, equal to:\n\nA. B. C. D. E.\n\n.01. .10. .15. .20. .60.\n\nA previous period's forecast error of 4 units would lead to a change in the forecast of 0.6 if alpha equals 0.15.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n78.\n\nGiven an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the forecast for the next period be using simple exponential smoothing?\n\nA. B. C. D. E.\n\n36.9 57.5 60.5 62.5 65.5\n\nMultiply the previous period's forecast error (-5) by alpha and then add to the previous period's forecast.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n79.\n\nGiven an actual demand of 105, a forecasted value of 97, and an alpha of .4, the simple exponential smoothing forecast for the next period would be:\n\nA. B. C. D. E.\n\n80.8. 93.8. 100.2. 101.8. 108.2.\n\nMultiply the previous period's forecast error (8) by alpha and then add to the previous period's forecast.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n80.\n\nWhich of the following possible values of alpha would cause exponential smoothing to respond the most quickly to forecast errors?\n\nA. B. C. D. E.\n\n0 .01 .05 .10 .15\n\nLarger values for alpha correspond with greater responsiveness.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n81.\n\nA manager uses the following equation to predict monthly receipts: Y t = 40,000 + 150t. What is the forecast for July if t = 0 in April of this year?\n\nA. B. C. D. E.\n\n40,450 40,600 42,100 42,250 42,400\n\nJuly would be period 3, so the forecast would be 40,000 + 150(3).\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n82.\n\nA. an exponentially smoothed forecast and a smoothed trend factor. B. an exponentially smoothed forecast and an estimated trend value. C. the old forecast adjusted by a trend factor. D. the old forecast and a smoothed trend factor. E. a moving average and a trend factor. Both random variation and the trend are smoothed in TAF models.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n83.\n\nIn the additive model for seasonality, seasonality is expressed as a ______________ adjustment to the average; in the multiplicative model, seasonality is expressed as a __________ adjustment to the average.\n\nA. B. C. D. E.\n\nquantity; percentage percentage; quantity quantity; quantity percentage; percentage qualitative; quantitative\n\nThe additive model simply adds a seasonal adjustment to the deseasonalized forecast. The multiplicative model adjusts the deseasonalized forecast by multiplying it by a season relative or index.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n84.\n\nWhich technique is used in computing seasonal relatives?\n\nA. B. C. D. E.\n\ndouble smoothing Delphi mean squared error centered moving average exponential smoothing\n\nThe centered moving average serves as the basis point for computing seasonal relatives.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n85.\n\nA persistent tendency for forecasts to be greater than or less than the actual values is called:\n\nA. B. C. D. E.\n\nbias. tracking. control charting. positive correlation. linear regression.\n\nBias is a tendency for a forecast to be above (or below) the actual value.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n86.\n\nWhich of the following might be used to indicate the cyclical component of a forecast?\n\nA. B. C. D. E.\n\nleading variable mean squared error Delphi technique exponential smoothing mean absolute deviation\n\nLeading variables, such as births in a given year, can correlate strongly with longterm phenomena such as cycles.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n87.\n\nThe primary method for associative forecasting is:\n\nA. B. C. D. E.\n\nsensitivity analysis. regression analysis. simple moving averages. centered moving averages. exponential smoothing.\n\nRegression analysis is an associative forecasting technique.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n88.\n\nWhich term most closely relates to associative forecasting techniques?\n\nA. B. C. D. E.\n\ntime series data expert opinions Delphi technique consumer survey predictor variables\n\nAssociative techniques use predictor variables.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n89.\n\nWhich of the following corresponds to the predictor variable in simple linear regression?\n\nA. B. C. D. E.\n\nregression coefficient dependent variable independent variable predicted variable demand coefficient\n\nDemand is the typical dependent variable when forecasting with simple linear regression.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n90.\n\nThe mean absolute deviation is used to:\n\nA. B. C. D. E.\n\nestimate the trend line. eliminate forecast errors. measure forecast accuracy. seasonally adjust the forecast. compute periodic forecast errors.\n\nMAD is one way of evaluating forecast performance.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n91.\n\nGiven forecast errors of 4, 8, and -3, what is the mean absolute deviation?\n\nA. B. C. D. E.\n\n4 3 5 6 12\n\nConvert each error into an absolute value and then average.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 1 Easy Topic: Forecast Accuracy\n\n92.\n\nGiven forecast errors of 5, 0, -4, and 3, what is the mean absolute deviation?\n\nA. B. C. D. E.\n\n4 3 2.5 2 1\n\nConvert each error into an absolute value and then average.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 1 Easy Topic: Forecast Accuracy\n\n93.\n\nGiven forecast errors of 5, 0, -4, and 3, what is the bias?\n\nA. B. C. D. E.\n\n-4 4 5 12 6\n\nSum the forecast errors.\n\nBlooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n94.\n\nWhich of the following is used for constructing a control chart?\n\nA. B. C. D.\n\nmean absolute deviation mean squared error tracking signal bias\n\nThe mean squared error leads to an estimate for the sample forecast standard deviation.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 2 Medium Topic: Approaches to Forecasting\n\n95.\n\nThe two most important factors in choosing a forecasting technique are:\n\nA. B. C. D. E.\n\ncost and time horizon. accuracy and time horizon. cost and accuracy. quantity and quality. objective and subjective components.\n\nMore accurate forecasts cost more but may not be worth the additional cost.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 2 Medium Topic: Choosing a Forecasting Technique\n\n96.\n\nThe degree of management involvement in short-range forecasts is:\n\nA. B. C. D. E.\n\nnone. low. moderate. high. total.\n\nShort-range forecasting tends to be fairly routine.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 1 Easy Topic: Choosing a Forecasting Technique\n\n97.\n\nWhich of the following is not necessarily an element of a good forecast?\n\nA. B. C. D. E.\n\nestimate of accuracy timeliness meaningful units low cost written\n\nA good forecast can be quite costly if necessary.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-03 List the elements of a good forecast. Level of Difficulty: 2 Medium Topic: Elements of a Good Forecast\n\n98.\n\nForecasting techniques generally assume:\n\nA. the absence of randomness. B. continuity of some underlying causal system. C. a linear relationship between time and demand. D. accuracy that increases the farther out in time the forecast projects. E. accuracy that is better when individual items, rather than groups of items, are being considered. Forecasting techniques generally assume that the same underlying causal system that existed in the past will continue to exist in the future.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-01 List features common to all forecasts. Level of Difficulty: 1 Easy Topic: Forecast Accuracy\n\n99.\n\nA managerial approach toward forecasting which seeks to actively influence demand is:\n\nA. B. C. D. E.\n\nreactive. proactive. influential. protracted. retroactive.\n\nSimply responding to demand is a reactive approach.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 1 Easy Topic: Using Forecast Information\n\n100. Customer service levels can be improved by better:\n\nA. B. C. D. E.\n\nmission statements. control charting. short-term forecast accuracy. exponential smoothing. customer selection.\n\nMore accurate short-term forecasts enable organizations to better accommodate customer requests.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-03 List the elements of a good forecast. Level of Difficulty: 3 Hard Topic: Operations Strategy\n\n101. Given the following historical data, what is the simple three-period moving average forecast for period 6?\n\nA. B. C. D. E.\n\n67 115 69 68 68.67\n\nAverage demand from periods 3 through 5.