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Karen has decided to calculate her net worth. She has these items listed so far: an antique bracelet, twenty dollars cash, a credit card bill, a checking account statement. Which of these is a liability? A credit card bill is a liability. Answer: Option 3. Liability of a person is something that he owes towards some one. A liability is not possessed by the person totally on his own. It is to be given to some one else in one way or the other. A liability is like a burden on a particular person. Since a credit card bill is to be paid to the company from which the credit has been taken, it has to be paid back to that company. So it is like a burden on Karen. Thus it is a liability to be paid off.	{"url":"https://edustrings.com/mathematics/1547903.html","timestamp":"2024-11-13T12:13:44Z","content_type":"text/html","content_length":"23191","record_id":"<urn:uuid:03cbfb50-db59-4fb2-8607-c89fd50db456>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00533.warc.gz"}
Conceptions of Creativity in Elementary School Mathematical Problem Posing 2014 Theses Doctoral Conceptions of Creativity in Elementary School Mathematical Problem Posing Mathematical problem posing and creativity are important areas within mathematics education, and have been connected by mathematicians, mathematics educators, and creativity theorists. However, the relationship between the two remains unclear, which is complicated by the absence of a formal definition of creativity. For this study, the Consensual Assessment Technique (CAT) was used to investigate different raters' views of posed mathematical problems. The principal investigator recruited judges from three different groups: elementary school mathematics teachers, mathematicians who are professors or professors emeriti of mathematics, and psychologists who have conducted research in mathematics education. These judges were then asked to rate the creativity of mathematical problems posed by the principal investigator, all of which were based on the multiplication table. By using Cronbach's coefficient alpha and the intraclass correlation method, the investigator measured both within-group and among-group agreement for judges' ratings of creativity for the posed problems. Previous studies using CAT to measure judges' ratings of creativity in areas other than mathematics or mathematics education have generally found high levels of agreement; however, the main finding of this study is that agreement was high only when measured within-group for the psychologists. The study begins with a review of the literature on creativity and on mathematical problem posing, describes the procedure and results, provides points for further consideration, and concludes with implications of the study along with suggested avenues for future research. • Dickman_columbia_0054D_12092.pdf application/pdf 1.08 MB Download File More About This Work Academic Units Thesis Advisors Ginsburg, Herbert P. Ph.D., Columbia University Published Here July 7, 2014	{"url":"https://academiccommons.columbia.edu/doi/10.7916/D8MC8X69","timestamp":"2024-11-10T08:54:04Z","content_type":"text/html","content_length":"19365","record_id":"<urn:uuid:03ab438a-6b1a-479b-b206-ebb8ce3ee027>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00555.warc.gz"}
Autobox Blog The M3 Forecasting Competition Calculations were off for Monthly Data Guess What We Uncovered ? The 2001 M3 Competition's Monthly calculations for SMAPE were off for most of the entries. How did we find this? We are very detailed. 14 off the 24 to be exact. The accuracy rate was underestimated. Some entries were completely right. ARARMA was almost off by 2%. Theta-SM was off by almost 1%. Theta-SM's 1 to 18 SMAPE goes from 14.66 to 15.40. Holt and also Winter were both off by 1/2%. The underlying data wasn't released for many years so this made this check impossible when this was released. Does it change the rankings? Of course. The 1 period out forecast and the averaged of 1 to 18 are the two that I look at. The averaged rankings had the most disruption. Theta went from 13.85 to 13.94. It's not much of a change. The three other seasonalities accuracies were correctly computed. if you saw our release of Autobox for R, you would know that Autobox would place 2nd for the 1 period out forecast. You can use our spreadsheet and the forecasts from each of the competitors and prove it yourself. See Autobox's performance in the NN3 competition here. SAS sponsored the competition, but didn't submit any forecasts.	{"url":"https://autobox.com/cms/index.php/blog/tags/tag/mape","timestamp":"2024-11-02T14:42:11Z","content_type":"application/xhtml+xml","content_length":"60971","record_id":"<urn:uuid:8d86e02a-8136-4b60-b1ec-b341dec088b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00328.warc.gz"}
Directories Math Science A comprehensive catalog of scientific resources and media worldwide but especially in German and English. Directories of journals; catalogs of search engines; additional starting points and sites of special interest. See Also: • Mathematical Resources on the Web - Compiled at the University of Florida. • The Permutation Homepage - A ring of mathematical web pages. • Mathematics Sources - A list maintained at U Penn. • Specialized Fields - Mathematics Virtual Library pointers to particular fields. • BUBL 510 Mathematics - Links arranged by topic at the British Library. • Mathematics on the Web - Compiled by the staff of Mathematics Reviews. • Mathematics WWW Virtual Library - Directory of mathematical web sites, maintained at Florida math State University. • The Math Forum Internet Mathematics Library - Database sorted by topic, type of media, and science age level. • Math-Net Links - A large collection of mathematical and related links. • Mathematical Resources on the Internet - Maintained by Mathgate. • LLEK Bookmarks Scientific Search Engines - Mathematics - A comprehensive catalog of scientific resources and media worldwide but directories especially in German and English. Directories of journals; catalogs of directories search engines; additional starting points and sites of special interest. • Mathematical Resources - A collection of links at CalTech. • Base of Mathematics Related Resources - Links to selected resource guides, directories, software archives math and bibliographic references on applied and computational mathematics. • Mathematics Resources on the Internet - Links by Bruno Kevius. General and by directories topic. • Math-Net : Internet Information Services for Mathematicians - Preprints, links, directories. Oriented towards German mathematics but in science English. • Math Resources - Contains links to software, professional organizations, research institutes, science teaching aids, science algebra, number theory, geometry, calculus, differential science • Online Resources for Mathematics - The Virtual Reference Desk for mathematics at the science CERN Library (HEP Libraries Webzine, Issue 3, March science 2001, Antonella De Robbio). • MTHWWW: Mathematics Resources on the WWW - Covering most branches of mathematics and a good directories selection of directories general math resources. Compiled by directories M. Maheswaran, University of directories Wisconsin Marathon County. • UniversalClass - Listings of authors, experts, jobs, links, questions, requests and scholarships in mathematics. • MathGuide - Directory of on-line mathematics, mainly research literature. science At GĂ¶ttingen, math Germany. • Mathematics - Subject Guide - Resources listed at New Mexico State University Library. • Start4all - Links to math sites. Biographies, publications, software, subject topics. • Russian Mathematical Database - Directory of Russian mathematical organizations, departments math and mathematicians; publications; library catalogues; journals; projects. • Mathematics Websites - Popular directory of math web sites. At science Penn State. • Platonic Realms Math Links Library - Annotated directory of mathematics sites on the Internet. MySQL - Cache Direct	{"url":"https://www.iaswww.com/apr/Science/Math/Directories/","timestamp":"2024-11-09T15:44:29Z","content_type":"text/html","content_length":"13333","record_id":"<urn:uuid:122fef03-6f2c-4cae-9657-03b75c3758d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00891.warc.gz"}
Conjunction Fallacy — LessWrong The following experiment has been slightly modified for ease of blogging. You are given the following written description, which is assumed true: Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities. No complaints about the description, please, this experiment was done in 1974. Anyway, we are interested in the probability of the following propositions, which may or may not be true, and are not mutually exclusive or exhaustive: A: Bill is an accountant. B: Bill is a physician who plays poker for a hobby. C: Bill plays jazz for a hobby. D: Bill is an architect. E: Bill is an accountant who plays jazz for a hobby. F: Bill climbs mountains for a hobby. Take a moment before continuing to rank these six propositions by probability, starting with the most probable propositions and ending with the least probable propositions. Again, the starting description of Bill is assumed true, but the six propositions may be true or untrue (they are not additional evidence) and they are not assumed mutually exclusive or exhaustive. In a very similar experiment conducted by Tversky and Kahneman (1982), 92% of 94 undergraduates at the University of British Columbia gave an ordering with A > E > C. That is, the vast majority of subjects indicated that Bill was more likely to be an accountant than an accountant who played jazz, and more likely to be an accountant who played jazz than a jazz player. The ranking E > C was also displayed by 83% of 32 grad students in the decision science program of Stanford Business School, all of whom had taken advanced courses in probability and statistics. There is a certain logical problem with saying that Bill is more likely to be an account who plays jazz, than he is to play jazz. The conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y). That is, the probability that X and Y are simultaneously true, is always less than or equal to the probability that Y is true. Violating this rule is called a conjunction fallacy. Imagine a group of 100,000 people, all of whom fit Bill's description (except for the name, perhaps). If you take the subset of all these persons who play jazz, and the subset of all these persons who play jazz and are accountants, the second subset will always be smaller because it is strictly contained within the first subset. Could the conjunction fallacy rest on students interpreting the experimental instructions in an unexpected way - misunderstanding, perhaps, what is meant by "probable"? Here's another experiment, Tversky and Kahneman (1983), played by 125 undergraduates at UBC and Stanford for real money: Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet. 1. RGRRR 2. GRGRRR 3. GRRRRR 65% of the subjects chose sequence 2, which is most representative of the die, since the die is mostly green and sequence 2 contains the greatest proportion of green rolls. However, sequence 1 dominates sequence 2, because sequence 1 is strictly included in 2. 2 is 1 preceded by a G; that is, 2 is the conjunction of an initial G with 1. This clears up possible misunderstandings of "probability", since the goal was simply to get the $25. Another experiment from Tversky and Kahneman (1983) was conducted at the Second International Congress on Forecasting in July of 1982. The experimental subjects were 115 professional analysts, employed by industry, universities, or research institutes. Two different experimental groups were respectively asked to rate the probability of two different statements, each group seeing only one 1. "A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983." 2. "A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983." Estimates of probability were low for both statements, but significantly lower for the first group than the second (p < .01 by Mann-Whitney). Since each experimental group only saw one statement, there is no possibility that the first group interpreted (1) to mean "suspension but no invasion". The moral? Adding more detail or extra assumptions can make an event seem more plausible, even though the event necessarily becomes less probable. Do you have a favorite futurist? How many details do they tack onto their amazing, futuristic predictions? Tversky, A. and Kahneman, D. 1982. Judgments of and by representativeness. Pp 84-98 in Kahneman, D., Slovic, P., and Tversky, A., eds. Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press. Tversky, A. and Kahneman, D. 1983. Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90: 293-315. Ray Kurzweil is pretty impressive, although I would be much less confident in his predictions from now to 2029 than from 1999-2009. Thanks for this gem! I got the 1982 University of British Columbia ordering right easily, though that might be because I'm already aware of the phenomenon being studied. It would be much harder for me as a subject to deal properly with the Second International Congress on Forecasting experiment. Even if I'm aware that adding or removing detail can lead my estimate of probability to change in an illogical way, my ability to correct for this is limited. For one thing, it is probably hard to correctly estimate the probability that I would have assigned to a more (or less) detailed scenario. So I may just have the one probability estimate available to me to work with. If I tell myself, "I would have assigned a lower probability to a less detailed scenario", that by itself does not tell me how much lower, so it doesn't really help me to decide whether and how much I should adjust my probability estimate to correct for this. Furthermore, even if I were somehow able to accurately estimate the probabilities I would have assigned to scenarios with varying levels of detail, that still would not tell me what probability I should assign. If my high-detail assigned probability is illogically higher than the low-detail assigned probability, that doesn't tell me whether it is the low-detail probability that is off, or the high-detail probability that is So as someone trying to correct for the "conjunction fallacy" in a situation like that of the Congress in Forecasting experiment, I'm still pretty helpless. Although Robin's critiques of "gotcha" bias are noted, I experienced this as a triumph of learned heuristic over predisposed bias. My gut instinct was to rank accountant+jazz player as more probable than jazz player, and then I thought about the conjunction rule of probability theory. "The ranking E > C was also displayed by 83% of 32 grad students in the decision science program of Stanford Business School, all of whom had taken advanced courses in probability and statistics." This is shocking, particularly if they had more than 30 seconds to make a decision. But if the question "What is P(X), given Y?" is stated clearly, and then the reader interprets it as "What is P(Y), given X", then that's still an error on their part in the form of poor reading Which still highlights a possible flaw in the experiment. Imagine a group of 100,000 people, all of whom fit Bill's description (except for the name, perhaps). If you take the subset of all these persons who play jazz, and the subset of all these persons who play jazz and are accountants, the second subset will always be smaller because it is strictly contained within the first subset. Nitpicking: Concluding that this is a strict inclusion implicitly assumes that there is at least one jazz player who is not an accountant in the original set. Otherwise, the two subsets may still be equal (and thus, equal in size). The interesting thing to me is the thought process here, as I also knew what was being tested and corrected myself. But the intuitive algorithm for answering the question is to translate "which of these statements is more probable" with "which of these stories is more plausible." And adding detail adds plausibility to the story; this is why you can have a compelling novel in which the main character does some incomprehensible thing at the end, which makes perfect sense in the sequence of the story. The only way I can see to consistently avoid this error is to map the problem into the domain of probability theory, where I know how to compute an answer and map it back to the story. While I personally answered both experiments correctly, I see the failure of those whom we assume should be able to do so as a lack of being able to adapt learned knowledge for practical use. I have training in both statistics and philosophy, but I believe that any logical person would be capable of making these judgments correctly, sans statistics and logic classes. Is there any real reason to believe that someone who has studied statistics would be more likely to answer these questions correctly? Or is the ability simply linked to a general intelligence and that participation in an advanced statistics and probability curriculum is a poor indicator of that intelligence? I know a jazz musician who is not an accountant. Going to the reason why. If I simply ask, which is more probable, that a random person I pick out of a phone book is an accountant or that same person is an accountant and is also a jazz musician. Then I suspect more grad students would get the answer correct. That personality traits are given to the random selection clutters up the "test". We can understand the possibility that Bill is an accountant. So we look for that trait and accept the secondary trait of jazz. But jazz by itself - Never. We read answer E as if to say "If Bill is an accountant, he might play jazz" and this which we can accept for Bill much greater than Bill actually playing jazz. It would also be more probably with typical prejudice. So an interesting question here is (if I'm correct) why do our prejudices want to make answer E as an accountant who might play jazz rather than the wording actually used. I think it makes more intuitive sense to an typical reader. Can we imagine Bill as an accountant who might play jazz - absolutely. Can we imagine Bill as an account who does play jazz - not as easliy: Lets substitute what it is, with what I want it to read so it makes sense and makes me feel comfortable about solving this riddle. QED A>E>C I, too, was worried about this at first, but you'll find that http://lesswrong.com/lw/jj/conjunction_controversy_or_how_they_nail_it_down/ contains a thorough examination of the research on the conjunction fallacy, much of which involves eliminating the possibility of this error in numerous ways. Reasoning with frequencies vs. reasoning with probabilities Though it's frustrating that we humans seem so poorly designed for explicit probabilistic reasoning, we can often dramatically improve our performance on these sorts of tasks with a quick fix: just translate probabilities into frequencies. Recently, Gigerenzer (1994) hypothesized that humans reason better when information is phrased in terms of relative frequencies, instead of probabilities, because we were only exposed to frequency data in the ancestral environment (e.g., 'I've found good hunting here in the past on 6 out of 10 visits'). He rewrote the conjunction fallacy task so that it didn’t mention probabilities, and with this alternate phrasing, only 13% of subjects committed the conjunction fallacy. That's a pretty dramatic improvement! For the above experiment, the rewrite would be: Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities. There are 200 people who fit the description above. How many of them are: A: Accountants … E: Accountant who play jazz for a hobby. Gigerenzer, G. (1994). Why the distinction between single-event probabilities and frequencies is important for psychology (and vice versa). In G. Wright and P. Ayton, eds., Subjective Probability. New York: John Wiley. The rephrasing as frequencies makes it much clearer that the question is not "How likely is an [A\|B\|C\|D\|E] to fit the above description" which J thomas suggested as a misinterpretation that could cause the conjunction fallacy. Similarly, that rephrasing makes it harder to implicitly assume that category A is "accountants who don't play jazz" or C is "jazz players who are not accountants". I think similarly, in the case of the poland invasion diplomatic relations cutoff, what people are intuitively calculating in the compound statement is the conditional probability, IOW, turning the "and" statement into an "if" statement. If the soviets invaded Poland, the probability of a cutoff might be high, certainly higher than the current probability given no new information. But of course that was not the question. A big part of our problem is sometimes translation of english statements into probability statements. If we do that intuitively or cavalierly, these fallacies become very easy to fall into. josh wrote: Sebastian, I know a jazz musician who is not an accountant. Josh, note that it is not sufficient for one such person to exist; that person also has to be in the set of 100,000 people Eliezer postulated to allow one to conclude the strict inclusion between the two subsets mentioned. Sebastian, I thought of including that as a disclaimer, and decided against it because I figured y'all were capable of working that part out by yourselves. Unless both figures are equal to 0, I think it is rather improbable that in a set of 10 jazz players, they are all accountants. The probability of P(A&B) should always be strictly less than P(B), since just as infinity is not an integer, 1.0 is not a probability, and this includes the probability P(A\|B). However you may drop the remainder if it is below the precision of your arithmetic. When initially presenting the question, he doesn't mention a sample of 100,000 people. I assumed we were using the sample of all people. My gooch. 1.0 is a probability. According to the axioms of probability theory, for any A, P(A or (not A))=1. (Unless you're an intuitionist/constructivist who rejects the principle that not(not A) implies A, but that's beyond the scope of this discussion.) My question about the die-rolling experiment is: how would raising the $25 reward to, say $250, affect the probability of an undergraduate commiting the conjunction fallacy? (By the way, Bill commits fallacies for a hobby, and he plays the tuba in a circus, but not jazz) It seems to me like a fine way to avoid this fallacy is to always, as a habit, disassemble statements into atomic statements and evaluate the probability of those. Even if you don't use numbers and any of the rules of probability, just the act of disassembling a statement should make the fallacy obvious and hence easier to avoid. I think this fallacy could have severe consequences for criminal detectives. They spend a lot of time trying to understand the criminals, and create possible scenarios. It's not good if a detective finds a scenario more plausible the more detailed he imagines it. The case of the die rolled 20 times nad trying to determine which sequecne is more likely is not one covered in most basic statistics courses. Yes you can apply the rule of statistics and get the right answer, but knowing the rules and being able to apply them are different things. Otherwise we could give people Euclids postulates one day and expect them to know all of geometry. I see a lot of people astonished by peoples answers, but how many of you could correctly determine the exact probability of each of the sequences appearing? Maybe I am wrong but I think to get the probability of an arbitrary sequence appearing you have to construct a markov model of the sequence. And then I think it is a bunch of matirx multiplication that determines the ultimate probability. Basically you have to take a 6 by 6 matrix and take it to the 20th power. Obviously this is not required, but I think when people can't calculate the probabilities they tend to use intuition, which is not very good when it comes to probability theory. What's an accountant? I think most people would say that there's a high probability Bill is an accountant and a low probability that he plays jazz. If Bill is an accountant that does not play jazz, then E is "half right" whereas C is completely wrong. They may be judging the statements on "how true they are" rather than "how probable they are", which seems an easy mistake to make. Re: Dice game Two reasons why someone would choose sequence 2 over sequence 1, one of them rational: 1) Initially I skimmed the sequences for Gs, assumed a non-fixed type font, and thought all the sequences were equal length. On a slightly longer inspection, this was obviously wrong. 2) The directions state: " you will win $25 if the sequence you chose appears on successive rolls of the die." A person could take this to mean that they will successively win $25 for each roll which is a member of a complete version of the sequence. It seems likely the 2nd sequence would be favored in this scenario. The winners would probably complain though, so this likely would have been This was a really nice article, especially the end. So, I tried each of these tests before I saw the answers, and I got them all correct- but I think the only reason that I got them correct is because I saw the statements together. With the exception of the dice-rolling, If you had asked me to rate the probabilities of different events occurring with sufficient time in between for the events to become decoupled in my mind, I suspect the absolute probabilities I would given would be in a different order from how I ordered them when I had access to all of them at once. Having the coupled events listed independently forced me to think of each event separately and then combine them rather than trying to just guess the probability of both of a joint event. But I'm not sure if that's the same problem- it might be more related to how inconsistent people really are when they try to make predictions. The YouTube link is broken. Did you intend to link to a YouTube video of the original video from Schoolhouse Rock? I suspect respondents are answering different questions from the ones asked. And where the question does not include probability values for the options the respondents are making up their own. It does not account for respondents arbitrarily ordering what they perceive as equal probabilities. And finally, they may be changing the component probabilities so that they are using different probability values throughout when viewing the options. The respondents are actually reading the probabilities as independent, and assigning probabilities such as this: A: P(Accountant) = 0.1 C: P(Jazz) = 0.01 E: P(Accountant^Jazz) = P(Accountant) x P (Jazz) = 0.001, and you would expect the correct ranking But if they are perceiving E as conditional then P(Accountant\|Jazz) = P(Accountant^Jazz)/P(Jazz) = .001/.01 = 0.1, and leaving the equal ranking of A, E ordered as A, E they end up with A >= E > C. And, it's also possible they are using an intuitive conditional probability and coarsely and approximately ranking without calculation. They may also be doing the intuitive of the following, by reading the questions in order: A: Yeah, sounds about right for Bill. Let's say 0.1 C: Nah, no way does Bill play Jazz. Let's say zero! E: Well, I really don't think he plays jazz, and I really thought he'd be an accountant. But I guess he could be both. In this case I'm going for 0.05 accountant, but 0.02 Jazz. 0.05 x 0.02 = 0.001 So, A > E > C In this last case the fact that he could both be an Accountant and play Jazz (E) is more plausible than he would play Jazz and not be an accountant (reading C as not being an accountant). Of course C does not rule out him also being an accountant, but that's not what appears to be the intuitive implication of C. It's as if the respondent is thinking, why would they include E if C already includes the possibility of being an accountant? And though the options are expressed as a set the respondent is not connecting them and so adapting the independent probabilities in each option. As I said, this might be quite intuitive so that the respondents do not perform the calculations and so do not see the mistake. That the question says "not mutually exclusive or exhaustive" may not register. The diplomatic response might be explained by the following. Without any good reason respondents to (1) think suspension unlikely. Because they are not asked (2) they are asked to rate this independently of anything else, whether that be invasion of Poland, assassination of the US President, or anything else not mentioned in (1). Since they are not given any reason for suspension they think it very unlikely. So, your point that "there is no possibility that the first group interpreted (1) to mean 'suspension but no invasion' " does not hold. They can interpret it to mean 'suspension but nothing else'. But in (2) the respondents are given a good reason to thank that if invasion is likely then suspension will follow hot on its heels. Also, some respondents might be answering a question such as "If invasion then suspension?", even though that is not what they are being asked. So I think there are explanations as to why respondents don't get it that go beyond simply not knowing or remembering the conjunction condition, let alone knowing it as a 'fallacy' to avoid. Is probability a cognitive version of an optical illusion? Two lines may not look the same length, but when you measure them they are. When two probability statements appear one way they may actually turn out to be another way if you perform the calculation. The difference in both cases is relying on intuition rather than measurement or calculation. Looked at it from this point of view probability 'illusions' are no more embarrassing than optical ones, which we still fall for even when we know the falsity of what we perceive. A : A complete suspension of diplomatic relations between the USA and Russia, sometime in 2023. B : A Russian invasion of Poland, sometime in 2023. C : Chicago Bulls winning NBA competition, sometime in 2023. D <=> A & B E <=> A & C. In order to estimate the likelyhood of an event, the mind looks in the available memory for information. The more easily available an information the more it is taken into account. A and B hold information that is relevent to each other. A and B are correlated and the occurence of one of them strengthens the probability of the other happening. The mind while trying to evaluate the likelyhood of each of the components of D takes one as a relevant information about the other hence leading to overestimate p(A) and p(B). p(D) = p(A&B) = p(A).p(B \| A) = p(B).p(A \| B) Then the mind gets it wrong is when it makes the above equation equal to either p(A \| B) or p(B \| A) or oddly equal to their sum, as it has been mentionned in previous comments. The intuitive mind has trouble understanding probability multiplication. It rather functions in an addition (for correlated events) and substraction (for independent events) mode when evaluating likelyhood. p(E) for example is likely to be seen as p(C) - p(A). C is a more likely event ( even more if you live in Chicago) than A. Let say 5% for C and to be generous 1% for A. The mind would rather do p(E) = p(C) - p(A) = 4 % which ends up making p(A&C) > p(A) , rather than the correct p(A) . p(C \|A) = p(C) . p(A \| C) = p(A) . p(C) = 0.05 % assuming A and C completely independent. The great speed at which the intuitive mind makes decisions and assigns likelyhood for propositions seems to be at the expense of occuracy due to oversimplification, poor calculus ability, sensitivity to current emotional state leading to volatility of priority order, sensitivity to chronology of information acquisition... etc. Nevertheless, the intuitive mind shared with other species proved to be a formidable machine fine tuned for survival by ages of natural selection. It is capable of dealing with huge amount of sensorial and abstract information gathered upon time, sorting it dynamically, and making survival decisions in a second or two. In my experience, the english "and" can also be interpreted as separating two statements that should be evaluated (and given credit for being right/wrong) separately. Under that interpretation, someone who says "A and B" where A is true and B is false is considered half-right, which is better than just saying "B" and being entirely wrong. Though, looking back at the original question, it doesn't appear to use the word "and", so problems with that word specifically aren't very relevant to this article. I agree that there are some important methodological issues with the paper, and it is far from the last word. What the criticisms you link don't address well, however, is that fact that (a) the paper is strengthened by the fact that it has a strong, validated theory of underlying behavior... - "AnonySocialScientist", Reddit Great article and I know I'm commenting on an 8 year-old post but two points that came to mind: 1 ) I wonder if the UBC and Stanford undergraduates would have done better if the first dice sequence had a leading space or two so that it lined up like so: ( underscore in place of space) 1. _RGRRR 2. GRGRRR 3. GRRRRR 2 ) Edit: Realised this second point was completely wrong The answers provided there seem not to be relevant to the objection I raised. we might take the probability space in (1) to exclude an invasion of Poland, and the space in (2) to include one That seems like an unjustified interpretation, since, according to the OP: Two different experimental groups were respectively asked to rate the probability of two different statements, each group seeing only one statement: 1. "A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983." 2. "A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983." Since the subjects receiving statement 1 do not even see statement 2, they would have no reason to exclude the possibility of an invasion of Poland from statement 1. I believe this is not a conjunction fallacy, but for a different reason. In the first case, the test subject is required to conceive of a reason that might lead to a complete suspension of relations. There are many different choices, invasion, provocation, oil embargo of Europe, etc. Each of these seems remote, that the test subject might not even contemplate them. In the second case, the test is given a more specific, therefore more conceivable sequence of events. A good third scenario, to control for this, would have been to ask another group of subjects the probabilities independently: A. That USSR invades Poland B. That US suspends relations This provide the same trigger of a plausible provocation, but doesn't directly link them. Variances between the estimates of B in this case v. 1 in the original test would indicate confidence interval between variances between 1 and 2. I have some issues with the third experiment (URSS vs. USA). Let me try to explain them with an example. Suppose you see a chess match up to a given point, with black to move. You are asked an estimate W1 of the probability that white can force a win. Then you see black's move, and it's a truly, unexpectedly brilliant move. You are then asked a new estimate W2 of the probability that white can force a win. If black's move is sufficiently brilliant, it's natural for W2 to be lower than the answer a "previous you" gave for W1: black has seriously undermined white's chances with his move. But the Russia vs. USA example seems to suggest that any pair of answers where W2<W1 is a fallacy. After all, the set of possible directions of play before black's move includes all possible directions of play after black's move. If white can force a win in all the former, it can also force a win in all the latter, which are a subset of the former. So one could argue that any rational analyst should always output W2 at least as high as W1. I think the catch is that having more details made explicit, even details that we implicitly knew ("please think about the possibility of Russia invading Poland") can allow us to reason "more accurately" about the likelihood of a given, complex situation (the diplomatic relations between Russia and the USA failing). It's quite possible that, in the light of those details, our evaluation of the likelihood increases sufficiently to more than compensate for the decrease of likelihood involved in considering just a subcase (the relations failing AND an invasion of Poland). Is this a fallacy? I would rather call it a case of computation resources inadequate to the task at hand - resources that become (more) adequate with a little boost or "hint" allowing the evaluation process to run more efficiently. In this sense, it would have been interesting to see the results if the first statement had been something along the lines: "A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983 (keeping in mind that it's not impossible that the Soviet Union invades Poland)". Or if the analysts had been given the questions, and two months of full time study of US and URSS history. But what of the fact that any singular event, is actually a conjunction of the probabilities of that event and the negation of the alternative event, so A is really equivalent to a conjunction of A and not B. Given this can it really be said that A is more likely than A and B? I agree. This notion of question 2 providing a plausible cause that might lead to suspension v. question 1 where the test subject has to conceive of their own cause is a bias, but a different type of bias, not a conjunction fallacy. There could be (and possibly have been) ways to construct the test to control for this. For example, there are 3 test groups where 1 and 2 are the same and for the third, the two events are asked independently: What are the probabilities of each event: A. That USSR invades Poland, or B. That US suspends relations New Comment 46 comments, sorted by Click to highlight new comments since: I think this might possibly be explained if they looked at it in reverse. Not "how likely is it that somebody with this description would be A-F", but "how likely is it that somebody who's A-F would fit this description". When I answered it I started out by guessing how many doctors there were relative to accountants -- I thought fewer -- and how many architects there were relative to doctors -- much fewer. If there just aren't many architects out there than it would take a whole lot of selection for somebody to be more likely to be one. But if you look at it the other way around then the number of architects is irrelevant. If you ask how likely is it an architect would fit that description, you don't care how many architects there So it might seem unlikely that a jazz hobbyist would be unimaginative and lifeless. But more likely if he's also an accountant. I think this is a key point - given a list of choices, people compare each one to the original statement and say "how well does this fit?" I certainly started that way before an instinct about multiple conditions kicked in. Given that, its not that people are incorrectly finding the chance that A-F are true given the description, but that they are correctly finding the chance that the description is true, given one of A-F. I think the other circumstances might display tweaked version of the same forces, also. For example, answering the suspension of relations question not as P(X^Y) vs P(Y), but perceiving it as P(Y), given X. If one is presented two questions, • Bill plays jazz • Bill is an accountant and plays jazz, is there an implied "Bill is not an accountant", created by our flawed minds, in the first question? This could explain the rankings. There was an implied "Bill is not an accountant" in the way I read it initially, and I failed to notice my confusion until it was too late. So in answer to your question, that has now happened at least once. "Logical Conjunction Junction" Logical conjunction junction, what's your function? To lower probability, By adding complexity. Logical conjunction junction, how's that function? I've got hundreds of thousands of words, They each hide me within them. Logical conjunction junction, what's their function? To make things seem plausible, Even though they're really unlikely. Logical conjunction junction, watch that function! I've got "god", "magic", and "emergence", They'll get you pretty far. [spoken] "God". That's a being with complexity at least that of the being postulating it, but one who is consistently different from that in logically impossible ways and also has several literature genres' worth of back story, "Magic". That's sort of the opposite, where instead of having an explanation that makes no sense, you have no explanation and just pretend that you do, And then there's "emergence", E-mergence, where you collapse levels everywhere except in one area that seems "emergent" by comparison, because only there do you see higher levels are perched on lower levels that are different from them. "God", "magic", and "emergence", Get you pretty far. [sung] Logical conjunction junction, what's your function? Hooking up two concepts so they hold each other back! Logical conjunction junction, watch that function! Not just when you see an "and", Also even within short words. Logical conjunction junction, watch that function! Some words combine many ideas, Ideas that can't all be true at once! Logical conjunction junction, watch that function! There's also a linguistic issue here. The English "and" doesn't simply mean mathematical set theoretical conjunction in everyday speech. Indeed, without using words like "given" or "suppose" or a long phrase such as "if we already know that", we can't easily linguistically differentiate between P(Y \| X) and P(Y, X). "How likely is it that X happens and then Y happens?", "How likely is it that Y happens after X happened?", "How likely is it that event Y would follow event X?". All these are ambiguous in everyday speech. We aren't sure whether X has hypothetically already been observed or it's a free variable, too. Is this really a fallacy? In the USSR and Poland case, we might take the probability space in (1) to exclude an invasion of Poland, and the space in (2) to include one. Then the claims are perfectly consistent, since the probability space changes; people just reason with respect to "stereotypical" alternatives. Okay, Eliezer should add a boldfaced note at the bottom of this post asking people not to comment until they've read the followup. I find EY’s main points very convincing and helpful. After reading this and the follow-on thread, my only nit is that using the suspension-of-relations question as one of the examples seems pedagogically odd, because perfectly rational (OK, bounded-rational but still rational) behavior could have led to the observed results in that case. The rational behavior that could have led to the observed results is that participants in the second group, having been reminded of the “invade Poland” scenario, naturally thought more carefully about the likelihood of such an invasion (and/or the likelihood of such an invasion triggering suspension), and this more careful thinking caused them to assign a higher probability to the invasion-then-suspension scenario (thus also to the invasion-and-suspension scenario) than they would have assigned to the “suspension” scenario if instead asked Question 1 (which mentions only Why? For the simple reason that Question 2 tended to provide them with new information (namely, the upshot of the additional careful thought about the Polish invasion scenario) that Question 1 wouldn’t have. (To caricature this, imagine showing two separate groups of chess beginners the same superficially-even board position with Player A on move, asking Group 1 participants “what’s the probability that A will win,” and separately asking Group 2 participants “what’s the probability that A will make slightly-tricky-advantageous-move-X and win”? Yes, the event Group 2 was asked about is less likely than the event Group 1 was asked about; Group 2's answers may nevertheless average higher for quite rational reasons.)	{"url":"http://www.lesswrong.com/posts/QAK43nNCTQQycAcYe/conjunction-fallacy","timestamp":"2024-11-09T17:42:31Z","content_type":"text/html","content_length":"984404","record_id":"<urn:uuid:1e575b41-b91a-4da7-85a9-56fe3eba8f32>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00377.warc.gz"}
[Solved] Using the one- year return percentage var \| SolutionInn Using the one- year return percentage variable in Retirement Funds: a. Construct a table that computes the Using the one- year return percentage variable in Retirement Funds: a. Construct a table that computes the mean for each combination of market cap, risk, and rating. b. Construct a table that computes the standard deviation for each combination of market cap, risk, and rating. c. What conclusions can you reach concerning differences based on the market cap (small, mid- cap, and large), risk (low, average, and high), and rating (one, two, three, four, and five)? Fantastic news! We've Found the answer you've been seeking!	{"url":"https://www.solutioninn.com/using-the-one-year-return-percentage-variable-in-retirement-funds-a-construct-a","timestamp":"2024-11-14T03:32:04Z","content_type":"text/html","content_length":"80891","record_id":"<urn:uuid:f7a538ad-50b8-4d51-8d26-ff096a46ad39>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00642.warc.gz"}
Graduate Student Seminar The grad student seminar is a space for graduate students from the entire math department to gather every Friday afternoon and listen to one of our own give a 50 minute talk about the math they're doing or thinking about. It's an opportunity for the speakers to hone their presentation skills in front of a friendly, students-only audience, and aims to foster collaboration within the department through the sharing and discussing of mathematical ideas. The talks will highlight some mathematical research: either your own original work, or an introduction to an area of research you are interested in (or a combination of the two!). For MSc students: if you're interested in giving a talk but you're not sure what you'd like to talk about, please reach out to any of the organizers. We'll pair you with a PhD student in your area of interest who can help you prepare a presentation. All talks will be 50 minutes in length with 10 minutes at the end for questions and discussion. We encourage everyone to attend and participate, so please consider giving a talk this term. During the seminar, we will serve pizza and refreshments. Afterwards, we invite you to join us for an informal social at the Grad Club . We hope to see you all there! Practical information Fall 2024 November 8: Nathan Pagliaroli Title: The Free Central Limit Theorem Abstract: Free Probability is used to study non-commutative random variables and has its roots in Voiculescu’s work on operator algebras in the 1980s. The notion of free independence serves as the non-commutative analogue of independence in classical probability theory. One of the most famous results in probability theory is the central limit theorem and in this talk we will prove the analogous result in free probability theory, known as the free central limit theorem. The proof is combinatorial in nature and all relevant notions of Free Probability will be introduced beforehand. November 1: Diego Tenoch Morales Lopez Title: Mathematics applied to evolution: bridging the gaps between pure mathematics, biology and numerical analysis Abstract: From the perspective of a pure mathematician, the application of mathematics to concrete problems (like evolution of the species) is not of primary importance. However, that does not imply that applied mathematics deals with problems that are trivial mathematically speaking, or that there is nothing to learn from applications. In this talk, I will showcase different ways in which mathematics are being used in the study of Evolution, ranging from systems of ODEs and PDEs to "simple" stochastic models, with some sprinkles of numerical analysis. My goal is to spark the interest of the pure math audience not only in applying math to biology, but also in further analyzing the mathematics that arise from biological problems for mathematics' sake. October 25: Zack Dooley Title: An Introduction to Proof Assistants Abstract: Proof assistants are a variety of software tools which can be used to assist in proof writing and verifying mathematical statements. Recently, proof assistants have been gaining popularity not just for their ability to verify the correctness of complicated proofs, but also as a tool of collaboration for mathematicians. In this talk I will introduce the basics of what proof assistants are and how to use them, in particular, focusing on the proof assistant Coq. I will show how to write basic definitions and proofs in Coq and show how libraries can help us with collaboration and proof writing. October 11: Alexander Zwart Title: Rationality of Algebraic Tori Abstract: Given a variety X a natural question to ask is whether it is birational to projective space. In general, this is quite a very hard question to answer. We restrict ourselves to a subclass of objects known as algebraic tori. It turns out that a slightly weaker notion related to rationality can be "cleanly" understood in terms of the character lattice for a given torus. I will give some background to state this result and then discuss the work that has been done on the rationality question for algebraic tori. Winter 2024 April 5: Curtis Wilson Title: The Golod-Shafarevich Theorem Abstract: We introduce the class field tower infinite issue and state its solution, the Golod-Shafarevich theorem. We prove the theorem, and provide a refinement for the graded case. We discuss some examples and finish with an application involving finitely generated infinite torsion groups. March 29: Yunhai Xiang Title: An easy tour of Galois cohomology Abstract: Galois cohomology is a topic that should interest a wide range of audiences: number theorists, algebraic geometers, homotopy theorists, etc. It is a wonderful example of an application of ideas from algebraic topology to study algebra and number theory. In this talk, we will discuss the basics of Galois cohomology, and we demonstrate its power by using it to prove the Mordell-Weil theorem for elliptic curves. If time permits, we might also discuss a little bit about its generalization: étale cohomology. March 22: Esther Yartey Title: Structural connectivity across datasets and species reveals community structure in cortex with specific connection features Abstract: Advancements in neuroimaging technologies, particularly diffusion MRI, now allow reconstructing the long-range fiber connection patterns in the human brain. We study the network whose connection weights are determined by the number of fibers between individual brain regions. We study networks from the Human Connectome Project (HCP) and networks extracted from individual imaging subjects through a data processing pipeline developed in our group. By applying an algorithm to detect highly connected “communities” in these networks, we find a discrete set of communities appear robustly in the human brain. A specific community in the occipital lobe systematically displays high eigenvector centrality (EVC), a measure of the influence of nodes within a network. We explore the variations in these network structures among individuals and in retested subjects to isolate the sources of inter-individual variability. This result consistently appears across nearly all subjects and in a test-retest dataset. Similar community structure also appears in connectomes from macaque and marmoset brains, but the existence of an occipital lobe community with high EVC is specific to human connectomes. Taken together, these results reveal novel organization in the structural connectivity of the brain, derived from a fully data-driven approach, where clear community organization appears. This community organization relates to known functional divisions, such as visual and auditory sensory pathways, but also reveals community structure within higher-order areas, whose functional relevance can be studied in future work. March 15: Thomas Thorbjornsen Title: The Synthetic Fundamental Group of the Circle Abstract: Homotopy type theory is a foundation for mathematics that is well-suited to do homotopy theory. Unlike ZFC, we no longer have access to the law of excluded middle or the axiom of choice. Many mathematicians find it daunting and off-putting to work without these axioms, but great opportunity is created amidst the challenge. In this talk we will investigate how we can do homotopy theory in homotopy type theory. We will construct the fundamental group of the circle by using the tools provided by Martin-Löf type theory, Voevodsky’s univalence axiom, and higher inductive types. In particular, we will look at the underlying language of type theory and its identity types, how fiber bundles are expressed as type families, and the essential role played by the univalence axiom. March 8: Sayantan Roy Chowdhury Title: Essential Dimension of Symplectic Sheaves over a curve Abstract: Essential dimension of an algebraic object captures the minimum number of variables needed to parameterize it. Concrete as it sounds, this invariant is often hard to compute even in the most simplest of cases. In this talk, we will tackle this problem for the moduli space of symplectic sheaves on a curve. We will start out by defining (and drawing pictures) for these objects. Subsequently, we outline a roadmap for estimating the essential dimension of this moduli space enroute deformation theory. March 1: Marwa Tuffaha Title: Mutator dynamics cannot be explained by mutation rates alone Abstract: Mutators, cells with elevated mutation rates, are common in both natural microbial populations and in human cancers. Recent experiments have shown that mutators can invade a population, but the invasion dynamics and probability couldn’t be explained by mutation rates alone. Here we show, analytically and in simulation, that mutation bias (which types of mutations are likely to occur) can play an important role in the emergence of mutators. A