Split (1)

train · 167k rows

url stringlengths 6 1.61k	fetch_time int64 1,368,856,904B 1,726,893,854B	content_mime_type stringclasses 3 values	warc_filename stringlengths 108 138	warc_record_offset int32 9.6k 1.74B	warc_record_length int32 664 793k	text stringlengths 45 1.04M	token_count int32 22 711k	char_count int32 45 1.04M	metadata stringlengths 439 443	score float64 2.52 5.09	int_score int64 3 5	crawl stringclasses 93 values	snapshot_type stringclasses 2 values	language stringclasses 1 value	language_score float64 0.06 1
http://forums.wolfram.com/mathgroup/archive/2004/Oct/msg00699.html	1,716,065,256,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971057494.65/warc/CC-MAIN-20240518183301-20240518213301-00504.warc.gz	11,054,271	7,531	Re: Matrix Problem • To: mathgroup at smc.vnet.net • Subject: [mg51688] Re: Matrix Problem • From: Roger Bagula <tftn at earthlink.net> • Date: Fri, 29 Oct 2004 03:39:18 -0400 (EDT) • References: <clpr7g\$a3d\$1@smc.vnet.net> • Sender: owner-wri-mathgroup at wolfram.com ```Reference the first level of the list N=Table[M[[7-n]],{n,0,6}] It is actually a permutation of the rows of the matrix. Columns would be harder, ha, ha... nilaakash wrote: >Suppose I have a m*n matrix like this type. > >M={{1, a1, b1, c1}, {2, a2, b2, c2}, {3, a3, b3, c3}, {4, a4, b4, c4}, {5, a5, > b5, c5}, {6, a6, b6, c6}, {7, a7, b7, c7}} > >I want to change my matrix in such a way, > >N={{7, a7, b7, c7}, {6, a6, b6, c6}, {5, a5, b5, c5}, {4, a4, b4, c4}, {3, a3, > b3, c3}, {2, a2, b2, c2}, {1, a1, b1, c1}} > >Could anybody tell me how to this. > >Thanks. > >nilaakash > > > -- Respectfully, Roger L. Bagula	388	901	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.515625	3	CC-MAIN-2024-22	latest	en	0.75889
https://www.topperlearning.com/doubts-solutions/if-x-seca-tana-y-coseca-cota-pt-xy-1-y-x-rnwpz3yy/	1,547,886,944,000,000,000	text/html	crawl-data/CC-MAIN-2019-04/segments/1547583662863.53/warc/CC-MAIN-20190119074836-20190119100836-00009.warc.gz	982,039,180	48,084	1800-212-7858 (Toll Free) 9:00am - 8:00pm IST all days or Thanks, You will receive a call shortly. Customer Support You are very important to us For any content/service related issues please contact on this toll free number 022-62211530 Mon to Sat - 11 AM to 8 PM # If x=SecA-TanA y=CosecA+CotA P.t x.y+1=y-x Asked by chethan U 22nd July 2012, 5:37 PM Answer : Given : If x=SecA-TanA y=CosecA+CotA To Prove : x.y + 1 = y - x LHS = xy + 1 = ((sec A - tan A ) (cosec A + cotA) )+ 1 = sec A cosec A + sec A cotA - tanA cosecA - tan A cotA + 1 { using cotA = 1/ tan A in 4th term , tanA =sinA / cosA and cosec A = 1/sinA in 3rd term, sec A = 1/cosA and cotA = cosA / sinA in 2nd term } =secA cosecA + ( (1/cosA) ( cos A / sin A ) ) - ( (1/sinA )(sinA / cos A ) ) - tan A (1/tanA) + 1 = secA cosecA +(1/sinA ) -(1/cosA ) - 1 +1 = secA cosecA +(1/sinA ) -(1/cosA ) { using 1/sinA = cosec A and 1/cos A = sec A , we get} = secA cosecA + cosecA - secA .......................(1) RHS = y- x = ( cosec A + cotA ) - ( sec A - tanA) = cosecA - sec A + ( cotA + tanA ) { using cotA = 1/tanA } = cosecA -secA +( (1/tanA) + tanA ) = cosecA -secA + ( (1+tan2A) /tanA ) using 1+tan2A = sec2 =cosecA -secA + ( sec2A /tanA) using secA = 1/cos A and tan A = sinA / cosA , we get = cosecA -secA + ( ( 1/cos2 A) * (cosA/sinA) ) = cosecA - secA + ( 1/ (cos A sinA ) ) = cosecA - secA + cosecA secA..............(2) { using 1/sinA = cosecA and 1/cosA=secA } from eq (1) and (2) , LHS =RHS Hence Proved Answered by Expert 22nd July 2012, 6:53 PM • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 You have rated this answer /10	737	1,623	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2019-04	latest	en	0.646535
http://www.gamedev.net/topic/64743-fidning-the-coordinates-of-the-edges-of-the-screen/	1,480,985,745,000,000,000	text/html	crawl-data/CC-MAIN-2016-50/segments/1480698541864.44/warc/CC-MAIN-20161202170901-00250-ip-10-31-129-80.ec2.internal.warc.gz	448,506,152	25,357	• Create Account ## Fidning the coordinates of the edges of the screen Old topic! Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic. 4 replies to this topic ### #1griffenjam Members 193 Like Likes Like Posted 25 October 2001 - 12:04 PM I know this sounds dumb, but I can''t figure out how to find out the coordinates of the edges of the screen. I initally thought that it was a set number, but I guess not. Is there a formula? Jason Mickela ICQ : 873518 E-Mail: jmickela@sbcglobal.net ------------------------------ "Evil attacks from all sides but the greatest evil attacks from within." Me ------------------------------ ### #2Shag Members 122 Like Likes Like Posted 25 October 2001 - 01:23 PM Are you talking 2D or 3D? "Evil attacks from all sides but the greatest evil attacks from within." Me Me = you or Windows ME edition? Edited by - Shag on October 25, 2001 8:26:52 PM ### #3a2k Members 124 Like Likes Like Posted 25 October 2001 - 01:55 PM the extents would be whatever you set for glViewport in 3d or in glOrtho in 2d ### #4Dactylos Members 122 Like Likes Like Posted 25 October 2001 - 02:26 PM a2k: glViewport should be called in the same way regardless of whether you''re going to setup an orthographic projection or a perspective projection. It provides the mapping from normalized device coordinates to window coordinates. griffenjam: It depends on how you setup your projection matrix. ### #5griffenjam Members 193 Like Likes Like Posted 26 October 2001 - 04:41 AM I set up my projection like this. gluPerspective(45.0f,(GLfloat)glWidth/(GLfloat) glHeight,0.1f,1000.0f); Now, given the current depth can you figure out the displayable width and height? Jason Mickela ICQ : 873518 E-Mail: jmickela@sbcglobal.net ------------------------------ "Evil attacks from all sides but the greatest evil attacks from within." Me ------------------------------ Old topic! Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.	597	2,196	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.59375	3	CC-MAIN-2016-50	latest	en	0.809586
http://www.dummies.com/how-to/content/ohms-law-in-electronics.html	1,448,562,187,000,000,000	text/html	crawl-data/CC-MAIN-2015-48/segments/1448398447769.81/warc/CC-MAIN-20151124205407-00201-ip-10-71-132-137.ec2.internal.warc.gz	402,336,803	15,478	Sometimes in electronics you have no alternative but to whip out your calculator and do a little math. The most likely reason for doing this is to calculate how much resistance you need for a given situation, how much current a circuit will pull, or how much voltage will be dropped between two points in a circuit. All of these calculations can be made by using one of the following formulas derived from Ohm’s Law: In the above formulas, V represents voltage (in volts, naturally), I represents current (in amperes), and R represents resistance in ohms.	114	556	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.796875	3	CC-MAIN-2015-48	longest	en	0.955672
https://playtaptales.com/numbers/duotrigintatrecentilliduoquinquagintasescentillion/	1,660,448,056,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882571993.68/warc/CC-MAIN-20220814022847-20220814052847-00180.warc.gz	430,681,622	4,461	## Duotrigintatrecentilliduoquinquagintasescentillion A Duotrigintatrecentilliduoquinquagintasescentillion (1 Duotrigintatrecentilliduoquinquagintasescentillion) is 10 to the power of 997959 (10^997959). This is a stupendously astronomical number! ## How many zeros in a Duotrigintatrecentilliduoquinquagintasescentillion? There are 997,959 zeros in a Duotrigintatrecentilliduoquinquagintasescentillion. ## What's before Duotrigintatrecentilliduoquinquagintasescentillion? A Duotrigintatrecentilliunquinquagintasescentillion is smaller than a Duotrigintatrecentilliduoquinquagintasescentillion. ## What's after Duotrigintatrecentilliduoquinquagintasescentillion? A Duotrigintatrecentillitresquinquagintasescentillion is larger than a Duotrigintatrecentilliduoquinquagintasescentillion. ## Duotrigintatrecentilliduoquinquagintasescentillionaire A Duotrigintatrecentilliduoquinquagintasescentillionaire is someone whos assets, net worth or wealth is 1 or more Duotrigintatrecentilliduoquinquagintasescentillion. It is unlikely anyone will ever be a true Duotrigintatrecentilliduoquinquagintasescentillionaire. If you want to be a Duotrigintatrecentilliduoquinquagintasescentillionaire, play Tap Tales! ## Is Duotrigintatrecentilliduoquinquagintasescentillion the largest number? Duotrigintatrecentilliduoquinquagintasescentillion is not the largest number. Infinity best describes the largest possible number - if there even is one! We cannot comprehend what the largest number actually is. ## Duotrigintatrecentilliduoquinquagintasescentillion written out Duotrigintatrecentilliduoquinquagintasescentillion is written out as: 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https://www.convertunits.com/from/pond/to/zettagram	1,695,908,165,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233510412.43/warc/CC-MAIN-20230928130936-20230928160936-00615.warc.gz	774,843,037	12,495	## Convert pond [Dutch] to zettagram pond zettagram How many pond in 1 zettagram? The answer is 2.0E+18. We assume you are converting between pond [Dutch] and zettagram. You can view more details on each measurement unit: pond or zettagram The SI base unit for mass is the kilogram. 1 kilogram is equal to 2 pond, or 1.0E-18 zettagram. Note that rounding errors may occur, so always check the results. Use this page to learn how to convert between pond [Dutch] and zettagrams. Type in your own numbers in the form to convert the units! ## Want other units? You can do the reverse unit conversion from zettagram to pond, or enter any two units below: ## Enter two units to convert From: To: ## Definition: Zettagram The SI prefix "zetta" represents a factor of 1021, or in exponential notation, 1E21. So 1 zettagram = 1021 grams-force. ## Metric conversions and more ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 70 kg, 150 lbs, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!	358	1,336	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.609375	3	CC-MAIN-2023-40	latest	en	0.833197
https://kriscosmos.com/online-bingo-join-the-fun-join-the-family-unit/	1,717,072,576,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971667627.93/warc/CC-MAIN-20240530114606-20240530144606-00479.warc.gz	293,318,788	19,530	Hi ZS, assuming that whether not really one wins or loses on one scratch ticket (what is that, anyhow?) is independent from winning or losing on additional scratch ticket, you treat each event as a self-sufficient event. Laws of probability tell us to multiply the various probabilities of independent ordeals. It appears that the probability of [losing] on any particular scratch ticket must be 2/3. Useful and obviously the probabilities of [losing] on 30 scratch tickets one after (if that’s what your issue is asking) should be (2/3)^30 = approximately simple.2 x 10^-6, which is about.0000052, or 52 the particular 10 million, which boils down to 1 chance out of 192,307. This software works for a lotto number generator. May well gather all the online lottery winning numbers in the past 30 days and nights. And then, it will observe its patterns and can build suggestions of what numbers or combinations which might be best to place a bet about. Here are six men or women combinations. These six combinations are six good reasons why an established lottery winning system including the Lotto Black Book is a sound investment that preserve you a lot of wasted money in doomed-to-failure lottery tickets. www.riceballhouse.com like taking risks in today’s world. Some feel that lotteries possess a different essence of pretty own just like adventure sports. Scientific studies have also risen to suffice to do this. Analysts in the medical field say that entering a online lottery website, creating a bet or taking a risk causes the adrenals to to liberate lot of adrenal within your body. This adrenal makes the metabolic activities faster as a result increases the motor senses of consume due to secondary relieve serotonin. Serotonin induces sense of excitement and well-being, the same as your can of beer or Red-Bull. Is there a method to get money from the lottery regardless if you don’t win the lottery? Well the way to that is yes. Yes, it may be accomplished for you to put cash in your pocket from the lottery by yourself. Have your old ticket? Players are also able to check past outcomes for tickets they found involving their coat pockets from weeks ago. One does are on Twitter, we even post results over there. Lotto players can follow us so these people able to get our feed right to the Twitter fan pages.	490	2,333	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.90625	3	CC-MAIN-2024-22	latest	en	0.950777
https://www.scienceforums.net/topic/126011-length-contraction-in-a-block-universe-must-be-an-illusion/page/3/?tab=comments#comment-1190129	1,653,128,174,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662539049.32/warc/CC-MAIN-20220521080921-20220521110921-00180.warc.gz	1,136,984,921	40,100	# Length contraction in a block universe must be an illusion ## Recommended Posts 2 minutes ago, 34student said: During the year 2050, was the train 1 meter or 100 meters? Who was doing the measuring? The answer is frame-dependent • Replies 139 • Created #### Posted Images 6 hours ago, studiot said: Modern derivations start with the idea that different observers will make different measurements on the same phenomenon. Now how is that logically possible. How can one length be two lengths? ##### Share on other sites Just now, 34student said: Now how is that logically possible. How can one length be two lengths? It's not one length. Go back to the example of kinetic energy you have ignored. In my frame, the train is at rest, and thus no kinetic energy. In your frame it is moving, and has a KE of, say 10^7 Joules. How can it have two values of kinetic energy? (answer: because KE is a frame-dependent quantity) ##### Share on other sites 31 minutes ago, 34student said: Now how is that logically possible. How can one length be two lengths? Does this mean you wish to follow / engage with my chaing of reasoning ? It is logically possible because the circumstances of different observers are well, ..... different. It is necessary to find quantities that are not different for different observers and equate them. This will enable you deduce the contribution made by the differing circumstances. Markus' muon example is one good example of this because 'number' is invariant. We can also look carefully at your depiction of Bob. Do you realise what 'speed' he is going to achieve a 100 : 1 length contraction ? It works out at 0.99995c to 5 decimal places. This is why I asked you how you think Bob can 'see' this contraction. Are you going to consider and answer this ? ##### Share on other sites 54 minutes ago, swansont said: It's not one length. Go back to the example of kinetic energy you have ignored. In my frame, the train is at rest, and thus no kinetic energy. In your frame it is moving, and has a KE of, say 10^7 Joules. How can it have two values of kinetic energy? (answer: because KE is a frame-dependent quantity) Interesting. So are you saying that the train does not have one absolute size in the past? 16 minutes ago, studiot said: Does this mean you wish to follow / engage with my chaing of reasoning ? It is logically possible because the circumstances of different observers are well, ..... different. It is necessary to find quantities that are not different for different observers and equate them. This will enable you deduce the contribution made by the differing circumstances. Markus' muon example is one good example of this because 'number' is invariant. We can also look carefully at your depiction of Bob. Do you realise what 'speed' he is going to achieve a 100 : 1 length contraction ? It works out at 0.99995c to 5 decimal places. This is why I asked you how you think Bob can 'see' this contraction. Are you going to consider and answer this ? It doesn't really matter if Bob can see the contraction or not. My argument is against the idea that the train contracts for Bob whether he can see the train or not. And as a thought experiment, if I go fast enough, the sun should turn into the shape of a dinner plate. Wouldn't that cause a lot more fusion and other changes to its gravity and overall behavior? Edited by 34student ##### Share on other sites 35 minutes ago, 34student said: It doesn't really matter if Bob can see the contraction or not. My argument is against the idea that the train contracts for Bob whether he can see the train or not. And as a thought experiment, if I go fast enough, the sun should turn into the shape of a dinner plate. Wouldn't that cause a lot more fusion and other changes to its gravity and overall behavior? Either you want to understand or you want to argue. Which is it ? Just remember that if you want to argue you are arguing with one of the two most experimentally tested theories in history. ##### Share on other sites 48 minutes ago, studiot said: Either you want to understand or you want to argue. Which is it ? Just remember that if you want to argue you are arguing with one of the two most experimentally tested theories in history. I want to learn, always. I always thought that the past was one 4 dimensional static block, literally a block of static particles that would look like like strings because of the dimension of time. But now I am being told that there is a number of different pasts. One past universe where the train is 1 meter long, another past universe where the train is 100 meters long, etc. This is quite shocking. ##### Share on other sites 22 minutes ago, 34student said: This is quite shocking. It is indeed pretty shocking. It shocked a lot of people all over the word when it was first proposed, including some very important thinkers. 22 minutes ago, 34student said: But now I am being told that there is a number of different pasts. One past universe where the train is 1 meter long, another past universe where the train is 100 meters long, etc. I don't think this is the case. There is only one train. But there are many points of view and I am offering a slightly different route into them than usual. Especially as I see you have actually been at SF for a long time so must have seen quite a few arguments about the subject. It is a pity you don't address my questions about the example you specified since that example is impossible to conduct in practice. This is why I would like to introduce and discuss a real world example, the first exerimental verification that the formulae actually correspond to observation. Edited by studiot ##### Share on other sites 12 minutes ago, studiot said: It is indeed pretty shocking. It shocked a lot of people all over the word when it was first proposed, including some very important thinkers. I don't think this is the case. There is only one train. But there are many points of view and I am offering a slightly different route into them than usual. Especially as I see you have actually been at SF for a long time so must have seen quite a few arguments about the subject. It is a pity you don't address my questions about the example you specified since that example is impossible to conduct in practice. This is why I would like to introduce and discuss a real world example, the first exerimental verification that the formulae actually correspond to observation. I addressed your issue about Bob observing the contraction. Why does it matter whether Bob sees the contraction or not. I do not know what this has to do with my issue. ##### Share on other sites 1 hour ago, 34student said: I want to learn, always. 3 hours ago, 34student said: So are you saying that the train does not have one absolute size in the past? From the viewpoint (inertial frame) of a muon traveling through the earth's atmosphere the length contraction effect of special relativity allows the muon to penetrate and reach earth. In the muon frame its lifetime is unaffected, but the length contraction causes distances through the atmosphere and Earth to be far shorter than these distances in the Earth rest-frame. The above is not an illusion; the distance through the atmosphere is, was, and will be different in different inertial frames and it has physical effects. (above text is based on https://en.wikipedia.org/wiki/Muon) Edited by Ghideon source & quote ##### Share on other sites 2 hours ago, studiot said: Just remember that if you want to argue you are arguing with one of the two most experimentally tested theories in history. And obviously if one wants to argue against some of the most tested theories in history, one should know all there is about such theories. I certainly don't and I doubt that our doubter in this thread does...why even some scientists fail to understand it properly. The point is that the experimentally shown interchangeability of space [length] and time, means that if one is affected then the other is also. My interval of time and space is your space...your space and time, is my time. In essence, they are the same thing...one cannot exist without the other. 6 minutes ago, Ghideon said: Agreed, your first example was convincing as I did mention earlier...our "doubting Thomas" appears to have overlooked it though. 3 hours ago, 34student said: Now how is that logically possible. How can one length be two lengths? Edited by beecee ##### Share on other sites From any valid frame, inside the universe, length is frame dependent, as so many people have repeatedly told you. There is NO valid frame outside the universe ( block or any other model; nor other dimensions in the way you describe ), from which to measure the train. The OP is ill-posed, and cannot be answered, or even considered, as such. ##### Share on other sites 1 hour ago, 34student said: I addressed your issue about Bob observing the contraction. Why does it matter whether Bob sees the contraction or not. I do not know what this has to do with my issue. Really? You said 4 hours ago, 34student said: It doesn't really matter if Bob can see the contraction or not. And yet you continue to describe your pet example to others as though it is of vital importance. 5 hours ago, 34student said: Now how is that logically possible. How can one length be two lengths? and you received my explanation to which you have made a defeaningly silent response. Yet you say cavalierly that you have dealt with my issue. You have also not responsed to my earlier offers of other simple explanations like invariants and Bob's speed etc and my proposal to discuss a different (and better) example since Bob's was irrelevant to you and impossible for me. In short you are just trolling. ##### Share on other sites 11 hours ago, 34student said: I always thought that the past was one 4 dimensional static block, literally a block of static particles that would look like like strings because of the dimension of time. Sure, that would be a block of events, and the "strings" are world lines of particles. Imagine you have a block of say wood, and you draw a small line on it to represent a world line. You can rotate the block in your hands, but the shape and length of the line remains invariant. Say that you rotate it through a fixed grid of coordinates in a room. Using just 2 dimensions for simplicity, you can make the line align with the x-axis, or the y-axis, or anything in between. By rotating it, you can change how long the line is in the x-dimension and how long it is in the y-dimension, without changing its length. But the block universe also has a time dimension, so instead of x and y, imagine it's t and x. The line could be a function x(t) representing how far x a particle moves in time t in a particular set of coordinates, and you get different x-lengths in different coordinates just by rotating the block. In this analogy the rotation is a spherical rotation which keeps r^2 = x^2 + y^2 invariant as you rotate it. With spacetime, the rotation is hyperbolic, which instead keeps s^2 = x^2 - t^2 invariant as you rotate it. So you can't just turn it upside down and reverse time etc. ##### Share on other sites 18 hours ago, 34student said: During the year 2050, was the train 1 meter or 100 meters? It depends on the observer of course - there will exist an observer in whose frame it is 1m, and there will also exist an observer in whose frame it is 100m. All of these observers are right - but only in their own frames. The length of the train is thus not an intrinsic property of it, any more than eg. its speed or kinetic energy. It is meaningful only in relation to a specific observer measuring it, so asking about its “real” length is meaningless. There are indeed quantities that all observers in spacetime agree on (tensors and invariants), but length is not one of them; it’s strictly a quantity that describes a relationship to a given observer only. ##### Share on other sites 19 hours ago, 34student said: Interesting. So are you saying that the train does not have one absolute size in the past? Yes. The length is relative (hence “relativity”). It depends on the observer. ##### Share on other sites 8 hours ago, md65536 said: Sure, that would be a block of events, and the "strings" are world lines of particles. That is what I thought, and this is why I am so confused. Going back to the example in my OP, let's think about a particle sitting at one end of the train. The train lies on the x axis. The particle has one "String"/world line for Bob and another for Earth observers, (and possible an infinite more creating a plane). Which world line is real, or do they both exist? If both particle locations exist, why can't one of the two observers detect both particles? ##### Share on other sites 25 minutes ago, 34student said: That is what I thought, and this is why I am so confused. Going back to the example in my OP, let's think about a particle sitting at one end of the train. The train lies on the x axis. The particle has one "String"/world line for Bob and another for Earth observers, (and possible an infinite more creating a plane). Which world line is real, or do they both exist? If both particle locations exist, why can't one of the two observers detect both particles? Did none of the rest of what I wrote make sense? The particle has only one world line. The world line is the events (the x and t locations) that the particle passes through over its lifetime. The world line is a fixed set of events, but the x and t values of those events change if you rotate it into other coordinate systems representing other observers. For example, put a stick on a grid with an x and t axis. If you align it with the t axis, it has the same x value at different t, representing the world line of a particle at rest. If you rotate the grid, the stick stays the same but now it has different x at different t, representing a different observer, for which the particle is moving. This is an oversimplified analogy and it uses the wrong rotation, so don't draw too many conclusions from it. If you rotate the stick or grid, the stick's x and t components change, but the length of the stick stays the same. It's the hypotenuse in the Pythagorean theorem r^2 = x^2 + t^2. With the correct hyperbolic rotation, s^2 = x^2 - t^2 stays the same. ##### Share on other sites 2 hours ago, 34student said: If both particle locations exist, why can't one of the two observers detect both particles? A specific observer can't move relative themselves; one observer is always in one inertial frame. So the observer will make observations, consistent with relativity theories, in that single frame. That means the observer will observer one and only one set of x and t coordinates (as described by @md65536). ) Multiple observers may share a single frame and any observer may change frame but as far as I know no observer can be simultaneously in two different frames. ##### Share on other sites 2 hours ago, Ghideon said: A specific observer can't move relative themselves; one observer is always in one inertial frame. So the observer will make observations, consistent with relativity theories, in that single frame. That means the observer will observer one and only one set of x and t coordinates (as described by @md65536). ) Multiple observers may share a single frame and any observer may change frame but as far as I know no observer can be simultaneously in two different frames. Ok, but objectively speaking, does more than one world line exist for the particle? 3 hours ago, md65536 said: Did none of the rest of what I wrote make sense? The particle has only one world line. The world line is the events (the x and t locations) that the particle passes through over its lifetime. The world line is a fixed set of events, but the x and t values of those events change if you rotate it into other coordinate systems representing other observers. For example, put a stick on a grid with an x and t axis. If you align it with the t axis, it has the same x value at different t, representing the world line of a particle at rest. If you rotate the grid, the stick stays the same but now it has different x at different t, representing a different observer, for which the particle is moving. This is an oversimplified analogy and it uses the wrong rotation, so don't draw too many conclusions from it. If you rotate the stick or grid, the stick's x and t components change, but the length of the stick stays the same. It's the hypotenuse in the Pythagorean theorem r^2 = x^2 + t^2. With the correct hyperbolic rotation, s^2 = x^2 - t^2 stays the same. I think that I understand what you are saying. Now here is my question that will either put my issue to rest for me or keep me wondering. If the particle is at x1 for Bob, and x2 for people on Earth, does the particle necessarily have to go through x1 and x2 (not necessarily in this sequence)? 7 hours ago, swansont said: Yes. The length is relative (hence “relativity”). It depends on the observer. Then that would seem to imply that the train does not have a single world line. 11 hours ago, Markus Hanke said: It depends on the observer of course - there will exist an observer in whose frame it is 1m, and there will also exist an observer in whose frame it is 100m. All of these observers are right - but only in their own frames. The length of the train is thus not an intrinsic property of it, any more than eg. its speed or kinetic energy. It is meaningful only in relation to a specific observer measuring it, so asking about its “real” length is meaningless. There are indeed quantities that all observers in spacetime agree on (tensors and invariants), but length is not one of them; it’s strictly a quantity that describes a relationship to a given observer only. With world lines in mind, this means that there are at least 2 trains in existence (and probably many more from other frames of reference). Do you agree? ##### Share on other sites 1 hour ago, 34student said: Then that would seem to imply that the train does not have a single world line. I don’t see how. The train isn’t in two places at once. ##### Share on other sites The system of the whole train is not. The individual cars are, in which case he’s mixing frames (pretending that the cars being in different places is equivalent to the train as a whole being in more than one place). He thinks he’s being clever. ##### Share on other sites 1 hour ago, 34student said: With world lines in mind, this means that there are at least 2 trains in existence (and probably many more from other frames of reference). Do you agree? No, there is one train. If you had two, one could crash and the other not, but every observer has to agree events, like whether it crashed. The observers will disagree on what time and how far away the event occurred. ##### Share on other sites 48 minutes ago, iNow said: The system of the whole train is not. The individual cars are, in which case he’s mixing frames (pretending that the cars being in different places is equivalent to the train as a whole being in more than one place). He thinks he’s being clever. So if there are 10 cars to the train, are you saying that there actually are 20 cars? ##### Share on other sites 38 minutes ago, 34student said: So if there are 10 cars to the train, are you saying that there actually are 20 cars? Nobody said anything like that. There’s nothing legitimate to be gained by making the problem more complex. There is nothing about relativity that says anything about the number of cars being relative. ##### Share on other sites This topic is now closed to further replies. ×	4,417	19,918	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.125	3	CC-MAIN-2022-21	latest	en	0.920412
http://gmatclub.com/forum/gmat-study-partner-in-philadelphia-159048.html	1,484,921,347,000,000,000	text/html	crawl-data/CC-MAIN-2017-04/segments/1484560280835.22/warc/CC-MAIN-20170116095120-00167-ip-10-171-10-70.ec2.internal.warc.gz	128,188,980	44,568	GMAT study partner in Philadelphia ? : GMAT Study Groups Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack It is currently 20 Jan 2017, 06:09 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track Your Progress every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # GMAT study partner in Philadelphia ? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message Intern Joined: 24 Feb 2011 Posts: 9 Location: United States GMAT 1: 570 Q44 V23 Followers: 0 Kudos [?]: 5 [0], given: 100 GMAT study partner in Philadelphia ? [#permalink] ### Show Tags 02 Sep 2013, 19:08 Please let me know if anybody is looking for a study partner. I have my exam coming up in 2 months and i am dead serious with my preparations. My aim is to aim to cross the 700 barrier. -J Intern Joined: 08 Aug 2013 Posts: 3 Followers: 1 Kudos [?]: 0 [0], given: 0 Re: GMAT study partner in Philadelphia ? [#permalink] ### Show Tags 09 Sep 2013, 13:07 Dude, If you want we can start a study group online because I am in Los Angeles. Anyway, I just started an intense Gmat study by myself (full time)this week and today I did schedule my test for this Nov 25th. The goal is a high score 700+ . Send me a line ,we can do skype or something else. later Intern Joined: 24 Feb 2011 Posts: 9 Location: United States GMAT 1: 570 Q44 V23 Followers: 0 Kudos [?]: 5 [0], given: 100 Re: GMAT study partner in Philadelphia ? [#permalink] ### Show Tags 16 Sep 2013, 16:27 thanks for the reply and sorry for the late response. even i also booked my exam date. please give me your email id. or you can even reach me at https://www.facebook.com/profile.php?id=100006508899729 -J Re: GMAT study partner in Philadelphia ? [#permalink] 16 Sep 2013, 16:27 Similar topics Replies Last post Similar Topics: Study Partner/Group in the Philadelphia area 0 11 Dec 2016, 16:22 GMAT Study partner(s) 2 09 Mar 2016, 19:23 Philadelphia Study Partner/Group 2 03 May 2015, 16:19 Looking for a study partner for GMAT 5 15 Jan 2013, 14:40 GMAT study partners at Bellevue 0 17 May 2011, 17:55 Display posts from previous: Sort by # GMAT study partner in Philadelphia ? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	810	2,975	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.796875	3	CC-MAIN-2017-04	latest	en	0.857686
https://discourse.julialang.org/t/objects-of-type-logtargetdensity-are-not-callable/100761	1,696,187,596,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233510924.74/warc/CC-MAIN-20231001173415-20231001203415-00279.warc.gz	227,582,688	5,827	Objects of type LogTargetDensity are not callable Hi, I’m trying to execute the following example for AdvancedHMC: ``````using AdvancedHMC, ForwardDiff using LogDensityProblems using LinearAlgebra # Define the target distribution using the `LogDensityProblem` interface struct LogTargetDensity dim::Int end LogDensityProblems.logdensity(p::LogTargetDensity, θ) = -sum(abs2, θ) / 2 # standard multivariate normal LogDensityProblems.dimension(p::LogTargetDensity) = p.dim LogDensityProblems.capabilities(::Type{LogTargetDensity}) = LogDensityProblems.LogDensityOrder{0}() # Choose parameter dimensionality and initial parameter value D = 10; initial_θ = rand(D) ℓπ = LogTargetDensity(D) # Set the number of samples to draw and warmup iterations # Define a Hamiltonian system metric = DiagEuclideanMetric(D) hamiltonian = Hamiltonian(metric, ℓπ, ForwardDiff) # Define a leapfrog solver, with initial step size chosen heuristically initial_ϵ = find_good_stepsize(hamiltonian, initial_θ) integrator = Leapfrog(initial_ϵ) # Define an HMC sampler, with the following components # - multinomial sampling scheme, # - generalised No-U-Turn criteria, and # - windowed adaption for step-size and diagonal mass matrix proposal = NUTS{MultinomialTS, GeneralisedNoUTurn}(integrator) # Run the sampler to draw samples from the specified Gaussian, where # - `samples` will store the samples # - `stats` will store diagnostic statistics for each sample `````` However, ForwardDiff does not seem equipped to handle the LogTargetDensity type. Here’s the stacktrace: ``````ERROR: MethodError: objects of type LogTargetDensity are not callable Stacktrace: [1] vector_mode_dual_eval!(f::LogTargetDensity, cfg::ForwardDiff.GradientConfig{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10, Vector{ForwardDiff.Dual{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10}}}, x::Vector{Float64}) @ ForwardDiff ~/.julia/packages/ForwardDiff/vXysl/src/apiutils.jl:24 [2] vector_mode_gradient!(result::DiffResults.MutableDiffResult{1, Float64, Tuple{Vector{Float64}}}, f::LogTargetDensity, x::Vector{Float64}, cfg::ForwardDiff.GradientConfig{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10, Vector{ForwardDiff.Dual{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10}}}) [3] gradient!(result::DiffResults.MutableDiffResult{1, Float64, Tuple{Vector{Float64}}}, f::LogTargetDensity, x::Vector{Float64}, cfg::ForwardDiff.GradientConfig{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10, Vector{ForwardDiff.Dual{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10}}}, ::Val{true}) [4] gradient!(result::DiffResults.MutableDiffResult{1, Float64, Tuple{Vector{Float64}}}, f::LogTargetDensity, x::Vector{Float64}, cfg::ForwardDiff.GradientConfig{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10, Vector{ForwardDiff.Dual{ForwardDiff.Tag{LogTargetDensity, Float64}, Float64, 10}}}) (repeats 2 times) [5] ∂ℓπ∂θ_forwarddiff(ℓπ::LogTargetDensity, θ::Vector{Float64}) [7] ∂H∂θ(h::Hamiltonian{DiagEuclideanMetric{Float64, Vector{Float64}}, LogTargetDensity, AdvancedHMC.var"#∂ℓπ∂θ#66"{LogTargetDensity}}, θ::Vector{Float64}) [8] phasepoint(h::Hamiltonian{DiagEuclideanMetric{Float64, Vector{Float64}}, LogTargetDensity, AdvancedHMC.var"#∂ℓπ∂θ#66"{LogTargetDensity}}, θ::Vector{Float64}, r::Vector{Float64})	936	3,319	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.53125	3	CC-MAIN-2023-40	latest	en	0.626629
http://stackoverflow.com/questions/16707521/extracting-words-from-observation-and-matching-them-with-others-in-data-table	1,409,151,292,000,000,000	text/html	crawl-data/CC-MAIN-2014-35/segments/1408500829421.59/warc/CC-MAIN-20140820021349-00392-ip-10-180-136-8.ec2.internal.warc.gz	189,485,352	15,891	# Extracting words from observation and matching them with others in data table I have data like this ``````Code Sx1 Sx2 Sx3 SX4 P1 XX AA ZZ HH P2 AA XX AA BB RR1 XA XX ZA HB RR2 AA XA ZA BH RR3 AX AX ZZ HH RR4 AA AA AA BB RR5 XX XX ZA HB RR6 AX XA ZZ HH RR7 AA AX AA BB RR8 XX AA AZ HH `````` I want to extract one letter from P1 (X) and another letter from P2 (A) and then I want to use those extracted letters (AX or XA) to match with observations from RR1 to RR8, if they match then 1, not then will be 0. I want to use arrays because my data contains over 8,000 observations. I want like this ``````SX1 SX2 SX3 SX4 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 `````` I am trying to solve this by using Substr function ``````Data x; Set y; var1 = SUBSTR (SX1,1,1); var2= SUBSTR (SX2,1,1); var3= SUBSTR (SX3,1,1); var4= SUBSTR (SX4,1,1); PACKAGE_SUBSTR = LETTER_1\|\| LETTER_2\|\| LETTER_3; RUN; `````` I will use arrays to store my variables from SX1-SX4 and then Here after I want to match this concatenated variable with the other observations in the array to reach my goal. I want to know I am proceeding in the right way or not? help needed to solve this problem. - You have 8000 observations and 4 SX1-SX4 variables, or do you have more than those 4 variables? – Joe May 23 '13 at 13:53 Thank you very much Joe for helping me, actually i have 44 variables here i metioned only 4, i just want to extract one letter from both P1 and P2 for one variables for example SX1 and if i get extracted letters like XA and then i want to match them with remaining observation of SX1 vars. If observation of var SX1 matched then 1 else 0.Thanking you very much – user2134713 May 24 '13 at 4:59 The solution I would use is to transpose your dataset so that instead of 44 variables (or whatever) you have one row per variable. Then merge them together. That way it is very flexible and can accommodate any number of variables. Something like this: ``````data have; input Code \$ Sx1 \$ Sx2 \$ Sx3 \$ SX4 \$; datalines; P1 XX AA ZZ HH P2 AA XX AA BB RR1 XA XX ZA HB RR2 AA XA ZA BH RR3 AX AX ZZ HH RR4 AA AA AA BB RR5 XX XX ZA HB RR6 AX XA ZZ HH RR7 AA AX AA BB RR8 XX AA AZ HH ;;;; run; data p1(drop=code rename=value=value1) p2(drop=code rename=value=value2) code; set have; array vars sx1-sx4; do _t = 1 to dim(vars); varname = vname(vars[_t]); value = vars[_t]; if code='P1' then output p1; else if code='P2' then output p2; else output code; end; keep code varname value; run; proc sort data=p1; by varname; run; proc sort data=p2; by varname; run; data type1 type2; merge p1 p2; by varname; value = cat(substr(value1,1,1),substr(value2,2,1)); output type1; value = cat(substr(value2,2,1),substr(value1,1,1)); output type2; keep varname value; run; proc sort data=code; by varname value; run; proc sort data=type1; by varname value; run; proc sort data=type2; by varname value; run; data code_match; merge code(in=c) type1(in=t) type2(in=t); by varname value; if c; if c and t then match=1; else match=0; run; proc sort data=code_match; by code varname; run; proc transpose data=code_match out=want; by code; id varname; var match; run; `````` You probably need to adjust the rules somewhat for your needs, I'm not sure if I implemented them correctly, but it does for this example get your desired dataset. The variables may not be in proper order in this case since you have SX4 capitalized (capitals sort before lower case by default). - Joe , Thank you very much for your help and your code is working fine and i made changes as per my requirement. – user2134713 May 30 '13 at 6:34	1,241	3,731	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.703125	3	CC-MAIN-2014-35	latest	en	0.87162
https://lists.launchpad.net/kicad-developers/msg03796.html	1,586,070,839,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585370529375.49/warc/CC-MAIN-20200405053120-20200405083120-00465.warc.gz	534,116,342	6,411	# kicad-developers team mailing list archive ## Re: where footprints come from (was Re: Re: Internal • From: Dick Hollenbeck <dick@...> • Date: Mon, 21 Dec 2009 10:56:36 -0600 • Scanners: none • User-agent: Thunderbird 2.0.0.23 (X11/20090817) Dick Hollenbeck wrote: Werner Almesberger wrote: Dick Hollenbeck wrote: How [ are measurements ] done? What software? That's with fped, the tool I wrote to make the footprints we use in gta02-core, and that's also my platform for trying concepts that should hopefully yield a better module editor: - partial documentation of the GUI: http://people.openmoko.org/werner/fped/gui.html - description of language and concepts: In fped, you use vectors to define all points that are used to construct pads, silk, etc. Vectors can be concatenated and their x/y size is a fairly arbitrary arithmetic expression. Vectors are also labeled, so that they can be referenced by name. (You don't have to do this manually - if you construct things through the GUI, you'll probably never label a vector. fped will generate suitable names.) To draw a pad, you define the two corners, then create the pad with these corners. The pad only has the corner references (and the pad name), no coordinate calculations. In the fped language, this looks like this: package "Example" unit mm ll: vec @(10mm, 10mm) ur: vec .(5mm, 10mm) E.g., the "lower left" (ll) vector starts at the origin (@) and goes to (10, 10). The "upper right" (ur) vector continues from the previous vector (.) and goes for another (5, 10). Then pad "1" is made between points ll and ur. Since we now have a nice and simple geometry, we can define measurements through the same set of references: measx ll >> ur -1mm measy ll >> ur 1mm measx means that we only measure along a parallel of the x axis, not the diagonal. >> means that we measure between (the minimum) ll and the maximum ur. In this case, there's only one of each anyway, but if you add repetition, there may be several instances of the same vector, and fped can distinguish between the "next" and the "last" instance. At the end, there's the distance the measurement is offset from the line being measured. (I'm not proud of this bit. I think fped should figure out such things on its own.) This is what the result looks like: Keeping all geometry tied to vectors helps to visualize things and keeps the number of "hidden parameters" low. I.e., the calculations that go into a vector may not be trivial, but you can readily see what it does. This model is a bit unconventional. For measurements, you could get the same functionality by, say, extracting coordinates from pads. To be able to also include silk screen objects in measurements, you'd have to be able to name/reference them as well. By simple, I mean the footprint editor has to be able to edit the file in a WYSIWIG mode, i.e. graphical editing mode as now. Yes, I think this is crucial. I would also say that it's important that the textual representation is sufficiently readable and efficient that people would choose it over making their own scripts. Otherwise, we'd once again end up with the problem that the common format isn't the format people actually use. How do you name pins in a loop without introducing a reasonably rich programming language? Then such language is not easily editable in a WYSIWYG mode. In fped, loops come in two flavours: you have the usual counting loop and you have tables. Tables define an iteration over multiple variables. Here's a fairly complex example: http://people.openmoko.org/werner/fped-table.png A simpler case would be the association between BGA row names and the row number (for calculating the position). To make all this GUI-friendly, iterations are declarative, not procedural. This means that fped iterates over all values and instantiates the respective objects for each of them. You control what is affected by the loops through so-called frames. The variables in a frame only affect what's going on in the frame and any child frames. So if you wanted to make a line of pads, you could make a "pads" frame, put a counter or table in it, then have a vector point to the pad "\$i" or maybe first set name = i+1, then use "\$name". Most of this is fairly natural if you structure things from the beginning. One of fped's current shortcomings is that the GUI doesn't make it easy to rearrange items if your structure turns out to have a problem. We cannot make commitments for anyone else. So writing a specification without a commitment to code it is pointless. I am trying to suggest things I suspect somebody can code in short order. My objective here is to find out where we want to go. Once the goal is set, we can look for ways to get there in reasonable steps, and recruit volunteers. I think now is a good time for considering this, because 1) you're revamping the module format anyway, 2) the natives are more restless on the library front than ever, so we may actually see something happen, and 3) we have a playground (fped) to experiment with crazy new ideas. Now, my C++ fu isn't strong enough for me to commit to any heroic deeds on the module editor in KiCad. But if we should consider a "bag of tools" approach (similar to what we have now for routing), at least temporarily, I could implement anything interesting we come up in fped. The external tool for footprint editing would be more practical if each footprint was in a separate file. I have supported this for some time, even in the face of disagreement from others. The notion of a footprint library can easily be done by using file system directories as the container. Even better: But there was also a good deal of discussion about putting a layer of abstraction between the actual file format and the client code of footprints. This layer of abstraction was an API. To do an API you eventually have to define in memory data structures, because these data structures are the goodies that you exchange with the service implementing the API. I have said before that if such an API is defined, (and this could be done using a *.h file and Doxygen without writing one line of actual code), then any library can be implemented using a file format in any way the implementor wants. It just has to provide a C++ plugin for the PCBNEW's client usage of said API. To fully implement this vision, it does suggest separating out the module editor, and making it responsible only for one implementation of a footprint library, but not responsible for all implementations. I would love for semi-conductor companies to provide us with source code implementing our API, to a library that they maintain. These libraries (and companies) need to compete with one another. The source code is necessary so we can be cross platform. The API is a read only service. It does not need to support modification of footprints. And as such, much of the complexities of how a footprint were made are long hidden. Loops, vectors, and whatnot need not be exposed at that point. We could even use some of the same data structures we are currently using in the API. Although we could also use a text string for passing some of this, using a format similar to what we have been talking about here, and also in the context of smart clipboard pasting. Dick	1,689	7,303	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.84375	3	CC-MAIN-2020-16	latest	en	0.918229
https://www.causal.app/formulae/var-p-excel	1,653,486,486,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00082.warc.gz	791,357,960	4,705	Excel ## How do you use VAR.P in Excel? There are a few ways to use VAR.P in Excel. You can use it to calculate variance of a population, to calculate variance of a sample, or to calculate standard error. You can also use it to calculate confidence intervals. In order to use VAR.P in Excel, you first need to type in the following formula: =VAR.P(number1,number2,number3,...) where number1, number2, and number3 are the numbers you want to calculate the variance for. You can also include a cells range in the formula, for example =VAR.P(A1:A10). ## What is the syntax of VAR.P in Excel? The syntax of VAR.P in Excel is as follows: =VAR.P(number,k) This function calculates the variance of a population, using the k-th moment about the mean. number is the population size, and k is the moment order. ## What is an example of how to use VAR.P in Excel? One example of how to use VAR.P in Excel is to calculate the variance of a population based on a sample. This can be done using the following steps: 1. Enter the data for the population and sample into two separate columns in Excel. 2. In the cell next to the population data, enter the formula =VAR.P(A2:A6). This will calculate the population variance based on the data in the column A2 through A6. 3. In the cell next to the sample data, enter the formula =VAR.P(B2:B6). This will calculate the sample variance based on the data in the column B2 through B6. 4. Compare the two values to see how close they are to each other. The closer the values are, the more accurate the sample variance is. ## When should you not use VAR.P in Excel? There are a few occasions when you should not use VAR.P in Excel. If you have a column of numbers that are not equally spaced, VAR.P will not give you an accurate result. Additionally, if your data set includes text values or logical values, VAR.P will not give you a valid result. ## What are some similar formulae to VAR.P in Excel? There are a few similar formulae to VAR.P in Excel. One is VAR.S, which calculates the variance of a sample. Another is VAR.I, which calculates the variance of an individual observation. Finally, there is VAR.F, which calculates the variance of a frequency distribution. ### Excel Get started with Causal today. Build models effortlessly, connect them directly to your data, and share them with interactive dashboards and beautiful visuals.	538	2,386	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.59375	4	CC-MAIN-2022-21	latest	en	0.884796
https://www.hackmath.net/en/example/391	1,563,873,427,000,000,000	text/html	crawl-data/CC-MAIN-2019-30/segments/1563195529175.83/warc/CC-MAIN-20190723085031-20190723111031-00368.warc.gz	714,951,900	6,777	# Car Car is going 95 km per hour. How long distance goes in 1 minute? Result l = 1583.3 m #### Solution: Leave us a comment of example and its solution (i.e. if it is still somewhat unclear...): Be the first to comment! #### To solve this example are needed these knowledge from mathematics: Do you want to convert length units? ## Next similar examples: 1. Train and walker Walker passes an hour 5 km. Express train per hour 100 km. Walker went 4 hours. How many minutes took the distance express train? 2. Avg speed of flight The students vice adventure had a 2,367 km flight. If they travel time was 2 hours and 56 minutes, what was their average speed in kilometres per hour? 3. Cyclist vs car Cyclist rode out of the city at 18 km/h. 1 hour 30 minutes behind him started car and caught up with the cyclist in 50 minutes. How fast was driving the car? Where (what kilometer) from the city car overtook a cyclist? 4. Cyclist 9 A cyclist travels at a a speed of 4.25 km per hour. At that rate, how far can he travel in 3.75 hours? 5. Walkers Walker, which makes 120 steps per minute, make distance from point A to point B for 55 minutes. The length of his step is 75cm. For how long does this distance go boy who will do 110 steps 60 cm long in a minute? 6. Two cars 2 Two cars started from two positions 87 km distant at the same time in opposite directions at speeds 81 km/h and 75 km/h. What was the distance between them after 2 hours 50 minutes of driving. 7. Geometric plan At what scale the building plan if one side of the building is 45m long and 12mm long on a plan? 8. Knight Knight passed 13 km long track in 26 minutes, what was his average speed? 9. Aircraft nose down How long will fall airliner from a height of 10000 m at speed 1,000 km/h? 10. Earth and Sun Earth is 150 million km away from the Sun, 300,000 km of sunlight travels in 1 second. How long will the light come from the Sun to the Earth? 11. Hours The lesson lasts 45 minutes. For the week, students have 18 lesson hours. How many are the actual hours? 12. Average speed Michal went out of the house by car at speed 98 km/h. He came into the goal place for 270 minutes. Determine the distance between the two places. 13. Speeds George pass on the way to school distance 200 meters in 165 seconds. What is the average walking speed in m/s and km/h? 14. Minute angle Determine size of angle, which takes minute hand for 75 minutes. 15. Pedestrian Dana started at 10:00 from the point A to point B. These points are distant 12 km. Determine how fast Daniel went when the place B arrived at 11:54. Speed express in km/h. 16. Bus 14 Boatesville is 65.35 kilometers from Stanton. A bus traveling from Stanton is 24.13 kilometers from Boatesville. How far has the bus traveled? 17. Wavelength Calculate the wavelength of the tone frequency 11 kHz if the sound travels at speeds of 343 m/s.	753	2,876	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.8125	4	CC-MAIN-2019-30	latest	en	0.946251
https://borenson.com/product/the-hands-on-equations-fractions-learning-system-hf-1000/	1,680,200,537,000,000,000	text/html	crawl-data/CC-MAIN-2023-14/segments/1679296949355.52/warc/CC-MAIN-20230330163823-20230330193823-00179.warc.gz	164,639,188	24,103	# The Hands-On Equations Fractions Learning System \$29.50 Complete program for use with one student. Includes manuals for Levels I and II, worksheets, answer key, and one student kit of game pieces, consisting of a flat laminated balance, a set of eight number cubes, and a set of 33 fraction blocks. Additional HOE Fractions student kits available for \$8.00 each (see next product item). Note: The set of fraction blocks used for the Hands-On Equations Fractions program is the same as the set used for the Developing Fractions Sense program. SKU: HF-1000-P ## The Hands-On Equations Fractions Learning System Each student needs the HOE Fractions Student Kit to represent and solve the equations concretely. This kit consists of a set of 8 numbered cubes, a flat laminated scale, and a set of 33 plastic fraction blocks. ## Materials needed for each student The Hands-On Equations Fractions Learning System is a complete program with one set of student manipulatives. The program consists of two manuals encompassing 12 lessons, a set of worksheets and an answer key, a flat laminated balance, a set of 8 number cubes, and a set of 33 fraction blocks. ## Materials needed for each teacher The class set of materials includes The Hands-On Equations Fractions Learning System, as well as a HOE Fractions student kit for each student. After providing the two instructional examples for a lesson, the teacher distributes the worksheet for that lesson. Each worksheet has 8 equations, 4 from the current lesson, and 4 from prior lessons. The teacher answer key enables the teacher to quickly check student solutions. ## Methodology #### Hands-On Equations Fractions Class Set This program consists of two levels encompassing a total of 12 lessons. As with the original Hands-On Equations program, students represent the fractional equation on a flat laminated scale using their game pieces, and then use “legal moves” to solve the equations. Each lesson comes with a worksheet containing 8 problems. Unique to HOE Fractions is that students solve first for the unit fraction of the unknown. Hence, if a fractional equation has been simplified to 7/3 y = 28, they first determine the value of the unit fraction 1/3 y, namely, 4. Hence y, consisting of 3 copies of the unit fraction 1/3 y, is 12. This methodology provides students with a sense-making approach to fractional linear equations. Each level ends with an abstract solution approach which may be presented to younger students as an enrichment lesson. For older students, the abstract solution approach provides a transition to solving these equations without manipulatives. Level II has fractional equations with negative constants. #### Hands-On Equations Fractions for the Interactive Whiteboard It is also highly recommended that the teacher obtain the HOE Fractions for the Interactive Whiteboard (Promethean, Mimio, etc.). A teacher without an interactive whiteboard can manipulate the digital game pieces from the computer keyboard. (This product is also available for the SMART Board.)	654	3,067	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.609375	3	CC-MAIN-2023-14	longest	en	0.903538
https://www.studentlance.com/solution/in-the-metro-area-there-are-four-major-supermarket-chains-5	1,519,595,082,000,000,000	text/html	crawl-data/CC-MAIN-2018-09/segments/1518891817437.99/warc/CC-MAIN-20180225205820-20180225225820-00224.warc.gz	991,019,732	5,287	# In the metro area there are four major supermarket chains. - 10961 Solution Posted by ## mareya Rating : No Rating Solution Detail Price: \$1.00 • From: , • Posted on: Thu 14 Jun, 2012 • Request id: None • Purchased: 0 time(s) • Average Rating: No rating Request Description In the metro area there are four major supermarket chains. To determine if there was a difference between these stores, regarding their pricing of food, a consumer group did a test. In the Eastgate area of town, each of the supermarket chains has a store. Twenty-five common household items were selected for the test. These items were purchased at each of the four stores and the prices were compared. To analyze this data, what would be the best statistical test to use? A. Two sample test B. Chi square C. ANOVA D. Regression correlation Solution Description In the metro area there are four major supermarket chains. To determine if there was a difference between these stores, regardi	230	984	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.5625	3	CC-MAIN-2018-09	latest	en	0.943215