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n102. Given the following historical data and weights of .5, .3, and .2, what is the threeperiod moving average forecast for period 5?\n\nA. B. C. D. E.\n\n144.20 144.80 144.67 143.00 144.00\n\nMultiply period 4 (144) by .5, period 3 (148) by .3, and period 2 (142) by .2, then sum these products.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n103. Use of simple linear regression analysis assumes that:\n\nA. variations around the line are nonrandom. B. deviations around the line are normally distributed. C. predictions can easily be made beyond the range of observed values of the predictor variable. D. all possible predictor variables are included in the model. E. the variance of error terms (deviations) varies directly with the predictor variable. That deviations conform to the normal distribution is a very important assumption underpinning simple linear regression.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 3 Hard Topic: Associative Forecasting Techniques\n\n104. Given forecast errors of -5, -10, and +15, the MAD is:\n\nA. B. C. D. E.\n\n0. 10. 30. 175. 225.\n\nConvert these errors into absolute value, then average.\n\nAACSB: Analytic Accessibility: Keyboard Navigation Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 1 Easy Topic: Forecast Accuracy\n\n105. The president of State University wants to forecast student enrollments for this academic year based on the following historical data:\n\nWhat is the forecast for this year using the naive approach?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,000 22,800\n\nThis year's forecast would be last year's enrollment.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n106. The president of State University wants to forecast student enrollments for this academic year based on the following historical data:\n\nWhat is the forecast for this year using a four-year simple moving average?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\nAverage enrollment from the last four years.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n107. The president of State University wants to forecast student enrollments for this academic year based on the following historical data:\n\nWhat is the forecast for this year using exponential smoothing with alpha = .5, if the forecast for two years ago was 16,000?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\nMultiply last year's forecast error by the smoothing constant, then add that adjusted error to last year's forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n108. The president of State University wants to forecast student enrollments for this academic year based on the following historical data:\n\nWhat is the forecast for this year using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\nTreat 5 years ago as period 0.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n109. The president of State University wants to forecast student enrollments for this academic year based on the following historical data:\n\nWhat is the forecast for this year using trend-adjusted (double) smoothing with alpha = .05 and beta = .3, if the forecast for last year was 21,000, the forecast for two years ago was 19,000, and the trend estimate for last year's forecast was 1,500?\n\nA. B. C. D. E.\n\n18,750 19,500 21,000 22,650 22,800\n\nSmooth both the trend and the forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n110. The business analyst for Video Sales, Inc. wants to forecast this year's demand for DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using the naive approach?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\nThis year's forecast is last year's demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n111. The business analyst for Video Sales, Inc. wants to forecast this year's demand for DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using a three-year weighted moving average with weights of .5, .3, and .2?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\nMultiply the last three periods of demand by the appropriate weights, then sum the resulting products.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n112. The business analyst for Video Sales, Inc. wants to forecast this year's demand for DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using exponential smoothing with alpha = .4, if the forecast for two years ago was 750?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\nFirst formulate last year's exponentially smoothed forecast, then proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n113. The business analyst for Video Sales, Inc. wants to forecast this year's demand for DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n163 180 300 420 510\n\nTreat the earliest period of demand as period 0, then formulate least squares estimates and proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n114. The business analyst for Video Sales, Inc. wants to forecast this year's demand for DVD decoders based on the following historical data:\n\nWhat is the forecast for this year using trend-adjusted (double) smoothing with alpha = .3 and beta = .2, if the forecast for last year was 310, the forecast for two years ago was 430, and the trend estimate for last year's forecast was -150?\n\nA. B. C. D. E.\n\n162.4 180.3 301.4 403.2 510.0\n\nSmooth both the trend and the forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-12 Prepare a trend-adjusted exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n115. Professor Very Busy needs to allocate time next week to include time for office hours. He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using the naive approach?\n\nA. B. C. D. E.\n\n45 50 52 65 78\n\nThis week's forecast is last week's demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n116. Professor Very Busy needs to allocate time next week to include time for office hours. He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using a three-week simple moving average?\n\nA. B. C. D. E.\n\n49 50 52 65 78\n\nAverage the three most recent weeks of demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n117. Professor Very Busy needs to allocate time next week to include time for office hours. He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using exponential smoothing with alpha = .2, if the forecast for two weeks ago was 90?\n\nA. B. C. D. E.\n\n49 50 52 65 77\n\nFormulate the forecast for last week, then use that to get this week's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n118. Professor Very Busy needs to allocate time next week to include time for office hours. He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n49 50 52 65 78\n\nTreat the earliest period as period 0, then formulate least squares coefficients and proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n119. Professor Very Busy needs to allocate time next week to include time for office hours. He needs to forecast the number of students who will seek appointments. He has gathered the following data:\n\nWhat is this week's forecast using trend-adjusted (double) smoothing with alpha = . 5 and beta = .1, if the forecast for last week was 65, the forecast for two weeks ago was 75, and the trend estimate for last week's forecast was -5?\n\nA. B. C. D. E.\n\n49.3 50.6 51.3 65.4 78.7\n\nSmooth both the trend and the forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-12 Prepare a trend-adjusted exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n120. A concert promoter is forecasting this year's attendance for one of his concerts based on the following historical data:\n\nWhat is this year's forecast using the naive approach?\n\nA. B. C. D. E.\n\n22,000 20,000 18,000 15,000 12,000\n\nThis year's forecast is last year's attendance.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n121. A concert promoter is forecasting this year's attendance for one of his concerts based on the following historical data:\n\nWhat is this year's forecast using a two-year weighted moving average with weights of .7 and .3?\n\nA. B. C. D. E.\n\n19,400 18,600 19,000 11,400 10,600\n\nMultiply the two most recent periods by the appropriate weights, then sum the resulting products.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n122. A concert promoter is forecasting this year's attendance for one of his concerts based on the following historical data:\n\nWhat is this year's forecast using exponential smoothing with alpha = .2, if last year's smoothed forecast was 15,000?\n\nA. B. C. D. E.