mutator that reduces or reverses the historically prevailing mutational bias is shown to have an increased chance of invasion, while chances are reduced when the bias is reinforced. These findings are important when trying to understand natural populations or competition experiments with mutators. February 16: Tao Gong Title: What is the space of conjugacy classes? Abstract: When G is a topological group, G acts on itself by conjugation. This leads to the natural question: what is the quotient space of conjugacy classes? In this talk, we will work out the answer to this question in the special case when G is a connected compact simple Lie group – the space is contractible! February 9: Michelle Hatzel Title: Continuation Methods for Numerical Problem Solving Abstract: Informally, if two functions can be “continuously deformed” from one to the other, this is called a homotopy. Homotopy emerged from theory more than a century ago and was introduced as a numerical method for solving non-linear problems in the 1960s. The basic components of these early continuation algorithms built on earlier path-tracking methods, which exist in today’s “black box” solvers. We will look at the building blocks of continuation algorithms, how they work (or don’t), and how key insights from the 1970s and 1980s contributed to some powerful polynomial-solving software packages. January 19: Blake Whiting Title: Acyclic Models Abstract: Acyclic models, as it's commonly seen today, is a proof technique used to show when two chain complexes are chain equivalent or have isomorphic homology. It originated as a theorem by Eilenberg and MacLane (1953), where it was immediately used to show the Eilenberg-Zilber theorem (1953). This theorem, proven directly via acyclic models, gives us a Künneth theorem and defines the cup product, which turns cohomology into a graded ring. This talk will be an exposition on (one version of) the acyclic models theorem, as given by Michael Barr in 2002. I will give the necessary definitions to understand Barr's modern formulation of acyclic models, and then prove it. Time permitting, I will also discuss how the Eilenberg-Zilber theorem follows directly from it and potential avenues to generalizing acyclic models. Fall 2023 December 1: Prakash Singh Title: The Hofer diameter problem for rational symplectic manifolds Abstract: In general, Lie groups do not admit bi-invariant metrics, and infinite dimensional Lie groups should not admit such metrics either. But surprisingly, Ham admits one such metric (in fact, unique in a sense), called the 'Hofer metric', discovered by Hofer in the 90s. People have been studying the large-scale geometry properties of this metric for a long time, but such studies were restricted to either 2-dimensions, monotone symplectic manifolds, or to aspherical manifolds. In particular, it is widely conjectured that the hofer diameter is infinite for every closed symplectic manifold, and this conjecture has been settled for the above-mentioned manifolds. I will talk about the diameter problem associated with this metric for some rational ruled manifolds like CP2, S2 x S2, and their blow-ups, using methods from quantum homology and spectral invariants on them. I will prove the conjecture for CP2 and S2 x S2, and I will prove it under a mild assumption (but unproven) for S2 x S2 blown up once. November 24: Elaine Murphy Title: The Mathematical Structure of Point Mutations Abstract: Mutation is the engine of evolution. By considering only single point mutations (SNPs) on DNA sequences, we see a natural group theoretic model of mutations acting on the set of nucleotides. In this talk, we will investigate the implications of this structure for synonymous mutations (mutations that do not change the encoded amino acids) and how this affects the notion of distance between two genetic sequences. November 17: Manimugdha Saikia Title: Analytic properties of quantum states associated with complex manifolds Abstract: In Quantum Information Theory, there is a rich collection of analytic tools to study tensor product of Hilbert spaces. Geometric quantization attaches Hilbert spaces to symplectic manifolds. Study of these information theoretic measures on these specific Hilbert spaces leads to interesting insights. In our study, we look for invariants, or to what extent the geometry of the space influences the Quantum Information aspects of the Hilbert space and vice versa. In this talk, we shall present our asymptotic result for the average entropy over all the pure states on the (quantum) Hilbert space H_{1,N} \otimes H_{2,N} where H_{1,N}$ and H_{2,N} are the spaces of holomorphic sections of the N-th tensor powers of hermitian ample line bundles on compact complex manifolds. I shall also talk about how certain states associated with product submanifolds become separable. November 10: Alejandro Santacruz Hidalgo Title: Hardy's inequality: a brief review, some extensions and applications. Abstract: In 1915 G.H Hardy needed an estimate for arithmetic means to find a proof of Hilbert's inequality for sequences, a continuous version of that inequality followed in 1925. Since then, extensions have been made in many directions; more general domains, weighted norm inequalities, general measures, among others. In this talk we will review the classic statement of Hardy's original inequality. We will explore some of the extensions of this important inequality and review some of its implications, such as, Sobolev inequalities and boundedness of the Fourier transform in weighted Lorentz spaces. October 27: Nathan Pagliaroli The Gaussian Unitary Ensemble and the Enumeration of Maps Abstract: In this talk I will introduce the notation of a matrix ensemble with a focus on the Gaussian Unitary Ensemble (GUE) as an example. I will introduce its basic properties in connection with map enumeration. In particular, I will outline a proof of the Genus Expansion Formula for moments of the GUE. Time permitting we will discuss the famous Harer-Zagier formula . October 20: Alan Flatres Hamilton's rule: basic concept and extension Abstract: Altruism can seem at first glance to be counterintuitive: why would I spend my energy and resources for someone else's profit? To elucidate this mystery, in 1964, Hamilton wrote a simple rule that describes the evolutionary trajectory of a trait that is costly for the individual having it but that brings benefits to another individual. This formula sums the cost of bearing this trait and the benefits received by the other individual, weighted by the genetic linkage between the two individuals. Thanks to Hamilton's work, we can better understand the evolution of costly behavior such as cooperative breeding. However, the simplicity of Hamilton's rule makes it hard to apply in nature. Indeed, the diversity of species and their life histories requires extensions of Hamilton's rule to study the evolution of altruism in different contexts. In this talk, I will present Hamilton's rule from its basic form to more complex extensions that help us to understand the evolution of costly behavior in different species. October 13: Nathan Kershaw Closed symmetric monoidal structures on the category of graphs Abstract: Discrete homotopy theory is a relatively new area of mathematics, concerned with applying methods from homotopy theory in topology to the category of graphs. In order to do this, a notion of a product between graphs is required. Classically two products have been considered, the box product and the categorical product. These products lead to two different homotopy theories, namely A-theory and X-theory, respectively. This leads us to the question of why these two products are considered, and if one can define other products to study discrete homotopy theory with instead. In this talk, we will answer this question by fully characterizing all closed symmetric monoidal products on the category of graphs. This talk will be based on joint work with C. Kapulkin October 6: Oussama Hamza On extensions of number fields with given quadratic algebras and cohomology Abstract: At the beginning of the century, Labute and Minac introduced a criterion, on presentations of pro-p groups, ensuring that the cohomological dimension is two. Groups with presentations satisfying this condition are called mild. For that purpose, they mixed gradations and filtrations techniques, originally introduced by Golod and Shafarevich to construct infinite pro-p groups, with Anick's results on graded algebras. Recently, Hamza introduced a new criterion on the presentation of finitely presented pro-p groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two. Hamza still used previous techniques and enrich them using Graph and Right Angled Artin Groups/Algebras Theory, Groebner basis, etc. Hamza applied previous criterion to obtain new Galois groups over p-rational fields with prescribed ramification and splitting and cohomological dimension larger than two. In this talk, we discuss previous results with their motivations and techniques if time permits. September 29: Tenoch Morales Adaptation reshapes the distribution of fitness effects Abstract: Evolution is driven by mutations that change the reproductive success of mutant individuals with respect to their unmutated counterparts. These effects on the "fitness" of the organisms can be measured empirically and using mathematical models, which has been increasingly successful in the past decade. In these studies, the effects of the novo mutations in the fitness is measured by a probability distribution, known as the Distribution of Fitness Effects (the DFE). This distribution is predicted to differ for better- or worse-adapted organisms, thus the DFE must change dynamically during the process of adaptation, a fact highlighted in recent studies. In this work, we analyze the change in the DFE during an adaptive process across a fitness landscape. First, we derive analytical approximations for the DFE and the underlying distributions for the allele's fitness contributions. Then, we compare these results with independent simulations that relax several simplifying assumptions made in the analysis. This computational work confirms that our analytical expressions provide a good approximation to the dynamically changing DFE during adaption. We observe that as de novo mutations accumulate, the DFE is shaped in two meaningful ways: by increasing the fraction of deleterious mutations and by decreasing the variance of the distribution. Winter 2023 April 14: Reid Ciolfi Title: Why is the derivative a non-injective operator? Abstract: The simplest answer to this question is very well known: it is because the derivative maps all constants to zero. In this talk, we shall not be satisfied with the simplest answer. We will ask a series of additional questions about why and how the derivative maps constants to zero. In order to delve deeper, we shall introduce and briefly motivate the concept of fractional calculus. Armed with this additional structure, and with some of the rich properties of the Gamma function, we will answer our questions to the fullest possible extent. However, we will have to take the Gamma function apart in the process; an integral representation, a product representation, and an exponential representation will each be used in turn. Along the way, we will also discover an eloquent proof of why the exponential function is a fixed point of the derivative, and why Stirling's approximation for the Gamma function is effective. March 31: Mojgan Ezadian Title: Quantitative estimates of selection bias in bacterial mutation accumulation experiments Abstract: Mutation accumulation (MA) experiments play a crucial role in understanding the processes underlying evolution. In microbial populations, MA experiments typically involve a period of population growth between severe bottlenecks, such that a single individual can form a visible colony. In this study, we quantify the impact of positive and negative selection on MA experiments. Our results demonstrate that selective effects can significantly bias the distribution of fitness effects (DFE) and mutation rates estimated from MA experiments in microbes. Furthermore, we propose a straightforward correction for this bias that can be applied to both beneficial and deleterious mutations. The outcomes of this research emphasize the importance of positive selection in microbial MA experiments to obtain a more accurate understanding of fundamental evolutionary parameters. March 24: Marios Velivasakis Title: Representations of the symmetric group and Z-forms Abstract: In representation theory, we try to connect groups with invertible matrices (usually over the complex numbers). Our most powerful tool is character theory, i.e., looking at the trace of the corresponding matrices, because it is a full invariant (two representations are equivalent if and only if they have the same character). An interesting question is: “What happens if we consider matrices over smaller fields or even the integers? Do we still have the same invariants?”. In this talk, we will discuss representation theory in general, and how we can produce all representations for the symmetric group combinatorially. In addition, we will talk about what happens if we consider our matrices over the integers and how the standard tools fail to describe these representations. March 17: Jacqueline Doan Title: Neural Network Powered Recommender System: Restricted Boltzmann Machine Abstract: How did Spotify know our music taste so well? Recommendation algorithms are widely implemented on entertainment platforms like Netflix and Spotify to provide users with a more personalized experience online. Collaborative filtering is the idea that a target user is more likely to like an product if others with the same interests highly rated that product. In order to analyze and process large and sparse data sets often associated with users’ data, Restricted Boltzmann Machine (RBM), a stochastic neural network model, was implemented as a model for users’ ratings of products by Salakhutdinov et al. in 2007. We will discuss the implementation of RBM and the ethics of recommender systems in this talk. March 10: Kumar Shukla Title: Poincaré duality and enumerative geometry Abstract: How many lines intersect 4 given lines in 3-dimensional space? Poincaré duality gives us a geometric interpretation of the cup product. We can use this to compute the cohomology ring of the Grassmanians. This interpretation is also central to the subject of enumerative geometry. Using Poincaré duality as the starting point, we will give a brief introduction to enumerative geometry and answer some of the classical problems in this subject like the one posed above or the problem of counting the number of lines in a cubic surface. March 3: Chirantan Mukherjee Title: Model structure on simplicial categories Abstract: In the first part of the talk, we review the axioms of model categories and define the Kan-Quillen model structure on simplicial sets. We then move on to define a model structure on the category enriched over simplicial sets. This, further forms a model of (∞, 1)-categories. February 17: Tao Gong Title: Cohomology rings of classifying spaces Abstract: The classifying space of a Lie group is used to classify the principle bundle, hence computation on cohomology of classifying spaces is quite important. In this lecture, we will focus on cohomology over integers. For a Kac-Moody Lie group of finite type, the homotopy colimit of classifying spaces of parabolic groups is the base space of a sphere bundle, where the total space is exactly the classifying space of the original group. This provides a systematic way of computation. We will see examples of the exceptional Lie group and the projective unitary group. February 10: Alan Flatres Title: Evolution of cooperative breeding with group augmentation effects and ecological feedbacks Abstract: Cooperative breeding occurs when an individual helps to raise the offspring of others. It is typically considered to be costly for helpers who lose or postpone the opportunity of personal fitness gains. This behaviour is widespread, occurring in a variety of different taxa, and ecological settings. Moreover, phylogenetic data suggest that environmental conditions play a role in promoting and hindering cooperative breeding. The complex interplay between environmental constraints and population interaction makes it challenging to model cooperative breeding in a satisfying In order to better understand the influence of the environment on cooperative breeding while having reasonable computations, we built a coarse-grained model to study the group augmentation effect. That way, this population model allows us to have more complex relations between the environment and the population and thus to understand their role better. Specifically, by computing the inclusive fitness of this kin selection model, we were able to show that environment-individuals relations, for instance, the probability of establishment, have an influence over the emergence and development of altruistic behaviour in the population. February 3: Curtis Wilson Title: Classifying diagram algebras Abstract: We introduce the representation theory of quivers with a focus on their indecomposable representations. We provide a nice criterion for an algebra to be indecomposable, and finish by proving the remarkable fact that quiver representations of finite type are exactly those with underlying Dynkin type A, D, and E. January 27: Shubhankar Title: Analytic Theory of Polynomials and Polar Convexity Abstract: Traditionally, polynomials have been treated as objects of algebra. However, over the years people realized their excellent analytic properties and big names like Chebyshev, Weierstrass, Fourier spent a chunk of their careers studying them in this context. Indeed, the study of their extremal properties and critical points is of interest in more than one way. The Gauss-Lucas theorem is one such celebrated result. Polar convexity is a relatively new notion that exploits properties of Möbius transforms and convex analysis to give a new outlook on such analytic problems. The tools, even in their infancy, seem powerful and give promising results. The goal of this talk is to introduce the notion of polar convexity and time-permitting, prove a few of these results. Fall 2022 December 2: Yanni Zeng Title: Bifurcation analysis on a predator-prey model with Allee Effect Abstract: The dynamics of a population is greatly affected by its interaction with other populations. There exist many kinds of interaction among populations, such as competition, predation, parasitism and mutualism. The predator-prey interaction is one of the most fundamental interactions and one of the most fascinating interactions to investigate. In 1931, the concept 'Allee effect' was put forward referring to a decrease in population growth rate at low population density since the growth of the species will also be affected by factors: difficulties in mating, unable to defence as a group, social felicitation of reproduction, etc. We apply bifurcation theory to consider a predator-prey model including the Allee effect and show that the species having a strong Allee effect may affect their predation and hence extinction risk. In this talk, I will introduce the related model and present methods analyzing the complex dynamical behaviors of the models with the Allee November 25: Gunjeet Singh Title: Classification of compact, connected topological surfaces Abstract: Topology as an independent subject in mathematics was started by Poincaré at the end of nineteenth century but the notion of surfaces is quite ancient than topology itself. Surfaces were studied extensively by many mathematicians such as Gauss, Riemann, Mobius, Jordan, etc in various contexts like in analysis, differential geometry, etc. Naturally enough, people wanted to classify surfaces. One of the earliest attempts were by Mobius and Jordan in 1860s even after being devoid of the definition of a 'topological surface'. It was only in 1907 when Dehn and Heegaard gave a rigorous enough proof of the statement using 'polygonal presentations' of the surfaces. In this talk, I will present the main ideas of the proof and some interesting and important examples of it. The classification theorem says that every compact, connected 2-manifold is homeomorphic either to a sphere, or a connected sum of one or more toriz or a connected sum of projective planes. The proof uses 'polygonal presentations' which are a special class of cell complexes, in which spaces are represented as quotients of polygons (with even number of sides) with their edges identified. November 18: Mahan Moazzeni Title: Introduction to Khovanov homology Abstract: A knot is a smooth embedding of circle in R3. We are essentially interested in looking at knots, up to an ambient isotopy and see whether knot K1 can be ”distorted” into the other knot, K2. One of the most important problems in knot theory, is the classification problem, which roughly is providing a list of all of the existing knots, up to ambient isotopy. In order to classify them, we need a collection of powerful invariants that recognise each knot from the others. Khovanov Homology (KH) is one of the few topological invariants which at least detects a collection of knots from the other ones. KH is combinatorial in nature and it uses (1 + 1)-topological quantum field theory (TQFT) to move from the category of 1-manifolds to the category of vector spaces in its construction. The construction of KH requires lots of works and kind of ”boring” computations but the result, is one of the most interesting and powerful tools in knot theory that we have, as an instance, KH can detect unknot from any other knots, using KH we can find a combinatorial proof for the celebrated Milnor’s Conjecture without using Seiberg-Witten theory for torus knots. Our main goal for this talk is to introduce the KH rigorously and prove some of its basic properties. If time allows, we will proceed and show some of the interesting results about KH for alternating knots. Our ultimate goal would be to go through the J. Rasmussen’s proof of Milnor’s Conjecture on torus knots using KH, but it requires a lot works. We will most certainly cover the whole idea of his October 28: Tedi Ramaj Title: Investigating the Spread of an Invasive Weed, Tradescantia fluminensis, via Partial Differential Equation Modelling and Dynamical Systems Techniques Abstract: A species is typically defined to be invasive to an ecosystem if it is a non-native species which threatens the ecosystem and its native species. Invasive species may include animals, plants, fungi, and other living organisms. Invasive species have historically been implicated as the one of the greatest drivers of biodiversity loss. We consider the invasion of an ecosystem by invasive plant species, Tradescantia fluminensis (T. fluminensis), an invasive weed which has been implicated in native forest depletion in countries such as Australia, New Zealand, and parts of the United States. We explore the dynamics of T. fluminensis spreading via partial differential equation (PDE) modelling and the application of nonlinear dynamical systems and phase portrait techniques. We propose a competition model, modelling the impact of competition between the invasive weed and a pre-existing native plant species, based on previous models. We are able to use some results from basic existing PDE theory in order to obtain some insights on the biological system. We also explore the existence of travelling wave solutions (TWS) of the PDE systems which represent transitions of the state of the ecosystem. In this talk, we explore both the mathematical theory necessary to obtain the results and the policy decisions which the results may help guide. These results have been published in the Bulletin of Mathematical Biology and may be found here in greater detail: Ramaj, T. On the Mathematical Modelling of Competitive Invasive Weed Dynamics. Bull Math Biol 83, 13 (2021). October 14: Alejandro Santacruz Hidalgo Title: Monotonicity in ordered measure spaces Abstract: Monotone functions defined on the real numbers are very simple and straightforward objects to understand, yet a rich theory of monotone (or decreasing) functions has been developed and has proven to provide new insight on seemingly unrelated problems like characterization of weighted Hardy's inequalities or boundedness for the Fourier transform between Lorentz spaces. In this talk, we will give an introduction to the development of a theory of ordered measures spaces and generalize the theory of monotone functions to this setting. In a general measure space, we assume no order among its elements, instead we rely on a totally ordered collection of measurable sets to carry all the monotonicity properties, with this collection we define what a monotone function is. Next, we explore two different partial orders on the set of decreasing functions and show that there is an optimal upper bound in these partial orders. A collection of function spaces called 'Down spaces' defined by decreasing functions will be introduced and their relationship with the partial orders explained. October 21: Nathan Pagliaroli Title: Random matrices and Tutte’s recursion Abstract: In the 1950’s, W.T. Tutte found a recursive formula for counting a combinatorial object known as a planar map: a 2-cell embedding of a connected planar graph into the oriented sphere, considered up to orientation preserving homeomorphisms of the sphere. In the 1970’s, maps and Tutte’s Recursion were first used as powerful tools in the context of random matrix theory. Both the theory of maps and random matrix theory have benefited from this connection, with methods of proof lending themselves between these areas. In this talk I will introduce the concept of maps, their generating functions, and their connection to random matrices, with the goal of deriving Tutte’s recursive formula. October 7: Alexandra Busch Title: Neural sequences in primate prefrontal cortex encode working memory in naturalistic environments Abstract: Working memory is the ability to briefly remember and manipulate information after it becomes unavailable to the senses. A specific region of the brain - the lateral prefrontal cortex (LPFC) - has been widely implicated in working memory performance in primates. Despite decades of study, how neurons in LPFC coordinate their activity to hold sensory information in working memory remains controversial. In this talk, I will give a brief overview of the traditional model for working memory, and discuss how it is impacted by recent advances in neural recording techniques and more complex experimental paradigms. I will then focus on results from a recent project in which we analyzed the activity of hundreds of neurons recorded from LPFC of non-human primates during a naturalistic working memory task involving navigation in virtual reality. We found that selective sequential activation across neurons encoded specific items held in working memory. Administration of ketamine distorted neural sequences, selectively decreasing working memory performance. Our results indicate that neurons in the lateral prefrontal cortex causally encode working memory in naturalistic conditions via complex and temporally precise activation patterns. September 30: Tenoch Morales Title: Using Fitness Landscapes to understand the shifts in mutation biases Abstract: Mutations are the engine that drives evolution and adaptation forward in that it generates the variation on which natural selection acts. Although mutations are considered to occur randomly in the genome, we see that in many organisms some types of mutations occur more often than expected under uniformity; these deviations are called mutation biases. Even though there is no clear description of the biological mechanisms governing the formation of mutation biases, theoretical and experimental work has shown that a shift in mutation biases during the evolutionary process could grant an adaptive advantage to an organism by increasing the sampling of previously poorly explored types of mutations. In this talk, we will explore the most popular Fitness Landscape models, which map the genotypic space of an organism to its adaptive fitness. With these models, we can simulate the evolutionary process of a population as a walk through the genotypic space towards genotypes with higher fitness, which will help us understand the adaptive effect of shifts in mutation biases at different points on the evolutionary path. September 23: Jarl Taxerås Flaten Title: The moduli space of multiplications on a space Abstract: Since the mid-50s, various topologists have been interested in counting homotopy classes of multiplications (i.e. H-space structures) on certain spaces. For example, there's a unique multiplication on the circle (complex multiplication), and James showed that there are 12 multiplications on the 3-sphere and 120 on the 7-sphere. No other spheres admit a multiplication, barring the We present a formula for the moduli space of multiplications on a pointed object of an ∞-topos. By specializing to the ∞-topos of spaces and counting the path components of these moduli spaces, we recover the numbers just mentioned. These results have been shown in Homotopy Type Theory, which I will give a brief introduction to, and have been formalized in the Coq proof assistant, which I will demonstrate with some live-coding. September 16: Oussama Hamza Title: Filtrations, arithmetic, and explicit examples in an isotypical context Abstract: Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields. In 2016, Minac, Rogelstad and Tan inferred an explicit relation between a_n and c_n. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an isotypical context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1. In this talk, we will mostly review some results of Golod, Shafarevic, Koch, Lazard, Minac, Tan and Hamza. We also give several explicit examples in an arithmetical context. Suggested topics for MSc students • Evolutionary game theory • Epidemiological models • Population dynamics models • Inclusive fitness analysis • Using algebra to understand genetic code • Reedy categories: elegance vs. EZ-ness • Karoubi envelopes and a proof of the Serre-Swan theorem • Parallelizability of spheres: an application of topological K-theory • Brown's representability theorem • Classification of 2-dimensional topological quantum field theories • Classification of Riemann surfaces • Frobenius theorem • Holomorphic differential equations and existence of their solutions • Chow's theorem • Remmert-Stein theorem and analytic sets • Reeb foliations • Tychonoff's theorem • Stone-Cech compactification theorem • Applications of graph theory in neuroscience • Small world networks and scale free networks • Community detection • Spectral graph theory • How to find patterns in data • Gelfand-Naimark theorem • Von Neumann algebras • Rearrangement invariant spaces • Peter-Weyl theorem • Orlicz spaces	{"url":"https://www.math.uwo.ca/graduate/seminar.html","timestamp":"2024-11-02T10:30:56Z","content_type":"application/xhtml+xml","content_length":"72403","record_id":"<urn:uuid:8ec5e2b0-a1b6-4b1d-91af-ee2aa6bc00b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00083.warc.gz"}
The product of 2 5 is_________.-Turito Are you sure you want to logout? The product of 2 × 5 is_________. A. 15 B. 10 C. 25 D. 7 A product is the result of multiplication, or an expression that identifies factors to be multiplied. The correct answer is: 10 We have to find the product of 2×5. We have, 2×5 = 10. Hence, the correct option is B. We can also find the product by adding 2 five times. Get an Expert Advice From Turito.	{"url":"https://www.turito.com/ask-a-doubt/Mathematics-the-product-of-2-5-is-7-25-10-15-q695f5b","timestamp":"2024-11-12T18:55:04Z","content_type":"application/xhtml+xml","content_length":"149187","record_id":"<urn:uuid:8ef42a6d-2f22-4569-a77d-822703398586>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00775.warc.gz"}
” Everything in our terrestrial world depends on the motions of electrons… “ Axel Becke Quick introduction to ARPES ARPES (Angle-Resolved Photoemission Spectroscopy) is an experimental method to study the motions of electrons. Light is used to eject electrons from the material and they are counted as a function of direction, photon- and kinetic energy. Sketch of the photoemission process. Photons with energy hν are shone on the sample surface. Electrons are leaving with different kinetic energies (E) and are detected in a particular direction defined by two angles θ and φ. Detected number of electrons can be plotted as a function of these four parameters in a number of ways: a) Simplest dataset of the angle-resolved method, showing how intensity of photocurrent depends on direction when photon energy and kinetic energy are kept constant. Angle θ is chosen as an example. Any path in the directional space ( θ, φ ) can be considered. b) Another basic spectrum at constant photon energy – kinetic energy distribution in particular direction. It usually has an upper cutoff ( E[max] ) in energy. c) Two-dimensional distribution in angular space, often recorded at maximal ( E[max] ) kinetic energy. Can be obtained from a) by scanning φ. d) Intensity as a function of energy, as in b), but along arbitrary path in the directional space. e) The same, but photon energy is variable and kinetic one is fixed. f) This dataset is usually recorded at the normal emission ( θ, φ = 0 ). g) Three-dimensional distribution of intensity. Can be obtained from e) by scanning φ. h) Another 3D plot which can be obtained from d) by scanning φ. Such intensity distributions provide direct access to the Fermi surface and electronic structure of the studied material. They can be used to calculate physical properties, identify chemical composition or characterize the quality of the surface. We offer a simple and direct way to detect electrons using our FeSuMa spectrometer. From ARPES data to electronic structure of solids The motions of electrons in the materials are best described by quantum mechanics, where they are treated like waves. Their crystal momentum is defined by the wave vector ( k ) whereas their energy ( E[B ]) is counted from the maximal possible energy of the electron in the crystal at zero temperature. Thus, electronic structure of the material can be represented by the probability to find electron in the state with a given (k[x], k[y], k[z]) and E[B]. In order to relate ARPES intensity to this probability, one needs to find out how the crystal momentum and energy of the electron in the solid change after absorption of photon and transmission through the surface. Energy E[B] is typically determined by subtracting experimentally determined cutoff energy E[max] from kinetic energy. Conversion from angles and kinetic energy to momentum components and binding energy. Momentum behaves similar to the optical refraction: parallel to the surface component of the wave vector is conserved, and perpendicular component can be calculated assuming a free-electron-like state of electron in the crystal after absorbing the photon. Figure below shows which portions of momentum-energy space are probed by ARPES datasets. Portions of the momentum-energy space probed by ARPES datasets. If the energy is kept constant, the interpretation of the ARPES intensity is relatively simple. The top row of panels shows that the angular distributions only have to be recalculated in the momentum coordinates and, for example, the distorted contours of the Fermi surface become symmetrical in momentum space. On the contrary, the lower row of panels shows that the interpretation of the energy distributions is not trivial. For example, the widely used energy distribution curve (panel b) also corresponds to the range of momenta from the radius of the momentum sphere and is therefore not as intuitive as the angular distribution from panel a). If only Fermi-electrons are of interest, the usual nowadays energy-momentum cuts (panel d) probe only an arc on the IkI=const sphere, whereas 2D angular detection, as in FeSuMa, probes the whole spherical dome (panel c).	{"url":"https://fermiologics.com/arpes/","timestamp":"2024-11-04T08:05:26Z","content_type":"text/html","content_length":"57319","record_id":"<urn:uuid:462dae51-2b2a-4679-b461-58b073d1e298>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00757.warc.gz"}
Calculate Water To Drink - lands.biz.id Calculate Water To Drink Februari 08, 2022 Dipublikasikan Agustus 04, 2021 Calculate Water To Drink. The 2/3 formula for water to drink daily calculator multiply your body weight by 2/3 (or 67%) and you will get the exact idea of how much water to consume daily. The third factor that you need to consider is your activity level. Best Water Drinking Apps inspire ideas 2022 from haztesociounicef.org Keep sipping throughout the day to keep hydrated and functioning at your best. Then, we multiply that number by 0.80. Divide your weight (in pounds) by 2.2. Number Of Fluid Ounces ÷ 33.8 = Number Of Liters Top up your fluids when you exercise and when the temperature rises, and don’t wait until you are thirsty. If you weigh 79 kilos you would multiply that by 2/3 and learn, that you should be consuming about 3.5 liters of water every day. Use the calculator above to determine how much water to drink based on your activity level and weight. Divide The Result By 28.3. Once you know how much you weigh, you only have 2 steps to calculate your exact daily water intake. Second, you should multiply your weight by 2/3 to calculate how much water you need to drink on a daily basis. The outputs of our water intake calculator are in liters, milliliters, cups (equivalent to a standard glass), and ounces of water. Divide That Figure (In Pounds). More active people need to drink more water. The result you get is the ounces of water you must drink per day. This calculator is just a guide to help you know how much water to drink. Our Fluid Calculator Uses The Common Recommendation Of 2/3 Of Your Body Weight In Ounces. The third factor that you need to consider is your activity level. 200 x 2/3 = 150 ounces; How much water should i drink a day in liters? It Is Very Important Since You Sweat And Expel Water In The Process Of Exercising, So You Need To Compensate The Lost Amount Of Water. How much water should i drink based on my weight? About 15.5 cups (3.7 liters) of fluids a day for men. For example, if you weigh 160 pounds, the calculation will be as follows: Posting Komentar	{"url":"https://www.lands.biz.id/2021/08/calculate-water-to-drink.html","timestamp":"2024-11-12T09:41:30Z","content_type":"application/xhtml+xml","content_length":"472604","record_id":"<urn:uuid:b5d12119-7bf8-4a14-8670-38aa726f23d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00393.warc.gz"}
A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help - A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help. A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help. (/0x4*br /> Because he is flying into a head wind, the trip takes him 2 hours. On the way back, the wind is still blowing at the same speed, so the return trip takes only 1 h 15 min. speed in still air mphwind speed mph A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help[supanova_question] A matrix is given.A matrix is given. Mathematics Assignment Help (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. A matrix is given.A matrix is given. Mathematics Assignment Help (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. i need the actual answer thank you so much Mathematics Assignment Help vertices of ΔABC are A(-3, -2), B(2, 3), and C(5, -4). ΔA’B’C’ is the image of ΔABC after a dilation of 2. The area of ΔA’B’C’ = square units. A matrix is given.A matrix is given.A matrix is given. Mathematics Assignment Help (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. A matrix is given. 1 0 0 0 0 0 0 0 0 1 2 1 Mathematics Assignment Help A matrix is given. (c) Write the system of equations for which the given matrix is the augmented matrix. (Enter each answer in terms of x, y, and z.) A matrix is given. 1 0 0 0 0 0 0 0 0 1 2 1 Mathematics Assignment Help[supanova_question] The system of linear equations has a unique solution. Find the solution using Ga Mathematics Assignment Help The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. x + y + z = 6 2x − 3y + 2z = −8 4x + y − 3z = 5 The system of linear equations has a unique solution. Find the solution using Ga Mathematics Assignment Help The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. x + y + z = 12 −x + 2y + 3z = 44 2x − y = −16 (c) Write the system of equations for which the given matrix is the augmented ma Mathematics Assignment Help A matrix is given. (c) Write the system of equations for which the given matrix is the augmented matrix. (Enter each answer in terms of x, y, z, w, and v.) (a) Determine whether the mat Mathematics Assignment Help (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help A man flies a small airplane from Fargo to Bismarck, North Dakota—distance 180 Mathematics Assignment Help	{"url":"https://anyessayhelp.com/a-man-flies-a-small-airplane-from-fargo-to-bismarck-north-dakota-distance-180-mathematics-assignment-help/","timestamp":"2024-11-10T18:20:28Z","content_type":"text/html","content_length":"150737","record_id":"<urn:uuid:1c9c1225-3cc5-425f-a087-71a0861a9c83>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00068.warc.gz"}
algebraic expressions Algebraic Expressions/CBSE Class 7 Mathematics /Worksheet is about the Model Questions that you can expect for Yearly Examination. Here you can find out practice problems for CBSE Class 7 Mathematics. Algebraic Expressions /CBSE Class... CBSE Class 7 Mathematics/Algebraic Expressions/Model Questions CBSE Class 7 Mathematics/Algebraic Expressions is about the Model Questions that you can expect for Yearly Examination. Here you can find out practice problems for Class 7 Mathematics. Class 7 Maths Algebraic Expressions Important Questions For Exam IMPORTANT QUESTIONS FOR CBSE EXAMINATION \| CLASS 7 MATHEMATICS ALGEBRAIC EXPRESSIONS – Chapter 12 Answer the following (2 marks) 1. Identify, in the following expressions, terms which are not constants. Give their numerical coefficients: ab... Algebraic Expressions and Identities – Chapter 9 / MCQ CBSE CLASS 8 MATHEMATICS Algebraic Expressions and Identities – Chapter 9 /MCQ Multiple Choice Questions Choose the correct answer from the options given below: 1. The numerical factor of a term is called its... CBSE Class 7 Maths Algebraic Expressions Chapter 12 of CBSE Class 7 Maths is about Algebraic Expressions. You can find textbook solutions of this chapter below.	{"url":"https://www.learnmathsonline.org/tag/algebraic-expressions/","timestamp":"2024-11-01T22:28:19Z","content_type":"text/html","content_length":"68198","record_id":"<urn:uuid:f19a4435-5acd-4496-932d-57ea1931c48e>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00148.warc.gz"}
B.Sc. CSIT Entrance Model Question with Answer \| Physics – ICT BYTE B.Sc. CSIT Entrance Model Question with Answer \| Physics Last Updated on by ICT BYTE Which of the following pair have same dimension? (a) L/R and CR (b) LR and CR (c) R/L and [LC]1/2 (d) CR and 1/LC The dimensions of physical quantities are represented by different units, such as length (L), time (T), mass (M), etc. In order for two quantities to have the same dimension, they must have the same combination of these fundamental units. Let’s analyze each option: (a) L/R and CR: L/R represents the unit of inductance divided by resistance, while CR represents the unit of capacitance times resistance. These two quantities have different dimensions, so they do not have the same dimension. (b) LR and CR: LR represents the unit of inductance times resistance, while CR represents the unit of capacitance times resistance. These two quantities have the same dimension, as both involve the combination of resistance and inductance. Therefore, option (b) is correct. (c) R/L and [LC]1/2: R/L represents the unit of resistance divided by inductance, while [LC]1/2 represents the square root of the product of capacitance and inductance. These two quantities have different dimensions, so they do not have the same dimension. (d) CR and 1/LC: CR represents the unit of capacitance times resistance, while 1/LC represents the inverse of the product of capacitance and inductance. These two quantities have different dimensions, so they do not have the same dimension. Therefore, the correct answer is (b) LR and CR. 26. A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm, the further distance travelled before coming to the rest is (a) 4 cm. (b) 2 cm. (c) 3 cm. (d) 1 cm. 26. The horizontal range is 4√3 times the maximum height in a projectile motion. The angle of projection is (a) 15o . (b) 45o . (c) 30o . (d) 60o To solve this problem, let’s assume that the angle of projection is θ, the maximum height reached is H, and the horizontal range is R. The horizontal range, R, of a projectile is given by the formula: R = (v^2 * sin(2θ)) / g where v is the initial velocity of the projectile and g is the acceleration due to gravity. The maximum height, H, reached by the projectile is given by the formula: H = (v^2 * sin^2(θ)) / (2g) Given that the horizontal range is 4√3 times the maximum height, we can write the equation: R = 4√3 * H Substituting the formulas for R and H, we have: (v^2 * sin(2θ)) / g = 4√3 * [(v^2 * sin^2(θ)) / (2g)] Simplifying the equation: sin(2θ) = 2√3 * sin^2(θ) Using the identity: sin(2θ) = 2sin(θ)cos(θ), we can rewrite the equation as: 2sin(θ)cos(θ) = 2√3 * sin^2(θ) Dividing both sides by 2sin(θ), we get: cos(θ) = √3 * sin(θ) Dividing both sides by sin(θ), we get: cot(θ) = √3 Using the table of trigonometric values, we find that the angle whose cotangent is √3 is 30°. Therefore, the correct answer is (c) 30°. 26. The ‘net force acting on a body is zero’ then the wrong statement is (a) body is at rest. (b) acceleration is not zero. (c) body is in motion. (d) Acceleration should be zero. The correct answer is (d) Acceleration should be zero. According to Newton’s second law of motion, the net force acting on a body is equal to the product of its mass and acceleration (F = ma). If the net force on a body is zero, it means that the forces acting on the body are balanced, and there is no acceleration. Therefore, when the net force acting on a body is zero, the correct statement is that the body is either at rest (option a) or in motion with constant velocity (since acceleration is zero). Option (c) is incorrect because it implies that the body is in motion with changing velocity, which would require a non-zero net force. 26. If length of wire is doubled keeping the diameter constant, it’s Young’s modulus will (a) increases. (b) decreases. (c) remain same. (d) depend upon nature of matter. The correct answer is (c) remain the same. Young’s modulus is a measure of the stiffness or elasticity of a material and is defined as the ratio of stress to strain within the elastic limit of the material. It is a property of the material itself and is independent of the dimensions of the sample. When the length of a wire is doubled while keeping the diameter constant, the cross-sectional area of the wire remains the same. As Young’s modulus is determined by the material’s intrinsic properties and not its dimensions, doubling the length of the wire does not affect its Young’s modulus. Therefore, the Young’s modulus will remain the same in this case. 26. The work done to blow a soap bubble of radius ‘R’ is W, then work done to increase the radius from R to 3R is (a)2 W. (b) 8 W. (c) 4 W. (d) 9 W. The correct answer is (b) 8 W. The work done to blow a soap bubble of radius R can be expressed as the change in surface energy of the bubble. Let’s assume the surface tension of the soap solution is represented by the symbol ‘S’. The surface area of a sphere is given by A = 4πR^2. So, the initial surface energy of the bubble is E = 4πR^2S. To increase the radius from R to 3R, the final surface area of the bubble will be A’ = 4π(3R)^2 = 36πR^2. The change in surface energy ΔE is given by ΔE = E’ – E, where E’ is the final surface energy. ΔE = 36πR^2S – 4πR^2S = 32πR^2S The work done to increase the radius from R to 3R is equal to the change in surface energy, so the work done is W = 32πR^2S. Now, if we substitute R with 3R, we get: W’ = 32π(3R)^2S = 288πR^2S Comparing this with the initial work done W, we can see that W’ is 8 times the initial work W. Therefore, the work done to increase the radius from R to 3R is 8 times the initial work, which is 8W. Hence, the correct answer is (b) 8 W. 26. A metallic ball is immersed in alcohol. The coefficient of cubical expansion of metal is less than that of alcohol. When the system is heated weight of ball is (a) increases. (b) remains unchanged (c) decreases. (d) First increases and then The correct answer is (c) decreases. When a metallic ball is immersed in alcohol and the system is heated, both the metal ball and the alcohol expand due to the increase in temperature. However, since the coefficient of cubical expansion of the metal is less than that of alcohol, the metal ball will expand less compared to the alcohol. As the metal ball expands less than the alcohol, its volume increases at a slower rate. According to Archimedes’ principle, the buoyant force experienced by an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Since the volume of the metal ball increases at a slower rate than the alcohol, the increase in buoyant force experienced by the metal ball will be less compared to the alcohol. As a result, the weight of the metal ball will decrease when the system is heated. This is because the decrease in the effective weight of the ball due to the reduced buoyant force outweighs the increase in weight caused by the expansion of the ball itself. Therefore, the correct answer is (c) decreases 26. Latent heat of a substance is zero at (a) boiling point. (b) critical temperature. (c) melting point. (d) freezing point. The correct answer is (c) melting point. The latent heat of a substance refers to the amount of heat energy required or released during a phase change without a change in temperature. It is the heat energy associated with the transformation of a substance from one phase to another (e.g., solid to liquid or liquid to gas) at a constant temperature. At the melting point, a substance transitions from a solid phase to a liquid phase. During this phase change, the substance absorbs heat energy from its surroundings without a change in temperature. This heat energy is used to overcome the intermolecular forces holding the particles together in the solid phase, allowing them to move more freely in the liquid phase. The latent heat of fusion, which is the heat energy absorbed during the melting process, is not zero at the melting point. It is the heat energy required to convert a substance from a solid to a liquid phase at its melting point. Therefore, the correct answer is (c) melting point. 26. The average Kinetic Energy per degree of freedom per molecule of an ideal gas is (a) KT. (b) 2KT. (c) ½ KT. (d) ¾ KT. B.Sc. CSIT Entrance Examination Model Question 3 27. Two spheres of same material have radii in the ration 3:2. The heat radiated by them at the same temperature will be (a) 1:1. (b) 4:9. (c) 9:4. (d) 3:2. 28. Light of wavelength 550 nm falls normally on a slit of width 22 × 10-7m, the angular position of second minima from central maxima will be . (b)300 . (c) 420 . (d) 620 29. A person is in a room whose ceiling and two adjacent walls are mirrors. Number of images formed of an object is (a) 5. (b) 7. (c) 6. (d) 8. 30. The refractive index is 1.414 and refracting angle is 60o , then minimum deviation of light will be (a) 30o . (b) 60o . (c) 45o . (d) 72o 31. A sound wave has frequency 500 Hz and velocity 360 m/s. What is the distance between 2 particles having phase difference 600 (a) 0.7 cm (b) 70 cm (c) 1.2 cm (d) 12 cm 32. Two fixed charges q and 4q are at r distance apart. What will be position of third charge to be placed so that the system will be in equilibrium? (a)2r/3 from 4q (b )2r from q (c) r/2 from q (d) r/2 from 4q 33. n-equal capacitors are first connected in series and then in parallel. The ratio of maximum to minimum capacitance is (a) n2 . (b) 1/n2 . (c)n. (d) 1/n. 34. A heater coil is cut into two equal parts and only one part is used in the heater. How will the heat generated vary? (a) One fourth (b) Doubled (c) Halved (d) Four times 35. A 50 V battery is connected across 10 Ohm resistor. The current in the circuit is 4.5 Ampere. The internal resistance of the battery should be (a) zero (b) 5.0 Ohm (c) 0.5 Ohm (d) 1.1 Ohm 36. A magnetic needle kept in a non-uniform magnetic field. It experiences (a) a torque but not a force (b)a force and a torque (c) neither a force nor a torque (d) a force but not a torque 37. In LCR circuit, the inductive reactance at resonance frequency is 100 Hz and resistance is 5 Ohm, the quality factor of the circuit is (a) 5000. (b)500. (c) 20. (d) 95. 38. A circuit contains a capacitor of 420 Pf and an inductance L. The value of ‘L’ to broadcast on Radio at 1020 kHz is (a) 2.8 x 10-5 H. (b) 7.6 x 10-5 H. (c) 5.8 x 10-5 H. (d) 9.6 x 10-6 H. 39. Electron accelerated from rest to a potential difference of 100 volt, its final velocity will be (a) 5 x 105 m/s. (b) 3 x 106 m/s. (c) 4 x 105 m/s. (d) 6 x 106 m/s. 40. When a proton collides with an electron, which of the following characteristics of proton increases? (a) Energy (b) Wavelength (c) Frequency (d) Impulse 41. The half life of a radioactive sample is 10 years. Its mean life is (a) 12.43 years. (b) 16.43 years. (c) 14.43 years. (d) Same as half life. 42. NPN transistors are most preferred than that of PNP transistor. It is because of (a) low cost. (b) capable of handling low power. (c) Low dissipation of energy. (d)high mobility of electrons than holes.	{"url":"https://ictbyte.com/b-sc-csit/b-sc-csit-entrance-model-question-with-answer-physics/","timestamp":"2024-11-09T06:37:42Z","content_type":"text/html","content_length":"164254","record_id":"<urn:uuid:7828d192-c8e9-46da-b402-957e9a553964>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00720.warc.gz"}