https://www.bartleby.com/questions-and-answers/questions-24-are-based-on-the-following-information-consider-the-following-population-model-for-hous/e3208e6f-00e4-42da-9dc9-ae28b86f57df	1,582,335,490,000,000,000	text/html	crawl-data/CC-MAIN-2020-10/segments/1581875145621.28/warc/CC-MAIN-20200221233354-20200222023354-00505.warc.gz	668,166,314	29,145	Questions 2-4 are based on the following information:Consider the following population model for household consumption:cons= a Binc+ Bzeduc +B,hhsize + u, where cons is consumption, inc is income, educis the education level of household head, hhsize is the size of a household.2. Suppose a researcher estimates the model and gets the predicted value, cons_hat,and then runs a regression of cons_hat on inc, educ, and hhsize. One cana) be certain that the R2 is equal to 1b) be certain that the R2 is equal to 0c) be certain that the R2 is less than 1 but greater than 0.d) not be certain3. Suppose that the variable for consumption is measured with error, so conss = cons +e, where conss is the mismeaured variable, cons is the true variable, e is random, i.e., eindependent of all the regressors. We would expect that:a) OLS estimators for the coefficients will all be biasedb) OLS estimators for the coefficients will all be unbiasedc) all the standard errors will be bigger than they would be without the measurementerror Question 314 views #2 check_circle star star star star star 1 Rating Step 1 2) If the no of independent variable increases, Coefficient of determination i.e. R2 increases and R2 value is... Want to see the full answer? See Solution Want to see this answer and more? Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.* See Solution *Response times may vary by subject and question. Tagged in	363	1,489	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.65625	3	CC-MAIN-2020-10	latest	en	0.892507
http://mathhelpforum.com/algebra/227806-increase.html	1,526,840,327,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794863662.15/warc/CC-MAIN-20180520170536-20180520190536-00557.warc.gz	178,871,560	10,654	1. ## % Increase I have made some modifications to my automobile to increase the fuel mileage. The EPA estimates my car's hwy fuel mileage (at 55 mph) to be about 39 mpg. In a recent road test at an average speed of 72.8 mph on the hwy (a 32% increase in velocity over the 55 mph the EPA tests at) I achieved 41.6 miles per gallon. It is estimated that the average car is 18% less efficient at that speed differential (72.8 mph - 55 mph Speed diff = 17.8 mph). With the fuel saving devices I installed, at a higher velocity, 17.8 mph faster, I achieved better fuel mileage than the EPA estimated 39 mpg at 55 mph. This is a 2.6 mpg increase over the EPA estimate while driving 72.8 mph (41.6 - 39 mpg = 2.6 mpg increase). Considering the 18% loss in efficiency of an average car traveling 17.8 miles faster than 55 mph, can you help me solve for the actual mpg % increase I get now vs. the EPA mpg sticker equivalent? ( I can't drive 55!) 2. ## Re: % Increase Originally Posted by Jamez I have made some modifications to my automobile to increase the fuel mileage. The EPA estimates my car's hwy fuel mileage (at 55 mph) to be about 39 mpg. In a recent road test at an average speed of 72.8 mph on the hwy (a 32% increase in velocity over the 55 mph the EPA tests at) I achieved 41.6 miles per gallon. It is estimated that the average car is 18% less efficient at that speed differential (72.8 mph - 55 mph Speed diff = 17.8 mph). With the fuel saving devices I installed, at a higher velocity, 17.8 mph faster, I achieved better fuel mileage than the EPA estimated 39 mpg at 55 mph. This is a 2.6 mpg increase over the EPA estimate while driving 72.8 mph (41.6 - 39 mpg = 2.6 mpg increase). Considering the 18% loss in efficiency of an average car traveling 17.8 miles faster than 55 mph, can you help me solve for the actual mpg % increase I get now vs. the EPA mpg sticker equivalent? ( I can't drive 55!) The EPA says avg mpg @ 55 mph is 39 mpg. At 72.8 mph there is a loss of 18% fuel efficiency, i.e. the estimated mpg is $(1-18\%)39=32$ mpg Your test achieved 41.6 mpg at 72.8 mph $\dfrac {41.6}{32}\approx 1.3$ So you achieved at 30% increase in fuel efficiency over the EPA rating. 3. ## Re: % Increase Awesomeness! Thx Romsek. 1.3 x 100 -100 = 30% Problem solved.	644	2,284	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.21875	3	CC-MAIN-2018-22	latest	en	0.905541
https://www.insidehighered.com/blogs/mama-phd/math-geek-mom-ambition-many-dimensions	1,519,347,395,000,000,000	text/html	crawl-data/CC-MAIN-2018-09/segments/1518891814300.52/warc/CC-MAIN-20180222235935-20180223015935-00474.warc.gz	887,597,867	18,953	# Math Geek Mom: Ambition in Many Dimensions What really matters? ## By October 15, 2015 I love teaching Multivariate Calculus for many reasons. It is the piece of math that most reflects my training as an economist, is the area that is most open to bringing up topics relating to economics, and is, in fact, the area that allows me to teach undergraduate students techniques I learned in my Ph.D. program. However, I most enjoy talking to my students about the idea of imagining the world in more than three dimensions, a world that can include “n” dimensions, with “n” becoming as large as one might want it to be. Indeed, this is where I mention that, in my doctoral dissertation, I dealt with twenty five equations in twenty five dimensions. I always enjoy the confused looks this brings on faces of students who are used to thinking of the world as consisting on only three dimensions. I found myself thinking of the many dimensions that are accessible through math when I read a recent article from Time magazine. Called “Why Ambition isn’t working for Women,” This article explores the issue of how women approach ambition, even proposing that to call a woman “ambitious” is to insult her, in a back-handed way. Instead, the author of this article proposes that women are just as ambitious as men, but that their ambition reflects different types of ambition, including a search for a balanced life. I was particularly intrigued by one line that claims it is not that women are less ambitious, but rather that “They are just unable to commit to a structure that was set up for 50% of the population.” Of course, a similar thing can be said about the academy, as women here are again a recent addition. Indeed, can it be that women, in academia and out, are solving maximization problems that include more than the usual number of dimensions? Of course, it is not just women who are seeking a balanced life. I recall one colleague I know who took paternal leave, a decision that probably slowed down his progress towards what eventually became multiple promotions and an administrative position. However, along the way, focusing on his baby daughter, now in college, took his attention away from his work as a professor. Even for him, it seems that the structures of the market for paid labor were indeed created by and for people who did not have to make such decisions in multiple dimensions. When I think of what it is that I am ambitious about, I think of my work and my students, but I also recall that I want to do what I can to help my daughter live the best life possible. I want her to learn what she needs to learn to pursue the career of her choice, whatever that may turn out to be, and to make decisions that will help her find meaning in her life. I want to create a loving home for my family, one that helps us all to become the best people possible. And I want to nurture my family in a way that allows us to all leave this world a little better than we found it. Of course, when I look at my daughter who is growing up before my eyes, I am left with one thought. As far as she is concerned, she has already made this world a better place. And so I leave my readers with the question; what are the many dimensions that you confront in making decisions about your own work choices, and how do you balance them? + -	704	3,345	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.671875	3	CC-MAIN-2018-09	latest	en	0.981113
https://physics.stackexchange.com/questions/681906/confusion-about-the-collision-of-a-ball-against-a-wall	1,702,127,513,000,000,000	text/html	crawl-data/CC-MAIN-2023-50/segments/1700679100909.82/warc/CC-MAIN-20231209103523-20231209133523-00337.warc.gz	507,253,974	43,042	# Confusion about the collision of a ball against a wall I'm facing a lot of confusion regarding the conservation of momentum when a ball deflects from an infinitely massive wall. Suppose, the ball makes an angle $$\alpha$$ with the normal, before the deflection. After it, it makes the angle $$\beta$$ against the normal. Now I know, in a perfectly elastic collision, the coefficient of restitution $$e$$ is equal to $$1$$. In that case, it turns out that the angle of incidence is equal to the angle of reflection, such that $$\alpha=\beta$$. My confusion comes, when we consider the non-elastic general cases. I want to know, what is the relation between $$\alpha$$ and $$\beta$$ when the collision is not perfectly elastic. According to some sources, we have : $$\alpha+\beta=\frac{\pi}{2}$$ In some other examples, this is not true. In those examples, $$\alpha+\beta$$ can take any value. I'm inclined to believe the latter, that the sum of the angles can take any random value depending on the coefficient of restitution and so on. However, what is the special case, when the sum of these angles is $$\pi/2$$, and how can I prove it ? • Inelastic is a general name for every case energy isn't conserved. This about the case that only a tiny fraction of the energy goes to waste, meaning energy is approximately conserved and the angles are almost the same. If that is the case, then surely it can't be that the general formula is $\alpha+\beta=\pi / 2$. Dec 10, 2021 at 19:23 • @OfekGillon yes, but I've seen examples when we had $\beta=\pi/6$ and we had to consider $\alpha=\pi/3$ just to proceed with the sum. It was a problem where we had been given the coefficient of restitution and the angle of deflection, and had to calculate percentage change in kinetic energy. Dec 10, 2021 at 19:28 • if you were given the angle of deflection, then what seems to be the problem? This was a specific case that you couldn't know the angle a priori without extra knowledge about "what happened during the collision" Dec 10, 2021 at 19:36 • What source states alpha + beta = pi/2? If the ball is sticky putty it will stick to the wall. Dec 10, 2021 at 19:40 • @JohnDarby is it true then that when two balls collide, but not along the line joining their COM, they are deflected in different directions, and the sum of angles of deflection is $\pi/2$ ? Dec 10, 2021 at 19:44 Suppose that the wall is parallel to the $$y$$ axis, and that the incoming velocity vector is $${\bf v}_{\rm in}= (v_x,v_y)$$ giving $$\alpha= \tan^{-1}(v_y/v_x)$$. With coefficient of restitution $$c$$, the outgoing velocity is $${\bf v}_{\rm out}= (-cv_x, v_y)$$ making $$\beta = \tan^{-1}(v_y/cv_x)$$ Then $$\tan(\alpha+\beta)=\frac{\tan \alpha +\tan\beta}{1-\tan\alpha\tan\beta},$$ is not going to be infinity, so $$\alpha+\beta\ne \pi/2$$. • with \begin{align} &\tan(\beta)=\frac{1}{\epsilon}\,\tan(\alpha)\\ &\alpha+\beta=\frac{\pi}{2}\\ \end{align} you obtain unique solution for $~\alpha~,\beta~$ so what is wrong with this solution ?? • @Eli Because $\alpha$ is an arbitrary angle between 0 and $\pi/2$. It's determined by how you throw the ball at the wall. Dec 11, 2021 at 22:56 • O.k I see , but if I know the coefficient of restitution i can choose $\alpha$ ? and obtain $\alpha+\beta=\pi\2$ For an elastic collision between balls of equal mass $$\alpha + \beta = {\pi \over 2}$$. This is not true in general. See a physics mechanics test, such as Mechanics by Symon for the general case.	950	3,482	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2023-50	longest	en	0.954413
http://politicalcalculations.blogspot.com/2010/01/unexpected-source-of-increasing-income.html	1,513,163,075,000,000,000	text/html	crawl-data/CC-MAIN-2017-51/segments/1512948522999.27/warc/CC-MAIN-20171213104259-20171213124259-00198.warc.gz	230,295,996	117,399	Unexpectedly Intriguing! January 14, 2010 Having now created a tool for estimating how many federal employees will fall within a given range of income for 2010, we thought we might go the extra mile and show approximately how many of the U.S. government's two-million strong civilian labor force fall into each \$100 interval from \$0 through \$300,000. Our chart visualizing the distribution of federal employee salaries for 2010 is below: We find that very few employees of the federal government are likely to make less than \$21,000, while the greatest concentration will be clustered between \$42,000 and \$43,000. The median income for federal employees in 2010 is roughly \$63,915 while the average income is \$72,377. Looking up the income scale, we find that based upon the 2008 sample of federal employees named "Smith" that we used to create our income distribution model and the average percentage raises of 3.9% for 2009 and 2.0% for 2010, we would project that just over 30,000 federal employees would make more than \$170,000 in 2010. But the real question is how many federal employees will actually make \$170,000 or more in 2010. If there is a significant difference between the actual figure and our projected figure of 30,000, that would be an indication that the federal government's pay scales are being significantly skewed to favor its top paid staff. After all, with just a 0.8% difference in the inflation-adjusted value of a U.S. dollar between 2008 and 2010, the large, progressively greater rightward shift in the distribution of the upper echelons of income for federal employees between 2008 to 2010 indicates a rather significant increase in the income inequality of both the federal government's employees and the United States. Who knew that the federal government's high salaries and regular raises were such a contributing factor to a perceived problem that former Federal Reserve Chairman Alan Greenspan described as a "disturbing trend"?! Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at: ironman at politicalcalculations.com Recent Posts Applications This year, we'll be experimenting with a number of apps to bring more of a current events focus to Political Calculations - we're test driving the app(s) below! Most Popular Posts Quick Index Site Data JavaScript The tools on this site are built using JavaScript. If you would like to learn more, one of the best free resources on the web is available at W3Schools.com.	607	2,773	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.828125	3	CC-MAIN-2017-51	latest	en	0.9553
http://ivyhawnschool.org/classroom-connect/sixth-grade/cornelius-c/october-2-6-2017-6th-grade-science/	1,531,908,548,000,000,000	text/html	crawl-data/CC-MAIN-2018-30/segments/1531676590127.2/warc/CC-MAIN-20180718095959-20180718115959-00261.warc.gz	193,107,503	8,849	# October 2-6, 2017 3 Explanation – Motion Graphing 3 Explanation – Balanced and Unbalanced Forces Teacher Crystal Cornelius Science 6 7 Force and Motion Standard(s) Taught SC.6.P.13.1 Investigate and describe types of forces including contact forces and forces acting at a distance, such as electrical, magnetic, and gravitational. SC.6.P.13.2 Explore the Law of Gravity by recognizing that every object exerts gravitational force on every other object and that the force depends on how much mass the objects have and how far apart they are. SC.6.P.13.3 Investigate and describe that an unbalanced force acting on an object changes its speed, or direction of motion, or both. SC.6.P.12.1 Measure and graph distance versus time for an object moving at a constant speed. Interpret this relationship. SC.6.N.1.1 Define a problem from the sixth grade curriculum, use appropriate reference materials to support scientific understanding, plan and carry out scientific investigation of various types, such as systematic observations or experiments, identify variables, collect and organize data, interpret data in charts, tables, and graphics, analyze information, make predictions, and defend conclusions. SC.6.N.1.4 Discuss, compare, and negotiate methods used, results obtained, and explanations among groups of students conducting the same investigation SC.6.N.1.2 Explain why scientific investigations should be replicable. Learning Targets and Learning Criteria Students will: • investigate and describe types of forces, including: o Forces at a distance: electrical, magnetic, and gravitational o Contact forces: normal, applied, friction Students will: • explore the Law of Gravity by recognizing that every object exerts gravitational force on every other object and that force depends on how much mass the objects have and how far apart they are Students will: • investigate and explain how an unbalanced force acting on an object changes its speed or direction of motion (or both) Students will: • measure and graph distance versus time for an object moving at a constant speed • analyze and interpret graphs and data tables of distance and time for an object moving at a constant speed Students will: • design an investigation to test the effect a variable (mass, tire size, ramp incline, etc.) has on the speed of an object using a minimum of 10 trials o collect data and organize it in a table o analyze the data through graphing o draw conclusions from the analysis about the speed and position of the object as well as the forces acting on it • discuss and compare methods used, results obtained, and proposed explanations among groups of students conducting the same investigation • explain why scientific investigation should be Classroom Activities Motion/Speed Graphs: Distance Vs. Time on a graph Law of Gravity Assignments Due	598	2,862	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2018-30	latest	en	0.89084
https://math.answers.com/Q/What_is_point_487_expressed_as_a_fraction_in_its_most_simple_from	1,660,698,655,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882572833.78/warc/CC-MAIN-20220817001643-20220817031643-00026.warc.gz	357,379,218	40,577	0 # What is point 487 expressed as a fraction in its most simple from? Wiki User 2010-07-02 13:51:51 0.487 = 487/1000. Since neither 2 nor 5 are factors of 487, this can't be simplified. Wiki User 2010-07-02 13:51:51 Study guides 20 cards ➡️ See all cards 3.74 1201 Reviews	101	281	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.828125	3	CC-MAIN-2022-33	latest	en	0.867637
https://www.bearnaiserestaurant.com/writing-advice/can-you-have-multiple-dependent-variables/	1,723,273,024,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640790444.57/warc/CC-MAIN-20240810061945-20240810091945-00093.warc.gz	519,509,079	11,641	Can you have multiple dependent variables? Can you have multiple dependent variables? Researchers in psychology often include multiple dependent variables in their studies. The primary reason is that this easily allows them to answer more research questions with minimal additional effort. Which analysis is done when you have two dependent variables? Multivariate analysis is needed when there are 2 or more Dependent Variables (DV) are in your research model. Can you run a regression with multiple dependent variables? Yes, it is possible. What you’re interested is is called “Multivariate Multiple Regression” or just “Multivariate Regression”. I don’t know what software you are using, but you can do this in R. Here’s a link that provides examples. What are multiple dependent variables? Multivariate Multiple Regression is the method of modeling multiple responses, or dependent variables, with a single set of predictor variables. For example, we might want to model both math and reading SAT scores as a function of gender, race, parent income, and so forth. How many dependent variables can a study have? A well-designed experiment normally incorporate one or two independent variables, with every other possible factor eliminated, or controlled. There may be more than two dependent variables in any experiment. Why use a MANOVA instead of ANOVA? The correlation structure between the dependent variables provides additional information to the model which gives MANOVA the following enhanced capabilities: Greater statistical power: When the dependent variables are correlated, MANOVA can identify effects that are smaller than those that regular ANOVA can find. How many dependent variables are typically taken in multiple regression? one dependent variable Multiple linear regression is used to estimate the relationship between two or more independent variables and one dependent variable. What is multivariate analysis in research? Multivariate analysis is based in observation and analysis of more than one statistical outcome variable at a time. In design and analysis, the technique is used to perform trade studies across multiple dimensions while taking into account the effects of all variables on the responses of interest. How many dependent variables can be used in a MANOVA? It can assess only one dependent variable at a time. This limitation can be an enormous problem in certain circumstances because it can prevent you from detecting effects that actually exist. MANOVA provides a solution for some studies. What is multivariate analysis in research methodology? How many variables is too many for regression? Many difficulties tend to arise when there are more than five independent variables in a multiple regression equation. One of the most frequent is the problem that two or more of the independent variables are highly correlated to one another. This is called multicollinearity. What does MANOVA do with multiple dependent variables? Multivariate ANOVA (MANOVA) extends the capabilities of analysis of variance (ANOVA) by assessing multiple dependent variables simultaneously. ANOVA statistically tests the differences between three or more group means. What are multivariate analysis techniques? There are two types of multivariate analysis techniques: Dependence techniques, which look at cause-and-effect relationships between variables, and interdependence techniques, which explore the structure of a dataset. • October 11, 2022	633	3,496	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.25	3	CC-MAIN-2024-33	latest	en	0.914415
https://community.powerbi.com/t5/Desktop/Measure-for-a-DistinctCount-by-Categories/m-p/773716	1,568,930,717,000,000,000	text/html	crawl-data/CC-MAIN-2019-39/segments/1568514573735.34/warc/CC-MAIN-20190919204548-20190919230548-00170.warc.gz	436,179,930	120,766	cancel Showing results for Did you mean: Highlighted Regular Visitor ## Measure for a DistinctCount by Categories Hi Power BI Users, I have a table similar to below and trying to create a new measure by using DAX to get the distinct count of prodid&customers. In short, we are trying to calculate how many orders we have for each of the productID under each Customer. Sampletable: Order ProductID Customers XX1 A X XX6 A X XX7 A Y XX10 A Y XX2 B Y XX3 B X XX4 B X XX5 B Y XX8 C X XX9 C X Desired output on 4th column: Order Product ID Customers Distinct Count of ProdID x Customers XX1 A X 2 XX6 A X 2 XX7 A Y 2 XX10 A Y 2 XX2 B Y 1 XX3 B X 3 XX4 B X 3 XX5 B Y 3 XX8 C X 2 XX9 C X 2 I've tried out with below DAX but it shown error below, Countitem = CALCULATE(distinctcount(Sampletable[Order]),(Sampletable[ProductId] && Sampletable[Customers])) Error message: The expression contains multiple columns, but only a single column can be used in a True/False expression that is used as a table filter expression. Hope someone could shed some lights on this. 1 ACCEPTED SOLUTION Accepted Solutions Regular Visitor ## Re: Measure for a DistinctCount by Categories @Vvelarde, this has been resolved by added in "KEEPFILTERS". DC Prod by Customers = CALCULATE ( DISTINCTCOUNT ( Table1[Order ] ), KEEPFILTERS(ALLEXCEPT ( Table1, Table1[ProductID ], Table1[Customers] )) ) THANK YOU! 3 REPLIES 3 Super Contributor ## Re: Measure for a DistinctCount by Categories Hi, try with this measure: ```DC Prod by Customers = CALCULATE ( DISTINCTCOUNT ( Table1[Order ] ), ALLEXCEPT ( Table1, Table1[ProductID ], Table1[Customers] ) )``` Regards Victor Lima - Peru Proud to be a Datanaut! Regular Visitor ## Re: Measure for a DistinctCount by Categories Thanks, @Vvelarde . This works pretty well if we are not filtering the data. However, lets say if there is a date column and we would love to filter by date and check what is the distinctcount, what can be @Vvelarde wrote: Hi, try with this measure: ```DC Prod by Customers = CALCULATE ( DISTINCTCOUNT ( Table1[Order ] ), ALLEXCEPT ( Table1, Table1[ProductID ], Table1[Customers] ) )``` Regards Victor done in order to sync with the filters? Regular Visitor ## Re: Measure for a DistinctCount by Categories @Vvelarde, this has been resolved by added in "KEEPFILTERS". DC Prod by Customers = CALCULATE ( DISTINCTCOUNT ( Table1[Order ] ), KEEPFILTERS(ALLEXCEPT ( Table1, Table1[ProductID ], Table1[Customers] )) ) THANK YOU! Announcements #### Microsoft Business Applications October Virtual Launch Event Join us for an in-depth look at the new innovations across Dynamics 365 and the Microsoft Power Platform. #### Power BI Helps Homeless and Trouble Youth We spoke with Power BI Super User, Greg Deckler, about his charity work #### Watch Sessions On Demand! Continue your learning in our online communities. Top Kudoed Authors Users Online Currently online: 186 members 1,841 guests Recent signins:	782	2,982	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.71875	3	CC-MAIN-2019-39	latest	en	0.863475
https://cracku.in/16-read-the-statements-given-below-1-atomic-bomb-deto-x-xat-2013	1,721,147,480,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763514759.37/warc/CC-MAIN-20240716142214-20240716172214-00880.warc.gz	162,627,453	28,659	Instructions Analyse the following passage and provide appropriate answers An example of a scientist who could measure without instruments is Enrico Fermi (1901-1954), a physicist who won the Nobel Prize in physics in 1938. He had a well-developed knack for intuitive, even casual-sounding measurements. One renowned example of his measurement skills was demonstrated at the first detonation of the atom bomb, the Trinity Test site, on July 16, 1945, where he was one of the atomic scientists observing the blast from base camp. While final adjustments were being made to instruments used to measure the yield of the blast, Fermi was making confetti out of a page of notebook paper. As the wind from the initial blast wave began to blow through the camp, he slowly dribbled the confetti into the air, observing how far back it was scattered by the blast (taking the farthest scattered pieces as being the peak of the pressure wave). Fermi concluded that the yield must be greater than 10 kilotons. This would have been news since other initial observers of the blast did not know that lower limit. After much analysis of the instrument readings, the final yield estimate was determined to be 18.6 kilotons. Like Eratosthenes, Fermi was aware of a rule relating one simple observationāthe scattering of confetti in the wind āto a quantity he wanted to measure. The value of quick estimates was something Fermi was familiar with throughout his career. He was famous for teaching his students skills at approximation of fanciful-sounding quantities that, at first glance, they might presume they knew nothing about. The best-known example of such a "Fermi question" was Fermi asking his students to estimate the number of piano tuners in Chicago, when no one knows the answer. His studentsāscience and engineering majorsāwould begin by saying that they could not possibly know anything about such a quantity. Of course, some solutions would be to simply do a count of every piano tuner perhaps by looking up advertisements, checking with a licensing agency of some sort, and so on But Fermi was trying to teach his students how to solve problems where the ability to confirm the results would not be so easy. He wanted them to figure out that they knew something about the quantity in question. Question 16 # Read the statements given below:1. Atomic bomb detonation was a result of Fermi's Nobel Prize contribution2. Fermi's students respected him as a scientist3. Yield of atomic bomb can only be measured in KilotonsWhich of the following statement(s) can be inferred from the passage? Solution All the statements cannot be inferred from the passage. 1 "Atomic bomb detonation was a result of Fermi's Nobel Prize contribution" is incoherent with the data. Though both the events occured in chronological order butĀ the inference is incorrect. 2. "Fermi's students respected him as a scientist' is not mentioned in the paragraph. 3. "Yield of atomic bomb can only be measured in Kilotons " cannot be inferred from the paragraph. ### Video Solution • All Quant Formulas and shortcuts PDF • 15 XAT previous papers with solutions PDF	672	3,148	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.203125	3	CC-MAIN-2024-30	latest	en	0.979498
https://gamedev.stackexchange.com/questions/53400/cave-generation-with-perlin-worms?noredirect=1	1,716,895,194,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971059085.33/warc/CC-MAIN-20240528092424-20240528122424-00706.warc.gz	227,986,042	40,324	# Cave generation with Perlin worms I'm currently trying to generate a Minecraft like voxel terrain with 3D Simplex Noise and also want to implement caves. I found the method of Perlin Worms in this thread, which generates really nice results. However, I have no clue on how to generate it on a chunk by chunk basis. Is this possible or are there any alternatives that produce a similar worm like cave on a chunk by chunk basis? Edit: This is the problem which I don't know how to solve. Edit2: This is the result from combining 2D Ridged Multifractal Noise with a Simplex Noise heightmap. Still needs some tweaking, but this is pretty much the result I wanted. Thanks to Byte56. • You should use direct formulas for terrain generation if possible. If you can calculate a voxel just based on its coordinates and without information about adjacent ones, chunks aren't a problem anymore. If you want to use perlin worms, which I guess can't be calculated for each voxel individually, take a look on the last paragraph in the accepted answer of the question you linked. Apr 6, 2013 at 12:15 • Basically, that is my question. The basic terrain is easy to calculate with just the coordinates given, but I don't know how to calculate, whether there is a cave or not. Apr 6, 2013 at 12:22 Most perlin noise algorithms will allow you to retrieve the noise value at any given location, with something like noise(x,y,z). This makes it fairly trivial to generate noise on a chunk by chunk basis. All you need to do is pass the global position, instead of the chunk position. for(int i = 0; i < CHUNKMAX_X; i++) for(int j = 0; j < CHUNKMAX_Y; j++) for(int k = 0; k < CHUNKMAX_Z; k++) if(isSolid(perlinNoise.get(chunkPosition.x + i, chunkPosition.y + j, chunkPosition.z + k)) thisChunk[i,j,k] = new Voxel(solid); else thisChunk[i,j,k] = new Voxel(air); So you can see, we're generating terrain for the chunk, by iterating over the chunk bounds, and checking to see if that global position is solid or not. This is likely the same methodology you're using to generate the terrain in general. perlinNoise.get takes a global position and returns its density. Where isSolid would just be a simple test to see if the voxel is "dense enough" to qualify for solid. perlinNoise.get can be more complex than just a simple noise algorithm. You can have checks based on the depth of the voxel in your world. For example, if the voxel is below what you've decided is "absolute base ground level" then it can use the perlin worms algorithm to return a density, if it's above the absolute base, it can use a normal density function to give you more varied terrain. I would recommend some blending between the two however. Combining different Perlin noise functions is just something you have to play with and see what works. It's best to set up your environment so that you can just change some values and hot-swap the terrain without needing to reload your game. Happy experimenting. • Thanks for the answer. However, I don't know how to use "regular" Perlin or Simplex Noise to create the worm-like caves. They are either spherical, scaled into one direction or not very long. And for the Perlin Worms algorithm, I don't know how to determine whether the current position is inside a worm, without knowing the head position. Apr 6, 2013 at 16:00 • The worms algorithm does not rely on knowing the "head position". From what you're saying, it appears your problem is not with generating noise on a chunk by chunk basis. There are tutorials on the question you linked for implementing perlin worms. How to generate "regular" terrain can be found here. Apr 6, 2013 at 16:10 • The basic terrain generation with 3D Simplex Noise works fine, but I don't know how to determine a worms position/segment inside a chunk. From what I understood, you give the worm a head position and n segments and calculate the angle between the segments with the noise function. Here is an image that may better explain, what my problem is: link. Apr 6, 2013 at 16:49 • There are different types of noise. Using a head position and following from there is one method. If you look near the bottom of this page, you'll find some noise that creates the kind noise you want without needing a head position first. Apr 6, 2013 at 17:19 • Thanks for the hint on that. I implemented the 3D Ridged Multifractal Noise, though, the result with it was rather unsatisfying. I will try to combine it in 2D with a 2D Heightmap and then report again. This may create more worm-like caves. Apr 6, 2013 at 19:35 I think this is how it works in Minecraft. Each worm has a maximum length (let's call it M). The heads of each worm are calculated based on the chunk position. When you render each chunk, you have to check all chunks within an M radius, and follow all of their worms. It's not ideal in terms of performance, but it does work.	1,146	4,883	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.28125	3	CC-MAIN-2024-22	latest	en	0.904936
https://mathematica.stackexchange.com/questions/246972/two-identical-functions-returning-different-values	1,716,827,712,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971059044.17/warc/CC-MAIN-20240527144335-20240527174335-00270.warc.gz	346,835,526	40,405	# Two "identical" functions returning different values When taking the norm of a complex number in Mathematica you get the same result when you use the Norm function or the Abs function. So why do the following two functions (kp and kpp) return different results? k[k1_, k2_] := {Sqrt[3]/2 k1, k1/2 + k2, 0}; kp[k1_, k2_] := (#/Norm[#]) &[k[k1, k2][[1]] + I k[k1, k2][[2]]] kpp[k1_, k2_] := (k[k1, k2][[1]] + I k[k1, k2][[2]])/Abs[k[k1, k2][[1]] + I k[k1, k2][[2]]] array = Range[-1.4, 1.4, 0.1]; kp[2 π (array + 0.5), 2 π (-array + 0.5)] kpp[2 π (array + 0.5), 2 π (-array + 0.5)] To me these functions should return the same list of values but instead, they return two different lists of values. Since you did not show the result of your execution, I think people misunderstood your situation. The problem is not in the inequality of the two function forms. The problem is that the functions kp and kpp, when operated on an array of complex numbers (actually, the real and complex parts, separated), behave differently. I could not find more detail reasons, but they operate on the list in different order. The results are exactly the same if you change the definition of kp to kp[k1_, k2_] := (#/Norm[#])& /@ (k[k1, k2][[1]] + I k[k1, k2][[2]]) Map is, surprisingly, much safer to use when your function may work on only one data point or a list of data points. Don't rely too much on square brackets to tell whether you are now working on scalar or vector. Things may go wrong at unexpected situations. Clear["Global"] k[k1_, k2_] := {Sqrt[3]/2 k1, k1/2 + k2, 0}; kp[k1_, k2_] := (#/Norm[#]) &[k[k1, k2][[1]] + I k[k1, k2][[2]]] kpp[k1_, k2_] := (k[k1, k2][[1]] + I k[k1, k2][[2]])/ Abs[k[k1, k2][[1]] + I k[k1, k2][[2]]] Test whether the functions are equivalent array = {a, b}; FullSimplify[kp[c (array + d), c (-array + d)]] === FullSimplify[kpp[c (array + d), c (-array + d)]] ( False ) They are not even equivalent for real values kp[c (array + d), c (-array + d)] // ComplexExpand // Simplify ( {(c ((-I + Sqrt[3]) a + (3 I + Sqrt[3]) d))/( 2 Sqrt[c^2 (a^2 + b^2 + 6 d^2)]), ( c ((-I + Sqrt[3]) b + (3 I + Sqrt[3]) d))/(2 Sqrt[c^2 (a^2 + b^2 + 6 d^2)])} ) kpp[c (array + d), c (-array + d)] // ComplexExpand // Simplify ( {(c ((-I + Sqrt[3]) a + (3 I + Sqrt[3]) d))/(2 Sqrt[c^2 (a^2 + 3 d^2)]), ( c ((-I + Sqrt[3]) b + (3 I + Sqrt[3]) d))/(2 Sqrt[c^2 (b^2 + 3 d^2)])} ) % === %% ( False ) Consider: kp[k1, k2] == kpp[k1, k2] As you see, the numerator are identical, therefore the difference must come from the denominator. Further note that array and also the arguments are vectors. Now investigate how Norm and Abs behave with vector arguments. E.g.: Norm[{1,1}] Abs[{1,1}] Norm gives a scalar, Abs a vector! Now if we simplify array to: array = Range[1.3, 1.4, 0.1]; The numerator is in both cases the same: ( {9.79452 + 0.628319 I, 10.3387 + 0.314159 I} ) But the denominator: ( 14.2588 ) ( {9.81465, 10.3434} *) In the first case with Norm we have: {9.79452 + 0.628319 I, 10.3387 + 0.314159 I} / 14.2588 ==> {0.686909 + 0.0440653 I, 0.725071 + 0.0220326 I} And in the second case with Abs: {9.81465, 10.3434} / {9.81465, 10.3434} ==> {0.997949 + 0.0640184 I, 0.999539 + 0.0303728 I} `	1,186	3,260	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.4375	3	CC-MAIN-2024-22	latest	en	0.717488
https://www.angelfire.com/planet/iconosphere-zix/zix/laws-of-form.html	1,624,050,624,000,000,000	text/html	crawl-data/CC-MAIN-2021-25/segments/1623487641593.43/warc/CC-MAIN-20210618200114-20210618230114-00410.warc.gz	566,998,299	2,324	``` [plase use the BACK key to return to your previous pageLaws of Form (iconosphere entry) Refer to: -[Laws of Form]- (in SCI: Maths-) The so-called "Laws of Form" were the brain child of G. Spenser Brown in attempting to resolve both the self-referential and Russell's paradoxes. Following a brief review of these two topics, a discussiojn will be attempted as to how the laws of form act and are used. When Bertrand Russell and Alfred North WHitehead set out to completely systemitise maths in the late 1800's, little did they realise how completely hopeless the whole problem would soon bcome. In their monumental task of "starting from scratch" and then carefully building theorem on theorem until (by #500 or so) they were finally able to show that indeed 1 + 1 = 2 In the meantime, a chap in Germany by the name of Otto Friche (?sp?) was doing the same thing for axiomatic set theory. He had finished the first volume and sent it off to Russell for commenary. Russell responded with a simple question which (so the story goes) drove Friche to suicide. This so-called "Russell's Paradox" goes something like this: Consider two catalogs (sets), the first catalog consists of all sets that CONTAIN THEMSELVES. the second catalog consists of all sets that do NOT contain themselves. All, fine and well, we take each set, look at it, and then place it the correct catalog. But, of course the two catalogs are themselves sets, so..... If catalog one contains itself (which it may or may not) then it is placed in its entries. If catalog one does NOT contain itself, then it is placed in catalog two. Well, and good. Now, for catalog 2. If catalog 2 DOES contain itself, then it gets listed in catalog one. But, wait, it can't contain itself, since it's the catalog that contains sets that do NOT contain themselves. Therefore, it must not contain itself. But. By definition, catalog 2 contains all sets that do NOT contain themselves, so by its own definition, it MUST list itself. This was all (sort of) taken care of by [Goedel's Theorem] But, still even introducing the "theory of types" as Russell attempted to do, not much could be done with that nasty paradox floating around like an albatros around the neck. Enger, G. Specner Brown. He postulated a new kind of meta-logic as well as classifying self-referential problems as equivalently to the imaingary number i. (The square root of -1). This helped imensely, although the work still "has problems". (THere's no escaping Goedel's Theorem). The Laws of Form ```	632	2,550	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.234375	3	CC-MAIN-2021-25	latest	en	0.958811
http://www.electronicdesign.com/test-amp-measurement/microcontroller-based-sine-wave-generator-has-crystal-accuracy	1,532,194,831,000,000,000	text/html	crawl-data/CC-MAIN-2018-30/segments/1531676592650.53/warc/CC-MAIN-20180721164755-20180721184755-00085.warc.gz	448,053,373	16,736	Electronic Design # Microcontroller-based sine-wave generator has crystal accuracy One way to generate a sine wave is to pass a square wave through a low-pass filter. The high-order harmonics will be filtered out, leaving only the fundamental. The higher the order of the filter, the purer the sine wave produced. The MAX292 is an 8th-order Bessel low-pass switched-capacitor filter. Its -3-dB corner frequency is controlled by its clock frequency in a 1:100 ratio. Using a 1:64 input-signal/clock ratio will produce a very clean sine wave. If this ratio is kept constant across the entire input frequency range, the amplitude of the filter output will be independent to the frequency. Here, a high-speed microcontroller (Atmel’s AT90S1200) is used to generate square-wave and clock signals (see the figure). Meanwhile, the microcontroller accepts 13-bit frequency-control signals. The 13 bits are divided into two parts—the lower eight bits (D7-D0) are named delay_1, and the upper five bits (D12-D8) are named delay_2. No matter what frequency is selected, the base square wave and the clock will maintain a 1:64 ratio. Thus, the circuit will generate both sine-wave and square-wave signals at the base frequency, as well as a higher (64x) frequency square wave. The frequency of the sine wave is given by: fOUT = fOSC / (98688 * (delay_2-1) + 384 * delay_1 + 1408) where fOUT is the sine-wave frequency and fOSC is the clock frequency of the microcontroller. To generate a specific frequency sine wave, the values of delay_1 and delay_2 can be calculated from the equation above. For instance, assume fOSC is 16 MHz and a 60-Hz sine wave is needed. By using this equation, you can determine that delay_1 and delay_2 should be 177 and 3, respectively. Plugging these numbers back into the equation, the calculated output frequency will be 59.98 Hz. The useful frequency range that can generated by this circuit is between 5.2282 Hz and 8928.6 Hz when the microcontroller is operating at a 16-MHz clock frequency. See associated figure	488	2,047	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.921875	3	CC-MAIN-2018-30	latest	en	0.905759
https://blog.lore.ai/convolutional-neural-networks-using-numpy-part-1/	1,611,341,115,000,000,000	text/html	crawl-data/CC-MAIN-2021-04/segments/1610703531335.42/warc/CC-MAIN-20210122175527-20210122205527-00578.warc.gz	249,479,636	29,026	# Convolutional Neural Networks using Numpy – Part 1 There are many powerful tools like Keras and Tensorflow out there to make convolutional neural networks (CNNs). However, unless I have opened the hood and peeked inside, I am not really satisfied that I know something. If you are like me read on to see how to build CNNs from scratch using Numpy (and Scipy). The code for this post is available in my repository There are many powerful tools like Keras and Tensorflow out there to make convolutional neural networks (CNNs). However, unless I have opened the hood and peeked inside, I am not really satisfied that I know something. If you are like me read on to see how to build CNNs from scratch using Numpy (and Scipy). I am making this post a multi part post. As of now I have planned two parts. In the first I will only do a two class classification problem on Fashion MNIST restricted to two classes. The images are monochromatic and the CNN will have only one convolutional filter followed by a dense layer. This simplicity will allow us to focus on the details of convolution without worrying about incidental complexity. In the second post I will tackle the full Fashion MNIST dataset (10 classes) and multiple convolution filters. I will also look at the CIFAR-10 dataset which has colored images in 10 classes. This is going to a technical post and certain familiarity with algebra and calculus is assumed. While I know that one can do a lot with tensorflow without knowing advanced math, I do not see how it is possible to peek under the hood without the same. Algebra is essential in the forward pass and algebra and calculus are useful to compute the derivative of the loss function with respect to the weights of the network which I derive in closed form. These are used to update the weights, something commonly known as back propagation. Readers unwilling to go through the derivation but able to follow the final expressions can still benefit by understanding the final results and implementing (or following my implementation) in numpy. # Convolution In this section we will discuss what exactly we mean by convolution in image processing and how it is related to the implementation in scipy. This is useful as scipy implementation is much faster than a naive numpy implementation. In the end we will consider an example where we compute the convolution by hand and by using scipy as a sanity check. For simplicity we will work in 1 dimension for this section but the arguments extend simply to any number of dimensions. Mathematically a convolution of a function f by a function g is defined as This definition is useful in physics and signal processing. For instance in electromagnetism g could be the charge density and f (called a Green’s function in that case) would help us compute the potential due to said charge distribution. (See books on electromagnetism like Jackson or particle physics like Peskin and Schroeder for more details). In image processing this definition is a bit backwards in that for a convolution window g , of length , we would want the convolution at to be the weighted average of such that the value at −/2+ is weighted by (). Thus, for image processing purposes the correct definition is Note that this will require value of f(x) for negative values of x and if this is put in as it is then numpy will interpret the negative numbers as indexing from the end of the array. Thus, while implementing this in numpy, we need to make sure that the original array is embedded in a bigger 0-padded one and negative indexes are understood appropriately. In our implementation of CNNs, we will use scipy.convolve as it will be faster than a naive implementation in numpy. So it is useful to understand the way scipy.convolve is related to what we want. Scipy’s convolve is for signal processing so it resembles the conventional physics definition but because of numpy convention of starting an array location as 0, the center of the window of g is not at 0 but at K/2. So scipy.convolve uses the definition Now, we if reverse the scipy convolution window we have y ->K-y and that makes the integral Thus we will get the result we want by giving the reversed array of the convolution window to scipy.convolve. As an example consider the signal and filter given below. We then implement the convolution by hand and using scipy in the following command `convolve(sig,win[::-1],'same')` The results are plotted below. The reader is advised to convince herself that the results are the same either by eyeballing the graphs or implementing the two methods herself. We are now ready to implement CNNs using numpy. # Two-class classification As our first example we implement a two class classification using numpy. I discuss this in full detail and for the other examples in latter posts I will expect that the reader has understood this example in detail and go a bit faster. For the data we will used the Fashion MNIST. We get this from Keras as it is easily available but do note that this is the only use of Keras in this post. `(X_train_full, y_train_full), (X_test, y_test) = keras.datasets.fashion_mnist.load_data()` We then restrict our dataset to just the category 1 and 3. ```cond=np.any([y_train_full==1,y_train_full==3],0)X_train_full=X_train_full[cond] y_train_full=y_train_full[cond] X_test=X_test[np.any([y_test==1,y_test==3],0)] y_test=y_test[np.any([y_test==1,y_test==3],0)]y_train_full=(y_train_full==3).astype(int) y_test=(y_test==3).astype(int)``` We then split our training set into a train and validation set ```X_train, X_valid = X_train_full[:-1000], X_train_full[-1000:] y_train, y_valid = y_train_full[:-1000], y_train_full[-1000:] ``` And we finally standardize the data as Neural Networks work best with data with vanishing mean and unit standard deviation. ```X_mean = X_train.mean(axis=0, keepdims=True) X_std = X_train.std(axis=0, keepdims=True) + 1e-7 X_train = (X_train - X_mean) / X_std X_valid = (X_valid - X_mean) / X_std X_test = (X_test - X_mean) / X_std``` Our data are monochrome images that come as matrices of dimension L x L (here L=28). For simplicity we have only one convolution layer (we will relax this restriction in later posts) which is a matrix of size K x K (we will take K=3 in our examples) whose weights can be learnt. In addition we have a dense layer which is a matrix of size (LL) x 1. Note that we have not kept bias terms and the reader can include them as an exercise once she has understood this example. We now describe the forward pass, the error and the back propagation in some detail. ## Forward Pass • We get the images and refer to it as the 0-th layer 0 • We embed the image centered into one of size (+,+) with zero padding • Then we pass it through a convolutional layer and an activation function f1 (that we will take to be Relu). This is the first layer l1. • Finally we make a dense layer wrapped in a function f2 (which in this case we take to be a sigmoid function). This is the second layer l2. Note that even though the weights W2 are for a dense layer we have written the indices here without flattening as it has an intuitive interpretation in that it takes each pixel of the convoluted image and makes a weighted sum. Nevertheless, flattening both and doing a numpy dot is more performant and that is what we do in the code below. For the loss function we take the usual log-loss where y is the true outcome. Backpropagation is just a fancy word for saying that all the learnable weights are corrected by the gradient of the loss function with respect to the weights that are being learned. It is straightforward to differentiate the loss function with respect to W1 and W2 using the chain rule. • The derivative of the loss function with respect to the layer 2 is • The derivative of the loss function with respect to the dense layer weights is where we have used the fact that l2 is the output of the sigmoid function s(x) and that s’(x)=s(x)(1-s(x)). • Similarly, the derivative of the loss function with respect to layer 1 is • The derivative of the loss function with respect to the convolution filter is Now we have closed form expressions for the derivative of the loss function with respect to the weights of the convolution filter as well as the final dense matrix so we can update the weights (with a hyperparameter the learning rate) as That’s it. That is all that is required to write the implementation of the CNN. Before doing that we first note the loss function and accuracy for initial random weights so as to have a benchmark. Before training starts, the loss on average can be obtained analytically from the expression for the loss and is 1. The accuracy is 0.5. We will run our code over five epochs and see the loss and accuracy on the test and validation set. First we write some custom functions for Relu, its derivative and the sigmoid function that we require for the main code and a function to just do the forward pass and compute loss and accuracy to do so on the validation set ```def relu(x): return np.where(x>0,x,0) def relu_prime(x): return np.where(x>0,1,0) def sigmoid(x): return 1./(1.+np.exp(-x)) def forward_pass(W1,W2,X,y): l0=X l0_conv=convolve(l0,W1[::-1,::-1],'same','direct') l1=relu(l0_conv) l2=sigmoid(np.dot(l1.reshape(-1,),W2)) l2=l2.clip(10-16,1-10-16) loss=-(ynp.log(l2)+(1-y)np.log(1-l2)) accuracy=int(y==np.where(l2>0.5,1,0)) return accuracy,loss``` Now we write the main part of the code ```# learning rate eta=.001 for epoch in range(5): # custom code to keep track of quantities to # keep a running average. it is not shown for clarity. # the reader can implement her own or ask me in the comments. train_loss, train accuracy=averager(), averager() for i in range(len(y_train)): # Take a random sample from train set k=np.random.randint(len(y_train)) X=X_train[k] y=y_train[k] ##### FORWARD PASS ###### # First layer is just the input l0=X # Embed the image in a bigger image. # It would be useful in computing corrections # to the convolution filter lt0=np.zeros((l0.shape[0]+K-1,l0.shape[1]+K-1)) lt0[K//2:-K//2+1,K//2:-K//2+1]=l0 # convolve with the filter # Layer one is Relu applied on the convolution l0_conv=convolve(l0,W1[::-1,::-1],'same','direct') l1=relu(l0_conv) # Compute layer 2 l2=sigmoid(np.dot(l1.reshape(-1,),W2)) l2=l2.clip(10-16,1-10-16) ####### LOSS AND ACCURACY ####### loss=-(ynp.log(l2)+(1-y)np.log(1-l2)) accuracy=int(y==np.where(l2>0.5,1,0)) # Save the loss and accuracy to a running averager train_loss.send(loss) train_accuracy.send(accuracy) ##### BACKPROPAGATION ####### # Derivative of loss wrt the dense layer dW2=(((1-y)l2-y(1-l2))l1).reshape(-1,) # Derivative of loss wrt the output of the first layer dl1=(((1-y)l2-y(1-l2))W2).reshape(28,28) # Derivative of the loss wrt the convolution filter f1p=relu_prime(l0_conv) dl1_f1p=dl1f1p dW1=np.array([[ (lt0[alpha:+alpha+image_size,beta:beta+image_size]\ dl1_f1p).sum() for beta in range(K)]\ for alpha in range(K)]) W2+=-etadW2 W1+=-etadW1 loss_averager_valid=averager() accuracy_averager_valid=averager() for X,y in zip(X_valid,y_valid): accuracy,loss=forward_pass(W1,W2,X,y) loss_averager_valid.send(loss) accuracy_averager_valid.send(accuracy) train_loss,train_accuracy,valid_loss,valid_accuracy\ =map(extract_averager_value,[train_loss,train_accuracy, loss_averager_valid,accuracy_averager_valid]) # code to print losses and accuracies suppressed for clarity``` On my trial run I got the following results. Your’s should be similar We can see that even this simple model reduces the loss from about 1 to 0.1 and increases the accuracy from 0.5 to about 0.96 (on the validation set). We can visualize the convolution by plotting a few (standardized) images and their convolution with the original random filter and the final trained one. We see that the convolution filter seems to have learned to identify the right edges somewhat. It should be better when we train multiple filters but we have kept things simple by having just one filter. (This choice itself is a hyperparameter). We can do one more thing before finishing off part 1 of this post. We can make two more models: 1) We freeze the convolution layer weights and 2) We freeze the dense layer weights. This would help us understand if these layers contribute to the process. This is very easy. For the former we comment out the update line for W1 ```W2+=-etadW2 # W1+=-etadW1``` and for the latter we comment out the update line for W2 ```# W2+=-etadW2 W1+=-eta*dW1``` The resulting loss and accuracy are and respectively. It is clear that most of the work is done by the dense layer but we can still get a loss of .5 and an accuracy of .8 by just the convolution layer. Presumably this performance will increase with a larger number of filters. We will explore this in later posts.	3,137	12,934	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.765625	3	CC-MAIN-2021-04	longest	en	0.953542
http://forum.allaboutcircuits.com/threads/derivative-of-a-polynomial-function-in-c-and.116228/	1,481,290,931,000,000,000	text/html	crawl-data/CC-MAIN-2016-50/segments/1480698542712.12/warc/CC-MAIN-20161202170902-00298-ip-10-31-129-80.ec2.internal.warc.gz	102,326,238	18,282	# derivative of a polynomial function in C and ... Discussion in 'Programmer's Corner' started by Eric007, Oct 3, 2015. 1. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Hi All, How y'all doing? 1. I need a to know if the there's a function in a C library that compute the derivative of a polynomial function? 2. Assuming there's no function that computes it, how would you write that? While you trying to answer me I am also trying to look at the analytical method that compute the derivative of a function. Many thanks! eric007 2. ### WBahn Moderator Mar 31, 2012 17,757 4,800 I don't believe that there is such a function, particularly not in the standard C libraries. C is focused much to much on the low level stuff to have something like that. As for how to write such a function, that depends in part on how you intend to represent your polynomial function. 3. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Ok. Writing that function would be more interesting anyway. That a good question. Thinking... any suggestions are welcome as well. Last edited: Oct 3, 2015 4. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 an example would be x^3 + 5x^2 - 1. This is just the form but it could be anything. 5. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 So basically, at program startup you enter f(x). There should be a function, let's say, df_x, which is the derivative of f_x. We should be able to compute f_x (x0) and df_x(x0) as well where x0 is an integer constant. Last edited: Oct 3, 2015 6. ### shteii01 AAC Fanatic! Feb 19, 2010 3,393 497 There is a reason we have Matlab, Maple and other math oriented packages. Eric007 likes this. 7. ### vpoko Member Jan 5, 2012 258 47 You just need to break the string up into terms, isolate the exponent and coefficients, subtract one from the exponent, and the coefficient becomes the old coefficient times the original exponent. Eric007 likes this. 8. ### joeyd999 AAC Fanatic! Jun 6, 2011 2,689 2,750 Symbolic derivative in C? Do you ever do anything easy? When I was about 12, I wrote a BASIC program that symbolically evaluated algebraic equations. That was hard enough. Eric007 likes this. 9. ### vpoko Member Jan 5, 2012 258 47 If you limit yourself to polynomials, differentials are trivial. Easier than arbitrary algebra for sure, only one rule to apply. 10. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Of course yes but I was thinking of being able to input a polynomial of any degree and compute the above. Well maybe for simplicity, I can only process polynomial of degree equal to 2 or 3 and that would be easy to parse it (excuse my french)... well any suggestions... this can actually become a complex thing but I want to keep it very simple. It would be so easy if the function is know already but that would be a useless program. Thx! 11. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Hahahahah... I like complicated things... I know you are a genius and you also like complex things too. I want to work with very simple polynomial functions... 12. ### vpoko Member Jan 5, 2012 258 47 It's not much easier if you limit the degree. You just need to tokenize the string, put each term into a linked list or other structure, and then apply the power law of derivatives to each term. Should be fairly easy. Eric007 likes this. 13. ### nsaspook AAC Fanatic! Aug 27, 2009 2,910 2,172 thumb2 likes this. 14. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Sounds good. Will try to come up with something... it's quite late here and need to sleep. Hope to see more comment when I wake up. Last edited: Oct 4, 2015 15. ### MrAl Well-Known Member Jun 17, 2014 2,438 492 Hello, Regular real valued polynomials already have a general form: y=a0+a1x+a2x^2+a3x^3+...+aNx^N The derivative with respect to x is: y'=a1+2a2x+3a3x^2+...+NaNx^(N-1) The main thing to do is store the coefficients, which the user could enter, they dont have to enter the real string. For a poly like: y=1+5x+6x^2 they would enter 1,5,6 and for: y=1+6*x^2 which is missing the 'x' term, they would enter 1,0,6 and that's all you really need. The coefficients would be stored in an array. Eric007 and vpoko like this. 16. ### WBahn Moderator Mar 31, 2012 17,757 4,800 Again, it comes down to how you want to represent the information. If you keep an array of the coefficients then you can compute the coefficients of the derivative very easily. You have a couple of options. The first is to maintain an array in which the index is the exponent -- though this only works if the exponents are non-negative integers. But if the order of your polynomial is large and/or arbitrary, then it might be better to maintain parallel arrays with the coefficients in one and the exponents in the other. If you want the person to enter the polynomial, then you have to decide how you want them to enter it. They could enter it as two lists containing coefficients in one and exponents in the other. Another would be to provide a list of coefficient/exponent pairs. Yet a third would be to enter the polynomial as a string and then parse it to generate your internal representation. Eric007 likes this. 17. ### Eric007 Thread Starter Senior Member Aug 5, 2011 1,044 33 Entering the polynomial as string then parsing is interesting (I like parsing) but for now as I want to keep everything simple and basic I think I will go for the option where the user enters only the coefficients as mentioned by @MrAI 18. ### WBahn Moderator Mar 31, 2012 17,757 4,800 Be sure to think about the limitations of that approach. From a practical standpoint, what if the person wants to enter f(x) = x^27 - 1 ? Then consider if you want to limit yourself to strict polynomial functions requiring that the exponents be a set of the whole numbers (non-negative integers). If you do (and if the order of the polynomial is going to be reasonably small) then this is a good way to go. 19. ### Papabravo Expert Feb 24, 2006 10,148 1,791 You can "evaluate" the value of the derivative of a polynomial without using symbolic differentiation: $f'(x)= \frac{f(x+\Delta h) - f(x)} {\Delta h}$ with tiny values of $\Delta h$ 1e-7 for single precision and 1e-15 four double precision for example. 20. ### WBahn Moderator Mar 31, 2012 17,757 4,800 I don't think he is wanting to do numeric estimation of the derivative. But even if he is, setting the delta to that value is inviting disaster. If x is greater than 1 then you will get zero for the numerator.	1,753	6,619	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 4, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.078125	3	CC-MAIN-2016-50	longest	en	0.929568