\n\n20,000 19,000 17,500 16,000 15,000\n\nMultiply last year's forecast error by the smoothing constant, then add that product to last year's forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n123. A concert promoter is forecasting this year's attendance for one of his concerts based on the following historical data:\n\nWhat is this year's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n20,000 21,000 22,000 23,000 24,000\n\nTreat the earliest year as period zero in formulating least squares coefficients.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n124. A concert promoter is forecasting this year's attendance for one of his concerts based on the following historical data:\n\nThe previous trend line had predicted 18,500 for two years ago, and 19,700 for last year. What was the mean absolute deviation for these forecasts?\n\nA. B. C. D. E.\n\n100 200 400 500 800\n\nConvert each period's forecast error into absolute value, then average.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecast Accuracy\n\n125. The dean of a school of business is forecasting total student enrollment for this year's summer session classes based on the following historical data:\n\nWhat is this year's forecast using the naive approach?\n\nA. B. C. D. E.\n\n2,000 2,200 2,800 3,000 4,300\n\nThis year's forecast would be last year's enrollment.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n126. The dean of a school of business is forecasting total student enrollment for this year's summer session classes based on the following historical data:\n\nWhat is this year's forecast using a three-year simple moving average?\n\nA. B. C. D. E.\n\n2,667 2,600 2,500 2,400 2,333\n\nAverage the most recent periods of enrollment.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n127. The dean of a school of business is forecasting total student enrollment for this year's summer session classes based on the following historical data:\n\nWhat is this year's forecast using exponential smoothing with alpha = .4, if last year's smoothed forecast was 2,600?\n\nA. B. C. D. E.\n\n2,600 2,760 2,800 3,840 3,000\n\nMultiply last year's forecast error by the smoothing constant. Add the product to last year's forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n128. The dean of a school of business is forecasting total student enrollment for this year's summer session classes based on the following historical data:\n\nWhat is the annual rate of change (slope) of the least squares trend line for these data?\n\nA. B. C. D. E.\n\n0 200 400 180 360\n\nTreat the earliest period as period 0, then formulate the least squares slope.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n129. The dean of a school of business is forecasting total student enrollment for this year's summer session classes based on the following historical data:\n\nWhat is this year's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n3,600 3,500 3,400 3,300 3,200\n\nTreat the earliest period as period 0, then formulate the least squares coefficients and proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n130. The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using the naive approach?\n\nA. B. C. D. E.\n\n100 160 130 140 120\n\nThis month's forecast is last month's demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n131. The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using a four-month weighted moving average with weights of .4, .3, .2, and .1?\n\nA. B. C. D. E.\n\n120 129 141 135 140\n\nMultiply the four most recent periods of demand by the appropriate weights, then sum the resulting products.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n132. The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using exponential smoothing with alpha = .2, if August's forecast was 145?\n\nA. B. C. D. E.\n\n144 140 142 148 163\n\nFirst calculate September's forecast, then use that to calculate this month's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n133. The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is the monthly rate of change (slope) of the least squares trend line for these data?\n\nA. B. C. D. E.\n\n320 102 8 -.4 -8\n\nTreat the earliest period as period 0, then formulate the least squares slope.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n134. The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data:\n\nWhat is this month's forecast using the least squares trend line for these data?\n\nA. B. C. D. E.\n\n1,250 128.6 102 158 164\n\nTreat the earliest period as period 0, then formulate the least squares coefficients and proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n135. Which of the following mechanisms for enhancing profitability is most likely to result from improving short-term forecast performance?\n\nA. B. C. D. E.\n\nincreased inventory reduced flexibility higher-quality products greater customer satisfaction greater seasonality\n\nShort-term forecast performance won't necessarily improve product quality, but it does allow firms to better satisfy their customers.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting 3-121 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.\n\ntechnique. Level of Difficulty: 2 Medium Topic: Operations Strategy\n\n136. Which of the following changes would tend to shorten the time frame for short-term forecasting?\n\nA. B. C. D. E.\n\nbringing greater variety into the product mix increasing the flexibility of the production system ordering fewer weather-sensitive items adding more special-purpose equipment investing in the production system to make it more task-specific\n\nAn increasingly flexible system permits more rapid responses to changing conditions, which allows firms to reduce their forecast time horizon.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-16 Describe the key factors and trade-offs to consider when choosing a forecasting technique. Level of Difficulty: 2 Medium Topic: Operations Strategy\n\n137. Which of the following helps improve supply chain forecasting performance?\n\nA. contracts that require supply chain members to formulate long-term forecasts B. penalties for supply chain members that adjust forecasts C. sharing forecasts or demand data across the supply chain D. increasing lead times for critical supply chain members E. increasing the number of suppliers at critical junctures in the supply chain Sharing forecasts and/or demand data is a means of ensuring that the supply chain's overall forecast is as accurate as it can be.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-03 List the elements of a good forecast. Level of Difficulty: 1 Easy Topic: Forecasting and the Supply Chain\n\n138. Which of the following would tend to decrease forecast accuracy?\n\nA. a reduction in demand variability B. a shortening of the forecast time horizon C. an attempt to forecast demand for a group of similar items rather than an individual item D. a change in the underlying causal system Forecasting techniques generally assume that the same underlying causal system that existed in the past will continue to exist in the future.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-02 Explain why forecasts are generally wrong. Level of Difficulty: 2 Medium Topic: Forecasting and the Supply Chain\n\n139. Which of the following is the most valuable piece of information the sales force can bring into forecasting situations?\n\nA. what customers are most likely to do in the future B. what customers most want to do in the future C. what customers' future plans are D. whether customers are satisfied or dissatisfied with their performance in the past E. what the salesperson's appropriate sales quota should be Knowledge about what customers are likely to do is much more valuable than information regarding what customers plan or want to do.\n\nAACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Learning Objective: 03-06 Describe four qualitative forecasting techniques. Level of Difficulty: 2 Medium Topic: Qualitative Forecasts\n\nEssay Questions\n\n140. Develop a forecast for the next period, given the data below, using a three-period moving average.\n\nFeedback: Average demand from periods 3 through 5.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n141. Consider the data below:\n\nUsing exponential smoothing with alpha = .2, and assuming the forecast for period 11 was 80, what would the forecast for period 14 be?\n\nFeedback: The forecast error in period 13 (2.84) is multiplied by the smoothing constant. This is then added to the period 13 forecast to get the period 14 forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n142. A manager is using exponential smoothing to predict merchandise returns at a suburban branch of a department store chain. Given a previous forecast of 140 items, an actual number of returns of 148 items, and a smoothing constant equal to .15, what is the forecast for the next period?