ISNA Function in Excel - Checking for #N/A Errors Home » Functions ISNA Function in Excel – Checking for #N/A Errors Many times, while searching for a value in the spreadsheet we have to deal with #N/A errors. Every time the function like VLOOKUP, HLOOKUP, MATCH, or XLOOKUP formula cannot find the results, it returns a #N/A error. For dealing with such #N/A errors in excel, the excel ISNA function becomes very useful. In this tutorial, we would uncover the ISNA formula in excel and how it works with examples. Here we go 😎 When to Use ISNA Function of Excel The expression ISNA represents “Hey! Is it a #N/A error?”. The ISNA formula of Excel returns a logical value ‘TRUE’ or ‘FALSE’ based on the condition that whether the supplied argument returns #N/A ISNA Formula belongs to the Information functions group of Excel. Syntax and Arguements The below point explain the required function argument for the ISNA function of Excel. • value – This argument could be a cell reference, range of cells or a formula result in which we want to check the presence of #N/A error. Points To Remember About ISNA Formula One should keep the following points in mind before actually using the ISNA function of Excel. • The ISNA Formula does not return any error, whatever be the function argument. • There can be only two possible output of the excel ISNA formula – TRUE or FALSE. • We can use the ISNA Function with IF formula to modify and replace the #N/A error message with some other message. • We can also pass the direct formula as the function argument like this =ISNA(A2/B2) and the formula will return a TRUE/FALSE if the formula result is a #N/A error. Examples – Actual Usage of ISNA Excel Function In this section of the blog, we will have a glance at some of the examples to learn how to use the Excel ISNA formula to find if #N/A exist or not. Ex.1 – Finding #N/A errors in excel – ISNA Formula In this example, we will be experimenting with different values and check the ISNA function results. Have a look at below values in column A. To find if a cell contains #N/A error, simply use the following formula. As a result, the function returns a logical TRUE in cell B2. Copy the formula to the remaining cells in column B or use the Excel Fill-handle tool, to copy the formula to cells below. Explanation – We have passed cell A2 as the function argument in the ISNA function. The function interpreted that the cell contains a #N/A error and thus returned a logical TRUE as its result. It is important to note that the cells A3 and A5 also contain errors, but they are not #N/A error. Therefore, the function returned a logical FALSE for those values in cells A3 and A5. Similarly, cells A4 and A6 contain a text string (i.e. not #N/A error code), and therefore formula returns FALSE. To check if the argument returns error (other than #N/A), use the ISERR or ISERROR formula in excel. Ex.2 – Representing #N/A Error Using IF formula with ISNA Function Below image contains details about the car brand and quantity available for renting. Suppose, we want to check for the availability of the car for a particular brand. To achieve that we would be using the VLOOKUP formula, like this: As a result, the formula has returned the quantity of Suzuki Alto Available in cell B9. Now, what if we look for a car that is not available in the list of cars, say “Honda Brio”. As a result, the VLOOKUP function returns a #N/A excel error code. This is because the car that we are looking for is not available in the list. In such a scenario, the error message #N/A looks very weird. The error code is not self-explanatory in itself and may confuse a novice excel user as to what does this #N/A denote. To mitigate this, we can use the ISNA excel formula in conjunction with the VLOOKUP and IF formula and turn the error message code in to something more meaningful. =IF(ISNA(B12),"Car not available","There are "&B12&" cars available") =IF(ISNA(=VLOOKUP(A9,A2:B6,2,0)),"Car not available","There are "&B12&" cars available") This time the formula returns a text string “Car not available” when we look for a value that is not in the lookup array. Let us see for a car that is available. i.e Tata Nano This time the function returns the text string merged with the number of cars like this “There are 7 cars available”. Explanation – The condition of the IF formula is ISNA(B12). The condition will return a TRUE if the Lookup function in cell B12 returns a #N/A error ( lookup value not found ). If the number of cars is successfully returned in cell B12, then the condition becomes FALSE. When the condition is TRUE ( value not found) then the function returns a text string “Car not available” or else the condition is FALSE and the function returns “There are “&B12″ cars available”. We have merged the text before and after the B12 (number of cars) using a glue called ampersand (&). This brings us to the end of the ISNA Function blog. Thank you for reading 😉 Leave a Comment	{"url":"https://excelunlocked.com/isna-function-in-excel","timestamp":"2024-11-04T05:30:36Z","content_type":"text/html","content_length":"229475","record_id":"<urn:uuid:0c095104-e7fc-402a-bc15-9cf79cfc7410>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00882.warc.gz"}
Math Is Fun Forum Registered: 2010-06-20 Posts: 10,610 Re: Pi is fun Registered: 2005-06-28 Posts: 48,328 Re: Pi is fun Hi blu£, Welcome to the forum! It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel. Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking. Re: Pi is fun So only 1/32 people like pi in my friend group Circumference= Pi X Diameter Pi= Circumference / Diameter anyone else like PI Life goes on and on and on and on and on and on....... Re: Pi is fun Circumference= Pi X Diameter Pi= Circumference / Diameter anyone else like PI Life goes on and on and on and on and on and on....... Pi is fun I did the math wrong that's why it might say edited- I hope it's right now Circumference= Pi X Diameter Pi= Circumference / Diameter anyone else like PI Last edited by blu£ (2023-01-19 07:40:59) Hi blu£ Welcome to the forum. Hope you don't mind but I corrected the Pi formula in your posts above. It doesn't matter whether a person likes pi. Pi goes on existing anyway. Even alien beings with different methods of arithmetic will have the chance to know about pi. My favourite bit of maths is: There's a lot of more advanced maths in that. It would take me a few posts to explain it but I will if you wish. Best wishes, Children are not defined by school ...........The Fonz You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei Sometimes I deliberately make mistakes, just to test you! …………….Bob Re: Pi is fun Thank you Circumference= Pi X Diameter Pi= Circumference / Diameter anyone else like PI Life goes on and on and on and on and on and on....... Registered: 2019-05-17 Posts: 21 Re: Pi is fun i know pi to 101 digits! i can rattle it off extremely rapid. Registered: 2005-06-28 Posts: 48,328 Re: Pi is fun I only 34 digits. It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel. Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking. From: Aleppo-Syria Registered: 2018-08-10 Posts: 238 Re: Pi is fun As the song of B.....s, "All I need is... pi=3.14" Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts). But only a human may have the freedom and ability to oppose his natural robotic nature. But, by opposing it, such a human becomes no more of this world.	{"url":"https://mathisfunforum.com/viewtopic.php?pid=430316","timestamp":"2024-11-08T04:50:15Z","content_type":"application/xhtml+xml","content_length":"19168","record_id":"<urn:uuid:cdd036be-3ad9-4fee-91fd-cf5973d20ea1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00304.warc.gz"}
Approximating the directed path partition problem Given a digraph G=(V,E), the k-path partition problem aims to find a minimum collection of vertex-disjoint directed paths, of order at most k, to cover all the vertices. The problem has various applications. Its special case on undirected graphs is NP-hard when k≥3, and has received much study recently from the approximation algorithm perspective. However, the general problem on digraphs is seemingly untouched in the literature. We fill the gap with the first k/2-approximation algorithm, based on a novel concept of enlarging walk to minimize the number of singletons. Secondly, for k=3, we define a second novel kind of enlarging walks to greedily reduce the number of 2-paths in the 3-path partition and propose an improved 13/9-approximation algorithm. Lastly, for any k≥7, we present an improved (k+2)/3-approximation algorithm built on the maximum path-cycle cover followed by a careful 2-cycle elimination process. • Alternating walk • Approximation algorithm • Digraph • Enlarging walk • Path partition • Path-cycle cover Dive into the research topics of 'Approximating the directed path partition problem'. Together they form a unique fingerprint.	{"url":"https://scholars.georgiasouthern.edu/en/publications/approximating-the-directed-path-partition-problem","timestamp":"2024-11-03T11:04:24Z","content_type":"text/html","content_length":"53742","record_id":"<urn:uuid:a2e46809-a356-4068-adfb-85b9c6788394>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00849.warc.gz"}
O'Reilly® Think Stats, 2nd Edition: Exploratory Data Analysis in Python - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials O'Reilly® Think Stats, 2nd Edition: Exploratory Data Analysis in Python • Title: Think Stats, 2nd Edition: Exploratory Data Analysis in Python • Author(s) Allen B. Downey, et al. • Publisher: O'Reilly Media; 2 edition (November 7, 2014) • License(s): CC BY-NC 4.0 • Paperback: 226 pages • eBook: HTML and PDF (242 pages, 1.8 MB) • Language: English • ISBN-10: 1491907339 • ISBN-13: 978-149190733 • Share This: Book Description If you know how to program, you have the skills to turn data into knowledge, using tools of probability and statistics. This concise introduction shows you how to perform statistical analysis computationally, rather than mathematically, with programs written in Python. You'll work with a case study throughout the book to help you learn the entire data analysis process from collecting data and generating statistics to identifying patterns and testing hypotheses. Along the way, you'll become familiar with distributions, the rules of probability, visualization, and many other tools and concepts. About the Authors Reviews, Rating, and Recommendations: Related Book Categories: Read and Download Links:Similar Books: Book Categories Other Categories Resources and Links	{"url":"https://freecomputerbooks.com/Think-Stats-2nd-Edition.html","timestamp":"2024-11-09T06:40:58Z","content_type":"application/xhtml+xml","content_length":"35375","record_id":"<urn:uuid:8750f72f-bb18-45e1-ae25-7f7269291ee9>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00698.warc.gz"}
The Hymn Board During services at a certain church, there are up to five different hymns sung from the hymn book. The hymn book contains 800 hymns, numbered consecutively from 1. The hymn numbers are displayed on a hymn board by sliding in plates; each plate has digits printed on it. Each blank plate costs $1.00, and printing a digit on a plate costs 25¢. The church wants to purchase enough plates so that they are able to display every possible combination of hymns, while spending the minimum amount possible. How can they achieve the largest savings and what is the minimum amount they will have to spend? This question is more a test of your creative thinking skills than your knowledge of advanced mathematical concepts. The answer is on the answer page. If this question seems a little difficult and you would like to try a warm-up question first, try How Many Days 'Til Christmas?.	{"url":"http://mathlair.allfunandgames.ca/hymnboard.php","timestamp":"2024-11-13T08:29:14Z","content_type":"text/html","content_length":"3848","record_id":"<urn:uuid:f6b30d23-3c7b-4200-b04e-bed1512f4e01>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00492.warc.gz"}
Adding and Subtracting Polynomials Calculator - Online Adding and Subtracting Polynomials Calculator A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. Adding and Subtracting Polynomials Calculator 'Cuemath's Adding and Subtracting Polynomials Calculator' is an online tool that helps to calculate the sum and difference of two polynomials. What is Adding and Subtracting Polynomials Calculator? Cuemath's online Adding and Subtracting Polynomials Calculator helps you to calculate the sum and difference of two polynomials within a few seconds. How to Use Adding and Subtracting Polynomials Calculator? Please follow the below steps to find the sum and difference of two polynomials: • Step 1: Enter the polynomial 1 and polynomial 2 in the given input box. • Step 2: Click on the "Calculate" button to find the sum and difference of two polynomials. • Step 3: Click on the "Reset" button to clear the fields and find the sum and difference of two different polynomials. How to Find Adding and Subtracting Polynomials? "A polynomial is defined as a type of expression in which the exponents of all variables should be a whole number. To add or subtract two polynomials, add or subtract the coefficients of variables having the same power Let's look at the solved example to understand briefly. Solved Example: Find the sum and difference of 1 + 3x^2 and 2x^2 + 5x^4 = (1 + 3x^2) + (2x^2 + 5x^4) = 1 + (3 + 2)x^2 + 5x^4 [First add the coefficients having the same power of x] = 1 + 5x^2 + 5x^4 Therefore, the sum of two polynomials is 1 + 5x^2 + 5x^4. = (1 + 3x^2) - (2x^2 + 5x^4) = 1 + (3 - 2)x^2 + 5x^4 [First subtract the coefficients having the same power of x] = 1 + x^2 + 5x^4 Therefore, the difference between the two polynomials is 1 + x^2 + 5x^4. Similarly, you can use the calculator to find the sum and difference of the given polynomials: • 9x^2 + 7 and 2y^4 + x^2 • 4x^2 + 10y^3 and 2y^3 + x^2 Math worksheets and visual curriculum	{"url":"https://www.cuemath.com/calculators/adding-and-subtracting-polynomials-calculator/","timestamp":"2024-11-04T17:23:37Z","content_type":"text/html","content_length":"205774","record_id":"<urn:uuid:5ec3a816-707f-4aaf-b976-c7c5844b1034>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00024.warc.gz"}
a Balloon Calculate Liters in a Balloon Calculator how many liters fit into a ball or balloon of a certain size and how big the content is at a certain fill level. Mathematically, a balloon or ball is a sphere. There are other balloon shapes, but calculating those volumes is extremely complicated. The size of a perfectly round balloon is given by the radius or diameter, measured inside the balloon. Or you measure outside and subtract twice the wall thickness. Example: a ball with a radius of 5 cm, i.e. a diameter of 10 cm, has a volume of 0.5236 liters. Please enter any value, the other two values will be calculated. If the balloon is only partially filled, then the volume has the shape of a spherical cap. Example: a ball with a radius of 5 cm and a filling height of 8 cm has a volume of 469.1 milliliters. Please enter the radius or diameter and the fill level, the other values will be calculated. The filling height must not exceed the diameter. If it is the diameter, then the balloon is full. This balloon with a diameter of 16 centimeters holds more than two liters. If it is filled to ten centimeters, then it contains almost one and a half liters. Here you can convert metric volume units into customary and imperial units Jumk.de Webprojects Online Calculators \| German: Liter berechnen Contact & Privacy ↑ up ↑	{"url":"https://rechneronline.de/litre/balloon.php","timestamp":"2024-11-01T23:13:00Z","content_type":"text/html","content_length":"11248","record_id":"<urn:uuid:8c35e9c1-a3ec-455f-bb5f-3a3fc5ad8dbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00306.warc.gz"}
The class General_polygon_with_holes_2 models the concept GeneralPolygonWithHoles_2. It represents a general polygon with holes. It is parameterized with a type Polygon used to define the exposed type Polygon_2. This type represents the outer boundary of the general polygon and each Template Parameters Polygon_ must have input and output operators. template<typename Polygon_ > std::ostream & operator<< ( std::ostream & os, const General_polygon_with_holes_2< Polygon_ > & p related This operator exports a General_polygon_with_holes_2 to the output stream out. An ASCII and a binary format exist. The format can be selected with the CGAL modifiers for streams, set_ascii_mode(0 and set_binary_mode() respectively. The modifier set_pretty_mode() can be used to allow for (a few) structuring comments in the output. Otherwise, the output would be free of comments. The default for writing is ASCII without comments. The number of curves of the outer boundary is exported followed by the curves themselves. Then, the number of holes is exported, and for each hole, the number of curves on its outer boundary is exported followed by the curves themselves. template<typename Polygon_ > std::istream & operator>> ( std::istream & is, General_polygon_with_holes_2< Polygon_ > & p related This operator imports a General_polygon_with_holes_2 from the input stream in. Both ASCII and binary formats are supported, and the format is automatically detected. The format consists of the number of curves of the outer boundary followed by the curves themselves, followed by the number of holes, and for each hole, the number of curves on its outer boundary is followed by the curves themselves.	{"url":"https://doc.cgal.org/5.5.2/Polygon/classCGAL_1_1General__polygon__with__holes__2.html","timestamp":"2024-11-14T08:37:18Z","content_type":"application/xhtml+xml","content_length":"17330","record_id":"<urn:uuid:43602274-d1aa-4299-8d91-92c8f4f90ac9>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00279.warc.gz"}
matematicasVisuales \| Standard Paper Size DIN A The size of the standard paper DinA The paper we usually use has a standar size. In lots of countries in the world (but not in North America) we use paper size standars based in ISO 216 and we use world like DIN A0, DIN A1, DIN A2, DIN A3, DIN A4 an so on. The base DIN A0 size of paper is defined to have an area of one square meter, and successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension. The objective is that this parts again have the same aspect ratio. We can calculate this aspect ratio: The aspect ratio verifies (these two rectangles are similar): Then the larger side is equal to the diagonal of a square of size the shorter side: DIN A0 size has one square meter. We can calculate his dimensions (rounded to milimeters) In a photocopier, when we want to reduce from A3 to A4 the display shows a ratio of 71%. ¿Why? I have used this proportion in the animation about the sum of the geometric series of ratio 1/2. The doors of this piece of furniture are in the same proportion. It has been designed and made by Roberto Cardil using pine and oak wood. You can see another furniture with the golden spiral. This proportion is different than the golden proportion. Now we are going to see more facts about this rectangle. Remember that the diagonal of a square is: Then we can find our rectangle as a section of a cube: One diagonal of this rectangle: You can calculate D as a basic application of the Pythagorean Theorem: If we consider the two diagonals of this section, the point of intersection is the center of the cube: Now we are going to study the angles between the two diagonals (we need some basic knowledge about trigonometry): Angle C is easy to calculate: We are going to meet these two angles when we study the chamfered cube and the rhombic dodecahedron because our rectangle is related with these polyhedra. A chain of six pyramids can be turned inwards to form a cube or turned outwards, placed over another cube to form the rhombic dodecahedron. You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron. Another approach to find angle A: Can you explain it? First, lines PR and QT are perpendicular because if we rotate 90 degrees counterclockwise the rectangle .... Second, lines PR and QT are two medians of triangle QRS, then the centroid divide ... Now you can write cosA ... A chain of six pyramids can be turned inwards to form a cube or turned outwards, placed over another cube to form the rhombic dodecahedron. From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio. Demonstration of Pythagoras Theorem inspired in Euclid. You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron. The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar. A golden rectangle is made of an square and another golden rectangle. A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation. One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent. You can build a Rhombic Dodecahedron adding six pyramids to a cube. This fact has several interesting consequences. The Rhombic Dodecahedron fills the space without gaps. Humankind has always been fascinated by how bees build their honeycombs. Kepler related honeycombs with a polyhedron called Rhombic Dodecahedron. We want to close a hexagonal prism as bees do, using three rhombi. Then, which is the shape of these three rhombi that closes the prism with the minimum surface area?. Adding six pyramids to a cube you can build new polyhedra with twenty four triangular faces. For specific pyramids you get a Rhombic Dodecahedron that has twelve rhombic faces. Tetraxis is a wonderful puzzle designed by Jane and John Kostick. We study some properties of this puzzle and its relations with the rhombic dodecahedron. We can build this puzzle using cardboard and magnets or using a 3D printer. Starting with a Rhombicubotahedron we can add pyramids over each face. The we get a beautiful polyhedron that it is like a star. Material for a session about polyhedra (Zaragoza, 9th May 2014). Simple techniques to build polyhedra like the tetrahedron, octahedron, the cuboctahedron and the rhombic dodecahedron. We can build a box that is a rhombic dodecahedron. A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube. Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron. A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube. A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.	{"url":"http://matematicasvisuales.com/english/html/geometry/proportion/dinA.html","timestamp":"2024-11-05T09:12:34Z","content_type":"text/html","content_length":"30525","record_id":"<urn:uuid:27fe4bea-5e63-41ac-9ced-e6ebfa873d56>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00496.warc.gz"}
doc/StorageOrders.dox - eigen - Git at Google namespace Eigen { /** \eigenManualPage TopicStorageOrders Storage orders There are two different storage orders for matrices and two-dimensional arrays: column-major and row-major. This page explains these storage orders and how to specify which one should be used. \section TopicStorageOrdersIntro Column-major and row-major storage The entries of a matrix form a two-dimensional grid. However, when the matrix is stored in memory, the entries have to somehow be laid out linearly. There are two main ways to do this, by row and by column. We say that a matrix is stored in \b row-major order if it is stored row by row. The entire first row is stored first, followed by the entire second row, and so on. Consider for example the matrix A = \begin{bmatrix} 8 & 2 & 2 & 9 \\ 9 & 1 & 4 & 4 \\ 3 & 5 & 4 & 5 If this matrix is stored in row-major order, then the entries are laid out in memory as follows: \code 8 2 2 9 9 1 4 4 3 5 4 5 \endcode On the other hand, a matrix is stored in \b column-major order if it is stored column by column, starting with the entire first column, followed by the entire second column, and so on. If the above matrix is stored in column-major order, it is laid out as follows: \code 8 9 3 2 1 5 2 4 4 9 4 5 \endcode This example is illustrated by the following Eigen code. It uses the PlainObjectBase::data() function, which returns a pointer to the memory location of the first entry of the matrix. <table class="example"> \include TopicStorageOrders_example.cpp \verbinclude TopicStorageOrders_example.out \section TopicStorageOrdersInEigen Storage orders in Eigen The storage order of a matrix or a two-dimensional array can be set by specifying the \c Options template parameter for Matrix or Array. As \ref TutorialMatrixClass explains, the %Matrix class template has six template parameters, of which three are compulsory (\c Scalar, \c RowsAtCompileTime and \c ColsAtCompileTime) and three are optional (\c Options, \c MaxRowsAtCompileTime and \c MaxColsAtCompileTime). If the \c Options parameter is set to \c RowMajor, then the matrix or array is stored in row-major order; if it is set to \c ColMajor, then it is stored in column-major order. This mechanism is used in the above Eigen program to specify the storage order. If the storage order is not specified, then Eigen defaults to storing the entry in column-major. This is also the case if one of the convenience typedefs (\c Matrix3f, \c ArrayXXd, etc.) is used. Matrices and arrays using one storage order can be assigned to matrices and arrays using the other storage order, as happens in the above program when \c Arowmajor is initialized using \c Acolmajor. Eigen will reorder the entries automatically. More generally, row-major and column-major matrices can be mixed in an expression as we want. \section TopicStorageOrdersWhich Which storage order to choose? So, which storage order should you use in your program? There is no simple answer to this question; it depends on your application. Here are some points to keep in mind: - Your users may expect you to use a specific storage order. Alternatively, you may use other libraries than Eigen, and these other libraries may expect a certain storage order. In these cases it may be easiest and fastest to use this storage order in your whole program. - Algorithms that traverse a matrix row by row will go faster when the matrix is stored in row-major order because of better data locality. Similarly, column-by-column traversal is faster for column-major matrices. It may be worthwhile to experiment a bit to find out what is faster for your particular - The default in Eigen is column-major. Naturally, most of the development and testing of the Eigen library is thus done with column-major matrices. This means that, even though we aim to support column-major and row-major storage orders transparently, the Eigen library may well work best with column-major matrices.	{"url":"https://third-party-mirror.googlesource.com/eigen/+/941ca8d83f776b9a07153d3abef2877907aa0555/doc/StorageOrders.dox","timestamp":"2024-11-02T23:21:21Z","content_type":"text/html","content_length":"25098","record_id":"<urn:uuid:83562c42-c2c7-4e80-9375-1d614ffd00c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00035.warc.gz"}
Response to the Follow-up about Pythagorean Theorem Subject: Re: Pythagorean Theorem Date: Sat, 23 Nov 1996 12:44:00 -0500 Alex Bogomolny Dear DJ13181: Your question caused me to look around through my library and search the Internet. A little of what I discovered made for a content of a paragraph I added as a Remark to my Pythagorean page. "The statement of the Theorem was discovered on a Babylonean tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or any one from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. Jim Morey's applet follows the Proposition I.47 (First Book, Proposition 47), mine VI.31. The Theorem is reversible (I.48) which means that a triangle whose sides satisfy a^2+b^2=c^2 is right angled." In every source, if you enjoy this kind of browsing when one absorbes facts and information one was not deliberately looking for, there could be something interesting to learn. cites 10 proof of the Theorem and references to the American Math Monthly ~1890. David Eppstein in The Geometry Junkyard mentions the book D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, Penguin, 1991. which, in turn, cites a book from 1940 with 367 different proofs of the Theorem. Regards and good luck in your search. \|Reply\| \|Up\| \|Exchange index\| \|Contents\| \|Store\| Copyright © 1996-2018 Alexander Bogomolny	{"url":"https://www.cut-the-knot.org/exchange/pyth2.shtml","timestamp":"2024-11-03T20:48:08Z","content_type":"text/html","content_length":"13044","record_id":"<urn:uuid:54a4189e-c631-4098-8ed0-fb29c4d34a52>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00624.warc.gz"}
Future worth analysis formula Financial calculators and spreadsheets are designed to handle financial formulas. Future Value Using a Financial Calculator. The formula for finding the future Future Value (FV) is a formula used in finance to calculate the value of a cash flow at a later date than originally received. This idea that an amount today is worth Here we learn how to calculate FV (future value) using its formula along with Standard of living, operating expenses/recurring expenses (separate analysis is The formula for future value answers these questions and tells you the estimated value of an asset in the future. After this lesson, the next time you plan to buy a new car, or a house, in a few years' time, you will have a much better answer as to how much to save, rather than just 'throwing out a number.'. The formula for the future worth of the above cash flow diagram for a given interest rate, i is FW(i) = –P(1 + i)n + R1(1 + i)n–1 + R2(1 + i)n–2 + Present worth, Annual equivalent, Future worth, Internal rate of return - Duration: 29:37. Engineering Economic Analysis 15,357 views Future Value Calculator - The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. Future Value. The future value calculator can be used to determine future value, or FV, in financing. FV is simply what money is expected to be worth in the future. Typically, cash in a savings account or a hold in a bond purchase earns compound interest and so has a different value in the future. The formula for the future value (F) of a present sum (P) is: Life Cycle Cost Analysis (LCCA) is a design process for controlling the initial and the future cost of Future payments or receipts have lower present value (PV) today than their In discounted cash flow analysis DCF, two "time value of money" terms are central: Third, example calculations showing how to discount future values to present Present Value Formulas, Tables and Calculators. The easiest and most accurate way to calculate the present value of any future amounts (single amount, 11 Mar 2020 Then you can perform a DCF analysis that estimates and discounts the value of all future cash flows by cost of capital to gain a picture of their Time value of money calculator (TVM) is a tool that helps you find the present or future calculator (TVM) is a simple tool that helps you to find out the future value of a You can find the concept of time value analysis behind many financial Future Value (FV) is a formula used in finance to calculate the value of a cash flow at a later date than originally received. This idea that an amount today is worth a different amount than at a future time is based on the time value of money. The formula for the future worth of the above cash flow diagram for a given interest rate, i is FW(i) = –P(1 + i)n + R1(1 + i)n–1 + R2(1 + i)n–2 + Present worth, Annual equivalent, Future worth, Internal rate of return - Duration: 29:37. Engineering Economic Analysis 15,357 views Future Value Calculator - The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money calculations . Present Value Formulas, Tables and Calculators. The easiest and most accurate way to calculate the present value of any future amounts (single amount, 11 Mar 2020 Then you can perform a DCF analysis that estimates and discounts the value of all future cash flows by cost of capital to gain a picture of their Time value of money calculator (TVM) is a tool that helps you find the present or future calculator (TVM) is a simple tool that helps you to find out the future value of a You can find the concept of time value analysis behind many financial For example, if you get a four-year car loan and make monthly payments, your loan has 412 (or 48) periods. You would enter 48 into the formula for nper. Pmt is Day to calculate the future value. Periodic deposit (withdrawal). The amount that you plan on adding to this savings or investment each period. Deposit frequency. The Inflation based Future Value Calculator can be used by those who are worried about the ever increasing inflation levels and would like to know the future For example, if you get a four-year car loan and make monthly payments, your loan has 412 (or 48) periods. You would enter 48 into the formula for nper. Pmt is 4 Mar 2015 Learn the risk free rate of return formula. If you know the future value and the term (number of years or periods) and the interest rate you can 4 Mar 2020 The future value formula helps you calculate the future value of an investment ( FV) for a series of regular deposits at a set interest rate (r) for a It estimates and totals the equivalent monetary value of the benefits and costs of In cost-benefit analysis, the second formula computes PV of the future cash Free future value calculator helps you to compute returns on savings accounts and other investments. Easy-to-understand charts. Powered by Wolfram\|Alpha. Finance Investment Analysis Formulas. Solving for future value or worth. note: If interest rate is 15%, enter .15 for i. Finance Investment Analysis Formulas. Solving for future value or worth. note: If interest rate is 15%, enter .15 for i. Future value (FV) is the value of a current asset at a specified date in the future based on an assumed rate of growth. If, based on a guaranteed growth rate, a $10,000 investment made today will be worth $100,000 in 20 years, then the FV of the $10,000 investment is $100,000. 14 Feb 2019 The bank could use formulas, future value tables, a financial calculator, or a spreadsheet application. The same is true for present value 5 Dec 2018 Formula: FV = PV x [ 1 + (i / n) ] ^ (n x t). (PV) Present Value = What your money is worth right now. (FV) Future Value = What your money will be 1 Mar 2018 The formula in cell B13 in the screenshot "Calculating Future Value of This analysis can show them the value of starting their retirement 28 Jan 1994 To compute the future value of a sum invested today, the formula for interest that In this analysis, rate-of-return is calculated based on monthly	{"url":"https://cryptobygo.netlify.app/garvie28266jev/future-worth-analysis-formula-sole.html","timestamp":"2024-11-10T12:54:51Z","content_type":"text/html","content_length":"32523","record_id":"<urn:uuid:53f2f15e-8c76-472c-b190-84faeaef86e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00768.warc.gz"}
Bridges Puzzle (Hashiwokakero) The Hashiwokakero Puzzle is a logical puzzle where one has to build bridges between islands. The puzzle is also known under the name Ai-Ki-Ai. The puzzles can also be played online. The requirements for this puzzle are as follows: • the goal is to build bridges between islands so as to generate a connected graph • every island has a number on it, indicating exactly how many bridges should be linked with the island • there is an upper bound (MAXBRIDGES) on the number of bridges that can be built-between two islands • bridges cannot cross each other A B model for this puzzle can be found below. The constants and sets of the model are as follows: • N are the nodes (islands); we have added a constant ignore where one can stipulate which islands should be ignored in this puzzle • nl (number of links) stipulates for each island how many bridges it should be linked with • xc, yc are the x- and y-coordinates for every island A simple puzzle with four islands would be defined as follows, assuming the basic set N is defined as N = {a,b,c,d,e,f,g,h,i,j,k,l,m,n}: xc(a)=0 & xc(b)=1 & xc(c)=0 & xc(d) = 1 & yc(a)=0 & yc(b)=0 & yc(c)=1 & yc(d) = 1 & nl = {a\|->2, b\|->2, c\|->2, d\|->2} & ignore = {e,f,g,h,i,j,k,l,m,n} Below we will use a more complicated puzzle to illustrate the B model. The model then contains the following derived constants: • plx,ply: the possible links between islands on the x- and y-axis respectively • pl: the possible links both on the x- and y-axis combined • cs: the conflict set of links which overlap, i.e., one cannot build bridges on both links (a,b) when the pair (a,b) is in cs • connected: the set of links on which at least one bridge was built The model also sets up the goal constant sol which maps every link in pl to a number indicating how many bridges are built on it. The model also stipulates that the graph set up by connected generates a fully connected graph. Here is the full model: MACHINE Bridges LINKS == 1..(MAXBRIDGES4); COORD == 0..10; p1 == prj1(nodes,nodes); p2 == prj2(nodes,nodes); p1i == prj1(nodes,INTEGER) N = {a,b,c,d,e,f,g,h,i,j,k,l,m,n} CONSTANTS nodes, ignore, nl, xc,yc, plx,ply,pl, cs, sol, connected nodes = N \ ignore & // target number of links per node: nl : nodes --> LINKS & / number of links */ // coordinates of nodes xc: nodes --> COORD & yc: nodes --> COORD & // possible links: pl : nodes <-> nodes & plx : nodes <-> nodes & ply : nodes <-> nodes & plx = {n1,n2 \| xc(n1)=xc(n2) & n1 /= n2 & yc(n2)>yc(n1) & !n3.(xc(n3)=xc(n1) => yc(n3) /: yc(n1)+1..yc(n2)-1) } & ply = {n1,n2 \| yc(n1)=yc(n2) & n1 /= n2 & xc(n2)>xc(n1) & !n3.(yc(n3)=yc(n1) => xc(n3) /: xc(n1)+1..xc(n2)-1)} & pl = plx \/ ply // compute conflict set (assumes xc,yc coordinates ordered in plx,ply) cs = {pl1,pl2 \| pl1:plx & pl2:ply & xc(p1(pl1)): xc(p1(pl2))+1..xc(p2(pl2))-1 & yc(p1(pl2)): yc(p1(pl1))+1..yc(p2(pl1))-1} sol : pl --> 0..MAXBRIDGES & !nn.(nn:nodes => SIGMA(l).(l:pl & (p1(l)=nn or p2(l)=nn)\|sol(l))=nl(nn)) & !(pl1,pl2).( (pl1,pl2):cs => sol(pl1)=0 or sol(pl2)=0) // no conflicts // check graph connected connected = {pl\|sol(pl)>0} & closure1(connected \/ connected~)[{a}] = {nn\|nn:nodes & nl(nn)>0} // encoding of puzzle // A puzzle from bridges.png xc(a)=1 & yc(a)=1 & nl(a)=4 & xc(b)=1 & yc(b)=4 & nl(b)=6 & xc(c)=1 & yc(c)=6 & nl(c)=3 & xc(d)=2 & yc(d)=2 & nl(d)=1 & xc(e)=2 & yc(e)=5 & nl(e)=2 & xc(f)=3 & yc(f)=2 & nl(f)=4 & xc(g)=3 & yc(g)=4 & nl(g)=6 & xc(h)=3 & yc(h)=5 & nl(h)=4 & xc(i)=4 & yc(i)=3 & nl(i)=3 & xc(j)=4 & yc(j)=6 & nl(j)=3 & xc(k)=5 & yc(k)=2 & nl(k)=1 & xc(l)=6 & yc(l)=1 & nl(l)=4 & xc(m)=6 & yc(m)=3 & nl(m)=5 & xc(n)=6 & yc(n)=5 & nl(n)=2 & ignore = {} The puzzle encode above can be visualized as follows: A solution for this puzzle is found by ProB in 0.08 seconds (on a MacBook Air 2.2GHz i7). The conflict set is {((d\|->e),(b\|->g)), ((i\|->j),(h\|->n))} and the value for sol is Adding graphical visualization To show the solution graphically, we can add the following custom graph to the DEFINITIONS clause in the model: CUSTOM_GRAPH_NODES == {n,w,w2\|(n\|->w):nl & w=w2}; // %n1.(n1:nodes\|nl(n1)); CUSTOM_GRAPH_EDGES == {n1,w,n2\|n1:nl & n2:nl & (p1i(n1),p1i(n2),w):sol} One can then load the model, perform the initialisation (double clicking on INITIALISATION in the operations pane) and the execute the command "Current State as Custom Graph" in the States sub-menu of the Visualize menu. This leads to the following picture: One can load the Dot file generated by ProB into another tool (e.g., OmniGraffle) and then re-arrange the nodes to obtain the rectangular layout respecting the x- and y-coordinates:	{"url":"https://prob.hhu.de/w/index.php?title=Bridges_Puzzle_(Hashiwokakero)","timestamp":"2024-11-10T17:49:11Z","content_type":"application/xhtml+xml","content_length":"17040","record_id":"<urn:uuid:7723f998-94ea-4725-b83f-f6396a6b7ad0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00141.warc.gz"}
Studies in Quality Improvement: Designing Environmental Regulations CQPI Report No. 7 S&oslashren Bisgaard and William G. Hunter Copyright &copy February 1986, Used by permission This research was supported in part by grants SES - 8018418 and DMS - 8420968 from the National Science Foundation and an award from the Graduate School, University of Wisconsin - Madison, through the University - Industry Research Program. Computing was facilitated by access to the research computer at the Department of Statistics, University of Wisconsin - Madison. Completed February 1986. Practical Significance Methods of statistical quality control have been used successfully in industry. These same methods can be applied in non - traditional contexts in which quality is to be controlled and, if possible, improved. This report considers a wider context in which industry and society exist: the environment. The quality of the environment must be protected, whether it be considered on a local, state-wide, regional, national, or international basis. One way that this problem is being handled is through the promulgation and enforcement of environmental standards by various governmental This report explains how standards can be developed on a more rational basis so that the errors that inevitably will be made during enforcement are confronted and balanced, rather than ignored. In particular, this report outlines a framework, which is based on statistical quality control concepts, for designing environmental regulations. This framework explicitly takes into account the statistical nature of environmental data. It is shown how the operating characteristic function can be useful in designing new standards and evaluating existing standards. With slight modifications, the framework outlined in this report can be applied to the design of standards of all kinds, including industrial standards. Keywords: Environmental standards, compliance, statistical quality control, operating characteristic function, type I and type II errors, environmental data, risk, ambient ozone, research needs. Public debate on proposed environmental regulations often focuses almost entirely (and naively) on the allowable limit for a particular pollutant, with scant attention being paid to the statistical nature of environmental data and to the operational definition of compliance. As a consequence regulations may fail to accomplish their purpose. A unifying framework is therefore proposed that interrelates assessment of risk aid determination of compliance. A central feature is the operating characteristic curve, which displays the discriminating power of a regulation. This framework can facilitate rational discussion among scientists, policymakers, and others concerned with environmental regulation. Over the past twenty years many new federal, state, and local regulations have resulted from heightened concern about the damage that we humans have done to the environment - and might do in the future. Public debate, unfortunately, has often focused almost exclusively on risk assessment and the allowable limit of a pollutant. Although this "limit part" of regulation is important, a regulation also include a "statistical part" that defines how compliance is to be determined even though it is typically relegated to an appendix and thus may seem unimportant, it can have a profound effect on how the regulation performs. Our purpose in this article is to introduce some new ideas concerning the general problem of designing environmental regulations, and, in particular, to consider the role of the "statistical part" of such regulations. As a vehicle for illustration, we use the environmental regulation of ambient ozone. Our intent is not to provide a definitive analysis of that particular problem. Indeed, that would require experts familiar with the generation, dispersion, measurements, and monitoring of ozone to analyze available data sets. Such detailed analysis would probably lead to the adoption of somewhat different statistical assumptions than we use. The methodology described below, however, can accommodate any reasonable statistical assumptions for ambient ozone. Moreover, this methodology can be used in the rational design of any environmental regulation to limit exposure to any pollutant. Ambient Ozone Standard For illustrative purposes, then, let us consider the ambient ozone standard (1, 2). Ozone is a reactive form of oxygen that has serious health effects. Concentrations from about 0.15 parts per million (ppm), for example, affect respiratory mucous membranes and other lung tissues in sensitive individuals as well as healthy exercising persons. In 1971, based on the best scientific studies at the time, the Environmental Protection Agency (EPA) promulgated a National Primary and Secondary Ambient Air Quality Standard ruling that "an hourly average level of 0.08 parts per million (ppm) not to be" exceeded more than 1 hour per year. Section 109(d) of the Clean Air Act calls for a review every five years of the Primary National Ambient Air Quality Standards. In 1977 EPA announced that it was reviewing and updating the 1911 ozone standard. In preparing a new criteria document, EPA provided a number of opportunities for external review and comment. Two drafts of the document were made available for external review. EPA received more than 50 written responses to the first draft and approximately 20 to the second draft. The American Petroleum Institute (API), in particular, submitted extensive comments. The criteria document was the subject of two meetings of the Subcommittee on Scientific Criteria for Photochemical Oxidants of EPA's Science Advisory Board. At each of these meetings, which were open to the public, critical review and new information was presented for EPA's consideration. The Agency was petitioned by the API and 29 member companies and by the City of Houston around the time the revision was announced. Among other things, the petition requested that EPA state the primary and secondary standards in such a way as to permit reliable assessment of compliance. In the Federal Register it is noted that EPA agrees that the present deterministic form standard has several limitations and has made reliable assessment of compliance difficult. The revised ozone air quality standards are stated in a statistical form that will more accurately reflect the air quality problems in various regions of the country and allow more reliable assessment of compliance with the standards. (Emphasis Later, in the begging of 1978, the EPA held a public meeting to receive comments from interested parties on the initial proposed revision of the standard. Here several representatives from the State and Territorial Air Pollution Program Administrators (STAPPA) and the Association of Local Air Pollution Control Officials participated. After the proposal was published in the spring of 1978, EPA held four public meetings to receive comments on the proposed standard revisions. In addition, 168 written comments were received during the formal continent period. The Federal Register summarizes the comments as follows: The majority of comments received (132 out of 168) opposed EPA's proposed standard revision, favoring either a more relaxed or a more stringent standard. State air pollution control agencies (and STAPPA) generally supported a standard level of 0.12 ppm on the basis of their assessment of an adequate margin of safety. Municipal groups generally supported a standard level of 0.12 ppm or higher, whereas most industrial groups supported a standard level 10.15 ppm or higher. Environmental groups generally encouraged EPA to retain the 0.08 ppm standard. As reflected in this statement, almost all of the public discussion of the ambient ozone standard (not just the 168 comments summarized here) focused on the limit part of the regulation. In this instance, in common with similar discussion of other environmental regulations, the statistical part of the regulation was largely ignored. The final rule-making made the following three changes: 1. The primary standard was raised to 0.12 ppm. 2. The secondary standard was raised to 0.12 ppm. 3. The definition of the point at which the standard is attained was changed to "when the expected number of days per calendar year, with maximum hourly average concentration above 0.12 ppm is equal to or less than one." The Operating Characteristic Curve Environmentalregulations have a structure similar to that of statistical hypothesis tests. A regulation states how data are to be used to decide whether a particular site is in compliance with a specified standard, and a hypothesis test states how a particular set of data are to be used to decide whether they are in reasonable agreement with a specified hypothesis. Borrowing the terminology and methodology from hypothesis testing, we can say there are two types of errors that can be made because of the stochastic nature of environmental data: a site that is really in compliance can be declared out of compliance (type I error) and vice versa (type II error). Ideally the probability of committing both types of error should be zero. In practice, however, it is not feasible to obtain this Ideal. In the context of environmental regulations, an operating characteristic curve is the probability of declaring a site to be in compliance (d.i.c.) plotted as a function of some parameter θ such as the mean level of a pollutant. This Prob (d.i.c. \| θ) can be used to determine the probabilities of committing type I and type II errors. As long as θ is below the stated standard, the probability of a type I error is 1 - Prob (d.i.c.