http://physics.bu.edu/ulab/prelabs/prelab_specheat_new.html	1,550,978,868,000,000,000	text/html	crawl-data/CC-MAIN-2019-09/segments/1550249578748.86/warc/CC-MAIN-20190224023850-20190224045850-00633.warc.gz	206,608,481	1,562	Specific Heat Pre-lab Assignment In this experiment you will investigate the ideas of specific heat and latent heat. Specific heat comes into play when a substance is being heated or cooled, while latent heat is important when a substance changes phase. 1. Run the simulation, which shows five graphs. If heat is being added to a sample of water at a constant rate, which graph best represents the plot of temperature vs. time over a two-minute period? Note that each graph shows the sample, which is initially solid, starting at -10°C and taking 10 seconds to reach 0°C. You may find the following data helpful when deciding which graph is correct: Specific heats for water: cliquid = 1.0 cal/g °C and cice = csteam = 0.5 cal/g °C Latent heats for water: heat of fusion Lf = 80 cal/g and heat of vaporization Lv = 540 cal/g Circle the correct graph below. Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 2. If heat is being added to the sample at a constant rate of 90 cal/s, what is the mass of the sample?	245	1,008	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.859375	3	CC-MAIN-2019-09	latest	en	0.935651
https://www.nag.com/numeric/nl/nagdoc_26/nagdoc_fl26/html/c09/c09aaf.html	1,628,178,488,000,000,000	text/html	crawl-data/CC-MAIN-2021-31/segments/1627046155925.8/warc/CC-MAIN-20210805130514-20210805160514-00130.warc.gz	894,372,683	5,607	# NAG Library Routine Document ## 1Purpose c09aaf returns the details of the chosen one-dimensional discrete wavelet filter. For a chosen mother wavelet, discrete wavelet transform type (single-level or multi-level DWT or MODWT) and end extension method, this routine returns the maximum number of levels of resolution (appropriate to a multi-level transform), the filter length, and the number of approximation coefficients (equal to the number of detail coefficients) for a single-level DWT or MODWT or the total number of coefficients for a multi-level DWT or MODWT. This routine must be called before any of the one-dimensional discrete transform routines in this chapter. ## 2Specification Fortran Interface Subroutine c09aaf ( mode, n, nf, nwc, Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: nwlmax, nf, nwc, icomm(100) Character (), Intent (In) :: wavnam Character (1), Intent (In) :: wtrans, mode #include nagmk26.h void c09aaf_ ( const char wavnam, const char wtrans, const char mode, const Integer n, Integer nwlmax, Integer nf, Integer nwc, Integer icomm[], Integer ifail, const Charlen length_wavnam, const Charlen length_wtrans, const Charlen length_mode) ## 3Description One-dimensional discrete wavelet transforms (DWT) or maximum overlap wavelet transforms (MODWT) are characterised by the mother wavelet, the end extension method and whether multiresolution analysis is to be performed. For the selected combination of choices for these three characteristics, and for a given length, $n$, of the input data array, $x$, c09aaf returns the dimension details for the transform determined by this combination. The dimension details are: ${l}_{\mathrm{max}}$, the maximum number of levels of resolution that that could be computed were a multi-level DWT/MODWT applied; ${n}_{f}$, the filter length; ${n}_{c}$ the number of approximation (or detail) coefficients for a single-level DWT/MODWT or the total number of coefficients generated by a multi-level DWT/MODWT over ${l}_{\mathrm{max}}$ levels. These values are also stored in the communication array icomm, as are the input choices, so that they may be conveniently communicated to the one-dimensional transform routines in this chapter. None. ## 5Arguments 1: $\mathbf{wavnam}$ – Character()Input On entry: the name of the mother wavelet. See the C09 Chapter Introduction for details. ${\mathbf{wavnam}}=\text{'HAAR'}$ Haar wavelet. ${\mathbf{wavnam}}=\text{'DB}\mathbit{n}\text{'}$, where $\mathbit{n}=2,3,\dots ,10$ Daubechies wavelet with $\mathbit{n}$ vanishing moments ($2\mathbit{n}$ coefficients). For example, ${\mathbf{wavnam}}=\text{'DB4'}$ is the name for the Daubechies wavelet with $4$ vanishing moments ($8$ coefficients). ${\mathbf{wavnam}}=\text{'BIOR}\mathbit{x}$.$\mathbit{y}\text{'}$, where $\mathbit{x}$.$\mathbit{y}$ can be one of 1.1, 1.3, 1.5, 2.2, 2.4, 2.6, 2.8, 3.1, 3.3, 3.5 or 3.7 Biorthogonal wavelet of order $\mathbit{x}$.$\mathbit{y}$. For example ${\mathbf{wavnam}}=\text{'BIOR3.1'}$ is the name for the biorthogonal wavelet of order $3.1$. Constraint: ${\mathbf{wavnam}}=\text{'HAAR'}$, $\text{'DB2'}$, $\text{'DB3'}$, $\text{'DB4'}$, $\text{'DB5'}$, $\text{'DB6'}$, $\text{'DB7'}$, $\text{'DB8'}$, $\text{'DB9'}$, $\text{'DB10'}$, $\text{'BIOR1.1'}$, $\text{'BIOR1.3'}$, $\text{'BIOR1.5'}$, $\text{'BIOR2.2'}$, $\text{'BIOR2.4'}$, $\text{'BIOR2.6'}$, $\text{'BIOR2.8'}$, $\text{'BIOR3.1'}$, $\text{'BIOR3.3'}$, $\text{'BIOR3.5'}$ or $\text{'BIOR3.7'}$. 2: $\mathbf{wtrans}$ – Character(1)Input On entry: the type of discrete wavelet transform that is to be applied. ${\mathbf{wtrans}}=\text{'S'}$ Single-level decomposition or reconstruction by discrete wavelet transform. ${\mathbf{wtrans}}=\text{'M'}$ Multiresolution, by a multi-level DWT or its inverse. ${\mathbf{wtrans}}=\text{'T'}$ Single-level decomposition or reconstruction by maximal overlap discrete wavelet transform. ${\mathbf{wtrans}}=\text{'U'}$ Multi-level resolution by a maximal overlap discrete wavelet transform or its inverse. Constraint: ${\mathbf{wtrans}}=\text{'S'}$, $\text{'M'}$, $\text{'T'}$ or $\text{'U'}$. 3: $\mathbf{mode}$ – Character(1)Input On entry: the end extension method. Note that only periodic end extension is currently available for the MODWT. ${\mathbf{mode}}=\text{'P'}$ Periodic end extension. ${\mathbf{mode}}=\text{'H'}$ Half-point symmetric end extension. ${\mathbf{mode}}=\text{'W'}$ Whole-point symmetric end extension. ${\mathbf{mode}}=\text{'Z'}$ Zero end extension. Constraints: • ${\mathbf{mode}}=\text{'P'}$, $\text{'H'}$, $\text{'W'}$ or $\text{'Z'}$ for DWT; • ${\mathbf{mode}}=\text{'P'}$ for MODWT. 4: $\mathbf{n}$ – IntegerInput On entry: the number of elements, $n$, in the input data array, $x$. Constraint: ${\mathbf{n}}\ge 2$. 5: $\mathbf{nwlmax}$ – IntegerOutput On exit: the maximum number of levels of resolution, ${l}_{\mathrm{max}}$, that can be computed when a multi-level discrete wavelet transform is applied. It is such that ${2}^{{l}_{\mathrm{max}}}\le n<{2}^{{l}_{\mathrm{max}}+1}$, for ${l}_{\mathrm{max}}$ an integer. 6: $\mathbf{nf}$ – IntegerOutput On exit: the filter length, ${n}_{f}$, for the supplied mother wavelet. This is used to determine the number of coefficients to be generated by the chosen transform. 7: $\mathbf{nwc}$ – IntegerOutput On exit: for a single-level transform (${\mathbf{wtrans}}=\text{'S'}$ or $\text{'T'}$), the number of approximation coefficients that would be generated for the given problem size, mother wavelet, extension method and type of transform; this is also the corresponding number of detail coefficients. For a multi-level transform (${\mathbf{wtrans}}=\text{'M'}$ or $\text{'U'}$) the total number of coefficients that would be generated over ${l}_{\mathrm{max}}$ levels and with ${\mathbf{keepa}}=\text{'A'}$ for MODWT. 8: $\mathbf{icomm}\left(100\right)$ – Integer arrayCommunication Array On exit: contains details of the wavelet transform and the problem dimension which is to be communicated to the one-dimensional discrete transform routines in this chapter. 9: $\mathbf{ifail}$ – IntegerInput/Output On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details. For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit. On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6). ## 6Error Indicators and Warnings If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf). Errors or warnings detected by the routine: ${\mathbf{ifail}}=1$ On entry, ${\mathbf{wavnam}}=〈\mathit{\text{value}}〉$ was an illegal value. ${\mathbf{ifail}}=2$ On entry, ${\mathbf{wtrans}}=〈\mathit{\text{value}}〉$ was an illegal value. ${\mathbf{ifail}}=3$ On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$ was an illegal value. On entry, ${\mathbf{wtrans}}=\text{'T'}$ or $\text{'U'}$ and ${\mathbf{mode}}\ne \text{'P'}$. Constraint: ${\mathbf{mode}}=\text{'P'}$ when ${\mathbf{wtrans}}=\text{'T'}$ or $\text{'U'}$. ${\mathbf{ifail}}=4$ On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{n}}\ge 2$. ${\mathbf{ifail}}=-99$ See Section 3.9 in How to Use the NAG Library and its Documentation for further information. ${\mathbf{ifail}}=-399$ Your licence key may have expired or may not have been installed correctly. See Section 3.8 in How to Use the NAG Library and its Documentation for further information. ${\mathbf{ifail}}=-999$ Dynamic memory allocation failed. See Section 3.7 in How to Use the NAG Library and its Documentation for further information. Not applicable. ## 8Parallelism and Performance c09aaf is not threaded in any implementation. None. ## 10Example This example computes the one-dimensional multi-level resolution for $8$ values by a discrete wavelet transform using the Haar wavelet with zero end extensions. The length of the wavelet filter, the number of levels of resolution, the number of approximation coefficients at each level and the total number of wavelet coefficients are printed. ### 10.1Program Text Program Text (c09aafe.f90) ### 10.2Program Data Program Data (c09aafe.d) ### 10.3Program Results Program Results (c09aafe.r) © The Numerical Algorithms Group Ltd, Oxford, UK. 2017	2,636	8,852	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 110, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.609375	3	CC-MAIN-2021-31	latest	en	0.767818
https://socratic.org/questions/how-do-you-write-14-2-5-in-radical-notation	1,721,571,651,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763517701.96/warc/CC-MAIN-20240721121510-20240721151510-00334.warc.gz	458,100,610	5,798	# How do you write 14^(2/5) in radical notation? ${14}^{\frac{2}{5}} = {\left({14}^{2}\right)}^{\frac{1}{5}} = {196}^{\frac{1}{5}} = {5}^{t h} \text{root of } 196$ Now, ${5}^{t h} \text{root of a number}$ is , in radical sign, written as root(5)("the number)". Hence, ${14}^{\frac{2}{5}} = \sqrt[5]{196}$.	132	306	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.03125	4	CC-MAIN-2024-30	latest	en	0.743868
https://community.topcoder.com/stat?c=problem_statement&pm=14297&rd=16839	1,501,029,167,000,000,000	text/html	crawl-data/CC-MAIN-2017-30/segments/1500549425737.60/warc/CC-MAIN-20170726002333-20170726022333-00498.warc.gz	613,759,017	8,783	JOIN Problem Statement Problem Statement for MinimumCutsAgain ### Problem Statement In this problem you will be constructing a simple weighted directed graph. A 0-1 cut in the graph is a subset S of vertices that does contain the vertex 0 but does not contain the vertex 1. Let T be the complement of S. The size of the cut is the total weight of all edges that lead from S to T. (The weight of edges from T to S does not matter.) A 0-1 cut is called a minimum 0-1 cut if its size is the smallest among all 0-1 cuts. You are given an int n. Construct any undirected graph with the following properties: • The number of vertices must be between 2 and 20, inclusive. • The vertices must be numbered 0 through v-1, where v is their number. • The graph must be simple - i.e., no self-loops and no multiple edges going in the same direction. (However, for any distinct x,y the graph may contain both an edge from x to y and an edge from y to x.) • All edge weights must be integers between 1 and 10^9, inclusive. • The graph must have exactly n different minimum 0-1 cuts. Suppose the graph you constructed has v vertices and e edges. Return a int[] with 1 + 3*e elements. Element 0 of the return value should be the number v of vertices. The rest of the return value should be a list of edges. For each edge i, give its source vertex si, its destination vertex ti, and its weight wi. That is, the return value should be formatted as follows: { v, s0, t0, w0, s1, t1, w1, ... }. If there are multiple correct solutions, you may output any of them. ### Definition Class: MinimumCutsAgain Method: construct Parameters: int Returns: int[] Method signature: int[] construct(int n) (be sure your method is public) ### Constraints -n will be between 1 and 1,000, inclusive. ### Examples 0) `1` `Returns: {2, 0, 1, 6 }` The return value describes a graph on 2 vertices. There is an edge from vertex 0 to vertex 1 of weight 6. This graph has exactly one minimum 0-1 cut: the cut {0}. The cost of this cut is 6. 1) `2` `Returns: {3, 0, 2, 1, 0, 1, 1, 2, 1, 1 }` There are two different minimum 0-1 cuts: the cut {0} and the cut {0,2}. Each of these cuts has cost 2. 2) `4` `Returns: {4, 0, 2, 4, 2, 1, 4, 0, 3, 2, 3, 1, 2 }` This time there are four 0-1 cuts: {0}, {0,2}, {0,3}, and {0,2,3}. The cost of each of them is 6. Thus, all four cuts are minimum 0-1 cuts. 3) `6` `Returns: {6, 0, 1, 1, 0, 5, 2, 1, 0, 4, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 3, 4, 2 }` The minimum 0-1 cuts in this graph have cost 1. There are six such cuts: the cuts {0,2,3,4,5}, {0,2,4,5}, {0,2,5}, {0,3,4,5}, {0,4,5}, and {0,5}. 4) `752` ```Returns: {20, 0, 16, 1, 1, 5, 2, 2, 11, 2, 3, 5, 2, 5, 2, 1, 5, 3, 1, 5, 11, 1, 7, 4, 2, 7, 18, 1, 9, 3, 2, 9, 17, 1, 11, 10, 2, 13, 1, 1, 13, 5, 2, 13, 15, 1, 14, 16, 2, 14, 17, 1, 16, 6, 2, 16, 13, 1, 16, 17, 1, 17, 12, 1, 18, 3, 2, 18, 6, 1, 19, 15, 1 }``` This problem statement is the exclusive and proprietary property of TopCoder, Inc. Any unauthorized use or reproduction of this information without the prior written consent of TopCoder, Inc. is strictly prohibited. (c)2010, TopCoder, Inc. All rights reserved. This problem was used for: 2016 TCO Algorithm Algo Semifinal 1 - Division I, Level Two	1,160	3,244	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.765625	4	CC-MAIN-2017-30	longest	en	0.90068
https://www.solve-variable.com/math-variable/logarithms/algebra1calulator.html	1,527,204,381,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794866894.26/warc/CC-MAIN-20180524224941-20180525004941-00047.warc.gz	839,459,300	11,666	Free Algebra Tutorials! Home Systems of Linear Equations and Problem Solving Solving Quadratic Equations Solve Absolute Value Inequalities Solving Quadratic Equations Solving Quadratic Inequalities Solving Systems of Equations Row Reduction Solving Systems of Linear Equations by Graphing Solving Quadratic Equations Solving Systems of Linear Equations Solving Linear Equations - Part II Solving Equations I Summative Assessment of Problem-solving and Skills Outcomes Math-Problem Solving:Long Division Face Solving Linear Equations Systems of Linear Equations in Two Variables Solving a System of Linear Equations by Graphing Ti-89 Solving Simultaneous Equations Systems of Linear Equations in Three Variables and Matrix Operations Solving Rational Equations Solving Quadratic Equations by Factoring Solving Quadratic Equations Solving Systems of Linear Equations Systems of Equations in Two Variables Solving Quadratic Equations Solving Exponential and Logarithmic Equations Solving Systems of Linear Equations Solving Quadratic Equations Math Logic & Problem Solving Honors Solving Quadratic Equations by Factoring Solving Literal Equations and Formulas Solving Quadratic Equations by Completing the Square Solving Exponential and Logarithmic Equations Solving Equations with Fractions Solving Equations Solving Linear Equations Solving Linear Equations in One Variable Solving Linear Equations SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA SOLVING LINEAR EQUATIONS Try the Free Math Solver or Scroll down to Tutorials! Depdendent Variable Number of equations to solve: 23456789 Equ. #1: Equ. #2: Equ. #3: Equ. #4: Equ. #5: Equ. #6: Equ. #7: Equ. #8: Equ. #9: Solve for: Dependent Variable Number of inequalities to solve: 23456789 Ineq. #1: Ineq. #2: Ineq. #3: Ineq. #4: Ineq. #5: Ineq. #6: Ineq. #7: Ineq. #8: Ineq. #9: Solve for: Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg: ### Our users: Thank you! Your software (Algebrator) was a great help with my algebra homework. Blaine Milham, MH If you are having problems with algebra and you cant find a simple way to learn, you must try Algebrator, you will never believe what its capable of doing. Katherine Tsaioun, MA What a great step-by-step explanations. As a father, sometimes it helps me explaining things to my children more clearly, and sometimes it shows me a better way to solve problems. C.P., Massachusetts WOW!!! This is AWESOME!! 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Use grouping to factor polynomials completely • system of equations calculator • solving second order differential equations substitution to get a first order • cube root x gragh • answers factoring and solving equations using square root • changing mix fraction to decimal • mcdougal littell geometry workbook answers • even answers to prentice hall algebra 2 • entering square terms Ti-89 solver • solving logarithmic and exponential simultaneous equations • addition and subtraction equations calculator • elementary algebra practice problems integers • how do you factor equations with negative exponents? • how to input function limits with square root in a graphic calculator • examples of percentages, fractions, addition, subtraction, multiplication,and division FOR COLLEGE STUDENT • online calculator to solve for x • solve radical rational square roots • how to do higher order derivatives using matlab • GCD calculator that shows steps • meaning of square root property • Method to take a fraction to a whole number in java • lcd finder math • finding slope of a quadratic equation • what happens if there is no slope in a liner equation • What Is the Hardest Math Problem • integers in order + least to greatest • writing fraction in order from least to greatest • solving variables and expressions 1.7 houghton mifflin • teacher's guide to algebra connections volume 1 • algebra equation to the fractional' degree • substitution and simplifying algebraic expressions calculator • converting exponents to fractions • Math Problem Solver • expression with square roots solver • solving prealegbra formula problems free • root of a number decimal • square root equations+worksheets • mixed fraction to percentage • nth term even number • sample lesso plan in dividing whole numbers by whole numbers with decimals • how to square a decimal • create factoring programs calculator • hot to add and subtract positive and negative • general linear equation of one variable in real life situation • year 7 maths worksheets algebra • function t-charts\ math notes • adding negative and positive fractions • pre algebra class review worksheets • fcat equasion worksheet • investigatory project in math 4 • Reducing Rational Expressions calculator • multiplying and dividing integers to the second power • math trivia questions and answers • practice worksheet A lesson 1.1 • lesson plan & equations solution by substitution & 8th grade • online graphing calculator and table • Where is the percent button on a ti-83 plus • how to solve cube quadratic functions that cant be factored • solve a= p(1+rt) for t • multiply rational fractions • adding positive and negative numbers free worksheets • exercise mathematics form 1 model questioan • mixed fractions to decimals • adding subtracting negative numbers worksheet • square decimal • algebrator.com • Algebra with Pizzazz Answer Key • lowest common denominator fractions calculator • how to solve exponential equations that are quadratic in form • positive and negative numbers pretest for middle school • calculate GCD • multiplying and dividing integers with square roots • polynomials calculator divide • solve third order equation • Factor -2x2+12x-11 algrebra • 92 least common factor • slope formula worksheet • very long algebra equations • how to make an equation a perfect square • worsheets math b factoring • calculating slope using TI 83 • how to square root on calculator • formula for percentage of a number	1,495	6,639	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.84375	4	CC-MAIN-2018-22	latest	en	0.83072
https://www.merlot.org/merlot/materials.htm?nosearchlanguage=&pageSize=&page=4&nosearchlanguage=&category=2605&userId=21266	1,529,653,708,000,000,000	text/html	crawl-data/CC-MAIN-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00356.warc.gz	855,133,901	19,490	# MERLOT Materials #### Filter by 73-96 of 840 results for Materials Submitted by John Walkup #### Vector Land This is the beginning of an online course to get to grips with 3D vectors for anyone who wants to learn. Most of the... see more #### Vectors and Motion in Two Dimensions This site contains an extensive set of notes on basic topics in physics. There are extensive illustrations and animations... see more #### Waves, Frequency, and Wavelength Applet allows user to change frequency, wavelength, and velocity and see the results. #### Work, Energy, and Power This site contains an extensive set of notes on basic topics in physics. There are extensive illustrations and animations... see more #### Writing Guidelines for Engineering... Writing Guidelines for Engineering and Science Students From the author: "These guidelines are designed to help you, the engineering or science student, perform writing... see more #### WWW Unit Converter An online utility for making unit conversions between SI and the English system. #### BJT Base Simulation Applet An interactive applet, with many of the parameters user-defined. #### pn Junction Diode A multimedia approach to the fabrication steps in making a pn junction diode. #### ComSpec3D - VRML Visualization of... ComSpec3D - VRML Visualization of Raman and Infrared Spectra of Molecules VRML utility for the calculation and visualization of IR and Raman Spectra and 3D visualization of corresponding normal... see more #### Some General Physics Java Applets Java applets to help understanding of some basic physics ideas. Applets deal with simple harmonic motion, wave motion,... see more #### Capacitance vs. Voltage of PN... Capacitance vs. Voltage of PN Junctions Applet and tutorial, with references, quiz, worksheet, and SPICE model. #### Eigenvectors and Eigenvalues Allows user to choose an energy for a quantum-mechanical system described by a user-defined potential energy function.... see more #### Magnetic Field from Two Current... Magnetic Field from Two Current Sources (Biot-Savart Law) Applet uses the Biot-Savart law to illustrate the magnetic field created by two parallel wires. Source code available in... see more #### Pendulum Simulation A Java applet simulation of basic pendulum motion. Source code available on main site. Foreign language versions... see more #### Projectile Motion Java applet plots the trajectory of an object undergoing projectile motion. Many parameters user-defined. Includes... see more #### Radioactive Decay An applet that demonstrates the rate of conversion from parent nucleus to daughter nucleus in radioactivity. #### Rolling and Slipping Allows the user to vary the coefficient of friction, angular speed, and linear velocity of a wheel. #### The Power House Shockwave simulation. Depicts a typical household and its appliances. Selecting an appliance displays important... see more #### Using Vectors Simple applet for demonstrating the adding of up to four vectors. #### Vector Math A Java applet for demonstrating vector addition and subtraction using the head-to-tail method of vector summation. #### Building a Transistor Explore how an individual Field Effect (FET) transistor is fabricated on a silicon wafer simultaneously with millions of... see more #### Accuracy and Precision Home Page A series of tutorials on basic measurement theory, including such topics as experimental errors, measurement, accuracy... see more #### Adding Vectors Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. #### Adding Vectors with Shockwave... Adding Vectors with Shockwave (7Stones.com) Applet uses Shockwave to demonstrate vector addition. Use can add up to 10 vectors.	773	3,793	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.734375	3	CC-MAIN-2018-26	latest	en	0.843008
mhf.k-sam.eu	1,675,219,244,000,000,000	text/html	crawl-data/CC-MAIN-2023-06/segments/1674764499899.9/warc/CC-MAIN-20230201013650-20230201043650-00063.warc.gz	421,166,449	44,073	1. vb 2. ci Solved problems on complex numberspdf By zf ni Solving Rational Equations 2001-2003www.beaconlearningcenter.com Rev.72503 SOLVING RATIONAL EQUATIONS EXAMPLES 1. Recall that you can solve equations containing fractions by using the least common denominator of all the fractions in the equation. Multiplying each side of. Roots of Quadratic Equation. a) The solution of the quadratic equation, ax2 bx c 0 is given by x b b2 4ac 2a. The expression D b2 4ac is called the discriminant of the quadratic equation. b) If and are the roots of the quadratic equation ax2 bx c 0, then. i) b a. 2022. 9. 14. &0183;&32;The complex number 2 4i is one of the root to the quadratic equation x 2 bx c 0, where b and c are real numbers. a) Find b and c b) Write down the second root and check it.. Section 1.3 Complex Numbers (pdf) Video Video Section 1.5 Solving Inequalities (pdf) Video Video Section 1.6 Equations and Inequalities with Absolute value (pdf) Video Video Section 1.7 Problem Solving Interest, Mixture, Constant Rate Apps (pdf) Video Video Practice Problems for Test 1 (MyMathLab website) (click on "Take A Test"). USING COMPLEX FORM Consider the following problem. Multiply 3 45o x 2 30o. The result is 675 o. Figure 5 To do this as complex numbers is more difficult as we shall now see. In the form A j B wehave the following. P1 has coordinates A1 3 cos 45 2.121 and B1 3 sin 45 2.121 P2 has coordinates A2 2 cos 30 1.732 and B2 2. vw This text constitutes a collection of problems for using as an additional learning resource for those who are taking an introductory course in complex analysis. The problems are numbered and allocated in four chapters corresponding to different subject areas Complex Numbers, Functions, Complex Integrals and Series. The majority of problems are. Answer (1 of 3) Modern math is a dense web of methods and ideas, and it&x27;s usually impossible to disentangle a proof and declare "see Topology was used here". Topology is part of the language, and it&x27;s used almost everywhere, in many fields. Still, there are some cool cases where a surprising l. 2016. 6. 8. &0183;&32;Chapter 3 Complex Numbers h A x B C 3 COMPLEX NUMBERS . solve the problem of finding two numbers, a and b, whose sum is 10 and whose product is 40; i.e. a b 10 . 2 10a 40 0. Solving this quadratic gives a 1 2 (10 &177;60) 5&177;15 This shows that there are no real solutions, but if it is agreed to continue using the. Here the conjugate of a Complex number can be used with multiplication, to get a Real number Conjugate notation is an &x27;overbar&x27; That is 78AB C4D8E TASK Before you look over the page, try and work out what form the Conjugate will take. Try and develop your problem solving powers as it is GREAT practice for. 2007. 1. 3. &0183;&32;Homework Set 1 Exercises on Complex Numbers Directions You are assigned the Calculational Problems 1(a, b, c), 2(b), 3(a, b), 4(b, c), 5(a, b), and the Proof-Writing Problems 8. 1 day ago &0183;&32;Simplify the complex expressions Find the absolute value of a complex number Find the sum, difference and product of complex numbers x and y Find the quotient of. Reason quantitatively and use units to solve problems. N-Q-1, 2, 3. Rockford International Airport. Rockford, IL Transportation Chapter 1 . Relationships Between Quantities. The Grass Is Always Greener. Objectives Analyze proportional relationships and percentages, and use them to solve real-world and mathematical problems. Here the conjugate of a Complex number can be used with multiplication, to get a Real number Conjugate notation is an &x27;overbar&x27; That is 78AB C4D8E TASK Before you look over the page, try and work out what form the Conjugate will take. Try and develop your problem solving powers as it is GREAT practice for. Calculations using Complex Numbers We will need to add two complex numbers z1x1y1and z2x2y2 z z1z2(20) but to do this you just as the real parts to get xx1x2and the imaginary parts to get yy1y2. It should be obvious how you subtract one complex number from another. There are a few geometric problems, and a few trigonometric identities, but most of the trigonometric questions ask for the values of particular trigonometric functions. Typical examples are (p. 53) is "Find sin 10 sin 50 sin 70 " and (p. 113) to solve 2 sin (3 x 4) 3. Most of the complex number problems are actually. Addition and subtraction of complex numbers Let (a bi) and (c di) be two complex numbers, then (a bi) (c di) (a c) (b d)i (a bi) -(c di) (a. To understand the notion of base numbers, we look at our own number system. We use the decimal, or base-10, number system. To help explain what this means, consider the number 2746. This number can be rewritten as. Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone).Due to the nature of the mathematics on this site it is best views in landscape mode. e z2H0T1 O2U IK tult xav vS3omfYtXw 4a MrHeC gLILfC3. o c 0AHlYlt 6r ji3g0h 6t9sp crce3sLeyr nvUeDdl.Z h qM 4a zd cew Mwqi WtBh7 KIhn ZfBi qn TigtmeS GAol3g4e Obnr ba4 G2I. j Worksheet by Kuta Software LLC. Please update your bookmarks Enjoy these free printable sheets focusing on the complex and imaginary numbers, typically covered unit in Algebra 2. Each worksheet has model problems. Supplementary Notes for Complex Variables, Differential Equations, and Linear Algebra. Prerequisite materials, detailed proofs, and deeper treatments of selected topics. Invented Number Systems (PDF - 2.9MB) Development of the Complex Numbers (PDF - 1.1MB) Linear Independance (PDF - 3.0MB) Some Notes on Differential Operators (PDF - 4.0MB). 2012. 10. 7. &0183;&32;Theorem (Complex numbers are weird) 1 1. Proof. The obvious identity p 1 p 1 can be rewritten as r 1 1 r 1 1 Distributing the square root, we get p 1 p 1 p 1 p 1 Finally, we. Two complex numbers abiand cdiare equal if and only if acand bd. Equality of Two Complex Numbers Find the values of xand ythat satisfy the equation 2x 7i 10 yi. SOLUTION Set the real parts equal to each other and the imaginary parts equal to each other. 2x 10 Equate the real parts. 7i Equate the imaginary parts.yi. Solves problems and attempts solutions others suggest Uses the words and skills heshe has in everyday setti ngs Understands and responds to directions Displays awareness of the dis tinc ons between things Interacts with books, pictures, and print Demonstrates an understanding of familiar scripts in play Does the child. This document is a short introduction to the relation between complex numbers and trigonometry, and shows how to approach trigonometrical problems using complex numbers. This is not com-pletely complete, maybe I&x27;ll add something else later. The targets of this document are beginner Olympiad problem solvers who have begun studying complex numbers. In particular, the existence and uniqueness theorem shows that there is a unique such function f(z) satisfying f0(z) f(z); f(0) 1 This function is called the complex exponential function ez. The number eis de ned as the value of ezat z 1. But it is the function ez, not the number e, that is truly important. Graph your problem using the following steps Type in your equation like y2x1. If you have a second equation use a semicolon like y2x1 ; yx3) . where a, b, c are real numbers and a 0. Now, in terms of graphing quadratic functions, we will understand a step-by-step procedure to plot the. Lesson 3 Function Transformations. Imaginary numbers can help us solve some equations Example Solve x 2 1 0. Using Real Numbers there is no solution, but now we can solve it Subtract 1 from both sides . But using complex numbers makes it a lot easier to do the calculations. And the result may have "Imaginary" current, but it can still hurt you Mandelbrot Set. Section 6.2 Operations on Complex Numbers. A2.1 Students analyze complex numbers and perform basic operations. A2.1.1 Define complex numbers and perform basic operations with them. A2.1.2 Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. 2007. 4. 4. &0183;&32;5.1 Constructing the complex numbers One way of introducing the eld C of complex numbers is via the arithmetic of 2&215;2 matrices. DEFINITION 5.1.1 A complex number is a matrix of the form x y y x , where x and y are real numbers. Complex numbers of the form x 0 0 x are scalar matrices and are called real complex numbers and are denoted by. Solving Complex Problems Review A subroutine is a set of instructions to perform a particular computation used to solve subproblems of more . contactmyassignmenthelp.com The coefficients of a, b, c and d are real or complex numbers with a not equals to zero (a 0). It must have the term x3 in it, or else it will not be a. RUBRIC-Math-problem-solving category mathematical concepts explanation shows complete understanding of the mathematical concepts used to solve the problem(s). Introducing Ask an Expert We brought real Experts onto our platform to help you even better. Children will complete math problems involving powers, roots, and complex numbers with this printable. WORKSHEETS. Real Numbers, Algebra, and Problem Solving, Set 2. The word problems in this printable will give children practice with their algebra and problem solving skills. WORKSHEETS. Systems of Equations and Problem Solving, Set 2. Separable Equations - Identifying and solving separable first order differential equations. We&x27;ll also start looking at finding the interval of validity from the solution to a differential equation. Exact Equations - Identifying and solving exact differential equations. We&x27;ll do a few more interval of validity problems here as well. Curriculum Standards CCSS.Math.Content.HSN.CN.A.1 Know there is a complex number i such that i2 -1, and every complex number has the form a bi with a and b real. CCSS.Math.Content.HSN.CN.A.2. It also includes a brief primer on complex numbers and their manipulations. A. Table of contents by sections 1. Abstract (you&x27;re reading this now) 2. Complex numbers Magnitude, phase, real and imaginary parts 3. Complex numbers Polar-to-Rectangular conversion and vice-versa 4. Complex numbers Addition, subtraction, multiplication, division 5. Problem Description A complex number is a number of the form (ab i), where a and b are real numbers and 1 is (sqrt -1). The numbers (a) and (b) are known as the real part and imaginary part of the complex number, respectively. You can perform addition, subtraction, multiplication, and division for complex numbers using. The Complex Numbers and Their Arithmetic Previously, we have discussed R, the set of all real numbers, and the funda-mental properties satis-ed by real number arithmetic. We have also discussed how the problem of solving x21 0, or equivalently, factoring x21, led to the creation (discovery) of the number ithat satis-es i2 (i)2 1. 6 Geometry of Complex Numbers 1.Explain why iz is perpendicular to z, without using DeMoivre&x27;s theorem. Let z a bi.Then iz i.a bi b ai, which is the transformation.a;b . b;a .Drawing this out on the 2D plane makes clear that the angle between the two points and the origin is 90 , simply by subtracting angles.90 90 .This is shown in Figure 1. Solve problems in new and unfamiliar situations using rational numbers, determine the reasonableness of solutions, and justify solutions. Complex Fraction. Set. Addding Rational Numbers.pdf (954k) Miss Richmond, Aug 23, 2018, 909 AM. v.1. Dividing Rational Numbers.pdf (1301k) Miss Richmond, Sep 10, 2018, 1005 AM. 2) the basic operations such as addition, subtraction, multiplication and division of complex numbers are easier to carry out and to program on a computer. Note. 1) Because the symbol i is used for currents in AC circuits, here we use j as the imaginary unit defined by j2 1 or j 1. 2) The symbol e represents the real part of a. Returns the smallest (closest to negative infinity) value that is not less than the argument and is an integer. conjugate of complex number. Example conj (23i) 2 3i. real part of complex number. Example re (23i) 2. imaginary part of complex number. Example im (23i) 3i. Excel has the capability to solve linear (and often nonlinear) programming problems with the SOLVER tool, which - May be used to solve linear and nonlinear optimization problems - Allows integer or binary restrictions to be placed on decision variables - Can be used to solve problems with up to 200 decision variables. Online tool Multiplying Complex Numbers Calculator is programmed to perform multiplication operation of complex numbers and gives the result in no time. All you need to do is enter the complex numbers and tap on the enter button to get the product of complex numbers. Ex (22i) (44i) or (42i) (44i) or (22i) (44i) (44i). Complex numbers extend the real numbers, adding functionality to solve prob- lems such as x24 0. Problems like this can not be solved with real num- bers since the solution involves taking the square root of a negative number; e.g., x p 4. The set of real numbers is a subset of the set of complex numbers. 1. Problems Can Be Solved in Different Ways. Although most math problems have only one answer, there may be many ways to get to that answer. Learning math is more than finding the correct answer; it&x27;s also a process of solving problems and applying what you&x27;ve learned to new problems. 2. Wrong Answers Sometimes Can Be Useful. mg pn Pure Mathematics, Vol. 2 (both). The material on complex numbers and equations occupies roughly the first half of the book. Challenging problems, similar to STEP papers or old S-levels. Ferrar (1943). Higher Algebra (both). About 60 pages on geometrictrigonometric applications and 100 on the theory of equations. Addition of Complex Numbers Add real parts, add imaginary parts. Subtraction of Complex Numbers Subtract real parts, subtract imaginary parts. Example 1- Addition & Subtraction a. 92;displaystyle &92;left (6 7 j&92;right) &92;left (3- 5 j&92;right) (6 7j)(3 5j). A complex number is any expression that is a sum of a pure imaginary number and a real number. A complex number usually is expressed in a form called the a bi form, or standard form, where a and b are real numbers. The expressions a bi and a - bi are called complex conjugates. Complex conjugates are used to simplify the denominator when. Evaluate the following by rst converting to polar form (Rei). Express your answer in Cartesian form (abi) (a) (1i)12 (1i)12 2ei 4 12 (2)12ei3 26(1) 64 (b) (i)13 i13 ei 2 13 ei 6 cos 6 isin 6 3 2 1 2 i 6. Find every complex root of the following. sb Mat104 Solutions to Problems on Complex Numbers from Old Exams (1) Solve z5 6i. Let z r(cos isin). Then z5 r5(cos5 isin5). This has modulus r5 and argument 5. We want this to match the complex number 6i which has modulus 6 and innitely many possible arguments, although all are of the form 2,22,2. 2016. 12. 30. &0183;&32;for any complex number zand integer n, the nth power zn can be de ned in the usual way (need z6 0 if n<0); e.g., z 3 zzz, z0 1, z 1z3. WarningAlthough there is a way. 2020. 6. 14. &0183;&32;Produced by The Open University of Sri Lanka 2015 1 Examples for Complex numbers Question (01) (i) Find the real values of x and y such that (1) 2 (2 3) 3 3 i x i i y i i i i. 2016. 6. 8. &0183;&32;Chapter 3 Complex Numbers h A x B C 3 COMPLEX NUMBERS . solve the problem of finding two numbers, a and b, whose sum is 10 and whose product is 40; i.e. a b 10 . 2 10a 40 0. Solving this quadratic gives a 1 2 (10 &177;60) 5&177;15 This shows that there are no real solutions, but if it is agreed to continue using the. An imaginary number is a complex number that can be written in the form of a real number multiplied by an imaginary part, named i. Now, let&x27;s see how to solve a more elaborate problem using the imaginary numbers. First, we have &92;(&92;sqrt-9&92;sqrt-47&92;) Let&x27;s rewrite the expression in a way that we can isolate. Definition 2.1 (Rectangular form of a complex number) A complex number is of the form x iy (or x yi) , where x and y are real numbers. x is called the real part and y is called the imaginary. 2021. 6. 28. &0183;&32;The complex number z represents the point P x y(,) in the Argand diagram. Given that z z 1 2 2 , show that the locus of P is given by ()x y 3 42 2. proof Question 4 The. Systems of Equations (Graphing & Substitution) Worksheet Answers. Solving Systems of Equations by Elimination Notes. System of Equations Day 2 Worksheet Answers. Solving Systems with 3 Variables Notes. p165 Worksheet Key. Systems of 3 Variables Worksheet Key. Linear-Quadratic Systems of Equations Notes. Learn. Dividing complex numbers polar & exponential form. Visualizing complex number multiplication. Powers of complex numbers. Complex number equations x1. Visualizing complex number powers. Complex number polar form review. The magnitude or absolute value of a complex number z x iyis r p x2 y2. Combine this with the complex exponential and you have another way to represent complex numbers. rsin rcos x r rei y z xiy rcos ir sin r(cos i) rei (36) This is the polar form of a complex number and x iyis the rectangular form of the same number. The. 2018. 3. 28. &0183;&32;14 Chapter 1. Complex Numbers 2.Verify that p 2jzj jRezjjImzj. Hint Reduce this inequality to (jxjj yj)2 0 Solution. Note that 0 (jRezjjImzj)2 jRezj2 2jRezjjImzjjImzj2 Thus. In mathematics there are operators and functions which can reduce a complex operation to a simple one. For example, logarithms can easy the multiplication by reducing it to addition using logarithmic tables, one may transform large factors to logarithm terms, add them and reverse again obtaining the multiplication result. Not in use anym. CONTENT S Introduction 3 Chapter 1 Natural Numbers and Integers 9 1.1 Primes 10 1.2 Unique Factorization 11 1.3 Integers 13 1.4 Even and Odd Integers 15 1.5 Closure Properties 18 1.6 A Remark on the Nature of Proof 19 Chapter 2 Rational Numbers 21 2.1 Definition of Rational Numbers 21 2.2 Terminating and Non-terminating Decimals 23 2.3 The Many Ways of Stating and Proving Propositions 26. Solving Complex Problems Review A subroutine is a set of instructions to perform a particular computation used to solve subproblems of more . contactmyassignmenthelp.com The coefficients of a, b, c and d are real or complex numbers with a not equals to zero (a 0). It must have the term x3 in it, or else it will not be a. 5. Multiplying a complex z by i is the equivalent of rotating z in the complex plane by 2. a). Verify this for z 22i (b). Verify this for z 43i (c). Show that zi z for all complex z. The easiest way is to use linear algebra set z x iy. Then zi ix y. This corresponds to the vectors x y and y x in the complex plane. complex) can bind to the metal with one or another atom as the donor atom, linkage isomers are formed. Is this a structural or geometric isomer Structural, bonding different . Suctural Isomers Some isomers differ in what ligands are bonded to the metal (coordination. xa . To solve problems that involve sinusoids (s uch as AC voltages and currents) we use the phasor transform. That is, we transform sinusoids into complex numbers in polar form, solve the problem using complex arithmetic (as described above), and then transform the result back to a sinusoid. Solve a quadratic equation over the set of complex numbers algebraically. Calculate the discriminant of a quadratic equation to determine the number and type of solutions. Solve equations and verify algebraic solutions using a graphing utility. Just in Time Quick Check Just in Time Quick Check Teacher Notes Supporting Resources. Powerful lightweight complex system for mathematical calculations, which brings to you opportunity to solve your problems, as fast as possible. More than 160 build-in functions, full support of complex numbers and matrix calculations. 2022. 9. 24. &0183;&32;Problem Solving, Abstraction, and Design using C (6th Edition) 6th (sixth) Edition by Friedman, Frank L., Koman, Elliot B. Paperback March 18, 2010 by Frank L. Friedman (Author) 4.2 out of 5 stars 16 ratings See all formats and editions Problem Solving, Abstraction, and Design using C (6th. A complex number zcan also be written in terms of polar co-ordinates (r;) where . because we are solving a cubic equation z3 1 i, so r 0;1;2, say. Check that r 3 gives the same value for zas r 0). This is a theorem which is useful in complex number problems - and elsewhere. A polyno-mial equation azn bzn 1 czn 2 f 0 has. Lets try these 4 problems. More Practice 1-10 Introduction to Complex Numbers What is a complex number To see a complex number we have to first see where it shows up Solve both of these Um, no solution does not have a real answer. It has an "imaginary" answer. To define a complex number we have to create a new variable. Choose one of the two functions from each group in Question 1. Use the Quadratic Formula and what you know about imaginary numbers to solve an equation of the form f(x) 0 for each function you choose. Group A h(x) Group B n(x) g(x) x2 4 m(x) -2 (Q 3.7 Quadratics and Complex Numbers 269. HANDOUT M. 1 - COMPLEX NUMBERS NOTE All the problems in this handout must be solved by hand using the analytical procedure. MATLAB must be used only to verify the result obtained. What are complex numbers The set of complex numbers is represented in the standard form as a bi where &x27;a&x27; and &x27;b&x27; are real numbers. 2010. 3. 23. &0183;&32;Complex Practice Exam 1 This practice exam contains sample questions. The actual exam will have fewer questions, and may contain questions not listed here. 1. Be. The expression exp(j) is a complex number pointing at an angle of and with a magnitude of 1. M 1). We can use this notation to express other complex numbers with M 1 by multiplying by the magnitude. Mexp(j) This is just another way of expressing a complex number in polar form. M same as z Mexp(j) Using Euler&x27;s formula. mathematics idaho content standards . idaho state department of education <<department>> <<program>> 650 w state street, 2nd floor . boise, idaho 83702. The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2x10 had had no solutions. Because of &x27;i&x27;, we can now extend the square root to be defined for negative numbers as well. Calculator Use. This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 bx c 0 for x, where a 0, using the quadratic formula. The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots.Uninstall and. Search Center Of Mass Calculator Symbolab. 1 Arithmetic of the complex plane complex numbers as "useless" in connection to solving quadratic equations. That complex numbers were introduced to solve quadratic equations is a lie repeated blindly by many ill-informedtextbookauthors The correct reason for introducing complex numbers is in connection with solving cubics. Section 6.2 Operations on Complex Numbers. A2.1 Students analyze complex numbers and perform basic operations. A2.1.1 Define complex numbers and perform basic operations with them. A2.1.2 Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. dh Solve a quadratic equation over the set of complex numbers algebraically. Calculate the discriminant of a quadratic equation to determine the number and type of solutions. Solve equations and verify algebraic solutions using a graphing utility. Just in Time Quick Check Just in Time Quick Check Teacher Notes Supporting Resources. Apply set operations to solve the word problems on sets 7. Each student in a class of 40 plays at least one indoor game chess, carrom and scrabble. 18 play chess, 20 play scrabble and 27 play carrom. 7 play chess and scrabble, 12 play scrabble and carrom and 4 play chess, carrom and scrabble. Find the number of students who play (i) chess and. Visualizing the complex numbers as two-dimensional vectors, it is clear how to add two of them together. If z 1 x 1 iy 1, and z 2 x 2 iy 2, then z 1 z 2 (x 1 x 2) i (y 1 y 2). The real parts and imaginary parts are added separately, just like vector components. Multiplying two complex numbers together does not have quite such. Infinite Algebra 2 - Multiplying Complex Numbers Practice Created Date 152016 114626 PM. One center (or pathway) would be a historicaltheoretical avenue that explores imaginary and complex numbers. There would be one center replicating ancient Greco-Roman artifacts from Heron of Alexandria, who is on record as the first to dabble with complex numbers. It would also have Italian Renaissance mathematicians Gerolamo Cardono, who. wz Advanced Subtraction Worksheets Large Numbers. These dynamically created multi-digit subtraction worksheets are recommended for students of Grades 4 and grade 5. An endless supply of large number subtraction problems that feature 4-digit, 5-digit, 6-digit and 7- digit numbers in the vertical format are displayed here. This Module aims to develop the performance of children in tackling mathematical problems of a more varied, more open and less standardised kind than isnormal on present examination papers. It emphasises anumber ofspecificstrategies which may help such problem solving. These include the following try some simple cases find a helpful diagram. (1.1) It is said to be exact in a regionRif there is a functionhdened on the region with dhpdxqdy. 1.2) Theorem. An exact form is closed. The converse is not true. Consider, for instance, the plane minus the origin. The form (ydxxdy)(x2y2) is not exact in this region. It is, however, exact in the plane minus the negative axis. Transcript. We&x27;re converting equations from rectangular form to polar form. Here is a slightly harder example. X plus y minus the square root of x plus y, minus y equals 0. So let&x27;s recall that x plus y is r. So this whole thing can be written as r minus, and then I have the square root of r minus, and then y is the same as r. ty The Process of Solving Complex Problems volume 4, no. 1 (Winter 2012) The Journal of Problem Solving 22 Andreas Fischer, Samuel Greiff, and Joachim Funke 1. 2The complexityof a system may be defined as the number of elements and rela - tions of the system (see Funke, 1985). As Drner (1989) stated, "the complexity of. Un buen ejemplo de Complex Problem Solving es cuando diseas procesos y departamentos (que stos al final son agrupaciones nominales de procesos) en tu empresa. Simplificando mucho, en realidad lo que ests diseando es el movimiento de los flujos de informacin en la empresa. Hay que verlo como un diseo de caeras que fluye por. Once we convert each negative exponent into a fraction, the problem solves exactly like the other complex fraction problems. Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons . 7.5 Practice - Complex Fractions Solve. 1) 1 1 x 1. These thorough worksheets cover concepts from expressing complex numbers in simplest form, irrational roots, and decimals and exponents. Moving on to quadratic equations, students will become competent and confident in factoring, completing the square, writing and solving equations, and more. Students will be excited to learn how to find the. COMPLEX NUMBERS AND QUADRATIC EQUATIONS 101 i2 11 1 1()() (by assuming ab ab for all real numbers) 1 1, which is a contradiction to the fact that i2 1. Therefore, ab ab if both a and b are negative real numbers. Further, if any of a and b is zero, then, clearly, ab ab 0. 5.3.7 Identities We prove the following identity. un First, we will take on numbers. Sequences have an ancient history dating back at least as far as Archimedes who used sequences and series in his &92;Method of Exhaustion" to compute better values of and areas of geometric gures. 6.1 The Symbols 1and1 We often use the symbols 1and1in mathematics, including courses in high school. complex numbers. probability. Euler. conics. vector. cyclic quadrilateral. Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. This can happen when you are logged in to Art of Problem Solving on multiple tabs of the same browser, and then sign out on one of those tabs. RUBRIC-Math-problem-solving category mathematical concepts explanation shows complete understanding of the mathematical concepts used to solve the problem(s). Introducing Ask an Expert We brought real Experts onto our platform to help you even better. Expected Learning Outcomes The students will be able to 1) Define and use the imaginary number i. 2) Add, subtract, and multiply complex numbers. 3) Rationalize the denominator of a complex quotient using the conjugate. 2022. 9. 22. &0183;&32;Complex Numbers Answer Key 5 18. What is the complex number shown on the complex plane below Answer 3 - i 19. What does a discriminant of -3 tell you, when solving a. 5. Multiplying a complex z by i is the equivalent of rotating z in the complex plane by 2. a). Verify this for z 22i (b). Verify this for z 43i (c). Show that zi z for all complex z. The easiest way is to use linear algebra set z x iy. Then zi ix y. This corresponds to the vectors x y and y x in the complex plane. Complex numbers incorporate all numbers. 2i 3 4i 1 0.41 -1 0 2 -3 f A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 3i is a complex number x iy is the general form of a complex number If x iy 6 - 4i then x 6 and y - 4 The &x27;real part&x27; of 6 - 4i is 6 f Powers of i i i i 2. This investigation is based on a problem that was fractal geometry. Components Read through examples and make notes. What are Complex Numbers httpsbit.ly3c4qCKM Addition & Subtraction of Complex Numbers Multiplication of Complex Numbers httpsbit.ly33zS2EB Powers of i httpsbit.ly3hzK7MF Complex Conjugates httpsbit.ly32CiTkd. Reason quantitatively and use units to solve problems. N-Q-1, 2, 3. Rockford International Airport. Rockford, IL Transportation Chapter 1 . Relationships Between Quantities. The Grass Is Always Greener. Objectives Analyze proportional relationships and percentages, and use them to solve real-world and mathematical problems. Returns the smallest (closest to negative infinity) value that is not less than the argument and is an integer. conjugate of complex number. Example conj (23i) 2 3i. real part of complex number. Example re (23i) 2. imaginary part of complex number. Example im (23i) 3i. Complex numbers are written in the form where . Complex numbers include the set of Real and Imaginary numbers. I. Model Problems In the following examples you will solve quadratic equations with the quadratic formula over the set complex numbers. Example 1 Solve . Write down the equation. Identify the values of a, b, and c. Using Complex Numbers to Solve Equations Solving Quadratic Equations with Real Coefficients. This problem demonstrates a type of problem that many students will shy away from one involving not 1, not 2, but n variables. The trick is to deal with each variable separately, or to work patiently through sums, preferably using summation. 1. Identify the numerator and the denominator of a fraction. 2. Write a fraction to represent parts of figures or real-life data. 3. Graph Fractions on a Number Line. 4. Review division properties for 0 and 1. 5. Write mixed numbers as improper fractions. 6. Write improper fractions as mixed numbers or whole numbers. 7. 5 3 2i 6i Real numbers together, i&x27;s together 8 8i Add real to real (5 3), i&x27;s to i&x27;s (2i 6i) (5 2i) - (3 6i) Replace and with the complex numbers 5 2i - 3 - 6i Remove brackets, careful with signs. pn (ii) The complex number zei is a rotation of z by radians in the counter clockwise direction about the origin. Example 2.34 Solve the equation z3 8i 0 , where z C. Solution Let z3 8i 0 . Then, we get z3 -8i Example 2.35 Find all cube roots of 3 i Solution We have to find (3 i)13. Let z3 3 i r (cos i sin). Adding and Subtracting Complex Numbers Write the expression as a complex number in standard form. a. 4i)(32i)b. 75i)(15i)c.6(29i)(84i) SOLUTION a. 4i)(32i) (43)(12)iDefinition of complex addition 7iStandard form b. 75i)(15i) (71)(55)iDefinition of complex subtraction 60iSimplify. 