\n\nFeedback: The forecast error in the previous period is multiplied by the smoothing constant. This is then added to the previous period's forecast to get the upcoming period's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data 3-125 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.\n\n143. A manager is using the equation below to forecast quarterly demand for a product: Yt = 6,000 + 80t where t = 0 at Q2 of last year Quarter relatives are Q1 = .6, Q2 = .9, Q3 = 1.3, and Q4 = 1.2. What forecasts are appropriate for the last quarter of this year and the first quarter of next year?\n\nFor Q4 of this year t = 6 For Q1 of next year t = 7\n\nFeedback: Adjust deseasonalized forecasts by the quarterly seasonal relatives.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n144. Over the past five years, a firm's sales have averaged 250 units in the first quarter of each year, 100 units in the second quarter, 150 units in the third quarter, and 300 units in the fourth quarter. What are appropriate quarter relatives for this firm's sales? Hint: Only minimal computations are necessary.\n\nFeedback: Since a trend is not present, quarter relatives are simply a percentage of average, which is 200 units.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n145. A manager has been using a certain technique to forecast demand for gallons of ice cream for the past six periods. Actual and predicted amounts are shown below. Would a naive forecast have produced better results?\n\nCurrent method: MAD = 3.67; MSE = 16.8; 2s control limits ± 8.2 (OK) Naive method: MAD = 4.40; MSE = 30.0; 2s control limits ± 11.0 (OK) Feedback: Either MSE or MAD should be computed for both forecasts and compared. The demand data are stable. Therefore, the most recent value of the series is a reasonable forecast for the next period of time, justifying the naive approach. The current method is slightly superior both in terms of MAD and MSE. Either method would be considered in control.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 3 Hard Topic: Forecast Accuracy\n\n146. A new car dealer has been using exponential smoothing with an alpha of .2 to forecast weekly new car sales. Given the data below, would a naive forecast have provided greater accuracy? Explain. Assume an initial exponential forecast of 60 units in period 2 (i.e., no forecast for period 1).\n\nExponential method: MAD = 1.70; MSE = 6.34 Naive method: MAD = 3.00; MSE = 15.25 Feedback: The exponential forecast method appears to be superior because both MAD and MSE are lower when it is used.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 3 Hard Topic: Forecast Accuracy\n\n147. A CPA firm has been using the following equation to predict annual demand for tax audits: Yt = 55 + 4t. Demand for the past few years is shown below. Is the forecast performing as well as it might? Explain.\n\nMSE = 11/6 and s = = 3.41. Even with ± 2s limits (6.82), all values are within the limits. It seems, then, that only random variation is present, so one might say that the forecast is working. One might also observe that the first three errors are negative and the last three are positive. Although six observations constitute a relatively small sample, it may be that the errors are cycling, and this would be a matter to investigate with additional data. Feedback: Either a tracking signal or a control chart is called for. To conduct these assessments, it is necessary to generate the forecasts so that errors can be determined.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n148. Given the data below, develop a forecast for period 6 using a four-period weighted moving average and weights of .4, .3, .2 and .1.\n\n.4(17) + .3(19) + .2(18) + .1(20) = 18.1 Feedback: Multiply demand observed in periods 2 through 5 by the appropriate weight, then sum these products.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-09 Prepare a weighted-average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n149. Use linear regression to develop a predictive model for demand for burial vaults based on sales of caskets.\n\nFeedback: Least-squares estimation leads to this regression equation.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 3 Hard Topic: Associative Forecasting Techniques\n\n150. Given the following data, develop a linear regression model for y as a function of x.\n\nFeedback: Least squares estimation leads to this regression equation.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 2 Medium Topic: Associative Forecasting Techniques\n\n151. Given the following data, develop a linear regression model for y as a function of x.\n\nFeedback: Least squares estimation leads to this regression equation.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 3 Hard Topic: Associative Forecasting Techniques\n\n152. Develop a linear trend equation for the data on bread deliveries shown below. Forecast deliveries for period 11 through 14.\n\nYt = 518.2 + 52.164t r = +.935\n\nFeedback: Formulate the regression equation using least squares estimation, then apply the result to periods 11 through 14.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-14 Compute and use regression and correlation coefficients. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n153. Demand for the last four months was:\n\nA) Predict demand for July using each of these methods: 1) a three-period moving average 2) exponential smoothing with alpha equal to .20 (use a naive forecast for April for your first forecast) B) If the naive approach had been used to predict demand for April through June, what would MAD have been for those months?\n\nFeedback: The naive approach leads to absolute forecast errors of two units in each period.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n154. A manager wants to choose one of two forecasting alternatives. Each alternative was tested using historical data. The resulting forecast errors for the two are shown in the table. Analyze the data and recommend a course of action to the manager.\n\nAlthough Alternative 1 has the smaller MSE, it appears to be cycling and steady; Alternative 2 errors after the first three periods are small or zero. For the last six periods, Alternative 2 was much better, suggesting that approach would be better. Feedback: Although Alternative 1 has the smaller MSE, it appears to be cycling and steady; Alternative 2 errors after the first three periods are small or zero. For the last six periods, Alternative 2 was much better, suggesting that approach would be better.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-05 Summarize forecast errors and use summaries to make decisions. Level of Difficulty: 3 Hard Topic: Forecast Accuracy\n\n155. A manager uses this equation to predict demand: Yt = 20 + 4t. Over the past eight periods, demand has been as follows. Are the results acceptable? Explain.\n\ns = 2.10; 2s control limits are ± 4.20. Although all values are within control limits, the errors may be exhibiting cyclical patterns, which would suggest nonrandomness. Feedback: s = 2.10; 2s control limits are ± 4.20. Although all values are within control limits, the errors may be exhibiting cyclical patterns, which would suggest nonrandomness.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-15 Construct control charts and use them to monitor forecast errors. Level of Difficulty: 2 Medium Topic: Approaches to Forecasting\n\n156. Data on demand over the last few years are available as follows:\n\nWhat would this year's forecast be if we were using the naive approach?\n\n49 Feedback: This year's forecast would be last year's demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-07 Use a naive method to make a forecast. Level of Difficulty: 1 Easy Topic: Forecasts Based on Time-Series Data\n\n157. Data on demand over the last few years are available as follows:\n\nWhat is this year's forecast using a four-year simple moving average?\n\n45.5 Feedback: Average the four most recent periods of demand.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-08 Prepare a moving average forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n158. Data on demand over the last few years are available as follows:\n\nWhat is this year's forecast using exponential smoothing with alpha = .25, if last year's smoothed forecast was 45?\n\n45.8 Feedback: Multiply last year's forecast error by the smoothing constant. Add the resulting product to last year's forecast to get this year's forecast.