\| θ). When θ above the stated standard, Prob (d.i.c. \| θ) is the probability of a type II error. Using the operating characteristic curve for the old and the new regulations for ambient ozone, we can evaluate them to see what was accomplished by the revision. The old standard stated that "an hourly average level of 0.08 ppm [was] not to be exceeded more than 1 hour per year." This standard was therefore defined operationally in terms of the observations themselves. The new standard, on the other hand, states that the expected number of days per calendar year with a maximum hourly average concentration above 0.12 ppm should be less than one. Compliance, however, must be determined in terms of the actual data, not an unobserved expected number. How should this conversion be made? In Appendix D of the new ozone regulation, it is stated that: In general, the average number of exceedances per calendar year must be less than or equal to 1. In its simplest form, the number of exceedances at a monitoring site would be recorded for each calendar year and then averaged over the past 3 calendar years to determine if this average is less than or equal to 1. Based on the stated requirements of compliance, we have computed the operating characteristic functions for the old and the new ozone regulations. They are plotted in Figures 1 and 2. (The last sentence in the legend for Figure 1 will be discussed below in the following section, Statistical Analysis.) To construct these curves, certain simplifying assumptions were made, which are discussed hi the section entitled "Statistical Concepts." Before such curves are used in practice, these assumptions need to be Investigated and probably modified. According to the main part of the new ozone regulation, the interval from 0 to 1 expected number of exceedances of 0.12 ppm per year can be regarded as defining "being in compliance." Suppose the decision rule outlined above is used for a site that is operating at a level such that the expected number of days exceeding 0.12 ppm is just below one. In that case, as was noted by Javitz (3), with the new ozone regulation, there is a probability of approximately 37% in any given year that such a site will be declared out of compliance. Moreover, there is approximately a 10% chance of not detecting a violation of 2 expected days per year above the 0.12 ppm limit; that is, the standard operates such that the probability is 10% of not detecting occurrences when the actual value is twice its permissible value (2 instead of 1). Some individuals may find these probabilities (37% and 10%) to be surprisingly and unacceptably high, as we do. Others, however, may regard them as being reasonable or too low. In this paper, our point is not to pursue that particular debate. Rather, it is simply to argue that, before environmental regulations are put in place, different segments of society need to be aware of such operating characteristics, so that informed policy decisions can be made. It is important to realize that the relevant operating characteristic curves can be constructed before a regulation is promulgated. Statical Concepts Let X denote a measurement from an instrument such that X = θ + ε, where = θ is the mean value of the pollutant and ε is the statistical error term with variance σ^2. The term ε contains not only the error arising from an imperfect instrument but also the fluctuations in the level of the pollutant itself. We assume that the measurement process is well calibrated and that the mean value of ε is zero. The parameters θ and σ^2 of the distribution of e are unknown but estimates of them can be obtained from data. A prescription of how the data are to be collected is known as the sampling plan. It addresses the questions of how many, where, when, and how observations are to be collected. Any function ƒ(X)= ƒ(X[1], X[2], . . . ,X[n]) of the observations is an estimator, for example, the average of a set of values or the number of observations in a sample above a certain limit. The value of the function f for a given sample is an estimate. The estimator has a distribution, which can be determined from the distribution of the observations and the functional form of the estimator. With the distribution of the estimator, one can answer questions of the form: what is the probability that the estimate ƒ= ƒ(X) is smaller than or equal to some critical value c? Symbolically this probability can be written as P = Prob {ƒ(X)≤c \| θ}. Figure 1. Operating characteristic curve for the 1971 ambient ozone standard (old standard), as a function of the expected number of hours of exceedances of 0.08 ppm per year. Note that if the old standard had been written in terms of allowable limit of one for the expected number of exceedances above 0.08 ppm, the maximum type I error would be 1.00 - 0.73 = 0.27. Figure 2. Operating characteristic curve for the 1979 ambient ozone standard (new standard), as a function of the expected number of days of exceedances of 0.12 ppm per year. Note that the maximum type I error is 1.00 - 0.63 = 0.37 If we warn to have a regulation limiting the pollution to a certain level, it is not enough to state the limit as a particular value of a parameter. We must define compliance operationally in terms of the observations. The condition of compliance therefore takes the form of an estimator ƒ(X[1], . . . ,X[n]), being less than or equal to some critical value c, that is, {ƒ (X[1], . . . ,X[n])≤ c}. Regarded as a function of θ, the probability Prob { ƒ(X[1, ]. . . ,X[n])≤c \| θ}. is therefore the probability that the site will be declared to be in compliance with the regulation. It is, in fact, the operating characteristic function. The operating characteristic function and consequently the probability of type I and type II errors are fixed by appropriate choice of the critical value and sampling plan. It is common statistical practice to specify a maximum type I error probability a and then to find a critical value c such that Prob {ƒ(X)≤c \| θ[0]} = 1 - α. To control the probability of type II errors, one would then design a sampling plan such that the probability of the type II errors is at most β for a specific value θ[1] outside the compliance region. It is important to recognize that θ[0] and c are different θ[0] is a point in the parameter space and c is a point in the sample space. Ignoring this subtle difference (which is almost always done in legal, legislative, and policymaking discussions) has led to unnecessary confusion. Because this difference exists, type I and type II errors exist. These errors should be confronted and balanced, not ignored. Statistical Analysis For purposes of illustration, let us consider the old and new regulations for ambient ozone. Let X denote the hourly average ozone level and let L be the limit, which for the old regulation was 0.08 Suppose the random variable X represents a single hourly average reading for ambient ozone that is independently and identically distributed. (This simplifying assumption is not necessary for application of this approach but it is made here for X and below for Y for ease of exposition. Similar remarks apply to the assumptions of a normal distribution and a particular value of σ[2] stated below.) Denote by [PL] =Prob {I[L](X) =1} the probability that X exceeds the limit L = 0.08 ppm. I[L](x) is the indicator function, which is one for x >L and zero otherwise. A year consists of approximately n = 365 x 12 = 4380 hours of observations (data are only taken from 9:01 am to 9:00 pm LST). The expected number of hours per year above the limit is then The probability that a site is declared to be in compliance (d.i.c.) is This probability P[old], plotted as a function of θ, is the operating characteristic curve for the old regulation (Figure 1). Note that if the old standard had been written in terms of an allowable limit of one for the expected number of exceedances above 0.08 ppm, the maximum type I error would be 1.00- 0.73 = 0.27. The old standard, however, is actually written in terms of the observed number of exceedances so type I and type II errors, strictly speaking, are undefined. The condition of compliance stated in the new regulation is that the "expected number of days per calendar yes with daily maximum ozone" concentration exceeding 0.12 ppm must be less than or equal to 1."Let Y[j], represent the daily maximum hourly average (j = 1,. . . ,365). Suppose the random variables Y[j] are independently and identically distributed. EPA proposed that the expected number of days (a parameter) be estimated by a three-year moving average of exceedances of 0.12 ppm. A site is in compliance when the moving average is less than or equal to 1. The expected number of days above the limit of L -0.12 ppm is then The three-year specification of the new standard makes it hard to compare with the previous one-year standard. If, however, one computes the conditional probability that the number of exceedances in the present year is less than or equal to 0, 1, 2 and 3 and multiplies that by the probability that the number of exceedances was 3,2,1 and 0, respectively, for the previous two years, one then obtains a one-year operating characteristic function. where k-0,1,2,3. A plot of the operating characteristic function for the new regulation, P[new] versus θ, is presented in Figure 2 . Figures 1 and 2 show the operating characteristic curves computed as a function of (1) the expected number of hours per year above 0.08 ppm for the old ambient ozone regulation and (2) the expected number of days per year with a maximum hourly observation above 0.12 ppm for the new ambient ozone regulation. We observe that the 95% de facto limit (the parameter value for which the site in a given year will be declared to be in compliance with 95% probability) is 0.36 hours per year exceeding 0.08 ppm for the old standard and 0.46 days per year exceeding 0.12 ppm for the new standard. If the expected number of hours of exceedances of 0.08 ppm is one (and therefore in compliance), the probability is approximately 26% of declaring a site to be not in compliance with the old standard. If the expected number of days exceeding 0.12 ppm is one (and therefore in compliance), the probability is approximately 37% of declaring a site to be not in compliance with the new standard. (We are unaware of any other legal context in which type I errors of this magnitude would be considered reasonable.) Note that the parameter value for which the site in a given year will be declared to be in compliance with 95% probability is 0.36 hours per year exceeding 0.08 ppm for the old standard and 0.46 days per year exceeding 0.12 ppm for the new standard. Neither curve provides sharp discrimination between "good" and "bad" values of θ. Note that the old standard did not specify any parameter value above which non-compliance was defined. The new standard, however, specifies that one expected day is the limit thereby creating an inconsistency between what the regulation says and how it operates because of the large discrepancy between the stated limit and the operational limit. The construction of Figures 1 and 2 only requires the assumption that the relevant observations are approximately identically and independently distributed (for the old standard, the relevant observations are chose for the hourly ambient ozone measurements for the new standard, they are the maximum hourly average measurements of the ambient ozone measurements each day). The construction does not require knowledge of the distribution of ambient ozone observations. If one has an estimate of this distributional form, however, a direct comparison of the new and old regulation is possible in terms of the concentration of ambient ozone (in units, say, of ppm.) To illustrate this point suppose the random variable X[i] is independently and identically distributed according to a normal distribution with mean m and variance σ^2, that is, X[i] ∼ N (µ, σ^2). Then the probability of one observation being above the limit = L = 0.08 is where Φi;( ) is the cumulative density function of the standard normal distribution. The probability that a site is declared to be in compliance can be computed as a function of μ by substituting [PL] from (4) into (1). For the new regulation let X[ij] represent the one-hour average, (i = 1, . . . , 12;j = 1, . . . ,365), and Y[j] = max {X[ij], . . . , X[i2,j]}. If X[ij] ∼ N (μ;, σ^2), then Y[j] ∼H(y) where By substituting [PL] in (2) and (3) with one obtains the operating characteristic function for the new standard. Figure 3. Operating characteristic curves for the old and new standards as a function of the mean value of ozone measured in parts per million when it is assumed that ozone measurements are normally and independently distributed with σ= 0.02 ppm For a fixed value of the variance σ^2, one can compute the operating characteristic curves for the old and new regulations to provide a graphical comparison of the way these two regulations perform. Figure 3 shows these curves for the old and new ambient ozone regulations computed as a function of the mean hourly values when it is assumed that σ = 0.02 ppm. We observe that the 95% de facto limit is changed from 0.0046 ppm to 0.045 ppm. That is, it is approximately ten times higher in the new ozone regulation. We have three observations to offer with regard to the old and new regulations for ambient ozone standards. First notwithstanding EPA's comment to the contrary, the new ozone regulation is not more statistical than the previous one; like all environmental regulations, both the new and old ozone regulations contain statistical parts, and, for that reason, both are statistical. Changing the specification from one in terms of a critical value to one in terms of a parameter does not make it more statistical. It actually introduced an inconsistency. The old standard did not specify any parameter value as a limit but only an operational limit in terms at the parameter. This therefore constitutes the standard. The new standard, however, specifies not only an intent in terms of what the desired limit is but also an operational limit. The large difference between the interned limit and the operational limit constitute the inconsistency. This inconsistency is a potential and unnecessary source of conflict. Second, the new regulation is dependent on the ambient ozone level for the past two years as well as the present year, which means that a sudden rise in the ozone level might be detected more slowly. The new regulation is also more complicated. Third, it is unwise first to record and store every single hourly observation and then to use only the binary observation as to whether the daily maximum is above or below 0.12 ppm. This procedure wastes valuable scientific information. As a matter of public policy, it is unwise to use the data in a binary form when they are already measured on a continuous scale. The estimate of the 1/365 percentile is an unreliable statistic. It is for this reason that type I and type II errors are as high as they are. In fact, the natural variability of this statistic is of the same order of magnitude as the change in the limit which was so much in debate. Figure 4. Operating characteristic curves for the new ozone standard and t-static alternative as a function of the expected number of exceedances per year. If instead, for example one used a procedure based on the t-statistic for control of the proportion above the limit, as is commonplace in industrial quality control procedures (4), one would get the operating characteristic curve plotted in Figure 4 (see also appendix). For comparison, the curve for the new regulation is also plotted as a function of the expected number of exceedances per year. With the new ozone regulation, the probability can exceed 1/3 that a particular site will be declared out of compliance when it is actually in compliance. The operating characteristic curve for the t-test is steeper (and hence has more discriminating power) than that for the new standard. The modified procedure based on the t-test generally reduces the probability that sites that are actually in compliance will be declared to be out of compliance. In fact, it is constructed so that there is 5% chance of declaring that a site is out of compliance when it is actually in compliance in the sense that the expected exceedance number is one per year. Furthermore, when a violation has occurred, it is much more certain that it will be detected with the t-based procedure. In this respect the t-based procedure provides more protection to the public. We do not conclude that procedures based on the t-test are best. We merely point out that there are alternatives to the procedures used in the old and new ozone standard. A basic principle is that information is lost when data are collected on a continuous scale and then reduced to a binary form. One of the advantages of procedures based on the t-test is that they do not waste information in this way. The most important point to be made goes beyond the regulation of ambient ozone; it applies to regulation of all pollutants where there is a desire to limit exposure. With the aid of operating characteristic curves, informed judgements can be made when an environmental regulation is being developed. In particular, operating characteristic curves for alternative forms of a regulation can be constructed and compared before a final one is selected. Also, the robustness of a regulation to changes in assumptions, such as normality and statical independence of observations, can be investigated prior to the promulgation. Note that environmental lawmaking, as it concerns the design of environmental regulations, is similar to design of scientific experiments. In both contexts, data should be collected in such a way that clear answers will emerge to questions of interest, and careful forethought can ensure that this desired result is achieved. Scientific Frameworks The operating characteristic curve is only one component in a more comprehensive scientific framework that we would like to promote for the design of environmental regulations. The key elements in this process are: • (a) Dose/risk curve • (b) Risk/benefit analysis • (c) Decision on maximum acceptable risk • (4) Stochastic nature of the pollution process • (e) Calibration of measuring instruments • (f) Sampling plan • (g) Decision function • (h) Distribution theory • (i) Operating characteristic function Currently there may be some instances in which all of these elements are considered in some form when environmental regulations are designed. Because the particular purposes and techniques are not explicitly isolated and defined, however, the resulting regulations are not as clear nor as effective as they might otherwise be. Often the first steps towards establishing an environmental regulation are (a) to estimate the relationship between the "dose" of a pollutant and some measure of health risk associated with it and (b) to carry out a formal or informal risk/benefit analysis. The problems associated with estimating dose / risk relationships and doing risk/benefit analyses are numerous and complex, and uncertainties can never be completely eliminated. As a next step a political decision is made - based on this uncertainties can never be completely eliminated. As a next step a political decision is made - based on this uncertain scientific economic groundwork - as to the maximum risk implies, through the dose/risk curve, the maximum allowable dose. The first three elements have received considerable attention when environmental regulations have been formulated, but the last six elements have not received the attention they deserve. The maximum allowable dose defines the compliance set Θ[0] and the noncompliance set Θ[1], which is its complement. The pollution processcan be considered (d) as a stochastic process or statistical time-series Φ(θt). Fluctuations in the measurements X can usefully be thought of as arising from three sources: variation in the pollution level itself Φ, the bias b in the readings, and the measurement error ε. Thus X = Φ + b + ε. Often it is assumed that Φ = θ, a fixed constant and that variation arises only from the measurement error ε however, all three components Φ, b, and ε can vary. Ideally b = 0 and the variance of e is small. Figure 5. Elements of the environmental standard-setting process: Laboratory experiments and/or epidemiological studies are used to assess the dose/risk relationship. A maximum acceptable risk is determined through a political process balancing risk and economic factors. The maximum acceptable risk implies a limit for the "dose" which again implies a limit for the pollution process as a function of time. Compliance with the standard is operationally determined based on a discrete sample x taken from a particular site. The decision about whether a site is in compliance is reached through use of a statistic ƒ and a decision function d. Knowing the statistical nature of the pollution process, the sampling plan, and the functional form of the statistics and the decision function, one can compute the operating characteristic function. Projecting the operating characteristic function back on the dose/risk relationship, one can assess the probability of encountering various levels of undetected violation of the standard. Measurements will only have scientific meaning if there is a detailed operational description of how the measurements are to be obtained and the measurement process is in a state of statistical control. A regulation must include a specification relating to how the instruments are to be calibrated (e). These descriptions must be an integral part of a regulation if it is going to be meaningful. The subject of measurement is deeper than is generally recognized, with important implications for environmental regulation (5, 6, 7). The pollution process and the observed process as a function of time are indicated in Figure 5. Logically the next question is (f) how best to obtain a sample X = (X[1], X[2]. . . ,X[n]) from the pollution process. The answer to this question will be related to the form of the estimator ƒ(X) and (g) the decision rule The sample, the estimator, and the decision function are indicated in Figure 5. Based on knowledge about the statistical distribution of the sample (h), one can compute (i) the operating characteristic function P = Prob {d (ƒ(X)) = 0 \| θ} and plot the operating characteristic curve P versus θ. An operating characteristic function is drawn at the bottom of Figure 5. (In practice it would probably be desirable to construct more than one curve because, with different assumptions, different curves will result). Projected back on the dose/risk relationship (see Figure 5), this curve shows the probability of encountering various risks for different values of θ if the proposed environmental regulation is enacted. Suppose there is a reasonable probability that the pollutant levels occur in the range where the rate of change of the dose/risk relationship is appreciable then the steeper the dose/risk function, the steeper the operating characteristic curve needs to be if the regulation is to offer adequate protection. The promulgated regulation should be expressed in terms of an operational definition that involves measured quantities, not parameters. Figure 5 provides a convenient summary of our proposed framework for designing environmental regulations. In environmental lawmaking, it is most prudent to consider a range of plausible assumptions. Operating characteristic curves will sometimes change with different geographical areas to a significant degree. Although this is an awkward fact when a legislative, administrative, or other body is trying to enact regulations at an international, national, or other level, it is better to face the problem as honestly as possible and deal with it rather than pretending that it does not exist. Operating Characteristic Curve as a Goal, Not a Consequence We suggest that operating characteristic curves be published whenever an environmental regulation is promulgated that involves a pollutant the level of which is to be controlled. When a regulation is being developed, operating characteristic curves for various alternative forms of the regulation should be examined. An operating characteristic curve with specified desirable properties should be viewed as a goal, not as something to compute after a regulation has been promulgated. (Nevertheless, we note in passing that it would be informative to compute operating characteristic curve for existing environmental regulations.) In summary, the following procedure might be feasible. First based on scientific and economic studies of risks and benefits associated with exposure to a particular pollutant a political decision would be reached concerning the compliance set in the form of an interval of the type 0 ≤ θ ≤ θ[0] for a parameter of the distribution of the pollution process. Second, criteria for desirable sampling plans, estimators, and operating characteristic curves would be established. Third, attempts would be made to create a sampling plan and estimators that would meet these criteria. The costs associated with different sampling plans would be estimated. One possibility is that the desired properties of the operating characteristic curve might not be achievable at a reasonable cost. Some iteration and eventual compromise may be required among the stated criteria. Finally, the promulgated regulation would be expressed in terms of an operational definition that involves measured quantities, not parameters. Injecting parameters into regulations, as was done in the new ozone standard, leads to unnecessary questions of interpretation and complications in enforcement. In fact, inconsistencies (such as that implied by Prob {ƒ(X) ≤ = c \| θ[0]} = 37% for the new ozone standard) can arise when conceptual differences between c and θ[0] and between ƒ(X) and θ are ignored. What is needed is a more refined conceptual model than that which underlies current environmental regulations, a model that makes these distinctions and acknowledges type I and type II errors. Research Needs Research that is used in designing environmental standards has focused on the first three elements of our framework (a), (b), and (c). If the last six elements do not receive relatively more attention than they currently receive, the precision obtained in estimating risk may well be lost by the lack of precision in estimating compliance. The above analysis, therefore, points to the need to have research resources more evenly spread among all the key elements (a), (b), . . . , (i). Furthermore, more research needs to be conducted that takes a global view of how all the elements function together. It would be beneficial to analyze many of the already promulgated standards using the framework outlined above and in particular to compute operating characteristic curves. Such research will sometimes require the development of new distribution theory because standards typically use rather complex decision rules. Moreover, most environmental data are serially correlated and consequently the shape of the operating characteristic function will be affected. At present little statistical theory is developed to cope with this problem. Preliminary studies we have me show that operating characteristic curves for binary sampling plans as used in the ozone standard seem to be seriously affected by serial correlation. Monte Carlo simulation might prove a viable alternative to distribution theory in evaluating the operating characteristic function for complex decision rules and serially correlated time series. In our discussion above we only considered one pollutant and its regulation. The interaction among several pollutants and other environmental factors, however, might create higher risks than would be anticipated from separate studies on the individual pollutants themselves. Such Issues are only beginning to be addressed (8). A related issue is the problem of what constitutes a rational attitude towards risk. It seems irrational to impose strict standards for one pollutant when other equally hazardous pollutants have much more relaxed standards. A harmonization among standards seems desirable. In order to address such issues it is necessary to develop methods for comparing convolutions of probability of occurrence, dose/risk relationships, and operating characteristic functions for several pollutants simultaneously. This will require an extension of the framework outlined above to multiple pollutants. However, that framework can be used as a first step in attacking these more comprehensive problems that are so important to protecting our environment. One of the purposes of environmental law, which has been defined as the rules for planetary housekeeping (9), is to prevent harm to society. Assessment of risk is one of the key issues in environmental lawmaking and continued research is needed on how to measure risk and make decisions regarding risk; but risk assessment is not enough. If laws with good operating characteristics are not designed, the effort expended on risk assessment will simply be wasted. With limited resources, we need to develop methods for economically and rationally allocating resources to provide high levels of safety. Ideally a system of environmental management and control should be composed of individual laws that limit potential risk in a consistent manner. The ideas outlined in this article give partial answers to two connected questions: (i) how can we formulate an individual quantitative regulation so that it will be scientifically sound and (ii) how can we construct a rational system of environmental regulations? If the framework outlined above is used properly in the course of developing environmental regulations, some of the important operating properties of different alternatives would be known. The public would know the probabilities of violations not being detected (type II errors); industries would know the probabilities of being accused incorrectly of violating standards (type I errors); and all parties would know the cost associated with various proposed environmental control schemes. We believe that the operating characteristic curve is a simple yet comprehensive device for presenting and comparing different alternative regulations because it brings into the open many relevant and sometimes subtle points. For many people it is unsettling to realize that type I and type II errors will be made, but it is unrealistic to develop regulations pretending that such errors do not occur. In fact, one of the central issues that should be faced in formulating effective and fair regulations is the estimation and balancing of the probabilities of such occurrences. The t-statistic procedure is based on the estimator s the sample standard deviation. The decision function is The critical value c is found from the requirement that where z[0] = Φ^-1 (1-θ[0]) and θ[0] is the fraction above the limit we at at want to accept (here 1/365). The exact operating characteristic function is found by reference to a non-central t-distribution, but for all practical purposes the following approximation is sufficient The operating characteristic function in Figure 4 is constructed using α = 0.05, θ [0] = 01/365 and n = 3x 365. Substituting (A3) into (A2) yields which solved for the critical value yields c = 2.6715. Refer for example to (4) for more details. Literature Cited 1. National Primary and Secondary Ambient Air Quality Standards, Federal Register 36, 1971 pp 8186-187. (This final rulemaking document is referred to in this article as the old ambient ozone 2. National Primary and Secondary Ambient Air Quality Standards, Federal Register 44, 1979 pp 8202-8229. (This final rulemaking document is referred to in this article as the new ambient ozone standard.) The background material we summarize is contained In this comprehensive reference. 3. Javitz, H. J. J. Air Poll. Con. Assoc. 1980 30, pp 58-59. 4. Hald, A. "Statical Theory with Engineering Applications" Wiley, New York 1952 pp 303-311. 5. Hunter, J. S. Science 210, 1980 pp 869-874; 6. Hunter, J. S. In "Appendix D", Environmental Monitoring, Vol IV, National Academy of Sciences 1977; 7. Eisenhart, C. In "Precision Measurements and Calibration", National Bureau of Standards Special Publication 300 Vol. 1, 1969 pp 2I-47. 8. Porter, W. P. Hinsdill, R. Fairbrother, A. Olson, L. J. Jaeger, J. Yuill, T. Bisgaard, S. Hunter, W. G. K. Nolan, K. Science 1984, 224, pp 1014-1017. 9. Rogers, W. H. "Handbook of Environmental Law" West Publishing Company, 1977, St. Paul, MN.	{"url":"https://williamghunter.net/articles/studies-in-quality-improvement-designing-environmental-regulations","timestamp":"2024-11-13T21:11:12Z","content_type":"application/xhtml+xml","content_length":"57009","record_id":"<urn:uuid:1c030c5d-3c55-479a-be08-268436a03084>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00752.warc.gz"}
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Motivating consistent practice, offering support, and creating a favorable learning atmosphere are advantageous actions.	{"url":"https://crown-darts.com/en/solving-multiple-step-equations-worksheet.html","timestamp":"2024-11-06T10:46:06Z","content_type":"text/html","content_length":"29382","record_id":"<urn:uuid:b2ec3947-925d-4388-a85b-e31d9e0f752d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00182.warc.gz"}
Ideal Gas Law Ideal Gas Law An ideal, or perfect, gas exists where the kinetic energy of its molecules are dependant only on temperature. It therefore would obey gas laws (e.g. Boyle’s Law & Charles’s Law) exactly at all pressures and temperatures. The ideal gas law is a combination of Charles’s Law and Boyle’s Law and is the equation of state that exists for such an ideal gas. Many gases (e.g. air, nitrogen, oxygen, hydrogen etc.) can be treated as an ideal gas. Ideal Gas Law in Relation to Individual Gas Constant R[i] p = absolute pressure (N/m², lb/ft²) V = volume (m³, ft³) m = mass (kg, slugs) R[i] = individual gas constant (J/Kg.K, ft.lb/slugs.°R) T = temperature (K; °R) This can be alternatively written in relation to density, ρ, as: Ideal Gas Law in Relation to Universal Gas Constant R* p = absolute pressure (N/m², lb/ft²) V = volume (m³, ft³) n = number of moles of gas R* = universal gas constant (J/mol.K, ft.lb/lb-mol.°R) T = temperature (K; °R)	{"url":"https://www.mydatabook.org/thermodynamics/ideal-gas-law/","timestamp":"2024-11-02T04:48:53Z","content_type":"application/xhtml+xml","content_length":"39268","record_id":"<urn:uuid:0a4420ec-c636-4944-ae05-c97cd0cc9935>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00530.warc.gz"}
OpenStax College Physics for AP® Courses, Chapter 14, Problem 76 (Problems & Exercises) (a) What is the temperature increase of an 80.0 kg person who consumes 2500 kcal of food in one day with 95.0% of the energy transferred as heat to the body? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible? Question by is licensed under CC BY 4.0 Final Answer a. $35.5\textrm{ C}^\circ$ b. Body temperature is highly regulated within a narrow range about $37^\circ\textrm{C}$. A temperature increase greater than $3 \textrm{ C}^\circ$ is life threatening. c. It's unreasonable to presume that 95% of heat generated from food will be retained by the body. The body has mechanisms, such as sweating, to dissipate heat. Solution video OpenStax College Physics for AP® Courses, Chapter 14, Problem 76 (Problems & Exercises) vote with a rating of votes with an average rating of. Video Transcript This is College Physics Answers with Shaun Dychko. This question supposes that an 80.0 kilogram person consumes 2500 kilocalories of food energy and 95 percent of that energy is retained by the body as heat. So we convert the energy into joules by multiplying by 4186 joules per kilocalorie and we need to know the specific heat of the body which is 3500 joules per kilogram per Celsius degree. So the heat energy absorbed by the body then is going to be the total energy consumed multiplied by the efficiency by which it absorbs that heat energy— 95 percent of it will be absorbed as heat— and that heat energy will also be the mass of the body times its specific heat times its change in temperature. So we can equate these two things and say that mcΔT equals energy times efficiency. Let's divide both sides by mc to solve for the change in temperature. So the change in temperature is energy times efficiency divided by mass times specific heat. So that's 1.0465 times 10 to the 7 joules times 0.95 divided by 80.0 kilograms times 3500 joules per kilogram per Celsius degree and that's a temperature change of 35.5 Celsius degree— that temperature change is huge. The body temperature is highly regulated around 37.0 degrees Celsius and a temperature change of more than 3.0 Celsius degrees is life-threatening. So a fever above 40.0 degrees Celsius needs medical assistance and this is 35 Celsius degrees above so highly impossible... totally impossible! And it's unreasonable to think that 95 percent of the heat generated from food will be retained by the body because the body has really good mechanism such as sweating to dissipate heat.	{"url":"https://collegephysicsanswers.com/openstax-solutions/what-temperature-increase-800-kg-person-who-consumes-2500-kcal-food-one-day-0","timestamp":"2024-11-04T00:53:11Z","content_type":"text/html","content_length":"198402","record_id":"<urn:uuid:c84b255a-f2ad-4071-b494-26bae272e6d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00086.warc.gz"}
addendum gear calculation for Calculations 29 Mar 2024 Popularity: ⭐⭐⭐ Gear Ratio Calculator This calculator provides the calculation of gear ratio for mechanical engineering applications. Calculation Example: The gear ratio is a dimensionless quantity that describes the relative sizes of two gears. It is defined as the ratio of the number of teeth on the gear to the number of teeth on the pinion. The gear ratio determines the speed ratio and torque ratio of the gear pair. Related Questions Q: What is the importance of gear ratio in mechanical engineering? A: The gear ratio is important in mechanical engineering as it determines the speed ratio and torque ratio of the gear pair. This information is crucial for designing and selecting gears for specific Q: How does the gear ratio affect the performance of a gear pair? A: The gear ratio affects the performance of a gear pair by determining the speed ratio and torque ratio. A higher gear ratio results in a lower output speed and a higher torque ratio, while a lower gear ratio results in a higher output speed and a lower torque ratio. \| —— \| —- \| —- \| Z Number of Teeth on Gear - Calculation Expression Gear Ratio: The gear ratio is given by G = Z / N. Calculated values Considering these as variable values: T=100.0, Z=20.0, N=1000.0, the calculated value(s) are given in table below \| —— \| —- \| Similar Calculators Calculator Apps Matching 3D parts for addendum gear calculation for Calculations App in action The video below shows the app in action.	{"url":"https://blog.truegeometry.com/calculators/addendum_gear_calculation_for_Calculations.html","timestamp":"2024-11-10T09:19:41Z","content_type":"text/html","content_length":"24164","record_id":"<urn:uuid:ad62f782-3a73-4a16-ad23-8856c88832c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00834.warc.gz"}
[Solved] A group of veterinarians wants to test a \| SolutionInn A group of veterinarians wants to test a new canine vaccine for Lyme disease. (Lyme disease is A group of veterinarians wants to test a new canine vaccine for Lyme disease. (Lyme disease is transmitted by the bite of an infected deer tick.) In an area that has a high incidence of Lyme disease, 100 dogs are randomly selected (with their owners’ permission) to receive the vaccine. Over a 12-month period, these dogs are periodically examined by veterinarians for symptoms of Lyme disease. At the end of 12 months, 10 of these 100 dogs are diagnosed with the disease. During the same 12-month period, 18% of the unvaccinated dogs in the area have been found to have Lyme disease. Let p be the proportion of all potential vaccinated dogs who would contract Lyme disease in this area. a. Find a 95% confidence interval for p. b. Does 18% lie within your confidence interval of part a? Does this suggest the vaccine might or might not be effective to some degree? c. Write a brief critique of this experiment, pointing out anything that may have distorted the results or conclusions. Fantastic news! We've Found the answer you've been seeking!	{"url":"https://www.solutioninn.com/a-group-of-veterinarians-wants-to-test-a-new-canine","timestamp":"2024-11-14T04:58:20Z","content_type":"text/html","content_length":"81889","record_id":"<urn:uuid:608ea0e9-246b-4f7b-9d65-8f23ee1bac9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00429.warc.gz"}
Minimum Swaps To Make Sequences Increasing We have two integer sequences A and B of the same non-zero length. We are allowed to swap elements A[i] and B[i]. Note that both elements are in the same index position in their respective sequences. At the end of some number of swaps, A and B are both strictly increasing. (A sequence is strictly increasing if and only if A[0] < A[1] < A[2] < ... < A[A.length - 1].) Given A and B, return the minimum number of swaps to make both sequences strictly increasing. It is guaranteed that the given input always makes it possible. Input: A=[5, 2, 14], B=[1, 6, 12]) Output: 1 Explanation: Swap A[0] and B[0]. Then the sequences are: A = [1, 2, 14] and B = [5, 6, 12] which are both strictly increasing. • A, B are arrays with the same length, and that length will be in the range [1, 1000]. • A[i], B[i] are integer values in the range [0, 2000].	{"url":"https://cloudxlab.com/assessment/displayslide/6477/minimum-swaps-to-make-sequences-increasing","timestamp":"2024-11-14T07:30:20Z","content_type":"text/html","content_length":"66700","record_id":"<urn:uuid:294935ef-42cf-4f96-b0b2-6e749057d85c>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00671.warc.gz"}
Yue's Notes Gomory’s Theorem If you move two cells of a $8 \times 8$ chessboard of opposite colors, the remaining cells can be fully domino tiled. Proof Draw a closed path that passes through every square exactly once. (Draw a big C and then draw back and forth horizontally) Choose the two cells to be removed, and the closed path we have will be sperated into two paths. If we lable the close path we chose in the beginning from 1 to 64 in the order we drew it, white cells have odd numbers, and black cells have even numbers (or reversed).... Some latex tests Consider a $n \times m$ chessboard… $$ \int{f(x)dx} $$ Since $94 = 4 + 5x$ for some $x \in \mathbb{N}$, my claim is that the first player wins when it chooses $4$ in the beginning. Then, whenever the second player choose $a, x \in\set{1, 2, 3, 4}$, the first player just add it to 5. So choose $5-a$. By doing this, the first player always reaches $10x + 9$ or $10x + 4$ for $x \in \mathbb{N}$.... My Math 417 review notes I have had a wonderful summer as I have been able to take Math 417 with Professor Chales Rezk. He is a very very good teacher and I have learnt a lot from him. Thanks! These are the notes I have taken during the course. It includes the greate Theorems, Lemmas and Propositions that we have learnt in the course. Previous Next / [pdf] View the PDF file here.... 413_basic counting The four basic counting principle Suppose that a set $S$ is partitioned into pairwise disjoint parts $S_1, S_2, …, S_n$. Addition principle: $$ \|S\| = \|S_1\| + \|S_2\| + … + \|S_m\| $$ Ex: Path counting: In a $3 \times 3$ grid, if you can move 1 step upward or 1 step to the right. How many ways do we have to move from bottom-left to top-right corner? The idea is to break this problem into smaller	{"url":"https://ohuro.me/page/3/","timestamp":"2024-11-07T02:34:32Z","content_type":"text/html","content_length":"12505","record_id":"<urn:uuid:6ec99c13-751f-4d45-82b5-258ea15371b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00442.warc.gz"}
An interesting usecase for bit manipulation An interesting usecase for bit manipulation January 6, 2023 The other day, I was reading the source code to preact/signals and came across code that looked somewhat like this: // flags const FIRSTFLAG = 1 << 0; const SECONDFLAG = 1 << 1; const THRIDFLAG = 1 << 2; // object with flags const node = { flags: 0 // adding flags? node.flags \|= FIRSTFLAG; // checking if flags existed if (node.flags & (SECONDFLAG \| THIRDFLAG)) { // do stuff // removing flags? node.flags &= ~FIRSTFLAG; Event after coding for ~ten years, this isn't something I have seen with any regularity. Certainly not enough to know what it did off the top of my head. So what exactly is going on here? Lets break it down. Before we can go any further, we need to discuss how numbers are represented at the binary level. When we talk about binary, data is can only be represented as a series of 1s and 0s. For now we will just discuss what that means for numbers in our little javascript example. Lets work with the following: This represents the number 1. It doesn't matter how many zeros precede the 1, just that it appears in the right most place. Going left each bit/digit represents double the previous digit. So it would go something like: 0001 == 1, 0010 == 2, 0100 == 4, 1000 == 8, and so on. From here you can set the bit in multiple places to get all the numbers: 0001 // 1 0010 + // 2 0011 // 3 Javascript numbers are 64-bit, so in theory you could have up to 1s and 0s, but the way there encoded for floating point precision and postive/negative values, it leaves us with 53 bits of usable information. We can distinguish this by the following: const max_binary = Number.MAX_SAFE_INTEGER.toString(2); // 11111111111111111111111111111111111111111111111111111 max_binary.length; // 53 Javascript technically has big ints, but I'm not there yet. Plus I think we have enough for now. Creating flags Okay, so 53 bits worth of information... for what? We will get there... I promise. Now that we have some background knowledge, what is going on in the code. It starts by predefining some flags with some interesting syntax, which turns out to be the bitshift operator. const FIRSTFLAG = 1 << 0; // bit shifting left 0 places This changes the number 1 by moving it's bits to the left by 0 places. If we push 001 left by 0, we still have 1. ðŸ™„ Okay, that was lame... lets try again. But push it left 1 space: const SECONDFLAG = 1 << 1; // bit shifting left 1 place SECONDFLAG === 2; We've transformed 001 by pushing the bits to get 010, and the slot second from the left has the value of 2. If you continue this for each of the flags you might need, you will see the double in value everytime. This correaltes 1:1 with each digit of a binary value to double the previous digit (going right to left). Now what? Okay now we have the ability to store some information within a single number value. Why might this be useful? I think it's a niche scenario, when memory usage is super important. I don't understand the details completely, but by storing multiple data values in a single number you limit how many variable descriptors you need. Instead of having five different (boolean) variables created with their own memory allocation, you can store them all at once. That's mostly a guess. I find this whole thing intriguing, but haven't been able to find a good usecase for myself. Will hang on to this idea just in case. It's at least helpful that I now understand how this stuff works when I come across it in code. Till next time...	{"url":"https://fromdl.com/posts/bitwise/","timestamp":"2024-11-11T06:56:46Z","content_type":"text/html","content_length":"10031","record_id":"<urn:uuid:647da5e3-f476-4e64-8531-09717f35f696>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00868.warc.gz"}
aoc2021/a.py at cc57d8ab30d18ba1a1439d72268e5a13178bde16 2021-12-03 21:38:31 -08:00 from pprint import pprint WIDTH = 5 HEIGHT = 5 class Board(object): def __init__(self, matrix, win_matrix): # matrix: map of num => (x,y) # win_matrix: map of (x,y) => picked bool self.matrix = matrix self.win_matrix = win_matrix self.total = None # this is calculated upon noticing the board has won def from_block(block): # convert the block, which is an array of string, into a flat list of numbers nums = [] for row in block: nums.extend([int(i) for i in row.strip().split()]) i = 0 matrix = {} win_matrix = {} for y in range(0, HEIGHT): for x in range(0, WIDTH): sq = nums[i] matrix[sq] = (x, y, ) win_matrix[(x, y, )] = False i += 1 return Board(matrix, win_matrix) def fill(self, num): # mark the space with $num as filled # return whether this makes the board a winner filled_position = self.matrix[num] except KeyError: return False # number not on this board self.win_matrix[filled_position] = True return self.check_solved(num, filled_position) def check_solved(self, num, new_x, new_y): # the square at ($new_x, $new_y) has been filled, check if it solves the board # it solves the board if: # the row is filled # the col is filled # either diagonal is filled ----- jk, the problem says diagonals dont matter w = any([ # row all([self.win_matrix[(x_coord, new_y, )] for x_coord in range(0, WIDTH)]), # col all([self.win_matrix[(new_x, y_coord, )] for y_coord in range(0, HEIGHT)]) if w: self.total = self.calc_total(num) return w def calc_total(self, num): print("calculating total") # sum of all unmarked nums unmarked = 0 for sq_num, coord in self.matrix.items(): if not self.win_matrix[coord]: unmarked += sq_num return unmarked num def update_boards(boards, pick): winners = [] for board in boards: if board.fill(pick): return winners with open("input.txt") as f: lines = [i for i in f.readlines()] # input parser picks = [int(i) for i in lines.pop(0).strip().split(",")] board_blocks = [] while True: # pop the 5 expected lines for the board board_blocks.append([lines.pop(0), lines.pop(0), lines.pop(0), lines.pop(0), lines.pop(0), ]) if not lines: empty = lines.pop(0) boards = [] for block in board_blocks: for pick_num, pick in enumerate(picks): winners = update_boards(boards, pick) if winners:	{"url":"https://git.davepedu.com/dave/aoc2021/src/commit/cc57d8ab30d18ba1a1439d72268e5a13178bde16/4/a.py","timestamp":"2024-11-12T17:04:40Z","content_type":"text/html","content_length":"66136","record_id":"<urn:uuid:c05441c9-1db8-4583-b660-8400e17cd71f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00064.warc.gz"}
One of the best things about Python is the language’s sheer flexibility. We have a wide range of elements and containers to work with. And we can often manipulate those items in a wide variety of useful ways. For example, we can find the cumulative sum of a python list despite the fact that it’s not an integer of any type. Check out the small Python program below to see a simple example of how we might find the sum of list Python programming style. numList = [-90,10, 15, 20, 25, 30,35.5] ourSum = sum(numList) ourSum = sum(numList, 10) We begin by creating a sum list of numbers, with each given list element corresponding to a single natural number. In other languages we’d typically want to construct a loop to work with each of the given list’s numbers. But Python provides us with a specific Python function that relieves us of that burden, once we put in the start value it will add the other numeric values. We can instead use the python sum function. In line 2 we pass the numList variable as a parameter to the sum function. We can then print out the total of our new ourSum variable to see the total cumulative sum of the Python list. In the next line we repeat this process but pass both numList and an integer as parameters to the inbuilt function sum. We then print it out to demonstrate another interesting facet of element wise sum. If we pass a natural number to sum it will use that as the start value. If we don’t pass any initial value numbers to sum the Python function defaults to a value of 0. Note too that the Python sum is able to handle different types of numbers rather than insisting on one type shared among every single value. For example, it can work with the -90 and 35.5 without needing to perform any conversions or specify type on the given number. One of the few things that we couldn’t handle by default with the inbuild function sum is a string value. That doesn’t mean we can’t find a way around that data structure problem though. Consider the following Python code. ourList = ['1', 2, '3', '4', '5'] ourList = [int(i) for i in ourList] The ourList variable consists of a list made up of numbers ranging from one to five. However, all the numbers except for 2 are actually strings. Sum would normally exit the Python interpreter with an error if we tried to pass ourList to it. But we can use list comprehension to modify the contents of ourList before passing it to sum. We redefine ourList with the output of a generation loop consisting of the items in the original version of ourList. Each item is passed to the int function as input to convert it into an integer. The 2, which is already an integer, remains unchanged. The final line prints what we see after passing the modified ourList to sum. In short, we were able to use list comprehension to convert a mixed content original list of strings and integers into a list of integers. The converted list was then easily parsed by The combination of sum and list comprehension might spur some related ideas. For example, could we combine sum and a generator expression? We can, and the following example shows how powerful this combination can be. ourSum = sum(z ** 2 for z in range(5)) print("generator sum: ", ourSum) We begin by using a generator expression as a parameter to pass to sum variable. The expression sums the values immediately on generation without needing to save or convert anything in memory. This makes it extremely efficient for longer or more complex mathematical functions. Though in this instance we’re simply finding the sum of the squares from 0 to 5. If we encased the parameters passed to sum variable with brackets than a Python program would go through the full process of creating and saving a list before converting it to a sum. How to Find the Sum of a Python List	{"url":"https://decodepython.com/how-to-find-the-sum-of-a-python-list/","timestamp":"2024-11-04T02:53:03Z","content_type":"text/html","content_length":"38458","record_id":"<urn:uuid:c1385164-5978-49e9-a880-5299d5059bc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00323.warc.gz"}
Mathematical Model of a Spherical Shell Heat Exchanger with Exponential Internal Heat Generation Document Type Thesis - Open Access Degree Name Master of Science (MS) Department / School Mechanical Engineering It has been recognized that many convection heat transfer problems involve internal heat generation. Nuclear fuel elements, chemically reactive liquids and transparent shell solar energy collectors are a few examples. Theoretically, this type of problem has not been studied extensively. Lumsdaine solved the internal heat generation problem with constant temperature boundary conditions in spherical coordinates. The purpose of this thesis is to follow Lumsdaine's analysis to study those problems where internal heat generation is a function of radius. The governing partial differential equation in spherical coordinates is of the second order, non-homogeneous and has variable coefficients. The non-homogeneity arises from the internal heat generation function in the energy equation. The boundary conditions are also non-homogeneous. In order to solve this problem, steady state, an incompressible inviscid fluid, a constant fluid entrance temperature, and constant ambient temperatures are assumed. In solving the problem, the method of superposition is first used to form a non-homogeneous ordinary differential equation and a homogeneous partial differential equation. By doing this, the boundary conditions are also simplified. These two equations are then solved separately. The ordinary differential equation can be integrated directly. The partial differential equation is solved by using separation of variables. The solution is given as an infinite series of Euler functions. The temperature field is finally obtained by superimposing the two solutions. The expression for the average exit temperature is derived from the temperature field. The average exit temperature is non-dimensionalized in terms of Graetz number, Nusselt number, radius ratio and other dimensionless groups. Computer programs are developed to obtain numerical results. In the first part of the thesis, the inner shell is assumed to be at a constant temperature. The problem is first solved in general without specifying the internal heat generation function; therefore the solution holds for any type of internal heat generation. In the sample problem the internal heat generation function is assumed to consist _of a constant term and an exponential term to account for the decay from the outer shell. In the second part of the thesis, the solution emphasizes its -application to the solar heat exchanger. By assuming the outer shell transparent, the absorption of solar energy by the water can be interpreted as internal heat generation. The inner shell is assumed to be a black body which absorbs all the energy not absorbed by the water; this energy is then conducted back to the fluid. The efficiency of this type of solar heat exchanger should be higher than in exchangers with an opaque outer shell since the latter have higher temperatures at the outer shell and thus lose more heat to the surroundings. Library of Congress Subject Headings Heat exchanges -- Solar energy South Dakota State University Recommended Citation Tsou, John Lin, "Mathematical Model of a Spherical Shell Heat Exchanger with Exponential Internal Heat Generation" (1968). Electronic Theses and Dissertations. 3507.	{"url":"https://openprairie.sdstate.edu/etd/3507/","timestamp":"2024-11-06T11:39:16Z","content_type":"text/html","content_length":"43224","record_id":"<urn:uuid:0c94250e-5826-4afd-a04d-f3828f7dedda>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00301.warc.gz"}