6Standard form. 6 Geometry of Complex Numbers 1.Explain why iz is perpendicular to z, without using DeMoivre&x27;s theorem. Let z a bi.Then iz i.a bi b ai, which is the transformation.a;b . b;a .Drawing this out on the 2D plane makes clear that the angle between the two points and the origin is 90 , simply by subtracting angles.90 90 .This is shown in Figure 1. The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard. Simplicity in linearity In Mathematics, we know that the distributive property states . Solving these two equations simultaneously for the variables sin() and cos() then immediately gives. as solving problems. Apply the algebra of complex numbers, using extended abstract thinking, in solving problems. Explanatory Notes 1 This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education, 2007; and is related to the achievement objectives Manipulate complex numbers and. Euler&x27;s Form of the complex number - Definition, Properties, Formulas, Solved Example Problems Exercise 2.7 Polar and Euler form of a Complex Number - Problem Questions with Answer, Solution de Moivre&x27;s Theorem and its Applications - Definition, Formula, Solved Example Problems. Useful Inequalities Among Complex Numbers. z 2 x 2 y 2 Re (z) 2 Im (z) 2 . z 2 Re (z) 2 and z 2 Im (z) 2 . z Re (z) and z Im (z). The above help prove the triangle inequality in a formal manner. z w) 2. z w z w. The equality holds if one of the numbers is 0 and, in a non-trivial case. 2022. 9. 14. &0183;&32;The complex number 2 4i is one of the root to the quadratic equation x 2 bx c 0, where b and c are real numbers. a) Find b and c b) Write down the second root and check it.. 2022. 1. 17. &0183;&32;Solving an Equation in the Complex Number System. In math, there are two types of equations that involve complex numbers, equations that contain complex numbers and equations that have complex. Useful Inequalities Among Complex Numbers. z 2 x 2 y 2 Re (z) 2 Im (z) 2 . z 2 Re (z) 2 and z 2 Im (z) 2 . z Re (z) and z Im (z). The above help prove the triangle inequality in a formal manner. z w) 2. z w z w. The equality holds if one of the numbers is 0 and, in a non-trivial case. 6 Geometry of Complex Numbers 1.Explain why iz is perpendicular to z, without using DeMoivre&x27;s theorem. Let z a bi.Then iz i.a bi b ai, which is the transformation.a;b . b;a .Drawing this out on the 2D plane makes clear that the angle between the two points and the origin is 90 , simply by subtracting angles.90 90 .This is shown in Figure 1. counting things or for solving equations such as x1 2, but a problem arises if we want to solve an equation such as x 2 1, for example. In order to remedy this and other problems, negative whole numbers and the number zero were introduced. The natural numbers together with these are called the integers and are denoted Z. Answer (1 of 3) PRACTICE IS ALL THAT IS REQUIRED. This answer is going to be lengthy. Even if you are talented in maths, but you do not do practice, you won&x27;t be able to solve problems quickly. Maths is a subject where even a less talented student can beat the more intelligent. Here hard work. Imaginary numbers can help us solve some equations Example Solve x 2 1 0. Using Real Numbers there is no solution, but now we can solve it Subtract 1 from both sides . But using complex numbers makes it a lot easier to do the calculations. And the result may have "Imaginary" current, but it can still hurt you Mandelbrot Set. Complex Numbers - Euler&x27;s Formula on Brilliant, the largest community of math and science problem solvers. Euler&x27;s Form of the complex number - Definition, Properties, Formulas, Solved Example Problems Exercise 2.7 Polar and Euler form of a Complex Number - Problem Questions with Answer, Solution de Moivre&x27;s Theorem and its Applications - Definition, Formula, Solved Example Problems. 10.1.2 Add, subtract and multiply complex numbers 10.1.3 Solve a quadratic equation where the discriminant t- v < r 10.1.4 Find the conjugate of a complex number 10.1.5 Divide complex numbers in Cartesian form 10.1.6 Represent complex numbers on the Argand diagram 10.1.7 Find the modulus of a complex number. Section 6.2 Operations on Complex Numbers. A2.1 Students analyze complex numbers and perform basic operations. A2.1.1 Define complex numbers and perform basic operations with them. A2.1.2 Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. It is therefore correct tom assert that imaginary numbers are non-zero complex numbers with zero as the real part. The following are some examples of complex numbers. Get your 100 original paper on any topic done. in as little as 3 hours. Learn More. 2 5 i, 6 22 i, 12 - i. The &x27;i&x27; in complex numbers are not real. It also includes a brief primer on complex numbers and their manipulations. A. Table of contents by sections 1. Abstract (you&x27;re reading this now) 2. Complex numbers Magnitude, phase, real and imaginary parts 3. Complex numbers Polar-to-Rectangular conversion and vice-versa 4. Complex numbers Addition, subtraction, multiplication, division 5. 2018. 1. 16. &0183;&32;Chapter 1 Sums and Products 1.1 Solved Problems Problem 1. The harmonic series can be approximated by Xn j1 1 j 05772 ln(n) 1 2n Calculate the left and rigt-hand side. sq 2002. 10. 30. &0183;&32;Most problems in sections B and C are from previous Putnam contests. The questions are mainly related to analysis, while the C questions deal with geometry. Many. Complex numbers are written in the form where . Complex numbers include the set of Real and Imaginary numbers. I. Model Problems In the following examples you will solve quadratic equations with the quadratic formula over the set complex numbers. Example 1 Solve . Write down the equation. Identify the values of a, b, and c. Get Complex Numbers Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. Download these Free Complex Numbers MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. Solve a quadratic equation over the set of complex numbers algebraically. Calculate the discriminant of a quadratic equation to determine the number and type of solutions. Solve equations and verify algebraic solutions using a graphing utility. Just in Time Quick Check Just in Time Quick Check Teacher Notes Supporting Resources. z2 ihas two roots amongst the complex numbers. The quadratic formula (1), is also valid for complex coecients a,b,c,provided that proper sense is made of the square roots of the complex number b2 4ac. Problem 7 Find all those zthat satisfy z2 i. Suppose that z2 iand z abi,where aand bare real. Then i(abi) 2 ab 2abi. Solving (1) and (2), we get x 4 and y 5. So, the required number is 45. Problem 5 One number is greater than thrice the other number by 2. If 4 times the smaller number exceeds the greater by 5, find the numbers. Solution Let "x" and "y" be the two numbers such that x > y Given One number is greater than thrice the other number by 2. Get NCERT Solutions of Chapter 5 Class 11 - Complex Numbers free. All questions, including examples and miscellaneous have been solved and divided into different Concepts, with questions ordered from easy to difficult.The topics of the chapter includeSolvingQuadratic equationwhere root is in negativ. PS I&x27;m not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. Well that&x27;s about what it is. It&x27;s just that the applications for complex numbers gets simpler and sometimes more elegant using them. Expected Learning Outcomes The students will be able to 1) Define and use the imaginary number i. 2) Add, subtract, and multiply complex numbers. 3) Rationalize the denominator of a complex quotient using the conjugate. corresponds to a point in the complex plane and every point in the complex plane corresponds to a complex number.When we represent a complex number as a point in the complex plane, we say that we are plotting the complex number. Section 6.5 Complex Numbers in Polar Form; DeMoivre&x27;s Theorem 687 Real axis O Imaginary axis b a z a bi Figure. bo Rate is distance per time, so its units could be mph, meters per second, or inches per year. Now you can solve the system of equations 50t 100 (t - 2) (Multiply both values inside the parentheses by 100.) 50t 100t - 200. 200 50t (Divide 200 by 50 to solve for t.) t 4. Substitute t 4 into train No. 1. Our subtraction word problem pdfs are suitable for kindergarten through grade 5 students. CCSS 1.OA, 2.OA, 2.NBT, 3.NBT, 4.NBT Subtraction Within 20 Word Problems Worksheets Check out our real-life scenarios that pique kids&x27; interest and celebrate their problem-solving skills. 10-5 Problem-Solving Investigation Make a Model .82 10-6 Volume of Rectangular Prisms.83 10-7 Surface Area of Rectangular Prisms.84 11-1 Ordering Integers.85 11-2 Adding Integers.86 CONTENTS 0ii0ivFM881033 11508 1014 AM Page iii epg ju104MHGL149Quark0Word ProblemApplication file0FMCourse 1. 2019. 11. 26. &0183;&32;4 Division of Complex Numbers The Conjugate Before we can divide complex numbers we need to know what the conjugate of a complex is. To find the conjugate of a. Understanding the basic concepts of algebra helps in handling real-life problems. Most of the real-life scenarios are related to large expenses, profits and other lengthy problems. Having the knowledge and a firm grasp of solving algebraic equations help find out the missing factors and ease the solution of lengthy problems. 2018. 11. 22. &0183;&32;Complex numbers - Exercises with detailed solutions 1. Compute real and imaginary part of z i&161;4 2i&161;3 2. Compute the absolute value and the conjugate of z (1 i)6; w i17 3. Write in the algebraic" form (aib) the following complex numbers z i5 i1; w (33i)8 4. Write in the trigonometric" form ((cos&181; isin&181;)) the following. rp Here we have given NCERT Exemplar Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations. NCERT Exemplar Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Short Answer Type Questions Q6. If a cos i sin , then find the value of (1a1-a) Sol a cos i sin Q10. 2007. 1. 3. &0183;&32;Homework Set 1 Exercises on Complex Numbers Directions You are assigned the Calculational Problems 1(a, b, c), 2(b), 3(a, b), 4(b, c), 5(a, b), and the Proof-Writing Problems 8. Score 4.15 (54 votes) . Yes, any problem that can be solved recursively can also be solved through the use of iteration.In case you don't know, iteration is the use of a looping construct like a while loop, for loop, etc in order to solve a problem, whereas recursion is the use of a function that calls itself to solve a problem. 4 The "problem" that leads to complex numbers concerns solutions of equations. The need to extend the real numbers was prompted by the desire to solve problems like the following, which appears in the 1545 book Ars Magna by Cardana Divide 10 into two parts whose product is 40. To solve this problem, Cardano needed to. E2 Appendix E Complex Numbers The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a bi is (a bi) a bi.Additive inverse So, you have (a bi) (a bi) 0i 0.Adding and Subtracting Complex Numbers a.) (3 i (2 3i) 3 i 2 3i Remove parentheses. 2015. 6. 8. &0183;&32;By dividing two complex numbers, their arguments (angles,) are subtracted. If w should be real, its argument has to be zero. That means (z) (1 z 2). Here's a little sketch of both those complex numbers drawn as arrows in the complex plane You can see z, z 2, 1 and 1 z 2 in green. I also added the arguments of z and z 2 in red. 2014. 11. 10. &0183;&32;5. Multiplying a complex z by i is the equivalent of rotating z in the complex plane by 2. a). Verify this for z 22i (b). Verify this for z 43i (c). Show that zi z for all. These two solutions are called complex numbers. 1.1 Some definitions Suppose that z is a complex number. Let z a ib The real part of z , written Re(z), is a The imaginary part of z , written Im(z), is b The complex conjugate of z , written z or z , is z a - ib Example 1 Let z 5 - 2i, w -2 i and u 7i. Consider the multiplication problem 10i2 10(-1) -10. Consequently, in the complex number system 16 has two square roots, namely, 4i and 4i. We call 4i the principal square root of 16. Check Point 5 Solve using the quadratic formula 2x2 o. EXAM PLE 5 A Quadratic Equation with Imaginary Solutions. 2 days ago &0183;&32;Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is. Evaluating the needs or identifying the problem is a key step in finding a solution for engineering problems. Recognize and describe the problem accurately by exploring it thoroughly. Define what question is to be answered and what outputs or results are to be produced. Also determine the available data and information about the problem in hand. by. Math Teachers Lounge. 5.0. 5) 2.00. PDF. This worksheet is a fun way for your students to practice multiplying complex numbers. Students match expressions and will know right away if they&x27;ve solved correctly because of the puzzle The file contains the student worksheet and teacher answer key. 1. Problems Can Be Solved in Different Ways. Although most math problems have only one answer, there may be many ways to get to that answer. Learning math is more than finding the correct answer; it&x27;s also a process of solving problems and applying what you&x27;ve learned to new problems. 2. Wrong Answers Sometimes Can Be Useful. Complex Numbers and Quadratic Equations Class 11 MCQs Questions with Answers. Question 1. Question 2. Question 4. Question 5. Question 6. Question 7. Question 9. Let z 1 and z 2 be two roots of the equation z az b 0, z being complex. 2022. 9. 22. &0183;&32;Complex Numbers Answer Key 5 18. What is the complex number shown on the complex plane below Answer 3 - i 19. What does a discriminant of -3 tell you, when solving a. A complex number can be written in the form a b i where a and b are real numbers (including 0) and i is an imaginary number. Therefore a complex number contains two 'parts' one that is. corresponds to a point in the complex plane and every point in the complex plane corresponds to a complex number.When we represent a complex number as a point in the complex plane, we say that we are plotting the complex number. Section 6.5 Complex Numbers in Polar Form; DeMoivre&x27;s Theorem 687 Real axis O Imaginary axis b a z a bi Figure. A complex number is a number , where and are the numbers you&x27;re familiar with (they&x27;re called real numbers). We can add two complex numbers to get a new complex number, . We can multiply them, . Exercise 1 Work out and . Do you see a pattern What do you think is Exercise 2 Factorise the polynomial (you don&x27;t need complex numbers for this). With Soda PDF 12's batch creation process, you can take any number of files,. Jan 26, 2022 &183; Diploma In Computer Engineering Books PDF. Here are all the books PDF copies available of computer technology according to Probidhan 2016. Below I have added all books PDF copy semester-wise. Download the books that you need. Computer 1st Semester. Non-coding variants have been shown to be related to disease by alteration of 3D genome structures. We propose a deep learning method, DeepMILO, to predict the effects of variants on CTCFcohesin-mediated insulator loops. Application of DeepMILO on variants from whole-genome sequences of 1834 patients of twelve cancer types revealed 672 insulator loops. The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2x10 had had no solutions. Because of &x27;i&x27;, we can now extend the square root to be defined for negative numbers as well. (1 pt) Now for some practice working with complex num- bers Calculate 4 7i3i 4 7i3i 4 7i3i The complex conjugate of (1 i) is (1-i). In general to obtain the complex conjugate reverse the sign of the imaginary part. Geometrically this corresponds to nding the mirror image point in the complex plane by reecting through the x axis. Get NCERT Solutions of Chapter 5 Class 11 - Complex Numbers free. All questions, including examples and miscellaneous have been solved and divided into different Concepts, with questions ordered from easy to difficult.The topics of the chapter includeSolvingQuadratic equationwhere root is in negativ. NS50DF710NA19 Insignia 50-Inch Class Led 4K Uhd Smart Fire Tv Edition Tv Replacement Parts. This model has multiple variations. Here is an example of how to locate your Revision. Please choose your Revision from the list below, so that we can take you to your specific list of parts for model NS50DF710NA19. Insignia manufactures LCD HDTVs, GPS. 2018. 6. 1. &0183;&32;The standard form of a complex number is. a bi a b i. where a a and b b are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it. Rate is distance per time, so its units could be mph, meters per second, or inches per year. Now you can solve the system of equations 50t 100 (t - 2) (Multiply both values inside the parentheses by 100.) 50t 100t - 200. 200 50t (Divide 200 by 50 to solve for t.) t 4. Substitute t 4 into train No. 1. The modulus of the complex number a bi is a bi a2 b2. This is the distance between the origin (0, 0) and the point (a, b) in the complex plane. For two points in the complex plane, the distance between the points is the modulus of the difference of the two complex numbers. Let (a, b) and (s, t) be points in the complex plane. The. sn Rarely is a problem solved totally the first time. Complex problems have a messiness about them that takes you back and forth between hypotheses, analysis, and conclusions, each time deepening. Opt out or md anytime. See our mn. Complex sequences and series An innite sequence of complex numbers, denoted by zn, can be considered as a function dened on a set of positive integers into the unextended complex plane. For example, we take zn n 1 2n so that the complex sequence is zn 1 i 2, 2 i 22, 3 i 23, . Convergence of complex sequences. 2018. 11. 22. &0183;&32;Complex numbers - Exercises with detailed solutions 1. Compute real and imaginary part of z i&161;4 2i&161;3 2. Compute the absolute value and the conjugate of z (1 i)6; w. 2021. 12. 7. &0183;&32;PUTNAM TRAINING, 2008 COMPLEX NUMBERS 2 9. Suppose that a, b, u, vare real numbers for which av bu 1. Prove that a2 b2 u 2 v au bv p 3. 10. Let P n be a regular polygon inscribed in a unit circle. Denote by S n the set of all chords whose endpoints are vertices of P n, and let A n be the average length of all chords in S n.Write a closed form expression for A. eh • uv ge A complex number is a number comprising area land imaginary part. It can be written in the form aib, where a and b are real numbers, and i is the standard imaginary unit with the property i2-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of. • il nv complex numbers are coming from. fQuadratic Equations Examplesof quadratic equations 2x2 3x 5 0 x2x 6 0 x 2 4 Therootsof an equation are thex- values that make it "work" We can find the roots of a quadratic equation either by using the quadratic formula or by factoring. fWe can have 3 situations when solving quadratic equations. Possible Answers Correct answer Explanation Solving that equation is equivalent to solving the roots of the polynomial . Clearly, one of roots is 1. Thus, we can factor the polynomial as. so that we solve for the roots of . Using the quadratic equation, we solve for roots, which are . This means the solutions to are. • fg xi Calculations using Complex Numbers We will need to add two complex numbers z1x1y1and z2x2y2 z z1z2(20) but to do this you just as the real parts to get xx1x2and the imaginary parts to get yy1y2. It should be obvious how you subtract one complex number from another. • jf qw In mathematics there are operators and functions which can reduce a complex operation to a simple one. For example, logarithms can easy the multiplication by reducing it to addition using logarithmic tables, one may transform large factors to logarithm terms, add them and reverse again obtaining the multiplication result. Not in use anym. ge Problem-Solving Steps 1. What are the four problem-solving steps 2. What can you do to help yourself understand a question 3. Name at least three plans you can use to solve math problems. 4. What should you do if your plan for solving a problem does not work 5. How can reviewing the problem after you have an answer help you in the future. This large collection of printable word problem worksheets, ideal for children in kindergarten through grade 4 features scenarios that involve single-digit subtraction, two-digit subtraction, three-digit subtraction, and subtraction of large numbers up to six digits. Give yourself a head-start with our free subtraction worksheets. 2.1 Synthesis summaryexpert proofthe condensed form of the analysis and the plan of problem solvingpolished solution. 2.2 Plan with guidelines of proof writing. 3. Reflection (optional) Reflection is the abstraction of problem solving and argumentation. 3.1 Generalize the problem and solve this problem. 3.2 Make an alternative solution. Calculus of Complex functions. Laurent Series and Residue Theorem Review of complex numbers. A complex number is any expression of the form xiywhere xand yare real numbers, called the real part and the imaginary part of x iy;and iis p 1 Thus, i2 1. Other powers of ican be determined using the relation i2 1For example, i3 i2i iand. ld eb >	13,924	54,927	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.03125	4	CC-MAIN-2023-06	latest	en	0.835146
https://infinitylearn.com/surge/question/mathematics/degree-of-x31x36is/	1,685,753,445,000,000,000	text/html	crawl-data/CC-MAIN-2023-23/segments/1685224648911.0/warc/CC-MAIN-20230603000901-20230603030901-00250.warc.gz	347,621,988	13,310	Degree of x3−1×3+6 is # Degree of ${x}^{3}-\frac{1}{{x}^{3}}+6$ is 1. A 3 2. B 6 3. C 0 4. D not defined Register to Get Free Mock Test and Study Material +91 Live ClassesRecorded ClassesTest SeriesSelf Learning Verify OTP Code (required) ### Solution: ${x}^{3}-\frac{1}{{x}^{3}}+6$ is multinomial for multinomial Degree is not deformed. ## Related content Distance Formula Perimeter of Rectangle Area of Square Area of Isosceles Triangle Pythagoras Theorem Triangle Formulae Volume of Cylinder Perimeter of Triangle Formula Area Formulae Volume Formulae	176	576	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.75	3	CC-MAIN-2023-23	latest	en	0.549343
http://soft-matter.seas.harvard.edu/index.php?title=Controlling_the_Fiber_Diameter_during_electrospinning&diff=prev&oldid=6880	1,586,222,960,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585371662966.69/warc/CC-MAIN-20200406231617-20200407022117-00277.warc.gz	175,598,641	7,119	# Difference between revisions of "Controlling the Fiber Diameter during electrospinning" Zach Wissner-Gross (April 21, 2009) ## Information Controlling the fiber diameter during electrospinning Sergey V. Fridrikh, Jian H. Yu, Michael P. Brenner, and Gregory C. Rutledge Physical Review Letters, 2003, 90, 144502 ## Soft matter keywords Electrospinning, surface tension, scaling ## Summary Figure 1: Log-log plot of $Q/I$ vs. $d_t$, the minimum fiber radius. The inset figure renormalizes the data based on polymer concentration and compares this to their theoretical prediction, which nicely falls below the experimental data and has a similar slope. Fridrikh et al. derive an analytical model of the forces involved in electrospinning [1]. In the process of electrospinning, a polymer fiber is created by ejecting the polymer in a solvent between two plates of opposite electric charge. When the plate voltages are high enough, electrostatic forces dominate forces due to surface tension, so that the polymer develops a very high aspect ratio (i.e., becoming a thin fiber) as it streams toward the opposing plate. Electrospinning is further complicated by the "whipping instability," in which the electrospun fiber coils around itself. In their model, the authors are able to calculate the minimum possible fiber radius (for a thin viscous jet) before the whipping instability breaks the jet up into droplets (as in electrospraying [2]). Their main result is that this terminal radius $h_t$ scales as: $h_t\sim\left(\frac{Q}{I}\right)^{2/3}$ where $Q$ is the flow rate of the polymer solution and $I$ is current (due to the net downward motion of the fiber, which is simultaneously whipping around in circles). Figure 1 shows a log-log plot of $Q/I$ vs. $h_t$, which we would expect to have a slope of 2/3. The fitted line, with a slope of 0.639, corresponds well with their theory, and the offset in the log-log plot suggests that the authors' electrospun fibers could be reduced in size by a factor of two but no more. ## Soft matter discussion Figure 2: Equation 1 from the article, representing the equation of motion of an electrospun fiber.	513	2,164	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.65625	3	CC-MAIN-2020-16	longest	en	0.917911
https://liu.kattis.com/sessions/fn2c3x/problems/guessthedatastructure	1,653,588,473,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662619221.81/warc/CC-MAIN-20220526162749-20220526192749-00329.warc.gz	429,950,894	6,911	OpenKattis #### Start 2020-03-18 16:00 AKDT ## Extrauppgifter [converted from problem group] #### End 2021-03-18 16:00 AKDT The end is near! Session is over. Not yet started. Session is starting in -798 days 18:07:54 8760:00:00 0:00:00 # Problem DI Can Guess the Data Structure! There is a bag-like data structure, supporting two operations: $1\ x$: Throw an element $x$ into the bag. $2$: Take out an element from the bag. Given a sequence of operations with return values, you’re going to guess the data structure. It is a stack (Last-In, First-Out), a queue (First-In, First-Out), a priority-queue (Always take out larger elements first) or something else that you can hardly imagine! ## Input There are several test cases. Each test case begins with a line containing a single integer $n$ ($1\leq n\leq 1000$). Each of the next $n$ lines is either a type-1 command, or an integer $2$ followed by an integer $x$. This means that executing the type-2 command returned the element $x$. The value of $x$ is always a positive integer not larger than $100$. The input is terminated by end-of-file (EOF). The size of input file does not exceed 1MB. ## Output For each test case, output one of the following: stack It’s definitely a stack. queue It’s definitely a queue. priority queue It’s definitely a priority queue. impossible It can’t be a stack, a queue or a priority queue. not sure It can be more than one of the three data structures mentioned above. Sample Input 1 Sample Output 1 6 1 1 1 2 1 3 2 1 2 2 2 3 6 1 1 1 2 1 3 2 3 2 2 2 1 2 1 1 2 2 4 1 2 1 1 2 1 2 2 7 1 2 1 5 1 1 1 3 2 5 1 4 2 4 1 2 1 queue not sure impossible stack priority queue impossible	530	1,683	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.71875	3	CC-MAIN-2022-21	latest	en	0.791983
https://brainly.com/question/211260	1,484,854,207,000,000,000	text/html	crawl-data/CC-MAIN-2017-04/segments/1484560280730.27/warc/CC-MAIN-20170116095120-00532-ip-10-171-10-70.ec2.internal.warc.gz	800,379,574	9,161	• CSM • Ambitious 2014-12-03T20:49:13-05:00 M-4=2m {Take away 4 from both sides} m-4-m=2m-m {Take away m from both sides} -4=m Hope it helps 2014-12-03T20:52:08-05:00 To solve this equation you would need to move the m to one side so then the m on the left will be subtracted and it will also subtract on the right since what you do on one side of equation you need to do to the other and your answer turns out to be -4=m but how did it get to -4? m-4=2m : : if i subtract 4 fro! each side .. having 2 subtracting 4 from the 2 it dosent get -4 You don't subtract 4 first because first you have to have the m's on one side of the equal sign. The easiest way to have the m on one side is to subtract the m from both sides so that it cancels out on the side with the -4 and 2m-m=m this makes -4=m	254	796	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2017-04	latest	en	0.898109
https://docs.hummingbot.org/quickstart/custom-pmm-5/	1,686,301,348,000,000,000	text/html	crawl-data/CC-MAIN-2023-23/segments/1685224655446.86/warc/CC-MAIN-20230609064417-20230609094417-00765.warc.gz	247,441,811	18,103	# Exercise 5 - Dynamic Price Ceiling/Floor Video: Finally, let’s learn how to customize our script by utilizing order book data, leveraging Hummingbot’s ability to synchronously stream real-time Level 2 order book data from multiple exchanges simultaneously. ## Dynamic calculation using Bollinger Bands¶ We will improve the last exercise by adding a dynamic calculation of the price ceiling/floor feature based on the Bollinger Bands. To do so, we will: • Use the CandlesFactory object that will create an instance of the candles of the trading pair and interval that we want. Is important to notice that we can create as many candles as we want and they are not related to the markets that you define on the class variable markets. This is handy for example if you want to trade on KuCoin with the information of Binance. • Once we get the class of the candles, we need to start them. The start will initialize the WebSocket connection to receive the most updated information on the current candle, and also request the historical candles needed to complete the records requested. We are going to do this on the __init__. • Also we need to add a method to stop the candle when the script is stopped. We can override the method on_stop which let’s us code a custom shutdown when the bot is stopped via the stop command. • To calculate the bounds we are going to create a method that will add the Bollinger Bands using as a value 100 periods and 2 std. Then the upper bound will be price_ceiling and the lower bound will be price_floor. Now, let’s implement the solution, extending the same file as the last example: quickstart_script_2.py. ## Let's code!¶ First, we will create an candle instance: import logging from decimal import Decimal from typing import List, Dict import pandas as pd import pandas_ta as ta # noqa: F401 from hummingbot.core.data_type.common import OrderType, PriceType, TradeType from hummingbot.core.data_type.order_candidate import OrderCandidate from hummingbot.core.event.events import OrderFilledEvent from hummingbot.data_feed.candles_feed.candles_factory import CandlesFactory from hummingbot.strategy.script_strategy_base import ScriptStrategyBase class QuickstartScript2(ScriptStrategyBase): bid_spread = 0.008 ask_spread = 0.008 order_refresh_time = 15 order_amount = 0.01 create_timestamp = 0 trading_pair = "ETH-USDT" exchange = "binance_paper_trade" # Here you can use for example the LastTrade price to use in your strategy price_source = PriceType.MidPrice price_ceiling = 1700 price_floor = 1600 eth_1m_candles = CandlesFactory.get_candle(connector="binance", trading_pair="ETH-USDT", interval="1m") markets = {exchange: {trading_pair}} • We import the CandlesFactory class and call the get_candle method to create a candle instance in the variable eth_1m_candles. • Note that we are importing pandas_ta, a library that we will use to generate technical indicators from the candle data. • While we still define initial values for price_ceiling and price_floor, we will override them later in the on_tick method. Add functions that override the __init__ and on_close methods: def __init__(self, connectors: Dict[str, ConnectorBase]): # Is necessary to start the Candles Feed. super().__init__(connectors) self.eth_1m_candles.start() The method above starts collecting data when the script starts. def on_stop(self): """ Without this functionality, the network iterator will continue running forever after stopping the strategy That's why is necessary to introduce this new feature to make a custom stop with the strategy. :return: """ self.eth_1m_candles.stop() The method above stops collecting data when the user runs the stop command. ### calculate_pricing_ceiling_floor¶ Add a method that uses the data in eth_1m_candles to calculate moving price ceiling/floor using Bollinger Bands and update the values of the price_ceiling and price_floor. def calculate_price_ceiling_floor(self): candles_df = self.eth_1m_candles.candles_df candles_df.ta.bbands(length=100, std=2, append=True) last_row = candles_df.iloc[-1] self.price_ceiling = last_row['BBU_100_2.0'].item() self.price_floor = last_row['BBL_100_2.0'].item() Now, let’s add it to on_tick: def on_tick(self): if self.create_timestamp <= self.current_timestamp and self.eth_1m_candles.is_ready: self.cancel_all_orders() self.calculate_price_ceiling_floor() proposal: List[OrderCandidate] = self.create_proposal() proposal_filtered = self.apply_price_ceiling_floor_filter(proposal) proposal_adjusted: List[OrderCandidate] = self.adjust_proposal_to_budget(proposal_filtered) self.place_orders(proposal_adjusted) self.create_timestamp = self.order_refresh_time + self.current_timestamp • We are adding a new condition that will block the execution if the candles are not ready. • Then we are adding the method calculate_price_ceiling_floor and the implementation is accessing to the dataframe of the Candles object by using the method self.eth_1m_candles.candles_df • Lastly, we are getting the last row and assigning the upper and lower bound to price_ceiling and price_floor Tip Check out the pandas-ta documentation to learn how to generate other types of technical indicators. ### Custom status¶ Before we run the script, let’s improve the status command and show the candles data, as well as the current values for the price_ceiling and price_floor. def format_status(self) -> str: if not self.ready_to_trade: return "Market connectors are not ready." lines = [] mid_price = self.connectors[self.exchange].get_price_by_type(self.trading_pair, PriceType.MidPrice) best_ask = self.connectors[self.exchange].get_price_by_type(self.trading_pair, PriceType.BestAsk) best_bid = self.connectors[self.exchange].get_price_by_type(self.trading_pair, PriceType.BestBid) last_trade_price = self.connectors[self.exchange].get_price_by_type(self.trading_pair, PriceType.LastTrade) custom_format_status = f""" \| Mid price: {mid_price:.2f}\| Last trade price: {last_trade_price:.2f} \| Best ask: {best_ask:.2f} \| Best bid: {best_bid:.2f} \| Price ceiling {self.price_ceiling:.2f} \| Price floor {self.price_floor:.2f} """ lines.extend([custom_format_status]) if self.eth_1m_candles.is_ready: lines.extend([ "\n############################################ Market Data ############################################\n"]) candles_df = self.eth_1m_candles.candles_df # Let's add some technical indicators candles_df.ta.bbands(length=100, std=2, append=True) candles_df["timestamp"] = pd.to_datetime(candles_df["timestamp"], unit="ms") lines.extend([f"Candles: {self.eth_1m_candles.name} \| Interval: {self.eth_1m_candles.interval}\n"]) lines.extend([" " + line for line in candles_df.tail().to_string(index=False).split("\n")]) lines.extend(["\n-----------------------------------------------------------------------------------------------------------\n"]) else: lines.extend(["", " No data collected."]) return "\n".join(lines) • We are using the list approach to add strings and then show all of them • First, we are adding to our original text the price ceiling and price floor • Then we are logging the information of the candles. • Check that when you run the code, the last candle is updated in real-time ;) ## Running the script¶ Run the script with start again. After it’s running, run status --live to see your dynamic price ceiling/floor along with the candles data streaming in real-time! You can display any variable you want here, so use it to analyze what is happening with your bot. ## Debugging scripts¶ Watch this video to learn how to debug Scripts at runtime with the PyCharm IDE: Congratulations on successfully building your own custom market making script! If you’d like to learn more about how to build custom strategies using Hummingbot, check out Botcamp!	1,769	7,802	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.640625	3	CC-MAIN-2023-23	latest	en	0.846226
http://mathhelpforum.com/math-software/170484-reading-mic-word-formula.html	1,526,841,160,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794863662.15/warc/CC-MAIN-20180520170536-20180520190536-00128.warc.gz	194,083,818	10,566	1. ## Reading Mic. Word Formula Could someone explain to me how to interpret "y = 670127x2 - 3E+06x + 8E+06"? What is E? 06? 2. Your equation is $\displaystyle \displaystyle y = 670127x^2 - 3000000x +8000000$ The E part represents to the power of 10. 3. Originally Posted by pickslides Your equation is $\displaystyle \displaystyle y = 670127x^2 - 3000000x +8000000$ The E part represents to the power of 10. Ok, thank you 4. edited. disregard the below post. i set the y = 0, 1, 2... instead of y = 1790, 1800, 1810 and it seems to work now. I'm running into a problem here, which i feel should be fairly obvious and i'm going to feel pretty stupid if someone's able to point it out. I have some data that i'm trying to find a model for with microsoft excel. I'm getting the formula y = 6701.3x2 - 2E+07x + 2E+10 which i interpret to be y = 6701.3x^2 - 210^7 + 2*10^10 However, my data shows that when y = 1800, x = 5,308,000 While the formula corresponds y = 1800 to x = 5,712,212,000 The other values are similarly off. I've triple checked the data and the trendline on microsoft word actually fits the data points really well. Can anyone think of where i may be dropping the ball? 5. You'll have to post the data itself for anyone to give you additional help.	396	1,278	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.375	3	CC-MAIN-2018-22	latest	en	0.905403
https://www.ibtimes.co.uk/death-by-halloween-sweets-how-many-does-it-take-kill-you-1588559	1,601,210,858,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600400279782.77/warc/CC-MAIN-20200927121105-20200927151105-00331.warc.gz	840,551,592	39,944	Obesity, diabetes, tooth decay – now there's another risk of eating too much sugar to add to the list: death by overdose. It's good to know your limits on the night of Halloween, and the American Chemical Society have helpfully worked out exactly how many trick-or-treat sweets you'd need to eat before suffering death by sugar. Each of us has a different maximum sweet capacity. One of the main factors is body weight. You can calculate your own personal limit using the box below. If you do decide to go overboard and pass your limit – or if you already have – then don't panic yet, for a few of reasons. First, this calculation is based on the amount of sugar per kilogram of bodyweight that it takes to kill rats, not humans. That kind of trial, surprisingly, hasn't been carried out on people so far. Second, the calculation only finds the dose that kills half of the rats in a test population. So lucky rats can still eat that much sugar and live to see another day. Step 1: multiply your weight in kg by 29.7 Step 2: divide this number by 9.3 This is the number of average-sized sweets you'd have to eat all in one go to stand a 50:50 chance of dying. Third, you have to eat the candy all at once for the calculation to be accurate. It is pretty difficult to eat hundreds of sweets instantaneously, however big your mouth. So how do your figures stack up? According to the American Chemical Society, the average number of sweets an American would have to eat to have a fifty-fifty chance of dying is 262. That's 22,000 calories – more than ten days' recommended energy intake for a woman.	362	1,607	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.78125	3	CC-MAIN-2020-40	latest	en	0.967929
http://doubleroot.in/lessons/circle/tangent-point-form-examples/	1,519,600,655,000,000,000	text/html	crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00365.warc.gz	87,751,319	11,800	# Tangent: Point Form – Examples This lesson will cover a few examples to illustrate the equation of the tangent to a circle in point form. Let’s begin. Example 1 Find the equation of the tangent to the circle x2 + y2 = 25, at the point (4, -3) Solution Note that the problem asks you to find the equation of the tangent at a given point, unlike in a previous situation, where we found the tangents of a given slope. Therefore, we’ll use the point form of the equation from the previous lesson. The required equation will be x(4) + y(-3) = 25, or 4x – 3y = 25. Example 2 Find the equation of the tangent to the circle x2 + y2 – 2x – 6y – 15 = 0 at the point (5, 6). Solution This one is similar to the previous problem, but applied to the general equation of the circle. We’ll use the point form once again. The required equation will be x(5) + y(6) + (–2)(x + 5) + (– 3)(y + 6) – 15 = 0, or 4x + 3y = 38. Example 3 Find the point where the line 3x + 4y = 25 touches the circle x2 + y2 = 25 Solution We’ve done a similar problem in a previous lesson, where we used the slope form. Here, I’m interested to show you an alternate method. The problem has given us the equation of the tangent: 3x + 4y = 25. But we know that any tangent to the given circle looks like xx1 + yy1 = 25 (the point form), where (x1, y1) is the point of contact. Therefore, to find the values of x1 and y1, we must ‘compare’ the given equation with the equation in the point form. Head over to this lesson, to understand what I mean about ‘comparing’ lines (or equations). On comparing the coefficients, we get x1/3 = y1/4 = 25/25, which gives the values of x1 and y1 as 3 and 4 respectively. The point of contact therefore is (3, 4). Example 4 Find the point where the line 4y – 3x = 20 touches the circle x2 + y2 – 6x – 2y – 15 = 0. Solution This problem is similar to the previous one, except that now we don’t have the standard equation. We’ll use the new method again – to find the point of contact, we’ll simply compare the given equation with the equation in point form, and solve for x1 and y1. The equation of the tangent in the point for will be xx1 + yy1 – 3(x + x1) – (y + y1) – 15 = 0, or x(x1 – 3) + y(y1 – 1) = 3x1 + y1 + 15. On comparing the coefficients, we get (x1 – 3)/(-3) = (y1 – 1)/4 = (3x1 + y1 + 15)/20. On solving the equations, we get x1 = 0 and y1 = 5. Therefore, the point of contact will be (0, 5). Note that in the previous two problems, we’ve assumed that the given lines are tangents to the circles. Comparing non-tangents to the point form will lead to some strange results, which I’ll talk about sometime later. One more. Example 5 Show that the tangent to the circle x2 + y2 = 25 at the point (3, 4) touches the circle x2 + y2 – 18x – 4y + 81 = 0. Also find the point of contact. Solution The following figure (inaccurately) shows the complicated situation: The problem has three parts – finding the equation of the tangent, showing that it touches the other circle and finally finding the point of contact. The equation can be found using the point form: 3x + 4y = 25 To prove that this line touches the second circle, we’ll use the condition of tangency, i.e. its distance from the center of the circle must be equal to its radius. The circle’s center is (9, 2) and its radius is 2. The distance of the line 3x + 4y – 25 = 0 from (9, 2) is \|3(9) + 4(2) – 25\|/5 = 2, which is equal to the radius. Almost done! Now to find the point of contact, I’ll show yet another method, which I had hinted in a previous lesson – it’ll be the foot of perpendicular from the center to the tangent. To find the foot of perpendicular from the center, all we have to do is find the point of intersection of the tangent with the line perpendicular to it and passing through the center. The required perpendicular line will be (y – 2) = (4/3)(x – 9) or 4x – 3y = 30. And the final step – solving the obtained line with the tangent gives us the foot of perpendicular, or the point of contact as (39/5, 2/5). Phew! We’re finally done. That’ll be all for this lesson. The next lesson cover tangents drawn from an external point.	1,204	4,154	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.96875	5	CC-MAIN-2018-09	longest	en	0.862919
https://jp.mathworks.com/matlabcentral/answers/1617770-read-and-use-txt-file	1,670,146,270,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446710968.29/warc/CC-MAIN-20221204072040-20221204102040-00553.warc.gz	367,002,573	33,809	# read and use .txt file 8 ビュー (過去 30 日間) francesco baldi 2021 年 12 月 26 日コメント済み: Walter Roberson 2021 年 12 月 27 日 hi, I was wondering if there's a way to read the .txt file i attached and use it to calculate other parameters. The .txt file reports the transfer function of my system with frequency, amplitude and phase respectively in the first, second and third column. Now, i need to calculate another parameter of the system in function of the transfer function. Assuming, for example, that my transfer function is called H, i need to calculate another parameter G = H100/(1-H). Is there anyone who can tell me how to do it? ##### 3 件のコメント表示非表示 2 件の古いコメント Star Strider 2021 年 12 月 26 日 Reference — how to use LTSpice values in Matlab and earlier posts by the same OP, how to plot transfer function exported from LTSpice, how to use LTSpice values in Matlab, how to find the transfer function from LTspice, and perhaps others, since I may have missed a few of these prolific posts, all asking essentially the same question. . サインインしてコメントする。 ### 回答 (2 件) Meg Noah 2021 年 12 月 26 日 There's a lot of extra characters in your text file. % frequency, amplitude and phase str = strrep(str,'{',''); str = strrep(str,'}',''); str = strrep(str,'(',''); str = strrep(str,')',''); str = strrep(str,'°',''); str = strrep(str,'''',''); str = regexprep(str, '\t', ' '); str = regexprep(str, ' ', ' '); str = regexprep(str, ' ', ','); str = regexprep(str, 'dB', ''); objrec = regexp(str, '\r\n\|\r\|\n', 'split'); % remove empty cells (blank lines) objrec(strlength(objrec)<1) = []; nPts = numel(objrec); frequency = zeros(nPts,1); amplitude = zeros(nPts,1); phase = zeros(nPts,1); for iPt = 1:nPts v = sscanf(objrec{iPt},'%f,%f,%f,'); frequency(iPt) = v(1); amplitude(iPt) = v(2); phase(iPt) = v(3); end omega = 2pifrequency; H = 1./(1 + 1i2omega); G = H100.0./(1-H); myTable = table(frequency,amplitude,phase,omega,H,G,'VariableNames', ... {'frequency','amplitude','phase','omega','H','G'}); writetable(myTable,'myTable.xlsx'); ##### 0 件のコメント表示非表示 -1 件の古いコメントサインインしてコメントする。 Walter Roberson 2021 年 12 月 26 日 It is not clear whether your second column is all negative, or if '(-' is the delimiter, the same way that the line ends in '-)' . The below code assumes that '(-' is the delimiter. Your file does not have any degree symbols in it. data = cell2mat(textscan(S, '%f (-%fdB,%f%*[^\n]')); whos data Name Size Bytes Class Attributes data 301x3 7224 double ##### 5 件のコメント表示非表示 4 件の古いコメント Walter Roberson 2021 年 12 月 27 日 Let's experiment: webopt = weboptions('CharacterEncoding', 'ISO-8859-1'); SW = webread(filename, webopt); SW(65:75) ans = '4e+000°) 1' ans+0 ans = 1×11 52 101 43 48 48 48 176 41 13 10 49 That seems to have done the trick. サインインしてコメントする。 ### カテゴリ Find more on Standard File Formats in Help Center and File Exchange ### Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting! Translated by	918	2,983	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.640625	3	CC-MAIN-2022-49	latest	en	0.522401
http://www.techwr-l.com/archives/0412/techwhirl-0412-00764.html	1,603,620,602,000,000,000	text/html	crawl-data/CC-MAIN-2020-45/segments/1603107888931.67/warc/CC-MAIN-20201025100059-20201025130059-00075.warc.gz	180,939,330	7,338	Re: Life Cycle PLanning Subject: Re: Life Cycle PLanning From: Tony Markos <ajmarkos -at- yahoo -dot- com> To: "TECHWR-L" <techwr-l -at- lists -dot- techwr-l -dot- com> Date: Thu, 16 Dec 2004 17:27:47 -0800 (PST) --- Ned Bedinger <doc -at- edwordsmith -dot- com> wrote: You (Tony) seem to say that DFD is like a system of equations that you can solve to "discover" what you Tony Markos: Yes that is what all the DFD gurus say - and I have experienced. Many modeling techniques (Use Cases for example) can used to capture what one already knows about a system. DFDs are unique in that they guide one in "flushing out" what one does not know about a system. No other technique does such - and that is what makes them a true analysis (i.e., discovery) tool. Ned Bedinger: It isn't a DFD until it is complete.... Tony Markos: Correct, and DFDs are designed to help us, through multiple iterations, move from a flawed understanding of requirements to a correct understanding. Ned Bedinger: A DFD would indeed tell you... how that top-level function is decomposed into sub-functions, but without the knowledge of those sub-functions, you don't have a DFD, . Tony Markos: Your right, "The devil is in [knowledge of] the details". While DFDs are meant to created in as top down a fashion as possible, the method fully supports the anlayst switching gears and going from top-down to bottom-up, and then switching back again to proceed in a top-down manner. Ned Bedinger: Functional decomposition: In Kovitz' P.11 sense, you start off knowing at a high level what function is needed. At that point, someone will need to break it down, and find the best way to make that function happen. That person breaks it down into sub-functions and decides on what the sub-functional requirements will be. Tony Markos: There are two kinds of DFDs: DFDs of the "As Is" condition, and DFDs of the "To Be" condition (system). It is not until we are working on the "To Be" condition that we consider functions needed to make the higher level functions happen. In the "As Is" model, we strictly stick with what currently is. Our main objective in creating the "As Is" model is to uncover the essential logical requirements. Essential logical requirements exist irregarless of any implementation (i.e., make things happen) considerations. When I refer to DFDs, I generally mean "As Is" DFDs. This is true with most DFDers as, generally, the greatest part of the DFDing effort is in creating the "As Is" model. I think, Ned, that you (and Korvitz)are thinking "To Be" DFDs. With "To Be" models we determine additional requirements needed to implement our essential requirements. While "To Be" DFDs are very much needed, be aware that a very common problem in projects is forgoing the "As Is" and going directly to the "To Be". The problem here is that discovery of the essential logical requirement is cut short. __________________________________ Do you Yahoo!? http://celebrity.mail.yahoo.com ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ROBOHELP X5 - SEE THE ALL NEW ROBOHELP X5 IN ACTION! RoboHelp X5 is a giant leap forward in Help authoring technology, featuring all new Word 2003 support, Content Management, Multi-Author support, PDF and XML support and much more! View an online demo: http://www.macromedia.com/go/techwrldemo --- You are currently subscribed to techwr-l as: archiver -at- techwr-l -dot- com To unsubscribe send a blank email to leave-techwr-l-obscured -at- lists -dot- techwr-l -dot- com Send administrative questions to lisa -at- techwr-l -dot- com -dot- Visit http://www.techwr-l.com/techwhirl/ for more resources and info. Follow-Ups: References: Re: Life Cycle PLanning: From: Ned Bedinger Previous by Author: Re: Life Cycle PLanning Next by Author: Agile Development And Data Flow Diagrams Previous by Thread: Re: Life Cycle PLanning Next by Thread: Re: Life Cycle PLanning What this post helpful? Share it with friends and colleagues:	1,035	3,980	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.671875	3	CC-MAIN-2020-45	latest	en	0.928026
https://www.physicsforums.com/threads/magnetic-quantum-number.696853/	1,508,439,156,000,000,000	text/html	crawl-data/CC-MAIN-2017-43/segments/1508187823360.1/warc/CC-MAIN-20171019175016-20171019195016-00082.warc.gz	1,014,253,534	17,108	# Magnetic quantum number 1. Jun 13, 2013 ### jbar18 Hi, I am going through the hydrogen atom at the moment, and I saw that the magnetic quantum number must run between ±L. I cannot find the reason in a simple enough form for me to understand, so I was hoping someone here could quickly (or slowly!) explain why ml can't be greater than L. Thanks 2. Jun 13, 2013 ### Staff: Mentor (Warning - some hand-waving follows) Suppose that I told you that an airplane was flying at 1000 km/hr, but didn't tell you in which direction. Even without knowing the direction, you would know that it couldn't be moving north at any more than 1000 km/hr (if the direction of travel was due north) nor at any less than -1000 km/hr (if the direction of travel was due south). And of course it could be anywhere between those two extremes. The L quantum number is the magnitude of the angular momentum. The magnetic quantum number is the component of the angular momentum around a particular axis, so can't be more than the total. 3. Jun 13, 2013 ### jbar18 Ah, that's very simple. I thought it was more complicated than that. However I am still a little confused, only because I was told that the magnitude of the angular momentum is given by sqrt(l(l+1)) (in units of hbar), which should be a little bit more than l. Is it just because ml has to be an integer, and so it has to be less than sqrt(l(l+1)) and so also l? 4. Jun 13, 2013 ### dextercioby Well, there's no integer between l and sqrt(l(l+1)), so if m is an integer, it can't be bigger than l, so that its maximum value is l. 5. Jun 13, 2013 ### Khashishi I suppose it has to do with Heisenberg Uncertainty Principle. For angular momentum, it says $$\sigma_{L_x} \sigma_{L_y} \ge \frac{\hbar}{2} \left< \| L_z \| \right>$$ So, you can never quite align the angular momentum all into the z direction because there's some uncertainty in the x and y angular momenta. Last edited: Jun 14, 2013 6. Jun 13, 2013 ### Staff: Mentor Hey, I did warn you that there was some hand-waving in my answer, didn't I? Dextercioby and Khashishi have given you good answers (although Khashishi could have appealed to the non-commutativity of the angular momentum components instead of the HUP). If you want to visualize what's going on... Imagine that the z-axis is pointing straight up. The angular momentum L cannot be pointing straight up as well ($L_z = L$) because then we'd be in an eigenstate of all three $L_i$ (with eigenvalues 0, 0, and L), and that's impossible because they don't commute. Instead, the angular momentum vector L lies in the surface of a cone with its apex at the origin and its axis around the z-axis. Each value of $L_z$ corresponds to a cone of different height $L_z$ and width; in any cone the value of L and $L_z$ is fixed and $L_z < L$. Last edited: Jun 13, 2013 7. Jun 13, 2013 ### Mandelbroth What do you know about differential equations? If you're good with the math, then the answer is fairly clear from the derivation of the quantum numbers from the Schrodinger equation. Eventually, in solving for the wave function, we get a Legendre polynomial, which is in terms of l and m. The series only converges if m is within ±l. 8. Jun 14, 2013 ### jbar18 Well that all seems fairly straightforward. Is there an answer reminiscent of for why l also has to be less than the principal quantum number? If it just pops out of the math that's fair enough, but that explanation was very enlightening (somehow I missed the physical connection between ml and l!) 9. Jun 14, 2013 ### Staff: Mentor Yes, although it comes with the same disclaimer about hand-waving. The principal quantum number is roughly analogous to the radius of the classical electron orbit. The angular momentum of an orbiting particle is greatest in a circular orbit, and smaller values of the angular momentum correspond to more elliptical orbits. Thus, there can be no angular momentum values greater than that corresponding to a circular orbit with the energy given by the principal quantum number. This is a completely heuristic hand-wavy argument... Useful only because it puts some intuition behind the results that just pop out of the math. 10. Jun 14, 2013	1,052	4,220	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.375	3	CC-MAIN-2017-43	longest	en	0.964781
https://www.nag.com/numeric/cl/nagdoc_cl26.2/html/g03/g03fcc.html	1,632,663,019,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780057861.0/warc/CC-MAIN-20210926114012-20210926144012-00252.warc.gz	885,292,768	7,112	# NAG C Library Function Document ## 1Purpose nag_mv_ordinal_multidimscale (g03fcc) performs non-metric (ordinal) multidimensional scaling. ## 2Specification #include #include void nag_mv_ordinal_multidimscale (Nag_ScaleCriterion type, Integer n, Integer ndim, const double d[], double x[], Integer tdx, double stress, double dfit[], Nag_E04_Opt options, NagError fail) ## 3Description For a set of $n$ objects, a distance or dissimilarity matrix $D$ can be calculated such that ${d}_{ij}$ is a measure of how ‘far apart’ objects $i$ and $j$ are. If $p$ variables ${x}_{k}$ have been recorded for each observation this measure may be based on Euclidean distance, ${d}_{ij}={\sum }_{k=1}^{p}{\left({x}_{ki}-{x}_{kj}\right)}^{2}$, or some other calculation such as the number of variables for which ${x}_{kj}\ne {x}_{ki}$. Alternatively, the distances may be the result of a subjective assessment. For a given distance matrix, multidimensional scaling produces a configuration of $n$ points in a chosen number of dimensions, $m$, such that the distance between the points in some way best matches the distance matrix. For some distance measures, such as Euclidean distance, the size of distance is meaningful, for other measures of distance all that can be said is that one distance is greater or smaller than another. For the former, metric scaling can be used, see nag_mv_prin_coord_analysis (g03fac), for the latter, a non-metric scaling is more appropriate. For non-metric multidimensional scaling, the criterion used to measure the closeness of the fitted distance matrix to the observed distance matrix is known as $\mathit{stress}$. $\mathit{stress}$ is given by, $∑ i=1 n ∑ j=1 i-1 d ^ ij - d ~ ij 2 ∑ i=1 n ∑ j=1 i-1 d ^ ij 2$ where ${\stackrel{^}{d}}_{ij}^{2}$ is the Euclidean squared distance between points $i$ and $j$ and ${\stackrel{~}{d}}_{ij}$ is the fitted distance obtained when ${\stackrel{^}{d}}_{ij}$ is monotonically regressed on ${d}_{ij}$, that is, ${\stackrel{~}{d}}_{ij}$ is monotonic relative to ${d}_{ij}$ and is obtained from ${\stackrel{^}{d}}_{ij}$ with the smallest number of changes. So $\mathit{stress}$ is a measure of by how much the set of points preserve the order of the distances in the original distance matrix. Non-metric multidimensional scaling seeks to find the set of points that minimize the $\mathit{stress}$. An alternate measure is squared $\mathit{stress}$, $SSTRESS$, $∑ i=1 n ∑ j=1 i-1 d ^ ij 2 - d ~ ij 2 2 ∑ i=1 n ∑ j=1 i-1 d ^ ij 4$ in which the distances in $\mathit{stress}$ are replaced by squared distances. In order to perform a non-metric scaling, an initial configuration of points is required. This can be obtained from principal coordinate analysis, see nag_mv_prin_coord_analysis (g03fac). Given an initial configuration, nag_mv_ordinal_multidimscale (g03fcc) uses the optimization function nag_opt_conj_grad (e04dgc) to find the configuration of points that minimizes $\mathit{stress}$ or $SSTRESS$. The function nag_opt_conj_grad (e04dgc) uses a conjugate gradient algorithm. nag_mv_ordinal_multidimscale (g03fcc) will find an optimum that may only be a local optimum, to be more sure of finding a global optimum several different initial configurations should be used; these can be obtained by randomly perturbing the original initial configuration using functions from the g05 Chapter Introduction. ## 4References Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press ## 5Arguments 1: $\mathbf{type}$Nag_ScaleCriterionInput On entry: indicates whether $\mathit{stress}$ or $SSTRESS$ is to be used as the criterion. ${\mathbf{type}}=\mathrm{Nag_Stress}$ $\mathit{stress}$ is used. ${\mathbf{type}}=\mathrm{Nag_SStress}$ $SSTRESS$ is used. Constraint: ${\mathbf{type}}=\mathrm{Nag_Stress}$ or $\mathrm{Nag_SStress}$. 2: $\mathbf{n}$IntegerInput On entry: the number of objects in the distance matrix, $n$. Constraint: ${\mathbf{n}}>{\mathbf{ndim}}$. 3: $\mathbf{ndim}$IntegerInput On entry: the number of dimensions used to represent the data, $m$. Constraint: ${\mathbf{ndim}}\ge 1$. 4: $\mathbf{d}\left[{\mathbf{n}}×\left({\mathbf{n}}-1\right)/2\right]$const doubleInput On entry: the lower triangle of the distance matrix $D$ stored packed by rows. That is ${\mathbf{d}}\left[\left(\mathit{i}-1\right)×\left(i-2\right)/2+\mathit{j}-1\right]$ must contain ${d}_{\mathit{i}\mathit{j}}$, for $\mathit{i}=2,3,\dots ,n$ and $\mathit{j}=1,2,\dots ,i-1$. If ${d}_{ij}$ is missing then set ${d}_{ij}<0$; For further comments on missing values see Section 9. 5: $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{tdx}}\right]$doubleInput/Output Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{tdx}}+j-1\right]$. On entry: the $i$th row must contain an initial estimate of the coordinates for the $i$th point, $i=1,2,\dots ,n$. One method of computing these is to use nag_mv_prin_coord_analysis (g03fac). On exit: the $i$th row contains $m$ coordinates for the $i$th point, $i=1,2,\dots ,n$. 6: $\mathbf{tdx}$IntegerInput On entry: the stride separating matrix column elements in the array x. Constraint: ${\mathbf{tdx}}\ge {\mathbf{ndim}}$. 7: $\mathbf{stress}$double Output On exit: the value of $\mathit{stress}$ or $SSTRESS$ at the final iteration. 8: $\mathbf{dfit}\left[2×{\mathbf{n}}×\left({\mathbf{n}}-1\right)\right]$doubleOutput On exit: auxiliary outputs. If ${\mathbf{type}}=\mathrm{Nag_Stress}$, the first $n\left(n-1\right)/2$ elements contain the distances, ${\stackrel{^}{d}}_{ij}$, for the points returned in x, the second set of $n\left(n-1\right)/2$ contains the distances ${\stackrel{^}{d}}_{ij}$ ordered by the input distances, ${d}_{ij}$, the third set of $n\left(n-1\right)/2$ elements contains the monotonic distances, ${\stackrel{~}{d}}_{ij}$, ordered by the input distances, ${d}_{ij}$ and the final set of $n\left(n-1\right)/2$ elements contains fitted monotonic distances, ${\stackrel{~}{d}}_{ij}$, for the points in x. The ${\stackrel{~}{d}}_{ij}$ corresponding to distances which are input as missing are set to zero. If ${\mathbf{type}}=\mathrm{Nag_SStress}$, the results are as above except that the squared distances are returned. Each distance matrix is stored in lower triangular packed form in the same way as the input matrix $D$. 9: $\mathbf{options}$Nag_E04_Opt Input/Output On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for nag_opt_conj_grad (e04dgc). These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. You are referred to the nag_opt_conj_grad (e04dgc) document for further details. The default values used by nag_mv_ordinal_multidimscale (g03fcc) when the options argument is set to the NAG defined null pointer, E04_DEFAULT, are as follows: • ${\mathbf{optim_tol}}=0.00001$; • ${\mathbf{print_level}}=\mathrm{Nag_NoPrint}$; • ${\mathbf{list}}=\mathrm{Nag_FALSE}$; • ${\mathbf{verify_grad}}=\mathrm{Nag_FALSE}$; • ${\mathbf{max_iter}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{n}}×{\mathbf{ndim}}\right)$. If a different value is required for any of these four structure members or if other options available in nag_opt_conj_grad (e04dgc) are to be used, then the structure options should be declared and initialized by a call to nag_opt_init (e04xxc) and supplied as an argument to nag_mv_ordinal_multidimscale (g03fcc). In this case, the structure members listed above except for ${\mathbf{list}}$ will have the default values as specified above; ${\mathbf{list}}=\mathrm{Nag_TRUE}$ in this case. 10: $\mathbf{fail}$NagError Input/Output The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation). ## 6Error Indicators and Warnings NE_2_INT_ARG_LE On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{n}}>{\mathbf{ndim}}$. NE_2_INT_ARG_LT On entry, ${\mathbf{tdx}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdx}}\ge {\mathbf{ndim}}$. NE_ALLOC_FAIL Dynamic memory allocation failed. On entry, argument type had an illegal value. NE_INT_ARG_LT On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{ndim}}\ge 1$. NE_INTERNAL_ERROR Additional error messages are output if the optimization fails to converge or if the options are set incorrectly, Details of these can be found in the nag_opt_conj_grad (e04dgc) document. An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance. NE_NEG_OR_ZERO_ARRAY All elements of array ${\mathbf{d}}\le 0.0$. Constraint: At least one element of d must be positive. ## 7Accuracy After a successful optimization, the relative accuracy of $\mathit{stress}$ should be approximately $\epsilon$, as specified by ${\mathbf{optim_tol}}$. ## 8Parallelism and Performance nag_mv_ordinal_multidimscale (g03fcc) is not threaded in any implementation. Missing values in the input distance matrix can be specified by a negative value and providing there are not more than about two thirds of the values missing, the algorithm may still work. However, the function nag_mv_prin_coord_analysis (g03fac) does not allow for missing values so an alternative method of obtaining an initial set of coordinates is required. It may be possible to estimate the missing values with some form of average and then use nag_mv_prin_coord_analysis (g03fac) to give an initial set of coordinates. ## 10Example The data, given by Krzanowski (1990), are dissimilarities between water vole populations in Europe. Initial estimates are provided by the first two principal coordinates computed. ### 10.1Program Text Program Text (g03fcce.c) ### 10.2Program Data Program Data (g03fcce.d) ### 10.3Program Results Program Results (g03fcce.r)	3,003	10,155	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 106, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.671875	3	CC-MAIN-2021-39	latest	en	0.656444