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-10 Prepare an exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n159. Data on demand over the last few years are available as follows:\n\nWhat are this and next year's forecasts using the least squares trend line for these data?\n\n62; 69 Feedback: Treat the earliest period as period 0 in formulating least squares coefficients, then proceed.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-11 Prepare a linear trend forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n160. Data on demand over the last few years are available as follows:\n\nWhat is this year's forecast using trend-adjusted (double) smoothing with alpha = . 2 and beta = .1, if the forecast for last year was 56, the forecast for two years ago was 46, and the trend estimate for last year's forecast was 7?\n\n61.76 Feedback: Smooth both the trend and the forecasts using the appropriate smoothing coefficients.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-12 Prepare a trend-adjusted exponential smoothing forecast. Level of Difficulty: 3 Hard Topic: Forecasts Based on Time-Series Data\n\n161. Data on the last three years of demand are available as follows:\n\nWhat is the centered moving average for spring two years ago?\n\n29 Feedback: First average the four periods beginning fall three years ago. Then average the four periods beginning spring two years ago. Then average these two averages.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n162. Data on the last three years of demand are available as follows:\n\nWhat is the spring's seasonal relative?\n\nSpring = 0.91 Feedback: Divide data points by centered moving averages where moving averages are available. Average the resulting values across the seasons to get the seasonal relatives.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data 3-143 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.\n\n163. Data on the last three years of demand are available as follows:\n\nWhat is the linear regression trend line for these data (t = 0 for spring, three years ago)?\n\ny = 17 + 2.33t Feedback: Used deseasonalized data points to formulate least squares coefficients.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-12 Prepare a trend-adjusted exponential smoothing forecast. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data\n\n164. Data on the last three years of demand are available as follows:\n\nWhat is this year's seasonally adjusted forecast for each season?\n\nSpring = 40.93; Summer = 29.81; Fall = 51.14; Winter = 74.37 Feedback: First forecast each period's deseasonalized value (e.g., Spring is period 12). Then multiply the deseasonalized forecast by the appropriate seasonal relative.\n\nAACSB: Analytic Blooms: Apply Learning Objective: 03-13 Compute and use seasonal relatives. Level of Difficulty: 2 Medium Topic: Forecasts Based on Time-Series Data 3-144 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87023336,"math_prob":0.8077189,"size":140758,"snap":"2022-40-2023-06","text_gpt3_token_len":31240,"char_repetition_ratio":0.20815039,"word_repetition_ratio":0.7573635,"special_character_ratio":0.2314824,"punctuation_ratio":0.17068101,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.9623595,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-03T11:14:17Z\",\"WARC-Record-ID\":\"<urn:uuid:4848fc77-6141-465c-a2bd-d16542663b1b>\",\"Content-Length\":\"180509\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f52171bb-a9f8-4720-83dc-85ee571f2833>\",\"WARC-Concurrent-To\":\"<urn:uuid:d514f0a4-082b-4390-b0cf-e27b46331e92>\",\"WARC-IP-Address\":\"104.21.11.145\",\"WARC-Target-URI\":\"https://kupdf.net/download/acct-4400-ch3_5966242fdc0d60686fa88e76_pdf\",\"WARC-Payload-Digest\":\"sha1:7POKA3YETMP6LPKO3NR7JCXO7L5TX2WV\",\"WARC-Block-Digest\":\"sha1:HYYAE4V6XWFJ4INQZZA2QAQVCCGDZFVB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500044.66_warc_CC-MAIN-20230203091020-20230203121020-00018.warc.gz\"}"}
https://socratic.org/questions/55819b21581e2a2f4054822a	[ "Question #4822a\n\nJun 18, 2015\n\nBoth $F e C {l}_{2}$ and $N {a}_{2} S {O}_{3}$ can act as reducing agents\n\nExplanation:\n\nOxidation is loss of electrons. Reduction is gain of electrons.\n\nAn oxidising agent takes in electrons. A reducing agent gives them out.\n\nBoth ozone and chlorine are powerful oxidising agents.\n\nIron(II) choride can lose electrons to form iron(III):\n\n$F {e}^{2 +} \\rightarrow F {e}^{3 +} + e$\n\nA solution of iron(II) chloride will slowly oxidise to iron(III) in the presence of air.\n\nSodium sulfite is also a reducing agent:\n\n$S {O}_{3}^{2 -} + {H}_{2} O \\rightarrow S {O}_{4}^{2 -} + 2 {H}^{+} + 2 e$\n\nIt is used as an anti - oxidant in food preservation." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.833681,"math_prob":0.9881192,"size":620,"snap":"2019-43-2019-47","text_gpt3_token_len":149,"char_repetition_ratio":0.13149351,"word_repetition_ratio":0.0,"special_character_ratio":0.21935484,"punctuation_ratio":0.10344828,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9738448,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-20T04:19:11Z\",\"WARC-Record-ID\":\"<urn:uuid:c39d35f1-465c-434a-8b23-6d1e0ed3ddc9>\",\"Content-Length\":\"33728\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c5e728b2-b15c-4f82-83e6-cc1fa9ba8e94>\",\"WARC-Concurrent-To\":\"<urn:uuid:bf6e18b7-94ee-44c6-972a-0047303c3b9d>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/55819b21581e2a2f4054822a\",\"WARC-Payload-Digest\":\"sha1:WIZVAYNEPCOD3UKKCHJNOL4NWW24ZDII\",\"WARC-Block-Digest\":\"sha1:J4573A7VNJ6OZITCZTSNX3ZFHUW2UWQM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986702077.71_warc_CC-MAIN-20191020024805-20191020052305-00194.warc.gz\"}"}
https://elephant.readthedocs.io/en/latest/reference/cell_assembly_detection.html	[ "CAD [cad1] is a method aimed to capture structures of higher-order correlation in massively parallel spike trains. In particular, it is able to extract patterns of spikes with arbitrary configuration of time lags (time interval between spikes in a pattern), and at multiple time scales, e.g. from synchronous patterns to firing rate co-modulations.\n\nCAD consists of a statistical parametric testing done on the level of pairs of neurons, followed by an agglomerative recursive algorithm, in order to detect and test statistically precise repetitions of spikes in the data. In particular, pairs of neurons are tested for significance under the null hypothesis of independence, and then the significant pairs are agglomerated into higher order patterns.\n\nGiven a list of discretized (binned) spike trains by a given temporal scale (bin_size), assumed to be recorded in parallel, the CAD analysis can be applied as demonstrated in this short toy example of 5 parallel spike trains that exhibit fully synchronous events of order 5.\n\n `cell_assembly_detection`(binned_spiketrain, ...) Perform the CAD analysis [cad1] for the binned (discretized) spike trains given in the input.\n\n## Visualization¶\n\nVisualization of CAD method is covered in Viziphant `viziphant.patterns.plot_patterns()`\n\n## Examples¶\n\n```>>> import quantities as pq\n>>> import numpy as np\n>>> from elephant.cell_assembly_detection import cell_assembly_detection\n>>> from elephant.spike_train_generation import compound_poisson_process\n>>> from elephant.conversion import BinnedSpikeTrain\n```\n\nGenerate correlated data and bin it with a bin_size of 10ms.\n\n```>>> np.random.seed(30)\n>>> spiketrains = compound_poisson_process(rate=15pq.Hz,\n... amplitude_distribution=[0, 0.95, 0, 0, 0, 0, 0.05], t_stop=5pq.s)\n>>> bst = BinnedSpikeTrain(spiketrains, bin_size=10 * pq.ms)\n>>> bst.rescale('ms')\n```\n\nCall of the method.\n\n```>>> patterns = cell_assembly_detection(bst, max_lag=2)\n>>> patterns\n{'neurons': [0, 2],\n'lags': array([0.]) * ms,\n'pvalue': [5.3848138041122556e-05],\n'times': array([ 90., 160., 170., 550., 790., 910., 930., 1420., 1470.,\n1480., 1650., 2030., 2220., 2570., 3130., 3430., 3480., 3610.,\n3800., 3830., 3930., 4080., 4560., 4600., 4670.]) * ms,\n'signature': [[1, 83], [2, 25]]}\n```\n\nRefer to the Viziphant documentation regarding the visualization of this example." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7377663,"math_prob":0.9464133,"size":2411,"snap":"2022-27-2022-33","text_gpt3_token_len":636,"char_repetition_ratio":0.10760283,"word_repetition_ratio":0.0,"special_character_ratio":0.3007051,"punctuation_ratio":0.2560175,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9605214,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-04T20:50:56Z\",\"WARC-Record-ID\":\"<urn:uuid:3d7aa11e-db2a-4fa1-a2f4-32e7fb0b30df>\",\"Content-Length\":\"16754\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:68f906e0-ee68-47cf-8c2b-1422621261f0>\",\"WARC-Concurrent-To\":\"<urn:uuid:1d51285f-02fd-45a4-b71e-fe9e67e577d9>\",\"WARC-IP-Address\":\"104.17.32.82\",\"WARC-Target-URI\":\"https://elephant.readthedocs.io/en/latest/reference/cell_assembly_detection.html\",\"WARC-Payload-Digest\":\"sha1:Y2SDX2Z5ZV4I4WMXR3QI5YANR6DRMQUE\",\"WARC-Block-Digest\":\"sha1:NCPAT5PANUSGRL5FP2U2NKJH7PN76HKS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104496688.78_warc_CC-MAIN-20220704202455-20220704232455-00538.warc.gz\"}"}