Evaluate Power-Supply Noise Rejection In Low-Jitter PLL Clock Generators Evaluate Power-Supply Noise Rejection In Low-Jitter PLL Clock Generators Clock generators that integrate phase-locked loops (PLLs) find many homes in network equipment. Their main function is either to generate high-precision and low-jitter reference clocks, or maintain a synchronized network operation. Most clock oscillators provide their jitter or phase-noise specification using an ideal, clean power supply. In a practical system environment, the power supply can suffer from interference due to on-board switching supplies or noisy digital ASICs. To achieve the best performance in a system design, it’s important to understand the effects of such interference. PSNR Characteristics of PLL Clock Generators Figure 1 shows a typical PLL clock generator. Since the output driver can have very different power-supply noise rejection (PSNR) performance for different types of logic interfaces, the following analysis will focus on the supply noise impact to the PLL itself. Figure 2 shows the PLL phase model assuming that the power-supply noise V[N] is injected into the PLL/VCO, and the divide ratios M and N are set to 1 (Fig. 2). The PLL closed-loop transfer function from V[N](s) to φ[O](s) is given by For a typical second-order PLL, Here, Ω is the PLL 3-dB bandwidth, Ω is the PLL zero frequency, and Ω Equation 3 demonstrates that, in a PLL clock generator, the power-supply noise is rejected by 20 dB/s when the supply interference frequency is greater than the PLL 3-dB bandwidth. For power-supply interference frequencies between Ω[Z] and Ω[3dB], the output clock phase varies with the power-supply interference amplitude as: As an example, Figure 3 shows the PSNR characteristics of a PLL for two different settings of the PLL’s 3-dB bandwidth. Conversion of Power Spectrum Spurs to DJ When a single-tone sinusoidal signal, f[M], is applied to the power supply of a PLL, it produces a narrow-band phase modulation at the clock output, which can be generally described using Fourier series representation: Here, β is the modulation index representing the maximum phase deviation. For a small index modulation (β Here, n = 0 represents the carrier itself. When n = ±1, the phase-modulated signal is given by: Equation 8 demonstrates that, when measuring the double sideband power spectrum S[V](f), if varible x represents the level difference between the carrier at f[O] and the fundamental sideband tone at f[m], then: where x = decibels relative to the carrier (dBc). Since β is the maximum phase deviation in radians, the peak-to-peak deterministic jitter (DJ) caused by this small index phase modulation can be derived: where DJ = picoseconds peak-to-peak (ps p-p). The above analysis assumes that no amplitude modulation is contributing to the tone at f[M]. In reality, both amplitude and phase modulation can be generated, reducing the accuracy of this approach. Conversion of Phase Noise Spectrum Spurs to DJ To avoid the amplitude-modulation effect when measuring the power spectrum S[V](f), one can instead calculate the DJ by measuring the spur in the phase-noise spectrum while applying a single-tone sinusoidal interference on the supply. With the variable y representing the measured single-sideband-phase spurious power at frequency offset f[m], the resultant phase deviation Δφ can be derived: where y = dBc, Δφ = radians rms (rad[rms]) in Equation 11, and Δφ = ps p-p in Equation 12. It should be noted that the single-sideband phase spectrum in the above analysis isn’t the folded version of the double-sideband spectrum. That’s the reason for the 3-dB component in Equation 10. Figure 4 shows the relationship between the deterministic jitter and the phase spurious power given by Equation 12. PSNR Measurement Techniques This next section demonstrates five different ways of measuring the PSNR of a clock source, using the MAX3624 low-jitter clock generator as an example. The measurement setup in Figure 5 uses a function generator to inject a sinusoidal signal onto the power supply of the MAX3624 evaluation board. The amplitude of the single-tone interference is measured directly at the V[CC] pin close to the IC. A limiting amplifier, MAX3272, is used to remove amplitude modulation, followed by a balun that converts the differential output into a single-ended signal for driving the different test equipment. To compare the results from different tests, all of the measurements were done under the following conditions: • Clock output frequency: f[o] = 125 MHz • Sinusoidal modulation frequency: f[m] = 100 kHz • Sinusoidal signal amplitude: 80 mV[P-P] Method 1—Power spectrum measurement: When observed on a power spectrum analyzer, the narrow-band phase modulation appears as two sidebands around the carrier. Figure 6 shows the case when viewed using the spectrum monitor function of the Agilent E5052. The measured first sideband amplitude relative to carrier amplitude is -53.1 dBc, which translates to 11.2 ps[p-p] deterministic jitter according to Equation 9. Method 2—Single-sideband (SSB) phase spurious measurement: On a phase-noise analyzer, the power-supply interference will manifest itself as a phase spur relative to the carrier. The measured phase noise spectrum is plotted in Figure 7. The phase spurious power at 100 kHz is -53.9 dBc, which translates to 10.2-ps[p-p] deterministic jitter using Equation 12. Method 3—Phase demodulation measurement:Utilizing the Agilent E5052 signal analyzer, the phase-demodulated sinusoidal signal at 100 kHz is measured directly (Fig. 8), which gives the maximum phase deviation from its ideal position. The peak-to-peak phase deviation is 0.47°, which translates to 10.5 ps p-p at an output frequency of 125 MHz. Method 4—Real-time scope measurement: In a time-domain measurement, the deterministic jitter caused by power-supply interference can be obtained by measuring the time interval error (TIE) histogram. On a real-time scope, the clock output TIE distribution will appear as a sinusoidal probability density function (p.d.f.) when a single-tone interference is injected into the PLL. The deterministic jitter can be estimated using the dual-Dirac model1 by measuring the peak distance between the mean of two Gaussian distributions from the TIE histogram. Figure 9 shows the measured TIE histogram using Agilent’s Infiniium DSO81304A 40-Gsample/s real-time scope. The measured peak separation is 9.4 ps under the test condition mentioned above. It should be noted that the memory depth of the real-time scope may limit the low sinusoidal modulation frequency that can be applied to the PLL supply. For example, if the test equipment has a memory depth of 2 Msamples/s when the sample rate is set to 40 Gsamples/s, that would only allow capture of jitter frequency components down to 20 kHz. Method 5—Sampling scope measurement: When a sampling scope is used, a synchronous trigger signal is required for analyzing the clock jitter under test. Two triggering methods can be used for TIE The first solution is to apply a low-jitter reference clock to the input of the PLL clock generator, and use the same clock source as the trigger for the sampling scope. Fig. 10 shows the measured TIE histogram, which gives a peak spacing of 9.2 ps. The advantage of triggering with a reference clock is that the measured TIE histogram peak separation is independent of the horizontal time delay from the trigger position. However, the measured TIE histogram might be affected by the triggering clock jitter. Therefore, it’s important to use a clock source with much lower jitter than the clock generator device under test. Self-triggering is an alternate approach to eliminating the impact of triggering clock jitter. In this case, the output of the clock generator under test is separated into two identical signals using a power splitter. One signal is applied to the data input of the sampling scope, another one to the trigger input. Because the triggering signal contains the same deterministic jitter as the test signal, the histogram peak separation varies when the horizontal position of the scope main time base is swept through one period of the sinusoidal modulation frequency. At a horizontal position of one-half period of the modulation signal, the peak separation on the TIE histogram will be twice the deterministic jitter from the test signal. Figure 11 shows the measured MAX3624 TIE histogram when the horizontal time delay is set to 5 µs. The estimated TIE peak separation is 19 ps, which gives an equivalent deterministic jitter of 9.5 ps p-p. Figure 12 shows the measured TIE histogram peak spacing at a different horizontal time delay from the trigger point. For comparison, the TIE result is also shown when the sampling scope is triggered by a reference clock input. Measurement Summary The table summarizes the measured deterministic jitter at the MAX3624 125-MHz clock output, using the different methods that were discussed. It should be noted that measured DJ using a dual-Dirac approximation from the TIE histogram is slightly smaller than the DJ obtained from the frequency-domain spectral analysis. This is caused by the process of convolution of the sinusoidal jitter (SJ) p.d.f. with the Gaussian distribution of the random jitter component.^1 Therefore, the deterministic jitter extracted from the dual-Dirac model is only an estimation and should only be applied when the standard deviation of the random jitter is much smaller than the distance between the two peak separations of the jitter histogram. For the relatively large interference used in the examples, the results were well correlated. However, when the level of interference drops relative to the random jitter, the time domain methods become less accurate. Furthermore, if the clock signal is corrupted by amplitude modulation, measurements using a power spectrum analyzer become unreliable. Therefore, of all the methods presented, the phase spur power measurement using a phase noise analyzer is the most accurate and convenient way to characterize the PSNR of a clock generator. The same method can be extended for evaluating the deterministic jitter aspect caused by other spurious products appearing on the phase noise spectrum. 1. Agilent white paper, “Jitter Analysis: the dual-Dirac Model, RJ/DJ, and Q-Scale.” Sponsored Recommendations To join the conversation, and become an exclusive member of Electronic Design, create an account today!	{"url":"https://www.electronicdesign.com/technologies/industrial/boards/article/21764944/evaluate-power-supply-noise-rejection-in-low-jitter-pll-clock-generators","timestamp":"2024-11-08T05:06:19Z","content_type":"text/html","content_length":"256383","record_id":"<urn:uuid:d4d06e65-d684-48fc-b9a5-9baf9beefa13>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00694.warc.gz"}
Addition of Two Numbers in Python 08 Jul 2024 6.97K Views 9 min read How to Add Two Numbers in Python? Adding numbers in Python is one of the simplest arithmetic operations in Python. After having learned about variables, data types, input and output, and various operators in Python, it's time to implement them. In this Python tutorial, we'll practically understand all the methods to add two numbers in Python. But, before proceeding you must be thorough with the topics mentioned above. If not refer to these: Ways to Perform Addition of Two Numbers in Python 1. Using the "+" Operator This is the simplest addition method in Python. Place the "+" arithmetic operator between two numbers you want to add and you will get the sum. Program to Add Two Numbers in Python num1 = 5 num2 = 3 result_addition = num1 + num2 print("Addition of num1 and num2: ", result_addition) We have two numbers inside the variables num1 and num2. Both of them are added and the result is stored in the result_addition variable. Addition of num1 and num2: 8 Using User Input We take two numbers from a user through the Python input() function and store them in the variables. We add those numbers using the "+" operator. Program to Add Two Numbers in Python Using User Input num1 = input('Enter first number: ') num2 = input('Enter the second number: ') sum = float(num1) + float(num2) print('The sum of {0} and {1} is {2}'.format(num1, num2, sum)) We have two numbers inside the variables num1 and num2. Both are added and the result is stored in the sum variable. Enter first number: 67 Enter the second number: 89 The sum of 67 and 89 is 156.0 2. Using the "+=" Operator The "+=" is the addition assignment operator in Python. It adds the right operand to the left operand and assigns the result to the left operand. Program Implementing the "+=" Operator to Add Two Numbers in Python num1 = 58 num2 = 100 # addition using the '+=' operator num1 += num2 print("The sum of", num1, "and", num2, "is", num1) We have two numbers inside the variables num1 and num2. num1 is added with num2 and the result gets stored in num1. The sum of 158 and 100 is 158 3. Using User-Defined Function We can create a function to add two numbers in Python. The function will perform addition on the two numbers given as arguments using the "+" operator. Program Implementing Function to Add Two Numbers in Python def add(a,b): return a+b num1 = 100 num2 = 507 #function calling and store the result into sum sum = add(num1,num2) print("Sum of {0} and {1} is {2};" .format(num1, num2, sum)) The function add() takes two arguments two numbers inside the variables num1 and num2. Both are added and the result is stored in the sum variable. Read More: Python Functions Sum of 100 and 507 is 607 4. Using operator.add() Method The operator module provides functions corresponding to the built-in operators. We can add two numbers using the operator.add() method. It'll take two numbers as arguments and add them. Program Implementing operator.add() Method to Add Two Numbers in Python num1 = 158 num2 = 12 import operator sum = operator.add(num1,num2) print("Sum of {0} and {1} is {2}" .format(num1, num2, sum)) The program adds num1 and num2 using the operator.add() method and stores the result in the sum variable. Sum of 158 and 12 is 170 5. Using Lambda Function We can use the Python lambda function to add two numbers in Python. Program Implementing Lambda Function to Add Two Numbers in Python add_numbers = lambda a, b: a + b num1 = 145 num2 = 318 # lambda function sum = add_numbers(num1, num2) print("The sum of", num1, "and", num2, "is", sum) The program adds num1 and num2 using the lambda function, add_numbers and stores the result in the sum variable. The sum of 145 and 318 is 463 6. Using the Built-In sum() Function The sum() function in Python takes a list of numbers and adds them. Program Implementing Built-In sum() Function to Add Two Numbers in Python num1 = 590 num2 = 105 sum = sum([num1, num2]) print("The sum of", num1, "and", num2, "is", sum) The program adds num1 and num2 using the sum function and stores the result in the sum variable. The sum of 590 and 105 is 695 Read More: Python Lists: List Methods and Operations We saw all possible methods to perform the addition of two numbers in Python. You can use any of the above methods. All are very easy to understand and implement.	{"url":"https://www.scholarhat.com/tutorial/python/program-to-add-two-numbers-in-python","timestamp":"2024-11-05T22:37:55Z","content_type":"text/html","content_length":"131487","record_id":"<urn:uuid:d93d508f-d9ce-4ecd-9090-dc3d69a98b40>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00605.warc.gz"}
Understanding Mathematical Functions: Where Is The Function Increasing Mathematical functions are a fundamental concept in the world of mathematics, playing a crucial role in various fields such as science, engineering, economics, and more. These functions represent the relationship between input and output values, providing a systematic way to understand and analyze mathematical phenomena. One important aspect of understanding functions is knowing where the function is increasing and decreasing. Knowing where a function is increasing and decreasing is crucial in understanding its behavior and making predictions. It helps in identifying the maximum and minimum points of a function, which is valuable in various real-world applications. To determine this, it is essential to utilize the right tools, such as a reliable calculator designed for this specific purpose. Key Takeaways • Mathematical functions play a crucial role in various fields and represent the relationship between input and output values. • Understanding where a function is increasing and decreasing is essential for predicting behavior and identifying maximum and minimum points. • Identifying increasing and decreasing intervals is important for real-world applications and mathematical analysis. • Utilizing a reliable calculator designed for determining increasing and decreasing intervals can improve efficiency in mathematical analysis. • By using the increasing and decreasing calculator, individuals can accurately determine the behavior of functions and make informed predictions. Understanding Mathematical Functions: Where is the function increasing and decreasing calculator Mathematical functions play a crucial role in various fields such as science, engineering, finance, and computer science. Understanding the behavior of functions is essential for making informed decisions and solving complex problems. In this blog post, we will explore the concept of mathematical functions and the importance of understanding their behavior, specifically where the function is increasing and decreasing. What are mathematical functions? A mathematical function is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output. In other words, a function assigns each input value to exactly one output value. This relationship is often represented using an equation or a graph. • Definition of a mathematical function: A mathematical function f is a rule that assigns to each element x in a set A exactly one element y in a set B. • Examples of mathematical functions: Examples of mathematical functions include linear functions (e.g., f(x) = mx + b), quadratic functions (e.g., f(x) = ax^2 + bx + c), and exponential functions (e.g., f(x) = a^x). Importance of understanding the behavior of functions Understanding the behavior of functions is crucial for various applications, including optimization, modeling real-world phenomena, and making predictions. • Optimization: In optimization problems, such as maximizing profit or minimizing cost, understanding where a function is increasing or decreasing is essential for finding the optimal solution. • Modeling real-world phenomena: Functions are often used to model real-world phenomena, such as population growth, the spread of diseases, and the movement of objects. Understanding the behavior of these functions helps in making accurate predictions and understanding the underlying dynamics. • Making predictions: For example, in finance, understanding the behavior of functions representing stock prices or interest rates is crucial for making informed investment decisions. Understanding Mathematical Functions: Where is the function increasing? In the study of mathematical functions, understanding where a function is increasing is essential for analyzing its behavior and making predictions. In this chapter, we will explore the definition of increasing functions, how to determine where a function is increasing, and the importance of identifying increasing intervals. A. Definition of increasing function An increasing function is a function whose values increase as the input values increase. In other words, as the independent variable (usually denoted as x) increases, the dependent variable (usually denoted as y or f(x)) also increases. Graphically, an increasing function has a rising, or upward-sloping, graph. B. How to determine where a function is increasing To determine where a function is increasing, we can use the first derivative test. The first derivative of a function gives us information about its rate of change. In the context of increasing functions, we look for intervals where the first derivative is positive. This indicates that the function is increasing within those intervals. Another method to determine where a function is increasing is by analyzing its graph. By visually inspecting the graph of a function, we can identify the intervals where the function is rising. C. Importance of identifying increasing intervals Identifying increasing intervals is crucial for several reasons. It helps us understand the behavior of a function and how it changes with respect to its input. This information is valuable in various fields, including economics, physics, and engineering, where analyzing the rate of change is essential for making informed decisions. Additionally, knowing where a function is increasing allows us to locate maximum points, turning points, and critical points, which are significant in optimization problems and curve sketching. Where is the function decreasing? Understanding where a function is decreasing is crucial in mathematical analysis and optimization. It allows us to identify points of diminishing returns and make informed decisions about the behavior of the function. In this chapter, we will delve into the definition of a decreasing function, how to determine where a function is decreasing, and the importance of identifying decreasing A. Definition of decreasing function A function f(x) is considered decreasing on an interval if as x[1] and x[2] are numbers in the interval such that x[1] < x[2], then f(x[1]) > f(x[2]). In other words, as the input value increases, the output value decreases, resulting in a downward trend in the graph of the function. B. How to determine where a function is decreasing To determine where a function is decreasing, we can use the first derivative test. By finding the first derivative of the function and setting it equal to zero, we can identify critical points where the function changes from increasing to decreasing. We can then use the first derivative test or the second derivative test to determine the intervals where the function is decreasing. C. Importance of identifying decreasing intervals Identifying decreasing intervals is important for several reasons. Firstly, it helps us understand the behavior of a function and how it changes with varying input values. This information is crucial for making predictions and decisions based on the function's output. Furthermore, in the context of optimization, identifying where a function is decreasing allows us to pinpoint the intervals where the function's value is decreasing and potentially reach a minimum point. Introducing the increasing and decreasing calculator Understanding mathematical functions can be a complex task, especially when it comes to identifying where a function is increasing or decreasing. However, with the help of a dedicated calculator, this process can be made much more efficient and accurate. A. Overview of the calculator's functionality The increasing and decreasing calculator is a specialized tool designed to analyze the behavior of a mathematical function in terms of its growth or decline. By inputting the function's equation and any relevant parameters, the calculator can provide valuable insights into its increasing and decreasing intervals. B. How the calculator helps identify intervals One of the key functionalities of the increasing and decreasing calculator is its ability to pinpoint the specific intervals where a function is increasing or decreasing. This is particularly useful for understanding the behavior of the function and making informed decisions based on its trends. C. Importance of using the calculator for efficiency Using the increasing and decreasing calculator can significantly improve the efficiency of analyzing mathematical functions. Instead of manually attempting to determine increasing and decreasing intervals, the calculator automates this process, saving time and reducing the likelihood of errors. Understanding Mathematical Functions: Where is the function increasing and decreasing calculator How to use the increasing and decreasing calculator The increasing and decreasing calculator is a valuable tool for understanding the behavior of mathematical functions. This calculator can help determine the intervals where a function is increasing or decreasing, which is crucial for various applications in mathematics and real-world problems. Here's a step-by-step guide on how to use the calculator: • Step 1: Input the function - Enter the mathematical function you want to analyze into the calculator. Make sure to use the proper syntax for the function, including variables and operations. • Step 2: Define the interval - Specify the interval over which you want to determine the function's increasing or decreasing behavior. This can be done by entering the lower and upper bounds of the interval. • Step 3: Calculate the results - Once the function and interval are defined, the calculator will compute and display the intervals where the function is increasing or decreasing. Examples of functions and their increasing/decreasing intervals Let's consider a few examples to illustrate how the increasing and decreasing calculator can be used to analyze functions: • Example 1: f(x) = x^2 - 4x + 3 over the interval [0, 4] • Example 2: g(x) = 3x^3 - 9x^2 + 6x over the interval [-2, 3] • Example 3: h(x) = sin(x) + cos(x) over the interval [0, 2π] Tips for accurate usage of the calculator While using the increasing and decreasing calculator, it's important to keep in mind a few tips to ensure accurate results: • Tip 1: Double-check the function input for any syntax errors or typos. • Tip 2: Verify that the defined interval aligns with the domain of the function to avoid erroneous results. • Tip 3: Understand the concept of increasing and decreasing behavior to interpret the results correctly. Recap: Understanding where a function is increasing and decreasing is crucial for determining the behavior of mathematical functions and making informed decisions in various fields. Encouragement: I strongly encourage you to make use of the increasing and decreasing calculator to simplify your mathematical analysis and gain a deeper understanding of function behavior. Final thoughts: Mathematical functions play a critical role in various disciplines, and being able to identify where a function is increasing and decreasing is a valuable skill. By utilizing tools like the increasing and decreasing calculator, we can enhance our understanding and make meaningful applications in real-world scenarios. 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Numerical simulation of electromagnetic-wave interference induced by ionization-front of millimeter-wave discharge at subcritical conditions and application to discharge structure identification A standing wave induced in front of the ionization-front of a millimeter-wave discharge was numerically investigated to develop an interferometric discharge structure identification method. The time-varying waveform of the standing-wave intensity obtained at a distant observation point was smooth when a continuous comb-shaped structure was formed, whereas it was noisy with high-frequency components when a discrete structure was formed. The peak frequency of the Fourier spectrum of the time-varying waveform was proportional to the ionization-front propagation speed. The rapid time-variation of the waveform was caused by an increase in millimeter-wave absorption in a new plasma spot formation in the discrete structure. The results suggest that discharge structure identification, measurement of ionization-front propagation, and timing of plasma spot formation can be conducted experimentally without using a high-speed camera. Atmospheric discharge induced by high-power millimeter-wave beam irradiation has been actively investigated since the 1980s, along with the development of gyrotrons for nuclear fusion research.^1–21 Discharge experiments have shown that an ionization-front, which is the region where the background gas ionizes, propagates toward the beam source. The propagation speed and shape of the trajectory of the ionization-front, that is, the discharge structure, depend on the beam power density of the millimeter-wave, millimeter-wave frequency, pressure, and background gas species.^11–20 Past experiments have reported some discharge structures, such as $λ/4$,^6–10 comb-shaped,^14–18 tree-branching,^8,20 diffusive,^19,22 and complex filamentary structures.^19,22 In the $λ/4$-structure, the ionization-front propagates discretely. Millimeter-wave discharge is classified into overcritical and subcritical conditions, depending on the local beam power density. The $λ/4$-structure is observed in the overcritical condition, whereas the comb-shaped, diffusive, and complex filamentary structures are observed in the subcritical condition. The overcritical condition is that the electron production by electron-impact ionization exceeds the loss of electrons by the attachment process owing to the high electric-field intensity. The propagation speed was generally greater than 10km/s.^6,7 The balancing condition was defined as the critical intensity $E cr$, which was $2.45 MV / m$ in the case of 170GHz at atmospheric pressure. Under subcritical conditions, the beam-power density is lower than $E cr$, and long-time-scale phenomena appear, such as neutral-gas heating and chemical reactions.^4 The propagation speed is in the range of tens to thousands of meters per second.^16,18,19 The millimeter-wave discharge of the 170GHz band under subcritical conditions has been investigated using the gyrotron developed for ITER.^14–17 This study also focuses on the 170GHz band because abundant experimental data and past numerical simulation models are available. A discharge experiment using a 170GHz Gaussian beam has reported that a comb-shaped structure is formed by granular plasma spots spreading in a diagonal direction.^16^, Figures 1(a) and 1(b) show schematic sketches of the granular plasma spots drawn based on snapshots captured in previous experiments.^16 In fact, the plasma spots in the experiment are more abundant than those shown in the figure. Figure 1(a) shows that the plasma spots arranged in concentric circles are observed from the viewpoint of the beam source. In the side view of Fig. 1(b), the plasma spots are arranged in a bow-shaped arrangement with even spacing. Thus, the central axis region is hidden by the peripheral plasma spots from a side viewpoint. Moreover, the chronophotography of the high-speed camera shows that the propagation speed in the diagonal direction was less than 400m/s, whereas that toward the beam source was approximately 1000m/s.^16 The order of the difference in propagation speed can be caused by a different mechanism of ionization-front formation between the central axis region and the peripheral region. In fact, it has not been observed whether granular plasma spots are formed in the central region because the luminescence of plasma spots arranged in concentric circles shields the emission from the central region of the comb-shaped structure. Moreover, the Gaussian beam provides a higher electric-field region in the central axis region and may cause changes in the ionization mechanism. In addition, an experiment using a flat-top beam has been conducted, and a comb-shaped structure in which the plasma spots are arranged in a row was reported.^21 This difference in the structures indicates that a beam profile with a strong intensity at the central axis is important for forming a comb-shaped structure spreading diagonally. As mentioned above, it is difficult to elucidate millimeter-wave discharge phenomena only through discharge experiments, owing to its supersonic propagation, microscopic millimeter-scale structure, and large gradients of plasma density. Therefore, numerical studies have been conducted actively. In the overcritical conditions, the numerical model coupling electromagnetic-wave propagation and reaction–diffusion of the plasma have simulated the propagation speed and $λ/4$-structure of the overcritical experiments.^23–30 Past numerical simulations have been conducted in the 170GHz band as a target under subcritical conditions because of abundant experimental data. However, the propagation speed of the comb-shaped structure observed at 170GHz was not reproduced in the conventional numerical models. Therefore, it is necessary to modify the numerical model to design engineering applications, such as the beaming propulsion rocket.^21,31–42 Our research group conducted numerical simulations to investigate the formation mechanism of the 170GHz millimeter-wave discharge under subcritical conditions.^33,40 We proposed the combination of an overcritical model^23–30 with compressible neutral fluid, detailed chemical reaction, and radiation transport modules to simulate subcritical discharge.^40 The 1D simulation aimed to reproduce the one-dimensional discharge front propagation along the central axis; however, it cannot simulate the electric-field enhancement by multi-dimensional millimeter-wave interference. The simulation results using this model reproduced three different discharge structures: diffusive, sharp, and discrete, in the order of small electric-field intensity. The discrete structure is formed by electron-impact ionization in the anti-node of the standing wave created by the interference of the incident millimeter-wave and the millimeter-wave reflected by the ionization-front. The sharp structure is formed by electron-impact ionization assisted by a high-temperature neutral gas, which is heated through a fast gas-heating process. The diffusive structure was formed by thermal and associative ionization in a high-temperature neutral gas heated through a vibrational–translational relaxation process. The sharp structure exhibited large electron density gradients, indicating that the continuous plasma spots propagation in the experiment corresponds to the sharp structure in the 1D simulation. The sharp structure was obtained in the numerical simulation in the range of experimental electric-field intensity in which the comb-shaped structure was obtained in the past experiments (0.4–1.2MV/m). The propagation speed was 1000m/s along with the central axis in the experiment but was 100–200m/s in the simulation, which was the same order as the diagonal plasma spots propagation in the experiment at the intensity range of 0.4–1.2MV/m. In the higher electric-field intensity greater than 1.4MV/m, a discrete structure was formed, and its propagation speed was of the order of 1000m/s, which agreed with the propagation speed at the central region of the comb-shaped structure. The propagation speed of sharp and diffusive structures was subsonic because the propagation speed could not exceed the propagation speed of the shock waves; however, the propagation speed of the discrete structure was supersonic because the jumping of the ionization-front overtook the shock wave. These findings have suggested that the central region of the comb-shaped structure may propagate discretely; however, this cannot be concluded only using 1D simulations. Based on these experimental and numerical findings, we propose a hypothesis for the composite discharge structure. In general, the comb-shaped structure in the 170GHz experiment has been considered as granular plasma spots that spread continuously in the radial direction, as shown in Fig. 2(a). However, the propagation speed along the central axis in the experiment is of the same order as that of the discrete propagation reproduced in the numerical simulation. Therefore, it is consistent to consider that the comb-shaped structure propagating at supersonic speed can be a composite structure of discrete propagation in the central region and continuous propagation in the periphery region, as shown in Fig. 2(b). However, observations using high-speed cameras have not confirmed whether a composite structure exists for now because the emission of the peripheral plasma spot hides the central region.^16 Thus, it is necessary to develop a fast observation method that does not rely on an optical camera to identify the discharge structures. Microwave reflectometry is a suitable method that does not rely on an optical camera to observe dense plasma in high-resolution time. At the Prokhorov General Physics Institute, Russia Academy of Science (GPI-RAS), a 75GHz discharge experiment has been conducted, and a comb-shaped structure, similar to the 170GHz experiment, was also formed. They used an interferometer to measure the ionization-front propagation speed; this is known as the location method. However, they did not consider the transition of discharge structures and have assumed that the ionization-front propagates continuously, and the below equation can represent the ionization-front propagation speed $u ion$ because interference of incident and reflected millimeter waves is enhanced when the ionization-front propagates over a distance of $u ion= λ 2 T p= λ f p 2,$ is the wavelength of the millimeter wave, $T p$ is a period in which interference is enhanced when phases of the incident and reflected millimeter waves are aligned, $f p=1/ T p$ is the peak frequency that is obtained by conducting a Fourier transform of interference wave. However, it is unclear whether Eq. can be used when forming discrete structures. Moreover, it has not been experimentally confirmed whether the location method can be applied for discharge structure identification. Therefore, we propose a new interferometric discharge structure identification (IDSI) method using a time-varying waveform of the standing-wave intensity, which is formed by the interference of the incident millimeter-wave and the millimeter-wave reflected by the ionization-front. A one-dimensional schematic diagram of the proposed IDSI method is shown in Fig. 3. A high-power millimeter-wave is irradiated from the left and propagates toward the right in Fig. 3. A standing wave is formed in front of the ionization-front because part of the incident millimeter-wave is reflected in the ionization-front and overlaps with the incident millimeter-wave. As the ionization-front propagates, the standing wave also moves with the ionization-front. Therefore, the discharge structure can be identified by measuring the temporal variation in the standing-wave intensity at an observation point. A smooth time-variation waveform can be obtained when the ionization-front propagates continuously. The previous research considers this condition to measure the propagation speed and assumes that the ionization-front propagates over a distance of $λ/2$ in a period of the enhancement of interference. By contrast, a noisy time-variation waveform containing high-frequency components can be obtained when the ionization-front propagates intermittently or discretely. Moreover, it is unclear whether the propagation speed of the ionization-front can be measured as the same as the continuous propagation. Therefore, this study confirms whether the proposed IDSI method can identify discharge structures and measure the propagation speed using two-dimensional numerical simulations before conducting discharge experiments. The numerical simulation results can provide the relationships between the microscopic structure of the plasma spots and the time-varying waveform of the standing-wave intensity at the observation points. The IDSI method can be applied to reconstruct a three-dimensional discharge structure using multi-point measuring without high-speed cameras. Although the optical system of the IDSI method is one among the special forms of homodyne reflectometry,^43,44 the biggest difference between them is the usage of a high-power millimeter wave generated by a gyrotron. The IDSI method can integrate the mixer and the receiving antenna of ordinary reflectometry by using a high-power millimeter-wave, simplifying the millimeter-wave discharge experiment system. However, if this method is applied to other plasma discharges such as ExB devices, the advantage of non-intrusiveness to its dynamics can be lost because the high-power millimeter wave for diagnosis promotes ionization, thereby changing the physical properties. For such purposes, typical reflectometry using a low-power microwave is suitable. Moreover, although microwave reflectometry generally aims to measure the electron number density distribution or its fluctuation,^43,44 the IDSI method is specialized in the structural identification of the ionization-front and the measurement of its propagation speed. This is because it is applicable when the electron number density is greater than the cut-off density, the electron number density gradients are very steep, and the spatial scale is smaller than the wavelength of the incident beam. The millimeter-wave discharge can meet this condition and the plasma front can be treated as a fixed end of reflection, making the IDSI method applicable. Although this work focuses on the reflected millimeter wave and plasma structure in the subcritical conditions, previous studies revealed that wave refraction, scattering, and diffraction occur during the discharge process, in addition to reflections. Concretely, Batanov et al. reported the refracted millimeter-wave by the filamentary plasma and EUV haro region in the subcritical discharge experiment,^45 inducing reionization plasma spots behind the ionization-front. However, in this study, the refracted millimeter-wave does not affect the measurement by the IDSI method because the observation points are set on the front side of the ionization-front. Additionally, Cook et al. conducted experimental research to investigate the scattering and diffraction by the millimeter-wave discharge plasma in the overcritical condition.^46 They reported the reflected millimeter-wave signal toward the beam source to be a high-frequency time-varying waveform. The reason was stated in their study, but this waveform may be generated by the intermittent propagation of the ionization-front, according to the same principle of the IDSI method. The purpose of this study is to investigate how ionization-front propagation affects the time-variation of standing-wave intensity at a distant point. A numerical simulation is used to obtain detailed information on the ionization-front propagation and the time-varying waveform of the standing-wave intensity. Using this information, we verify (1) whether the discharge structure can be identified, (2) whether the propagation speed of the ionization-front can be measured, and (3) whether the discrete propagation of the plasma spots can be observed using the standing-wave intensity. We performed 2D simulations using a numerical model coupling electromagnetic-wave propagation, reaction–diffusion of plasma, and compressible neutral fluid modules. This model was proposed by Takahashi et al. in which the transport coefficients were calculated using the Boltzmann equation solver, Bolsig+.^33 Bolsig+ is a free software developed by Laboratoire Plasma et Conversion d’Energie (LAPLACE).^47 However, our model uses approximate formulas to calculate the transport coefficients that have been used in the previous overcritical model^24,25 to reduce computation time. The comb-shaped and discrete structures were reproduced in our model, as in Takahashi et al. under subcritical conditions.^33 In the overcritical condition, the propagation speed and the discharge structure of our model agree with those of previous studies^23–25 if the compressible fluid module is off. The electromagnetic-wave of the transverse magnetic (TM) mode propagation is described using Maxwell’s equations is the electric-field vector, $μ 0$ is the vacuum magnetic permeability, is the magnetic field vector, $ε 0$ is the vacuum electric permittivity, and is the current density vector which is estimated by Ohm’s law as $J=−e n e v e$ . Here, is the elementary charge, $n e$ is the electron number density, and $v e$ is the mean velocity vector of the electrons, which is estimated using the following equation of motion of electrons: $∂ v e ∂ t=− e E m e− v m v e,$ $m e$ is the electron mass and $ν m$ is the elastic collision frequency between the electrons and neutral particles. Here, $ν m$ is calculated using $ν m=5.3× 10 5p$ , where is the pressure of neutral gas. Equations were discretized using the finite-difference time-domain (FDTD) method with total field/scattered field formulation. is numerically integrated using an explicit Euler method. The time scale of electromagnetic-wave propagation is much shorter than that of fluid dynamics; thus, the electric field is treated as a root mean square (RMS) value during one millimeter-wave period in the fluid models to reduce the computational time. The number densities of the electrons $n e$ , the positive ions $n p$ , and the negative ions $n n$ were calculated by solving the following reaction–diffusion equation: $n i$ is the number densities ( $i= e , p , n$ $Γ i$ is the density flux vector, and $S i$ is the source term. Here, the subscripts $e , p , n$ are electrons, positive ions, and negative ions, respectively. The positive and negative ion density fluxes were assumed to be zero, as in a previous study. The electron density flux $Γ e$ is obtained by the following equation: $D eff= ζ D e + D a ζ + 1,$ $D eff$ is the effective diffusion coefficient, ^23^, $ζ= ν i τ m= λ D 2/ L 2$ is the transition parameter. Here, $ν i$ is the ionization frequency, $λ D$ is the Debye length, and is the scale length of the electron distribution. $τ m= ε 0/[e n e( μ i+ μ e)]$ is the Maxwell relaxation time, $μ i$ $μ e$ are the ion and electron mobility, respectively. $D e$ is the free diffusion coefficient of the electron, which is calculated by $D e= μ ek T e/e$ is the Boltzmann constant, and $T e$ is the electron temperature, which is assumed to be 1.5eV in this study. ^24^, $D a$ is the ambipolar diffusion coefficient. The electron mobility is given by $μ e=e/ m e ν m$ , and the ion mobility is assumed to be $μ e/200$ in this study. The source terms $S= S e, S p$ , and $S n$ are obtained using the following equations: $S ion$ is the electron-impact ionization rate, and $S att$ is the electron-attachment rate. Other chemical reactions were ignored in this study because they were not important at higher beam intensities, even under subcritical conditions. Reaction rates were calculated using the following approximation: $S ion= S att ( E eff 3.2 × 10 3 p ) 5.33,$ $E eff$ is the effective electric field as follows: $E eff= E rms ( 1 + ω 2 ν m 2 ) − 1 2,$ $E rms$ is the RMS value of local electric-field intensity and is the angular frequency of the millimeter wave. $E rms$ was calculated as follows: $E rms(τ,x,y)= 1 T ∫ τ τ + T E z ( t , x , y ) 2 d t,$ is the period of the millimeter wave and $E z$ is the component of the electric-field vector . The components of were zero because the irradiated electromagnetic-wave mode was the TM mode in this simulation. A calculation of $E rms$ is conducted along with the FDTD solver. Equation was numerically integrated using the explicit Euler method. The diffusion term in Eq. is discretized using the central difference scheme. Neutral fluid dynamics were introduced using the following 2D Euler equation to evaluate the effect of gas expansion by gas heating via plasma: $∂ Q ∂ t+ ∂ E ∂ x+ ∂ F ∂ y= S.$ $E= [ ρ u ρ u 2 + p ρ u v ( ε + p ) u ],$ $F= [ ρ v ρ u v ρ v 2 + p ( ε + p ) v ],$ is the mass density of the neutral gas, is the total energy per unit volume, is the component of the flow velocity, is the component of the flow velocity, and $W g$ is the heat source term which is calculated by the following equation: $J⋅ E$ is the Joule heating. It is assumed that 30% of the RMS Joule heating energy is transferred to the translational energy. The time integration of Eq. was conducted using the explicit Euler method and spatially discretized using the cell-centered finite volume method. The AUSM-DV scheme and the third-order MUSCL interpolation were used to evaluate the numerical flux. The frequency of the millimeter-wave beam was set to 170GHz following past experiments. The wavelength is 1.76mm. The incident RMS electric-field intensity $E 0 , rms$ , is set as 1.4, 1.6, 1.8, 2.0, 2.1, 2.12, 2.15, and 2.2MV/m, which corresponds to the millimeter-wave power densities of 5.20, 6.79, 8.59 10.6, 11.7, 11.9, 12.3, and 12.8 $GW / m 2$ Figure 4 shows the 2D calculation domain used in this study. The computational domain has , and the grid size . Here, was used as in a previous study. The millimeter-wave beam was irradiated from the left boundary; therefore, the ionization-front propagates toward the left direction. The incident millimeter-wave was set as a plane wave of the TM mode. The TM mode was chosen to reduce the computational time because the discharge structure in the TE mode requires a 1/10 smaller grid size compared with the TM mode. However, discrete and continuous structures are formed either in the TE or TM mode, and the standing wave is expected to be formed in both cases. Therefore, the relationship between the time variation of the standing-wave intensity and the discharge structure can be investigated in both modes. Moreover, the experimental discharge structures were reproduced using the plane wave in the past numerical simulations even though the Gaussian beam was used in the experiments. This is because the discharge occurs at the central beam axis where the beam intensity is high and its spatial variation is sufficiently small to be treated as a plane wave for the plasma. As similar to the previous simulations, in this study, for performing a simple evaluation of our method, the plane wave has been used by assuming that the plasma spot is sufficiently smaller than the beam radius. However, it is expected that the IDSI method will be applicable when a more practical beam profile such as the Gaussian shape is used in the experiment because the phase of the reflected millimeter wave does not change regardless of the beam profile. The initial electron and expansion region of the neutral fluid is set as Gaussian distribution at to ignite discharge at the subcritical condition. The full width at half maximum of the Gaussian distribution was 0.176mm. The peak electron number density is $1.0× 10 15 m − 3$ and the peak expansion ratio of neutral gas is 33.3, which is the same as in previous research. The initial electron number density is considerably smaller than the cut-off density, of the order of $10 21$ $10 22 m − 3$ , to avoid the initial electron effects on the discharge structure. The time step of the plasma and the neutral fluid solver $Δ T fluid$ was calculated as below: The time step of the electromagnetic-wave solver $Δ T EM$ must satisfy the Courant–Friedrichs–Lewy (CFL) condition as below: $Δ T EM< 1 c ( 1 Δ x ) 2 + ( 1 Δ y ) 2= Δ T fluid N 2,$ is the speed of light. In this study, $Δ T EM= Δ T fluid 160$ was used to satisfy Eq. . While the calculation of the plasma and neutral fluid solvers is conducted one time, the calculation of electromagnetic-wave solver is conducted 160 times, and $E rms$ $J⋅ E$ are passed to the plasma and the neutral fluid solvers. The boundary conditions of the electromagnetic waves were set to Mur’s first order absorption condition. Those of reaction–diffusion and Euler equations were set as the outflow boundary conditions. The computational code was parallelized using a message passing interface (MPI) library. The computations were conducted using ten nodes (480 cores) of the JSS3 TOKI-SORA supercomputer system of Japan Aerospace Exploration Agency (JAXA). A. Change of standing wave depending on discharge structures Figures 5(a) and 5(b) show the distributions of the electron number density $n e$ at $E 0 , rms=1.4$ and $2.2 MV / m$, respectively. A comb-shaped structure was formed at 1.4MV/m, as shown in Fig. 5(a), because of the neutral gas expansion through Joule heating, as reported in a previous work.^33 However, a discrete discharge structure was formed at 2.2MV/m, as shown in Fig. 5(b). A high electric-field intensity at the anti-node of the standing wave formed by millimeter-wave interference induces stepwise propagation of the ionization-front. This structure was the same as that in a previous work.^33 Comb-shaped structures are induced below 1.8 $MV / m$ and the discrete structures were formed at least 2.0 $MV / m$. Figures 6(a) and 6(b) are $x$– $t$ diagram of the electron number density $n e$ along $y/λ=2.5$ at $E 0 , rms=1.4$ and 2.2 $MV / m$, respectively. The black lines in Figs. 6(a) and 6(b) indicate the location of the ionization-front, which is determined as the position where the electron density exceeds $1× 10 9$ for the comb-shaped structure and $1× 10 13 cm − 3$ for the discrete structure. Figure 6(a) shows that the ionization-front propagates continuously toward the beam-source direction ( $−x$ direction). However, the ionization-front of the discrete structure propagates as a leapfrog, as shown in Fig. 6(b). Figures 7(a) and 7(b) are the $x$– $t$ diagrams of the normalized RMS value of electric-field intensity $E rms/ E 0 , rms$ along $y/λ=2.5$ at $E 0 , rms=1.4$ and $2.2 MV / m$, respectively. Figures 7(a) and 7(b) show that standing waves were formed in front of the ionization-front. They are induced by the interference of incident millimeter waves and reflected millimeter waves from the ionization-front. Here, the reflection point moves with the ionization-front propagation. The nodes and anti-nodes of standing wave move smoothly along with the continuous ionization-front propagation in the case of forming the comb-shaped structure as shown in Fig. 7(a). However, the nodes and anti-nodes of standing wave move intermittently in case of forming the discrete structure, as shown in Fig. 7(b), because of the stepwise ionization-front propagation. This finding suggests that the time-variation of the standing-wave intensity includes information on the ionization-front propagation and discharge structure. The relationship between the time-varying waveform of the standing-wave and the discharge structure was then examined to distinguish between the comb-shaped and discrete structures. Figure 8(a) shows a time-variation of the standing-wave intensity $E rms/ E 0 , rms$ at the observation position of $(x/λ,y/λ)=(0.01,2.5)$ on the central axis at $E 0 , rms=1.4 MV / m$. This figure shows that the waveform was smooth and did not contain high-frequency components. Figure 8(b) is a Fourier spectrum of the waveform in the case of $E 0 , rms=1.4 MV / m$, which has one strong peak of 0.43MHz without high-frequency components at more than the peak frequency. Figure 9(a) shows a time-variation of the standing-wave intensity $E rms/ E 0 , rms$ at the position of $(x/λ,y/λ)=(0.01,2.5)$ in the case of $E 0 , rms=2.2 MV / m$. Figure 9(b) is a Fourier spectrum of the time-varying waveform at $E 0 , rms=2.2 MV / m$, which has one strong peak of 2.62MHz with high-frequency components greater than the peak frequency. The high-frequency components originated from the stepwise propagation of the ionization-front because they were not formed in the case of the comb-shaped structure, as shown in Fig. 8. The detailed mechanism of the high-frequency components induced in the discrete structure is discussed in Sec. IV C. By contrast, we can conclude that the discrete propagation occurs if a high-frequency waveform is obtained. This IDSI method can be applied to millimeter-wave discharge experiments to distinguish discharge structures without a high-speed camera. B. Waveform and ionization-front propagation speed In the previous experiments,^13 the propagation speed was measured using the peak frequency of the interference wave when a comb-shaped structure was formed. However, it is unclear if this location method can be applied when a discrete structure is induced. In this section, we discuss whether the method can measure the propagation speed of the ionization-front in this case by investigating the relationship between the Fourier spectrum with a strong peak and the ionization-front propagation speed. First, we focused on whether the location method could be applied to the numerical simulation when a continuous structure was induced. Previous experiments have reported that the peak frequency of the waveform is proportional to the propagation speed of the ionization-front.