http://www.ques10.com/p/1202/principles-of-analog-digital-comm-question-paper-9/	1,558,374,624,000,000,000	text/html	crawl-data/CC-MAIN-2019-22/segments/1558232256082.54/warc/CC-MAIN-20190520162024-20190520184024-00069.warc.gz	321,121,239	7,763	Question Paper: Principles of Analog & Digital Comm. : Question Paper Dec 2013 - Information Technology (Semester 3) \| Mumbai University (MU) 0 ## Principles of Analog & Digital Comm. - Dec 2013 ### Information Technology (Semester 3) TOTAL MARKS: 80 TOTAL TIME: 3 HOURS (1) Question 1 is compulsory. (2) Attempt any three from the remaining questions. (3) Assume data if required. (4) Figures to the right indicate full marks. ### Attempt any four of the following: 1 (a) What is double spotting?(5 marks) 1 (b) Explain Eye pattern in detail(5 marks) 1 (c) Explain under water acoustic channel(5 marks) 1 (d) Explain low level modulated AM transmitter(5 marks) 1 (e) Compare FDM and TDM(5 marks) 2 (a) State and prove sampling theorem for low pass band limited signal(8 marks) 2 (b) Explain balanced modulator using FETs(6 marks) 2 (c) In an AM radio receiver the loaded Q of the antenna circuit at the input to the mixer is 100. If the intermediate frequency is 455 KHz, calculate the image frequency and its rejection at 1 MHz.(6 marks) 3 (a) Explain what is meant by quantization noise. Explain in detail.(8 marks) 3 (b) Find the mathematical expression of FM Signal(6 marks) 3 (c) Draw block diagram of BPSK generation with waveform(6 marks) 4 (a) Calculate the maximum bitrate for a channel having BW 3100 Hz and signal to noise ration 10db.(4 marks) 4 (b) State and prove the following properties of Fourier transform (i) Time Shifting (ii) Convolution in time domain (8 marks) 4 (c) Explain the term: (i) Shot noise and equivalent noise temperature (ii) Friis transmission formula (8 marks) 5 (a) Draw and explain PCM-TDM system(10 marks) 5 (b) What are the disadvantages of tune RF receiver? Draw the circuit of a superheterodyne receiver and explain the same.(10 marks) 6 (a) Draw neat block diagram of delta modulator and explain its working. What are the draw backs of delta modulator? How are they overcome by ADM?(8 marks) ### Write Short notes on: 6 (b) (i) Noise Triangle(3 marks) 6 (b) (ii) Pre-emphasis and De-emphasis(3 marks) 6 (b) (iii) White noise(3 marks) 6 (b) (iv) Properties of line codes.(3 marks)	581	2,131	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.59375	3	CC-MAIN-2019-22	latest	en	0.812583
https://math.stackexchange.com/questions/606557/volume-of-a-spherical-tetrahedron	1,566,349,789,000,000,000	text/html	crawl-data/CC-MAIN-2019-35/segments/1566027315695.36/warc/CC-MAIN-20190821001802-20190821023802-00210.warc.gz	549,851,367	29,814	# Volume of a spherical tetrahedron In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific example where the dihedral angles are $\theta_1=\pi/4$, $\theta_2=\pi/4$, $\theta_3=\pi/4$, $\theta_4=\pi/6$, $\theta_5=\pi/6$ and $\theta_6=\pi/6$. I get lost in a sea of root threes. Could someone help ? • While "spherical triangle" has a clear meaning (at least when three points are given within the same hemisphere of a sphere), I'm not certain what "spherical tetrahedron" should mean. Four points on a sphere might be considered to "enclose" the entire sphere, for example (and give a spherical volume). – hardmath Dec 14 '13 at 14:16 • From the abstract spherical tetrahedron, which is in a $3$-dimensional spherical space of constant curvature $+1$ – Sigur Dec 14 '13 at 14:50 • What exactly is it you want to know? – Igor Rivin Dec 14 '13 at 15:34 • What is $q_0$, $q_1$, $q_2$ and $z_0$ would be a good start ... thank you Sigur for sorting the link. – Herman Dec 14 '13 at 15:38	340	1,122	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.875	3	CC-MAIN-2019-35	latest	en	0.869267
http://openstudy.com/updates/50dd53b6e4b069916c862603	1,516,342,244,000,000,000	text/html	crawl-data/CC-MAIN-2018-05/segments/1516084887746.35/warc/CC-MAIN-20180119045937-20180119065937-00490.warc.gz	260,675,264	8,582	• anonymous Exercises on transposes, permutations, spaces 5.1 etc. Can someone please explain the homework questions, because I have not seen some of the notation in the MIT videos, lecture summary nor in recitation. Problem 5.1a): What is P3? We have learned P12 etc but I haven't seen this superscript. Problem 5.1b): What does the hat (^) over the P mean? Problem 5.2: What does it mean for an entry of A to be "chosen independently"? Thank you. MIT 18.06 Linear Algebra, Spring 2010 • Stacey Warren - Expert brainly.com Hey! We 've verified this expert answer for you, click below to unlock the details :) SOLVED At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat. Looking for something else? Not the answer you are looking for? Search for more explanations.	370	1,404	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.53125	3	CC-MAIN-2018-05	latest	en	0.496591
https://math.answers.com/Q/What_are_all_even_numbers_divisible_by_2_and_4	1,643,139,767,000,000,000	text/html	crawl-data/CC-MAIN-2022-05/segments/1642320304872.21/warc/CC-MAIN-20220125190255-20220125220255-00058.warc.gz	422,137,852	57,915	0 # What are all even numbers divisible by 2 and 4? Wiki User 2012-10-09 17:16:18 Every even number is divisible by two (2, 4, 6, 8, etc.) but every other even number is divisible by four (4, 8, 12, 16, etc.). Wiki User 2012-10-09 17:16:18 🙏 0 🤨 0 😮 0 Study guides 20 cards ➡️ See all cards 3.72 373 Reviews	130	315	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.734375	3	CC-MAIN-2022-05	latest	en	0.895346
http://www.perlmonks.org/?node_id=761366	1,502,898,387,000,000,000	text/html	crawl-data/CC-MAIN-2017-34/segments/1502886102307.32/warc/CC-MAIN-20170816144701-20170816164701-00425.warc.gz	619,289,882	9,058	### Re^2: Learning Fundamentals: Data Structures and Algorithm Analysis in Perl by tilly (Archbishop) on May 01, 2009 at 18:54 UTC ( #761366=note: print w/replies, xml ) Need Help?? Very mild disagreement here with For the most part, the monks I see using big O do in fact know what it means and are aware of all or most of the issues involved in it. I would say that most of the monks I see using it are aware of how it is used informally by hackers, but a very large fraction are unaware of the formal definition, and/or are unaware of several subtleties that come up fairly often. (I rather strongly suspect that most who are aware that the formal definition probably couldn't come up with a reasonable version of it on demand...) For instance if someone says, "That's O(1), not O(n)" they are really thinking of big-Omega, not big-O. Because by the formal definition, anything that is O(1) is also O(n). However that's not how hackers informally use the term. And lots of them don't know that their informal usage is technically incorrect. Furthermore hackers usually have an implicit "amortized average time, if things work like they are supposed to" thrown in that they are not even conscious of. Worse yet, if they were asked to really explain what they meant, most probably couldn't. For instance most of us accept that, "hash lookup is O(1)." Well actually that's the average case if your hashing algorithm is working properly. The worst case with most hash implementations is O(n). If your hashing algorithm isn't a good fit, you'll hit that worst case fairly often. And what about that "amortized" bit? Well if you accept that pushing an element is O(1), you've implicitly accepted the amortized bit. If you're doing Perl programming that's OK. If you're doing real-time programming however... Then of course there is the glaring issue that scalability is often not what matters. For instance which is faster? Scanning a sorted list for an element, or doing a binary search? Despite O(n) vs O(log(n)), in a low-level language on modern CPUs, scanning is probably faster. Very few of us have a good sense for this kind of detail. As you see there are a lot of issues and complications that come up in a more formal discussion. My suspicion is that a large fraction (and possibility the majority) of Perl hackers who throw around the term aren't really aware of those. They could learn it pretty easily, but don't know it off of the top of their heads. However I'd absolutely agree that their understanding is more than sufficient for a practical analysis of the kinds of problems that we programmers are usually likely to encounter. • Comment on Re^2: Learning Fundamentals: Data Structures and Algorithm Analysis in Perl Replies are listed 'Best First'. Re^3: Learning Fundamentals: Data Structures and Algorithm Analysis in Perl by BrowserUk (Pope) on May 01, 2009 at 21:33 UTC but a very large fraction are unaware of the formal definition I think far too much is made about "formal definitions". Back when I did my formal CS education, I chose a multi-part long questionon big-O as my optional question. I passed the exam with distinction. Of course they don't tell you anything more than the overall scores, but I think I would probably have had to have scored perfectly on the four compulsuries to achieve that if I'd totally flunked the optional. Back then I could have, and probably did, give a formal definition of big-O and big Omega. But over a 25 year career, I never once needed to repeat it. And nor has it ever been useful to me. But the education means that I do have a way of informally discussing algortihmic complexity in terms that many other programmers recognise and understand. So in that sense the knowledge of it is useful. But in the real-world, when, as you point out, an O(n^2) algorithm in one language can out perform an O(n log n) in another; when memory allocation/reallocation costs can render a formally faster algorithm slower than a formally slower algorithm, because the latter is more cache friendly (eg. Mergesort versus QuickSort); and given multi-tasking OSs and mutli-core hardware renders static formal analysis iffy at best and an outright lie at worst; the usefulness of big-O is pretty much limited to discussion purposes only. A benchmark run many times under real-world or realistic conditions renders it mostly redundant. A starting point, and a means of informal discussion, but little more. Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error. "Science is about questioning the status quo. Questioning authority". In the absence of evidence, opinion is indistinguishable from prejudice. When I speak of a significant portion of hackers, I am not speaking of any particular one. I am not surprised that you'd have learned the formal definition. Although I have to disagree on how extreme your dismissal is. Absolutely big-O is useful as a starting point and means of informal discussion. However it also provides a framework for how to think about performance, and that can go a lot farther. Being aware of big-O has helped me predict how well real systems will scale, understand bottlenecks, and multiple times has given me a sense as to whether a given implementation's performance is reasonable, or whether a performance mistake needs to be tracked down. My point wasn't that I know it. More that I knew it once, but probably could not have recalled it accurately for most of of my career and never missed the absence of the recall. I've never had the need to establish the formal, algorithmic complexity of any given algorithm in the real-world, since I learnt the formal methods involved. But the informal appreciation of what the notation means has allowed me to discuss algorithms without worrying too much about the fine distinctions. Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error. "Science is about questioning the status quo. Questioning authority". In the absence of evidence, opinion is indistinguishable from prejudice. Create A New User Node Status? node history Node Type: note [id://761366] help Chatterbox? [jedikaiti]: 'elo Monks [perldigious]: Good times. :-) How do I use this? \| Other CB clients Other Users? Others drinking their drinks and smoking their pipes about the Monastery: (6) As of 2017-08-16 15:42 GMT Sections? Information? Find Nodes? Leftovers? Voting Booth? Who is your favorite scientist and why? Results (269 votes). Check out past polls. Notices?	1,434	6,571	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.625	3	CC-MAIN-2017-34	longest	en	0.968799
https://gitlab.inria.fr/why3/why3/-/commit/158af05acc54a231960dfdf02f4f962006c166b3	1,675,361,339,000,000,000	text/html	crawl-data/CC-MAIN-2023-06/segments/1674764500035.14/warc/CC-MAIN-20230202165041-20230202195041-00735.warc.gz	300,586,861	35,952	Commit 158af05a by Jean-Christophe Filliâtre ### debuggage typage parent 4cebacb3 theory Test type 'a t type u = t end theory Test type 'a t type u = (int, int) t end theory Test logic c : int logic c : int end theory Test type t = t end theory Test type t = u type u = t end theory Test type ('a, 'a) t end theory Test type 'a t = 'b end ... ... @@ -99,6 +99,34 @@ theory Array end theory Set type 'a t logic mem ('a, 'a t) logic empty : 'a t axiom Empty_def1 : forall x : 'a. not mem(x, empty) logic add ('a, 'a t) : 'a t axiom Add_def1 : forall x, y : 'a. forall s : 'a t. mem(x, add(y, s)) <-> (x = y or mem(x, s)) logic union('a t, 'a t) : 'a t axiom Union_def1 : forall s1, s2 : 'a t. forall x : 'a. mem(x, union(s1, s2)) <-> (mem(x, s1) or mem(x, s2)) logic equal(s1 : 'a t, s2 : 'a t) = forall x : 'a. mem(x, s1) <-> mem(x, s2) end (* Local Variables: compile-command: "make -C ../.. test" ... ... theory Set type 'a t logic mem ('a , 'a t) logic empty : 'a t axiom empty_def1 : forall x:'a. not mem(x, empty) logic add ('a, 'a t) : 'a t axiom add_def1 : forall x,y:'a. forall s:'a t. mem(x, add(y, s)) <-> (x = y or mem(x, s)) end theory Set_list use open List logic add(x:'a,s:'a t) : 'a t = Cons(x,s) ... ...	449	1,230	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.546875	3	CC-MAIN-2023-06	latest	en	0.156133
https://dsp.stackexchange.com/questions/30637/gps-signal-acquisition?noredirect=1	1,718,366,550,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198861546.27/warc/CC-MAIN-20240614110447-20240614140447-00383.warc.gz	187,824,894	41,205	# GPS signal acquisition I have a set of GPS satellite signal data, the set is 1 ms long and has the information of two satellites (satellite number, code phase and frequency) buried in noise. This data has a carrier frequency of 4.309 Mhz and the sampling rate is 5.714 Mhz.I've been reading but I don't really have pretty clear what to do: • which are the options to make the acquisition? As far as I've read, there are two options: time and frequency based acquisition. which is the best option to use? • The set of data is noisy and continuous: I have generated the C/A gold sequencies sampled at 5.714 Mhz in order to perform the correlation with the satellite data. How can I design a narrowband filter to get rid of the noise? The local coded I generated are sequencies of +1's and -1's. Do I need to convert this data also into -1's and +1's to perform the acquisition? If the GPS signal is centered (carrier) at 4.309 MHz, and this is sampled at 5.714 MHz, then there would be an alias copy at 5.714-4.309 = 1.405 MHz. To acquire this signal, you would first "down-convert" the signal to baseband, or 0 frequency by multiplying by $exp(-j\omega t)$ where $\omega$ is $2\pi(1.405e6)$ Hz. The reason to use an exponential and complex downconverter such as this process instead of just multiplying with a sine, is that it will facilitate fine adjustments to carrier frequency and phase as you do your acquisition (which you can do manually to get a feel for how a carrier tracking system would work). After you multiply the signal as such to move it to 0 frequency, then correlate with the code for your specific SV of interest. When correlating to a known code, the magnitude of the correlation will be proportional to the residual frequency offset of your carrier as a Sinc function with the first nulls located at $F= \frac1T$ where $T$ is the length in time of your correlation. (For example, if you choose to correlate over a full PRN which is 1 ms, the null to null width will be ± 1kHz, meaning you should be able to see an SV even if it is still ±500 Hz off. For this reason if I was doing a manual search with a full length PRN, I would step my frequency in steps of 500 Hz and do a correlation in each "frequency bin" until I stepped over the full possible Doppler range for my system. For your case to give a specific example, I would do a first correlation after multiplying by -1.405 MHz ($e^{-j2\pi1.405e6t}$), and know that I would see and SV if it was within 500 Hz (the null extends to 1kHz offset, but beyond 500 Hz and the correlated signal would start to compete with noise). If I didn't see anything for an SV that I knew was present, I would then step my frequency up 1 kHz (to -1.406 MHz) and try there, and I would continue this over all possible Doppler offsets (typically up to ±5 kHz). If I knew my signal was strong, I could correlate over 1/2 a sequence to double the "window" in which I would see a signal in Doppler offset. But this would reduce the processing gain by 3 dB but useful for an initial acquisition search. Note, if needed see my code generator available here: https://www.mathworks.com/matlabcentral/fileexchange/14670-gps-c-a-code-generator which includes the ability to resample the code to any arbitrary frequency (in your case 5.7144 MHz). And if you really want to get fancy (at the expense of resources associated with a 2D FFT), see my implementation for "one-shot" code delay and carrier phase here: https://www.mathworks.com/matlabcentral/fileexchange/26035-joint-frequency-and-delay-correlation This shows how to determine code delay and frequency offset from one block of data in one matrix operation, using the cross-correlation theorem to relate the fft to a circular correlation; $\text{xcorr}= \text{ifft}\left(\text{fft}\left(a\right)\text{fft}\left(b\right)^*\right)$ combined with the property that a circularly rotated FFT of a time domain signal is the same as the time domain signal in a different Doppler bin. (So once one FFT is computed for the CDMA PRN, you have the FFT's for the PRN at all other Doppler bins just by rotating that one FFT!). The code provides a similar plot as shown below which is the correlation for an SV vs code and frequency offset. • Any chance you have a python implementation of your matlab code? Commented Oct 4, 2018 at 15:28 • No but wouldn’t be too difficult to do using SciPy/Numpy given the similarities of the functions used. Are you referring to the joint code and delay acquisition or just a simple acquisition within a single Doppler frequency range l? Commented Oct 4, 2018 at 15:30 • I was thinking for either one. Commented Oct 4, 2018 at 15:31 • If you do code it I would start with the former (correlate in a Doppler bin, then step to the next Freq and search again —- or optionally correlate with use of a relatively slow frequency ramp (modulate the C/A code onto a ramp that goes over your full Doppler range over many PRN cycles and do a sliding correlation), it will provide more insight. Watch what I and Q do from one complex correlation peak to the next (will rotate at the Doppler offset rate). Commented Oct 4, 2018 at 15:42	1,250	5,163	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.46875	3	CC-MAIN-2024-26	latest	en	0.959532
https://web2.0calc.com/questions/simple-math-question-a-boat-travels-30-km-in-still	1,603,418,773,000,000,000	text/html	crawl-data/CC-MAIN-2020-45/segments/1603107880519.12/warc/CC-MAIN-20201023014545-20201023044545-00252.warc.gz	597,738,609	6,509	+0 # Simple math question: a boat travels 30 km in still water. If this boat goes upstream in a river flowing at a rate of 12 km/h for 5km, and t 0 99 1 a boat travels 30 km in still water. If this boat goes upstream in a river flowing at a rate of 12 km/h for 5km, and then returns back downstream to its starting point, what is the average speed for the whole trip in km/h? Oct 13, 2020 #1 0 Total distance is 5 + 5 km = 10 km tu = 5 / ( 30-12) = 5/18 td = 5 / (30+12) = 5/42 total distance / total time = average km/hr 10 /( 5/18 + 5/42) = 10 / (300/756) = 7560/300 = 25.2 km/hr Oct 13, 2020	222	611	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.09375	4	CC-MAIN-2020-45	longest	en	0.877219
http://www.velocityreviews.com/forums/t439863-can-you-tell-that-y-the-code-is-work-like-that.html	1,386,249,450,000,000,000	text/html	crawl-data/CC-MAIN-2013-48/segments/1386163045148/warc/CC-MAIN-20131204131725-00099-ip-10-33-133-15.ec2.internal.warc.gz	597,236,038	11,146	Velocity Reviews > can you tell that y the code is work like that # can you tell that y the code is work like that venkatesh Guest Posts: n/a 10-21-2005 int p=20; printf("%d",p); printf("%d",p); the above code prints the output as somegarbage value //for p(may be address 20 //for p why can you explain thanks for that Christopher Benson-Manica Guest Posts: n/a 10-21-2005 venkatesh <(E-Mail Removed)> wrote: > int p=20; Broken. What makes you suspect that 20 is an appropriate value for a pointer to an integer? > printf("%d",p); Broken. What exactly do you think a pointer with the value 20 might point to? > printf("%d",p); Broken. The size of a pointer to an integer need not be equal to sizeof(int). printf( "%p", (void)p ); > why can you explain Pure random chance, as far as the C standard is concerned. -- Christopher Benson-Manica \| I should know what I'm talking about - if I ataru(at)cyberspace.org \| don't, I need to know. Flames welcome. Pierre Maurette Guest Posts: n/a 10-21-2005 venkatesh, le 21/10/2005, a écrit : > int p=20; > printf("%d",p); > printf("%d",p); > > > > the above code prints the output as > somegarbage value //for p(may be address > 20 //for p > why can you explain > thanks for that int dummy = 20; int p = &dummy; printf("%d\n",p); printf("%p\n",p); or int p = malloc(sizeof int); p = 20; printf("%d\n",p); printf("%p\n",p); -- Pierre Maurette Emmanuel Delahaye Guest Posts: n/a 10-21-2005 venkatesh a écrit : > int p=20; This is kinda nonsense. What to you think it meant ? The portable way of initializing a pointer to an object are - NULL - The address of an object of the same type - the value returned by malloc() - the value returned by fopen() if the type is void or FILE* and similar cases. But initialising a pointer with a plain integer is undefined by the language. Actually, it may work on a specific platform. It's called an implementation-dependent behaviour. -- C is a sharp tool vire Guest Posts: n/a 10-21-2005 yes, exactly what you wrote means: int* p; p = 20; Default User Guest Posts: n/a 10-21-2005 vire wrote: > yes, > exactly what you wrote means: > int* p; > p = 20; Brian -- Please quote enough of the previous message for context. To do so from Google, click "show options" and use the Reply shown in the expanded Keith Thompson Guest Posts: n/a 10-21-2005 Christopher Benson-Manica <(E-Mail Removed)> writes: > venkatesh <(E-Mail Removed)> wrote: >> int p=20; > > Broken. What makes you suspect that 20 is an appropriate value for a > pointer to an integer? What's surprising is that his compiler thought it was appropriate. A compiler might allow assigning an integer value to a pointer as an extension, but if the code is compiled in strict mode (might be called "ansi" or "iso", depending on the compiler), the compiler is required to issue a diagnostic. [...] >> printf("%d",p); > > Broken. The size of a pointer to an integer need not be equal to > sizeof(int). Size matters not. Using "%d" to print a pointer value invokes undefined behavior (though it will often do what you expect anyway). It can fail even if int and int happen to have the same size. The correct way to print an int* value is: printf("%p", (void)p); Also, the original code, if it doesn't blow up, will probably print the values adjacent to each other; you won't be able to tell where one ends and the other begins. Rather than int p=20; printf("%d",p); printf("%d",p); you might try this: int p = (int)20; printf("p = %d\n", p); printf("p = %p\n", (int)p); Of course this could still invoke undefined behavior, since (int)20 is unlikely to be a valid address. Here's a program that actually works: #include <stdio.h> #include <stdlib.h> int main (void) { int p = malloc(sizeof p); if (p == NULL) { printf("malloc failed\n"); } else { p = 20; printf("p = %d\n", p); printf("p = %p\n", (void)p); } return 0; } -- Keith Thompson (The_Other_Keith) http://www.velocityreviews.com/forums/(E-Mail Removed) <http://www.ghoti.net/~kst> San Diego Supercomputer Center <> <http://users.sdsc.edu/~kst> We must do something. This is something. Therefore, we must do this. Christopher Benson-Manica Guest Posts: n/a 10-21-2005 Keith Thompson <(E-Mail Removed)> wrote: > The correct way to print an int* value is: > printf("%p", (void)p); (which I did mention, FWIW) > Of course this could still invoke undefined behavior, since (int)20 > is unlikely to be a valid address. Is it not UB regardless, since 20 is not a value obtained from malloc()? -- Christopher Benson-Manica \| I should know what I'm talking about - if I ataru(at)cyberspace.org \| don't, I need to know. Flames welcome. Tim Rentsch Guest Posts: n/a 10-21-2005 Emmanuel Delahaye <(E-Mail Removed)> writes: > venkatesh a écrit : > > int p=20; > > This is kinda nonsense. What to you think it meant ? > > The portable way of initializing a pointer to an object are > > - NULL > - The address of an object of the same type > - the value returned by malloc() > - the value returned by fopen() if the type is void or FILE* > and similar cases. And 0 (or any 0-valued constant expression). > But initialising a pointer with a plain integer is undefined by the > language. Unless the integer is 0. Keith Thompson Guest Posts: n/a 10-21-2005 Christopher Benson-Manica <(E-Mail Removed)> writes: > Keith Thompson <(E-Mail Removed)> wrote: >> The correct way to print an int* value is: > >> printf("%p", (void)p); > > (which I did mention, FWIW) Ok. >> Of course this could still invoke undefined behavior, since (int)20 >> is unlikely to be a valid address. > > Is it not UB regardless, since 20 is not a value obtained from > malloc()? Valid pointers don't have to be obtained from malloc(). The conversion itself yields an implementation-defined result: An integer may be converted to any pointer type. Except as previously specified, the result is implementation-defined, might not be correctly aligned, might not point to an entity of the referenced type, and might be a trap representation. with a footnote: The mapping functions for converting a pointer to an integer or an integer to a pointer are intended to be consistent with the addressing structure of the execution environment. If the result happens to be an invalid pointer, dereferencing it invokes undefined behavior. If the implementation happens to guarantee that 20 is a valid address for an int, the behavior of dereferencing (int)20 is merely implementation-defined. -- Keith Thompson (The_Other_Keith) (E-Mail Removed) <http://www.ghoti.net/~kst> San Diego Supercomputer Center <> <http://users.sdsc.edu/~kst> We must do something. This is something. Therefore, we must do this.	1,869	6,773	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.546875	3	CC-MAIN-2013-48	longest	en	0.817351
http://www.mathworks.com/matlabcentral/answers/60201-how-can-i-delete-horizontal-vector-from-output-in-optical-flow	1,454,960,283,000,000,000	text/html	crawl-data/CC-MAIN-2016-07/segments/1454701153998.27/warc/CC-MAIN-20160205193913-00053-ip-10-236-182-209.ec2.internal.warc.gz	503,573,039	12,557	## how can i delete horizontal vector from output in optical flow? on 29 Jan 2013 hi i need to delete horizontal velocity vector from output in optical flow is there any way to show special vector from output of optical flow function in matlab r2012a for example vector with special angles can you help me??? ## Products No products are associated with this question. ### Image Analyst (view profile) on 29 Jan 2013 ```yourOutputArray(rowToDelete, :) = []; ``` on 31 Jan 2013 i cant understand your answer i use vision.optical flow and work with demo which exist in matlab for optical flow. the output array is out?? and what is rowToDelete? can you explain more pleas? Image Analyst ### Image Analyst (view profile) on 31 Jan 2013 I thought it was rather self explanatory. See where you say " i need to delete horizontal velocity vector from output"? And where you say "...from output of optical...."? Well what you called "output" is what I called yourOutputArray. And a vector is a 1D array of numbers,in other words a row or a column. Let's say you wanted to delete row #4, then rowToDelete would be equal to 4. Let's try an example. Let's say your array was really called myOpticalFlowMatrix and you wanted to get rid of row #18. So you'd do this: ```myOpticalFlowMatrix(18, :) = []; ``` I think that should explain it very well. on 2 Feb 2013 ok i explain my problem badly. i don't write any code i only use the demo in matlab r2012a to learn what optical flow work. this demo exist in help of vision.OpticalFlow the output of this demo are some matrices one of them is of which said in demo that is the optical flow for video.this matrix has the same size as video frame(for each frame)i need to know what is it? is it the velocity for each pixel and i need to consider vertical velocity. how can i do this. how can i find for example vertical velocity in this function of matlab for optical flow? what are these lines that shown in the output of demo?? they show the magnitude of velocity. i want to use optical flow for motion detection in fire video. i hope explain good! can you help me? MATLAB and Simulink resources for Arduino, LEGO, and Raspberry Pi test	510	2,186	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.546875	3	CC-MAIN-2016-07	latest	en	0.944301
https://www.hackmath.net/en/problem/1341	1,566,798,292,000,000,000	text/html	crawl-data/CC-MAIN-2019-35/segments/1566027330968.54/warc/CC-MAIN-20190826042816-20190826064816-00005.warc.gz	851,620,449	7,164	# Chicken I threw three chicken for a handful of grains and I noticed that it eat in a ratio of 8:7:6 and two of them grabbed 156 grains. How many grains have fought that chicken? Result x = 96 y = 84 z = 72 #### Solution: y+z =156 7 x = 8 y 6 y = 7 z y+z = 156 7x-8y = 0 6y-7z = 0 x = 96 y = 84 z = 72 Calculated by our linear equations calculator. Leave us a comment of example and its solution (i.e. if it is still somewhat unclear...): Be the first to comment! #### To solve this verbal math problem are needed these knowledge from mathematics: Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation? ## Next similar examples: 1. Rabbits 3 Viju has 40 chickens and rabbits. If in all there are 90 legs. How many rabbits are there with Viju? 2. Three friends The three friends spent 600 KC in a teahouse. Thomas paid twice as much as Paul. Paul a half less than Zdeněk. How many each paid? 3. Cows and calves There are 168 cows and calves in the cowshed. Cows are in nine stalls and calves in four stalls. Same count cows are in every cow stall and three more in each calf stall than in a cow stall. What is the capacity of the stalls for cows and what for calves? 4. Meat Half a kilogram of pork and three-quarters of a kilogram of beef cost total 5.1 USD. One quarter of a kilogram of beef and 1 kilogram of pork cost 4.9 USD. Determine the price of 1 kg of each type of meat. 5. Unknown number Identify unknown number which 1/5 is 40 greater than one tenth of that number. 6. ATC camp The owner of the campsite offers 79 places in 22 cabins. How many of them are triple and quadruple? 7. Theatro Theatrical performance was attended by 480 spectators. Women were in the audience 40 more than men and children 60 less than half of adult spectators. How many men, women and children attended a theater performance? 8. Ski lessons On the ski course leaves 80 pupils. They live in triple and quadruple rooms. Count if rooms is the 23. How many are triple and quadruple rooms? 9. Football league In the 2nd football league plays each of the 16 participants with each opponent twice. For every victory gets 3 points, 1 point for a draw, a defeat none. After the competition was team with 54 points and 12 defeats at 9th place. How many times have won? 10. Mom and daughter Mother is 39 years old. Her daughter is 15 years. For many years will mother be four times older than the daughter? 11. Boys and girls There are 48 children in the sports club, boys are 10 more than girls. How many girls go to the club? 12. Equations Solve following system of equations: 6(x+7)+4(y-5)=12 2(x+y)-3(-2x+4y)=-44 13. The larger The larger of two numbers is nine more than four times the smaller number. The sum of the two numbers is fifty-nine. Find the two numbers.	759	2,830	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.21875	4	CC-MAIN-2019-35	latest	en	0.960344
https://studylib.net/doc/8085092/hw-31---helix-algebra	1,600,777,120,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600400205950.35/warc/CC-MAIN-20200922094539-20200922124539-00546.warc.gz	647,388,046	13,603	# HW 31 - Helix Algebra ``` Assignment #31 Name:________________________________ Plotting Points; Vertical and Horizontal Lines *Use the coordinate planes provided on a separate sheet of paper àο Plot the four points on the coordinate plane. Connect the points to form a rectangle. 1.) Find the PERIMETER: (−4 , 0) (−4 , 6) (5 , 0) ( 5, 6) Perimeter: ___________ 2.) Find the AREA: (0 , 0) (−4 , 0) (0 , 7) ( −4, 7) Area: ___________ àο Plot the three points on a coordinate plane. Connect the points to form a triangle, then find the ! AREA. π΄πππ ππ π π‘πππππππ = (πππ π β βπππβπ‘) ! ! ! ! 3.) (1 , 2) (7 , 2) (1 , 6) − 4.) ! −4 , ! ! !−4 , !! !0 , ! ! Area: __________ Area: __________ àο Graph and label the three lines on the same coordinate plane. If the lines cross, label the coordinates of the point where the lines meet. 5.) π¦ = 5 , π₯ = 4 , π¦ = −1 6.) π¦ = 3, π¦ = 0 , π₯ = −2 π¦ = −4 , π₯ = 0 , π₯ = 2 7.) àο Graph the four lines on the same coordinate plane. From the rectangle formed by the lines, find: 8.) Perimeter: π¦ = 0, π₯ = 4, π¦ = 6, π₯ = 0 Perimeter: ___________ 9.) Area: π¦ = 6, π₯ = −4 , π¦ = −1 , π₯ = 5 Area: _____________ àο Use the graph below to answer questions 10-β14: Marks Bank Account Balance: \$20 \$15 Marks Money \$10 \$5 Jan Feb. Mar. Apr. May Jun. July Aug. Sep. Oct. Nov. Dec. -β \$5 -β\$10 -β\$15 -β\$20 10.) In which month did Mark have the most money? _____________________ 11.) In which months did Mark owe the bank money? ______________________ 12.) Plot points to represent the following: In May, Mark had \$3. In April, Mark had \$12. In June, Mark had \$0. 13.) What month did Mark have \$18? ___________________ 14.) Predict how much money you think Mark will have in December then justify your reasoning. àο Write the rule for each table (Hint: how can you get from x to y) 15.) π₯ π¦ 16.) π₯ π¦ 17.) 3 10 8 38 3 24 5 5 50 494 14 14 14 6 1 9 9 6 7 2 2 2 8 7 64 π₯ π¦ 0 1 0 0 -β4 -β7 -β37 -β4 -β10 -β29 -β9 -β10 -β106 π¦ = __________________ π¦ = __________________ π¦ = __________________ ```	978	2,321	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.21875	4	CC-MAIN-2020-40	latest	en	0.711631
https://gmatclub.com/forum/students-in-the-metropolitan-school-district-lack-math-79092.html	1,495,828,276,000,000,000	text/html	crawl-data/CC-MAIN-2017-22/segments/1495463608676.72/warc/CC-MAIN-20170526184113-20170526204113-00212.warc.gz	957,238,888	65,052	Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack It is currently 26 May 2017, 12:51 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track Your Progress every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # Students in the metropolitan school district lack math new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Intern Joined: 02 Jun 2009 Posts: 24 Followers: 0 Kudos [?]: 8 [1] , given: 0 Students in the metropolitan school district lack math [#permalink] ### Show Tags 02 Jun 2009, 03:40 1 This post received KUDOS 10 This post was BOOKMARKED 00:00 Difficulty: 65% (hard) Question Stats: 46% (01:52) correct 54% (01:13) wrong based on 806 sessions ### HideShow timer Statistics Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming [Reveal] Spoiler: OA If you have any questions you can ask an expert New! Manager Joined: 12 Apr 2006 Posts: 216 Location: India Followers: 1 Kudos [?]: 29 [0], given: 17 Re: students [#permalink] ### Show Tags 02 Jun 2009, 08:07 Not sure about the reasoning but I feel C is correct. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (Seem awkward to me) (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (Seem awkward to me) (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (My choice for correct answer) (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (Meaning changed) (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming (Meaning changed) Manager Joined: 01 May 2009 Posts: 106 Followers: 2 Kudos [?]: 47 [0], given: 11 Re: students [#permalink] ### Show Tags 02 Jun 2009, 09:51 Options A, B and E have pronoun reference errors Option D, meaning changed Option C seems right OA please? Senior Manager Joined: 11 Dec 2008 Posts: 479 Location: United States GMAT 1: 760 Q49 V44 GPA: 3.9 Followers: 40 Kudos [?]: 208 [0], given: 12 Re: students [#permalink] ### Show Tags 02 Jun 2009, 10:02 dtaneja wrote: Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming I go with option E. As before, PLEASE UNDERLINE THE SECTION NEEDING TO BE CHANGED. Manager Joined: 12 Apr 2006 Posts: 216 Location: India Followers: 1 Kudos [?]: 29 [0], given: 17 Re: students [#permalink] ### Show Tags 02 Jun 2009, 11:25 bipolarbear wrote: I go with option E. As before, PLEASE UNDERLINE THE SECTION NEEDING TO BE CHANGED. Option E says students are lacking in math skills itself, whereas the question somewhat says they have math skills but it is not upto mark to make them compititive in city economy. Not sure if it makes sense but you can say it's more GMATish So I strongly believe E is incorrect choice. Senior Manager Joined: 11 Dec 2008 Posts: 479 Location: United States GMAT 1: 760 Q49 V44 GPA: 3.9 Followers: 40 Kudos [?]: 208 [0], given: 12 Re: students [#permalink] ### Show Tags 02 Jun 2009, 11:49 humans wrote: bipolarbear wrote: I go with option E. As before, PLEASE UNDERLINE THE SECTION NEEDING TO BE CHANGED. Option E says students are lacking in math skills itself, whereas the question somewhat says they have math skills but it is not upto mark to make them compititive in city economy. Not sure if it makes sense but you can say it's more GMATish So I strongly believe E is incorrect choice. I stand by my answer choice of E. There isn't much of a difference between are so lacking math skills and lack math skills to such a large degree. A is way too complicated and uses unnecessary words. O.A please? Senior Manager Joined: 15 Jan 2008 Posts: 285 Followers: 2 Kudos [?]: 42 [0], given: 3 Re: students [#permalink] ### Show Tags 03 Jun 2009, 08:09 So shud always be accompained by That.. hence teh answer is A Senior Manager Joined: 11 Dec 2008 Posts: 479 Location: United States GMAT 1: 760 Q49 V44 GPA: 3.9 Followers: 40 Kudos [?]: 208 [0], given: 12 Re: students [#permalink] ### Show Tags 03 Jun 2009, 08:31 A is too wordy while E is much more concise and conveys the exact same point. Hence E... still waiting for OA Manager Joined: 31 Oct 2007 Posts: 55 Followers: 1 Kudos [?]: 9 [1] , given: 0 Re: students [#permalink] ### Show Tags 03 Jun 2009, 14:14 1 This post received KUDOS E. Them only refers to people. E avoids pronoun reference error created by they and wordiness of choice A. Manager Joined: 12 Apr 2006 Posts: 216 Location: India Followers: 1 Kudos [?]: 29 [0], given: 17 Re: students [#permalink] ### Show Tags 03 Jun 2009, 19:50 bipolarbear wrote: I stand by my answer choice of E. There isn't much of a difference between are so lacking math skills and lack math skills to such a large degree. A is way too complicated and uses unnecessary words. O.A please? You are right . OA is E, found this in OG11. Intern Joined: 03 Jun 2009 Posts: 49 Followers: 1 Kudos [?]: 12 [0], given: 7 Re: students [#permalink] ### Show Tags 04 Jun 2009, 11:06 The Sentence resembles an action happening (economy becoming )so the use of present progressive tense "are lacking " is preferred to rather simple present "lack" Also the usage of right idiomatic expression "so x that" as in option E makes it a preferable choice. Intern Joined: 02 Jun 2009 Posts: 24 Followers: 0 Kudos [?]: 8 [0], given: 0 Re: students [#permalink] ### Show Tags 05 Jun 2009, 00:41 OA is "E" Intern Joined: 30 Jul 2009 Posts: 20 Followers: 0 Kudos [?]: 4 [0], given: 3 Re: students [#permalink] ### Show Tags 26 May 2010, 08:24 Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming- (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes-- (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes---"Student in the metropolitan school district" does not go with "lack of math skills" (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming---Correct Intern Joined: 29 Mar 2010 Posts: 43 Followers: 0 Kudos [?]: 11 [0], given: 1 Re: students [#permalink] ### Show Tags 27 May 2010, 10:08 I would also go with E. Unfortunate but true I could get the answer on this SC in under 30 secs. And mostly do on all questions of SC on the club. But whenever I attempt some standardized CAT test my answers never reach this accuracy level. Why? Are these questions that are posted here easy? Posted from my mobile device _________________ ------------------------------------------------------------------------- Ros. Nice Post + Some help + Lucid solution = Kudos The greatest pleasure in life is doing what people say you cannot do \| Great minds discuss ideas, average minds discuss events, small minds discuss people. ------------------------------------------------------------------------- SVP Joined: 17 Feb 2010 Posts: 1511 Followers: 19 Kudos [?]: 630 [0], given: 6 Re: students [#permalink] ### Show Tags 31 May 2010, 15:21 can someone please explain why D is wrong? I thought "city's economy" is better than "city economy". Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming Intern Joined: 17 May 2010 Posts: 19 Followers: 0 Kudos [?]: 4 [0], given: 0 Re: students [#permalink] ### Show Tags 12 Jun 2010, 20:41 Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming >>> correct answer SVP Joined: 16 Jul 2009 Posts: 1538 Schools: CBS WE 1: 4 years (Consulting) Followers: 44 Kudos [?]: 1198 [0], given: 2 Re: students [#permalink] ### Show Tags 28 Nov 2010, 09:25 I think that it is all about the idiom so X that Y _________________ The sky is the limit 800 is the limit GMAT Club Premium Membership - big benefits and savings Manager Joined: 13 Jul 2010 Posts: 168 Followers: 1 Kudos [?]: 88 [1] , given: 7 Re: students [#permalink] ### Show Tags 28 Nov 2010, 16:17 1 This post received KUDOS seekmba wrote: can someone please explain why D is wrong? I thought "city's economy" is better than "city economy". Students in the metropolitan school district lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming ever more dependent on information-based industries. (A) lack math skills to such a large degree as to make it difficult to absorb them into a city economy becoming (B) lack math skills to a large enough degree that they will be difficult to absorb into a city’s economy that becomes (C) lack of math skills is so large as to be difficult to absorb them into a city’s economy that becomes (D) are lacking so much in math skills as to be difficult to absorb into a city’s economy becoming (E) are so lacking in math skills that it will be difficult to absorb them into a city economy becoming D is wrong because "are lacking so much" is unidiomatic correct way of saying it is "so lacking in" also "as to be" is not idiomatic either. Also when you using "so" to suggest exten - ot needs to be followed by "that" to complete the thought. Hope it helps. SVP Joined: 16 Nov 2010 Posts: 1666 Location: United States (IN) Concentration: Strategy, Technology Followers: 34 Kudos [?]: 533 [0], given: 36 Re: students [#permalink] ### Show Tags 30 May 2011, 04:36 The answer is E - the Idiom "So X that Y" has been used here. _________________ Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant) GMAT Club Premium Membership - big benefits and savings Intern Joined: 17 Dec 2010 Posts: 12 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: students [#permalink] ### Show Tags 30 May 2011, 06:20 Correct answer E Bcz of GMATish way of thinking Posted from my mobile device Re: students [#permalink] 30 May 2011, 06:20 Go to page 1 2 Next [ 31 posts ] Similar topics Replies Last post Similar Topics: 17 Students in the metropolitan school district lack math 27 07 May 2017, 07:47 2 Students in the metropolitan school district lack math 7 12 Sep 2014, 18:12 4 Students in the metropolitan school district lack math 4 02 Oct 2014, 06:00 3 Students in the metropolitan school district lack math 5 13 Feb 2016, 00:23 13 Students in the metropolitan school district lack math 12 26 Aug 2016, 06:54 Display posts from previous: Sort by # Students in the metropolitan school district lack math new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	3,762	14,380	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.8125	4	CC-MAIN-2017-22	longest	en	0.943484
https://www.ambrosiospa.com/what-are-the-3-examples-of-acceleration/	1,660,746,709,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882572908.71/warc/CC-MAIN-20220817122626-20220817152626-00394.warc.gz	547,738,818	22,270	# What are the 3 examples of acceleration? ## What are the 3 characteristics of acceleration? Acceleration is the rate at which an object changes its velocity. Acceleration has to have speed and direction and a rate of change of speed. Constant acceleration is when an object is changing its velocity at the same rate each second. ## What are 5 examples of acceleration? Examples • An object was moving north at 10 meters per second. • An apple is falling down. • Jane is walking east at 3 kilometers per hour. • Tom was walking east at 3 kilometers per hour. • Sally was walking east at 3 kilometers per hour. • Acceleration due to gravity. ## What are the five examples of acceleration? Examples • An object was moving north at 10 meters per second. • An apple is falling down. • Jane is walking east at 3 kilometers per hour. • Tom was walking east at 3 kilometers per hour. • Sally was walking east at 3 kilometers per hour. • Acceleration due to gravity. ## What are 10 examples of Newton’s second law? 10 Examples of Newton's Second Law of Motion in Everyday Life • Pushing a Car and a Truck. • Pushing a Shopping Cart. • Two People Walking Together. • Hitting a Ball. • Rocket Launch. • Car Crash. • Object thrown from a Height. • Karate Player Breaking Slab of Bricks. ## What are the three types of acceleration in driving? Aggressive acceleration – Firm pressure on the accelerator to increase speed of the vehicle rapidly. Backing – Moving the vehicle in reverse direction. Coasting – Level of braking in which releasing the accelerator stops the vehicle's forward propulsion. ## What are the different types of acceleration? In physics, the three types of acceleration are changes in speed, direction and both simultaneously. The word "velocity" is often used in place of speed. A person can calculate the acceleration of an object by determining its velocity and the length of time it accelerates. ## What is acceleration Class 9 answer? Acceleration: The time rate of change of velocity of a body is called acceleration. Acceleration=Velocity(Δv)time(Δt) If in a given time interval the velocity of a body changes from u to v , then acceleration a is expressed as. a=v−ut. ## What are 5 examples of law of acceleration? Examples • An object was moving north at 10 meters per second. • An apple is falling down. • Jane is walking east at 3 kilometers per hour. • Tom was walking east at 3 kilometers per hour. • Sally was walking east at 3 kilometers per hour. • Acceleration due to gravity. ## What are 3 examples of Newton’s 3rd law? Newton's third law of motion examples • Pulling an elastic band. • Swimming or rowing a boat. • Static friction while pushing an object. • Walking. • Standing on the ground or sitting on a chair. • The upward thrust of a rocket. • Resting against a wall or tree. • Slingshot. ## What are the two kinds of acceleration? Much like velocity, there are two kinds of acceleration: average and instantaneous. ## What are the types of acceleration Class 9? There are three types of accelerated motions : uniform acceleration, non-uniform acceleration and average acceleration.	689	3,145	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.828125	4	CC-MAIN-2022-33	latest	en	0.946079
https://math.stackexchange.com/questions/1141310/proof-of-a-equivalence-relation	1,569,246,195,000,000,000	text/html	crawl-data/CC-MAIN-2019-39/segments/1568514576965.71/warc/CC-MAIN-20190923125729-20190923151729-00243.warc.gz	575,625,827	31,406	# Proof of a equivalence relation A set $A$ is equipotent to a set $B$ $(A\sim B)$, if a bijection $f: A \rightarrow B$ exists. How to prove, that $\sim$ is a equivalence relation? EDIT: I understand the concept of reflexivity, symmetry and transitivity. I just don't know how to write it formally as a proof. • I don't understand how to start. Can I somehow use the elements of the sets? – Arthur Feb 9 '15 at 22:53 • First, an equivalence relation is symmetric, reflexive and transitive. Check this properties. For example: $A ~ B$ and $B~C$ implies $A~C$? – Diego Robayo Feb 9 '15 at 22:55 • In my example, is composition of bijections a bijection? For the other two is $A$ equipotent to itself? Does $A~B$ implies $B~A$? These three properties define an equivalence relation. – Diego Robayo Feb 9 '15 at 22:59 • @ArthurD.: Please pay no attention to this. In the most standard formal set theory (ZFC), this is not an equivalence relation, even though the usual criteria (reflexivity, symmetry, transitivity) are satisfied. The problem is that the relation is too large to be a set. – André Nicolas Feb 9 '15 at 23:02 The point of the exercise seems to be to expose and exploit the following properties of bijections: • The identity map is a bijection. • The inverse of a bijection is a bijection. • The composition of two bijections is a bijection. These properties are easy to prove and correspond to reflexivity, symmetry, and transitivity. However, there is an important technical detail: an equivalence relation is defined on a set. You cannot use the set of all sets because that is not a set. You need to fix a universe. Bottom line, the relation given is an equivalence relation on (any subset of) the set of all subsets of a fixed set $U$. • Thanks, but I don't understand, what do you mean by "set of all subsets"? There are two sets given. A and B. They are connected by a bijection, which implies they are equipotent. Every element in A is in relation with itself, so that's reflexivity. I think the bijection implies also, that the elements in A are in symmetric relation to B. But what's with transitivity? I would have to have another set C to prove that, or am I wrong? – Arthur Feb 10 '15 at 23:53	572	2,229	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.859375	4	CC-MAIN-2019-39	latest	en	0.923299
https://www.instasolv.com/question/the-value-of-5-2-dt-rodrig-a-1b-c-1d-none-ehd5ka	1,610,764,270,000,000,000	text/html	crawl-data/CC-MAIN-2021-04/segments/1610703499999.6/warc/CC-MAIN-20210116014637-20210116044637-00142.warc.gz	874,593,431	10,500	The value ∫_2^3f(5−x)dx−∫_2^3f(x)dx... Question # The value intlimits^3_2 {f} , (5-x)dx-intlimits^3_2 {f(x)} , dx is (a) 1 (b) 0 (c) -1 (d) None Other Exams Maths Solution 91 4.0 (1 ratings) @intlimits^3_2 f{(5-x)} , dx -intlimits^3_2 {f(x)} , dx =intlimits^3_2 {f(5-x)} , dx-intlimits^3_2 {f(3+2-x)} , dx =intlimits^3_2 f{(5-x)} , dx-intlimits^3_2 f{(5-x)} , dx =0 Ans Option b	190	382	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.296875	3	CC-MAIN-2021-04	latest	en	0.265826
https://www.codeproject.com/articles/11719/cyclomatic-code-complexity-analysis-for-microsoft?fid=218002&df=10000&mpp=25&sort=position&spc=relaxed&select=1232712&noise=3&prof=true&view=thread&fr=1	1,480,968,005,000,000,000	text/html	crawl-data/CC-MAIN-2016-50/segments/1480698541783.81/warc/CC-MAIN-20161202170901-00220-ip-10-31-129-80.ec2.internal.warc.gz	922,785,763	27,372	12,625,646 members (35,022 online) alternative version 150.8K views 38 bookmarked Posted # Cyclomatic Code Complexity Analysis for Microsoft .NET Applications , 21 Sep 2005 Rate this: Code Complexity is a measure that provides a single ordinal number which can be compared to the complexity of other programs. It is one of the most widely accepted static software metrics and is intended to be independent of language and language format. ## Introduction Cyclomatic Code Complexity was first introduced by Thomas McCabe in 1976. In 1976, Thomas McCabe published a paper arguing that code complexity is defined by its control flow. Since that time, others have identified different ways of measuring complexity (e.g. data complexity, module complexity, algorithmic complexity, call-to, call-by, etc.). Although these other methods are effective in the right context, it seems to be generally accepted that control flow is one of the most useful measurements of complexity, and high complexity scores have been shown to be a strong indicator of low reliability and frequent errors. ## Overview This measure provides a single ordinal number that can be compared to the complexity of other programs. It is one of the most widely accepted static software metrics and is intended to be independent of language and language format. Code Complexity is a measure of the number of linearly-independent paths through a program module and is calculated by counting the number of decision points found in the code (if, else, do, while, throw, catch, return, break etc.). ### Technical Specification Cyclomatic Complexity for a software module calculated based on graph theory is based on the following equation: `CC=E-N+p` Where • CC = Cyclomatic Complexity • E = the number of edges of the graph • N = the number of nodes of the graph • p = the number of connected components Further academic information on the specifics of this can be found here. From a layman’s perspective the above equation can be pretty daunting to comprehend. Fortunately there is a simpler equation which is easier to understand and implement by following the guidelines shown below: • Add 1 for each of the following keywords or their equivalent: `if`, `while`, `repeat`, `for`, `and`, `or`. • Add 1 for each `case` in a `switch` statement. Let’s look at a few examples to understand how the code complexity is calculated. #### Example 1 ```public void ProcessPages() { while(nextPage !=true) { if((lineCount<=linesPerPage) && (status != Status.Cancelled) && (morePages == true)) { //.... } } }``` In the code above, we start with 1 for the routine, add 1 for the `while` loop, add 1 for the `if`, and add 1 for each `&&` for a total calculated complexity of 5. #### Example 2 ```public int getValue(int param1) { int value = 0; if (param1 == 0) { value = 4; } else { value = 0; } return value; }``` In the code above, we start with 1 for the routine, add 1 for the `if`, and add 1 for the `else` for a total calculated complexity of 3. Members that have high code complexity should be reviewed for possible refactoring. The SEI provides the following basic risk assessment based on the value of code: Cyclomatic Complexity Risk Evaluation 1 to 10 a simple program, without very much risk 11 to 20 a more complex program, moderate risk 21 to 50 a complex, high risk program > 50 an un-testable program (very high risk) ## Tools There are several free tools available which help one analyze the code complexity: • devMetrics by Anticipating minds have a free community edition available for analyzing metrics for C# projects. • Reflector Add-In: Code Metrics can be used to analyze .NET assemblies and show design quality metrics. This add-in is to be used in conjunction with Lutz Roeder’s Reflector. • It is very easy to compute as illustrated in the example. • Unlike other complex measurements, it can be computed immediately in the development cycle (which makes it agile friendly). • It provides a good indicator of the ease of code maintenance. • It can help focus testing efforts. • It makes it easy to find complex code for formal review. A list of licenses authors might use can be found here ## Share India No Biography provided ## You may also be interested in... Pro Pro First Prev Next My vote of 1 DaveBlack4-Jan-11 10:06 DaveBlack 4-Jan-11 10:06 Update link to CodeMetrics .NET Reflector Add-In Michael Freidgeim30-Jan-09 11:45 Michael Freidgeim 30-Jan-09 11:45 Few questions on calculating CC of program Shiva Prasad Sheela22-Apr-08 2:58 Shiva Prasad Sheela 22-Apr-08 2:58 Great freeware tool to display Cyclomatic Complexity Hannes Pavelka19-Mar-06 4:14 Hannes Pavelka 19-Mar-06 4:14 Re: Great freeware tool to display Cyclomatic Complexity DSK Chakravarthy24-Nov-14 23:40 DSK Chakravarthy 24-Nov-14 23:40 Visual Studio .NET 2005 ssaSpot28-Sep-05 9:14 ssaSpot 28-Sep-05 9:14 VB.NET?? Drew Berkemeyer27-Sep-05 4:30 Drew Berkemeyer 27-Sep-05 4:30 Re: VB.NET?? alfalfa27-Sep-05 5:14 alfalfa 27-Sep-05 5:14 Thanks `leppie`21-Sep-05 8:08 `leppie` 21-Sep-05 8:08 I will be checking them out xacc-ide 0.0.15 now with C#, MSIL, C, XML, ASP.NET, Nemerle, MyXaml and HLSL coloring - Screenshots Last Visit: 31-Dec-99 19:00 Last Update: 5-Dec-16 5:00 Refresh 1	1,325	5,285	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.765625	3	CC-MAIN-2016-50	latest	en	0.919884