https://en.commtap.org/taxonomy_description/2way-term/86/256	[ "### Quick start\n\nChoose a section:\nChoose something to look for:\n...and choose a categorisation:\n\nSign me up to:\n\n# Category: Maths Number and Algebra L8\n\nAbbreviation: MN L8\n\n## Description\n\nPupils solve problems involving calculating with powers, roots and numbers expressed in standard form, checking for correct order of magnitude. They choose to use fractions or percentages to solve problems involving repeated proportional changes or the calculation of the original quantity given the result of a proportional change. They evaluate algebraic formulae, substituting fractions, decimals and negative numbers. They calculate one variable, given the others, in formulae such as V = πr2h. Pupils manipulate algebraic formulae, equations and expressions, finding common factors and multiplying two linear expressions. They know that a2 - b2 = (a + b)(a - b). They solve inequalities in two variables. Pupils sketch and interpret graphs of linear, quadratic, cubic and reciprocal functions, and graphs that model real situations.\n\n## \"Maths Number and Algebra L8\" pages\n\nThere are currently no \"Maths Number and Algebra L8\" pages. Would you like to create a new page?\n\nAds on this page are provided by Google Adsense - and their presence does not imply any endorsement by Commtap. Report a problem with an ad on this page. Log in (for free) to avoid seeing ads." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8787797,"math_prob":0.9619271,"size":1021,"snap":"2021-21-2021-25","text_gpt3_token_len":209,"char_repetition_ratio":0.100294985,"word_repetition_ratio":0.0,"special_character_ratio":0.19000979,"punctuation_ratio":0.14367816,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9896935,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-14T03:34:44Z\",\"WARC-Record-ID\":\"<urn:uuid:93a165d0-5128-4f83-b403-d8d9c1a6413c>\",\"Content-Length\":\"28273\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f347cca0-653c-41fb-9eba-4ac937bfe6cc>\",\"WARC-Concurrent-To\":\"<urn:uuid:4aa5c5b7-f4e7-43a9-834c-684cf74cf66d>\",\"WARC-IP-Address\":\"77.68.13.222\",\"WARC-Target-URI\":\"https://en.commtap.org/taxonomy_description/2way-term/86/256\",\"WARC-Payload-Digest\":\"sha1:DNUBPZ3AUB5Y3M4L4QD3K6QIGFLQ3AAN\",\"WARC-Block-Digest\":\"sha1:7Z25IXLHDOKQR7IK7PYYZKYW7KGC36TA\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487611320.18_warc_CC-MAIN-20210614013350-20210614043350-00415.warc.gz\"}"}
https://answers.opencv.org/question/75510/how-to-make-auto-adjustmentsbrightness-and-contrast-for-image-android-opencv-image-correction/	[ "# How to make auto-adjustments(brightness and contrast) for image Android Opencv Image Correction", null, "i'm using OpenCV for Android.\nI would like to know,how to make image correction(auto adjustments of brightness/contrast) for image(bitmap) in android via OpenCV or that can be done by native ColorMatrixFilter from Android!??\n\nI tried to google,but didn't found good tutorials/examples.\nSo how can i achieve my goal? Any ideas?\nThanks!\n\nedit retag close merge delete\n\n1\n\nSort by » oldest newest most voted", null, "Brightness and contrast is linear operator with parameter alpha and beta\n\nO(x,y) = alpha * I(x,y) + beta\n\n\nIn OpenCV you can do this with convertTo.\n\nThe question here is how to calculate alpha and beta automatically ?\n\nLooking at histogram, alpha operates as color range amplifier, beta operates as range shift.\n\nAutomatic brightness and contrast optimization calculates alpha and beta so that the output range is 0..255.\n\ninput range = max(I) - min(I)\nwanted output range = 255;\nalpha = output range / input range = 255 / ( max(I) - min(I) )\n\n\nYou can calculate beta so that min(O) = 0;\n\nmin(O) = alpha * min(I) + beta\nbeta = -min(I) * alpha\n\n\nHistogram Wings Cut (clipping)\n\nTo maximize the result of it's useful to cut out some color with few pixels.\n\nThis is done cutting left right and wings of histogram where color frequency is less than a value (typically 1%). Calculating cumulative distribution from the histogram you can easly find where to cut.\n\nmay be sample chart helps to understand:", null, "EDIT: Histogram Normalization vs Equalization\n\nBy the way BrightnessAndContrastAuto could be named normalizeHist because it works on BGR and gray images stretching the histogram to the full range without touching bins balance. If input image has range 0..255 BrightnessAndContrastAuto will do nothing.\n\nHistogram equalization and CLAE works only on gray images and they change grays level balancing. look at the images below:", null, "EDIT2: Add support for BGRA images too\n\nEDIT3: Correct error in \"restore alpha channel from source \"\n\nhowever here is the code:\n\n/*\n \\brief Automatic brightness and contrast optimization with optional histogram clipping\n* \\param [in]src Input image GRAY or BGR or BGRA\n* \\param [out]dst Destination image\n* \\param clipHistPercent cut wings of histogram at given percent tipical=>1, 0=>Disabled\n* \\note In case of BGRA image, we won't touch the transparency\n/\nvoid BrightnessAndContrastAuto(const cv::Mat &src, cv::Mat &dst, float clipHistPercent=0)\n{\n\nCV_Assert(clipHistPercent >= 0);\nCV_Assert((src.type() == CV_8UC1) \|\| (src.type() == CV_8UC3) \|\| (src.type() == CV_8UC4));\n\nint histSize = 256;\nfloat alpha, beta;\ndouble minGray = 0, maxGray = 0;\n\n//to calculate grayscale histogram\ncv::Mat gray;\nif (src.type() == CV_8UC1) gray = src;\nelse if (src.type() == CV_8UC3) cvtColor(src, gray, CV_BGR2GRAY);\nelse if (src.type() == CV_8UC4) cvtColor(src, gray, CV_BGRA2GRAY);\nif (clipHistPercent == 0)\n{\n// keep full available range\ncv::minMaxLoc(gray, &minGray, &maxGray);\n}\nelse\n{\ncv::Mat hist; //the grayscale histogram\n\nfloat range[] = { 0, 256 };\nconst float histRange = { range };\nbool uniform = true;\nbool accumulate = false;\ncalcHist(&gray, 1, 0, cv::Mat (), hist, 1, &histSize, &histRange, uniform, accumulate);\n\n// calculate cumulative distribution from the histogram\nstd::vector<float> accumulator(histSize);\naccumulator = hist.at<float>(0);\nfor (int i = 1; i < histSize; i++)\n{\naccumulator[i] = accumulator[i - 1] + hist.at<float>(i);\n}\n\n// locate points that cuts at required value\nfloat max = accumulator.back();\nclipHistPercent = (max / 100.0); //make percent as absolute\nclipHistPercent /= 2.0; // left and right wings\n// locate left cut\nminGray = 0;\nwhile (accumulator[minGray] < clipHistPercent)\nminGray++;\n\n// locate right cut\nmaxGray = histSize - 1;\nwhile (accumulator[maxGray] >= (max - clipHistPercent))\nmaxGray--;\n}\n\n// current range\nfloat inputRange = maxGray ...\nmore\n\n1\n\nNice answer @pklab. Well done. +1 from me ☺\n\nFunny, it seems that the Gimp white balance command uses the same principle.\n\nI also forgot to mention that the histogram equalization is a built-in functionnality in OpenCV...\n\nThere is also the CLAHE function (Contrast Limited Adaptive Histogram Equalization) that could be used but there is some parameters to tune to use it.\n\n@pklab Thanks for great answer,mate! But i have some problem with reproduce in Java version.\n\nThe code is plain... it's should be easy to translate in Java, may be you sould use ArrayList instead of std::vector@theodore , @VeTaLio thanks for apreciation ... if it works you could accept the answer ;=)\n\n@pklab yes,that is not a big problem. Ill mark your question as solution,but little bit later,ok:) ? So have some questions about how to reproduce in java,because some stuff doesn't clear for me :(!\n\n@VeTaLio you could do it yourself in Java... it's a simple exercise, feel free to ask about stuffs doesn't clear for you\n\n@pklab as i understand you correct,that wingsCutPercent i need to put manually(as you said,1% => 0.01f value or i need to extract) ? Also,i got a problem,when i'm doing picture,converting via BitmapToMat,my source Mat have type CV_8UC4. So i didn't use your block of code if /else if(where you put CvtColor). I'm doing this directly => CvtColor(source,gray, CV_BGR2GRAY); and the problem occurs here: calcHist - first elemnt,in java there is arraylist of Mat. So via converters i'm trying to convert into ArrayList<mat> and i getting this exception :\njava.lang.IllegalArgumentException: CvType.CV_32SC2 != m.type() \|\| m.cols()!