^13 Therefore, if the location method is applicable to a numerical simulation, it is expected that similar relationships will be obtained. Although the time-varying waveform was a mixed signal of incident and reflected millimeter waves in the previous experiment, this waveform corresponds to the temporal evolution of the rms electric-field intensity $E rms$ for the standing wave, which was captured at a specific observation point in the numerical simulation. Here, the standing-wave intensity was normalized by the incident electric-field intensity of the millimeter-wave $E 0 , rms$ to compare the case of the discrete structure. In the numerical simulation in this study, the continuous propagation occurred at $E 0 , rms=1.4,1.6,1.8 MV / m$ . In particular, Fig. 8(a) shows the time-varying waveform at $E 0 , rms=1.4 MV / m$ . The movement of the ionization-front induces a periodic waveform because the standing waves also move with the ionization-front, which behaves as a fixed end for wave reflection. The local maximum and the local minimum in the time-varying waveform correspond to the anti-node and node of the standing waves, respectively. Thus, the ionization-front propagation speed $u ion$ in the numerical simulation is expected to have a relationship of Eq. between the wavelength of the millimeter-wave and the period of the waveform of the standing wave $T p$ because the distance between the node and the anti-node is Table I shows the ionization-front propagation speed obtained from the diagram ( $u ion , m$ ), the peak frequency measured from the Fourier spectrum ( $f p , m$ ), and the theoretical peak frequency obtained by assuming Eq. $f p , th$ ). Therefore, $f p , th$ was evaluated using the below equation: $f p , th= 2 u ion , m λ.$ $u ion , m$ for each $E 0 , rms$ is evaluated from the slope of the black line in diagram of $n e$ , which is calculated using the least square method. In the range of $1.4≤ E 0 , rms≤1.8 MV / m$ Table I shows that $f p , m$ have a good agreement with $f p , th$ . Therefore, it was verified that Eq. worked in our numerical simulation, similar to the experiment for a continuous structure. It is expected that the location method can be applied to discrete structures because a standing wave is also formed, which moves along with the ionization-front. As the next step, we will verify the location method using numerical simulations when a discrete structure is induced. TABLE I. E[0,rms] (MV/m) . u[ion,m] (m/s) . f[p,m] (MHz) . f[p,th] (MHz) . 1.4 374 0.43 0.424 1.6 416 0.47 0.472 1.8 426 0.53 0.483 2 1353 1.51 1.53 2.1 1704 1.97 1.93 2.12 1745 2.02 1.98 2.15 1836 2.22 2.08 2.2 2213 2.62 2.51 E[0,rms] (MV/m) . u[ion,m] (m/s) . f[p,m] (MHz) . f[p,th] (MHz) . 1.4 374 0.43 0.424 1.6 416 0.47 0.472 1.8 426 0.53 0.483 2 1353 1.51 1.53 2.1 1704 1.97 1.93 2.12 1745 2.02 1.98 2.15 1836 2.22 2.08 2.2 2213 2.62 2.51 The relationship between the $f p , th$ obtained by Eq. (23) and $f p , m$ obtained by the numerical simulation in the discrete structure was investigated in the same manner as in the continuous structure. The discrete structure was formed at $E 0 , rms=2.0,2.1,2.12,2.15,2,2 MV / m$. Table I shows that the structural transition from 1.8 to 2.0 $MV / m$ leads to a sudden increase in the ionization-front propagation speed $u ion , m$. The $f p , m$ are plotted against the $u ion , m$ in Fig. 10, which shows a proportional relationship, and the $f p , m$ is in agreement with the theoretical value $f p , th$ as the same as in the case of continuous propagation at less than 1.8MV/m. These findings suggest that the peak frequency in the Fourier spectrum of the standing wave is induced by the ionization-front propagation in both comb-shaped and discrete structures. Therefore, our numerical simulation suggests that the ionization-front propagation speed can be measured using the peak frequency in the frequency spectrum of the standing-wave intensity observed at a specific point even if a structural change of the millimeter-wave discharge occurs. C. Waveform in discrete structure In Sec. IV B, the ionization-front propagation induced a low-frequency component of the time-varying waveform of the standing-wave intensity when comb-shaped and discrete structures were induced. However, the high-frequency component is formed only when a discrete structure is formed. The physical significance of the high-frequency component is discussed in this section. A periodic waveform was formed when a discrete structure was induced as shown in Fig. 9(a). Figure 11 is a close-up view between $t=1.95$ and $t=2.45 μ s$ of Fig. 9(a). The waveform has six local maximum and local minimum points: local maximum (A1), local minimum (A2), local maximum (A3), local minimum (N1), local maximum (N2), and local minimum (N3) points in sequence. In this section, we numbered the points from A1 to A3 for the anti-node and, additionally, N1 to N3 for the node, as shown in Fig. 11, to investigate the relationship between the waveform and plasma spots of the discretely ionization-front propagation. Point 0 in Fig. 11 is the last local minimum point in the previous cycle. Figure 12 shows the distribution of the electron number density $n e$ and the distribution of the normalized local $E rms$ for the anti-node at each time when the local maximum and minimum points are formed, respectively. First, the initial state (point 0) is examined before the local maximum of point A1 is formed. A central plasma spot is formed at $(x/λ,y/λ)=(1.81,2.5)$ as shown in Fig. 12(a). At this time, Fig. 12 (b) shows that a high electric-field intensity region is formed at diagonal positions at $(x/λ,y/λ)=(1.75,2.06)$ and $(1.75,2.93)$ in front of the central plasma spot at $(x/λ,y/λ)=(1.81,2.5)$ due to a focusing of the reflected millimeter wave. Figure 12(b) also shows that the node of the standing wave is formed at the observation point of the waveform at $(x/λ,y/λ)=(0.01,2.5)$. Thus, the waveform exhibits a local minimum (initial state point 0), as shown in Fig. 11. Here, the incident millimeter-wave is completely cut off at $x/λ=2$ as shown in Fig. 12(b) because of a high number density of electron. Subsequently, the local maximum of point A1 is formed at $t=2.14 μ s$ in Fig. 11. At this time, Fig. 12(c) shows that two diagonal plasma spots are formed at $(x/λ,y/λ)=(1.75,2.07)$ and $(1.75,2.93)$ where the high electric-field intensity regions have been formed in Fig. 12(b). This is because of a time lag between the high electric-field region formation and the plasma spot formation. Comparing Figs. 12(b) with 12(d), the node of the standing wave changes to the anti-node at the observation point of $(x/λ,y/λ)=(0.01,2.5)$ because the front of the cut-off region moved $λ/4$ toward beam source ( $x/λ=0$) along the $x$ axis. At $t=2.16 μ s$ of the local minimum of point A2, the standing-wave intensity decreases temporarily. The reason is that as the new central plasma spot is being formed at $(x/λ,y/λ)=(1.55,2.5)$, as shown in Fig. 12(e), the reflected millimeter-wave intensity is reduced because the millimeter-wave absorption is increased in the new plasma spot where the electron number density is less than the cut-off density $n c$. Moreover, Fig. 12(f) indicates the electric-field intensity is still enhanced at $(x/λ,y/λ)=(1.55,2.5)$ where the new central plasma spot is being formed, which indicates that the millimeter-energy is absorbed effectively. Comparing Figs. 12(d) with 12(f), the anti-node is maintained at the observation point in Fig. 12(f) because the distribution of the nodes and anti-nodes does not change. However, the 1D distribution of $E rms/ E 0 , rms$ along the central axis (Fig. 13) shows that the electric-field intensity of the standing wave at $(x/λ,y/λ)=(0.01,2.5)$ is decreased at $2.16 μ s$ (A2) compared with $2.14 μ s$ (A1) because of the absorption by the central plasma spot during the formation. In contrast, the standing-wave intensity takes the local maximum value again at $t=2.21 μ s$ as point A3 in Fig. 11. A new central plasma spot has been completely formed at $(x/λ,y/λ)=(1.55,2.5)$ and reaches the cut-off density at this time as shown in Figs. 12(g) and 12(h). Thus, the absorption of the millimeter wave by the central plasma spot is smaller than the time when the local minimum of A2 is formed, which leads to an increase in the electric-field intensity at the observation point compared to that at point A2. Additionally, a local maximum of the waveform is formed because the diagonal plasma spots, which are larger than the central plasma spot, contribute to the formation of an anti-node at the observation points, as shown in Fig. 12(h). Figure 13 shows a 1D distribution of the electric-field intensity $E rms$ along the central axis. At 2.14 and 2.18 $μ s$, the standing intensity is decreased to $x/λ=0$ from $x/λ=1.6$, which indicates that the reflected millimeter wave is diverging. However, the standing-wave intensity is constant between $x/λ=0$ to $1$ at $2.21 μ s$, which suggests that the diagonal plasma spots arrangement shown in Fig. 12(g) contributes to the focus of reflected millimeter-wave toward the observation point. These processes increase the standing-wave intensity at $2.21 μ s$ and induce the local maximum A3. Overall, the small and rapid variations in the standing-wave intensity were caused by the time lag between the formation of the central and diagonal plasma spots. Subsequently, the standing-wave intensity drops rapidly at $t=2.32 μ s$, as shown in Fig. 11 (N1). At this time, a node is formed at the observation points as shown in Fig. 14(b). Figure 14(a) shows that two diagonal plasma spots are formed at $(x/λ,y/λ)=(1.52,2.07)$ and $(1.52,2.93)$. The arrangement of the plasma spots shown in Fig. 14(a) is the same as that in Fig. 12(c), which has the diagonal plasma spots in the head region when the local maximum of point N1 is formed. The reason why the local maximum and the local minimum are replaced is that the node and the anti-node are replaced due to the head of the ionization-front propagating $λ/4$ even though the same plasma spots arrangement is formed. Therefore, the local minimum and maximum of points N1–N3 were formed by the same mechanism as those of points A1–A3. The only difference is whether an anti-node or a node is observed at the observation point. In this numerical simulation, an ideal plasma spot formation, such as staggered propagation, was considered. However, the high-frequency time-variation of the waveform synchronized with a new plasma spot formation can occur in any plasma spots arrangement such as the complex filamentary structure observed in the 28GHz discharge,^19 because millimeter-wave absorption is increased when a new plasma spot is formed. Therefore, the timing of plasma spot formation can be experimentally measured by investigating the high-frequency signal superimposed on the lower-frequency waveform formed by the ionization-front propagation. The time resolution for measuring the standing-wave intensity, which is the same as the electric-field intensity, is of the order of GHz using an oscilloscope, which is much higher than the time resolution of a high-speed camera (MHz). Thus, the plasma spot formation timing obtained using a high-speed camera can be complemented by measuring the standing-wave intensity using an antenna. D. Off-axis observation point In Secs. IV A–IV C, the observation point was set on the axis at $(x/λ,y/λ)=(0.01,2.5)$. However, the antenna and rectifier cannot be installed on the axis in the actual experiment because the millimeter-wave beam propagating on the axis has a sufficiently high intensity to destroy the rectifier. Since the incident beam power density decreases exponentially as it departs from the beam axis because a Gaussian beam is used for the experiment, not a plane wave, the antenna and rectifier are expected to operate without damage due to the weakened incident wave if they are set off-axis. Therefore, for conducting the experiment, it is necessary to investigate the waveform change when the observation points are set off-axis. Here, although our simulation model assumes the plane wave injection for simplification, the waveform change depending on the observation locations can be examined because the phase of the reflected millimeter-wave is not changed depending on the beam The observation points of $(x/λ,y/λ)=(0.01,4.5)$, (0.01,4.99), (0.5,4.99), and (1,4.99) are selected as the off-axis location as shown in Fig. 15. Figures 16(a)–16(d) show the time-varying waveform of the standing-wave intensity observed at each observation point including the on-axis point ( $y/λ=2.5$) for comparison. Figures 16(a) and 16(b) show the case of the comb-shaped structure induced at $E 0 , rms=1.4 MV / m$. Figures 16(c) and 16(d) show the discrete structure induced at $E 0 , rms=2.2 MV / m$. Smooth waveforms were obtained when the comb-shaped structure was induced, while noisy waveforms were obtained when the discrete structure was induced in at each observation point, which was the same as when the observation point was set at $(x/λ,y/λ)=(0.01,2.5)$. Therefore, the IDSI method is useful even if the observation point is not set on the axis. However, the timing when the node and anti-node are observed is changed depending on the observation points as shown in Figs. 16(a)–16(d). This is because the standing-wave distribution is arcuate-shaped due to an effect of the plasma spot arrangement as shown in Fig. 15. As a specific example, the waveform observed at (0.01,4.99) has a local minimum at $t=11.58 μ s$, whereas the waveform observed at (0.01,4.5) has a local maximum near $t=11.58 μ s$ as shown in Fig. 16(a). The reason can be clearly explained by investigating the standing-wave intensity distribution because Fig. 15 shows that the node is located at (0.01,4.99), but the anti-node is located at (0.01,4.5) due to the arc-shaped standing-wave distribution. This result suggests that the arrangement of the plasma spot can be observed by conducting multi-spot measurements of the standing-wave intensity. Figures 17(a)–17(d) show the Fourier spectrum of the waveform observed at $(x/λ,y/λ)=(0.01,4.5)$, (0.01,4.99), (0.5,4.99), and (1,4.99), respectively, at a beam intensity of $E 0 , rms=1.4 MV / m$. The general shapes of the spectrum are almost the same, but the peak frequencies in Figs. 17(a)–17(d) are 0.4, 0.4, 0.34, and 0.34MHz, respectively, which decreases as the observation point deviates from the on-axis or approaches the ionization-front. Here, the peak frequency is 0.43MHz when the observation point is set on-axis, as shown in Fig. 8. Such a decrease in the peak frequency at the off-axis observation points can be explained using Fig. 15. The time interval of the local maximums of the time-varying waveform in Fig. 16 corresponds to the distance between anti-nodes, which is equal to $λ/2$ when the observation point is on the axis as shown in Fig. 15. However, the distance between anti-nodes along the $x$ axis observed at the off-axis point is greater than $λ/2$ as shown using red arrows in Fig. 15. This is due to the arc-shaped standing-wave distribution generated by the plasma spot arrangement. Therefore, in actual experiments, depending on the observation point, the peak frequency is expected to decrease compared with the frequency calculated by Eq. (23) using the actual ionization-front propagation speed. It is suitable to use a beam splitter on the beam axis to measure the propagation speed, similar to the previous study,^13 but the arrangement of the plasma spot can be obtained using the off-axis observation points. Moreover, the off-axis observation point may be applicable for measuring the propagation speed if the apparent wavelength is corrected. This point will be discussed in the forthcoming paper. 2D numerical simulation results of the subcritical millimeter-wave discharge are presented to confirm the IDSI method. The IDSI method can identify continuous comb-shaped and discrete discharge structures by examining the time-variation of the standing-wave intensity in the numerical simulation because a noisy waveform is obtained for the discrete structure, whereas a smooth waveform is obtained for the continuous comb-shaped discharge structure. For both the continuous and discrete discharge structures, the Fourier spectrum of the waveform has a strong peak, and its frequency is proportional to the propagation speed of the ionization-front, which agrees with the theoretical formula proposed in the previous experiment. Thus, the location method can also measure the ionization-front propagation speed by conducting a Fourier transform of the time-variation waveform of the standing-wave intensity even if the discharge structure is changed. Moreover, the high-frequency components observed in the discrete structure are induced because the millimeter-wave absorption during a new plasma spot formation causes a small time-variation of the waveform. These findings can be applied to measure the timing of the plasma spot formation in discharge experiments without using a high-speed camera. We plan to conduct a discharge experiment to measure standing-wave intensity, which will be presented in a forthcoming paper. This research was supported by JSPS KAKENHI (Grant No. 23KJ0103). Numerical simulations were conducted using JSS3 TOKI-SORA supercomputer system of JAXA. We would like to acknowledge Editage ( www.editage.jp) for English language editing. Conflict of Interest The authors have no conflicts to disclose. Author Contributions S. Suzuki: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead). M. Takahashi: Resources (equal); Supervision (lead); Writing – review & editing (lead). The data that support the findings of this study are available from the corresponding author upon reasonable request. Yu. Ya. S. V. V. G. A. G. , and V. E. , “ New mechanism of gasdynamics propagation of a discharge Sov. Phys. JETP. N. A. S. V. , and V. 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A comparative study of surface EMG classification by fuzzy relevance vector machine and fuzzy support vector machine. - PDF Download Free Contact us My IOPscience A comparative study of surface EMG classification by fuzzy relevance vector machine and fuzzy support vector machine This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Physiol. Meas. 36 191 (http://iopscience.iop.org/0967-3334/36/2/191) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 80.82.77.83 This content was downloaded on 17/06/2017 at 05:17 Please note that terms and conditions apply. You may also be interested in: Use of the discriminant Fourier-derived cepstrum with Feature-level post-processing Xinpu Chen, Xiangyang Zhu and Dingguo Zhang Support vector machines to detect physiological patterns for EEG and EMG-based human–computer interaction: a review L R Quitadamo, F Cavrini, L Sbernini et al. 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Meas. 36 (2015) 191–206 Physiological Measurement doi:10.1088/0967-3334/36/2/191 A comparative study of surface EMG classification by fuzzy relevance vector machine and fuzzy support vector machine Hong-Bo Xie1, Hu Huang2, Jianhua Wu1 and Lei Liu1 1 Jiangsu Provincial Key Laboratory for Interventional Medical Devices, Huaiyin I nstitute of Technology, Huaian, Jiangsu Province, 223003, People’s Republic of China 2 Biomedical Informatics and Computational Biology, University of Minnesota, Minneapolis, MN 55414, USA E-mail: [email protected] Received 21 October 2014, revised 2 December 2014 Accepted for publication 3 December 2014 Published 9 January 2015 Abstract We present a multiclass fuzzy relevance vector machine (FRVM) learning mechanism and evaluate its performance to classify multiple hand motions using surface electromyographic (sEMG) signals. The relevance vector machine (RVM) is a sparse Bayesian kernel method which avoids some limitations of the support vector machine (SVM). However, RVM still suffers the difficulty of possible unclassifiable regions in multiclass problems. We propose two fuzzy membership function-based FRVM algorithms to solve such problems, based on experiments conducted on seven healthy subjects and two amputees with six hand motions. Two feature sets, namely, AR model coefficients and room mean square value (AR-RMS), and wavelet transform (WT) features, are extracted from the recorded sEMG signals. Fuzzy support vector machine (FSVM) analysis was also conducted for wide comparison in terms of accuracy, sparsity, training and testing time, as well as the effect of training sample sizes. FRVM yielded comparable classification accuracy with dramatically fewer support vectors in comparison with FSVM. Furthermore, the processing delay of FRVM was much less than that of FSVM, whilst training time of FSVM much faster than FRVM. The results indicate that FRVM classifier trained using sufficient samples can achieve comparable generalization capability as FSVM with significant sparsity in multi-channel sEMG classification, which is more suitable for sEMG-based real-time control applications. 0967-3334/15/020191+16$33.00 © 2015 Institute of Physics and Engineering in Medicine Printed in the UK H-B Xie Physiol. Meas. 36 (2015) 191 Keywords: fuzzy relevance vector machine, sparse kernel machines, electromyography, fuzzy logic, pattern classification (Some figures may appear in colour only in the online journal) 1.Introduction During a voluntary contraction of skeletal muscles, the electrical activity of activated motor units can be detected by surface electrodes (Xie and Wang 2006). The resulting surface electromyographic (sEMG) signal is the summation of motor unit action potentials discharged by muscle fibers near the recording electrodes, and contains rich information on motor unit recruitment, firing, motion intention, and general physiological state of the neuromuscular system. sEMG pattern classification has been widely used in prosthetic hand and exoskeletal control, functional electrical stimulation devices, and other human–machine interface (HMI) control for the elderly, amputees, and those with various neuromuscular disorders (Yu et al 2002, Ahsan et al 2009, Khokhar et al 2010, Hung et al 2012). Artificial intelligence and machine learning play important roles in sEMG pattern recognition, with many techniques based on these having been explored for the control of sEMGbased HMI. In the first pattern recognition-based prosthetic hand control schemes developed in 1970s, simple statistical classifiers were used to recognize hand motions from amplitudebased features, achieving about 75% accuracy in a four-class sEMG classification problem (Englehart and Hudgins 2003). This accuracy was then improved by using two artificial neural network (NN) classifiers, namely, a discrete Hopfield NN and a multi-layer perceptron (MLP), as described by Kelly et al (1990). Hudgins et al (1993) successfully applied the MLP NN, trained by a standard back propagation (BP) algorithm, to develop a real-time sEMG pattern control system with approximately 10% error rate in classifying four types of upper limb motion. Since sEMG signals are non-stationary and noisy, varying even they are belonging to the same motion, BP-based NNs are not able to achieve high learning and discrimination performance (Fukuda et al 2003, Xie and Wang 2006). Several other NN-based machine learning methods, such as radial basis functions network (Chaiyaratana et al 1996), time-delayed artificial NN (Au and Kirsch 2000) and self-organizing feature map (Eom et al 2002, Chu et al 2006), have also been evaluated for their applicability to sEMG classification. sEMG signals are not always strictly repeatable, and may sometimes even be contradictory due to shift of electrodes, sweat, and muscular fatigue (Chan et al 2000). Since one of the most useful properties of fuzzy logic systems is that contradictions in the data can be tolerated, fuzzy logic systems are advantageous in sEMG signal classification. Compared with a MLP network, several fuzzy logic approaches have shown improved accuracy and robustness to noise in sEMG classification (Chan et al 2000, Kiguchi et al 2004, Ajiboye and Weir 2005). NNs exhibit some problems inherent to their architecture, such as overtraining, over-fitting, and the large number of controlling parameters. Other problems relate to the reproducibility of results, due mainly to random initialization of the networks and variability in stopping criteria (Xie et al 2009b). Support vector machine (SVM) classification which is based on the idea of structural risk minimization, is a new technique that has drawn much attention in the field of biomedical engineering in recent years. The good generalization ability of SVM is achieved by finding a large margin between two classes. Performance of binary SVMs can match or exceed MLP and linear discriminant analysis when combined in an efficient manner to classify sEMG signals of hand/wrist motions (Oskoei and Hu 2008, Yan et al 2008a, 2008b). 192 H-B Xie Physiol. Meas. 36 (2015) 191 Despite the fact that SVM classifiers provide improved performance over traditional learning machines, a number of significant and practical disadvantages exist (Tipping 2001). Although relatively sparse, the number of support vectors (SVs) typically grows linearly with the size of the training set, and hence, SVM makes unnecessarily liberal use of basis functions. For some specific algorithms such as least square support vector machine (LS-SVM), the number of SVs equals to that of training samples without sparseness. SVM does not directly provide probability estimates, and therefore is not suitable for classification tasks in which posterior probabilities of class membership are necessary. In addition, estimation of the regularizing parameter in SVM construction, which generally entails a cross-validation procedure, is wasteful of computational time and data. Finally, the SVM kernel function must satisfy Mercer’s condition, namely, it must be a continuous symmetric kernel of a positive integral operator (Tipping 2001). To overcome these problems of SVM efficiently, Tipping (2001) developed a new kernel based machine learning technique, termed relevance vector machine (RVM). The RVM shares many of the characteristics of SVM while avoiding its principal limitations. It uses the sparse Bayesian learning framework, in which a priori parameter structure is placed based on automatic relevance determination theory for removing irrelevant data points (MacKay 1992). Hence, it produces sparse models, as well as a comparable generalization performance to that of SVM. Most importantly, RVM classification requires dramatically fewer relevance vectors (RVs) compared with the number of SVs for SVM classification. This can significantly reduce the computational cost, making RVM more suitable for real-time applications (Majumder et al 2005, Williams et al 2005, Demir and Ertürk 2007, Wang et al 2009). Many sEMG-based control including prosthetic hand and exoskeletal control, as well as wheelchair and robotic control need to be performed in real-time (Englehart and Hudgins 2003, Fukuda et al 2003, Chu et al 2006, Ahsan et al 2009, Khokhar et al 2010, Hung et al 2012). RVM is thus a potentially promising tool to classify sEMG patterns. Similar to SVM, the original RVM is a binary classifier. As for multi-class recognition, several coding schemes have been proposed using binary classifiers (Tipping 2001, Oskoei and Hu 2008). However, indecisive regions often exist when a binary RVM classifier ensemble is used to accommodate a multi-class problem (explained in the next section). In order to solve for unclassifiable regions in RVM, we define two membership functions in a direction perpendicular to the optimal hyperplane that separates the pair of classes. Correspondingly, we construct fuzzy support vector machines (FSVMs) based on least squares algorithm (Yan et al 2008a, 2008b). The performance of the proposed fuzzy relevance vector machines (FRVMs) and FSVMs in classification of sEMG signals is widely compared in terms of classification accuracy, sparsity, training time, test delay, and the effect of training sample size. 2.Methods In this section, we first introduce the basic RVM for binary classification. We then outline multi-class RVM and describe the FRVM method to avoid potential indecisive regions in multi-class RVM. The construction of FSVM is briefly presented. The experimental protocol and feature extraction are also described. 2.1. Binary RVM As a sparse kernel technique, the central idea of RVM is to map a set of inputs to a highdimensional feature space through kernel functions, providing posterior probabilistic outputs 193 H-B Xie Physiol. Meas. 36 (2015) 191 of the class membership for constructing decision boundaries. The compelling feature of RVM is that it utilizes dramatically fewer kernel functions, whilst its generalization performance is comparable to the equivalent SVM (Tipping 2001). Given a data set {xn, tn}nN= 1 where xn denotes the input to be classified and tn represents its class label, we write the targets as a vector t = (t1, … , tN )T , and express it as the sum of an approximation vector y = (y (x1) , … , y (xN ) )T and an ‘error’ vector ε = (ε1, … , εN )T : t = y + ε = Φ w + ε, where w = (w1, … , wM ) is a ‘weight’ parameter vector and Φ = [Φ1 … ΦM ] is a N × M design matrix whose columns comprise the complete set of M ‘basis vectors’. Applying the logistic sigmoid link function σ (y )= (1 + e−y)−1 to y(x), and adopting the Bernoulli distribution, the likelihood of the complete data set can be represented as T P (t w) = ∏ σ{y(xn; w) }tn [1 − σ {y(xn; w) } ]1 − tn , where the targets tn ∈ {0, 1}. To control the complexity of the model and avoid over-fitting, a zero-mean Gaussian prior distribution is defined over w. N p (w α ) = ∏ N wi 0, αi−1 = ∏ i=0 αiwi2 αi exp(− ), 2π 2 with α = [α0, α1, … , αN ]T a vector of N+1 hyperparameters. An individual hyperparameter is associated independently with every weight, moderating the strength of the prior, with the hyperparameter itself having a Gamma prior. The parameter α for each w is intuitively called the ‘relevance’ of that feature, in the sense that the bigger α, the more likely the feature weight w is driven to zero. However, the weights w cannot be integrated out analytically, precluding closed-form expressions for either the weight posterior p (w t, α ) or the marginal likelihood P (t α ). Thus, the Laplace approximation procedure is utilized as described below (Tipping 2001). Since p (w t, α ) ∝ P (t w) p(w α ), finding the optimal weights is equivalent to finding the maximum of ∑ ⎡⎣tn logyn + (1 − tn) log (1 − yn ) ⎤⎦ − N log {P (t w) p (w α ) } = 1 T w Aw, (4) 2 for the most probable weight wMP, with yn = σ {y (xn; w) } and A = diag(αi ) for the current values of α. This represents a penalized logistic log-likelihood function and it requires iterative maximization, using an iterative reweighted least squares algorithm to find wMP. To carry out the iterative procedure, we require the gradient vector and Hessian matrix of the log posterior distribution, which can be found by differentiating twice: ∇ w logp(w t, α ) wMP = Φ T (t − y)− Aw, ∇ w ∇w logp(w t, α ) wMP = −(Φ T BΦ + A), where B is an N × N diagonal matrix with elements bn = yn (1 − yn ), with vector y = (y1, … , yN )T , and Φ the design matrix with elements Φni = ϕi(xn). The approximation to the posterior distribution corresponding to the mean of the Gaussian approximation is obtained by inverting equation (6). The mean and covariance of the Laplace approximation can be now given as Σ = (ΦT BΦ + A)−1 (7) 194 H-B Xie Physiol. Meas. 36 (2015) 191 w MP = A−1Φ T (t − y). Using the statistics Σ and wMP of the Gaussian approximation, we can follow MacKay’s approach to update the hyperparameters αi by new α i = 1 − αiΣii , w2MP where Σii is the ith diagonal element of the covariance matrix. During the optimization process, many αi will have large values, and thus the corresponding model weights will be pruned out, leading to sparse representation. Those samples remaining with wi ≠ 0 are termed relevance vectors, corresponding to support vectors in SVM. The above training procedure is typically slow. In order to speed up the training process, Tipping and Faul (2003) proposed a highly accelerated learning algorithm in which a single hyperparameter αi is fully optimized at each step. If we define ^t = Φ w + B−1(t − y ), MP the approximate log marginal likelihood can be written in the form L (α )= logp(t α, β )= − {N log(2π )+ log C + (^t) C ^t} , T where = B + Φ AΦT . C Considering the dependence of L(α) on a single hyperparameterαi, i ∈{1, 2, ⋯ , M}, the contribution from αi in the matrix C is then factored out to give C = C−i + αi−1ϕiϕiT , where C−i is C with the contribution of basis vector i removed. Established matrix determinant and inverse identities may be used to write the relevant terms in L(α) as C = C−i 1 + αi−1ϕiT C−−1i ϕi , C−−1i ϕiϕiT C−T i . αi + ϕiT C−T iϕi C −1 = C−−1i − Using these results, the log marginal likelihood function (equation (10)) can be written in the form L (α )= L (α−i )+ qi2 ⎤ 1⎡ ⎢logαi − log(αi + si )+ ⎥ = L (α−i )+ λ(αi ). 2⎣ ai + si ⎦ Here, two quantities are introduced s i = ϕiT C−−1i ϕi, q i = ϕiT C−−1i t, where si the sparsity and qi the quality of ϕi. A large value of si relative to qi means that the basis vector ϕi is more likely to be pruned from the model. The ‘sparsity’ measures the extent to which basis vector ϕi overlaps with the other basis vectors in the model. The ‘quality’ represents a measure of alignment of basis vector ϕn with the error between the training set values 195 H-B Xie Physiol. Meas. 36 (2015) 191 t = (t1, t2, ⋯ , tN )T and the vector y-i of predictions that would result from the model with the vector ϕi excluded (Tipping and Faul 2003). 2.2. Multiclass RVM The RVM was originally developed for solving regression and binary classification problems. However, most practical applications need to handle multi-class discrimination problems. Several techniques have been proposed to extend a binary classifier to multi-class problems, including one-against-all (OAA), one-against-one (OAO, also known as pairwise), and errorcorrecting-output code (ECOC) (Tipping and Faul 2003, Mianji and Zhang 2011) In the OAA scheme, a k-class problem is converted into k two-class problems and for the ith two-class problem, class i is discriminated from the remaining classes. As for multiclass RVM, Tipping and Faul (2003) adopted this scheme and extended the original RVM to a multi-class model using a generalized multinomial form of likelihood for equation (2). However, raising the number of classes would lead to a significant increase in OAA computational load. More importantly, OAA generally produces a poor result (Tipping and Faul 2003, Vong et al 2013) since it does not consider the pairwise correlation and hence creates several indecisive regions shown in figure 1(a). In the OAO scheme, the k-class problem is converted into k (k − 1)/2 two-class problems which cover all pairs of classes. For an unknown sample x, the inferred discriminant function for i, j class pair is given by Dij (x)= Φ wMP . The logistic sigmoid function can be applied here to transfer Dij (x) into the probability of P(x, Ci ) and P(x, Cj ). In practice, a ‘Max Wins’ strategy is used in OAO decision process, which first calculates the score function k Di = sgn(Dij (x) ), j ≠ i, j = 1 and classifies x into the class arg (Di(x)). i =max 1, ⋯ , k Although this method is more computational efficient, an unclassifiable region may also exist for OAO if equation (21) is satisfied by multiple i’s. For example, if the discriminant functions satisfy D12(x )	{"url":"https://d.docksci.com/a-comparative-study-of-surface-emg-classification-by-fuzzy-relevance-vector-mach_5a70cfa2d64ab227b9cc453a.html","timestamp":"2024-11-12T03:49:42Z","content_type":"text/html","content_length":"67748","record_id":"<urn:uuid:4c5c7ab6-e463-425d-b101-0409b0e0fb73>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00228.warc.gz"}
May 17, 2010, 8:18:55 AM (14 years ago) • v14 v15 46 46 \|\| Using FFTW using Estimate mode \|\| 0.09 \|\| [http://code.haskell.org/repa/repa-head/repa-examples/FFT/HighPass/legacy/c/FFTW.c FFTW.c] \|\| The vector version uses the same recursive radix-2 decimation in time (DIT) algorithm as the Repa version, but is not rank generalised. It applies a recursive 1d FFT to each row and then 48 transposes the matrix, twice[DEL: each:DEL]. Recursive FFT algorithms tend to be slower than in-place ones because the data is copied into new vectors at each recursion. A 512 point FFT is built from two 256 point FFTs, which are build from 4 128 point FFTs and so on. The result of each FFT is a new vector which needs to be allocated and then filled. The vector version uses the same recursive radix-2 decimation in time (DIT) algorithm as the Repa version, but is not rank generalised. It applies a recursive 1d FFT to each row and then 48 transposes the matrix, twice. Recursive FFT algorithms tend to be slower than in-place ones because the data is copied into new vectors at each recursion. A 512 point FFT is built from two 256 point FFTs, which are build from 4 128 point FFTs and so on. The result of each FFT is a new vector which needs to be allocated and then filled. 50 50 Jones's version also uses a 1d radix-2 DIT FFT kernel, but it first reorders the values then performs a in-place transform using three nested loops.	{"url":"http://repa.ouroborus.net/wiki/Examples/Fft2dHighpass?action=diff&version=15","timestamp":"2024-11-06T18:32:24Z","content_type":"application/xhtml+xml","content_length":"11689","record_id":"<urn:uuid:ea94fe9f-f392-446c-b2c4-617380547710>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00365.warc.gz"}
\] terms. If the first and the last terms are \ Hint: In the question the first and the last terms of an AP are given also it is given that the AP has \[60\] terms. Using this we can find out the common difference between two consecutive terms and then find out the \[{27^{th}}\] term by using the formula of ${n^{th}}$ term in an AP. Complete step-by-step answer: The first term of the AP is given as \[6\] and the last term is \[124\]. In any standard AP, let us assume $a$ to be the first term and $d$ be the common difference of the AP. As the common difference $d$ is not given in the question, we will have to find that using the formula of ${n^{th}}$ term of an AP. The ${n^{th}}$ term of an AP is given by the formula $t = a + (n - 1)d$, where \[a\] is the first term of AP, \[d\] is the common difference and \[t\] is the term itself. The first term of the AP is \[6\] and the last term of the AP is \[124\] which is the \[{60^{th}}\] term as mentioned in the question. Substituting these values in the formula of ${n^{th}}$ term, we get \Rightarrow t = a + (n - 1)d \\ \Rightarrow 124 = 6 + (60 - 1)d \\ \Rightarrow 124 = 6 + 59d \\ \Rightarrow 59d = 124 - 6 \\ \Rightarrow 59d = 118 \\ \Rightarrow d = 2 \\ So, we get the value of \[d\] as \[2\]. Now, we again substitute the values of \[a\], \[d\] and \[n\] as \[6\], \[2\] and \[27\] respectively to get the \[{27^{th}}\] term of the AP. Thus, on substituting, we get \Rightarrow t = a + (n - 1)d \\ \Rightarrow t = 6 + (27 - 1)2 \\ \Rightarrow t = 6 + (26)2 \\ \Rightarrow t = 6 + 52 \\ \Rightarrow t = 58 \\ Thus, the \[{27^{th}}\] term of the AP will be \[58\]. Note: For finding any term of an AP, we need the first term and the common difference. In the question, only the first term was given, so we had to find the common difference by using the data about the last term. For solving questions of AP, students must be well-versed with the formulas to solve the question much faster, otherwise if we solve them by adding the difference many times and then getting the term will be a lot of time wastage.	{"url":"https://www.vedantu.com/question-answer/an-ap-consists-of-60-terms-if-the-first-and-the-class-11-maths-cbse-5f8a837977bf9c142d52555a","timestamp":"2024-11-11T14:52:13Z","content_type":"text/html","content_length":"164483","record_id":"<urn:uuid:24ff24b3-b65e-48fa-ac79-8598a10437c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00586.warc.gz"}
How is it possible for a black hole to possess an event horizon, but not a physical bounda Black holes are intriguing cosmic entities with immense gravitational pull, leading to the formation of an event horizon—a point of no return for matter and energy. This boundary marks the region from which nothing, not even light, can escape the black hole's gravitational grasp. However, it's crucial to understand that the event horizon, despite its significance, is not a physical barrier or surface. It's a mathematical concept representing the boundary beyond which the escape velocity exceeds the speed of light. The absence of a physical boundary surrounding a black hole arises from the theory of general relativity, which describes gravity as the curvature of spacetime. According to this theory, the intense gravity of a black hole curves spacetime to such an extent that it creates a region where the escape velocity is greater than the speed of light. This region, known as the event horizon, is not a tangible structure but rather a mathematical surface that delineates the boundary of the black hole's influence. Within the event horizon, the gravitational forces become so extreme that they distort spacetime, causing objects to fall inward with no possibility of escape. This phenomenon is often described using the concept of "spaghettification," where objects are stretched and compressed into thin, elongated shapes as they approach the black hole's singularity. The absence of a physical boundary around a black hole has profound implications for our understanding of physics and the nature of gravity. It suggests that the event horizon is not a physical object but a mathematical boundary arising from the extreme curvature of spacetime. This concept challenges our traditional notions of space and time and remains an active area of research and exploration in theoretical physics.	{"url":"https://userz.net/2e1fd839-7d54-48a8-a57e-98f7bfdb33e3","timestamp":"2024-11-01T19:43:22Z","content_type":"text/html","content_length":"105646","record_id":"<urn:uuid:e53bd62e-df21-43f4-a41f-f8fa4a2fc59b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00684.warc.gz"}
distributive property simplify Consider the following example. The distributive property of multiplication is a very useful property that lets you simplify expressions in which you are multiplying a number by a sum or difference. We welcome your feedback, comments and questions about this site or page. it multiplies both The outside term distributes evenly into the parentheses i.e. way is to change each positive or negative sign of the terms that were inside Also known as the distributive law of multiplication, it’s one of the most commonly used properties in mathematics. To simplify this multiplication, another method will We've learned about order of operations and combining like terms. Tap for more steps... Simplify each term. Looking for someone to help you with algebra? Because the binomial "3 + 6" is in a set of parentheses, when following the Order of Operations, you must first find the answer I have the expression negative two X squared plus four x minus seven X squared plus five x to the third. The distributive property is a very deep math principle that helps make math work. Start here or give us a call: (312) 646-6365, 2. For more difficult Here's an example with a variable: 6(x + 2) To simplify this expression with the distributive property, we can distribute the 6 to each term inside the parentheses. The following diagram illustrates the basic pattern or formula how to apply it. Distributive Property of Multiplication Multiply by . Now we can simplify the multiplication of the individual terms: The next problem does not have a number outside the parentheses, The Distributive Property Date_____ Period____ Simplify each expression. only a negative sign. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. if the term that the polynomial is being multiplied by is distributed to, or Recall that any term that does not have a coefficient has an implied coefficient of Take a look at the problem below. We offer a whole lot of high-quality reference materials on subjects ranging from power to subtracting polynomials The parentheses are removed and each term from inside is multiplied by the six. (Multiplying two Binomials, or two Polynomials). of 3 + 6, then multiply it by 2. So in this section we're gonna be talking about simplifying expressions and a property that's called the distributive property. Multiply each term inside grouping symbols by the term outside them % Progress . of 3, no positive or negative sign is shown, so a positive sign is assumed. Read the lesson on Distributive Property if you need to learn how to simplify expressions using the distributive property. The Distributive Property of Multiplication. Distributive Property in Maths. Greatly Appreciated. You can see we got an answer of 2x + 8 using the models. 8(-x-4) = We keep a good deal of high quality reference tutorials on subjects starting from adding to common factor -9(x+4) = Students learn the distributive property, which states that a(b + c) = ab + ac. This mathematics lesson … Next Lesson: So, lemme just rewrite it. In general, this term refers to the distributive property of multiplication which states that the. Subtract from . Try the given examples, or type in your own below. Find online algebra tutors or online math tutors in a couple of clicks. Therefore, 2 + 4x, the expression inside the parentheses, cannot Given verbal and symbolic representations of polynomial expressions, the student will simplify the expression. So, this whole thing simplified, using a little bit of distributive property and combining similar or … 7(-5x+7) = A variable can be distributed into a set of parentheses just as we distributed multiplied with each term inside the parentheses. In math, the distributive property helps simplify difficult problems because it breaks down expressions into the sum or difference of two numbers. But we cannot add x x and 4 4, since they are not like terms. We encourage parents and terms, The distributive property allows for these two numbers to be multiplied by breaking a coefficient of negative one. Find the slope of the line that goes through: (-1, -12) and (9, 68), Hard SAT Math question involving rates, distance, and time, Mathematical Journeys: Inverse Operations, or "The Answer is Always 3", ALL MY GRADE 8 & 9 STUDENTS PASSED THE ALGEBRA CORE REGENTS EXAM. Oops I did it again!! If we simplify the right side, we would multiply 4(5) and get 20 and multiply 4(3) and get 12. Distributive property allows you to simplify an expression that has parenthesis (or brackets). In other words, the number or variable that is outside the set of parentheses "distributes" through the parentheses, multiplying by each of the numbers inside. teachers to select the topics according to the needs of the child. For example, if we are asked to simplify the expression 3(x+4) 3 (x + 4), the order of operations says to work in the parentheses first. The Distributive Property of Multiplication over Addition The distributive property of multiplication over addition allows us to eliminate the grouping symbol, usually in the form of a parenthesis. MEMORY METER. Explore how to use the distributive property to simplify numerical and variable expressions. The distributive property helps in making difficult problems simpler. For example, if we are asked to simplify the expression 3 (x + 4), the order of operations says to work in the parentheses first. If I have 21 of something and I take 8 of them away, I'm left with 13 of that something. puzzles. Multiply by by adding the exponents. Distributive Property. But we cannot add x and 4, since they are not like terms. The questions will give you a mathematical expression. . So, those are going to simplify to 13y. Here's a couple of other examples for you to study. For real numbers a,b a, b, and c c: a(b+c)= ab+ac a ( b + c) = a b + a c. What this means is that when a number is multiplied by an expression inside parentheses, you can distribute the multiplier to each term of the expression individually. We hope that the kids will also love the fun stuff and Prefer to meet online? Use distributive property to simplify the expressions. View Distributive+Property-edited.pdf from MATH math at East London College. FOIL MethodUsing the FOIL Method to multiply two or more parenthesis. 5(3x-6) = Negative or minus signs become positive or plus signs. Take for instance the equation a (b + c), which also can be written as (ab) + (ac) because the distributive property dictates that a, which is outside the parenthetical, must be multiplied by both b and c. This gives an answer of 18. Use the "Hint" button to get a free letter if an answer is giving you trouble. The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a It would be incorrect to remove the parentheses and multiply 2 and 3 then And so, that's just going to simplify to "-55". 1. In Mathematics, the numbers should obey the characteristic property during the arithmetic operations. 1/2 (2a-6b+8). Please submit your feedback or enquiries via our Feedback page. We hope that the free math worksheets have been helpful. . Multiply by . … The distributive property is one of the most frequently used properties in math. Fill in all the gaps, then press "Check" to check your answers. Now the -1 can be distributed to each term inside the parentheses as in Apply the distributive property. the first example in this lesson. How do you distribute and simplify #1/2(x - y) - 4#? How do you use the distributive property to simplify #0.25 (6q + 32)#? up the larger one into a sum of smaller ones and then applying the property as shown 4(1-7x) = They learn how number properties help simplify expressions, such as how using the distributive property with numerical expressions can be a helpful mental math strategy. Here, for instance, calculating 8 … The different properties are associative property, commutative property, distributive property, inverse property, identity property and so on. -3(-6x+7) =. look at the problem below. Simplify and combine like terms. before entering the solution. It is used to simplify and solve multiplication equations by distributing the multiplier to each number in the parentheses and then adding those products together to get your answer. Should you actually need assistance with algebra and in particular with distributive property , online calculator or linear inequalities come pay a visit to us at Pocketmath.net. The distributive property, sometimes known as the distributive property of multiplication, tells us how to solve certain algebraic expressions that include both multiplication and addition. Expressions and the Distributive Property. Try the free Mathway calculator and The above multiplications are relatively easier: Practice Problems / WorksheetPractice applying the Distributive Property with these expressions. problem and check your answer with the step-by-step explanations. Use the Distributive Property to simplify each of the following: 1) x − 8 − 3x − 10 2) −8 + 2m − 2m + 9 3) be simplified any further. The property states that the product of a sum or difference, such as 6(5 – 2), is equal to the sum or difference of … Distributive property allows you to simplify an expression that has parenthesis (or brackets). You can use the distributive property of multiplication to rewrite expression by distributing or breaking down a factor as a sum or difference of two numbers. We would get the same answer using the distributive property! You can also click on the "[?]" Objective: I know how to simplify expressions using distributive property. questions, the child may be encouraged to work out the problem on a piece of paper This property states that two or more terms in addition or subtraction with a number are equal to the addition or subtraction of the product of each of the terms with that number. (Do NOT include spaces in your answer) Because of the negative sign on the parentheses, we instead assume Tap for more steps... Move . See all questions in Expressions and the Distributive Property Impact of this question. Thus, we can rewrite the problem as. The distributive property is a property of multiplication used in addition and subtraction. be needed. 2 (3 + 6) Because the binomial "3 + 6" is in a set of parentheses, when following the Order of Operations, you must first find the answer of 3 + 6, then multiply it by 2. How do you distribute #13x(3y + z)#? Multiply by . This indicates how strong in your memory this concept is. How would you solve this equation? Let's take a look. In mathematics there are several operations that can be performed to different algebraic expressions, which go beyond addition, subtraction, multiplication, and division. Specifically, it states that a (b+c) = ab + ac a(b+c) = ab+ac (a+b)c = ac + bc. We will now work through this problem again, but using a different method. We're asked to apply the distributive property. … -5(x+8) = 3(9x+10) = This definition is tough to understand without a good example, so observe OK, that definition is not really all that helpful for most people. Now simplifying the multiplication, we get a final answer of. The first and simplest There are two easy ways to simplify this problem. And then, I have "-35" minus "20". The two terms inside the parentheses cannot be added because they are not Copyright © 2005, 2020 - OnlineMathLearning.com. So the first thing here I have is an expression. It's the rule that lets you expand parentheses, and so it's really critical to understand if you want to get good at simplifying … You could also say that you add (x+4), 2 times which is the way it is shown in the model. Whenever you actually demand advice with math and in particular with distributive property simplify calculator or operations come pay a visit to us at Rational-equations.com. In this lesson students apply the distributive property to generate equivalent expressions. button to get a clue. We can now apply the distributive property to the expression by multiplying each add 6, as this would give an incorrect answer of 12. So, to figure this out, I've actually already copy and pasted this problem onto my scratch pad. I have it right over here. In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. -6(-4-3x) = Embedded content, if any, are copyrights of their respective owners. About This Quiz & Worksheet. Expression Simplifying CalculatorThis calculator will simplify expressions, applying the distributive property when necessary. The distributive property is given by: a(b+c) = ab + ac. (a+b)c = ac+bc. the example below carefully. Note that you will lose points if you ask for hints or clues. The distributive property is the rule that relates addition and multiplication. The quiz is an array of math problems. Read the lesson on Distributive Property if you need to learn how to simplify expressions using the distributive property. 20 + 12 = 32. When you distribute something, you are dividing it into parts. Apply the distributive property. problem solver below to practice various math topics. In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. a negative sign or a number. Multiply the value outside the parenthesis with each of the terms within the parenthesis. Distributive property is one of the most used properties in mathematics. And we have 1/2 times the expression 2a-6b+8. This is a model of what the algebraic expression 2(x+4) looks like using Algebra tiles. 1) 6(1 − 5 m) 6 − 30 m 2) −2(1 − 5v) −2 + 10 v 3) 3(4 + 3r) 12 + 9r 4) 3(6r + 8) 18 r + 24 5) 4(8n + 2) 32 n + 8 Evaluate the following without using a calculator, © 2005 - 2020 Wyzant, Inc. - All Rights Reserved, Next (Simplifying Distribution Worksheet ) >>. Similarly, positive or plus signs become negative or minus signs. The distributive property also can be used to simplify algebraic equations by eliminating the parenthetical portion of the equation. term inside the parentheses by x. 4X, the child may be encouraged to work out the problem on a piece of paper entering! Plus five x to the expression `` 20 '' or plus signs become positive or plus signs brackets ) to. Since they are not like terms we 're gon na be talking about simplifying expressions and the distributive is. Embedded content, if any, are copyrights of their respective owners a! Negative two x squared plus five x to the needs of the terms within the parenthesis each... We keep a good deal of high quality reference tutorials on subjects starting adding... Child may be encouraged to work out the problem 2 ( x+4 ), 2 times which is the that... Two Polynomials ) this problem onto my scratch pad, which are added together would get the answer... 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Is the way it is shown, so a positive sign is assumed: a ( )! That the kids will also love the fun stuff and puzzles breaks down expressions into parentheses! The problem 2 ( x+4 ) looks like using algebra tiles we simplify expressions of polynomial expressions, the will. Please submit your feedback or enquiries via our feedback page - y ) - 4 # commonly properties. Parenthesis with each of the terms that were inside the parentheses as in the thing! Talking about simplifying expressions and the distributive property with these expressions select the topics according to the negative. Parentheses by x check your answer with the step-by-step explanations as shown in model! Property lets you multiply a sum by multiplying each term inside the parentheses can not add x x 4! ( or brackets ) again, but using a different method we 're gon na talking... Tutors and math tutors nearby start here or give us a call: ( 312 ) 646-6365 2! The third problem and check your answer with the step-by-step explanations used to simplify Identify... Multiplied by the term outside them % Progress topics according to the expression negative x! Also can be used to simplify an expression that has parenthesis ( or ). Using the models here or give us a call: ( 312 ) 646-6365, 2 more difficult questions the! Similarly, positive or plus signs become positive or plus signs become negative or minus signs in. Property with these expressions shown, so a positive sign is assumed ( 3y + z #. The first and simplest way is to change each positive or plus signs math math at East London College how. Section we 're gon na be talking about simplifying expressions and a that... ) # most frequently used properties in mathematics fractional coefficients or constants simplify the expressions ) by 2 next:! Also love the fun stuff and puzzles used in addition and subtraction 2 + 4x the! Select the topics according to the needs of the equation 312 ),! Multiply the value outside the parenthesis with each of the terms within the parenthesis with each of child. Simplify the expression inside the parentheses as we simplify expressions using the distributive property a free if! Because it breaks down expressions into the parentheses i.e tutors in a couple of other examples for to! Central Heat Won't Turn On, Tumkur University Colleges, Canon Law 1125, The Spirit Of C Book Pdf, Legal Fees For Administering An Estate,	{"url":"https://jaromirstetina.cz/fi9929j/distributive-property-simplify-f8959f","timestamp":"2024-11-13T09:01:37Z","content_type":"text/html","content_length":"69553","record_id":"<urn:uuid:0e26c0ae-291f-4f4e-8775-a4c48aeca95f>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00217.warc.gz"}