http://www.derkeiler.com/Newsgroups/sci.crypt/2007-03/msg00661.html	1,386,766,756,000,000,000	text/html	crawl-data/CC-MAIN-2013-48/segments/1386164036407/warc/CC-MAIN-20131204133356-00012-ip-10-33-133-15.ec2.internal.warc.gz	306,629,542	4,768	# Re: Truncated multiplication (is it secure???) Kiuhnm <"kiuhnm03["@]yahoo.it> writes: A (brand new, I think) company named Xor Labs has proposed a very simple cryptographic system called XEVRON. The method is very simple. 1) Alice and Bob choose a common value Sh. 2) Alice generates a private random number idA. 3) Bob generates a private random number idB. 4) Alice computes idASh and removes the n least significant digits (LSD) and the n most significant digits (MSD) from it obtaining midA. Alice sends the truncated value to Bob. 5) Bob computes idBSh and removes the n LSD and the n MSD from it obtaining midB. Bob sends the truncated value to Alice. 6) Alice computes SecA = midBidA. 7) Bob computes SecB = midAidB. The middle parts of SecA and SecB are (or should be) identical, therefore, Bob and Alice have successfully exchanged a secret sequence of digits. For instance, Sh = 31356540235810673346618362866804034368776251178676 (public) idA = 4466022725645872080780142446 (Alice's) idB = 3223756751453228576175422353 (Bob's) midA = Trunc(ShidA, n=11) = 07596398517187085436866841011511082698521273905990903873 (public) midB = Trunc(ShidB, n=11) = 74094702438805240690212977889295113208465707248692447576 (public) SecA = midBidA = 330908624941672826410522480259254687282490036289143524193970577756316934433667410896 SecB = midAidB = 24488941006511161091119055359254687282490036289143524195846203343459100001998473169 ^^^^^^^^^^ Common subsequence = 9254687282 (We can, of course, use bigger numbers.) Let <x> be the truncated version of x and x ~ y mean that x and y have a common central subsequence. Basically, this method is based on the fact that <idASh>idB ~ idA<ShidB>. I think it can be cracked with lattice reduction. What do you think? Unfortunately, the documentation of XEVRON is written in Italian, but an English version should be made available soon (http://www.xorlabs.net/). I couldn't get GP's LLL to reverse that particular example, but I'm no expert at driving it. I keep thinking that continued fraction convergents might also point to a solution - have you considered that at all? Phil -- "Home taping is killing big business profits. We left this side blank so you can help." -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. . ## Relevant Pages • Re: Truncated multiplication (is it secure???) ... Alice generates a private random number idA. ... Bob generates a private random number idB. ... (sci.crypt) • Re: Is this a hoax or real? ... Alice is the bad guy. ... location of a file that Bob, the good guy, can get to. ... Alice has her own Web server. ... Alice can pilfer more than a file. ... (microsoft.public.security) • Is this a hoax or real? ... Coan's free Hidden File Detector software. ... Alice is the bad guy. ... location of a file that Bob, the good guy, can get to. ... Alice has her own Web server. ... (microsoft.public.security) • Whats the problem ... Alice is the bad guy. ... location of a file that Bob, the good guy, can get to. ... Alice has her own Web server. ... Alice can pilfer more than a file. ... (microsoft.public.security) • Does Microsoft listen or care? ... Alice is the bad guy. ... location of a file that Bob, the good guy, can get to. ... Alice has her own Web server. ... Alice can pilfer more than a file. ... (microsoft.public.win2000.security)	936	3,382	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.328125	3	CC-MAIN-2013-48	latest	en	0.799795
https://fliphtml5.com/tags/integers	1,618,688,355,000,000,000	text/html	crawl-data/CC-MAIN-2021-17/segments/1618038464045.54/warc/CC-MAIN-20210417192821-20210417222821-00444.warc.gz	371,061,375	11,025	CHP 1 Webegin with two very important properties of positive integers in Sections 1. Tags: numbers Section 1-8 Multiplying and Dividing Integers Multiplying and Dividing Integers Section 1-8 Multiplying Integers• The product of two integers with the same sign is positive. Tags: integers CSCI 124/224 Discrete Structures II: Homomorphisms and ... Let f : R → R, f (x) = xn for any n ∈ Z+, the positive integers. Tags: example 1 Optimal Interleaving on Tori - Paradise ... is deﬁned by the property that any connected subgraph with t or fewer vertices in the torus is labelled by all distinct integers. Tags: torus 1 Optimal Interleaving on Tori - Paradise ... tori, which is deﬁned by the property that every connected subgraph of order t in the torus is labelled by t distinct integers. Tags: torus Mathematics (Linear) 1MA0 SURDS Express 6 in the form a b , where a and b are positive integers. Tags: marks Surds 25 marks - Kamal Banat Math \| Enjoy Mathing (a) Express 6 in the form a b , where a and b are positive integers. Tags: form On the quotient sequence of sequences of integers 2 (1999) On the quotient sequence of sequences of integers byRudolf Ahlswede (Bielefeld), Levon H. Tags: theorem Dividing Integers - Warrick County School Corporation Page 1 of 5 Dividing Integers BEFORE Now WHY? You multiplied integers. Tags: degrees DEFINABILITY IN PHYSICS - PhilSci-Ar In ZF, a set U of finite integers isdefinable if and only if there exists a formula ΦU(n) from which we can unequivocally determinewhether a given finite integer ... Tags: finite	417	1,585	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.03125	3	CC-MAIN-2021-17	longest	en	0.791433
https://www.clutchprep.com/physics/practice-problems/45837/a-mass-m-15-kg-hangs-off-the-end-of-a-150-g-1-2-m-string-as-shown-in-the-figure-	1,619,115,390,000,000,000	text/html	crawl-data/CC-MAIN-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00435.warc.gz	795,825,223	31,159	Standing Waves Video Lessons Concept # Problem: A mass m = 15 kg hangs off the end of a 150 g, 1.2 m string, as shown in the figure below. (a) What is the fundamental harmonic frequency of the string? (b) What is the first overtone frequency of the string? ###### FREE Expert Solution 94% (121 ratings) ###### Problem Details A mass m = 15 kg hangs off the end of a 150 g, 1.2 m string, as shown in the figure below. (a) What is the fundamental harmonic frequency of the string? (b) What is the first overtone frequency of the string?	141	541	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3	3	CC-MAIN-2021-17	latest	en	0.870154
https://www.sscadda.com/ssc-cgl-reasoning-questions-for-tier-1-exam/	1,721,588,580,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763517768.40/warc/CC-MAIN-20240721182625-20240721212625-00315.warc.gz	861,378,733	112,814	Home » SSC CGL » SSC CGL Reasoning Question # SSC CGL Reasoning Questions Asked in SSC CGL 2021 Tier 1 Exam SSC CGL Previous Year Reasoning: Staff Selection Commission will release an official notification of SSC CGL 2023 for the recruitment of eligible aspirants for different Group B and Group C posts like income tax inspector, excise inspector, auditor, Assistant Section Officer, Assistant, Tax Assistant, etc in various departments on 1st April 2023. For all the eligible aspirants the commission is going to conduct a computer-based examination in the upcoming days this will be very helpful. To help our readers in their preparation and overcome a very steep competition in SSC CGL recruitment, we are providing you with a free pdf of all the questions that were asked in the SSC CGL 2020 Tier 1 examination from the General Intelligence section, which was conducted from 13th August to 24 August 2021. The General Intelligence section includes 25 Questions for 50 marks in the examination. This pdf will help you to assess the examination pattern of the General Intelligence section and the type of questions that are going to be asked in the coming recruitment examination. ## SSC CGL General Intelligence Questions Asked in SSC CGL 2021 Tier 1: As our goal is to help our readers in every way possible to achieve success, we are providing all the questions asked in SSC CGL 2020 Tier 1 examination in different shifts in a single pdf. Download this free pdf and attempt all the questions, it will help you to know about the examination pattern of the recruitment, there are more than 525 questions included in this pdf. In the following table, we are also providing the number of questions asked from different topics of the subject in the examination. S.No. Topics No. Of Questions asked in SSC CGL 2020-21 1 Analogy 3 2 Odd One Out 4 3 Series 3 4 Statement & Conclusions 1 5 Directions 6 Sequence (Acc. to Dictionary) 1 7 Word Formation 8 Coding-Decoding 3 9 Mathematical Operations 2 10 Matrix 11 Blood Relation (Age calculation) 1 12 Mirror Image 1 13 Venn Diagram 1 14 Paper Folding Image 1 15 Missing Term 16 Calendar 17 Cube 1 18 Counting Figure 1 19 Complete Figure 1 20 Hidden Figure 1 Total Questions 25 ## FAQs ### Is there any negative marking in SSC CGL Tier 1 exam? There is a negative marking of 0.5 marks in all sections in Tier 1. ### Is there any negative marking in CGL Tier 2 Exam? There is a negative marking of 1 mark for each wrong answer in Paper-I, and 0.50 marks for each wrong answer on Paper II and Paper-III. ### What is the time duration for each paper? The time duration for each paper is given below: Paper 1- 2 hours 30 minutes Paper 2 – 120 minutes Paper 3 – 120 minutes	652	2,737	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.65625	3	CC-MAIN-2024-30	latest	en	0.92699
https://b-ok.org/book/3141614/dcc6d8	1,571,258,145,000,000,000	text/html	crawl-data/CC-MAIN-2019-43/segments/1570986669546.24/warc/CC-MAIN-20191016190431-20191016213931-00443.warc.gz	395,440,424	25,068	Main Ten Digit Problems # Ten Digit Problems Oxford University Mathematical Institute, eprint, 2011, 18 р.Most quantitative mathematical problems cannot be solved exactly,but there are powerful algorithms for solving many of them numerically to a specified degree of precision like ten digits or ten thousand. In this article three difficult problems of this kind are presented, and the story is told of the SIAM 100-Dollar, 100-Digit Challenge. The twists and turns along the way illustrate some of the flavor of algorithmic continuous mathematics.По моему, каждому , имеющему дело с численными расчётами, будет полезна эта статьяНа сайте есть книга автора: /file/418153/ Trefethen L.N. Spectral Methods in MATLAB, SIAM, 2001, 184 p. Language: english Pages: 18 File: PDF, 437 KB You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. 1 ### Проективная диагностика в сказкотерапии Language: russian File: PDF, 2.01 MB 2 ### Україна: історичний атлас. 8 клас Language: ukrainian File: RAR, 57.60 MB ```Ten Digit Problems Lloyd N. Trefethen Abstract. Most quantitative mathematical problems cannot be solved exactly, but there are powerful algorithms for solving many of them numerically to a specified degree of precision like ten digits or ten thousand. In this article three difficult problems of this kind are presented, and the story is told of the SIAM 100-Dollar, 100-Digit Challenge. The twists and turns along the way illustrate some of the flavor of algorithmic continuous mathematics. 1 Introduction I am a mathematician who spends his time working with numbers, real numbers like 0.3233674316 . . . and 22.11316746 . . . . If I can compute a quantity to ten digits of accuracy, I am happy. Most mathematicians are not like this! In fact, sometimes it seems that the further you go in mathematics, the less important actual numbers become. But some of us develop algorithms to solve problems quantitatively, and we are called numerical analysts. I am the head of the Numerical Analysis Group at Oxford. Like all mathematicians, I enjoy having a concrete problem to chew on. For example, what is the value of the integral ! 1 x−1 cos(x−1 log x)dx ? (1) 0 You won’t find the answer in a table of integrals, and I don’t think anybody knows how to derive an exact formula. But even though an exact formula does not exist, the integral still makes sense. (More precisely, it makes sense Lloyd N. Trefethen Oxford University Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK. e-mail: trefethen@maths.ox.ac.uk 1 2 Lloyd N. Trefethen if we define (1) as the limit ε → 0 of an integral from ε to 1.) The only way to evaluate it is by some kind of numerical algorithm, and this is hard, for the integrand (i.e., the function under the integral sign) oscillates infinitely often as x approaches 0 while swinging between larger and larger values that diverge to infinity. To ten digits, the answer turns out to be the first number listed above. Each October in Oxford, four or five new graduate students arrive to begin a PhD in numerical analysis, and in their first term they participate in a course called the Problem Solving Squad. Each week I give them a problem like (1) whose solution is a single real number. Working in pairs, their job is to compute this number to as many digits of accuracy as they can. I don’t give any hints, but the students are free to talk to anybody and use the library and the web. By the end of six weeks we always have some unexpected discoveries — and some tightly bonded graduate students! given me pleasure, which I’ll call “two cubes”, “five coins”, and “blowup”. Though this is the first article I’ve written about it, Oxford’s Numerical Analysis Problem Solving Squad has been well known since the SIAM 100-Dollar, 100-Digit Challenge was organized in 2002. This involved ten problems selected from the early years of the Squad, and the challenge for contestants was to try to solve each problem to as many digits of accuracy as possible, up to ten digits for each. Teams from around the world entered the race, and twenty of them achieved perfect scores of 100 and won \$100. Afterwards cover picture illustrating an ingenious method of solving (1) using complex numbers [1]. I’ll say more about the 100-Digit Challenge at the end. 2 Two Cubes Our first problem is motivated by a simple question from physics. Isaac Newton discovered that if two point masses of magnitude m1 and m2 are separated by a distance r, then they are attracted towards each other by a gravitational force of magnitude Gm1 m2 F = , r2 where G is a constant known as the gravitational constant. If you have masses that are not points but spheres or other objects, then each point in one mass is attracted to each point in the other by the same formula. We now pose the following idealized problem: Problem 1. Two objects of mass 1 attract each other gravitationally according to Newton’s law with G = 1. Each object is a unit cube with its mass uniformly distributed. The centers of the cubes are one unit apart, so they are in contact along one face. What is the total force, F ? Ten Digit Problems 3 You can think of the cubes as suns or planets if you like. No mathematician will be troubled by the idea of cubic planets. That’s what mathematics is best at, reasoning about any kind of precisely defined situation, no matter unusual for most mathematicians. It is so trivial! We all know the formula for gravity, so where’s the interest here? Working out the force between these particular bodies should be just a matter of bookkeeping. We are not in this But some of us are in this business to design algorithms, and this innocentlooking problem is a killer. Let’s try to solve it, and you’ll see what I mean. The first thought that may occur to you is, can’t we replace each cube by a unit mass at the center and get the answer F = 1? Isn’t that what Newton showed so many years ago, that as far as gravity is concerned, planets are equivalent to points? Well yes, Newton did show that, but only for spherical planets. If the shape is a cube, we have to investigate more carefully. Let’s say that cube 1 consists of points (x1 , y1 , z1 ) with 0 < x1 , y1 , z1 < 1, and cube 2 consists of points (x2 , y2 , z2 ) in the same range except 1 < x2 < 2. For unit point masses at (x1 , y1 , z1 ) and (x2 , y2 , z2 ), the force would be 1 1 = , 2 2 r (x1 − x2 ) + (y1 − y2 )2 + (z1 − z2 )2 aligned in the direction between the points. Our job is to add up these forces over all pairs of points (x1 , y1 , z1 ) and (x2 , y2 , z2 ). That is, we need to evaluate a six-dimensional integral. The y and z components of the total force will cancel to zero, by symmetry, so it’s the x component we need to integrate, which is equal to (x2 − x1 )/r times the expression above. That is, the x component of the force between unit masses at (x1 , y1 , z1 ) and (x2 , y2 , z2 ) is f (x1 , y1 , z1 , x2 , y2 , z2 ) = [(x1 − x2 )2 x2 − x1 . + (y1 − y2 )2 + (z1 − z2 )2 ]3/2 The number F we are looking for is thus ! 1! 1! 2! 1! 1! 1 F = f (x1 , . . . , z2 )dx1 dy1 dz1 dx2 dy2 dz2 . 0 0 1 0 0 (2) (3) 0 This is an integral over a six-dimensional cube. How do we turn it into a number? 4 Lloyd N. Trefethen I must be honest and confess that I am always behind schedule and usually make up these problems the night before giving them to the students. I aim to make sure I can compute a couple of digits at least, trusting that more powerful and beautiful ideas will come along during the week. For this problem, the night before, I tried the most classical numerical method for evaluating integrals, Gauss quadrature. The idea of Gauss quadrature in one dimension is to sample the integrand at n precisely defined values of x called nodes, multiply the sampled values by n corresponding real numbers called weights, and add up the results. (The nodes and weights are determined by the condition that the estimate comes out exactly correct if the integrand happens to be a polynomial of degree no greater than 2n − 1.) For smooth integrands, such as those defined by functions that can be differentiated several times, this gives amazingly accurate approximations. And by squaring or cubing the grid, you can evaluate integrals in two or three dimensions. Here are pictures of 10, 102 , and 103 Gauss nodes for integration over an interval, a square, and a cube: For our integral (3) the same idea applies, though it’s not so easy to draw a picture. Here is what I found with this method of “Gauss quadrature raised to the 6-th power”. The number of nodes is N = n6 with n = 5, 10, . . . , 30, FN is the Gauss quadrature approximation to F , and time is the amount of time each computation took on my computer. N N N N N N = 15625 = 1000000 = 11390625 = 64000000 = 244140625 = 729000000 FN FN FN FN FN FN = = = = = = 0.969313 0.947035 0.938151 0.933963 0.931656 0.930243 time time time time time time = 0.0 = 0.3 = 3.2 = 17.6 = 66.7 = 198.2 secs. secs. secs. secs. secs. secs. This is awful! We can see that the answer looks like F ≈ 0.93, 7% less than if the cubes were spheres. But that is all we can see, and it has taken minutes of computing time. Computing 10 digits would take pretty much forever. In fact, these results from Gauss quadrature with its special nodes and weights are worse than what you get if you set all the weights equal to 1/N and place the nodes at random in the six-dimensional cube! This kind of randomized computation is called the Monte Carlo method. Here are typical sets of 10, 100, and 1000 random nodes in one, two and three dimensions: Ten Digit Problems 5 And here’s a set of Monte Carlo results for the same values of N as before. N N N N N N = 15625 = 1000000 = 11390625 = 64000000 = 244140625 = 729000000 FN FN FN FN FN FN = = = = = = 0.906741 0.927395 0.925669 0.925902 0.926048 0.925892 time time time time time time = 0.1 = 0.5 = 4.4 = 22.7 = 88.0 = 257.0 secs. secs. secs. secs. secs. secs. It seems that we now have three or four digits, F ≈ 0.9259 or 0.9260. In this collection of results, and indeed for all the numbers reported in this article, it is a very interesting matter to try to make more precise statements about the accuracy of a computation. This is an important aspect of the field of numerical analysis, but to keep things as simple as possible, we shall settle for experimental evidence here and not attempt such estimates. So, the world’s slickest method for numerical integration is losing out to the world’s simplest! Actually this often happens with high-dimensional integrals. √ The errors with Monte Carlo decrease in proportion to 1/ N , the inverse of the square root of the number of samples, more or less independently of the number of dimensions, whereas Gauss quadrature slows down greatly with increasing dimension. This is a widespread theme in numerical algorithms, and one speaks of “the curse of dimensionality”. But even Monte Carlo hits a wall at 4 or 5 digits, or maybe 6 or 7 if we run overnight or use a parallel computer. How can we get more? The students worked hard and came up with many good ideas. Let’s focus on one of these which eventually turned into a ten digit solution. If you’re familiar with Gauss quadrature, you can quickly spot why it has done so badly. The problem is that the integrand (2) is not smooth but singular because the cubes are right up against each other. The denominator goes to zero whenever x1 = x2 = 1, y2 = y1 , and z2 = z1 , so the fraction goes to ∞. This isn’t bad enough to make the values of the integral infinite, but it slows down the convergence terribly. We would like to eliminate the singularity. One way to do it would be to change the problem by separating the cubes, say, by a distance 1. 6 Lloyd N. Trefethen The convergence of Gauss quadrature changes completely, giving 14 digits in a fraction of a second: N N N N = = = = 15625 1000000 11390625 64000000 F F F F = = = = 0.24792296453612 0.24792296916638 0.24792296916638 0.24792296916638 time time time time = 0.0 = 0.3 = 3.2 = 17.6 secs. secs. secs. secs. Notice that the answer is close to 1/4, which is what the force would be if the cubes were spheres with centers separated by distance 2. So we can accurately solve a modified problem, with the cubes separated. What about the original problem? Let F (ε) denote the force between cubes separated by a distance ε ≥ 0. We want to know F (0), but we can only evaluate F (ε) accurately for values of ε that are not too small. A good idea is to perform some kind of extrapolation from ε > 0 to ε = 0. Extrapolation is a well-developed topic in numerical mathematics, and some of the important methods in this area are known as Richardson extrapolation and Aitken extrapolation. The students and I tried a number of strategies like these and got. . . well, we were disappointed. We got another digit or two. Alex Prideaux, which finally nailed the two cubes problem. Prideaux’s idea was, let’s break each cube into eight pieces, eight sub-cubes of size 1/2. Now the number F will be the sum of 64 pairwise contributions. Four of these pairs meet along a face. Eight pairs meet along an edge, and four meet at a vertex: In the other 48 cases the sub-cubes are well separated. Ten Digit Problems 7 Having started with one kind of six-dimensional quadrature problem, we now have four! — face, edge, vertex, and separated. Let’s define Face(d) to be the x component of the force between two cubes of size d touching along a face, and similarly Edge(d) and Vertex(d) for cubes of size d touching along edges and at vertices. If you think about the picture with 16 sub-cubes above, you will see that we can write equations for the forces at scale 1 in terms of the forces at scale 12 like this: Vertex(1) = Vertex( 12 ) + well-separated terms, Edge(1) = 2Edge( 12 ) + 2Vertex( 12 ) + well-separated terms, Face(1) = 4Face( 12 ) + 8Edge( 12 ) + 4Vertex( 12 ) + well-separated terms. This may look like dubious progress, until we note a basic fact of scaling: Vertex( 12 ) = 1 16 Vertex(1), Edge( 12 ) = 1 16 Edge(1), Face( 12 ) = 1 16 Face(1). The factors of 16 come about as follows. If you halve the scale of a cubes problem, each mass decreases by a factor of 8, so the product of masses decreases by 64. On the other hand the distance between the cubes also cuts in half, so 1/r2 increases by a factor of 4. Thus overall, the force changes by the factor 4/64 = 1/16. Putting these observations together, we find Vertex(1) = 1 16 Vertex(1) + well-separated terms, Edge(1) = 2 16 Edge(1) + Face(1) = 4 16 Face(1) + 2 16 Vertex(1) 8 16 Edge(1) + + well-separated terms, 4 16 Vertex(1) + well-separated terms. We can calculate the well-separated terms to high accuracy in a second or two by Gauss quadrature, and these formulas give us first Vertex(1), then Edge(1), and then the number we care about, Face(1). The answer is F ≈ 0.9259812606 . 3 Five Coins The second problem involves no physics, just geometry and probability. Problem 2. Non-overlapping coins of radius 1 are placed at random in a circle of radius 3 until no more can fit. What is the probability p that there are 5 coins? We shall see that this story, so far at least, has a less happy ending. 8 Lloyd N. Trefethen We can illustrate the game in a picture. We put a red coin down at random, then a green one, then a blue one. I’ll leave it to you to prove that fitting in at least three coins is always possible. Here’s an example that gets stuck at three, with no room for a fourth. (Incidentally, to be precise the problem must specify what “at random” means. The meaning is pretty obvious; the trick is how to say it mathematically. If k coins are down, consider the set S of points at which the center of another coin could be placed. If S is nonempty, we put the center of coin k + 1 at a point in S selected at random with respect to area measure.) Quite often a fourth coin can fit too. Here’s an example. At four coins, we are usually finished. But occasionally a fifth can fit too: Five coins is the limit. (Well, not quite. Six or seven are also possible, but the probabilities of these events are zero, meaning that no matter how many times you play the game randomly, you will never see them. Can you prove it? Think of where the centers of a 7-coin configuration would have to fall.) So the question is, how often do we get to five coins? This problem has something in common with the two cubes. Since it is posed in terms of probability, one approach to a solution should be Monte Carlo simulation. We can write a computer program and see what happens. It’s not obvious how best to organize the computation, but one reasonable approach is to replace the big disk by a fine grid, then pick points at random in that grid. Every time you pick a point, you remove it from further consideration along with all its neighbors at distance less than 2. For convergence to the required number, you must refine the grid and also increase the number of samples. By following this Monte Carlo approach we find approximately these frequencies: 3 coins: 18% 4 coins: 77% 5 coins: 5% Ten Digit Problems 9 This level of accuracy can be achieved in 5 minutes. Running overnight gives possibly one more digit: p ≈ 0.053 . You may sense a bit of foreshadowing in the fact that we’ve printed this number in large type. There’s a scientific context to the five coins: this is known as a “parking problem”. In one dimension, imagine a curb of length L with k cars parking at random, one after another, along the curb. How many cars will fit? Problems like this are of interest to chemists and physicists investigating aggregation of particles, and have been studied in 1, 2, and 3 dimensions. A question often asked is, in the limit of an infinite parking area, what fraction of the space can one expect will be filled by randomly arriving cars or coins or particles? In the one-dimensional case, the answer is known in the form of an integral that evaluates to 0.7475979202 . . . . For circular disks (“coins”) in two dimensions, or spheres in three, we speak of a “Tanemura parking problem”. So far as I am aware, no formula is known for the infinite-size limit in either of these situations. But in any case our Problem 2 concerns not a limit but a very concrete setting of 3, 4, and 5 coins. And do you know something? Despite hard work, the Problem Squad never improved on 0.053. We found variations on the theme of Monte Carlo, but none that helped decisively. Yet this problem is one of finite-dimensional geometry, equivalent in fact to another multiple integral. There must be a way to solve it to high accuracy! Sometimes a problem has no slick solution. In this case, I think a slick solution is still waiting to be found. 4 Blowup Our final problem involves a partial differential equation (PDE). Since this may be unfamiliar territory, let me explain a little. One of the best known PDEs is the heat or diffusion equation: ∂2u ∂u . = ∂t ∂x2 (4) We have here a function u(x, t) of a space variable x and a time variable t. The equation says that at each point in space-time, the partial derivative of u with respect to t is equal to its second partial derivative with respect to x. Physically, the idea is that at a particular point x and time t, the temperature will increase (∂u/∂t > 0) if the temperature curves upward as a function of x (∂ 2 u/∂x2 > 0), since this means that heat will flow in towards x from nearby hotter points. For example, one solution to (4) would be the function 10 Lloyd N. Trefethen u(x, t) = e−t sin(x) , for it is not hard to verify that for this choice of u, ∂u = −u , ∂t ∂2u = −u . ∂x2 In the time of Napoleon, Joseph Fourier discovered that equation (4) governs diffusion of heat in a one-dimensional body. For example, if a function u0 (x) describes the temperature distribution in an infinitely long bar at time t = 0, then a solution u(x, t) to (4) with initial condition u(0, x) = u0 (x) tells the temperature at times t > 0. This was a first-class scientific discovery, and it is bad luck for Mr. Fourier that for whatever accidental reason of history, we talk about the Laplace and Poisson equations but not the Fourier equation. Most PDE problems are posed on bounded domains, and then we must prescribe boundary conditions to determine the solution. For example, the heat equation might apply on the interval x ∈ [−1, 1], corresponding to a finite bar, with boundary conditions u(−1, t) = u(1, t) = 0, corresponding to zero temperature at both ends. Here’s an illustration of a solution to this problem at different times. Notice how the sharp edges diffuse instantly, whereas the larger structure decays more slowly. That makes sense, since strong temperature differences between nearby points will quickly equalize. t=0 t = 0.001 t = 0.1 Eventually, all the heat flows out the ends and the signal decays to zero. (If you are troubled by how to take the second derivative of the jagged initial function, you are right to be troubled! We may imagine that u(x, 0) is a smooth function that happens to match the jagged curve to high accuracy.) Equation (4) depends linearly on the variable u: it is a linear PDE. Our third problem involves a nonlinear PDE which consists of this equation plus an additional term, the exponential of u: ∂u ∂2u + eu . = 2 ∂t ∂x (5) Whereas the heat equation just moves heat around, conserving the total heat apart from any inflow or outflow at the boundaries, this nonlinear term adds heat. You can think of eu as a model of a chemical process like combustion, a temperature-dependent kind of combustion that accelerates exponentially as the temperature goes up. Suppose we apply equation (5) on an interval [−L, L] with initial condition u(x, 0) = 0 and boundary conditions u(−L, t) = u(L, t) = 0. For t > 0, Ten Digit Problems 11 the exponential term adds heat, and the derivative term diffuses it out the boundaries. There’s a competition here. If L is small, diffusion wins and the solution converges to a fixed limit function u∞ (x) as t → ∞, with combustion exactly balancing diffusion. If L is larger, combustion wins. The heat can’t diffuse away fast enough, and the solution explodes to infinity at a finite time t = tc . In particular, this happens for the case L = 1. Here are the solutions at times 0, 3, and 3.544, by which point the amplitude has reached about 7.5. Soon afterwards, the curve will explode to infinity. t=0 t=3 t = 3.544 Physically, this blowup is related to the phenomenon of spontaneous combustion. Imagine a heap of grass cuttings or compost. Chemical processes may generate some heat, but if it’s a small heap, the heat escapes and all is stable. In a larger pile, however, the heat may be unable to get away. Eventually, the heap catches fire. Much the same mathematics explains why a quantity of Uranium 235 has a critical mass above which it explodes in a fission chain reaction, the original principle behind atomic bombs. Here then is our mathematical problem. Problem 3. At what time tc does the solution u(x, t) to the problem ∂2u ∂u + eu , = 2 ∂t ∂x u(x, 0) = 0, u(−1, t) = u(1, t) = 0 (6) blow up to infinity? The numerical solution of PDEs on computers goes back to von Neumann and others in the 1940s. It is as important a topic as any in numerical analysis, and a huge amount is known. For Problem 3, the geometry is an interval and the equation is a simple one with a single variable. Other problems of interest to scientists and engineers may be much more complicated. The shapes of wings and airplanes are designed by solving PDEs of fluid and structural mechanics in complicated three-dimensional geometries. Weather forecasts come from solutions of PDE problems involving variables representing air 12 Lloyd N. Trefethen velocity, temperature, pressure, humidity and more, and now the geometry is nothing less than a chunk of planet Earth with its oceans and islands and mountains. Most numerical solutions of PDEs depend on discretizing the problem, replacing partial derivatives by finite approximations. The grid around an airplane may be eye-poppingly complicated, but for (6), one might begin by using a simple regular grid like this one, with the horizontal direction corresponding to x and the vertical direction to t. Quite a good solution strategy for Problem 3 is to discretize (6) on grids like this, shrink the step sizes ∆x and ∆t systematically, and then use some kind of extrapolation to estimate the blowup time. For example, one way to discretize this equation from t = 0 to t = 3.544 is to divide [−1, 1] into N space intervals and [0, 3.544] into 2N 2 time intervals and then approximate the PDE on this grid in a manner whose details we won’t go into. Here are the approximate values u(0, 3.544) produced by this method for a succession of values of N : N N N N N N = 32 = 64 = 128 = 256 = 512 = 1024 u(0,3.544) u(0,3.544) u(0,3.544) u(0,3.544) u(0,3.544) u(0,3.544) = = = = = = 9.1015726 7.8233770 7.5487013 7.4823971 7.4659568 7.4618549 time time time time time time = 0.0 = 0.1 = 0.6 = 3.3 = 21.2 = 136.2 secs. secs. secs. secs. secs. secs. It seems clear that the true value u(0, 3.544) must be about 7.46, and by applying Richardson extrapolation to the data, one can improve this estimate to 7.460488. Using methods like this, with some ingenuity and care, one can estimate the blow-up time for Problem 3 to six or seven digits. It seems wasteful, however, to use a regular grid for a problem like this where all the action happens in a narrow spike near x = 0 and t = 3.5. It is tempting to try to take advantage of this structure by using some kind of uneven grid, one which gets finer as the spike gets narrower, like this: Ten Digit Problems 13 Adapting grids like this to the solution being computed is a big topic in numerical PDE. When flows are computed over airplanes, the grids may be thousands of times finer near the surface than further out. The ten digit solution that I know of for Problem 3 makes use of a highly nontrivial adaptive-gridding algorithm due to my former student Wynn Tee. Tee’s method starts from the observation that although (6) is posed for x in the interval [−1, 1], we can extend the solution to complex values of x too, that is, to values of x with an imaginary as well as a real part. As t approaches tc , it turns out that the solution u(x, t) has singularities in the complex xplane that approach the real axis. By monitoring this situation and using what is known as a conformal map to distort the grid systematically, one can maintain extremely high accuracy with a small number of grid points even as the spike grows very tall and narrow. In fact it is possible to calculate a solution to ten digits of accuracy with only 100 grid points in the x direction: tc ≈ 3.544664598 . This solution also takes advantage of advanced methods of time discretization, and it is really a tour de force of clever computation, illustrating that some very abstract mathematics may be useful for concrete problems. 5 The 100-Digit Challenge What does it mean to solve a mathematical problem? That’s too big a question, for the solution to a mathematical problem might be a “yes” or “no”, a proof, a counterexample, a theorem, who knows what. More specifically, then: what does it mean to solve one of those mathematical problems whose solution, in principle, is a number? Must we find an exact formula — and if we do, is that good enough regardless of the formula’s complexity? Must we write down the number in decimal form — and how many digits are enough? Is it sufficient to find an algorithm that can generate the number — and does it matter how quickly it runs? There are plenty of sand traps in this discussion. Even the notion of an exact formula is elusive. For example, in the theory of roots of polynomials, √ an exact formula is traditionally allowed to include n-th roots, like 3 2, but not trigonometric or other special functions, like sin(2). A computer, however, 14 Lloyd N. Trefethen √ doesn’t see much difference between 3 2 and sin(2). Both are calculated by fast iterative algorithms. For that matter, even a fraction like 2/3 is calculated by an iterative algorithm on some computers, but nobody would question the credentials of 2/3 as an exact solution! What about a more complicated "1 4 expression like 0 e−x dx? That’s easy for your computer too, but it will probably have to call a piece of software rather than use something built-in to the microprocessor hardware. Is it an exact solution? For me, solving a numerical problem means finding an algorithm that can calculate the answer to high accuracy on a computer, whether or not there’s an explicit formula, and this brings us to the question: what’s so special about ten digits? Why not three digits, or a hundred? I’d like to suggest two reasons why ten digits is indeed a good goal to aim for. One reason is that in science, many things are known to more than three digits of accuracy, but hardly anything is known to more than ten. If a quantity is known to a hundred or a million digits, you can be sure it is a mathematical abstraction like π rather than a physical constant like the speed of light or Planck’s constant. So in science, you might say that ten digits is more or less the same as infinity. Ultimately this is why computers normally compute with 16-digit precision, not 160. (And since 10 digits is comfortably less than 16, you usually don’t have to worry too much about computer precision when tackling a ten-digit problem.) The second reason is illustrated by the five coins. To exaggerate a little bit, you can solve just about any problem to three digits of accuracy by brute force. But a brute force algorithm doesn’t encode much insight, and it often fails if you try to push it much further. This is just what happened with the five coins, where we got stuck at three digits with Monte Carlo. Ten digits is a very different achievement. To get to ten digits, you need a good understanding of your problem and a well targeted algorithm. In fact, if you can get this far, the chances are pretty good that you could get ten thousand The Challenge was launched in January 2002, and its ten problems involved the integral (1), some chaotic dynamics, the norm of an infinite matrix, global optimization in two dimensions, the approximation of the gamma function in the complex plane, a random walk on a lattice, inverting a 20 000 × 20 000 matrix, the heat equation on a square plate, parametrized optimization, and Brownian motion. Each team was allowed to have up to six members, and 94 teams entered from 25 countries. Twenty of them got perfect scores! That surprised me. I had planned to spend \$100 rewarding whoever got the most digits, but with twenty perfect scores, I was unsure what to do. To my great pleasure, a donor came forward to plug the gap — William Browning, founder of Applied Mathematics, Inc. in Connecticut. You might think that for a member of a team of six to give nights and weekends to a mathematical project and be rewarded with \$16.67 must be some kind of sour joke. But it turned out that receiving a little bit of cash meant a great deal to the winners as a symbolic recognition of their achievement. Ten Digit Problems 15 The winners also got certificates, like this one awarded to Folkmar Bornemann of the Technical University of Munich, one of the authors of the book The SIAM 100-Digit Challenge [1]. This is to certify that Folkmar Bornemann was a First Prize 100-digit Winner in this competition entered by hundreds of contestants around the world. Lloyd N. Trefethen Oxford University May 2002 0.99526291941.274224153-3.3068686470.21433523460.061913954470.72507834630.42401138700.78593367440.00000038375879790.32336743170.99526291941.274224153 9526291941.274224153-3.3068686470.21433523460.061913954470.72507834630.42401138700.78593367440.0000003837587979 The S IAM 100-dollar, 100-digit challenge 1.274224153-3.3068686470.21433523460.061913954470.72507834630.42401138700.78593367440.00000038375879790.3233674317 0.32336743170.99526291941.274224153-3.3068686470.21433523460.061913954470.72507834630.42401138700.78593367440.00000038375879790.32336743170.9952629194 I published an article in SIAM News telling the story and outlining a solution to each problem. The article ended like this: If you add together our heroic numbers, the result is τ = 1.497258836 . . . . I wonder if anyone will ever compute the ten thousandth digit of this fundamental constant? Now I wrote this to be funny, and to get people thinking. What’s funny is that this number τ is the most unfundamental constant you could imagine. The sum of the answers to ten arbitrary unrelated problems — what nonsense! I chose the Greek letter tau because privately I was thinking of this as “Trefethen’s Constant”. With good British modesty I felt it was OK to name something after oneself, provided the item was sufficiently ridiculous. In the book [1], τ took on a life of its own. The authors amazed us all by finding ways to solve nine of the problems, one after another, to ten thousand digits! The variety of mathematical, algorithmic, and computational tools they used was striking. Indeed, the book emphasizes throughout that there is no “right way” to solve a problem, and your bag of tools can never be too big. By one beautiful chain of reasoning making use of ideas of the Indian mathematician Ramanujan, Bornemann found an exact formula for the solution to Challenge Problem 10 (Brownian motion). By another remarkable method based on ideas related to the field of number theory, Jean-Guillaume Dumas together with 186 computer processors running for four days were able to compute exactly the solution to Challenge Problem 7 (inverting a matrix): the task was to find a particular element of this inverse, and they 16 Lloyd N. Trefethen discovered that the answer was a rational number equal to the quotient of two 97,389-digit integers. Bornemann et al. wrote an appendix called “Extreme Digit-Hunting”, which reported their super-accurate results in this format: 0.32336 74316 77778 76139 93700 <<9950 digits>> 42382 81998 70848 26513 96587 27 The listing goes to digit 10,002, so that if you added up the ten numbers, you’d be confident that the 10,000-th digit of the sum would be correct. But Challenge Problem 3 proved intractable. (This was to determine what is called the “2-norm” of a matrix with infinitely many rows and columns, with entries a11 = 1, a12 = 1/2, a21 = 1/3, a13 = 1/4, a22 = 1/5, a31 = 1/6, . . . .) With a month of computer time the authors computed 273 digits: 1.2742 24152 82122 81882 12340 <<220 digits>> 75880 55894 38735 33138 75269 029 And that’s where Trefethen’s constant is stuck as of today, at 273 digits. I decided at age 20 to devote my career to number crunching, and it has given me unending satisfaction since then. My knowledge and confidence have advanced with the years, and so have the computers and software tools available. What a feeling, to be working on algorithms related to those that systems — yet still be so close to elegant mathematics! The field of numerical analysis can be defined like this: Numerical analysis is the study of algorithms for the problems of continuous mathematics. This means problems involving real or complex variables, not just integers. “Continuous” is the opposite of “discrete”, and algorithms for discrete problems have quite a different flavor and a different community of experts. Like any scientific field, numerical analysis stretches over a pure–applied range, with some people spending most of their time inventing algorithms or applying them to scientific problems, while others are more interested in rigorous analysis of their properties. In earlier centuries the leading pure mathematicians were the same people as the leading numerical ones, like Newton and Euler and Gauss, but mathematics has grown a lot since then, and now the two groups are rather separate. If you look at numbers of specialists, numerical analysis is nowadays one of the biggest branches of mathematics. Let’s finish with another ten digit problem from my file. Suppose you have three identical regular tetrahedra, each of volume 1. What’s the volume of the smallest sphere you can fit them inside? Every problem is different. This is the only one so far for which I’ve found myself playing with cardboard models! I made three tetrahedra, then juggled and jiggled till I thought I knew roughly what the shape of the optimal configuration must be. By numerically minimizing a function whose derivation took hours of tricky trigonometry, I got the estimate Ten Digit Problems 17 22.113167462973 . . . . Now by a curious coincidence, my computer tells me √ √ 256π( 12 − 10)3 = 22.113167462973 . . . . Have we stumbled upon the exact answer? I think so, but I’m not sure, and I certainly don’t √ have a √ proof. And what in the world gave me the idea of calculating 256π( 12 − 10)3 ? 1 6 Epilogue Stop Press! We’ve had an unexpected development on Problem 1, the two cubes. I showed a draft of this article to Prof. Bengt Fornberg of the University of Colorado, one of the best numerical problem-solvers in the world. He got hooked. The problem is so simple, yet so devilishly hard! Working with pencil and paper and the symbolic computing system Mathematica, Fornberg managed to shrink the dimensionality from six to five, then four. Then three, then two. That is, he managed to reduce Problem 1 to a two-dimensional integral to be evaluated numerically. As he peeled off one dimension after another, the formulas kept getting more complicated, and he fought hard to keep the complexity under control. Then one morning Fornberg reported he was down to one dimension. This meant that the problem could be five-sixths solved analytically, leaving just a one-dimensional integral to be evaluated numerically. We were startled. The next morning, Fornberg had the exact solution! It was preposterously long and complicated. He kept working, making more and more simplifications of trigonometric functions and logarithms and hyperbolic functions and their inverses, combining some terms together and also splitting some terms into two to make the result more elementary. And here is what he found: 1 F = 3 1 # √ √ √ √ 26π − 14 + 2 2 − 4 3 + 10 5 − 2 6 + 26 log(2) − log(25) 3 √ √ √ + 10 log(1 + 2) + 20 log(1 + 3) − 35 log(1 + 5) √ √ \$ √ −1 + 6 log(1 + 6) − 2 log(4 + 6) − 22 tan (2 6) . OK, I’ll tell you. My calculation led to the estimate R ≈ 0.85368706 for the radius of the smallest sphere that can enclose three tetrahedra each of side length 1. I tried typing this number into the Inverse Symbolic Calculator at http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html, and back came the sug√ √ and multiplying by 4π/3 gestion that R might be 4( 6 − 5 ). Cubing this number √ 2/12, gives the volume of the sphere, and dividing that result by √ √ which is the volume of a regular tetrahedron with side length 1, gives 256π( 12 − 10 )3 . 18 Lloyd N. Trefethen So now we can have as many digits as we want: F ≈ 0.9259812605572914280934366870 . . . . Most computational problems don’t have exact solutions, but when I cook up challenges for the Problem Squad, the drive for elegance keeps me close to the edge of tractability. In this case we were lucky. [1] Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, The SIAM 100-digit challenge: a study in high-accuracy numerical computing. 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https://mathematica.stackexchange.com/questions/39200/is-there-a-difference-between-dividea-b-and-a-b?noredirect=1	1,718,299,333,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198861480.87/warc/CC-MAIN-20240613154645-20240613184645-00129.warc.gz	345,547,448	45,300	# Is there a difference between Divide[a,b] and a/b? In this comment it was asserted that Divide[a,b] and a/b are different, though the documentation indicates that they are the same. In particular, it was asserted that a/b is evaluated as a * 1/b, whereas Divide[a,b] performs the division directly. It was further asserted that this could result in an observable difference in behavior for machine-precision numbers. The question is: is there a difference? If so, what is it exactly? How does it manifest itself? Is there an example where Divide gives a different result than /? I tried a few edge cases with $MaxMachineNumber and $MinMachineNumber, but didn't find any differences in behavior between the two. • The answer is still the same as it was here. The prior post in that thread has examples that show different behavior. Commented Dec 21, 2013 at 20:47 • Ah, someone's here from Wolfram. That's good. I would assert that this behavior of / in a numerical expression is a bug. / should always give the most accurate result on machine numbers, but it currently does not. As it stands, I now have to use Divide wherever / used to be in order to get the correct result. That is a giant pain, since I suddenly lose 800 years of development of a convenient mathematical notation. Commented Dec 21, 2013 at 21:16 • And yes, I will also submit a bug report in the official manner. Commented Dec 21, 2013 at 21:19 • I made inquiries about the feasibility of changing the evaluation parsing/semantics of '/' to be like Divide. I was informed that this was considered some time ago, and rejected, because it would be backwardly incompatible for situations where HoldXXX are in play. I guess more generally it would create trouble if '/' was not an infix surrogate for '' (with inversion of the right hand argument). Reason being, a chain like ab/cde/fg would no longer parse to something with head of Times and seven arguments. Commented Dec 22, 2013 at 23:03 • The classic "too hard to fix" response to a bug. C'mon, you guys are smart. Suppose you retained information about the split of the original divide operation so that the multiplication and reciprocal could be recombined into a division when the time came to actually do the operation? Commented Dec 23, 2013 at 1:14 Oleksandr is correct about the way evaluation works. a/b seems to be interpreted (parsed) directly as Times[a, Power[b,-1]], or more readably: $a\times b^{-1}$. Divide[a,b] is interpreted as is. Evaluation then proceeds from these forms, and the arithmetic is carried out differently for the two cases: either $a\times (1/b)$ or $a/b$. Here are some examples that illustrate the evaluation sequence: In[95]:= On[] Divide[4,8] Off[] During evaluation of In[95]:= On::trace: On[] --> Null. >> During evaluation of In[95]:= Divide::trace: 4/8 --> 1/2. >> Out[96]= 1/2 In[98]:= On[] 4/8 Off[] During evaluation of In[98]:= On::trace: On[] --> Null. >> During evaluation of In[98]:= Power::trace: 1/8 --> 1/8. >> During evaluation of In[98]:= Times::trace: 4/8 --> 4/8. >> During evaluation of In[98]:= Times::trace: 4/8 --> 1/2. >> Out[99]= 1/2 This can indeed theoretically lead to different machine precision results. Let's find out if it really does! We are going to compare the complete binary representation of the results, and we won't use == or === (which both have some tolerance). Table[{k, RealDigits[k/137., 2] === RealDigits[Divide[k, 137.], 2]}, {k, 1, 20}] ( ==> {{1, True}, {2, True}, {3, True}, {4, True}, {5, True}, {6, True}, {7, True}, {8, True}, {9, True}, {10, True}, {11, True}, {12, True}, {13, True}, {14, True}, {15, False}, {16, True}, {17, True}, {18, True}, {19, True}, {20, True}} ) So Divide[15, 137.] and 15/137. really do lead to different results. Conclusion: yes, there is an observable difference. Again, keep in mind that even === has some tolerance (Internal$SameQTolerance) when comparing machine precision numbers (though less than ==, Internal$EqualTolerance) and 15/137. === Divide[15, 137.] returns True. The difference is there though as evidenced by the full 53-bit binary representation. So will you ever see the effects of this in practice? Theoretically the error may accumulate, and there are some functions which do not honour these tolerances and perform strict comparisons (try e.g. Union[{15/137., Divide[15, 137.]}], which returns a list of length 2). • Sure enough. Divide[15, 137.] - 15/137. gives 1.3877810^-17. Fascinating. Commented Dec 21, 2013 at 3:10 • @Silvia If you compare the binary digits directly, using RealDigits[..., 2], you'll see that all binary digits of Divide[15,137.] are correct, but the last two binary digits of 15/137. are wrong. Generally the Divide form is more likely to be correct because it is a result of a single division. The / form is the result of two operations: a division and a multiplication, so roundoff errors may accumulate. Commented Dec 21, 2013 at 3:22 • The documentation (which we have already established as incorrect) indicates that $a\div b$ (a[esc]div[esc]b) is another way to write Divide[a,b]. However it is actually interpreted as a/b. I think that documentation needs some work. Commented Dec 21, 2013 at 3:31 • @MarkAdler Some more observations: HoldComplete[Divide[a,b]] prints as HoldComplete[$\frac{a}{b}$], but it does actually keep the internal representation as Divide (as shown by FullForm). However, typing HoldComplete[Divide[a,b]] and then pressing Command-Shift-N to convert to StandardForm destroys Divide, and replaces it with /. Evaluating it and using FullForm gives Times[a,Power[b,-1]]. Commented Dec 21, 2013 at 3:55 • Hmm.. That is different from my observation.. I compared all Divide[k, n] vs k/n for k and n both from 1 to 200. Looks like the chance is 50-50. Commented Dec 21, 2013 at 3:58 I have just a little to add to @Szabolcs' answer, and this seemed an appropriate place rather than a separate Q&A. Since there is a theorem involved (see below), it should be pointed out that this theorem tends to fail to hold in Mathematica. Well, wait, it's a theorem: it must true! Then let us say that it tends to be difficult to apply this theorem to Mathematica code. As pointed out already, a / b is short for Times[a, Power[b, -1]], which involves two roundings. It also happens that sometimes Divide[a, b] evaluates to Times[a, Power[b, -1]], for example, symbolically and in some cases presented below. In fact, I sometimes find it difficult to get Divide to Divide[] and not to multiply by the reciprocal. There's a remarkable theorem for base-10 addicts due to W. Kahan (see David Goldberg, "What every computer scientist should know about floating-point arithmetic", ACM Computing Surveys, Vol 23, No 1, March 1991, 5–48, Theorem 7): Suppose $$x$$ is an integer with $$\|x\|<2^{p-1}$$ in a binary floating-point format with precision $$p$$ bits and floating-point operations that are exactly rounded. Then if $$n$$ is an integer of the form $$n = 2^i + 2^j$$ (such as $$n=10$$), then $$(x \,/\, n) \times n = x$$ (exactly). Unsurprisingly, this theorem does not apply if $$x \,/\, n$$ is computed as Times[x, Power[n, -1]]. According to the theorem, all the totals below should be zero. (* N.B.: Vectorized FAILS ) (Divide[#, 10.]10. - #) &[ N@Range@32] // Unitize // Total (* WORKS on short packed arrays? ) (Divide[#, 10.]10. - #) &[ N@Range@31] // Unitize // Total (* No, wait! Vectorized sometimes WORKS!! ) (Divide[#, ConstantArray[10., Length@#]]10. - #) &[ N@Range@32] // Unitize // Total (* WORKS on unpacked arrays? ) (Divide[#, 10.]10. - #) &[ N@Range@32 // DeveloperFromPackedArray] // Unitize // Total (* Nope, FAILS on large arrays! Maybe gets repacked/compiled? ) (Divide[#, 10.]10. - #) &[ N@Range@251 // DeveloperFromPackedArray] // Unitize // Total (* 9 -- 32 numbers: FAILURE 0 -- 31 numbers: SUCCESS 0 -- Double array: SUCCESS 0 -- Unpacked 32: SUCCESS 65 -- Unpacked >250: FAILURE ) Perhaps we're seeing some patterns, but the Trace the command changes the computational environment and results: ReleaseHold@Last@Trace[Divide[#, 10.]] - Divide[#, 10.] &[ N@Range@32] // Unitize // Total ( 11 ) ReleaseHold@Last@Trace[Divide[#, 10.]10. - #] &@[ N@Range@32] // Unitize // Total (* 0 ) These are not consistent with the first result, although both are computed from the same input N@Range@32. Apparently Trace prevents Divide from converting to multiplying by the reciprocal. On my system (Intel i7/Macbook Pro), we can address the two failures with system options: SetSystemOptions[ "ParallelOptions" -> ( originally {128, 32, 32} ) "VectorVendorLengthThresholds" -> {128, 32, Infinity}] SetSystemOptions[ "PackedArrayOptions" -> ( originally 250 ) "ListableAutoPackLength" -> Infinity] It's been noted elsewhere that Divide[a, b] is faster than a/b: First@AbsoluteTiming[Divide[##]] & @@ RandomReal[1, {2, 10^7}] ( 0.026914 ) First@AbsoluteTiming[#1/#2] & @@ RandomReal[1, {2, 10^7}] ( 0.05279 ) It seems completely consistent with Divide[a, b] having half as many FLOPS as a/b. It's too bad there isn't a reliable way to implement it. Addendum: A criticism of x (1/y). The recently well-publicized flaw in floating-point division on the Pentium chip has prompted the question whether $$\tt x/y$$ may simply be replaced by $$\tt x(1/y)$$. This note focuses on the example $$y=x$$ illustrating that this proposed fix would no longer conform to the IEEE standard. --- Edelman, "When is $$x(1/x)\ne1$$" (1994)	2,634	9,562	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.765625	3	CC-MAIN-2024-26	latest	en	0.965131