=1 I need to convert this for CvType.CV_32SC2?\n\n@pklab once i ask this question and didn't get answer => my old post http://answers.opencv.org/question/74... is similiar problem with Converters! Thanks!!!\n\nOK, 2nd edit...check new code, it supports BGRA images too. wingsCutPercent (now clipHistPercent) is a percent than if you want to cut 1% use 1.0. Howhever it's optional.\n\n@pklab so i've translated to java and problem is that i'm getting too much brightness and contrast. I will upload some example to show you.\n\nfor python users, here two functions create the same result.\n\ndef autoAdjustments_with_convertScaleAbs(img):\nalow = img.min()\nahigh = img.max()\namax = 255\namin = 0\n\n# calculate alpha, beta\nalpha = ((amax - amin) / (ahigh - alow))\nbeta = amin - alow alpha\n# perform the operation g(x,y)= α * f(x,y)+ β\nnew_img = cv2.convertScaleAbs(img, alpha=alpha, beta=beta)\n\nreturn [new_img, alpha, beta]\n\n\ndef autoAdjustments(img):\n# create new image with the same size and type as the original image\nnew_img = np.zeros(img.shape, img.dtype)\n\n# calculate stats\nalow = img.min()\nahigh = img.max()\namax = 255\namin = 0\n\n# access each pixel, and auto adjust\nfor x in range(img.shape):\nfor y in range(img.shape):\na = img[x, y]\nnew_img[x, y] = amin + (a - alow) * ((amax - amin) / (ahigh - alow))\n\nreturn new_img\n\n\nmore\n\nDoes anyone has been able to convert this to C# or VB.NET? I am having a hard time getting this converted.\n\nmore\n\nOfficial site\n\nGitHub\n\nWiki\n\nDocumentation" ]	[ null, "https://www.gravatar.com/avatar/3c4215f4c7840ac17739d3dd9bec904b", null, "https://www.gravatar.com/avatar/e4c22badd585dde21aa776c42466135c", null, "https://answers.opencv.org/upfiles/14474311256968999.png", null, "https://answers.opencv.org/upfiles/1447431096990625.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.648762,"math_prob":0.9598598,"size":3434,"snap":"2020-24-2020-29","text_gpt3_token_len":910,"char_repetition_ratio":0.12507288,"word_repetition_ratio":0.007561437,"special_character_ratio":0.27781013,"punctuation_ratio":0.17213115,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97352135,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,7,null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-03T04:10:18Z\",\"WARC-Record-ID\":\"<urn:uuid:639d0e41-80cc-450c-9532-0da1a9a3f368>\",\"Content-Length\":\"90584\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b56df4e7-caaf-4c84-b6d1-3977867fd749>\",\"WARC-Concurrent-To\":\"<urn:uuid:666e9c57-dd83-443d-a2aa-75eaed9227e3>\",\"WARC-IP-Address\":\"5.9.49.245\",\"WARC-Target-URI\":\"https://answers.opencv.org/question/75510/how-to-make-auto-adjustmentsbrightness-and-contrast-for-image-android-opencv-image-correction/\",\"WARC-Payload-Digest\":\"sha1:RKTLBP3WIRICU4DCG5GKZB3SBQVR7ZXE\",\"WARC-Block-Digest\":\"sha1:DQTR5655A6EDM6LYBQQ4SIIOBZS6O4YV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347428990.62_warc_CC-MAIN-20200603015534-20200603045534-00461.warc.gz\"}"}
https://stats.stackexchange.com/questions/251600/quantile-regression-loss-function?noredirect=1	[ "# Quantile regression: Loss function\n\nI am trying to understand the quantile regression, but one thing that makes me suffer is the choice of the loss function.\n\n$\\rho_\\tau(u) = u(\\tau-1_{\\{u<0\\}})$\n\nI know that the minimum of the expectation of $\\rho_\\tau(y-u)$ is equal to the $\\tau\\%$-quantile, but what is the intuitive reason to start off with this function? I don't see the relation between minimizing this function and the quantile. Can somebody explain it to me?\n\nI understand this question as asking for insight into how one could come up with any loss function that produces a given quantile as a loss minimizer no matter what the underlying distribution might be. It would be unsatisfactory, then, just to repeat the analysis in Wikipedia or elsewhere that shows this particular loss function works.\n\nLet's begin with something familiar and simple.\n\nWhat you're talking about is finding a \"location\" $x^{}$ relative to a distribution or set of data $F$. It is well known, for instance, that the mean $\\bar x$ minimizes the expected squared residual; that is, it is a value for which\n\n$$\\mathcal{L}_F(\\bar x)=\\int_{\\mathbb{R}} (x - \\bar x)^2 dF(x)$$\n\nis as small as possible. I have used this notation to remind us that $\\mathcal{L}$ is derived from a loss, that it is determined by $F$, but most importantly it depends on the number $\\bar x$.\n\nThe standard way to show that $x^{}$ minimizes any function begins by demonstrating the function's value does not decrease when $x^{}$ is changed by a little bit. Such a value is called a critical point of the function.\n\nWhat kind of loss function $\\Lambda$ would result in a percentile $F^{-1}(\\alpha)$ being a critical point? The loss for that value would be\n\n$$\\mathcal{L}_F(F^{-1}(\\alpha)) = \\int_{\\mathbb{R}} \\Lambda(x-F^{-1}(\\alpha))dF(x)=\\int_0^1\\Lambda\\left(F^{-1}(u)-F^{-1}(\\alpha)\\right)du.$$\n\nFor this to be a critical point, its derivative must be zero. Since we're just trying to find some solution, we won't pause to see whether the manipulations are legitimate: we'll plan to check technical details (such as whether we really can differentiate $\\Lambda$, etc.) at the end. Thus\n\n\\eqalign{0 &=\\mathcal{L}_F^\\prime(x^{})= \\mathcal{L}_F^\\prime(F^{-1}(\\alpha))= -\\int_0^1 \\Lambda^\\prime\\left(F^{-1}(u)-F^{-1}(\\alpha)\\right)du \\\\ &= -\\int_0^{\\alpha} \\Lambda^\\prime\\left(F^{-1}(u)-F^{-1}(\\alpha)\\right)du -\\int_{\\alpha}^1 \\Lambda^\\prime\\left(F^{-1}(u)-F^{-1}(\\alpha)\\right)du.\\tag{1} }\n\nOn the left hand side, the argument of $\\Lambda$ is negative, whereas on the right hand side it is positive. Other than that, we have little control over the values of these integrals because $F$ could be any distribution function. Consequently our only hope is to make $\\Lambda^\\prime$ depend only on the sign of its argument, and otherwise it must be constant.\n\nThis implies $\\Lambda$ will be piecewise linear, potentially with different slopes to the left and right of zero. Clearly it should be decreasing as zero is approached--it is, after all, a loss and not a gain. Moreover, rescaling $\\Lambda$ by a constant will not change its properties, so we may feel free to set the left hand slope to $-1$. Let $\\tau \\gt 0$ be the right hand slope. Then $(1)$ simplifies to\n\n$$0 = \\alpha - \\tau (1 - \\alpha),$$\n\nwhence the unique solution is, up to a positive multiple,\n\n$$\\Lambda(x) = \\cases{-x, \\ x \\le 0 \\\\ \\frac{\\alpha}{1-\\alpha}x, \\ x \\ge 0.}$$\n\nMultiplying this (natural) solution by $1-\\alpha$, to clear the denominator, produces the loss function presented in the question.\n\nClearly all our manipulations are mathematically legitimate when $\\Lambda$ has this form.\n\nThe way this loss function is expressed is nice and compact but I think it's easier to understand by rewriting it as $$\\rho_\\tau(X-m) = (X-m)(\\tau-1_{(X-m<0)}) = \\begin{cases} \\tau \|X-m\| & if \\; X-m \\ge 0 \\\\ (1 - \\tau) \|X-m\| & if \\; X-m < 0) \\end{cases}$$\n\nIf you want to get an intuitive sense of why minimizing this loss function yields the $$\\tau$$th quantile, it's helpful to consider a simple example. Let $$X$$ be a uniform random variable between 0 and 1. Let's also choose a concrete value for $$\\tau$$, say, $$0.25$$.\n\nSo now the question is why would this loss function be minimized at $$m=0.25$$? Obviously, there's three times as much mass in the uniform distribution to the right of $$m$$ than there is to the left. And the loss function weights the values larger than this number at only a third of the weight given to values less than it. Thus, it's sort of intuitive that the scales are balanced when the $$\\tau$$th quantile is used as the inflection point for the loss function.\n\n• Shouldn't it be the other way? Under-guessing will cost three times as much? – Edi Bice Apr 1 '19 at 15:01\n• Thanks for catching that. The formula is right but I initially worded it incorrectly in my explanation. – jjet Aug 17 '19 at 21:03" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8445982,"math_prob":0.99943376,"size":3143,"snap":"2019-51-2020-05","text_gpt3_token_len":860,"char_repetition_ratio":0.12105766,"word_repetition_ratio":0.0,"special_character_ratio":0.2857143,"punctuation_ratio":0.08637874,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999348,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-20T01:19:11Z\",\"WARC-Record-ID\":\"<urn:uuid:c4fd010d-cb38-4b85-a07c-6b22b2ed1c9c>\",\"Content-Length\":\"151146\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:33ed1034-3984-4368-8e02-35a4e77213eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:e98f73d4-f3e5-4df2-9b2a-325829266dd1>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/251600/quantile-regression-loss-function?noredirect=1\",\"WARC-Payload-Digest\":\"sha1:B3UMUSGO4KUW47NKZCFCS5WF75UTUQ4C\",\"WARC-Block-Digest\":\"sha1:HQHRSDEZZJDAMRYJT7TKWU5FRGOPZRB4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250595787.7_warc_CC-MAIN-20200119234426-20200120022426-00449.warc.gz\"}"}