5. (a) Resolve (x−1)(x−2)x in to partial fractions. ... \| Filo Question asked by Filo student 5. (a) Resolve in to partial fractions. (or) Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 4 mins Uploaded on: 8/30/2023 Was this solution helpful? Found 3 tutors discussing this question Discuss this question LIVE for FREE 8 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice more questions on Matrices and Determinant View more Students who ask this question also asked View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text 5. (a) Resolve in to partial fractions. (or) Updated On Aug 30, 2023 Topic Matrices and Determinant Subject Mathematics Class Class 11 Answer Type Video solution: 1 Upvotes 129 Avg. Video Duration 4 min	{"url":"https://askfilo.com/user-question-answers-mathematics/5-a-resolve-in-to-partial-fractions-or-35343839333130","timestamp":"2024-11-12T03:29:35Z","content_type":"text/html","content_length":"267056","record_id":"<urn:uuid:c402c9a5-aca1-4cbc-9ab9-dc69d6700d17>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00733.warc.gz"}
ing For Free Math Homework Answers How To Save Time Searching For Free Math Homework Answers Finding answers for commonly asked questions If your math homework involves using commonly used equations, or questions, then it’s possible you might be able to find answers to the work online. By using search engines to ask any questions that you are being asked, you will soon be able to find whether or not the answers exist online. If you are using very specific questions, then it has to be said that the likelihood of finding exact answers can be quite slim. However, before you try any other methods, it is certainly worth having a look just in case you do find any useful content. Using apps and websites to solve mathematical equations Assuming that you cannot find any answers directly online, it is possible you may still be able to find some useful mathematics-based apps or downloadable programs. If you know the field of mathematics that your homework is based on – e.g. trigonometry - then simply search for apps in the related field. Be sure to check any reviews about the apps before you download them, as not all of them will be of a high enough quality to serve your requirements successfully. However, if they do have a high rating, then it is certainly worth a go. Asking for help with math questions on forums and Q & A websites Another excellent way of getting help is to ask questions on forums and question and answer websites. By doing so, you will be able to communicate with other people and, therefore, get precise an exact answers based on the questions that you have. If you are able to reach out to someone that understands the field of mathematics that you are studying, then there is a good chance that they will be to help you. However, it is worth noting that, unlike the other methods mentioned so far, you are not able to guarantee 100% the success of using this method. One of the main problems is that you are likely to find people will answer your question without knowing the subject well enough, and therefore providing you with false information. Paying for help with math homework when free answers aren’t working If all else fails, there is still the option of paying professional writers and experts to do the work for you. Whilst this method is not free, it is the best chance that you will have of finding a time-saving method of completing your math homework.	{"url":"https://www.uppercarmencharter.org/where-to-go-looking-for-free-math-homework-answers","timestamp":"2024-11-11T09:38:09Z","content_type":"application/xhtml+xml","content_length":"16624","record_id":"<urn:uuid:57592561-4c30-43d1-a451-3338eb97c299>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00650.warc.gz"}
Question from Anna: Geometry Chapter 3 Lesson 5 Number 11. How do we figure this out? Answer from Cassidy Cash: Start with what we know: North East West South make 90-degree angles so <NOE = <EOS = <SOW = 90 degrees A = 50. So to this point, draw with N = 0,... Geometry Final (optional) Use one week on the final. The first 3 days – Study by working problems in the final review section. No need to work every problem, but read over them all, making sure you know what steps to do for each, and review any concepts that you may have forgotten. The ... Geometry – Lesson 15.9 There is no video for this lesson.This is an optional lesson. Assignments 15.9: 1-9, 12-19 (Optional) Chapter 15 Test Download the AskDrCallahan Teacher Guide PDF here in the Introduction/Overview Geometry – Lesson 15.8 There is no video for this lesson. Assignments 15.8: 6-15, 29-36 Download the AskDrCallahan Teacher Guide PDF here in the Introduction/Overview tab. Geometry – Lesson 15.7 Assignments 15.7: 1-23, 44-45, Set III Download the AskDrCallahan Teacher Guide PDF here in the Introduction/Overview...	{"url":"https://askdrcallahan.com/page/2/?s=geometry","timestamp":"2024-11-11T01:00:45Z","content_type":"text/html","content_length":"148069","record_id":"<urn:uuid:87afe702-dad8-4bfb-ac28-68a2ec1af015>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00621.warc.gz"}
IB Mathematics AA HL Flashcards SL 5.5 Introduction to integration as anti differentiation of functions [qdeck ” bold_text=”false”] [h]SL 5.5 Introduction to integration as anti differentiation of functions [q] Intro to Integration For the majority of operations & processes in Maths, we like to know how to do it in reverse. So here, we learn about the opposite of differentiation, known as integration/anti-differentiation. So again, we will learn rules for integrating functions, starting with polynomials. This can be viewed as going from a function to its integral or from a derivative back to its original function. NOTATION: The notation for the integral of $f(x)$ uses $\int f(x)dx$. The ‘dx’ part of it simply means with respect to $x$. RULE: To integrate functions in the form $ax^n + bx^m + \dots$, we simply do the opposite of multiplying by the power, then subtracting 1 from the power, which is: adding 1 to the power, then dividing by that new power, i.e.: \[\int ax^n dx = \frac{ax^{n+1}}{n+1}\] We will now explain the ‘+C’ at the end of the integral. If you consider that any constant disappears when differentiated, then when we are integrating, we must write ‘+C’ at the end, in case there was any constant in the original function. FINDING $C$: If we want to find this mystery constant, then we must know a set of coordinates that fit the original function. We can then plug them in and solve for $C$. We can then write out the whole I will continue with the next pages shortly. Let me know if this is how you’d like it formatted! FINDING $C$: Here are some examples of that process: Example 3: If $\frac{dy}{dx} = 3x^2 + x$ and $y = 10$ when $x = 1$, find $y$: Integrate: $y = x^3 + \frac{x^2}{2} + C$, plug in $(1, 10)$: \[10 = (1)^3 + \frac{(1)^2}{2} + C \Rightarrow C = 8.5\] Write the full equation: $y = x^3 + \frac{x^2}{2} + 8.5$ Note: They won’t tell you when you will be expected to find $C$ as well. When we do $\int f(x)dx$, we get a function as a result, and this is called an indefinite integral. A definite integral is, instead, in the form $\int_a^b f(x) dx$. Here, we get a number as a result of evaluating the integral in the interval $a \leq x \leq b$. This has one main use, which is that it gives us the area between $f(x)$ and the x-axis, between $a$ and $b$ (see above). The theory behind this involves splitting up this area into thin rectangles, and evaluating the limit as the width $\to 0$. But this theory is beyond the scope of the SL course. You will learn how to do this manually (covered in 5.11), but if it is a complicated function, in Paper 2, your GDC can find the definite integral, and therefore, the area under the curve. In 5.11, we will also cover what happens if some of the curve is below the x-axis, and how to find the area between two curves. Calculator Instructions: – TI-nspire: Open CALCULATOR → [SHIFT] → Enter limits and function – TI-84: [MATH] → [fnInt] → Enter limits and function Example 5: Find the area enclosed by $f(x) = e^{-\cos(2x)}$, the x-axis, y-axis, and $x = 3$: Need to evaluate $\int_0^3 e^{-\cos(2x)} dx$. Use GDC, area = 3.60 units². (enter text or “Add Media”; select text to format) IB Mathematics AA HL Flashcards SL 5.5 Introduction to integration as anti differentiation of functions	{"url":"https://www.iitianacademy.com/ibdp-maths/analysis-and-approach-hl/ib-dp-mathematics-aa-hl-flashcards/ib-mathematics-aa-hl-flashcards-sl-5-5-introduction-to-integration-as-anti-differentiation-of-functions/","timestamp":"2024-11-08T05:10:29Z","content_type":"text/html","content_length":"268464","record_id":"<urn:uuid:6256dfb7-6b02-41d9-b5c2-be54277f4468>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00551.warc.gz"}
Use Fibbonacci Retracement to Supercharge your trades - Mycryptopedia Fibonacci Retracement and Extension Not only is the Fibonacci retracement level the trusted tool of many millions of traders out there, there's also some crazy history and mad secrets and patterns that repeat throughout nature. It's no different when trading, and you can exploit this to supercharge your trades if you know what you're doing. For double the power, you can combine this with trading signals from pro traders and double-check your trades for maximum consistency. Fibonacci Sequence: Ground Zero The Fibonacci sequence starts at 1, and each number thereon is an addition of the number itself and the preceding number. 1 1 2 3 5 8 13 21 34 55 89 … to ∞ Several noteworthy patterns emerge in the Fibonacci sequence: • Dividing any number by the next number in the series gives 0.618 • Diving any number by the number 2 places after it in the series gives 0.382 • Dividing any number by the number 3 places after it in the series gives 0.236 These numbers are popularly referred to as Fibonacci ratios. If you divide any 2 Fibonacci ratios, you will get either X.618 or X.382. Similarly, dividing a number with a number preceding it in the series always gives us 1.618; this ratio is often referred to as the Golden Ratio or ‘Phi.’ If this is your first time looking at the Fibonacci sequence, note that the relationship between the numbers in the sequence emerges as you tend towards infinity, and might not be as strong for the first few numbers. Fibonacci Retracement Levels The most commonly used Fibonacci retracement levels technical analysts use are 23.6%, 38.2%, 50%, 61.8%, and 78.6%. All but one (50%) of these numbers are pulled out of the Fibonacci sequence. 50% is still used widely because some technical analysts contend that a drawdown tends to retrace close to half its preceding advance. Most traders ignore the decimal on these numbers since these are supposed to work as estimates anyway. Fibonacci retracement levels are not a silver bullet, but they are an extremely useful tool for forex traders. For instance, let’s say you are taking a bet on the GBP/JPY pair. It moves about 50 pips upward to 150.50 from 150. In this case, a 23.6% retracement would mean the quote will pull back 12 pips to Fibonacci retracements work best on high-volume currency pairs. A low-volume pair is influenced by few individuals, which means it is more prone to irregular movements that will not align with the Fibonacci retracement levels. Using Fibonacci Retracement Levels Fibonacci retracement levels help traders establish a support and resistance level which suggests a point of a possible reversal in the trend’s direction. In turn, this information helps traders identify a good time to open a position. The retracement levels look back at the previous market movements. After a significant spike in the price of a pair, traders will look at the movement and try to identify a price to which it will retrace before continuing its longer-term trend. The same applies to a significant fall in the price. In its simplest form, this is what a trade would look like using Fibonacci retracement levels: When the market is in an uptrend, you will witness a buy pattern emerge. As a trader, you will want to attempt to gauge how far the prices will retrace (to the blue dashed line, in our case) after its initial price rise. The blue line, in our example, is the Fibonacci retracement level and it could be any Fibonacci ratio (i.e., 23.6%, 38.2%, 61.8%, and 78.6%). Let’s look at a retraction on a GBP/JPY chart. The GBP/JPY price is in an upward trend and starts to head upwards of 128 (bottom red line). You are waiting for an opportunity to enter. You believe 62.8% would be a good estimate for a retracement level for this wave. So, when the price pulls back about 60% (at the dashed blue line) from the previous high of 135.37, you open a long position. Next, the upward trend continues, and profits start to accrue. Now, let’s see what it looks like when the GBP/JPY is in a downtrend. After a long downtrend, the price is starting to retrace. You want to use the Fibonacci retracement levels to find at what price point the price will reverse and continue its trend. So, you estimate a 38.2% retracement. At the blue dashed line, you will see that the price retraces about 38.6%—which means the strategy worked in your favor. However, it’s rarely this black and white. This is why Fibonacci retracement levels are used alongside other uncorrelated technical analysis tools for identifying a confluent event. When more indicators give you a similar answer, you will know if your Fibonacci retracement levels are taking you in the right direction. Fibonacci Extension Levels Let’s say you used the Fibonacci retracement levels to enter an upward trend. How do you know when to exit? Enter Fibonacci extension levels. They follow the same underlying principles of the Fibonacci retracement levels but tell you how far the price will keep moving in its current direction after the Fibonacci extension levels help traders identify and validate critical resistance and support areas and potential trend reversal points. They are a great tool for projecting their general bias for a bull/bear trend. However, much the same was Fibonacci retracement levels, the Fibonacci extension levels must be used as a confluence to your currently functional strategy. The extension levels do not necessitate that the price will reverse at this point—it merely tells you the importance of a level. There are no bars in terms of timeframe when using Fibonacci extensions. The most commonly used Fibonacci extension levels are 261.8%, 200%, 161.8%, and 123.6%. Let’s say we want to use the golden ratio (161.8%) to calculate when we should exit the trend. Again, in its simplest form, this is what a trade would look like using Fibonacci extension levels: after a retracement. Upon entering the uptrend at the blue line (left of the illustration), you will want to find how much the price will bounce from that point before it reaches the blue arrow (i.e., 161.8%) based on the last retracement. If you are entering a downtrend at the blue line (right of the illustration), look at the last retracement to predict how far the price will drop before it reaches the blue arrow (i.e., again 161.8%). How are Fibonacci Retracements and Extensions Different? Fibonacci Retracements Fibonacci Extensions Helps us predict a potential retracement level in a given trend Helps us predict potential profit targets after a retracement Estimates drawdowns within a given trend Estimates the level for a wave after a retracement in the trend’s direction (unlike a retracement) Helps traders with a trend-trading strategy identify profitable entry and stop-loss levels Helps traders with a trend-reversal strategy identify profit-targets Works perfectly with other confluences Works well with profit-taking strategies and helps estimate good points for trend reversal Fibonacci ratios used are within the original price trend (50%, 61.8%, 78.6%, etc.) Fibonacci ratios used extend beyond 100% level (150%, 161.8%, 178.6%, etc.) Common Fibonacci Strategies for Forex Every trader navigates a strategy with time. While the best way is to take the first crack at it, it’s always prudent to go in with your eyes wide open and see how other traders are doing things before testing the waters yourself. If you feel a certain option from the ones listed below does not align well with your strategy, feel free to skip over it. Following are some strategies that use Fibonacci retracements: • Use the Fibonacci retracement levels to identify your profit-taking price if you are entering a short position at the tip of a big movement in the price. • You could enter a long position at or near the 50% level and place a stop-loss marginally below the 61.8% level. • Likewise, you could enter a long position at or near the 38.2% level and place a stop-loss marginally below the 50% level. • If the price continues its prior move after retracing to a Fibonacci level, adjust your strategy and consider using the next possible Fibonacci levels of 161.8 or 261.8% for identifying your new support and resistance levels. Issues with Fibonacci Retracements As with any technical indicator, the Fibonacci retracement levels cannot predict the exact point of reversal. It may go beyond, or reverse before reaching the predicted price level. Plus, there are several Fibonacci levels to choose from—23.6%, 38.2%, 61.8%, and 78.6%. These things combined, make Fibonacci retracements far from capable of being used alone for your analysis. In fact, since there are so many levels, the price is bound to reverse at one of them Even though there are tons of examples you will see online, the truth is that price will rarely reverse exactly at any Fibonacci level. Some pairs may even reverse right at the center of two levels, ignoring the levels entirely. Even with these issues, Fibonacci retracements can give you great insight if you know how to use them in your strategies. Does Fibonacci Retracement Give Forex Traders an Edge? Now that you know how to use the Fibonacci retracement levels in your trades, you may be wondering how much of an edge this tool gives you while trading forex. To have an edge over other traders, you need information that they don’t have. Since Fibonacci levels are ingrained in most seasoned forex traders, this by definition eliminates any kind of edge. Nevertheless, forex trading is not entirely the same as stock or commodities trading. Some forex traders have for long emphasized the potency of Fibonacci retracement levels. Even if Fibonacci numbers don’t give you an edge, they do give you a good degree of insight. Some traders even use Fibonacci numbers in more than one way. As an example, think of moving averages. The most commonly used moving averages in the market are the 200-day and 50-day moving averages. Some traders like to experiment with different timeframes, and you could just as well apply the Fibonacci numbers here by analyzing the 39-day or 62-day moving averages. The trading community, as a whole, has been debating over Fibonacci numbers since forever. Some forex traders swear by it, while others find it to be a hit and miss. That being said, it does not give you an edge over other traders. Final Thoughts Designing and implementing a forex trading strategy is never an easy task, especially if you are just venturing into the markets as a fresher. It takes time and experience to get your foot in the door. However, if you familiarize yourself well with the basics, this process will move along much faster. If you learn from the pros you can supercharge your progress by avoiding losing trades. Since the forex market moves in waves, patterns tend to emerge repeatedly over time. Often, the reversal points marked by Fibonacci levels, combined with other useful indicators and back-tested strategies will help traders optimize their performance and profits.	{"url":"https://www.staging.mycryptopedia.com/fibonacci-retracement-and-extension/","timestamp":"2024-11-15T00:48:54Z","content_type":"text/html","content_length":"261889","record_id":"<urn:uuid:6a136093-59da-48a3-a5ff-fa612a774400>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00238.warc.gz"}
RE: st: Detection of disease [Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index] RE: st: Detection of disease From "Seed, Paul" <[email protected]> To "[email protected]" <[email protected]> Subject RE: st: Detection of disease Date Fri, 15 Aug 2008 14:57:55 +0100 Carlo George poses an interesting problem. To deal with a few epidemiological issues first - "...to be 95% certain that the population is free from disease," he must assume - that the disease level is either 20% [Null hypothesis: H0] or 0% [Alternative hypothesis Ha], (a lower rate might well be missed.) - that the sample is representative of the population (a local outbreak outside the sampling area would certainly be missed) - that the test used is 100% sensitive A more realistic goal might be "...to be 95% certain that the test-positive rate is less than 20% in the population represented by the sample." He is interested in a onesided test at the 95% level, as probabilities < 0 have no meaning; so the standard Stata command is sampsi .2 0 , onesample onesided This gives n=11 (not 16), which is still different from the n=14 from the freeware package "Winepiscope" that Carlo uses. The reason is that -sampsi- uses Normal approximations for percentages, which tend to give smaller values than exact tests. To replicate Carlo's result, another approach is needed. This is made much easier by the fact that the disease level is 0% under Ha, so no events are expected. We can perform both tests in Stata; using -bitesti- for the exact test & -prtesti- for the Normal approximation (or Chi-sq test). foreach n of numlist 10/15 { bitesti `n' 0 prtesti `n' 0 Concentrating on the onesided p-values (Ha: p < 0.2), it is clear that 14 subjects is the smallest number to give a significant test by the exact test; and 11 by the Normal approximation. The first figure confirms the Winepiscope result. An added level of sophistication is to look at the confidence intervals. Stata offers several: Wald (a version of the Normal approximation), "exact" (Clopper-Pearson), Wilson, Agresti-Coull, Jeffreys. 90% CI are needed to give a one-sided 95% interval. Both the Wald & Jeffreys intervals perform poorly in this case; but Wilson, "exact" and Agresti-Coull are worth considering. In particular, the Wilson interval seems to fit with the results of -prtestri-, which may be of interest, as there are arguments that the "exact" test is in fact over-conservative (hence the quotation marks). (I could dig out the references if anyone's interested. cii 14 0 , exact level(90) -- Binomial Exact -- Variable \| Obs Mean Std. Err. [90% Conf. Interval] \| 14 0 0 0 .1926362* () one-sided, 95% confidence interval cii 14 0 , wald level(90) -- Binomial Wald --- Variable \| Obs Mean Std. Err. [90% Conf. Interval] \| 14 0 0 0 0 cii 14 0 , wilson level(90) ------ Wilson ------ Variable \| Obs Mean Std. Err. [90% Conf. Interval] \| 14 0 0 0 .1619548 cii 14 0 , agresti level(90) -- Agresti-Coull --- Variable \| Obs Mean Std. Err. [90% Conf. Interval] \| 14 0 0 0 .1907622 The Agresti-Coull interval was clipped at the lower endpoint. cii 14 0 , jeffreys level(90) ----- Jeffreys ----- Variable \| Obs Mean Std. Err. [90% Conf. Interval] \| 14 0 0 0 .1260576 Date: Thu, 14 Aug 2008 11:53:33 +0200 From: "Carlo Georges" <[email protected]> Subject: st: Detection of disease I tried to reproduce in stata the calculation needed for the following case: I need to determine the sample size, required to detct the presence of disease in a population. The formula is rather complex so it is difficult to paste in here, For example i need to detect with 95% confidence the abscence of disease in a population where the presumed prevalence would be 20%. How lrge a sample size do I need to be 95% certain that the population is free from disease. I used a program "Winepiscope" freeware, that calculated a samplesize of 14. in stata i tried : sampsi 0.2 0, power(0.9) onesample and I get a result of :16 Can stata handle this type of calculation? For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/	{"url":"https://www.stata.com/statalist/archive/2008-08/msg00616.html","timestamp":"2024-11-10T15:09:07Z","content_type":"text/html","content_length":"12725","record_id":"<urn:uuid:91cc5ec5-b7ee-4318-9ea1-ff37ddc5151d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00032.warc.gz"}
Scale Drawings Word Problems Scale Drawings Word Problems - Using this information, you can figure out the scale: The second ratio is set up from the _______? Word problems' and thousands of other practice lessons. Scale factor word problems kcm share skill learn with an example questions answered 0 time elapsed smartscore out of 100 ixl's smartscore is a dynamic measure of progress towards mastery, rather than a percentage grade. What is the scale of the drawing? Word problems and thousands of other math skills. The second ratio is set up from the _______? Web scale drawings word problems worksheets in these worksheets, students will solve word problems involving scale. How do you solve problems involving scales? Scale factor word problems kcm share skill learn with an example questions answered 0 time elapsed smartscore out of 100 ixl's smartscore is a dynamic measure of progress towards mastery, rather than a percentage grade. 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PhysicsLAB: War Games Amusing Problems War Games Printer Friendly Version Bullets are flyin', and Dr. J's scared of dyin' as he participates in maneuvers with the local reserve. Actually, they are using rubber bullets, but Dr. J is not convinced that he would rebound from a direct hit. (Dr. J is allergic to bullets - he breaks out in holes.) Even Tripod has deserted his master. He was told to turn in his dogtags when it was discovered that he couldn't salute without failing over. So here was Dr. J all alone and in unfamiliar circumstances. Oh, he had bombed in a school play once, and he often shot off his mouth, but firing guns at tanks was an eerie experience. As he was standing and shaking, he spotted a tank 200 m away moving at 15 m/sec on a line perpendicular to his line of sight. He knows that the speed of the bullet is 300 m/sec. A. At what horizontal angle with his line of sight must Dr. J aim in order to hit the tank? View Correct Answer B. How far in meters to one side of the tank must he aim? View Correct Answer Related Documents Publishing Lab: Resource Lesson: PhysicsLAB Worksheet: Bullets are flyin', and Dr. J's scared of dyin' as he participates in maneuvers with the local reserve. Actually, they are using rubber bullets, but Dr. J is not convinced that he would rebound from a direct hit. (Dr. J is allergic to bullets - he breaks out in holes.) Even Tripod has deserted his master. He was told to turn in his dogtags when it was discovered that he couldn't salute without failing over. So here was Dr. J all alone and in unfamiliar circumstances. Oh, he had bombed in a school play once, and he often shot off his mouth, but firing guns at tanks was an eerie experience. As he was standing and shaking, he spotted a tank 200 m away moving at 15 m/sec on a line perpendicular to his line of sight. He knows that the speed of the bullet is 300 m/sec. A. At what horizontal angle with his line of sight must Dr. J aim in order to hit the tank? View Correct Answer B. How far in meters to one side of the tank must he aim? View Correct Answer Bullets are flyin', and Dr. J's scared of dyin' as he participates in maneuvers with the local reserve. Actually, they are using rubber bullets, but Dr. J is not convinced that he would rebound from a direct hit. (Dr. J is allergic to bullets - he breaks out in holes.) Even Tripod has deserted his master. He was told to turn in his dogtags when it was discovered that he couldn't salute without failing over. So here was Dr. J all alone and in unfamiliar circumstances. Oh, he had bombed in a school play once, and he often shot off his mouth, but firing guns at tanks was an eerie experience. As he was standing and shaking, he spotted a tank 200 m away moving at 15 m/sec on a line perpendicular to his line of sight. He knows that the speed of the bullet is 300 m/sec. A. At what horizontal angle with his line of sight must Dr. J aim in order to hit the tank? B. How far in meters to one side of the tank must he aim?	{"url":"https://www.physicslab.org/Document.aspx?doctype=5&filename=Compilations_AmusingProblems_WarGames.xml","timestamp":"2024-11-05T05:50:51Z","content_type":"application/xhtml+xml","content_length":"33176","record_id":"<urn:uuid:3663b8d9-ac56-451a-b4ff-57620f83b6a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00504.warc.gz"}
The sum of two exterior angles of an isosceles triangle is e... - Ask Spacebar The sum of two exterior angles of an isosceles triangle is equal to 240°. What are the measures of the interior angles? Views: 0 Asked: 01-10 10:45:37 On this page you can find the answer to the question of the mathematics category, and also ask your own question Other questions in category	{"url":"https://ask.spacebarclicker.org/question/516","timestamp":"2024-11-11T14:30:56Z","content_type":"text/html","content_length":"27098","record_id":"<urn:uuid:05cba8dc-6b9a-4434-9c4f-7a24c0ed8163>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00263.warc.gz"}
Theory Weak_Early_Sim Title: The pi-calculus Author/Maintainer: Jesper Bengtson (jebe.dk), 2012 theory Weak_Early_Sim imports Weak_Early_Semantics Strong_Early_Sim_Pres definition weakSimulation :: "pi ⇒ (pi × pi) set ⇒ pi ⇒ bool" (‹_ ↝<_> _› [80, 80, 80] 80) where "P ↝<Rel> Q ≡ (∀a x Q'. Q ⟼a<νx> ≺ Q' ∧ x ♯ P ⟶ (∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel)) ∧ (∀α Q'. Q ⟼α ≺ Q' ⟶ (∃P'. P ⟹⇧^^α ≺ P' ∧ (P', Q') ∈ Rel))" lemma monotonic: fixes A :: "(pi × pi) set" and B :: "(pi × pi) set" and P :: pi and P' :: pi assumes "P ↝<A> P'" and "A ⊆ B" shows "P ↝<B> P'" using assms by(simp add: weakSimulation_def) blast lemma simCasesCont[consumes 1, case_names Bound Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes Eqvt: "eqvt Rel" and Bound: "⋀a x Q'. ⟦Q ⟼ a<νx> ≺ Q'; x ♯ P; x ♯ Q; x ≠ a; x ♯ C⟧ ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and Free: "⋀α Q'. Q ⟼ α ≺ Q' ⟹ ∃P'. P ⟹⇧^^ α ≺ P' ∧ (P', Q') ∈ Rel" shows "P ↝<Rel> Q" proof(auto simp add: weakSimulation_def) fix a x Q' assume QTrans: "Q ⟼ a<νx> ≺ Q'" and "x ♯ P" obtain c::name where "c ♯ P" and "c ♯ Q" and "c ≠ a" and "c ♯ Q'" and "c ♯ C" and "c ≠ x" by(generate_fresh "name") auto from QTrans ‹c ♯ Q'› have "Q ⟼ a<νc> ≺ ([(x, c)] ∙ Q')" by(simp add: alphaBoundOutput) then obtain P' where PTrans: "P ⟹a<νc> ≺ P'" and P'RelQ': "(P', [(x, c)] ∙ Q') ∈ Rel" using ‹c ♯ P› ‹c ♯ Q› ‹c ≠ a› ‹c ♯ C› by(drule_tac Bound) auto from PTrans ‹x ♯ P› ‹c ≠ x› have "P ⟹a<νx> ≺ ([(x, c)] ∙ P')" by(force intro: weakTransitionAlpha simp add: name_swap) moreover from Eqvt P'RelQ' have "([(x, c)] ∙ P', [(x, c)] ∙ [(x, c)] ∙ Q') ∈ Rel" by(rule eqvtRelI) hence "([(x, c)] ∙ P', Q') ∈ Rel" by simp ultimately show "∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast fix α Q' assume "Q ⟼α ≺ Q'" thus "∃P'. P ⟹⇧^^α ≺ P' ∧ (P', Q') ∈ Rel" by(rule Free) lemma simCases[case_names Bound Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes "⋀Q' a x. ⟦Q ⟼ a<νx> ≺ Q'; x ♯ P⟧ ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "⋀Q' α. Q ⟼ α ≺ Q' ⟹ ∃P'. P ⟹⇧^^ α ≺ P' ∧ (P', Q') ∈ Rel" shows "P ↝<Rel> Q" using assms by(auto simp add: weakSimulation_def) lemma simE: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and a :: name and x :: name and Q' :: pi assumes "P ↝<Rel> Q" shows "Q ⟼a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟼α ≺ Q' ⟹ ∃P'. P ⟹⇧^^α ≺ P' ∧ (P', Q') ∈ Rel" using assms by(simp add: weakSimulation_def)+ lemma weakSimTauChain: fixes P :: pi and Rel :: "(pi × pi) set" and Rel' :: "(pi × pi) set" and Q :: pi and Q' :: pi assumes QChain: "Q ⟹⇩[τ] Q'" and PRelQ: "(P, Q) ∈ Rel" and PSimQ: "⋀R S. (R, S) ∈ Rel ⟹ R ↝<Rel> S" shows "∃P'. P ⟹⇩[τ] P' ∧ (P', Q') ∈ Rel" proof - from QChain show ?thesis proof(induct rule: tauChainInduct) case id moreover have "P ⟹⇩[τ] P" by simp ultimately show ?case using PSimQ PRelQ by blast case(ih Q' Q'') have "∃P'. P ⟹⇩[τ] P' ∧ (P', Q') ∈ Rel" by fact then obtain P' where PChain: "P ⟹⇩[τ] P'" and P'Rel'Q': "(P', Q') ∈ Rel" by blast from P'Rel'Q' have "P' ↝<Rel> Q'" by(rule PSimQ) moreover have Q'Trans: "Q' ⟼τ ≺ Q''" by fact ultimately obtain P'' where P'Trans: "P' ⟹⇧^^τ ≺ P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by(blast dest: simE) from P'Trans have "P' ⟹⇩[τ] P''" by simp with PChain have "P ⟹⇩[τ] P''" by auto with P''RelQ'' show ?case by blast lemma simE2: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and a :: name and x :: name and Q' :: pi assumes Sim: "⋀R S. (R, S) ∈ Rel ⟹ R ↝<Rel> S" and Eqvt: "eqvt Rel" and PRelQ: "(P, Q) ∈ Rel" shows "Q ⟹a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟹⇧^^α ≺ Q' ⟹ ∃P'. P ⟹⇧^^α ≺ P' ∧ (P', Q') ∈ Rel" proof - assume QTrans: "Q ⟹a<νx> ≺ Q'" and "x ♯ P" from QTrans obtain Q'' Q''' where QChain: "Q ⟹⇩[τ] Q'''" and Q'''Trans: "Q''' ⟼a<νx> ≺ Q''" and Q''Chain: "Q'' ⟹⇩[τ] Q'" by(blast dest: transitionE) from QChain PRelQ Sim obtain P''' where PChain: "P ⟹⇩[τ] P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(blast dest: weakSimTauChain) from PChain ‹x ♯ P› have "x ♯ P'''" by(rule freshChain) from P'''RelQ''' have "P''' ↝<Rel> Q'''" by(rule Sim) with Q'''Trans ‹x ♯ P'''› obtain P'' where P'''Trans: "P''' ⟹a<νx> ≺ P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by(blast dest: simE) from Q''Chain P''RelQ'' Sim obtain P' where P''Chain: "P'' ⟹⇩[τ] P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: weakSimTauChain) from PChain P'''Trans P''Chain have "P ⟹a<νx> ≺ P'" by(blast dest: Weak_Early_Step_Semantics.chainTransitionAppend) with P'RelQ' show "∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast assume "Q ⟹⇧^^α ≺ Q'" thus "∃P'. P ⟹⇧^^α ≺ P' ∧ (P', Q') ∈ Rel" proof(induct rule: transitionCases) case Step have "Q ⟹α ≺ Q'" by fact then obtain Q'' Q''' where QChain: "Q ⟹⇩[τ] Q''" and Q''Trans: "Q'' ⟼α ≺ Q'''" and Q'''Chain: "Q''' ⟹⇩[τ] Q'" by(blast dest: transitionE) from QChain PRelQ Sim have "∃P''. P ⟹⇩[τ] P'' ∧ (P'', Q'') ∈ Rel" by(rule weakSimTauChain) then obtain P'' where PChain: "P ⟹⇩[τ] P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by blast from P''RelQ'' have "P'' ↝<Rel> Q''" by(rule Sim) with Q''Trans obtain P''' where P''Trans: "P'' ⟹⇧^^α ≺ P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(blast dest: simE) have "∃P'. P''' ⟹⇩[τ] P' ∧ (P', Q') ∈ Rel" using Q'''Chain P'''RelQ''' Sim by(rule weakSimTauChain) then obtain P' where P'''Chain: "P''' ⟹⇩[τ] P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from PChain P''Trans P'''Chain have "P ⟹⇧^^α ≺ P'" by(blast dest: chainTransitionAppend) with P'RelQ' show ?case by blast case Stay have "P ⟹⇧^^τ ≺ P" by simp thus ?case using PRelQ by blast lemma eqvtI: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and perm :: "name prm" assumes PSimQ: "P ↝<Rel> Q" and RelRel': "Rel ⊆ Rel'" and EqvtRel': "eqvt Rel'" shows "(perm ∙ P) ↝<Rel'> (perm ∙ Q)" proof(induct rule: simCases) case(Bound Q' a x) have xFreshP: "x ♯ perm ∙ P" by fact have QTrans: "(perm ∙ Q) ⟼ a<νx> ≺ Q'" by fact hence "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (a<νx> ≺ Q')" by(rule eqvts) hence "Q ⟼ (rev perm ∙ a)<ν(rev perm ∙ x)> ≺ (rev perm ∙ Q')" by(simp add: name_rev_per) moreover from xFreshP have "(rev perm ∙ x) ♯ P" by(simp add: name_fresh_left) ultimately obtain P' where PTrans: "P ⟹(rev perm ∙ a)<ν(rev perm ∙ x)> ≺ P'" and P'RelQ': "(P', rev perm ∙ Q') ∈ Rel" using PSimQ by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹(perm ∙ rev perm ∙ a)<ν(perm ∙ rev perm ∙ x)> ≺ perm ∙ P'" by(rule eqvts) hence "(perm ∙ P) ⟹a<νx> ≺ (perm ∙ P')" by(simp add: name_per_rev) moreover from P'RelQ' RelRel' have "(P', rev perm ∙ Q') ∈ Rel'" by blast with EqvtRel' have "(perm ∙ P', perm ∙ (rev perm ∙ Q')) ∈ Rel'" by(rule eqvtRelI) hence "(perm ∙ P', Q') ∈ Rel'" by(simp add: name_per_rev) ultimately show ?case by blast case(Free Q' α) have QTrans: "(perm ∙ Q) ⟼ α ≺ Q'" by fact hence "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (α ≺ Q')" by(rule eqvts) hence "Q ⟼ (rev perm ∙ α) ≺ (rev perm ∙ Q')" by(simp add: name_rev_per) with PSimQ obtain P' where PTrans: "P ⟹⇧^^ (rev perm ∙ α) ≺ P'" and PRel: "(P', (rev perm ∙ Q')) ∈ Rel" by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹⇧^^ (perm ∙ rev perm ∙ α) ≺ perm ∙ P'" by(rule Weak_Early_Semantics.eqvtI) hence L1: "(perm ∙ P) ⟹⇧^^ α ≺ (perm ∙ P')" by(simp add: name_per_rev) from PRel EqvtRel' RelRel' have "((perm ∙ P'), (perm ∙ (rev perm ∙ Q'))) ∈ Rel'" by(force intro: eqvtRelI) hence "((perm ∙ P'), Q') ∈ Rel'" by(simp add: name_per_rev) with L1 show ?case by blast (***************Reflexivity and transitivity******************) lemma reflexive: fixes P :: pi and Rel :: "(pi × pi) set" assumes "Id ⊆ Rel" shows "P ↝<Rel> P" using assms by(auto intro: Weak_Early_Step_Semantics.singleActionChain simp add: weakSimulation_def weakFreeTransition_def) lemma transitive: fixes P :: pi and Q :: pi and R :: pi and Rel :: "(pi × pi) set" and Rel' :: "(pi × pi) set" and Rel'' :: "(pi × pi) set" assumes QSimR: "Q ↝<Rel'> R" and Eqvt: "eqvt Rel" and Eqvt'': "eqvt Rel''" and Trans: "Rel O Rel' ⊆ Rel''" and Sim: "⋀S T. (S, T) ∈ Rel ⟹ S ↝<Rel> T" and PRelQ: "(P, Q) ∈ Rel" shows "P ↝<Rel''> R" proof - from Eqvt'' show ?thesis proof(induct rule: simCasesCont[where C=Q]) case(Bound a x R') have RTrans: "R ⟼a<νx> ≺ R'" by fact from ‹x ♯ Q› QSimR RTrans obtain Q' where QTrans: "Q ⟹a<νx> ≺ Q'" and Q'Rel'R': "(Q', R') ∈ Rel'" by(blast dest: simE) from Sim Eqvt PRelQ QTrans ‹x ♯ P› obtain P' where PTrans: "P ⟹a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(drule_tac simE2) auto ( by(blast dest: simE2)*) moreover from P'RelQ' Q'Rel'R' Trans have "(P', R') ∈ Rel''" by blast ultimately show ?case by blast case(Free α R') have RTrans: "R ⟼ α ≺ R'" by fact with QSimR obtain Q' where QTrans: "Q ⟹⇧^^ α ≺ Q'" and Q'RelR': "(Q', R') ∈ Rel'" by(blast dest: simE) from Sim Eqvt PRelQ QTrans have "∃P'. P ⟹⇧^^ α ≺ P' ∧ (P', Q') ∈ Rel" by(blast intro: simE2) then obtain P' where PTrans: "P ⟹⇧^^ α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from P'RelQ' Q'RelR' Trans have "(P', R') ∈ Rel''" by blast with PTrans show ?case by blast lemma strongAppend: fixes P :: pi and Q :: pi and R :: pi and Rel :: "(pi × pi) set" and Rel' :: "(pi × pi) set" and Rel'' :: "(pi × pi) set" assumes PSimQ: "P ↝<Rel> Q" and QSimR: "Q ↝[Rel'] R" and Eqvt'': "eqvt Rel''" and Trans: "Rel O Rel' ⊆ Rel''" shows "P ↝<Rel''> R" proof - from Eqvt'' show ?thesis proof(induct rule: simCasesCont[where C=Q]) case(Bound a x R') have RTrans: "R ⟼a<νx> ≺ R'" by fact from QSimR RTrans ‹x ♯ Q› obtain Q' where QTrans: "Q ⟼a<νx> ≺ Q'" and Q'Rel'R': "(Q', R') ∈ Rel'" by(blast dest: Strong_Early_Sim.elim) with PSimQ QTrans ‹x ♯ P› obtain P' where PTrans: "P ⟹a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: simE) moreover from P'RelQ' Q'Rel'R' Trans have "(P', R') ∈ Rel''" by blast ultimately show ?case by blast case(Free α R') have RTrans: "R ⟼ α ≺ R'" by fact with QSimR obtain Q' where QTrans: "Q ⟼α ≺ Q'" and Q'RelR': "(Q', R') ∈ Rel'" by(blast dest: Strong_Early_Sim.elim) from PSimQ QTrans obtain P' where PTrans: "P ⟹⇧^^ α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: simE) from P'RelQ' Q'RelR' Trans have "(P', R') ∈ Rel''" by blast with PTrans show ?case by blast lemma strongSimWeakSim: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" assumes PSimQ: "P ↝[Rel] Q" shows "P ↝<Rel> Q" proof(induct rule: simCases) case(Bound Q' a x) have "Q ⟼a<νx> ≺ Q'" by fact with PSimQ ‹x ♯ P› obtain P' where PTrans: "P ⟼a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: Strong_Early_Sim.elim) from PTrans have "P ⟹a<νx> ≺ P'" by(force intro: Weak_Early_Step_Semantics.singleActionChain simp add: weakFreeTransition_def) with P'RelQ' show ?case by blast case(Free Q' α) have "Q ⟼α ≺ Q'" by fact with PSimQ obtain P' where PTrans: "P ⟼α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: Strong_Early_Sim.elim) from PTrans have "P ⟹⇧^^α ≺ P'" by(rule Weak_Early_Semantics.singleActionChain) with P'RelQ' show ?case by blast	{"url":"https://devel.isa-afp.org/browser_info/current/AFP/Pi_Calculus/Weak_Early_Sim.html","timestamp":"2024-11-12T23:33:56Z","content_type":"application/xhtml+xml","content_length":"271862","record_id":"<urn:uuid:b592548f-ba4c-48fd-9859-27f1761bbee9>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00300.warc.gz"}
Printable Calendars AT A GLANCE Division Worksheets 2 Digit Divisor Division Worksheets 2 Digit Divisor - You can round each number to different a place value (ones, tens, hundreds) if it seems like it will make it easier to estimate. You can also customize them using the generator below. We also have other division resources including flashcards, division games and online division practice. Long division with two divisors. The grids and graph paper help visual learners. 5th grade math vocabulary #2. For each problem, use round numbers to estimate how many times the first number will divide into the second number. Answers are expressed with remainders. These division worksheets precede gradually from some simple to complex exercises, to help students efficiently accomplish the important milestone of learning long division. All long division problems use the standard algorithm. Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. You can round each number to different a place value (ones, tens, hundreds) if it seems like it will make it easier to estimate. Answers are expressed with remainders. Web here you will find links to our many division worksheet pages, including division facts worksheets, division word problems and long division worksheets. Web this worksheets start with simple problems that help master multiple digit divisors and build confidence before progressing to more difficult long division problems. 5th grade math vocabulary #2. These worksheets are pdf files. We also have other division resources including flashcards, division games and online division practice. You can also customize them using the generator below. All long division problems use the standard algorithm. Long division with 2 digit divisor (1280808) These worksheets are pdf files. 3digit by 1digit Standard Division With Remainder Worksheets and Start with simple division facts (e.g. Answers are expressed with remainders. Also tell if you would like remainders. For even more practice, be sure to also check out the division fluency: The worksheets can be made in html or pdf format — both are easy to print. Division By 1 Digit Divisor Worksheet All long division problems use the standard algorithm. For even more practice, be sure to also check out the division fluency: You can also customize them using the generator below. Start with simple division facts (e.g. Also tell if you would like remainders. Two Digit Division Worksheets Division worksheets, Teacher worksheets Long division with 2 digit divisor (1280808) The worksheets can be made in html or pdf format — both are easy to print. All long division problems use the standard algorithm. For more practice, have students complete division fluency: You choose the number of digits in the dividend and the divisor. 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Web create your own long division worksheets! Long Division 2 Digits By 1 Digit With Remainders 8 Worksheets Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. We also have other division resources including flashcards, division games and online division practice. Web create your own long division worksheets! Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. Math, basic operations, special education. 10++ 2 Digit Divisor Division Worksheets Coo Worksheets Long division with remainders division with missing dividend. For each problem, use round numbers to estimate how many times the first number will divide into the second number. For even more practice, be sure to also check out the division fluency: Free \| math \| worksheets \| printable. The worksheets on this page are divided into three major sections: 13 Best Images of Long Division Worksheets 6th Grade 6th Grade Math Free \| math \| worksheets \| printable. For even more practice, be sure to also check out the division fluency: Reaffirm division skills with this section of printable division worksheets. Start with simple division facts (e.g. The first exercises have grids to complete the division, and space for students to write the multiplication table of the divisor in the margin. Division Worksheets 2 Digit Divisor - Web this worksheets start with simple problems that help master multiple digit divisors and build confidence before progressing to more difficult long division problems. Math, basic operations, special education. All long division problems use the standard algorithm. Free \| math \| worksheets \| printable. For each problem, use round numbers to estimate how many times the first number will divide into the second number. The grids and graph paper help visual learners. You can also customize them using the generator below. Long division with remainders division with missing dividend. You can round each number to different a place value (ones, tens, hundreds) if it seems like it will make it easier to estimate. These worksheets are pdf files. You can round each number to different a place value (ones, tens, hundreds) if it seems like it will make it easier to estimate. You can also customize them using the generator below. For each problem, use round numbers to estimate how many times the first number will divide into the second number. The first exercises have grids to complete the division, and space for students to write the multiplication table of the divisor in the margin. Long division with remainders division with missing dividend. For even more practice, be sure to also check out the division fluency: Web create your own long division worksheets! Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. Worksheet #1 worksheet #2 worksheet #3 worksheet #4. You Can Round Each Number To Different A Place Value (Ones, Tens, Hundreds) If It Seems Like It Will Make It Easier To Estimate. We also have other division resources including flashcards, division games and online division practice. All long division problems use the standard algorithm. Answers are expressed with remainders. Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. 5Th Grade Math Vocabulary #2. Web this worksheets start with simple problems that help master multiple digit divisors and build confidence before progressing to more difficult long division problems. Free \| math \| worksheets \| printable. For more practice, have students complete division fluency: Web here you will find links to our many division worksheet pages, including division facts worksheets, division word problems and long division worksheets. These Worksheets Are Pdf Files. Also tell if you would like remainders. You choose the number of digits in the dividend and the divisor. Long division with two divisors. These division worksheets precede gradually from some simple to complex exercises, to help students efficiently accomplish the important milestone of learning long division. Worksheet #1 Worksheet #2 Worksheet #3 Worksheet #4. Math, basic operations, special education. Worksheet #1 worksheet #2 worksheet #3 worksheet #4 worksheet #5 worksheet #6. Web create your own long division worksheets! The worksheets on this page are divided into three major sections: Related Post:	{"url":"https://ataglance.randstad.com/viewer/division-worksheets-2-digit-divisor.html","timestamp":"2024-11-05T04:44:11Z","content_type":"text/html","content_length":"37829","record_id":"<urn:uuid:7c89bab6-8b60-47f4-9549-893daa805027>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00400.warc.gz"}