https://math.stackexchange.com/questions/4453530/how-to-see-that-two-manifolds-are-diffeomorphic-when-you-can-give-them-each-a-c	1,716,807,739,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971059039.52/warc/CC-MAIN-20240527083011-20240527113011-00049.warc.gz	320,166,116	36,906	# How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps" This question is about the diffeomorphism of $$\mathbb{C}P^1$$ and $$S^2$$. At the end of youler's answer, we read "the general fact that two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps." I am not sure why this is true. I have not even a geometric intuition. In my lecture notes, we were given the same example, up to showing that the two manifolds have the same transition maps. I think we are expected to understand that the transition maps imply that the two manifolds must be diffeomorphic to each other, but I don't see why. If possible, I would like an intuitive reasoning, followed by a rigorous answer. Suppose that $$X, Y$$ are manifolds with atlases $$\{\phi_\alpha: \alpha\in A\}$$ and $$\{\psi_\alpha: \alpha\in A\}$$ such that for any two pairs of indices $$\alpha, \beta\in A$$, $$\phi_\beta\circ \phi_\alpha^{-1}= \psi_\beta\circ \psi_\alpha^{-1}.$$ Define the map $$f: X\to Y$$ by the formula $$f(x)=y$$ whenever $$\phi_\alpha(x)= \psi_\alpha(y)$$ for any pair of charts with common index whose domains contains $$x$$ and $$y$$ respectively. The fact that the transition maps of the two atlases are the same implies that this map is well-defined (i.e. are independent of the choice of the above charts). To see that this map is a diffeomorphism, one needs to check smoothness of this map (and of its inverse) in local coordinates, i.e. smoothness of compositions $$\psi_\alpha \circ f \circ \phi_\alpha^{-1}$$ and $$\phi_\alpha \circ f^{-1} \circ \psi_\alpha^{-1}.$$ But these compositions are equal the identity maps for every $$\alpha\in A$$. Hence, $$f$$ and its inverse are diffeomorphisms. Thus, $$f: X\to Y$$ is a diffeomorphism. The converse statement holds as well: Suppose that there is a diffeomorphism $$f: X\to Y$$. Then $$X, Y$$ can be equipped with atlases that have the same transition maps. Indeed, given an atlas $$\{\psi_\alpha: \alpha\in A\}$$ on $$Y$$, define charts on $$X$$ by the formula $$\phi_\alpha= \psi_\alpha\circ f.$$ Direct computations shows that transition maps for the two atlases are the same. Consider two manifolds $$M$$ and $$N$$, with atlases $$\{(U_{\alpha}, \phi_{\alpha})\}$$ and $$\{(V_{\alpha}, \psi_{\alpha})\}$$ respectively. Let $$\tau = \phi_{\beta} \circ\phi_{\alpha}^{-1}:\phi_{\alpha}(U_{\alpha}\cap U_{\beta})\rightarrow\phi_{\beta}(U_{\alpha}\cap U_{\beta})$$ and $$\xi=\psi_{\beta} \circ\psi_{\alpha}^{-1}:\psi_{\alpha}(V_1\cap V_{\beta})\rightarrow\psi_{\beta}(V_{\alpha}\cap V_{\beta})$$ be the transition maps associated with each atlas. If $$\tau = \phi_{\beta} \circ\phi_{\alpha}^{-1} = \psi_{\beta} \circ\psi_{\alpha}^{-1} = \xi$$, we then have $$\phi_{\beta}(U_{\alpha}\cap U_{\beta})=\psi_{\beta}(V_{\alpha}\cap V_{\beta}) \implies M \supset U_{\alpha}\cap U_{\beta}=V_{\alpha}\cap V_{\beta} \subset N,$$ since both $$\phi_{\beta}$$ and $$\psi_{\beta}$$ are homeomorphisms. Let's denote this open set with $$S=U_{\alpha}\cap U_{\beta}=V_{\alpha}\cap V_{\beta}$$. As stated in youler's answer, consider the map $$\Phi:M \rightarrow N$$, defined by $$\Phi(\phi_{\alpha}^{-1}(S))=\psi_{\alpha}^{-1}(S)$$. Its inverse $$\Phi^{-1}:N \rightarrow M$$, also exists and is defined by $$\Phi(\psi_{\alpha}^{-1}(S))=\phi_{\alpha}^{-1}(S)$$. If we can somehow show that $$\Phi$$ and $$\Phi^{-1}$$ are both smooth... then indeed the two manifolds are diffeomorphic.	1,093	3,526	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 40, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.421875	3	CC-MAIN-2024-22	latest	en	0.925713
http://slideplayer.com/slide/4238905/	1,521,667,061,000,000,000	text/html	crawl-data/CC-MAIN-2018-13/segments/1521257647692.51/warc/CC-MAIN-20180321195830-20180321215830-00205.warc.gz	269,282,117	39,959	# Spatial and Temporal Data Mining ## Presentation on theme: "Spatial and Temporal Data Mining"— Presentation transcript: Spatial and Temporal Data Mining Clustering II Vasileios Megalooikonomou (based on notes by Jiawei Han and Micheline Kamber) Agenda What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary Density-Based Clustering Methods Clustering based on density (local cluster criterion), such as density-connected points Major features: Discover clusters of arbitrary shape Handle noise One scan Need density parameters as termination condition Several interesting studies: DBSCAN: Ester, et al. (KDD’96) OPTICS: Ankerst, et al (SIGMOD’99). DENCLUE: Hinneburg & D. Keim (KDD’98) CLIQUE: Agrawal, et al. (SIGMOD’98) Density-Based Clustering: Background Two parameters: Eps: Maximum radius of the neighborhood MinPts: Minimum number of points in an Eps-neighborhood of that point NEps(p): {q belongs to D \| dist(p,q) <= Eps} Directly density-reachable: A point p is directly density-reachable from a point q wrt. Eps, MinPts if 1) p belongs to NEps(q) 2) core point condition: \|NEps (q)\| >= MinPts p q MinPts = 5 Eps = 1 cm Density-Based Clustering: Background Density-reachable: A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi (assymetric relationship) Density-connected: A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts (symmetric relationship). p p1 q p q o DBSCAN: Density Based Spatial Clustering of Applications with Noise Density-based cluster: A maximal set of density-connected points; points not contained in the cluster are considered to be noise Discovers clusters of arbitrary shape in spatial databases with noise The user selects certain parameters Core Border Outlier Eps = 1cm MinPts = 5 DBSCAN: The Algorithm Arbitrary select a point p Retrieve all points density-reachable from p wrt Eps and MinPts. If p is a core point, a cluster is formed. If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database. Continue the process until all of the points have been processed and no new point can be added to any cluster. O(n2) ->O(nlogn) with spatial indexing OPTICS: A Cluster-Ordering Method (1999) OPTICS: Ordering Points To Identify the Clustering Structure Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99) Produces a special order of the database wrt its density-based clustering structure This cluster-ordering contains info equiv to the density-based clusterings corresponding to a broad range of parameter settings Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure Can be represented graphically OPTICS: Some Extension from DBSCAN Index-based: k = number of dimensions N = 20 p = 75% M = N(1-p) = 5 Complexity: O(kN2) Core Distance: the smallest Eps that makes an object a core object Reachability Distance: D p1 o p2 o Max (core-distance (o), d (o, p)) r(p1, o) = 2.8cm. r(p2,o) = 4cm MinPts = 5 e = 3 cm Reachability-distance undefined Cluster-order of the objects DENCLUE: using density functions DENsity-based CLUstEring by Hinneburg & Keim (KDD’98) Major features Solid mathematical foundation Good for data sets with large amounts of noise Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significantly faster than existing algorithm (faster than DBSCAN by a factor of up to 45) …but needs a large number of parameters Denclue: Technical Essence Uses grid cells but only keeps information about grid cells that do actually contain data points and manages these cells in a tree-based access structure. Influence function: describes the impact of a data point within its neighborhood. Overall density of the data space can be calculated as the sum of the influence functions of all data points. Clusters can be determined mathematically by identifying density attractors. Density attractors are local maxima of the overall density function. Gradient: The steepness of a slope Example Density Attractor Center-Defined and Arbitrary Agenda What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary Grid-Based Clustering Method Using multi-resolution grid data structure Several interesting methods STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’98) A multi-resolution clustering approach using wavelet method CLIQUE: Agrawal, et al. (SIGMOD’98) STING: A Statistical Information Grid Approach Wang, Yang and Muntz (VLDB’97) The spatial area area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution STING: A Statistical Information Grid Approach Each cell at a high level is partitioned into a number of smaller cells in the next lower level Statistical info of each cell is calculated and stored beforehand and is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell count, mean, s, min, max type of distribution—normal, uniform, etc. Use a top-down approach to answer spatial data queries Start from a pre-selected layer—typically with a small number of cells For each cell in the current level compute the confidence interval STING: A Statistical Information Grid Approach Remove the irrelevant cells from further consideration When finish examining the current layer, proceed to the next lower level Repeat this process until the bottom layer is reached Advantages: Query-independent (summary info of data in grid cell independent of the query) Easy to parallelize, incremental update To generate clusters (computing the statistical parameters of the cells): O(n), where n is the total # of objects To process queries: O(g), where g is the number of grid cells at the lowest level (g << n) Disadvantages: All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected (the method does not consider the spatial relationship between the children and their neighboring cells for construction of a parent cell) WaveCluster (1998) Sheikholeslami, Chatterjee, and Zhang (VLDB’98) A multi-resolution clustering approach which applies wavelet transform to the feature space A wavelet transform is a signal processing technique that decomposes a signal into different frequency sub-bands. Both grid-based and density-based Input parameters: # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform. What is Wavelet (1)? WaveCluster (1998) How to apply wavelet transform to find clusters Summaries the data by imposing a multidimensional grid structure onto data space These multidimensional spatial data objects are represented in a n-dimensional feature space Apply wavelet transform on feature space to find the dense regions in the feature space Apply wavelet transform multiple times -> results in clusters at different scales from fine to coarse What Is Wavelet (2)? Quantization Transformation Wavelet transformation at different resolutions from a fine scale to a coarse scale (for each one four subbands are shown: Avg. neighborhood, hor. edges, vertical edges, and corners ) WaveCluster (1998) Why is wavelet transformation useful for clustering Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary Effective removal of outliers Multi-resolution Cost efficiency Major features: Complexity O(N), can be parallelized Detect arbitrary shaped clusters at different scales Not sensitive to noise, not sensitive to input order Only applicable to low dimensional data Clustering High-Dimensional Data Many applications: text documents, DNA micro-array data Major challenges: Many irrelevant dimensions may mask clusters Distance measure becomes meaningless—due to equi-distance Clusters may exist only in some subspaces Methods Feature transformation: only effective if most dimensions are relevant PCA & SVD useful only when features are highly correlated/redundant Feature selection: wrapper or filter approaches useful to find a subspace where the data have nice clusters Subspace-clustering: find clusters in all the possible subspaces CLIQUE, ProClus, and frequent pattern-based clustering The Curse of Dimensionality (graphs adapted from Parsons et al The Curse of Dimensionality (graphs adapted from Parsons et al. KDD Explorations 2004) Data in only one dimension is relatively packed Adding a dimension “stretches” the points across that dimension, making them further apart Adding more dimensions will make the points further apart—high dimensional data is extremely sparse Distance measure becomes meaningless—due to equi-distance Why Subspace Clustering. (adapted from Parsons et al Why Subspace Clustering? (adapted from Parsons et al. SIGKDD Explorations 2004) Clusters may exist only in some subspaces Subspace-clustering: find clusters in all the subspaces CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98). Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space CLIQUE can be considered as both density-based and grid-based It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into non-overlapping rectangular units A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter A cluster is a maximal set of connected dense units within a subspace CLIQUE: The Major Steps Partition the data space and find the number of points that lie inside each cell of the partition. Identify the subspaces that contain clusters using the Apriori principle: If a k-dim unit is dense, then so are its projections in (k-1)-dim space Identify clusters: Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests Generate minimal description for the clusters Determine maximal regions that cover a cluster of connected dense units for each cluster Determine minimal cover (logic description) for each cluster  = 3 20 30 40 50 60 age 5 4 3 1 2 6 7 Vacation(week) Salary (10,000) Vacation(week) Salary (10,000) 7 6 5 4 3 2 1 age 20 30 40 50 60 age Vacation Salary 30 50  = 3 Strength and Weakness of CLIQUE Strengths It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces It is insensitive to the order of records in input and does not presume some canonical data distribution It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness The accuracy of the clustering result may be degraded at the expense of simplicity of the method (requires tuning of the grid size (fixed) and density threshold Difficult to find clusters of different density within different dimensional subspaces Agenda What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary Model-Based Clustering What is model-based clustering? Attempt to optimize the fit between the given data and some mathematical model Based on the assumption: Data are generated by a mixture of underlying probability distribution Typical methods Statistical approach EM (Expectation maximization), AutoClass Machine learning approach COBWEB, CLASSIT Neural network approach SOM (Self-Organizing Feature Map) EM — Expectation Maximization EM — A popular iterative refinement algorithm An extension to k-means Assign each object to a cluster according to a weight (prob. distribution) New means are computed based on weighted measures (no strict boundaries between clusters) General idea Starts with an initial estimate of the parameter vector (parameters of mixture model) Iteratively rescores the patterns against the mixture density produced by the parameter vector The rescored patterns are used to update the parameter estimates Patterns belong to the same cluster, if they are placed by their scores in a particular component Algorithm converges fast but may not be in global optima The EM (Expectation Maximization) Algorithm Initially, randomly assign k cluster centers and make guesses for the other parameters Iteratively refine the parameters (clusters) based on two steps Expectation step: assign each data point Xi to cluster Ci with the following probability (form expected cluster memberships of Xi ) Maximization step: Estimate the model parameters using the probability estimates from above Conceptual Clustering Conceptual clustering (clustering + characterization) A form of clustering in machine learning Produces a classification scheme for a set of unlabeled objects Finds characteristic description for each concept (class) COBWEB (Fisher’87) A popular, simple method of incremental conceptual learning Creates a hierarchical clustering in the form of a classification tree Each node refers to a concept and contains a probabilistic description of that concept –> main difference from decision trees COBWEB Clustering Method A classification tree More on Conceptual Clustering Limitations of COBWEB The assumption that the attributes are independent of each other is often too strong - correlation may exist Not suitable for clustering large database data – skewed tree and expensive probability distributions (time and space complexity depends not only on the number of attributes but also on the number of values for these attributes) CLASSIT an extension of COBWEB for incremental clustering of continuous data suffers similar problems as COBWEB AutoClass (Cheeseman and Stutz, 1996) Uses Bayesian statistical analysis to estimate the number of clusters Popular in industry Neural Network Approach Neural network approaches Represent each cluster as an exemplar, acting as a “prototype” of the cluster New objects are distributed to the cluster whose exemplar is the most similar according to some distance measure Typical methods SOM (Soft-Organizing feature Map) Competitive learning Involves a hierarchical architecture of several units (neurons) Neurons compete in a “winner-takes-all” fashion for the object currently being presented Self-Organizing Feature Map (SOM) SOMs, also called topological ordered maps, or Kohonen Self-Organizing Feature Map (KSOMs) It maps all the points in a high-dimensional source space into a 2 to 3-d target space, such that, the distance and proximity relationship (i.e., topology) are preserved as much as possible A constrained version of k-means clustering: cluster centers tend to lie in a low-dimensional manifold in the feature space Clustering is performed by having several units competing for the current object The unit whose weight vector is closest to the current object wins The winner and its neighbors learn by having their weights adjusted SOMs are believed to resemble processing that can occur in the brain Useful for visualizing high-dimensional data in 2- or 3-D space Web Document Clustering Using SOM The result of SOM clustering of Web articles The picture on the right: drilling down on the keyword “mining” Based on websom.hut.fi Web page Agenda What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary What Is Outlier Discovery? What are outliers? The set of objects are considerably dissimilar from the remainder of the data Example: Sports: Michael Jordan, Wayne Gretzky, ... Problem Find top n outlier points Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) Use discordancy tests depending on data distribution distribution parameters (e.g., mean, variance) expected number of outliers Drawbacks most tests are for single attributes (not suitable for outlier detection in multidimensional spaces) in many cases, data distribution may not be known No guarantee that all outliers will be found when observed distributions cannot be modeled with standard distributions Outlier Discovery: Distance-Based Approach Introduced to overcome the main limitations of statistical methods We need multi-dimensional analysis without knowing data distribution. Distance-based outlier: an object that does not have “enough” neighbors: A DB(p, D)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than D from O Algorithms for mining distance-based outliers Index-based algorithm (uses SAMS to search for neighbors) Nested-loop algorithm (tries to minimize # of I/Os) Cell-based algorithm (partitions the space into cells) Density-Based Local Outlier Detection Distance-based outlier detection is based on global distance distribution Difficult to identify outliers if data is not uniformly distributed Ex. C1 contains 400 loosely distributed points, C2 has 100 tightly condensed points, 2 outlier points o1, o2 Distance-based method cannot identify o2 as an outlier Need the concept of local outlier Local outlier: outlier relative to its local neighborhood (w.r.t. density of neighborhood) Consider the degree to which an object is outlier Local outlier factor (LOF) Assume outlier is not crisp Each point has a LOF Outlier Discovery: Deviation-Based Approach Identifies outliers by examining the main characteristics of objects in a group Objects that “deviate” from this description are considered outliers sequential exception technique simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects OLAP data cube technique uses data cubes to identify regions of anomalies in large multidimensional data Agenda What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary Problems and Challenges Considerable progress has been made in scalable clustering methods Partitioning: k-means, k-medoids, CLARANS Hierarchical: BIRCH, CURE Density-based: DBSCAN, CLIQUE, OPTICS Grid-based: STING, WaveCluster Model-based: Autoclass, Denclue, Cobweb Current clustering techniques do not address all the requirements adequately Constraint-based clustering analysis: Constraints exist in data space (bridges and highways) or in user queries Constraint-Based Clustering Analysis Clustering analysis: less parameters but more user-desired constraints, e.g., an ATM allocation problem Summary Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be computed for various types of data Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches There are still lots of research issues on cluster analysis, such as constraint-based clustering	4,133	20,162	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.875	3	CC-MAIN-2018-13	latest	en	0.755998
https://www.handlebar-online.com/popular-questions/what-is-an-example-of-factored-form/	1,696,040,561,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233510575.93/warc/CC-MAIN-20230930014147-20230930044147-00691.warc.gz	851,001,446	10,453	# What is an example of factored form? ## What is an example of factored form? A fully factored form means the given number or polynomial is expressed as a product of the simplest possible form. For example, if we write 12y2−27=3(4y2−9) 12 y 2 − 27 = 3 ( 4 y 2 − 9 ) , then it is not considered as fully factored form as (4y2−9) ( 4 y 2 − 9 ) can be factored further. What is simplified factored form? In algebra, simplifying and factoring expressions are opposite processes. Simplifying an expression often means removing a pair of parentheses; factoring an expression often means applying them. The two forms of this expression — 5x(2×2 – 3x + 7) and 10×2 – 15×2 + 35x — are equivalent. ### What is standard factored form? The quadratic expression is called the standard form, the sum of a multiple of and a linear expression ( in this case). When the quadratic expression is a product of two factors where each one is a linear expression, this is called the factored form. What does it mean to be in factored form? Factored form refers to the form of a number or algebraic expression when it has been broken down into a product of its factors. ## How do you factor the expression completely? To factor an expression, you have to start by factoring out the GCF, or Greatest Common Factor. List the factors of each component of the expression. Here we are interested in finding the natural number factors. The expression x ^2 + 6x + 8 would have factors that look like this: x^ 2: 1 6x: 1, 2, 3,… standard form: y = x[math]2 + 8x + 15[/math] In order to find the factored form, you need to look at the 8x and the 15, you need to find combination of two number that can be used as terms and factors; and that apply to this standard form. The 8x is the result of a addition and the 15 is the result of a multiplication. How do you write a polynomial in factored form? Polynomial Functions In Factored Form. Polynomials are generally written in standard form , such as f(x) = x3 +4 x2 + x 6. A more useful way to write a polynomial function’s equation is to use factored form , such as f(x) = ( x 1)( x +2)( x +3). Each factor corresponds to an x-intercept of the function.	572	2,184	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.78125	5	CC-MAIN-2023-40	latest	en	0.938782
https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Motion+in+A+Plane&q_type=&q_topic=Motion+In+A+Plane&q_category=&question_id=PHEN11039635	1,553,557,750,000,000,000	text/html	crawl-data/CC-MAIN-2019-13/segments/1552912204736.6/warc/CC-MAIN-20190325234449-20190326020449-00071.warc.gz	969,924,819	15,683	A man can swim with a speed of 4.0 km/hr in still water. How long does he take to cross the river 1.0 km wide if the river flows steadily at 3.0 km/hr and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank? from Physics Motion in A Plane Class 11 Manipur Board ## Book Store Currently only available for. `CBSE` `Gujarat Board` `Haryana Board` ## Previous Year Papers Download the PDF Question Papers Free for off line practice and view the Solutions online. Currently only available for. `Class 10` `Class 12` A man can swim with a speed of 4.0 km/hr in still water. How long does he take to cross the river 1.0 km wide if the river flows steadily at 3.0 km/hr and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank? Rate of flow of swimmer in the still water = 4.0 km/hr Width of the river = 1.0 km Rate of flow of river = 3.0 km/hr Since the swimmer dives in the river normal to the flow of river, therefore time taken by swimmer to cross the river is, In this time the swimmer will also go down the river by distance BC due to river current. ∴ Distance of the river travelled by the swimmer when he reaches the other bank is given by, 144 Views Give three examples of scalar quantities. Mass, temperature and energy 769 Views Give three examples of vector quantities. Force, impulse and momentum. 865 Views What is a scalar quantity? A physical quantity that requires only magnitude for its complete specification is called a scalar quantity. 1212 Views What are the basic characteristics that a quantity must possess so that it may be a vector quantity? A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 814 Views What is a vector quantity? A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity. 835 Views	466	2,027	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2019-13	latest	en	0.952338
https://socratic.org/questions/how-do-you-write-the-expression-for-the-sum-of-twice-a-number-and-half-the-numbe	1,575,646,617,000,000,000	text/html	crawl-data/CC-MAIN-2019-51/segments/1575540488870.33/warc/CC-MAIN-20191206145958-20191206173958-00323.warc.gz	533,488,211	5,776	# How do you write the expression for: the sum of twice a number and half the number is 10? Apr 22, 2018 $2 x + \frac{x}{2} = 10$ $2 \left(x\right) + \frac{1}{2} \left(x\right) = 10$ $2 x + \frac{x}{2} = 10$	86	209	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.765625	4	CC-MAIN-2019-51	latest	en	0.698077
http://encyclopedia2.thefreedictionary.com/Junctional+escape+beat	1,532,140,517,000,000,000	text/html	crawl-data/CC-MAIN-2018-30/segments/1531676592150.47/warc/CC-MAIN-20180721012433-20180721032433-00586.warc.gz	124,291,966	11,082	# beat (redirected from Junctional escape beat) Also found in: Dictionary, Thesaurus, Medical, Legal, Wikipedia. ## beat 1. Physics the low regular frequency produced by combining two sounds or electrical signals that have similar frequencies 2. Prosody the accent, stress, or ictus in a metrical foot 3. Nautical a course that steers a sailing vessel as close as possible to the direction from which the wind is blowing 4. a. the act of scouring for game by beating b. the organized scouring of a particular woodland so as to rouse the game in it c. the woodland where game is so roused 5. Fencing a sharp tap with one's blade on an opponent's blade to deflect it ## Beat A variation in the intensity of a composite wave which is formed from two distinct waves with different frequencies. Beats were first observed in sound waves, such as those produced by two tuning forks with different frequencies. Beats also can be produced by other waves. They can occur in the motion of two pendulums of different lengths and have been observed among the different-frequency phonons in a crystal lattice. One important application of beat phenomena is to use one object with an accurately known frequency to determine the unknown frequency of another such object. The beat-frequency or heterodyne oscillator also operates by producing beats from two frequencies. ## beat [bēt] (physics) The periodic variation in amplitude of a wave that is the superposition of two simple harmonic waves of different frequencies. ## beat i. A low-frequency vibration produced when two sources of vibration act on the same object at the same time. For example, in a multiengine airplane, if two engines have slightly different RPM, airframe vibrations produced by these engines will produce a very noticeable beat. ii. When two waves are combined or superimposed, a beat occurs if two frequencies are not the same. Waves beat together to create the appearance of either a change in amplitude, if the frequencies differ by a few hertz (Hz), or new frequencies, called beat frequencies or heterodynes, if the original frequencies are far apart. Site: Follow: Share: Open / Close	457	2,160	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.5625	3	CC-MAIN-2018-30	latest	en	0.942878
http://clay6.com/qa/4199/using-matrices-solve-the-following-system-of-equations-large-frac-frac-frac	1,545,005,808,000,000,000	text/html	crawl-data/CC-MAIN-2018-51/segments/1544376828018.77/warc/CC-MAIN-20181216234902-20181217020902-00291.warc.gz	59,040,389	9,193	Comment Share Q) # Using matrices, solve the following system, of equations : $\large\frac{1}{x}-\frac{1}{y}+\frac{1}{z}$$=4: \: \large\frac{2}{x}+\frac{1}{y}-\frac{3}{z}$$=0: \: \large\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$=2 \qquad x \neq 0, y \neq 0, z \neq 0$ Comment A) Toolbox: • If the value of the determinant of a $3 \times 3$ square matrix is not equal to zero, then it is a non-singular matrix • If it is a nonsingular matrix, then inverse exists • $A^{-1}=\frac{1}{\|A\|} (adjoint A)$ • $A^{-1}B=X$ Step 1: Given $\large\frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4: \: \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0: \: \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$ Let $\large\frac{1}{x}=a,\frac{1}{y}=b\;and\; \frac{1}{z}=c$ Therefore the system of equations can be written as $a-b+c=4$ $2a+b-3c=0$ $a+b+c=2$ This is of the form $AX=B$ $(ie)\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}\begin{bmatrix} a \\ b \\ c \end{bmatrix}=\begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ where $A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix},\qquad X=\begin{bmatrix} a \\ b \\ c \end{bmatrix}\;and \; B=\begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ Let us first find the determinant value of A $\|A\|=1(1+3)+1(2+3)+1(2-1)$ $=4+5+1$ $=10$ $\|A\| \neq 0$ Hence it is a non singular matrix and $A^{-1}$ exists Step 2: Let us now find the cofactors of A $A_{11}=(-1)^{1+1} \begin{vmatrix} 1 & -3 \\ 1 & 1 \end{vmatrix}=1+3=4$ $A_{12}=(-1)^{1+2} \begin{vmatrix} 2 & -3 \\ 1 & 1 \end{vmatrix}=-(2+3)=-5$ $A_{13}=(-1)^{1+3} \begin{vmatrix} 2 & 1 \\ 1 & 1 \end{vmatrix}=2-1=1$ $A_{21}=(-1)^{2+1} \begin{vmatrix} -1 & 1 \\ 1 & 1 \end{vmatrix}=-(-1-1)=2$ $A_{22}=(-1)^{2+2} \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix}=1-1=0$ $A_{23}=(-1)^{2+3} \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix}=-(1+1)=-2$ $A_{31}=(-1)^{3+1} \begin{vmatrix} -1 & 1 \\ 1 & -3 \end{vmatrix}=3-1=2$ $A_{32}=(-1)^{3+2} \begin{vmatrix} 1 & 1 \\ 2 & -3 \end{vmatrix}=-(-3-2)=5$ $A_{33}=(-1)^{3+3} \begin{vmatrix} 1 & -1 \\ 2 & 1 \end{vmatrix}=1+2=3$ Therefore Adj $A=\begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}$ $=\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ $A^{-1}=\frac{1}{\|A\|} adj (A),$ we know $\|A\|=10$ Hence $A^{-1}=\frac{1}{10}\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ Step 3: $AX=B \qquad => X=A^{-1}B$ Now Substituting for X, $A^{-1}$ and B we get, $\begin{bmatrix} a \\ b \\ c \end{bmatrix}=\frac{1}{10}\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ $\begin{bmatrix} a \\ b \\ c \end{bmatrix}=\frac{1}{10}\begin{bmatrix} 16+0+4 \\ -20+0+10 \\ 4+0+6 \end{bmatrix}=\begin{bmatrix} \frac{20}{10} \\ \frac{-10}{10} \\ \frac{10}{10} \end{bmatrix}$ $\begin{bmatrix} a \\ b \\ c \end{bmatrix}=\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$ Therefore $a=2,b=-1,and \; c=1$ => $x=\large\frac{1}{2},y=-1,and \; z=1$	1,463	2,967	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.65625	5	CC-MAIN-2018-51	latest	en	0.240663
http://excel.bigresource.com/Extract-numbers-from-text-para-and-sum-extracted-numbers--2iHvZTHb.html	1,597,164,986,000,000,000	text/html	crawl-data/CC-MAIN-2020-34/segments/1596439738816.7/warc/CC-MAIN-20200811150134-20200811180134-00102.warc.gz	36,099,924	11,314	Extract Numbers From Text Para And Sum Extracted Numbers? Jan 27, 2012 I have a text string in cell A1 as below: repairs booked in Dec,11 (INR 37k) training fees Dec,11 (INR 42k)test Fee Reimb for 03 emp Dec,11 (INR 56k) skill fees booked in Jan,12 INR 52k Reimb for 01 employee in Jan,12 INR 8k fee accounted- xyz INR 250k, Quick solutions INR 52K, ABCD India INR 272K, Layer Technologies INR30K and complex mgt INR 21K Note: (INR 37K) means negative number i.e. -37 Now my task is to manually total the above amounts in calculator, i.e. -37-42-56+52+8+250+52+272+30+21 = =550 like this there are 100s of cells, could derive a formula for this task. Formatting Numbers Extracted With Web Query Jun 30, 2014 I have a simple macro which extracts EUR rates from a website but the number format as appears in Excel is a disaster. I am attaching the file : Querry EUR Rates.xlsm How to format the rates accordingly. Also posted here: [URL] ..... Extract Numbers Before Text Nov 20, 2009 I need some code that extracts everything after the last number in a cell. For example, in A1 I have and need to rearrange it to get Flat 3 45 into another cell. I've found this code; Extract Numbers From Text? Jan 16, 2012 I require a formula to extract minutes and seconds from the below cells and place them in the cells beneath the required data heading as shown. Code: ABCD1 REQUIREDREQUIREDREQUIRED2 DATA DATA DATA3DATAMINUTESSECONDS 1SECONDS 246m02.4s62453m58.s358061m11.98s1119870m58.54s05854812m58.04s1258491m12.56s11256105m08.s580115m.01s501121m00.17s1017 How To Extract Numbers From Text Aug 30, 2012 I need an automated way to extract the dollar amount from B and then from C and put them into separate columngs. So the dollar amount in column B is extracted out of the text and then put into a separate column. Then, the dollar amount in column C is extracted and put into a separate column. Extract Numbers From Text Sep 29, 2007 Cell A1 = RJ46512263-F Is there a formula wherein i can extract the numbers from the above text? Cell B1 should give the result "46512263" Extract Text And Numbers Nov 28, 2009 I want extract numbers and text from LAXMI GANPATI ENTERPRISES19.475 In column one LAXMI GANPATI ENTERPRISES In column Two 19.475 BALAMURLI ENTERPRISES 19.11 Extract Numbers Only From Cell Containing Text Also? Jan 24, 2008 I have a cell that I need to clean up. It contains text and numbers. What I want to know is this, can I have a formula that will extract just the numbers and not the text? For example if cell B2 = 'Ref No. 123456'. Could I just extract the '123456'? Extract Numbers From Text String Sep 24, 2009 I have several lines with text strings containting three numbers in each line. I need a code to extract all three numbers from each text string. The numbers can be placed on the following columns in each row. Extract Numbers From Text Into A Matrix Jul 9, 2007 I have a large number of text strings representing chemical formulas. They include letters for the element names and numbers for the number of atoms of each type. For example: C18H35NO3 C4H7S C11H16O2Na etc. The element name has either one or 2 letters (one capital, one small), if there is no number and no small letter after the capital letter that means that there is only one atom of this sort(like in C18H35NO3 - there is one N atom). If the element is not listed in the text string it means that it is not found in that particular formula (i.e. the numerical value is 0). Is there any function that could help converting such a vector (say A1:A3) into a matrix that will have the following form: C H N O S Na 18 35 1 3 0 0 4 7 0 0 1 0 11 16 0 2 0 1 Extract Numbers From Alphanumeric Text Feb 19, 2008 I'm looking for a way to extract numbers from a string. The string can potentially have as few as one number or as many as 25. Each number is separated by a comma. Here is the code that creates the string: ... Extract Numbers With Specific Text From Right Or Left Aug 15, 2009 i use this code to get the value from the cell that contains "Ink"., and i got the codes from reading other problems: =IF(SEARCH("Ink",a1),LOOKUP(99^99,--("0"&MID(a1,MIN(SEARCH({0,1,2,3,4,5,6,7,8,9},a1&"0123456789")),ROW(\$1:\$10000)))),"")+0 like this in a1 -> Ink 253.00 and totally working! but the problem is if the word "ink" in the left of the value --> 253.00 ink and the result is #NA, is there any way that i can get the value whether the word Ink is in the left side or right side of the value? also bothered why is it if the word is not "ink" in the cell and return -> #value since i put ("") in the last part of If function(value if false)? Extract Numbers From Specified Place In Text String Feb 19, 2008 I have got cell A1 containing this text string: =IF(SUM('SL-001 - AT-001-001'!R[852]C:R[856]C)=0,SUMPRODUCT('SL-001 - AT-001-001'!R[826]C:R[830]C, 'SL-001 - AT-001-001'!R[840]C:R[844]C,'SL-001 - AT-001-001'!R[846]C:R[850]C), SUMPRODUCT('SL-001 - AT-001-001'!R[826]C:R[830]C,'SL-001 - AT-001-001'!R[840]C:R[844]C, 'SL-001 - AT-001-001'!R[846]C:R[850]C,'SL-001 - AT-001-001'!R[852]C:R[856]C)) 'SL-001 - AT-001-001'!R992CR3C9 and I would like a macro that will extract the numbers between each instance of the letters R and C , i.e. 852, 856, 826 etc etc. in cells A2, A3, A4 respectively. Extract Specific Numbers From Text Strings Jun 26, 2008 Need to write a Macro to only extract numbers from text, characters and numbers fields. I would like the numbers to be extracted in column B, C & D. I am only interested in numbers that begins with 200's, 800's and 4500's. see the attached file. Extract Numbers And Text From Listbox Separated By Delimiter Feb 14, 2008 I have a listbox populated with a number followed by description. How do I write code to extract the number and description to different cells. Listbox example: "1234ES - Project Description". Now I want "1234ES" in one cell and "Project Description" in cell left of number. Extract Two Separate Numbers Of Varying Length From Text Field May 24, 2014 a formula to extract the numbers into two separate fields. The text may vary in length and the numbers vary in length also from 1 digit to 2,000,000. Data in A1 is as follows:- Extract Numbers After Text In Text String Mar 12, 2014 I am trying to extract numbers after a specific text in a text string, for eg :abc SN 12345 xyzedf SN No. 456 mnoAs per above, i want to extract any numbers afters "SN". the numbers can be vary in digits i.e. it can be 3 digit numbers or 4 or 6. Also, at times there is some other text in between (like SN No.) numbers and search word (i.e. SN)Any formula to get result as "12345" and "456". Excel 2010 :: Formatting Numbers Written In Text To Numeric And Pulling Only Numbers Out Feb 20, 2013 I am extracting data from a website to excel 2010. my problem is the data contains both text, numbers, and sometimes a combination of both. e.g. hi im ron for more information you can reach me at 6 five 6 four 5 seven 7 three 2 two.. I need it to look like this 6564577322 or 7 * 0 * 2 * 4 * 5 * 6 * 8 * 6 * 2 * 1...i need to look look the same How do i first format the numbers written in text into numbers and then show only the numbers in a cell minus the text? Import Large Text File Of Numbers Without Excel Splitting Numbers? Jul 19, 2014 I have a text file with rows and columns of numbers ranging from 1-4 digits that I'd like to import/copy into Excel with each number being in its own cell. But whenever I copy/try to import, Excel splits all of the 3-4 digit numbers up into single digit numbers. The text file has 10,000+ columns (each number occupies two columns so I have half of that amount in numbers) and 300+ lines. Is the file simply too large for Excel to handle or is there a way I can do this? Convert Numbers Stored As Text To Numbers (attached File) Oct 22, 2009 In the ID column of the attached excel file whenever I convert the numbers stored as text to numbers it results in a weird transformation. e.g. an ID of 480610141001 becomes 4.8061E+11. When I do the same process in the name column, which has similar numbers, it will give the correct result following the same process. Extracting Numbers From Text And Turning The Numbers Into Rows Of Data Feb 23, 2010 The format of the text in which I need to extract numbers is as follows: 23411268 - 23411270 Need to extract the following: 23411268 23411269 23411270 These numbers have to be listed in three seperate rows. Macro To Convert Numbers Stored As Text To Numbers? May 23, 2014 I have been looking through the forums and found the below code, but I have both text and numbers in the same range. I have attached an example of what I need to have converted. Book1.xlsx VLookup For Numbers Contained Within Cells With Numbers And Text Sep 26, 2012 A B C D E F G 23445566894 Brwn pdc aft Cat1 45687930596 \$5 Cat2 23445566894 \$6 In the example above, I would like to do a vlookup in column C that matches the digits in column A to those in column F, with the output being the corresponding value in column G. I believe it involves the left function but not sure how to really use it here. Making Sure Numbers In Column Are Formatted As Numbers And Not As Text Jun 27, 2013 I'm working on a spreadsheet with about 400 rows and 10 columns. All the numbers in the columns are formatted as text, which doesn't allow me to use AutoSum. I have selected the cells and formatted them as number with no decimals, but AutoSum still doesn't work because the formatting hasn't changed and they're still formatted as text. I know if I double click each cell, the cell changes the format from text to number, but that will take a lot of time to change 400 cells in each of the 10 columns. What can I do to quickly reformat the cells from text to number? Sort Mixture Of Numbers And Text Using Numbers And Decimals Nov 13, 2013 I am looking for a VBA to sort rows which include actual numbers and text representing decomposed CTQs (or procedures in IT development) Code: Col A Col B 1Billing Accuracy 2Billing Time 3Credit Check Accuracy 4Credit Check Time 2.1Bill preparation [Code].... This is the order in which the data is copied and saved from worksheets in which they are developed. Note that 3 rows (8.1.2.1 through 8.1.2.3) are below 8.1.3 (because the three come from Worksheet 8.1.2 which came after worksheet 8.1). The first four rows came from a Top Level Worksheet. I would like to see them intermixed but in proper order. Compare 2 Columns For Numbers In Mixed Text & Numbers May 3, 2006 I need to compare two colums by number decription for example m344 in one column and fsh344-1 in another. All I want to match is 344. In column a I want to indcate the match by placing an X by each match. View my attachment for reference. I don't know if it makes a difference but the columns are centered in my original spreadsheet. Converting Numbers Stored As Text To Numbers Via Macro Feb 9, 2007 I'm looking for the VBA command for this function. I tried just recording a macro in where I perform the task but it didn't record anything. Tried it several times even. Jan 16, 2008 See attached file. A colleague is downloading rows of data from a website which contains a number field Excel is currently treating as Text after being pasted in. My spreadsheet includes just a sample of the many rows of data however as you can see the VALUE function refuses to convert these text values to numbers. How these might be converted and why the VALUE function refuses to work in this case? Copy Only Numbers Within A Cell With Text And Numbers Mar 25, 2014 I am looking for a formula to grab only the numbers withing a cell that has text in it i.e (John Peters142 lp) or (Sally Jones54) or (Terry Cram310 jkb) sometimes there will be text after the number and sometimes not, the format is as shown with no space between name and number but a space after the number. I added a sheet for visual example. Converting Decimal Numbers To Text With Dot Numbers May 27, 2006 we work with both Lotus 123 and Excel 2003. Lotus will be gone next year, but for now, the official mean to publish our reports is Lotus. With my work, I copy/paste a Lotus page to Excel. I use the following macro to convert Lotus format numbers (which Excel considers as text) to real numbers: Sub ForceToNumber() Dim wSheet As Worksheet For Each wSheet In Worksheets With wSheet . Range("IV65536") = vbNullString .Range("IV65536").Copy End With Next wSheet End Sub Source : http://www.ozgrid.com/forum/showthre...087#post184087. The problem is that I need to send back this data in Lotus. Excel considers decimal numbers with a coma as real numbers and numbers with a dot as a text. This previous macro fixes that. However, Lotus works the other way. Only numbers with a dot are considered real numbers. So I would need to find a way to code a macro that converts any numbers in the Excel sheet to a number with a dot. It's a bit like doing the opposite operation.	3,570	13,065	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.65625	3	CC-MAIN-2020-34	latest	en	0.852704
https://www.techtud.com/example/host-sending-data-host-b-over-f-gate-2003	1,632,312,358,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780057347.80/warc/CC-MAIN-20210922102402-20210922132402-00189.warc.gz	1,029,654,980	16,805	##### Host A is sending data to host B over a f : GATE 2003 GATE 2003 Host A is sending data to host B over a full duplex link. A and B are using the sliding window protocol for flow control. The send and receive window sizes are 5 packets each. Data packets (sent only from A to B) are all 1000 bits long and the transmission time for such a packet is 50 microsec Acknowledgement packets (sent only from B to A) are very small and require negligible transmission time. The propagation delay over the link is 200 microsec. What is the maximum achievable throughput in this communication? (A) 7.69 × 106 bps (B) 11.11 × 106 bps (C) 12.33 × 106 bps (D) 15.00 × 106 bp Please first try yourself this question and then look at the answer and explanation. First , please write down the data and convert it into simplest unit possible. it eliminates calculation mistakes and give always uniform answer ,which you can convert later in required quantity. My personal approach in to convert 1) bandwidth from mbps , gbps etc to bps, 2) Data size ( ack or frame) from bytes to bits , 3) time ( PT,Transmission time, RTT etc ) from Microsec , millisec to sec. Sliding window protocol, Window size W = 5, frame size F = 1000 bits , F/C that is transmission time of frame over link of capacity/bandwidth C = 50 * $10^{-6}$sec , ack A = 0 bits ( given negligible) , PT = 200$10^{-6}$sec. we dont have bandwidth C, but we can find out it by using Transmission time TT= 50 $10^{-6}$sec F/C= 50* $10^{-6}$ sec=> 1000 bits/C= 50* $10^{-6}$sec => C = 1000 / 50 * $10^{-6}$ bps = 20 * $10^6$ bps => C = 20mbps Efficiency of anything is defined as useful time out of total time. Useful time will be transmission time TT of the frame which is 50 microsec * Window size = 50 * 5 * $10^{-6}$sec total time will be transmission time + RTT + Ack transmission time, not W TT + RTT+ A/C. A/C is 0 as A = 0. In sliding window protocol, the transmission of 2nd and subsequent frames can be overlapped with propagation time of 1st frame and so on. so total time does not include Window size transmission time. Rather it is just transmission of 1st frame and thats it. So efficiency of the protocol will be => TT/ ( TT+RTT+A/C) => TT / TT + RTT => 550 $10^{-6}$sec /( 50$10^{-6}$+ 2200$10^{-6}$) sec => 250/(50+400) => 250 /450 => 0.5555 => 55.55 % Throughput = Efficiency of link bandwidth = 0.5555* 20 * $10^6$ bps => 11.11.$10^6$ bps. answer : 11.11 $10^6$bps. Target Gate 3 Nov 2014 02:10 am why are we ignoring tt of remaining packets??? if number of packets be increased so much that they cannot be transmitted within the transmission time of first packet...then how will we solve such question???? Vivek Vikram Singh 5 Nov 2014 02:18 am Sliding window protocol was introduced to overcome the drawback of Stop and Wait Protocol,which was that Sender is idle after transmission of single frame. Means out of total time of F/C+ 2PT (ignoring ack), sender was idle for 2PT time. So the idea was to utilize this time to send more frames in this duration of 2PT to increase the utilization. Now if you observe this,new frames are sent but they are overlapping with 2PT time of 1st frame. Let me take this example. Suppose at t=0, sender starts sending frame. At t=50us, the whole 1 frame is sent from senders side. At t=100, 1/4th of the link is traveled by 1st frame(as 200 is total delay between ends) and meanwhile 2nd frame is totally sent from senders side,simultaneously. Can you observe this? Yes? At t=150 (50 TT of 1st frame+ 100 PT of 1st frame) , the 1st frame has traveled half of the link,whereas 3rd frame is simultaneously sent from sender side.It does not take any extra time.Right? At t=200 (50 TT of 1st frame+ 150 PT of 1st frame) , the 1st frame has traveled 1/3rd of the link,whereas 4th frame is simultaneously sent from sender side.It does not take any extra time also. At t=250 (50 TT of 1st frame+ 200 PT of 1st frame) , the 1st frame has traveled full link and reached to the receiver,whereas 5th frame is simultaneously sent from sender side.It does not take any extra time also. As here example says that W=5,so we stop here. But now can you see the overlapping? This overlapping of transmission of frames with PT of other frames make utilization of link go high. I hope you got the point.Please let me know If I am unclear and wrong. Jagmeet 5 Jan 2017 03:48 pm Nice explanation! Also here is some animation for GBN and SR. May be this is helpful too! http://www.ccs-labs.org/teaching/rn/animations/gbn_sr/ Target Gate 13 Nov 2014 11:32 am if in this question window size is 8 then what will be total time for communication?? Vivek Vikram Singh 13 Nov 2014 02:32 pm well, you need to check whether it is possible to send 8 frames ( or any number of frames) within that total time. WF/C <= F/C + 2PT 8* 50 * 10 ^ -6 <= 50 + 2* 200 * 10^-6 40010^-6 <= 450 10^-6 Yeah, these 8 frames can be sent. Total time for communication will still be 45010^-6 sec. In this system, maximum window size can be => W <= (F/C + 2PT )/ (F/C) W<= ( F + 2PT C ) /F. Now you can calculate maximum window size supported by this system. rachapalli vinay kumar 29 Dec 2016 07:55 pm is RTT=2PT or RTT=TT+2PT, In some sources it is 2PT,in some other resources TT+2PT. Jagmeet 5 Jan 2017 05:27 pm Where you studied RTT = TT+2PT ? As per give in Kurose and Ross it is RTT defined as time it takes for a small packet to travel from client to server and then back to the client. The RTT includes packet-propagation delays, packet-queuing delays in intermediate routers and switches, and packet-processing delays. As we consider other delays zero so it is only PT+PT. Ashish Kumar Goyal 21 Jan 2017 02:17 pm @vinayrachapalli The TT u might have seen is the transmission time of ACK from the receiver. Go through the RTT definition then u will understand. Just type "RTT definition" in google and u will get that. Utkarsh 7 Jan 2017 02:19 am Not Understood can anyone explain with all the terms used in detail?? 12 Jan 2017 08:55 am The channel is full duplex so shouldn't RTT be PT reather than 2PT?? Ashish Kumar Goyal 21 Jan 2017 02:11 pm but full duplex means that u can send and recieve at same time. the bandwidth of channel is shared here, the packet first need to be received by the receiver then only it can send Ack. Both actions can't happen simultaneously. so it is always 2PT Vivek Vikram Singh 12 Jan 2017 01:55 pm RTT is always 2PT. Harshita Sharma 7 Feb 2017 08:51 am Sliding Window uses full duplex only.	1,933	6,579	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.859375	3	CC-MAIN-2021-39	latest	en	0.845527
http://essayswriting.info/interactice-lessons-on-equal-and-non-equal-aprts/	1,670,404,613,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00401.warc.gz	15,047,941	15,905	Interactice Lessons On Equal And Non Equal Aprts Interactice Lessons On Equal And Non Equal Aprts. Worksheets are equal and unequal parts, practice and homework name lesson equal or unequal, identifying equal parts work, equal parts are the same size and, grade 1 fractions work, equal. Use this interactive math lesson to help your students practice fractions and equal parts. This google slides math lesson is perfect for teaching whole,. Equal and unequal parts by abingdon on vimeo, the home for high quality videos and the people who love them. Look at each set of shapes and color the : Explore More Than 10,000 'Equal And Unequal Parts' Resources For Teachers, Parents And Pupils As Well As Related Resources On 'Making Equal Parts'. Build student's concept of equal/unequal parts prior to. Use this interactive math lesson to help your students practice fractions and equal parts. Unit quiz view in classroom curriculum download (pdf) presentation. Worksheets Are Equal And Unequal Parts, Practice And Homework Name Lesson Equal Or Unequal, Identifying Equal Parts Work, Equal Parts Are The Same Size And, Grade 1 Fractions Work, Equal. Use this fractions with equal parts lesson to help students become more confident and proficient in their basic geometry skills, including determining fractions with equal parts. This year 2 recognise equal groups lesson covers the prior learning of making equal groups of 2, 5 and 10, before moving onto the main skill of recognising equal groups of 2, 3, 5 and 10. This math activity features shapes that are divided into equal and nonequal parts. In Part A, Observe The. These digital fraction activities will give students practice with partitioning shapes into equal parts.this digital resource uses google slidesâ„˘ and can be used on google class Two parts of a whole are said to be unequal when they do not match each other in terms of size and value. Equal and unequal parts by abingdon on vimeo, the home for high quality videos and the people who love them. Worksheets Are Equal And Unequal Parts, Practice And Homework Name Lesson Equal Or Unequal, Making Equal Groups, Identifying Equal. Explore that size of part is relative to. 16 pictures about equal or unequal fractions? This additional service allows tracking the writing process of big orders as the paper will be sent to you for approval in parts/drafts* before the final deadline. Game To Distribute Food, Cutting In Equal Parts To Distribute It Equally. Here the section is one third of the whole and the white section is two third. We will be using those circles to move on to the next math lesson plan. Our software turns any ipad or web browser into a recordable, interactive whiteboard, making it easy for teachers and experts to create engaging video lessons and.	576	2,798	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.75	4	CC-MAIN-2022-49	latest	en	0.865982
https://www.ademcetinkaya.com/2022/12/duneu-dune-acquisition-corporation-unit.html	1,713,011,854,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296816734.69/warc/CC-MAIN-20240413114018-20240413144018-00785.warc.gz	603,765,409	59,582	Outlook: Dune Acquisition Corporation Unit assigned short-term Ba3 & long-term Baa2 forecasted stock rating. Dominant Strategy : Sell Time series to forecast n: 09 Dec 2022 for (n+16 weeks) Methodology : Deductive Inference (ML) ## Abstract Security indices are the main tools for evaluation of the status of financial markets. Moreover, a main part of the economy of any country is constituted of investment in stock markets. Therefore, investors could maximize the return of investment if it becomes possible to predict the future trend of stock market with appropriate methods. The nonlinearity and nonstationarity of financial series make their prediction complicated. This study seeks to evaluate the prediction power of machine-learning models in a stock market. (Khan, W., Ghazanfar, M.A., Azam, M.A., Karami, A., Alyoubi, K.H. and Alfakeeh, A.S., 2020. Stock market prediction using machine learning classifiers and social media, news. Journal of Ambient Intelligence and Humanized Computing, pp.1-24.) We evaluate Dune Acquisition Corporation Unit prediction models with Deductive Inference (ML) and Stepwise Regression1,2,3,4 and conclude that the DUNEU stock is predictable in the short/long term. According to price forecasts for (n+16 weeks) period, the dominant strategy among neural network is: Sell ## Key Points 1. What are main components of Markov decision process? 2. Can neural networks predict stock market? 3. Nash Equilibria ## DUNEU Target Price Prediction Modeling Methodology We consider Dune Acquisition Corporation Unit Decision Process with Deductive Inference (ML) where A is the set of discrete actions of DUNEU stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4 F(Stepwise Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Deductive Inference (ML)) X S(n):→ (n+16 weeks) $∑ i = 1 n a i$ n:Time series to forecast p:Price signals of DUNEU stock j:Nash equilibria (Neural Network) k:Dominated move a:Best response for target price For further technical information as per how our model work we invite you to visit the article below: How do AC Investment Research machine learning (predictive) algorithms actually work? ## DUNEU Stock Forecast (Buy or Sell) for (n+16 weeks) Sample Set: Neural Network Stock/Index: DUNEU Dune Acquisition Corporation Unit Time series to forecast n: 09 Dec 2022 for (n+16 weeks) According to price forecasts for (n+16 weeks) period, the dominant strategy among neural network is: Sell X axis: Likelihood% (The higher the percentage value, the more likely the event will occur.) Y axis: Potential Impact% (The higher the percentage value, the more likely the price will deviate.) Z axis (Yellow to Green): Technical Analysis% ## Adjusted IFRS Prediction Methods for Dune Acquisition Corporation Unit 1. An entity has not retained control of a transferred asset if the transferee has the practical ability to sell the transferred asset. An entity has retained control of a transferred asset if the transferee does not have the practical ability to sell the transferred asset. A transferee has the practical ability to sell the transferred asset if it is traded in an active market because the transferee could repurchase the transferred asset in the market if it needs to return the asset to the entity. For example, a transferee may have the practical ability to sell a transferred asset if the transferred asset is subject to an option that allows the entity to repurchase it, but the transferee can readily obtain the transferred asset in the market if the option is exercised. A transferee does not have the practical ability to sell the transferred asset if the entity retains such an option and the transferee cannot readily obtain the transferred asset in the market if the entity exercises its option 2. An entity that first applies IFRS 17 as amended in June 2020 at the same time it first applies this Standard shall apply paragraphs 7.2.1–7.2.28 instead of paragraphs 7.2.38–7.2.42. 3. In addition to those hedging relationships specified in paragraph 6.9.1, an entity shall apply the requirements in paragraphs 6.9.11 and 6.9.12 to new hedging relationships in which an alternative benchmark rate is designated as a non-contractually specified risk component (see paragraphs 6.3.7(a) and B6.3.8) when, because of interest rate benchmark reform, that risk component is not separately identifiable at the date it is designated. 4. If there is a hedging relationship between a non-derivative monetary asset and a non-derivative monetary liability, changes in the foreign currency component of those financial instruments are presented in profit or loss. International Financial Reporting Standards (IFRS) are a set of accounting rules for the financial statements of public companies that are intended to make them consistent, transparent, and easily comparable around the world. ## Conclusions Dune Acquisition Corporation Unit assigned short-term Ba3 & long-term Baa2 forecasted stock rating. We evaluate the prediction models Deductive Inference (ML) with Stepwise Regression1,2,3,4 and conclude that the DUNEU stock is predictable in the short/long term. According to price forecasts for (n+16 weeks) period, the dominant strategy among neural network is: Sell ### Financial State Forecast for DUNEU Dune Acquisition Corporation Unit Options & Futures Rating Short-Term Long-Term Senior OutlookBa3Baa2 Operational Risk 7788 Market Risk7276 Technical Analysis3664 Fundamental Analysis4874 Risk Unsystematic8159 ### Prediction Confidence Score Trust metric by Neural Network: 78 out of 100 with 524 signals. ## References 1. Tibshirani R. 1996. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58:267–88 2. Athey S, Bayati M, Doudchenko N, Imbens G, Khosravi K. 2017a. Matrix completion methods for causal panel data models. arXiv:1710.10251 [math.ST] 3. J. Filar, D. Krass, and K. Ross. Percentile performance criteria for limiting average Markov decision pro- cesses. IEEE Transaction of Automatic Control, 40(1):2–10, 1995. 4. Belloni A, Chernozhukov V, Hansen C. 2014. High-dimensional methods and inference on structural and treatment effects. J. Econ. Perspect. 28:29–50 5. Çetinkaya, A., Zhang, Y.Z., Hao, Y.M. and Ma, X.Y., When to Sell and When to Hold FTNT Stock. AC Investment Research Journal, 101(3). 6. Knox SW. 2018. Machine Learning: A Concise Introduction. Hoboken, NJ: Wiley 7. R. Sutton and A. Barto. Reinforcement Learning. The MIT Press, 1998 Frequently Asked QuestionsQ: What is the prediction methodology for DUNEU stock? A: DUNEU stock prediction methodology: We evaluate the prediction models Deductive Inference (ML) and Stepwise Regression Q: Is DUNEU stock a buy or sell? A: The dominant strategy among neural network is to Sell DUNEU Stock. Q: Is Dune Acquisition Corporation Unit stock a good investment? A: The consensus rating for Dune Acquisition Corporation Unit is Sell and assigned short-term Ba3 & long-term Baa2 forecasted stock rating. Q: What is the consensus rating of DUNEU stock? A: The consensus rating for DUNEU is Sell. Q: What is the prediction period for DUNEU stock? A: The prediction period for DUNEU is (n+16 weeks)	1,865	7,525	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.59375	3	CC-MAIN-2024-18	latest	en	0.852431