https://www.hep.ucl.ac.uk/undergrad/1b40/excel.shtml	[ "# PHAS1240 Computing", null, "ExcelSome aspects of the Excel course will be discussed in the lecture slots, but the main thrust of the computing elements are carried out in ten weekly sessions during the first term with each student attending one afternoon per week from 2-5.30 pm. You should have been informed during Induction week as to which session you have been assigned. This may be checked by viewing the file 1B40_allocations-06.xls. If your name does not appear, please go to the sessions on Tuesdays.\n\nFor each week detailed instructions and some additional explanation of the mathematical background are given in task files. (These are in WORD or PDF format.) The starting positions for some spreadsheets are provided in some of the exercises, in others you create them from a description of the problem given in the task file. Samples of the results are shown in some of the files.\n\nFor some exercises the final result e.g. a graph, is well defined but the precise details of the spreadsheet layout are not. This is left to you to design. While good practice, elegance and clarity of the spreadsheets will be encouraged it is recognised that to many physicists and astronomers the final outcome is all that matters. Consequently spreadsheets that 'work' may get 50% of the marks, but well constructed ones will attract a higher reward!\n\nSome exercises get you to implement a simple task discussed in the lectures or task files. In other cases the problems involve the use of spreadsheet functions, methods or tools NOT covered in the notes. This is to encourage you to make use of the Excel help facility (or colleagues) - the way you will need to work on problems outside the classroom environment. At the end of some of the exercises there is a question asking you to give some explanation for the results obtained in the spreadsheet. These questions may not have a unique answer!\n\nIn the first session you will work through a set of short exercises, which will not be assessed.\n\nIn the next session, (the first of the assessed sessions) task 1 is an extension of the basic material. In this exercise (medical physics module) you have to plot several histograms of marks, normalised to the same sample size. The medics are also to be graded on a scale with some \"compensation\" of the grade if the mark is a \"fail\" mark. You need to find and look up the usage of various Excel worksheet functions (e.g. the IF function) using the Excel Help facility. (This exercise uses the spreadsheet Med_mod_results.xls.)\n\nIn task 2 we create the first spreadsheet dealing with the material given in the data analysis lectures. It is a simple example of the properties of the normal distribution and the idea of error on the mean.\n\nThe lectures on data analysis discuss how the errors on the sum or product of two independent variables is related to the individual errors. In task 3 you apply the error propagation formulae to data satisfying different forms, i.e. linear, logarithmic and sine functions from (a) a compound pendulum experiment, (b) a chemical reaction, (c) an X-ray scattering experiment. You will need to download the spreadsheets comp_pend_data.xls and reaction_data.xls.\n\nOne of the major practical procedures discussed in the data analysis lectures is the least squares fitting of a straight line to a set of data points. Task 4 is a guided spreadsheet application. (You will need the spreadsheet Least_squares_fit.xls.)\n\nMany physical systems can be modelled using an Excel spreadsheet. Modelling a system such as the motion of a body under gravity, involves differential equations. Therefore task 5 looks at how we can approximate a mathematical derivative of a continuous function by discrete quantities in cells of the spreadsheet. You examine two slightly different approximations for dy/dx.\n\nIn task 6 we consider a two-dimensional problem of the motion of a particle the Earth's gravitational field. You may well have met a simple projectile problem before - this was most likely the \"drag-free\" case - and solved it analytically. In this simulation you model the more realistic case of a projectile subjected to a drag force proportional to its speed. The aim (sic) is to find, by computer simulation, the angle of firing to achieve maximum range.\n\nThe use of random numbers (really pseudo-random numbers) in a computer is a powerful tool for the simulation of physical processes. In task 7 we first investigate some properties of random numbers and meet the central limit thereom - the saving grace for many physicists! You then simulate the exponential decay of nuclei. While this can be described by a simple mathematical expression (the exponential law) it is instructive to replicate this as close as possible in the discrete cells of a spreadsheet. The general ideas can be extended to cases which are not amenable to an analytical solution. This exercise also introduces you to some sophisticated use of the Excel functions to save tedious typing of formuale. Finally there is the option to investigate the 'Drunkard's walk' - a classical random process!" ]	[ null, "https://www.hep.ucl.ac.uk/undergrad/1b40/logos/ucl0004.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91386807,"math_prob":0.8626823,"size":4567,"snap":"2021-21-2021-25","text_gpt3_token_len":916,"char_repetition_ratio":0.12732852,"word_repetition_ratio":0.0,"special_character_ratio":0.19553317,"punctuation_ratio":0.080047786,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98498726,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T22:20:51Z\",\"WARC-Record-ID\":\"<urn:uuid:1c30badf-0849-4c98-b2dd-6fd78abeaff0>\",\"Content-Length\":\"6869\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1755000b-4b63-48ce-a9dc-fe770463bbab>\",\"WARC-Concurrent-To\":\"<urn:uuid:b2be3b6c-d250-46d7-afb2-980c9d8b59b7>\",\"WARC-IP-Address\":\"128.40.4.8\",\"WARC-Target-URI\":\"https://www.hep.ucl.ac.uk/undergrad/1b40/excel.shtml\",\"WARC-Payload-Digest\":\"sha1:EWTPXOTVYGBGUB6EPSCNQ7FAJNQZ2TTJ\",\"WARC-Block-Digest\":\"sha1:WMGGZH36BIHGDUJXC7UVFVGYE4BFL7IK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488559139.95_warc_CC-MAIN-20210624202437-20210624232437-00461.warc.gz\"}"}
https://community.wongcw.com/posts/213912	[ "What is Algebra? Everything You Need To Know\nAlgebra is a branch of mathematics that helps represent problems or situations using mathematical expressions. To form a meaningful mathematical expression, variables such as x, y, and z are combined with mathematical operations such as addition, subtraction, multiplication, and division. Algebra is used in all branches of mathematics, including trigonometry, calculus, and coordinate geometry....\n0 Comments 0 Shares 53 Views" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8985395,"math_prob":0.95807946,"size":540,"snap":"2022-40-2023-06","text_gpt3_token_len":109,"char_repetition_ratio":0.13992538,"word_repetition_ratio":0.0,"special_character_ratio":0.21666667,"punctuation_ratio":0.18367347,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9984354,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-27T08:41:04Z\",\"WARC-Record-ID\":\"<urn:uuid:0c0ba6bd-a377-4b88-bfd5-efeb40788eaa>\",\"Content-Length\":\"85903\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:884d8628-9bd8-45bd-ad1b-307d35e8111a>\",\"WARC-Concurrent-To\":\"<urn:uuid:964ca2e8-6141-40bf-b0d7-aa154584eed1>\",\"WARC-IP-Address\":\"178.170.9.206\",\"WARC-Target-URI\":\"https://community.wongcw.com/posts/213912\",\"WARC-Payload-Digest\":\"sha1:HN66GB3TKKBRIH3D4EWL5TOJRADHZKUS\",\"WARC-Block-Digest\":\"sha1:YN5HM7AAXEHFIN4PLLKXM4TVV2DMHKBQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764494974.98_warc_CC-MAIN-20230127065356-20230127095356-00084.warc.gz\"}"}