https://byjus.com/question-answer/the-volume-of-a-cube-is-64-cm-3-then-find-the-surface-area-1/	1,718,779,890,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198861806.64/warc/CC-MAIN-20240619060341-20240619090341-00865.warc.gz	136,118,515	23,063	1 You visited us 1 times! Enjoying our articles? Unlock Full Access! Question # The volume of a cube is 64 cm3, then find the surface area. Open in App Solution ## Let “a” be the side of the cube Given that, the volume of cube is 64 cm3 It means that a3=64 cm3 Hence, a = 4 cm Therefore, the surface area of a cube =6a2 =6×16 =96 cm2 Suggest Corrections 0 Join BYJU'S Learning Program Explore more Join BYJU'S Learning Program	129	430	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.453125	3	CC-MAIN-2024-26	latest	en	0.828274
https://www.rogerogreen.com/tag/esl/	1,725,867,839,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651092.31/warc/CC-MAIN-20240909071529-20240909101529-00058.warc.gz	917,214,308	24,763	## English in Math I can yield in my pedantry, but only so far. English in Math, part 1: Recently, several media outlets noted that the US women’s soccer team was subjected to wage discrimination, and that “the World Cup winners were paid four times less than their male counterparts last year.” One can argue the numbers, but there is a clear disparity. What struck me, though, was the phrase “four times less”, which to my ears, seemed incorrect. I asked my spouse, who is a teacher of English as a New Language (ENL), the new designation for what had been traditionally referred to as English as a Second Language (ESL), in part because those learners may be taking on English as a third language, or fourth, or more. She agreed it “sounded wrong.” We both would have said the men made four times as much, or the women made a quarter (or a fourth) as much. Professor Milo Schield, from the Department of Business, Accounting and MIS at Augsburg College in Minneapolis, MN, would agree with us. In COMMON ERRORS IN FORMING ARITHMETIC COMPARISONS, he writes of Using ‘times less’ as an inverted form of ‘times as much’: Since six is three times as much as two, it is tempting to say that two is three times less than six. Two is definitely less than six and their ratio is definitely that of three to one. But if two were three times less than six, then six should be three times more than two. Recall that six is three times as much as two – two times more than two. ‘Times less than’ is an inverted form of ‘times more than’ – not ‘times as much’. This error is more common in speech than in writing. This error is a variation on… Confusing ‘times as much’ with ‘times more than’. Got that? Of COURSE, you do. But after reading this language log, and this observation, I’m willing to cede that, while my thought process is technically correct, I may be willing to give this one a pass. I KNOW what they mean, and explaining the “error” is far too exhausting. ##### Percentage increase On the other hand (English in Math, part 2): Our tax accountant gave us an interesting tidbit, citing our cash charitable contributions as 320% higher than others who earn the same amount and noncash contributions as 40% lower. So, I surmised that if the AVERAGE person gave \$100, we would have given \$420. Ah, but that’s not what he meant. We have given \$320 versus \$100. That is 220% higher than OR 320% of the average. Quoting the professor: The essential feature is the difference is between ‘as much as’ and ‘more than.’ ‘As much as’ indicates a ratio; ‘more than’ indicates a difference. ‘More than’ means ‘added onto the base’. This essential difference is ignored by those who say that ‘times’ is dominant so that ‘three times as much’ is really the same as ‘three times more than.’ I saw this same error on The Daughter’s First in Math, where there was a 700% increase shown, but the choices were increases of 100%, 200%, 400% and 800%. We picked the 800%, since it was the closest, and it registered as correct. This all goes to show that I can yield in my pedantry, but only so far. ## The Wife turns…another year older Lots of people ask if my wife speaks another language besides English. She does – Spanish – but it isn’t used much since almost everything in ESL is taught in English. It’s always interesting, talking about other people while endeavoring to respect their boundaries. The Wife has never said, “Don’t put my age in your blog.” But I’ve been reluctant to anyway. I have noted that she is younger than I (which is far less revealing than if I were to say that someone was older than I.) One CAN assume she’s over 31 since we’ve been married for over 13 years. Every year on this date, I write something about her, but I have no idea whether she ever reads it. And I used to TELL her I was writing something. One of the things I have alluded to is the fact that she is a teacher of English as a Second Language. She works for an entity called BOCES which provides all sorts of training to several school districts in a given area. For five years, she was teaching entirely in two schools in one school district. This year, however, that district decided to hire its own ESL teacher, which means that the Wife had a new assignment, which ended up being three schools in two school districts in two different counties. Suffice to say, taking public transportation for her job has become impossible, unlike mine, which is at one place almost every day. ESL seems to be misunderstood. Lots of people ask if my wife speaks another language besides English. She does – Spanish – but it isn’t used much since almost everything is taught in English, the lingua franca. It is often assumed that the first language for most of her students is Spanish, when in fact she’s had a lot of kids who speak Urdu (pictured, via Wikipedia) or Chinese. The Wife went back to school in 1999 and graduated in 2002. Going back to school was scary, I imagine (it was for me!), but she excelled at it. I suspect that one day she’ll be an administrator – she’s taken subsequent courses to that end – though I suspect she’d miss the day-to-day activity of the classroom. Well, that’s enough for this year. Happy birthday, dear.	1,196	5,241	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.15625	3	CC-MAIN-2024-38	latest	en	0.987399
http://www.thestudentroom.co.uk/showthread.php?t=4144011	1,477,498,443,000,000,000	text/html	crawl-data/CC-MAIN-2016-44/segments/1476988720962.53/warc/CC-MAIN-20161020183840-00400-ip-10-171-6-4.ec2.internal.warc.gz	732,344,650	35,662	You are Here: Home >< Maths # S1 (edexcel) question. Announcements Posted on Why bother with a post grad? Are they even worth it? Have your say! 26-10-2016 1. I am very good at probabilties questions, but the language of the question confuses me sometimes, e.g.: 1. A group of office workers were questioned for a health magazine and 2/5 were found to take regular exercise. When questioned about their eating habits 2/3 said they always eat breakfast and, of those who always eat breakfast 9/25 also took regular exercise. 2. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn’t own any cats or dogs. In Q1. 9/25 was P(E I B), whereas in Q2. 16/120 was P(D n C). Both probablities were introduced with the same language, but they meant very different things. How could I know from such language whether it's P(A l B) or P(A n B)??? 2. In the first question, they have said that within those 2/3 of people who eat breakfast, 9/25 (of that 2/3) also do regular exercise. You could work out 9/25 of 2/3 and that would be your intersection overall. As P(E l B) is P(E n B) / P(B), which works out to (2/39/25)/(2/3), the 2/3 cancels out and you're left with 9/25. It's only because they give it as a probability based on the initial 2/3. For question two, they're telling you that 57 own a dog, but within these 57 people, 16 of them also own a cat, so they're just telling you the intersection is 16 people out of the total 120. It's not always as simple as if its a probability it must be conditional, and if numbers it is intersection. Just pay attention to how it's worded and what exactly they are telling you. 3. (Original post by Dominator1) I am very good at probabilties questions, but the language of the question confuses me sometimes, e.g.: 1. A group of office workers were questioned for a health magazine and 2/5 were found to take regular exercise. When questioned about their eating habits 2/3 said they always eat breakfast and, of those who always eat breakfast 9/25 also took regular exercise. 2. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn’t own any cats or dogs. In Q1. 9/25 was P(E I B), whereas in Q2. 16/120 was P(D n C). Both probablities were introduced with the same language, but they meant very different things. How could I know from such language whether it's P(A l B) or P(A n B)??? I see what you're saying, but in Q2, P(has a cat given has a dog) would be 16/57, rather than something out of 120, if you see what I mean, whereas for the first one the question implies that the [robability of someone taking regular exercise depends on whether they eat breakfast or not. I think.. it just takes practice with such questions until you consistently get it right. ## Register Thanks for posting! You just need to create an account in order to submit the post 1. this can't be left blank 2. this can't be left blank 3. this can't be left blank 6 characters or longer with both numbers and letters is safer 4. this can't be left empty 1. Oops, you need to agree to our Ts&Cs to register Updated: June 5, 2016 TSR Support Team We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out. This forum is supported by: Today on TSR ### Who is getting a uni offer this half term? Find out which unis are hot off the mark here Poll Useful resources ### Maths Forum posting guidelines Not sure where to post? Read here first ### How to use LaTex Writing equations the easy way ### Study habits of A students Top tips from students who have already aced their exams	999	3,844	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.6875	4	CC-MAIN-2016-44	latest	en	0.987712
http://mathhelpforum.com/trigonometry/118540-show-me-how-reduce-into-single-term-print.html	1,506,083,769,000,000,000	text/html	crawl-data/CC-MAIN-2017-39/segments/1505818688940.72/warc/CC-MAIN-20170922112142-20170922132142-00409.warc.gz	204,698,250	3,106	# Show me how to reduce into a single term • Dec 4th 2009, 04:10 PM KarlosK Show me how to reduce into a single term tan (x+y) - tan y --------------------- 1+ tan(x+y) tan y Thanks! • Dec 4th 2009, 04:17 PM bigwave Quote: Originally Posted by KarlosK tan (x+y) - tan y --------------------- 1+ tan(x+y) tan y Thanks! do you know the difference formula for tan(x - y) from the formula given just reverse back to this.. • Dec 4th 2009, 04:22 PM KarlosK Very vaguely. I am taking an online course and it is difficult to learn without a book. Sometimes seeing it written out helps me to understand. Otherwise I have to scour the internet looking at examples and piecing things together. Last time I ever take an online class that is for sure • Dec 4th 2009, 04:37 PM bigwave Quote: Originally Posted by KarlosK Very vaguely. I am taking an online course and it is difficult to learn without a book. Sometimes seeing it written out helps me to understand. Otherwise I have to scour the internet looking at examples and piecing things together. Last time I ever take an online class that is for sure $tan(U-V) = \frac{tanU - tanV}{1 + tanUtanV}$ if you look at it closely it is similiar to the to the given formula so that the U value is $(X+Y)$ and the $V$ value is $Y$ so then $tan(U-V)$ is $tan((X+Y) - Y)$ or just $tan(X)$ I have taken distance learning also, but with forums like this you get lots of help some times better than a class room	407	1,454	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.0625	3	CC-MAIN-2017-39	longest	en	0.907023
https://www.english-marathi.net/english-to-marathi-meaning-graph	1,721,858,776,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763518454.54/warc/CC-MAIN-20240724202030-20240724232030-00712.warc.gz	648,074,903	16,943	# English to Marathi Meaning of graph - आलेख Graph : आलेख आलेख, पत्र, डिझाइन, योजना, रेखाटन, आकृती, मॉडेल आलेखgraphedग्राफिक्सआलेख Definitions of graph in English Noun(1) a visual representation of the relations between certain quantities, represented as points, plotted with reference to a set of axes(2) a visual display of information(3) a written symbol that is used to represent speech Verb(1) represent by means of a graph(2) plot upon a graph Examples of graph in English (1) the graph illustrates unemployment rates(2) In the present invention, the lexical graph has phoneme branches.(3) Indeed, an accompanying bar graph illustrated what many of us have suspected all along.(4) In fact, all the real values are already in the graph along the x axis also called the real axis.(5) A directed graph is a collection of nodes and edges.(6) The inset graph shows the initial rate of water loss from these samples.(7) In general the text is slowed by frequent resort to lists of statistics that would be conveyed more effectively in a simple graph or diagram.(8) The graph above shows the ├ö├ç├┐survival curves├ö├ç├û, which shows how long patients survived in each group.(9) draw a graph of the results(10) The initial step is the conversion of the word sequences to a phoneme transcription graph .(11) You can chart variables on a graph and look at speed, power (a calculated estimate), temperature and altitude.(12) It is easier to see what is happening if we plot the ratios on a graph .(13) Otherwise, the implication is that the use of coordinate graphs simply adds to the learner's syntactic translational problem.(14) He notes that the pair provide graphs but no statistical analysis of their data.(15) These personnel also undertook much of the preparation of the visual evidential aids such as isographs, histograms, graphs , bar charts, photographs, tables, as built programmes and overlays.(16) The tool should also be able to generate a report easily through graphs and diagrams of the data and their relationships. Related Phrases of graph (1) line graph :: रेषीय आलेख (2) bar graph :: बार आलेख (3) graph paper :: आलेख कागद (4) pie graph :: पाय आलेख (5) graph theory :: आलेख सिद्धांत (6) directed graph :: दिग्दर्शित आलेख Synonyms Noun 1. chart :: चार्ट 2. diagram :: आकृती 3. bar chart :: बार चार्ट 4. pie chart :: पाय चार्ट 5. histogram :: स्तंभालेख 6. scatter diagram :: स्कॅटर आकृती Verb 7. plot :: प्लॉट 8. trace :: शोध काढूण 9. draw up :: बाहेर काढणार 10. delineate :: चित्र रेखाटणे Different Forms graph, graphed, graphing, graphs Word Example from TV Shows I-I bet we can use GRAPH theory to determine who Wil knows The Big Bang Theory Season 12, Episode 16 English to Marathi Dictionary: graph Meaning and definitions of graph, translation in Marathi language for graph with similar and opposite words. Also find spoken pronunciation of graph in Marathi and in English language. Tags for the entry 'graph' What graph means in Marathi, graph meaning in Marathi, graph definition, examples and pronunciation of graph in Marathi language.	796	3,068	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.5625	3	CC-MAIN-2024-30	latest	en	0.914671
https://pdf.capital/virtual-instrumentation-using-labview-256614.html	1,638,245,218,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964358903.73/warc/CC-MAIN-20211130015517-20211130045517-00123.warc.gz	537,424,248	5,131	Title Virtual Instrumentation Using Labview Array Data Structure Data Acquisition String (Computer Science) Electrical Engineering Digital & Social Media 13.3 MB 414 ``` Page 1 Page 1 ``` ##### Document Text Contents Page 2 VIRTUAL INSTRUMENTATION USING LabVIEW New Delhi-110001 2010 Jovitha Jerome Department of Instrumentation and Control Systems Engineering PSG College of Technology Page 207 Structures 185 Figure P8.4 Problem 8.5 Build a VI to find the roots of a quadratic equation. Input the coefficients of x2, x and constant as a, b and c respectively. Display the roots and the message if the roots are real or imaginary. Solution The front panel and the block diagram to solve the problem are shown in Figures P8.5(a) and P8.5(b). Figure P8.5 (Contd.) Page 208 186 Virtual Instrumentation Using LabVIEW Figure P8.5 Problem 8.6 Build a VI to find the roots of a quadratic equation with complex roots. Input the coefficients of x2, x and constant as A, B and C respectively. The LED should glow when roots are real and different or roots are real and equal or when the roots are complex. Display the message ‘A’ should not be zero for quadratic expression” is the coefficient of x2 is zero. Solution The front panel and the block diagram to solve the problem are shown in Figures P8.6(a) and P8.6(b). Figure P8.6 (Contd.) Page 413 392 Index VISA, 223, 230 Vision, 293, 313 Vision assistant, 313 Wait function, 55 Wait until next ms multiple function, 77 Waveform chart, 133 Waveform graph, 131, 132 While loop, 65, 67, 160 Wires, 35 Write LabVIEW measurement file, 208 Write to spread sheet file, 207 Write to/read from measurement file, 207 Xmath script, 179 XY graph, 131, 135 Z plane, 149	472	1,750	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.265625	3	CC-MAIN-2021-49	latest	en	0.747988
https://uk.mathworks.com/matlabcentral/cody/problems/28-counting-money/solutions/1387689	1,606,903,523,000,000,000	text/html	crawl-data/CC-MAIN-2020-50/segments/1606141706569.64/warc/CC-MAIN-20201202083021-20201202113021-00209.warc.gz	523,683,477	17,069	Cody Problem 28. Counting Money Solution 1387689 Submitted on 20 Dec 2017 by Tan Shuwen is pig This solution is locked. To view this solution, you need to provide a solution of the same size or smaller. Test Suite Test Status Code Input and Output 1 Pass a = {'\$12,001.87','\$0.04','\$103,887.55','\$0.32'}; b = 115889.78; assert(abs(moneySum(a)-b) < 1e-4) f = 1×4 cell array {1×2 cell} {1×1 cell} {1×2 cell} {1×1 cell} b = 1.2002e+04 b = 1.2002e+04 b = 1.1589e+05 b = 1.1589e+05 2 Pass a = {'\$0.02'}; b = 0.02; assert(abs(moneySum(a)-b) < 1e-4) f = 1×1 cell array {1×1 cell} b = 0.0200 3 Pass a = {'\$81.47','\$12.69','\$91,337.60'}; b = 91431.76; assert(abs(moneySum(a)-b) < 1e-4) f = 1×3 cell array {1×1 cell} {1×1 cell} {1×2 cell} b = 81.4700 b = 94.1600 b = 9.1432e+04 Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting!	376	915	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.390625	3	CC-MAIN-2020-50	latest	en	0.706823
www.centre-d-entraide-de-jette.be	1,590,668,260,000,000,000	text/html	crawl-data/CC-MAIN-2020-24/segments/1590347396089.30/warc/CC-MAIN-20200528104652-20200528134652-00423.warc.gz	698,265,728	5,991	calculation of impact force # calculation of impact force ### Fall g-Force Calculator - VaultCanada - Equestrian Vaulting This page provides a technical overview of how to calculate the force a vaulter ... recommendations specific to vaulting. ... Measure the impact dent with ... 247 support Online ### Re: How do I calculate the impact force on a wall from a ... Message: Tommy, As is the case in many situations, the answer is, "It depends." From Newton's second law, we know that force is equal to mass times acceleration (F=ma). 247 support Online ### Forces in Car Crashes - HyperPhysics Concepts Clicking outside the box will then initiate a calculation of the impact force and conversion of the data value to the other types of units. Discussion of car crash ... 247 support Online ### Impact Force & Fall Factor Calculations (Pictures) Impact Force Calculations - 3 The function simplifies so that impact force only depends on: - your mass - the fall factor (fall distance divided by length of rope out) 247 support Online ### Analysis of Impact Force Equations Pigeon Mountain Industries, Inc. ITRS 2002 Page 1 of 7 Analysis of Impact Force Equations Prepared for the International Technical Rescue Symposium, 247 support Online ### calculation of impact force in impact crusher 8 Dec 2013 . calculation of impact force in impact crusher - Basalt Crusher. impact jaw crusher, the mobile parameters and the changing rules of inertial force. 247 support Online ### Shock Load Calculator \| Impact Forces \| Kinetic ... 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The time duration is so small during imp... 247 support Online ### calculation of pendulum impact force - amo-plast.biz impact force from falling object. impact force from falling object note that the above calculation of impact force is accurate only if the height h includes the ... 247 support Online ### What is Impact Force and How to Protect Against It - … Impact force is a force that delivers a ... let me answer your question—just what is impact force and how to protect against it? Impact force transfers the ... 247 support Online ### Spray \| Nozzle Design \| Impact Force Measurement \| … Tactilus Spray Sensor Reveals Contact Impact ... proposed a model to calculate the ... the model considers droplet velocities estimated from a force ... 247 support Online Impact Load Factors for Static ... In a static stress analysis the static force ... then released that calculation gives the vertical Impact Factor equal to ... 247 support Online ### THE STANDARD EQUATION FOR IMPACT FORCE - … THE STANDARD EQUATION FOR IMPACT FORCE RICHARD GOLDSTONE 1. A simple rope model The spring model is an idealized model for the tension developed in a … 247 support Online ### How to calculate impact force in the simplest way? \| … Jan 16, 2009 · For example a car (weighing 500 kg) hits a concrete wall at a constant speed of 30 km/hr. What is the impact force when it hits the wall? 247 support Online ### How Do You Estimate Impact Force? \| WIRED How do you calculate the force of a falling piano? How do you calculate the force of a falling piano? ... How Do You Estimate Impact Force? DCL. 247 support Online ### Calculation Of Impact Force In Impact Crusher How to calculate impact force of a weighted calculation of impact force in impact crusher_Crusher How to Calculate the Force of a Falling Object eHow.com . 247 support Online ### Force of Impact - Force of Impact \| HowStuffWorks Force of Impact - Crumple zones reduce the force of impact by creating a buffer zone around the perimeter of the car. Learn how crumple zones can reduce the force of ... 247 support Online ### momentum - Calculating force of impact - Physics … Since \$text{force = mass}timestext{acceleration}\$, is it right to say that an object traveling at a high constant velocity (zero acceleration), exerts zero force ... 247 support Online ### calculation of impact force Calculating impact force for a falling, - Stack Exchange. 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Clicking outside the box will then initiate a calculation of the impact force and conversion of the data value to the other types of units. 247 support Online ### How do I determine an impact force from a signal of ... The challenging task is how to determine a magnitude of impact force in ... How do I determine an impact force from a ... do further calculation to determine the ... 247 support Online ### calculation of impact force – Grinding Mill China Calculating the impact force of a mass on an,Calculating the impact force of a mass falling on an elastic structure. T. J. Mackin. Dept of M&IE. 247 support Online	1,528	7,042	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.859375	3	CC-MAIN-2020-24	latest	en	0.779594
https://cpep.org/mathematics/1651025-factor-completely-3x-x-2-4-x-2-a-x-2-3x-4-b-x-2-12x-c-x-2-7x-d-prime.html	1,726,336,562,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651580.73/warc/CC-MAIN-20240914161327-20240914191327-00130.warc.gz	158,823,821	7,820	23 March, 14:59 # Factor completely: 3x (x + 2) + 4 (x + 2)A. (x + 2) (3x + 4)B. (x + 2) (12x)C. (x + 2) (7x)D. Prime +3 1. 23 March, 15:24 0 3 x (x + 2) + 4 (x + 2) = = (x + 2) (3 x + 4)	122	190	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.1875	3	CC-MAIN-2024-38	latest	en	0.665803
https://finally.com/blog/accounting/gross-profit-vs-net-profit/	1,713,977,592,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296819668.74/warc/CC-MAIN-20240424143432-20240424173432-00141.warc.gz	228,487,370	26,022	# Gross Profit vs Net Profit: Decoding Key Financial Metrics for Businesses Search In the world of business and finance, understanding different types of profit metrics is essential for assessing a company’s performance. Two critical concepts to grasp are gross profit and net profit, as they represent different aspects of a company’s profitability. By comparing these figures, one can gain insights into the overall efficiency and financial health of the business. Gross profit is calculated as sales revenue minus the cost of goods sold (COGS), which represents the direct costs related to producing the goods or services sold by a company. In contrast, net profit takes into account all expenses of the business, including operating expenses, taxes, and interest, making it a more comprehensive measure of a company’s financial health. Understanding the distinction between gross and net profit can help businesses make informed decisions about cost management and revenue generation. ### Key Takeaways • Gross profit focuses on direct costs, while net profit accounts for all expenses. • Analyzing both gross and net profit can provide insights into a company’s financial performance. • Comparing these profit metrics helps guide decision-making related to cost management and revenue generation. ## Understanding Gross Profit ### Components of Gross Profit Gross profit is a financial metric that represents the difference between a company’s total revenue and the cost of goods sold (COGS). It measures how well a company generates profit from its direct labor and raw materials. The primary components of gross profit are revenue and cost of goods sold. Revenue refers to the income generated from a company’s sales, services, or other business activities. COGS, on the other hand, represents the expenses directly associated with producing a product or delivering a service, including raw material costs and direct labor expenses. ### Gross Profit Formula To calculate gross profit, one can use the following formula: ``````Gross Profit = Revenue - Cost of Goods Sold (COGS) `````` By subtracting COGS from the revenue, the resulting value represents the gross income a company generates before accounting for other operational costs, taxes, and interest payments. ### Interpreting Gross Profit Margin Another important metric related to gross profit is the gross profit margin, which is expressed as a percentage. It is calculated using the following formula: ``````Gross Profit Margin = (Gross Profit / Revenue) * 100 `````` The gross profit margin helps gauge a company’s efficiency in using its resources to produce goods or deliver services. A higher margin indicates that a larger portion of the revenue is retained as gross profit, implying a more effective management of raw materials and labor costs. It is essential to be cautious when comparing gross profit margins between different companies and industries, as this metric can vary depending on factors such as pricing strategies, production costs, and market conditions. ## Demystifying Net Profit ### Net Profit Explained Net profit, also known as net income, is the amount of money a company has left after subtracting all its expenses. This includes both operating and non-operating expenses. Operating expenses are the costs associated with the day-to-day operations, such as salaries, rent, and utilities. Non-operating expenses include items like taxes, interest, and one-off costs that are not directly related to the core business activities. ### Calculating Net Profit To calculate net profit, one must follow these steps: 1. Determine total revenue: Add up all the money earned from sales, services, and any other income sources during a specific period. 2. Subtract operating expenses: Calculate the total cost associated with running the business, including salaries, rent, utilities, and other day-to-day costs. 3. Subtract non-operating expenses: Deduct other expenses not directly tied to the business operations, such as taxes, interest, and one-time costs. 4. Apply the formula: Net profit = Total revenue – Operating expenses – Non-operating expenses Here is a table to better illustrate the formula for net profit: Revenue `+All income sources` Operating expenses `- Salaries, rent, etc.` Non-operating expenses `- Taxes, interest, etc.` Net profit `Net income` ### Net Profit Margin Significance The net profit margin is a key performance indicator that measures the percentage of net profit made from total revenue. It helps assess a company’s financial health by revealing how much money it retains after covering all its expenses. A higher net profit margin indicates the company is managing its expenses well and has a better capacity to withstand economic downturns or competition. The formula for calculating the net profit margin is: Net Profit Margin = (Net Profit / Total Revenue) * 100 By understanding the net profit and net profit margin, businesses can gain insights into their overall profitability and make informed decisions to optimize their financial performance. ## Cost Analysis ### Direct vs Indirect Costs When analyzing gross profit and net profit, it’s essential to understand the difference between direct and indirect costs. Direct costs are expenses that can be traced directly to the production of a specific product or service. Examples of direct costs are raw materials, labor, and other production expenses directly impacting the cost of goods sold (COGS). On the other hand, indirect costs are expenses that cannot be directly attributed to a specific product or service. These costs support the overall operation of a business and are spread across multiple products or departments. Common examples of indirect costs include rent, utilities, marketing, and administrative expenses. Indirect costs do not directly impact the COGS but may influence the company’s net profit. ### Fixed and Variable Costs Another distinction in cost analysis is between fixed and variable costs. Fixed costs are expenses that remain constant regardless of the level of production or revenue. Examples of fixed costs include rent, salaries, and insurance premiums. They are considered indirect costs, as they do not directly contribute to the production process but are necessary for the overall functioning of the business. Variable costs, on the other hand, are expenses that fluctuate based on the volume of production or sales. These costs impact the COGS and can be both direct and indirect. Examples of variable costs are raw materials, labor costs (overtime pay), and shipping fees. For a better understanding, consider the following table: Type Direct Indirect Fixed Cost Rent, Insurance Variable Cost Raw Materials, Labor Shipping, Utilities When analyzing gross profit and net profit, accounting for these classifications is crucial for accurately assessing the financial health of a business. In summary, understanding the differences between direct vs indirect, and fixed vs variable costs is crucial for effective cost analysis. ## Revenue Streams ### Primary Revenue Generation Sales revenue, also known as gross sales, is the income generated by a business through the sale of its products or services. It represents the money earned before any deductions, such as returns, discounts, or allowances. To calculate sales revenue, multiply the number of units sold by the price per unit. Net sales is the income generated after accounting for deductions like returns, allowances, and discounts. These deductions are subtracted from the gross sales to arrive at the net sales figure. This metric gives a more accurate picture of a business’s true earnings since it considers all aspects of the sales process, including any losses incurred. Here’s an example: Units Sold Price per Unit Gross Sales Deductions Net Sales 1000 \$50 \$50,000 \$5,000 \$45,000 ### Secondary Sources of Income Apart from sales revenue, businesses may also earn income from secondary sources. These can include: 1. Commissions and fees: Income earned from providing services or access to platforms. 2. Investment income: Earnings from investing in stocks, bonds, or other financial instruments. 3. Rental income: Money earned from leasing or renting out property, equipment, or vehicles. 4. Royalties: Payments received for granting the right to use intellectual property, such as patents, trademarks, and copyrights. 5. Subsidies: Financial support from government agencies or other organizations. When analyzing a business’s financial performance, it’s essential to consider both primary and secondary revenue sources. This comprehensive view helps stakeholders to better understand the company’s overall health and growth potential. ### Operating vs Non-operating Expenses In the business world, expenses can be broadly classified into two categories: operating expenses and non-operating expenses. Operating expenses are the costs associated with the day-to-day operations of a business. These include rent, utilities, labor costs, and administrative costs. On the other hand, non-operating expenses are not directly tied to the daily operations and may involve costs like interest, depreciation, and one-time events such as lawsuit settlements. It’s essential to differentiate between these two types of expenses to evaluate a company’s financial performance accurately. Operating expenses have a direct impact on the company’s gross and net profit, whereas non-operating expenses only affect the net profit. ### Labor and Production Costs Labor and production costs are crucial components of operating expenses that contribute to determining a company’s gross profit and net profit. Let’s briefly examine these components: • Labor costs: These refer to the wages and salaries paid to employees who are involved in the production, sales, and administration of a business. Labor costs also encompass benefits like health insurance, paid time off, and retirement contributions. • Production costs: These include the direct materials and resources required to manufacture or produce the goods or services offered by a business. For instance, raw materials, production equipment, and machinery would all be considered production costs. An example of how these costs can be itemized is shown in the table below: Expense Type Category Wages and Salaries Operating Labor Cost Raw Materials Operating Production Cost Depreciation Non-operating N/A Interest Non-operating N/A By understanding and managing labor and production costs effectively, businesses can optimize their operations to maximize gross and net profits. ## Profitability and Financial Health ### Assessing Profitability Metrics Profitability is a key indicator of a company’s financial health, and it can be analyzed using two primary metrics: gross profit and net profit. Gross profit is the difference between a company’s revenue and cost of goods sold (COGS). It reflects the efficiency of resource usage in production and sales. On the other hand, net profit is the remaining income after deducting all the operating expenses, taxes, and other costs from the gross profit. It signifies the company’s overall financial performance. A company with a high gross profit margin and a low net profit margin might indicate efficient production processes but high operating costs. Whereas, a company with low gross margins and high net profit margins might suggest high efficiency in managing costs but inefficiencies in production. Data to pay attention to: Metric Formula Gross Profit Revenue – Cost of Goods Sold (COGS) Gross Profit Margin (Gross Profit / Revenue) x 100 Net Profit Gross Profit – All Operating Expenses, Taxes, and costs Net Profit Margin (Net Profit / Revenue) x 100 ### Implications for Cash Flow and Investment The financial health of a company is also determined by its cash flow and potential for future growth. A business with consistent net profits may attract investors, as this shows the company’s ability to generate income through operations. However, a net loss over an extended period might indicate unfavorable financial health and pose risks for investors. Cash flow is another crucial aspect of financial health. A company with negative cash flow might struggle to meet its financial obligations, even if it has net profits. This is because net profit is an accounting figure, whereas cash flow reflects the actual cash inflow and outflow in a business. It is essential for companies to manage both profitability and cash flow efficiently to remain viable in the long run and attract potential investors. In conclusion, assessing profitability metrics, such as gross profit and net profit, along with cash flow analysis, can provide valuable insights into a company’s financial health. These figures are instrumental for investors, lenders, and management in making informed decisions and ensuring the business’s long-term growth and sustainability. ## The Income Statement ### Deciphering the Income Statement An income statement, also called a profit and loss statement, reveals a company’s financial performance over a certain period. It is a crucial financial document that reports the company’s revenues, costs, and expenses. The main components of an income statement are: • Sales/Revenue • Cost of goods sold (COGS) • Gross Profit • Operating expenses • Operating income • Non-operating income and expenses • Taxes • Net Profit Gross profit represents the profit made after deducting the costs of producing and distributing its products. It is calculated as: ``````Gross Profit = Sales/Revenue - Cost of goods sold (COGS) `````` Gross profit helps to evaluate how well a company can generate profit from the direct costs involved in production. Net profit, also known as the bottom line, takes into account all expenses and revenues. It indicates the company’s overall profitability after considering operating costs, taxes, and non-operating items. It is calculated as: ``````Net Profit = Gross Profit - Total Expenses `````` ### Connection Between Income Statement and Profit Both gross profit and net profit are integral parts of the income statement. The income statement serves as a comprehensive record of a company’s financial operations, helping to evaluate its efficiency, profitability, and growth. The connection between the income statement and the two types of profits (gross and net) is essential in understanding a company’s financial position. Gross profit provides insights into the company’s ability to cover the cost of producing goods, while net profit reflects the overall profitability after accounting for all expenses. In summary, income statements contain vital information, including gross profit and net profit, to help stakeholders analyze a company’s financial performance. By assessing both types of profit, investors and management can make informed decisions and strategies to improve business performance. ### How do you calculate gross profit and net profit? Gross profit is calculated by subtracting the cost of goods sold (COGS) from total revenue. The formula is: `Gross Profit = Revenue – COGS` Net profit, on the other hand, is determined by subtracting all expenses, including taxes and operating costs, from the gross profit. The formula is: `Net Profit = Gross Profit – Operating Expenses – Taxes` ### What are the primary differences between gross and net income? Gross profit represents a company’s earnings after accounting for the costs of producing and distributing its products, while net income (net profit) takes into account all expenses, such as operating costs, taxes, and interest. Gross profit assesses profitability at the production level, while net profit considers the company’s overall financial health. ### Which financial statement components impact gross versus net profit? Gross profit primarily depends on the income statement’s revenue and cost of goods sold. Net profit, however, relies on a wider range of income statement components, including operating expenses, taxes, and interest. ### How can one distinguish between gross profit margin and net profit margin? Gross profit margin is the percentage of each sale that remains after subtracting the cost of goods sold. It reflects the profitability of goods produced or services offered. Net profit margin, on the other hand, indicates the percentage of each sale that remains after accounting for all expenses. Both margins help reveal an organization’s financial performance, with gross profit margin focusing on operational efficiency and net profit margin reflecting overall profitability. ### In what ways do gross profit and net profit impact financial analysis? Gross profit and net profit are crucial for financial analysis, as they shed light on a company’s profitability and overall financial health. Assessing gross profit helps determine a company’s production efficiency, while evaluating net profit reveals its financial performance upon accounting for all expenses. Comparing these metrics during different accounting periods can identify trends and potential areas for improvement. ### Can you give examples illustrating the contrast between gross and net profit? Suppose a company generates \$500,000 in revenue and has a cost of goods sold (COGS) of \$250,000. The gross profit would be: `Gross Profit = Revenue – COGS = \$500,000 - \$250,000 = \$250,000` Let’s assume the company has additional operating expenses of \$100,000, and taxes of \$40,000. The net profit would be: `Net Profit = Gross Profit – Operating Expenses – Taxes = \$250,000 - \$100,000 - \$40,000 = \$110,000` This example illustrates the distinction between gross profit, which reflects earnings solely from production, and net profit, which accounts for all expenses leading to the company’s overall earnings.	3,365	17,915	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.53125	3	CC-MAIN-2024-18	longest	en	0.921957
https://www.fimfiction.net/blog/884452/why-douglas-adams-is-the-best-part-one	1,597,288,782,000,000,000	text/html	crawl-data/CC-MAIN-2020-34/segments/1596439738950.61/warc/CC-MAIN-20200813014639-20200813044639-00220.warc.gz	678,052,256	9,899	• Member Since 9th Aug, 2019 • offline last seen Sunday # TheLonelyDraconequus I don’t need people to grow up! I drink milk! TheLonelyDraconequus Joined 9th Aug, 2019 · Offline # More Blog Posts7 • ## 30 weeks So I thought, "Why not make a blog post?" Just noticed I've somehow picked up a second follower. Hi Ann. Not sure how it happened, but I'm not complaining. Now there's two people who will randomly get a notification about a random blog post containing nothing particularly useful or informative. Unless I'm wrong about how FIMfiction's Groups system works (which is entirely possible, and given the number of views I've had and Dave's • ## 30 weeks Why Douglas Adams is the Best (Part Two) “The major problem—one of the major problems, for there are several—one of the many major problems with governing people is that of whom you get to do it; or rather of who manages to get people to let them do it to them. To summarize: it is a well-known fact that those people who must want to rule people are, ipso facto, those least suited to do it. • ## 31 weeks Why Math is Depressing Sometimes I'm a little weird. I occasionally do math I don't have to. I once took the concept of literary world-building way too far. I calculated the density of the "planet". And the percentage of elements in the crust (by weight), the atmospheric composition, distance from the star, and a bunch of other unnecessary details. So it should come as no surprise that I decided to find out the total • ## 31 weeks Why Douglas Adams is The Best (Part One) “It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. • ## 33 weeks This is a Title Just finished watching Rise of Skywalker. Not going to spoil it, but I will say this: none of the tearjerker moments got to me. Instead, only three things got me- "Hey kid"; a small gold object on a cord, and Carrie Fisher got first billing in the credits. Jan 2nd 2020 # Why Douglas Adams is The Best (Part One) · 8:48am January 2nd “It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.”	672	2,900	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.6875	3	CC-MAIN-2020-34	latest	en	0.963541
https://www.weegy.com/?ConversationId=HBQ2IWTC	1,537,879,167,000,000,000	text/html	crawl-data/CC-MAIN-2018-39/segments/1537267161638.66/warc/CC-MAIN-20180925123211-20180925143611-00034.warc.gz	933,220,437	9,282	If f(x) = x2 + x, find f(-3). Question Asked 12/16/2013 10:11:18 AM Updated 5/3/2014 7:19:18 PM Original conversation User: If f(x) = x2 + x, find f(-3). Weegy: The answer is 12 frostydc4\|Points 19\| User: Which of the following equations is the translation 2 units left of the graph of y = \|x\|? Weegy: Please list down the options. Question Asked 12/16/2013 10:11:18 AM Updated 5/3/2014 7:19:18 PM Rating 3 f(x) = x^2 + x, f(-3) = (-3)^2 + (-3) = 9 - 3 = 6 Added 5/3/2014 7:19:18 PM This answer has been confirmed as correct and helpful. Confirmed by alfred123 [5/3/2014 8:43:50 PM] There are no comments. Questions asked by the same visitor TrueFalse Score: Adults do not need to make decisions. Question Updated 5/18/2016 3:54:36 PM Adults do not need to make decisions. FALSE. Added 5/18/2016 3:54:36 PM This answer has been confirmed as correct and helpful. Confirmed by yumdrea [5/18/2016 3:57:46 PM] High Middle Ages Weegy: In European history, the Middle Ages, or Medieval period, lasted from the 5th to the 15th century. It began with the collapse of the Western Roman Empire and merged into the early modern period. [ The Middle Ages is the middle period of the traditional division of Western history into Antiquity, Medieval, and Modern periods. ] (More) Question Updated 12/30/2014 6:47:51 AM The High Middle Ages is a period in the European history that followed the period of the Early Middle Ages lasting from the 10th to end of the 13th centuries. Added 12/30/2014 6:47:51 AM This answer has been confirmed as correct and helpful. Confirmed by jeifunk [12/30/2014 9:25:07 AM] 27,373,405 * Get answers from Weegy and a team of really smart live experts. Popular Conversations Government _____ and all of the rules associated with driving anger ... Weegy: Anger is an emotion characterized by antagonism toward someone or something you feel has deliberately done you ... 9/23/2018 4:41:28 PM\| 2 Answers Synonym for poor: Weegy: A synonym is a word or phrase that means exactly or nearly the same as another word or phrase in the same ... 9/16/2018 4:15:32 PM\| 2 Answers What are the means of the following proportion? 3/15 = 12/60 ... Weegy: The extremes of the proportion 3/15 = 12/60 are 3 and 60. 9/17/2018 12:13:43 PM\| 2 Answers What is a command staff? Weegy: COMMAND is The act of directing and/or controlling resources by virtue of explicit legal, agency, or delegated ... 9/17/2018 5:30:24 PM\| 2 Answers S L Points 1302 [Total 1613] Ratings 16 Comments 1142 Invitations 0 Online S R L R P R P R R R R Points 471 [Total 1355] Ratings 4 Comments 341 Invitations 9 Offline S L R P R P R P R Points 451 [Total 1226] Ratings 3 Comments 391 Invitations 3 Offline S L P P P Points 363 [Total 1239] Ratings 0 Comments 363 Invitations 0 Offline S L Points 286 [Total 286] Ratings 2 Comments 266 Invitations 0 Offline S L R Points 77 [Total 194] Ratings 0 Comments 77 Invitations 0 Offline S Points 23 [Total 23] Ratings 0 Comments 23 Invitations 0 Offline S Points 10 [Total 10] Ratings 0 Comments 0 Invitations 1 Offline S L 1 R Points 2 [Total 1453] Ratings 0 Comments 2 Invitations 0 Offline S Points 2 [Total 2] Ratings 0 Comments 2 Invitations 0 Offline * Excludes moderators and previous winners (Include) Home \| Contact \| Blog \| About \| Terms \| Privacy \| © Purple Inc.	1,074	3,294	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.03125	3	CC-MAIN-2018-39	longest	en	0.930579
https://www.gmattutor.com/practice/?qbid=29	1,632,093,159,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780056902.22/warc/CC-MAIN-20210919220343-20210920010343-00087.warc.gz	833,850,495	5,937	# Free GMAT Test Questions Question 1 of 1 ID: GMAT-DS-8 Section: Quantitative Reasoning - Data Sufficiency Topics: Quadratic Equation; Computation of Value Difficulty level: Challenging (Practice Mode: Single selected Question » Back to Overview) If $a^2-4a=0,$ and $b\neq 0,$ is $\dfrac{a}{b}=0?$ 1. $b^2-16=0$ 2. $a^2+8a=0$ AStatement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient to answer the question asked. BStatement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient to answer the question asked. CBOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.	207	701	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.953125	3	CC-MAIN-2021-39	longest	en	0.850882
https://hpmuseum.org/forum/thread-3696.html	1,695,934,092,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233510454.60/warc/CC-MAIN-20230928194838-20230928224838-00175.warc.gz	348,364,496	5,884	Precedence Rule with Function App 04-22-2015, 07:55 PM (This post was last modified: 04-22-2015 07:56 PM by John Colvin.) Post: #1 John Colvin Member Posts: 173 Joined: Dec 2013 Precedence Rule with Function App When plotting the function x^2/2 - 1/2 in the function app, the plot appears as (x/2)^2 - 1/2 which is incorrect according to precedence rules. This error does not occur on the HP50G. I can get the correct plot if I enter 1/2(x)^2-1/2. Is there a reason that powers have the same or less precedence than division? 04-22-2015, 08:09 PM Post: #2 Han Senior Member Posts: 1,882 Joined: Dec 2013 RE: Precedence Rule with Function App (04-22-2015 07:55 PM)John Colvin Wrote: When plotting the function x^2/2 - 1/2 in the function app, the plot appears as (x/2)^2 - 1/2 which is incorrect according to precedence rules. This error does not occur on the HP50G. I can get the correct plot if I enter 1/2(x)^2-1/2. Is there a reason that powers have the same or less precedence than division? Did you press the right arrow one too many times prior to the [x^y] button (in textbook mode)? Otherwise, I am unable to replicate your observations whether in algebraic or textbook entry mode. Graph 3D \| QPI \| SolveSys 04-22-2015, 08:12 PM Post: #3 John Colvin Member Posts: 173 Joined: Dec 2013 RE: Precedence Rule with Function App After deleting and reentering the function x^2/2-1/2, I got the correct plot, so I have no idea what is happening. I entered the function exactly the same way when I reentered it. Has anyone else encountered this issue? 04-22-2015, 08:19 PM Post: #4 John Colvin Member Posts: 173 Joined: Dec 2013 RE: Precedence Rule with Function App (04-22-2015 08:09 PM)Han Wrote: (04-22-2015 07:55 PM)John Colvin Wrote: When plotting the function x^2/2 - 1/2 in the function app, the plot appears as (x/2)^2 - 1/2 which is incorrect according to precedence rules. This error does not occur on the HP50G. I can get the correct plot if I enter 1/2(x)^2-1/2. Is there a reason that powers have the same or less precedence than division? Did you press the right arrow one too many times prior to the [x^y] button (in textbook mode)? Otherwise, I am unable to replicate your observations whether in algebraic or textbook entry mode. No, I didn't, however I was in RPN mode. I doubt if you will be able to reproduce this. Sorry I posted this since it seems to be a one time occurrence. 04-22-2015, 10:00 PM Post: #5 Han Senior Member Posts: 1,882 Joined: Dec 2013 RE: Precedence Rule with Function App (04-22-2015 08:19 PM)John Colvin Wrote: (04-22-2015 08:09 PM)Han Wrote: Did you press the right arrow one too many times prior to the [x^y] button (in textbook mode)? Otherwise, I am unable to replicate your observations whether in algebraic or textbook entry mode. No, I didn't, however I was in RPN mode. I doubt if you will be able to reproduce this. Sorry I posted this since it seems to be a one time occurrence. In textbook mode, typing X / 2 and then pressing the right arrow (twice?) makes it so that the [x^y] key applies to the entire expression currently displayed. That's one way to reproduce what you got. Graph 3D \| QPI \| SolveSys « Next Oldest \| Next Newest » User(s) browsing this thread: 1 Guest(s)	969	3,246	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	2.65625	3	CC-MAIN-2023-40	latest	en	0.872729
http://math.stackexchange.com/questions/174175/proof-of-the-divisibility-rule-of-17?answertab=oldest	1,444,009,528,000,000,000	text/html	crawl-data/CC-MAIN-2015-40/segments/1443736676547.12/warc/CC-MAIN-20151001215756-00112-ip-10-137-6-227.ec2.internal.warc.gz	201,479,268	21,098	Proof of the divisibility rule of 17. Rule: Subtract 5 times the last digit from the rest of the number, if the result is divisible by 17 then the number is also divisible by 17. How does this rule work? Please give the proof. - Is this homework? The imperative at the end sounds like it. – celtschk Jul 23 '12 at 10:36 @celtschk: I don't think so, the question probably comes from math.stackexchange.com/questions/174114/… – Generic Human Jul 23 '12 at 10:44 @celtschk no this is not a homework.This question comes from math.stackexchange.com/questions/174114/… (same link as provided by @generichuman). – abhinav8 Jul 23 '12 at 10:52 @celtschk if he wanted to mask it as non homework, i'm sure he is clever enough. why would it matter? – IIIIIllllllllIlllllIIIIIIIIlll Jul 23 '12 at 17:31 @АртёмЦарионов: On one hand, it would matter because it should then get the [homework] tag, on the other hand it would mean that the tone of the last sentence shouldn't bother you because it was just copied from a context where it was appropriate. – celtschk Jul 23 '12 at 17:41 Write your number $10a+b$. Then because 10 and 17 are relatively prime, $$17\mid a-5b \iff 17\mid 10a-50b \iff 17\mid 10a+b$$ The last equivalence is because $10a+b-(10a-50b) = 51b$ is always a multiple of 17. - 10x+y will be divisible by an odd prime p iff a.p.x ~ b(10x+y) is divisible by p. Now by Euclid's GCD algorithm, we can find integers a,b such that a.p ~ b.10=1 where (p,10)=1. If p=17, by observation 3.17 - 5.10=1. So, 3.17.x-5(10x+y)=x-5y. If this is divisible by 17, so will be 10x+y. - Let $$n=\sum_{k=0}^N 10^k a_k$$ be the number we want to test for divisibility, where the $a_k$s are the digits in the decimal expansion of $n$. We form the second number $m$ by the process you describe, $$m = \sum_{k=1}^N 10^{k-1} a_k - 5 a_0 = \frac{n-a_0}{10}- 5 a_0$$ Now suppose $17\|m$. Then there exists a natural number $b$ such that $17b = m$. We then have $$17 b = \frac{n-a_0}{10}- 5 a_0$$ $$10 * 17 b = n-a_0- 50 a_0 \implies n= 17(10b + 3a_0)$$ and so $n$ is divisible by 17. - $17 = 2^3 + 3^2$ the sum of the exponents of $2$ and $3$ is $(3 + 2)$ i.e. the sum of the exponents is $5$ Now a number can be expressed as $abcde$ i.e. $abcde = (10^4)a + (10^3)b + (10^2)c + (10^1)d + e$ remove the units digit from $abcde$ to get $$abcd = (10^3)a + (10^2)b + (10^1)c + d$$ Now, let $A = abcd - (3 + 2)(e)$ Assume $A$ is divisible by $17$ i.e. $A = abcd - (5)e = (17)k$ i.e. $abcde = (10)(abcd) + e$ i.e. $abcde = (10)((17)k + (5)e) + e$ i.e. $abcde = (170)k + (50)e + e$ i.e. $abcde = (17)(10)k + (51)e$ i.e. $abcde = (17)(10)k + (17)(3)e$ i.e. $abcde = (17)((10)k + (3)e)$ i.e. $abcde$ is indeed divisible by $17$ - How are you using $\, 17 = 2^3+3^2\,$ to compute $\: 10^{-1}\equiv -5\pmod{17}$ ? $\ \$ – Bill Dubuque Jul 23 '12 at 13:56 rajesh do you have an answer for the above question? – IIIIIllllllllIlllllIIIIIIIIlll Jul 23 '12 at 17:29 The modulus of a congruence a=b (mod m) is the number m. It is the "base" with respect to which a congruence is computed (i.e., m gives the number of multiples of a that are "thrown out"). For example, when computing the time of day using a 12-hour clock obtained by adding four hours to 9:00, the answer, 1:00, is obtained by taking 9+4=1 (mod 12) (i.e., adding the hours with modulus 12). – Rajesh K Singh Jul 27 '12 at 5:58 please clarify what is $10^{-1}$ – Rajesh K Singh Jul 29 '12 at 10:38 is it $\frac{1}{10}$ – Rajesh K Singh Jul 29 '12 at 10:39 Write the original number as 10x+y to separate out the last digit (y) from the number without the last digit (x). Recognize that divisibility by a 17 means you can write the number as 17n for a positive integer n. So our statement is if x-5y=17n then 10x + y = 17m, where x,y,m,n are positive integers. Assume x - 5y = 17n. Now we try to get the left-hand side to look like our original number. First multiply both sides by 10. (10x - 50y = 170n), and then add 51y to both sides. (10x+ y = 170n + 51y). Factor 17 from the right hand side. (10x+y=17(10n+3y)). Recognize that 10n+3y is a positive integer as n and y are integers, let's define it as m, proving that 10x+y=17m meaning that the original number is divisible by 17 if x-5y was divisible by 17. -	1,512	4,280	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.25	4	CC-MAIN-2015-40	longest	en	0.879191

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