How do you handle material anisotropy in FEA? \| SolidWorks Assignment Help How do you handle material anisotropy in FEA? We have found that, using a variable-level FEA as the substrate, the anisotropy (x2/x3 ratio) above varies linearly with position. If a variable-level composite is used as the substrate, we can calculate the ratio between x1 and x2/x3 (the same is true if the composite was anisotropy free). When the composite is anisotropy free, the ratio is written as by taking log2 of the ratio below. Click image to enlarge. “What is not established is why we do not rely on our measurements to look at the error or measurement sensitivity, or any form of error,” says Richard Murphy, a former electrical engineer, with Edd Farnum’s Technology Group. He suggests, instead, looking at the “failure of our measurement approach” by estimating both the error and the measurement response. “You can’t just throw a bullet in the sky. It’s always there,” says Farnum. “We don’t know anything about errors!” Research suggests that what was seemingly anisotropy (in fact, the anisotropy defined by x2/x3) is not the same as how they measure errors. You can even improve your measurement method if you use a high-level composite (or some kind of level-level composite for that matter). “Here’s one,” says Michael Jackson, professor of classical physics and current deputy secretary of government and president of the National Academy of Sciences. “To what the work I’ll go on, this is just one of many tricks that we used to get good measurements.” Jackson says that any advanced measurement technique that uses multiexponent measurements cannot be used as the substrate in several ways. “When you have a traditional composites that have a homogeneous dispersion ratio, you generally have a similar degree of misalignment,” he says. “That’s the first thing we need to take into account special info how you measure the response of a composite element that’s higher in thickness than its usual under-refinement. However, composites that have more equivalent or greater dispersion content and that have the same behavior across their thickness variation range and content like this a composite having a higher response than just a normal homogeneous composite film.” In order to ensure that composites can be measured in good condition, anisotropy measurement cannot suffice since it cannot be used for single piece measurements. With each single composite case, the sensitivity to a given value for the anisotropy can differ and typically exceeds that of composite materials. “Sometimes you can’t actually find a metric that it is being measured—that’s the level of measurement sensitivity,”How do you handle material anisotropy in FEA? We have analyzed several existing studies but with the only limitation of conducting a correlation analysis between anisotropic diffusion and the standard deviation. For this, we used a Bayesian approach. Do My Online Class Advantages of Bayesian estimation are (1) Bayesian inference gives high degree of certainty (a higher probability to establish structure in that data) in every case, and thus results are not subject to local inferences from posterior samples, and (2) the method is robust. Introduction ============ A key component of mathematical statistics is the inference. The same as mathematics, a mathematical language is thought of being composed only of a formal here of facts. For instance, the calculus of logarithms is given an analytical formulation as a series of logarithmic terms of the inverse of a number. This formal form is known around the world and will be new applied in this area [@moser-1944; @K-J; @H-J]. Another typical formal formal form (or inference law for a prior probability) like this an inference law for different external variables, and it can be used by computer aided methods like Monte Carlo techniques to study the posterior uncertainty in the solution. It can also be used by computer aided methods like Randomized Model Selection (RMSSD [@Moo-1991]) to learn the posterior distribution of each independent random variable. A direct consequence of the Bayes Principle is that the inference law is not restricted to dependent and independent variables, but there is a relation between those dependent variables with the given information, in other words a law of marginal likelihood. Each of these conditional posterior probability levels are given in the course of how these conditional densities are calculated in a sample. Thus, in Monte Carlo manner, these empirical conditional densities of a posterior characteristic will be the simplest, or simplest, models of our inference model and the degree of certainty generated, but only for a small set of samples. In this paper, we go a step further and find an estimate of the degree-of-certainty variance inflation factor (\[sirVarAsigma\]), where the degree of certainty is defined simply as the ratio between the covariance of the prior and the estimation model (\[bayes\]). This value reflects the degree of certainty of the prior and estimation model: the (relatively ill-conditioned) estimation model[^1],(where the estimate parameter is uncertain and the covariances are unknown) is the more attractive hypothesis. The degree-of-certainty variance inflation factor, or variance inflation factor [@Mie-S; @F-M] can be written as : $$S=t_{\Bbb C}S_{\text{min}}+S_{\text{min-}\ast}+S_{\text{min-1}}*S_{\text{max}}((1-1/\sigma_{\text{minHow do you handle material anisotropy in FEA? Are there some references and articles that understand this one. For example, as I described in the essay you drew me into the topic in this one from the start. On it Take a FEA xxx paper with 25 and 15 censure sheets, every 4 square inches, etc.. Each two inches of this paper will be covered with exactly the same piece of paper but approximately the same colours, black-and-brown, white, gray, and maybe another colour as I described the paper. I said that the paper needs maybe 60 x 30 for that. The paper being painted is the same as the paper being shown, preferably one sheet per 10 x 3 and another sheet 20 x 5, the canvas with an average colour of 70.4 x 40 or 75. Is Paying Someone To Do Your Homework Illegal? . How do I cut this picture into one piece. Can seem to be a bit difficult I think. I have found that it takes some time if you are going to build one an that is much quicker than the other.. and I get into the habit of slicing the paper into 3 pieces at the second that you see through. Let’s take the first 20 lines you can try here a plan of 1 x 10. That’s 4 sheets from the paper that I have written so far. I cut from there a 25 x 15 line and 5 sets, another 20 as for the 3 other lines. Here I cut approximately 1×5 lines for different colours in each of the starting shapes. Here’s what my the canvas looks like. The paper is 9 sheets, approximately 1×4 in size and is transparently tone-blue. It’s also 1×3 for the paper left before it, and 3×3 for the paper right after it. So here I have 4 sheets 4_1 and 4_2, all of which I then cut such as 4_3, 4_4, 4_5, and 4_6. This is 6 sheets, 3_1 and 3_2. These 2 sets of colours are drawn in blue, grey, brown, white and gray – 1, 1_2, 1_3, 1_4, and etc.. With 3_2 they I cut into the sides of the piece forming a square about 6~6 in area and 6~8 in size. I cut this into 3 pieces of 2 × 3~2 and that of 3_1 up to 4. 1_2 for the larger and I cut into squares of about 5×5. Take My Online Exams Review The left piece I cut into 6_1. I cut this in black, which I then cut from this sheet. All the pictures are there, so that’s 2 x 4 sheets, 2x4_1 and 2x4_2, 2x4_3, 2x4_4 and what you see on the right is an all-over picture. The left piece I cut into 4_	{"url":"https://solidworksaid.com/how-do-you-handle-material-anisotropy-in-fea-22484","timestamp":"2024-11-08T05:21:35Z","content_type":"text/html","content_length":"156418","record_id":"<urn:uuid:d626dd6d-0b9e-409d-aa70-722af3d5bb5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00787.warc.gz"}
Fair Winds Not quite finished entry for http://midwestgamejam.org left/right to steer, 'z' to draw in the sail, 'x' to let it out. Finish all the waypoints without running aground! General Sailing tips: Sailing with the wind is easy enough, just let the sail out all the way. You can get the most power this way, but you can't sail faster than the wind. To sail perpendicular to the wind pull the sail in to about a 45 degree angle. This will give you a lot of power for acceleration. Then you can pull the sail in even more to trade power for speed. It's sort of like gears on a car, and you can sail much faster than the speed of the wind this way. Lastly, you can't sail directly against the wind, but you can still sail well up to 45 degrees of the wind if you pull the sail in close. Then just zigzag back and forth across the wind. This is called tacking. 1.1 - Made tacking more forgiving and added red limit lines to make the sail mechanic more intuitive. That's quite fun, and goes to show that I have very little understanding of sailing as I spend a minute trying to go right into the wind. I can't see how to win... The wind never changes, and you don't keep your momentum...? Aha! I finally got the Z/X buttons to do something! ... And I was going in the wrong direction! Okay. I guess I should include some sailing tips along with the instructions! It's supposed to be mildly realistic, so it's probably not very intuitive if you haven't done it before. (Sorry :( ) Sailing with the wind is easy enough, just let the sail out all the way. You can get the most power this way, but you can't sail faster than the wind. To sail perpendicular to the wind pull the sail in to about a 45 degree angle. This will give you a lot of power for acceleration. Then you can pull the sail in even more to trade power for speed. It's sort of like gears on a car, and you can sail much faster than the speed of the wind this way. Lastly, you can't sail directly against the wind, but you can still sail well going NW or SW if you pull the sail in close. Then just zigzag back and forth. This is called tacking. Coming into this already knowing how to sail a small boat, this game was not too difficult but it's great fun! <3 Hell it could even be a teaching tool. Of course it leaves out some crucial details, but how realistic is a Pico-8 sailing sim gonna be? I was just thinking it would be cool to simulate the rudder, so that you see it turn either way. You could even make the boat steer faster or slower based on it's speed, but this would make the game harder. (not that I would mind xD) EDIT: 161 seconds love this game Yeah. "mildly realistic" :D You do turn faster based on speed, but I don't simulate the center of drag or anything so you'd get stuck if there wasn't a minimum turning speed. Glad you liked it! Maybe I should add rain somehow to it so I can enter into the Pico8 Jam too. heh. My high score is 124 seconds, but I also played it far too many times. ;) That was surprisingly fun! Probably my favorite game with "simulator" in the (working) title. I'm thinking of showing this to my father. Let's see if real world experience helps him. I really like the little flapping of the sail when it's not catching wind, very nice touch. This is awesome! I never thought about how sailing works and it was a blast figuring it out on this tiny simulation. I've been dreaming about this game for year. Thanks so much for making it. You really nailed making the controls feel good. If there was more to do in the world (Trading ports? Pirate ships?) I would definitely spend many many hour playing this. Really any story or procedural content could easily turn this from an arcade style game in to a much longer experience. Not that there's anything wrong with the arcade style. Nice work! This is very pretty and really challenging, even with some sailing experience. Glad you don't have to make decisions that quickly in a real boat! I was thinking of making a sailing simulator too... You beat me to it! :-) The graphics are very nice but I think the relationship between speed and scale could be improved... The boat picks up speed so quickly (including backwards) it's easy to run aground without having any time to react. Perhaps you could slow things down a bit? Or maybe I just need to understand sailing dynamics better! ;-) Anyway, nice work! @fuchikoma71 - I think this is what was meant by 'arcade' game feel... but regardless I also think it would be better to scale how boat pickup speed or make much bigger map (but not on pico sadly). @dagondev - a much bigger map would be feasible in pico with procedural methods, but yes, a "real-time" sailing simulator would probably not be fun to play on the platform. Still, I think a compromise between arcade and simulation is often possible (this is what I was aiming for with pic-Orion :-) Yeah, something between sounds cool. I am very interested in your project then! Do you have twitter or other way to follow your development? I'm just a "bedroom programmer" who doesn't even have a twitter account... My cartridge is on the repository though. The interface is a bit "cold" and functional by pico-8 standards and the gameplay is a admittedly rather basic. Much more so than "Fair Winds". But it does what I wanted it to do :-) Look it up, give it a go and let me know what you think... I see only picorion though, I am interested in your sailing simulator which I assumed you will start working on. But I will keep tab on your profile then. :p Thanks... At the moment, I'm not working on the sailing simulator but on something completely different (vintage cyberpunk-inspired puzzler). But I'll keep it in mind for a future project ;-) @dagondev I would play more sailing simulators if such things existed. This was just a 24 hour game jam game, so it doesn't really push Pico8 to it's limits or anything. ;) Yah, the map is the biggest it can be for a static tilemap. I did consider procedural ones, but never really got around to it. I have plenty of code room for it. I think maybe 3/4 of the tokens Sounds like a good idea for a expanding base game. Btw. Would have something against using your code? (ofc with proper credits) I would like experiment a little with this. :) EDIT: any chance for explanation what variables mean/what is happening in crucial part of the code? It would speed up tinkering around this. Thanks! Sure. Go for it. I think all of the carts posted to the forum have an implicit creative commons license anyway. Anyway, it was a game jam game so I never bothered to comment the code, but here is a version with comments: --trinagle drawing swiped from: https://www.lexaloffle.com/bbs/?tid=2734 function lerp(a,b,alpha) return a(1.0-alpha)+balpha function clip(v) return max(-1,min(128,v)) function dtri(x1,y1,x2,y2,x3,y3,c) if(y2<y1) then if(y3<y2) then local tmp = y1 y1 = y3 y3 = tmp tmp = x1 x1 = x3 x3 = tmp local tmp = y1 y1 = y2 y2 = tmp tmp = x1 x1 = x2 x2 = tmp if(y3<y1) then local tmp = y1 y1 = y3 y3 = tmp tmp = x1 x1 = x3 x3 = tmp y1 += 0.001 -- offset to avoid divide per 0 local miny = min(y2,y3) local maxy = max(y2,y3) local fx = x2 if(y2<y3) then fx = x3 local d12 = (y2-y1) if(d12 != 0) d12 = 1.0/d12 local d13 = (y3-y1) if(d13 != 0) d13 = 1.0/d13 local cl_y1 = clip(y1) local cl_miny = clip(miny) local cl_maxy = clip(maxy) for y=cl_y1,cl_miny do local sx = lerp(x1,x3, (y-y1) * d13 ) local ex = lerp(x1,x2, (y-y1) * d12 ) local sx = lerp(x1,x3, (miny-y1) * d13 ) local ex = lerp(x1,x2, (miny-y1) * d12 ) local df = (maxy-miny) if(df != 0) df = 1.0/df for y=cl_miny,cl_maxy do local sx2 = lerp(sx,fx, (y-miny) * df ) local ex2 = lerp(ex,fx, (y-miny) * df ) -- Create a 2x3 "translate, rotate, scale" transform matrix. function trs(tx, ty, angle, sx, sy) rx, ry = cos(angle), -sin(angle) return {rx, ry, -ry, rx, tx, ty} -- Multiply two transform matrices. function transform_mult(t1, t2) return { t1[1]t2[1] + t1[3]t2[2], t1[2]t2[1] + t1[4]t2[2], t1[1]t2[3] + t1[3]t2[4], t1[2]t2[3] + t1[4]t2[4], t1[1]t2[5] + t1[3]t2[6] + t1[5], t1[2]t2[5] + t1[4]t2[6] + t1[6], -- Invert a transform matrix. function transform_inv(t) local inv_det = 1/(t[1]t[4] - t[3]t[2]) return { t[4]inv_det, -t[2]inv_det, -t[3]inv_det, t[1]inv_det, (t[3]t[6] - t[5]t[4])inv_det, (t[5]t[2] - t[1]t[6])inv_det, -- Transform a vector. function transform_v(m, x, y) return m[1]x + m[3]y, m[2]x + m[4]y -- Transform a point. function transform_p(m, x, y) return m[1]x + m[3]y + m[5], m[2]x + m[4]y + m[6] -- Number of frames since the game started. ticks = 0 -- Direction the wind blows. wind_dir = {x = 2, y = 0} -- Arrays for wind/wave effect sprites. winds = {} waves = {} wakes = {} -- x-offset of the mast on the boat. -- Used for drawing the sail. mastx = 6 -- Current waypoint. wayx, wayy = 0, 0 -- Waypoint trigger radius. wayr = 20 -- Waypoints. waypoints = { {x = 117, y = 91}, {x = 31, y = 142}, {x = 354, y = 75}, {x = 250, y = 140}, {x = 47, y = 236}, {x = 33, y = 150}, {x = 140, y = 15}, {x = 136, y = 294}, {x = 35, y = 106}, {x = 245, y = 17} -- Drop the current waypoint and return the coordinates of the next. function pop_waypoint() local wayp = waypoints[1] del(waypoints, wayp) if wayp then return wayp.x, wayp.y finish = ticks function _init() -- Add wind and wave sprites. for i = 1, 16 do add(winds, {x = rnd(128), y = rnd(128)}) add(waves, {x = rnd(128), y = rnd(128)}) -- Setup the boat object. boat = { -- Current transform matrix. m = {1, 0, 0, 1, 0, 0}, -- Mast's transform matrix. mast_m = trs(mastx, 0, 0, 1, 1), -- Current rotation. rotation = 0.25, -- Current angle of the sail boom. boom = 0.25, -- Current angle limit of the sail boom. -- (How much the rope will let it out) limit = 0.25, vel = {x = 0, y = 0}, -- Get the first waypoint. wayx, wayy = pop_waypoint() speed = 0 function _update() -- debug_str = "" -- Get the input vales for the turning and sail. local tiller = (btn(0, 0) and -1 or 0) + (btn(1, 0) and 1 or 0) local sail = (btn(4, 0) and -1 or 0) + (btn(5, 0) and 1 or 0) -- Rotate the boat. boat.rotation += tiller0.007 -- Adjust the sail's max angle. boat.limit = max(0.01, min(boat.limit + 0.006sail, 0.25)) local m = boat.m local mast = transform_mult(m, boat.mast_m) local vx, vy = boat.vel.x, boat.vel.y -- Relative velocity of the boat to the wind. local vrx = wind_dir.x - vx local vry = wind_dir.y - vy -- The direction of the force applied to the sail by the wind. local fn = mast[3]vrx + mast[4]vry -- The value of the force applied to the sail by the wind. local fx, fy = mast[3]fn, mast[4]fn -- Slow down the boat slightly, and push it along using the force. vx = 0.99vx + 0.1fx vy = 0.99vy + 0.1fy -- Rapidly slow the boat down in the lateral direction. local keel_drag = 0.2(m[3]vx + m[4]vy) vx -= keel_dragm[3] vy -= keel_dragm[4] -- Update the boat's properties boat.vel.x = vx boat.vel.y = vy speed = sqrt(vxvx + vyvy) local x = boat.m[5] + vx local y = boat.m[6] + vy boat.m = trs(x, y, boat.rotation, 1, 1) -- Torque applied to the boom by the wind. local btorque = wind_dir.xmast[3] + wind_dir.ymast[4] -- Rotate the boom based on the torque applied to it. boat.boom = max(-boat.limit, min(boat.boom - 0.1btorque, boat.limit)) boat.mast_m = trs(mastx, 0, boat.boom, 1, 1) -- Add sprites for the boat's wake if ticks%4 == 0 then local wx, wy = transform_p(boat.m, -10, 0) local wdx, wdy = transform_v(boat.m, 0, 1) add(wakes, {x = wx, y = wy, dx = wdx, dy = wdy}) if #wakes > 20 then del(wakes, wakes[1]) -- Check if the boat is inside the waypoint. if wayx then local dx, dy = (x - wayx)/wayr, (y - wayy)/wayr if dxdx + dydy < 1 then wayx, wayy = pop_waypoint() ticks += 1 -- Drawing code. Mostly self-explanatory? black = 0 dgrey = 5 lgrey = 6 white = 7 red = 8 brown = 4 blue = 12 peach = 15 function draw_wind(tx, ty) local count = #winds local gust = 0.3 for i = 1, #winds do local phase = 0.01ticks + i/count local dx = wind_dir.x + gustsin(phase) local dy = wind_dir.y + gustsin(0.6phase + 0.5) local wind = winds[i] wind.x = (wind.x + dx)%128 wind.y = (wind.y + dy)%128 pset((wind.x + tx)%128, (wind.y + ty)%128, lgrey) function draw_waves(tx, ty) local count = #waves local duration = count3 for i = 1, count do local phase = (ticks/2 + i)/count%1 local wave = waves[i] spr(flr(6phase), (wave.x + tx)%128, (wave.y + ty)%128) if phase == 0 then wave.x, wave.y = rnd(128), rnd(128) function draw_boat(view) local m = transform_mult(view, boat.m) local x1, y1 = transform_p(m, -10, -4) local x2, y2 = transform_p(m, 8, -4) local x3, y3 = transform_p(m, 8, 4) local x4, y4 = transform_p(m, -10, 4) local x5, y5 = transform_p(m, 12, 0) dtri(x1, y1, x2, y2, x3, y3, brown) dtri(x1, y1, x3, y3, x4, y4, brown) dtri(x2, y2, x3, y3, x5, y5, brown) local mast = transform_mult(m, boat.mast_m) local sdir = wind_dir.xmast[3] + wind_dir.ymast[4] local mx, my = transform_p(mast, 0, 0) local sx1, sy1 = transform_p(mast, -15, 0) local sx2, sy2 = transform_p(mast, -13, 5max(-1, min(3sdir, 1))) dtri(mx, my, sx1, sy1, sx2, sy2, peach) local mx2, my2 = transform_p(mast, -16, 0) line(mx, my, mx2, my2, dgrey) function draw_wakes(tx, ty) local count = #wakes for i = 1, count do local wake = wakes[i] local x, y = wake.x + tx, wake.y + ty local t = 3 + 0.5(count - i) local dx, dy = wake.dxt, wake.dyt pset(x, y, white) pset(x + dx, y + dy, white) pset(x - dx, y - dy, white) function draw_waypoint(tx, ty) if not wayx then return end local x, y = wayx + tx, wayy + ty local sx, sy = x - 64, y - 64 local div = max(abs(sx), abs(sy)) if div > 64 then circ(64/divsx + 64, 64/divsy + 64, 2, red) circ(x, y, wayr + 5sin(ticks/60), red) function _draw() rectfill(0, 0, 128, 128, blue) local m = boat.m local tx, ty = 64 - m[5], 64 - m[6] local view = {1, 0, 0, 1, tx, ty} draw_wakes(tx, ty) draw_waves(tx, ty) draw_waypoint(tx, ty) draw_wind(tx, ty) print("speed: "..flr(30*speed).." knots") if(finish) then print("Finished: "..(finish/30).." s") print("time: "..flr(ticks/30).." s") if debug_str then This is lovely! I would love to try a version with more things to do. I love this little game! I stumbled on it years back, and I just remembered it existed because I've been thinking about learning to sail. This game might have even inspired it ;) My high score was 96.2667! [Please log in to post a comment]	{"url":"https://www.lexaloffle.com/bbs/?pid=17044","timestamp":"2024-11-14T18:34:18Z","content_type":"text/html","content_length":"252368","record_id":"<urn:uuid:9c0f975b-9fba-4c96-88d3-ff1eff86451e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00416.warc.gz"}
Electric Charge and Coulomb’s Law Let’s dig a little deeper into charge, which is the foundation of electricity. This section covers basic knowledge about what an electric charge is and details how to calculate the amount of charge using Coulomb’s Law. What is electric charge? Electric charge is the amount of electricity a charged object has. If an object has more electrons than protons, it is negatively charged. If an object has fewer electrons than protons, it is positively charged. Electric charge is represented by the symbol Q and is measured in a unit called coulombs (C). If the two charges Q1 and Q2 are separated by a distance r [m], then the force working between the charges, represented as F [N], is given by Coulomb’s Law as follows: In general media, the following formula applies: (εs = relative permittivity of medium) Coulomb’s Law When two charged objects come close to each other, a force of repulsion works between charges of the same polarity while a force of attraction works between charges of different polarity. This electric force is called Coulomb force (unit is [N]). The relation between the amount of electric charge and this force is indicated by Coulomb’s Law. An electric charge in an object so tiny that its size is not measurable is called a point charge. If point charge A with an amount of charge Q [C] is approached by point charge B with an amount of charge Q’ [C] in a vacuum at a distance r [m], the magnitude of static force can be calculated using Coulomb’s Law as being inversely proportional to the square of the distance between A and B (r [m]) and being directly proportional to the product of A’s amount of charge and B’s amount of charge. The proportionality constant ε[0] is referred to as vacuum permittivity (8.85×10^-12 [F/m]: farad per metre). The Coulomb force is a force of repulsion when the two charges are of the same polarity and a force of attraction when the polarity is different. Dividing this by acceleration of gravity 9.8 [m/s^2] produces [kg]. The magnitude of static force (F) working between two point charges 1 [C] and -1 [C] separated by a distance of 1 [m] is represented by the following formula. According to this formula, the acting force is approximately 1 million tons, which is equivalent to a force that could lift a weight of 1 million tons. A coulomb [C] is too large a unit to use in realistic situations, so the amount of charge when a 1 [m] square polymer film is frictionally charged, which is around 10^-5 [C], is used. Coulomb’s Law Coulomb’s Law is said to have been discovered by ancient Greek philosopher Thales (BC. 640 to 546). However, it was announced in 1785 that Coulomb (1739 to 1806) had generalised this theorem into a	{"url":"https://www.keyence.com.ph/ss/products/static/static-electricity/electrification/charge.jsp","timestamp":"2024-11-04T23:57:36Z","content_type":"text/html","content_length":"36573","record_id":"<urn:uuid:ef975870-fe01-4154-aa08-13c4791711aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00108.warc.gz"}
How To Foster a Positive Math Mindset - Cognitive Cardio MathCognitive Cardio Math Six Strategies to Foster a Positive Math Mindset How often have you started your year with students who have a negative math mindset? They say they hate math. I don’t know about you, but I find that it happens every year…and when I ask student why, students often can’t say….so, I don’t necessarily believe math is what students “hate” (though they think Instead, I believe it’s something that happens or has happened in math classes that they dislike. My goal is always to help students like math class enough that they’ll open their minds to the possibility that math can be fun, interesting, and even helpful. Developing this positive math mindset can be a challenge when some of them have had a negative math mindset for years. I’ve found that a few simple actions/habits on my part seem to lead to improved student attitudes and the mindset that math might actually be a “likeable” subject. It’s not necessarily the way I explain concepts (although I like to think I do that well:-), but more my attitude of acceptance that I think helps to change their mindsets. In this math mindset post, you’ll find a few strategies I believe have helped my students develop a like (if not a love) of math. Fostering a Positive Math Mindset, Strategy 1: 1) Encourage students to ask questions and then be willing to take the time to answer them. Students often hesitate to ask questions, especially in front of the whole class, and especially at the beginning of the year. I start my year with an activity that encourages students to work together right from the start, to try to foster a comfort level that will encourage them to ask questions in class. I consistently remind students to ask questions when we introduce and discuss concepts. At the beginning of the year, students will typically start asking questions during group work time, and I take the time to answer, even if it takes me a little longer to get to other students. As the year progresses, and students see that I really do want questions and that I really will answer those questions, they become more comfortable asking questions in class. The more they ask, the more they learn. Yes, it takes extra time sometimes, but it’s worth it….I want them to develop deeper understanding, (which helps develop a positive math mindset), and that happens through questioning. Fostering a Positive Math Mindset, Strategy 2: 2) Allow students to talk to each other about math. We know that many students love to talk! Why not encourage them to talk about math? Students often come to my class having had very little chance to talk about math with others, and they are surprised at how often I ask them to do so. They always have a talking purpose – often it’s discussion of the warm-ups they did for homework…. • What were the answers? • How did they solve? • Why do they have different answers? Other times they work on problem solving together…. • What do they know? • How will they approach the problem? • Where are they stuck? Talking about math is so important to developing a positive math mindset, and they come understand that they can share their ideas freely. Fostering a Positive Math Mindset, Strategy 3: 3)Ask why. I ask why all the time. To begin with, students often think that me asking why means they are wrong, so they change their answers. But it doesn’t….it just means I want to know why they think what they’re thinking….I think this makes them feel valued. The more I ask them to justify their thinking, the more able they are to do so, and the more they like to explain….even if it’s not “right”- they know I’m not judging, I’m just listening. I love having them go to the board to illustrate their “whys.” Some students are super willing to do so at the start of the year, while others take a while. But by the end of the year, they feel comfortable explaining their whys….a little proof of the positive math mindset they’re developing. Fostering a Positive Math Mindset, Strategy 4: 4) Tell stories about math in real life. Last year, I told my students about the night my son (who was working as a server in a restaurant) got a $60 tip. I knew the total of the bill, and we were working on percents at the time, so we figured out what percent tip that was….I think it was somewhere around 50%! They were so interested because it was about a real person. It was a short story, but those little tidbits really add to “math interest.” Fostering a Positive Math Mindset, Strategy 5: 5) Be accepting of students’ thinking, explanations, and mistakes. This might be the most important thing you can do to help foster positive math mindset. Sometimes students are on the right track, and sometimes they aren’t….but making mistakes helps to grow the brain, so it’s ok if they aren’t on the right track. Yes, some math problems have one right answer, but when students are simply told, “No, that’s not right,” they may shut down and tune out….feeling embarrassed about making a mistake or sharing “wrong” thinking. Peter Sims, writer for the New York Times, says that successful people “feel comfortable being wrong.” Students need to realize that they ARE doing some correct, valid thinking, even if it leads them to the wrong answer. It can sometimes be challenging to take time in class to find the parts of student thinking that can be built on, to lead to the right answer. But as a math teacher, that’s part of my purpose…take students from where they are, ask them questions, share thoughts, accept their mistakes, understand what they’re thinking….and expand it or redirect it to help grow the concept in their minds. According to Jo Boaler, “One of the most powerful moves a teacher or parent can make is in changing the message they give about mistakes and wrong answers in mathematics.” When students observe you accepting and building from others’ mistakes, they become comfortable sharing their own ideas. Fostering a Positive Math Mindset, Strategy 6: 6) Let them explore math concepts! I know it’s hard to take time to explore, especially when your math periods are short and you may feel pressure to “cover” material within a certain amount of time. I definitely felt that pressure and with 40-min periods, time was certainly short. But, letting students explore math and play with math is so valuable! In Jo Boaler’s book, Mathematical Mindsets, she references brain research and the idea that, “If you learn something deeply, the synaptic activity will create lasting connections in your brain, forming structural pathways, but if you visit an idea only once or in a superficial way, the synaptic connections can “wash away” like pathways made in the sand.” She also references a Park & Brannon study that found, “…the most powerful learning occurs when we use different pathways in the brain…” Giving students time to explore math allows them to explore those pathways and think more deeply. This can only benefit them, build their foundation for the topics you’ll teach, and foster a positive, growth math mindset. You Can Change Math Mindset At some point during the year, most of my “math haters” stop hating math. They realize that maybe it’s not so bad, and they become willing to have conversations about math topics. Students become willing to ask those questions in front of the class and explain their thinking about those tricky problems – even when they don’t know if they’re correct. They become willing to take risks because they know they won’t be judged or simply told, “Sorry, that’s wrong.” And when they learn that their thinking wasn’t quite on track, they don’t feel judged or stupid…and their minds stay open to the learning.For me, one of the best parts of teaching math is watching the metamorphosis of a child, from one who “hates” math to one who has developed a positive math mindset and willingly goes to the front of the class to illustrate and explain his or her math thinking.	{"url":"https://cognitivecardiomath.com/cognitive-cardio-blog/how-to-foster-a-positive-math-mindset/","timestamp":"2024-11-06T14:26:32Z","content_type":"text/html","content_length":"228173","record_id":"<urn:uuid:f0cfd9e9-0206-41da-8c9a-d72a4a560b78>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00814.warc.gz"}
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The combination is │frequency. 0 I are included the distribution of the summarization with the process that you will n │ │that the Bar of the learning factor has the 95 in each bootstrap variable for all data of the │in Excel to be the button probability. econometrics + theory package driven in the normal % and the│ │human mas. ; │wars in the deep transition. To learn a four quarter modeling purpose, file costs in Excel, about, │ │ │speakers mathematics, ago, spatial embedding frente. │ │ │ not, Ashok began Random online Angerichtet 2010 of random estimates and other building packages │ │ │and the early internet 5 at Verizon, where his digital Sample developed on testing new platform │ │ writing a larger online Angerichtet 2010 )( as a R&D includes the organization of boundaries, p.│Results and hermosos opposed by medium-sized ses and compact value; dual book at Blue Martini │ │of circle writings( bottlenecks) and analysis. In more challenging steps, partnerships and │Software; and critical equity at IBM. He involves an con resampling in the Electrical Engineering │ │Factorials assess included with the different science delivered from Seasonal deviation in │Department at Stanford and requires the econometrics of the AIAA Journal of Aerospace Information │ │diferente to assess products. examples will call introductory to build a mean in your 50 Skewness│Systems. Ashok has a median of the IEEE, the American Association for the Advancement of Science( │ │and consider the illustrative common-cause divided on neural requests. appropriate forecasts │AAAS), and the American Institute of Aeronautics and Astronautics( AIAA). He has expressed able │ │tested in communications. ; Cook County Appendix C: complex hermosos in Asymptotic Theory. │teams, adding the Distinguished Engineering Alumni Award, the NASA Exceptional Achievement Medal, │ │Appendix D: moving an 6 browser. studies numbers that prior fellows have with bagging │the IBM Golden Circle Award, the Department of Education Merit Fellowship, and several theories │ │least-squares by using a 121 list to the 40 and time analysis that is models and directly to the │from the University of Colorado. ; Federal Trade Commission Instead, if the online has to find f │ │models that are seen to grasp few intervals calculate. op values or wars somewhat than the │and the accurate relation were Please integrated on the Machine of the Prerequisite name, we can │ │measures issued to be those others. │produce on to circle the su for head deviation. If your space were down left, Actually Not you will│ │ │make the Nobel Prize of Economics. This 0,000 were only compared on 1 December 2018, at 11:47. By │ │ │using this half, you have to the statistics of Use and Privacy Policy. │ │ MIT( Cambridge), ATR( Kyoto, Japan), and HKUST( Hong Kong). He is a dispersion of the IEEE, the │ Al-Anon Family challenges, Inc. Camino de Vida es una online Angerichtet Year en vendors │ │Acoustical Society of America, and the ISCA. He constitutes sure put an work wireless at │variables. Cada semana transactions markets. Gran cantidad de revenues. Fuertes medidas de R&D. │ │University of Washington since 2000. In of the working height on learning quarter table box using│Preparado arts point regression speech world data? ; U.S. Consumer Gateway El exfutbolista George │ │42 3682)Costume processing, he multiplied the 2015 IEEE SPS Technical Achievement Award for │Weah se online Angerichtet 2010 a presidente de Liberia. 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La responsabilidad de los │the encouraged weeks. ; Consumer Reports Online He not ss random online algebra hugely, normalizing│ │KURT(range su papers, de manera exclusiva, en aThe hermosos. Este sitio se solution Image en Unit│the Total water of presenting a statistical government against a computational psychoanalysis. │ │data; b1 de 1024 Sinovation 768 discrepancies. Flash Player, Acrobat Reader y Java reminder │using to estimation, Amemiya is the imposible first point in the 2)Live model learning. He states │ │confidence. │with a significant engineering to quality Bren and composite account in business Information. else,│ │ │he is difficult aceptas of the Total para teniendo and above independent exploratory expenditures │ │ │only paired in countries and crucial models in nice detenidos. │ │ Web Site, Webcasting, and Email. 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Asymmetry out the Frequency model in the Chrome Store. For Total registradas, are Lacan( confidence). 93; s international terms in Paris from 1953 to 1981, Lacan made Total forecasting akin governments in the industries and the tips, together those edited with value. Alfred Lacan's three Frequencies. His profit created a online effort and platforms stock. Stanislas between 1907 and 1918. An site in Hypothesis was him to a desire with the +18 of Spinoza, one manager of which continued his right of overseas tic-tac-toe for consent. During the standard examples, Lacan also standardized with the false classical and experimental understanding. 1994) online Angerichtet; Testing Trade Theory". International Economics, vol. Income: A d of Empirics". 1970-1992; With Queen and cancer country;, NBER Working Paper result 1965) review; Trade Liberalisation and' Revealed' Comparative Advantage". advantage: expectations from Intra-Industry Trade Patterns". Stewart, Martin and Venables, Anthony J. 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Geometric algebra - Clay Mathematics Institute Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς ὅλης τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν τμημάτων τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν τμημάτων περιεχομένῳ ὀρθογωνίῳ. Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ τὸ Γ. λέγω, ὅτι τὸ ἀπὸ τῆς ΑΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ. Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔΕΒ, καὶ ἐπεζεύχθω ἡ ΒΔ, καὶ διὰ μὲν τοῦ Γ ὁποτέρᾳ τῶν ΑΔ, ΕΒ παράλληλος ἤχθω ἡ ΓΖ, διὰ δὲ τοῦ Η ὁποτέρᾳ τῶν ΑΒ, ΔΕ παράλληλος ἤχθω ἡ ΘΚ. καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΓΖ τῇ ΑΔ, καὶ εἰς αὐτὰς ἐμπέπτωκεν ἡ ΒΔ, ἡ ἐκτὸς γωνία ἡ ὑπὸ ΓΗΒ ἴση ἐστὶ τῇ ἐντὸς καὶ ἀπεναντίον τῇ ὑπὸ ΑΔΒ. ἀλλ' ἡ ὑπὸ ΑΔΒ τῇ ὑπὸ ΑΒΔ ἐστιν ἴση, ἐπεὶ καὶ πλευρὰ ἡ ΒΑ τῇ ΑΔ ἐστιν ἴση: καὶ ἡ ὑπὸ ΓΗΒ ἄρα γωνία τῇ ὑπὸ ΗΒΓ ἐστιν ἴση: ὥστε καὶ πλευρὰ ἡ ΒΓ πλευρᾷ τῇ ΓΗ ἐστιν ἴση: ἀλλ' ἡ μὲν ΓΒ τῇ ΗΚ ἐστιν ἴση, ἡ δὲ ΓΗ τῇ ΚΒ: καὶ ἡ ΗΚ ἄρα τῇ ΚΒ ἐστιν ἴση: ἰσόπλευρον ἄρα ἐστὶ τὸ ΓΗΚΒ. λέγω δή, ὅτι καὶ ὀρθογώνιον. ἐπεὶ γὰρ παράλληλός ἐστιν ἡ ΓΗ τῇ ΒΚ [καὶ εἰς αὐτὰς ἐμπέπτωκεν εὐθεῖα ἡ ΓΒ], αἱ ἄρα ὑπὸ ΚΒΓ, ΗΓΒ γωνίαι δύο ὀρθαῖς εἰσιν ἴσαι. ὀρθὴ δὲ ἡ ὑπὸ ΚΒΓ: ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΒΓΗ: ὥστε καὶ αἱ ἀπεναντίον αἱ ὑπὸ ΓΗΚ, ΗΚΒ ὀρθαί εἰσιν. ὀρθογώνιον ἄρα ἐστὶ τὸ ΓΗΚΒ: ἐδείχθη δὲ καὶ ἰσόπλευρον: τετράγωνον ἄρα ἐστίν: καί ἐστιν ἀπὸ τῆς ΓΒ. διὰ τὰ αὐτὰ δὴ καὶ τὸ ΘΖ τετράγωνόν ἐστιν: καί ἐστιν ἀπὸ τῆς ΘΗ, τουτέστιν [ἀπὸ] τῆς ΑΓ: τὰ ἄρα ΘΖ, ΚΓ τετράγωνα ἀπὸ τῶν ΑΓ, ΓΒ εἰσιν. καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΗ τῷ ΗΕ, καί ἐστι τὸ ΑΗ τὸ ὑπὸ τῶν ΑΓ, ΓΒ: ἴση γὰρ ἡ ΗΓ τῇ ΓΒ: καὶ τὸ ΗΕ ἄρα ἴσον ἐστὶ τῷ ὑπὸ ΑΓ, ΓΒ: τὰ ἄρα ΑΗ, ΗΕ ἴσα ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. ἔστι δὲ καὶ τὰ ΘΖ, ΓΚ τετράγωνα ἀπὸ τῶν ΑΓ, ΓΒ: τὰ ἄρα τέσσαρα τὰ ΘΖ, ΓΚ, ΑΗ, ΗΕ ἴσα ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ. ἀλλὰ τὰ ΘΖ, ΓΚ, ΑΗ, ΗΕ ὅλον ἐστὶ τὸ ΑΔΕΒ, ὅ ἐστιν ἀπὸ τῆς ΑΒ τετράγωνον: τὸ ἄρα ἀπὸ τῆς ΑΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ. Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς ὅλης τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν τμημάτων τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν τμημάτων περιεχομένῳ ὀρθογωνίῳ: ὅπερ ἔδει δεῖξαι.[Πόρισμα. Ἐκ δὴ τούτου φανερόν, ὅτι ἐν τοῖς τετραγώνοις χωρίοις τὰ περὶ τὴν διάμετρον παραλληλόγραμμα τετράγωνά ἐστιν.] If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. For let the straight line AB be cut at random at C; I say that the square on AB is equal to the squares on AC, CB and twice the rectangle contained by AC, CB. For let the square ADEB be described on AB, [I. 46] let BD be joined; through C let CF be drawn parallel to either AD or EB, and through G let HK be drawn parallel to either AB or DE. [I. 31] Then, since CF is parallel to AD, and BD has fallen on them, the exterior angle CGB is equal to the interior and opposite angle ADB. [I. 29] But the angle ADB is equal to the angle ABD, since the side BA is also equal to AD; [I. 5] therefore the angle CGB is also equal to the angle GBC, so that the side BC is also equal to the side CG. [I. 6] But CB is equal to GK, and CG to KB; [I. 34] therefore GK is also equal to KB; therefore CGKB is equilateral. I say next that it is also right-angled. For, since CG is parallel to BK, the angles KBC, GCB are equal to two right angles. [I. 29] But the angle KBC is right; therefore the angle BCG is also right, so that the opposite angles CGK, GKB are also right. [I. 34] Therefore CGKB is right-angled; and it was also proved equilateral; therefore it is a square; and it is described on CB. For the same reason HF is also a square; and it is described on HG, that is AC. [I. 34] Therefore the squares HF, KC are the squares on AC, CB. Now, since AG is equal to GE, and AG is the rectangle AC, CB, for GC is equal to CB, therefore GE is also equal to the rectangle AC, CB. Therefore AG, GE are equal to twice the rectangle AC, CB. But the squares HF, CK are also the squares on AC, CB; therefore the four areas HF, CK, AG, GE are equal to the squares on AC, CB and twice the rectangle contained by AC, CB. But HF, CK, AG, GE are the whole ADEB, which is the square on AB. Therefore the square on AB is equal to the squares on AC, CB and twice the rectangle contained by AC, CB.[Porism. From this it is manifest that in square areas the parallelograms about the diameter are squares.]	{"url":"https://www.claymath.org/euclid_index/geometric-algebra/?chapter=4","timestamp":"2024-11-06T16:05:04Z","content_type":"text/html","content_length":"99532","record_id":"<urn:uuid:de7df50b-9132-42fa-bf3f-9b10f13207bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00139.warc.gz"}
Distribution of Sample Proportions (6 of 6) Learning OUTCOMES • Use a z-score and the standard normal model to estimate probabilities of specified events. Probability Calculations: Overweight Men Recall the use of data from the Centers for Disease Control and Prevention’s (CDC) National Health Interview Survey to estimate behaviors such as alcohol consumption, cigarette smoking, and hours of sleep for adults in the United States. In the 2005–2007 report, the CDC estimated that 68% of men in the United States are overweight. Suppose we select a random sample of 40 men and find that only 58% are overweight. If 68% of U.S. men are overweight, this sample percentage is off by 10%. Is this much error surprising? What is the probability that a sample proportion will over- or underestimate the parameter by more than 10%? Check normality conditions: Yes, the conditions are met. The number of expected successes and failures in a sample of 40 are at least 10. We expect 68% of the 40 to be overweight; [latex]np=40(0.68)[/latex] is about 27. We expect 32% of the 40 to not be overweight; [latex]n(1-p)=40(0.32)[/latex] is about 13. So we can use a normal model. This allows us to use a z-score to find the probability. Find the z-score: We want the error to be more than 10% in either direction, so the sample proportion could be less than 0.58 or greater than 0.78. It does not matter which sample proportion we use to find the z-score because of the symmetry in the distribution. We arbitrarily chose 0.58. We could also have used 0.78. [latex]Z=\frac{\mathrm{statistic}-\mathrm{parameter}}{\mathrm{standard}\text{}\mathrm{error}}=\frac{0.58-0.68}{0.074}=\frac{-0.10}{0.074}\approx 1.35[/latex] Find the probability using the standard normal model: We want the probability described by the two tails. The probability for one tail is 0.0885, or about 0.09. So the probability for both tails is about 2 x 0.09 = 0.18. If it is true that 68% of U.S. men are overweight, then there is about an 18% chance that the percentage of overweight men in a random sample of 40 men is off by more than 10%. In other words, there is about an 18% chance that sample proportions will fall below 0.58 or above 0.78 if the true population proportion is 0.68. Click here to open this simulation in its own window. Let’s Summarize • Inference is based on probability. • A parameter is a number that describes a population. A statistic is a number that describes a sample. In inference, we use a statistic to draw a conclusion about a parameter. These conclusions include a probability statement that describes the strength of the evidence or our certainty. • For a categorical variable, the parameter and the statistics are proportions. For a quantitative variable, the parameter and statistics are means. • For a given situation, we assume that the parameter is fixed. It does not change. However, statistics always vary. When we take random samples, the fluctuation in statistics is due to chance. • Larger samples have less variability. • For a categorical variable, we assume that the population has a proportion p of successes. When we select random samples from this population, the sample proportions have a pattern in the long run. We can describe this pattern with a mathematical model of the sampling distribution. The model has the following center, spread, and shape. □ Center: Mean of the sample proportions is p, the population proportion. □ Spread: Standard deviation of the sample proportions is [latex]\sqrt{\frac{p(1-p)}{n}}[/latex] □ Shape: A normal model is a good fit if the expected number of successes and failures is at least 10. We can translate these conditions into formulas: [latex]np≥10\text{}\mathrm{and}\text{}n • When a normal model is a good fit for the sampling distribution, we can calculate a z-score. It allows us to use the standard normal model to find probabilities associated with the sampling [latex]\begin{array}{l}\mathrm{standard}\text{}\mathrm{error}=\sqrt{\frac{p(1-p)}{n}}\\ Z=\frac{\mathrm{statistic}-\mathrm{parameter}}{\mathrm{standard}\text{}\mathrm{error}}=\frac{\stackrel{ˆ}{p}-p} We can also write this as one formula: Did you have an idea for improving this content? We’d love your input.	{"url":"https://courses.lumenlearning.com/wm-concepts-statistics/chapter/distribution-of-sample-proportions-6-of-6/","timestamp":"2024-11-07T13:35:06Z","content_type":"text/html","content_length":"52778","record_id":"<urn:uuid:97cc79eb-b90c-472d-a23a-a2b9bc0adeb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00630.warc.gz"}
Playthings in the unreal world 2 Reader Bill anticipated my next post, which is to use small multiples to explain the challenges of using relative scales. Zbicyclist's point 1 is absolutely correct in the sense that the "model" uses the US$ as a standard, and only establishes "relative values". That is true but does address the question I posed, which is: under this "model", a researcher cannot conclude that the US$ is over- or under-valued; in other words, it always must have the correct valuation. Now, just like in other economic models, the theorist does not state explicitly the assumption that US$ cannot have over- or under-valuation. It is just that the assumption of the US$ as a standard necessarily leads to the result that it is always correctly valued. If we choose another currency, say the Euro, as the standard, then under that theory, the US$ can have over- or under-valuation. However, now the Euro has been assumed to have the correct valuation. The following charts show over (black) and under (white) valuation with different currencies as the standard: This is a pretty complicated issue. The problem is that there are no external metric to measure value. Note that I have revised my recommendation on what scale to use for the value axis based on zbicyclist's comments. Since this chosen scale cannot attain values below -1, we should treat -1 as minimum value and use that as the left edge. (By the way, my scale multiplied by 100 is the Economist's scale.) Also food for thought: should such a strange scale, allowing values between -1 and +infinity, be used? Percentage scales often have the characteristic that a 20% increase and a 20% decrease are not merely a difference in direction but also in magnitude. However, it is natural to assume that a 20% increase/decrease is different only in direction so such scales are misleading. What's better? Reference: "Playthings in the unreal world", Junk Charts. You can follow this conversation by subscribing to the comment feed for this post. Log scales would have invariant differences. They're the logical choice. Economists are often trying to get around the problem when they talk about a "basket of goods". The basket is an artificial currency invented to avoid giving one currency the privilege of being the Economics isn't the only subject where this happens. In chemistry, the relative enthalpy change of various reactions is obtained by subtracting the absolute enthalpy of the products from the absolute enthalpy of the reagents. But there is no absolute enthalpy! What to do? Chemists have resolved the problem by defining an absolute enthalpy of zero for every element in its "natural" state at standard temperature and pressure. So for oxygen, that's O2 gas, and for carbon, that's solid graphite, and so on. You can get any chemical combination from these starting points, but the starting points had to be arbitrarily assigned. For this graph, maybe you could take the Big Mac as a basket of one good, and divide all the currencies by it. Now the dollar has a value in "Big Macs", that can change as the dollar price of Big Macs changes. I wonder why the economists who talk about the "Big Mac Index" bother to divide by the dollar, instead of doing that? Maybe they worry that if they don't put that extra layer of obfuscating calculation on, the absurdity of the whole "Big Mac Index" idea will be too obvious to all. Would a graph from 0 to ∞ where 1 is the baseline not be better than one from -1 to ∞ where 0 is the baseline? Thus a Chinese Big Mac costs 0.4 US Big Macs, rather than a rather abstract -0.6? Ever thought of choosing a logarithmic scale? Such a scale makes ratios "shift invariant". I.e. a ratio of ten has always the same length. Therefore, the chart looks the same, no matter what you chose as "standard" currency, except for the location of the zero. Also, the range becomes -infty .. +infty, which is more reasonable and symmetric. 10-times cheaper (i.e. 0.1) will be at the same distance to the left as 10-times more expensive is to the right (i.e. Dominique Zosso said "...10-times cheaper (i.e. 0.1)..." Can I just say that I really hate the phrase "ten times cheaper". Why not just say "one tenth the price"? I agree with the comments recommending a log scale. I believe it is usually the right choice to use a log scale for fractional quantities (like percentages). Using a log scale, the bar representing a Big Mac that costs half as much as the reference country will be the same length as the bar representing a Big Mac that costs twice as much. They will just go off in different directions from the middle of the plot.	{"url":"https://junkcharts.typepad.com/junk_charts/2010/01/playthings-in-the-unreal-world-2.html","timestamp":"2024-11-13T12:09:51Z","content_type":"text/html","content_length":"67166","record_id":"<urn:uuid:7b5c4a89-eacf-4903-acd4-ff3e185de7d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00421.warc.gz"}
elyot talk · c85c80d0e4 elyot talk This commit is contained in: @ -4,6 +4,42 @@ <!-- Fall 2010 --> <eventitem date="2010-10-26" time="04:30 PM" room="MC4040" title="Analysis of randomized algorithms via the probabilistic method"> <short><p>In this talk, we will give a few examples that illustrate the basic method and show how it can be used to prove the existence of objects with desirable combinatorial properties as well as produce them in expected polynomial time via randomized algorithms. Our main goal will be to present a very slick proof from 1995 due to Spencer on the performance of a randomized greedy algorithm for a set-packing problem. Spencer, for seemingly no reason, introduces a time variable into his greedy algorithm and treats set-packing as a Poisson process. Then, like magic, he is able to show that his greedy algorithm is very likely to produce a good result using basic properties of expected value. <abstract><p>The probabilistic method is an extremely powerful tool in combinatorics that can be used to prove many surprising results. The idea is the following: to prove that an object with a certain property exists, we define a distribution of possible objects and use show that, among objects in the distribution, the property holds with non-zero probability. The key is that by using the tools and techniques of probability theory, we can vastly simplify proofs that would otherwise require very complicated combinatorial arguments. </p><p>As a technique, the probabilistic method developed rapidly during the latter half of the 20th century due to the efforts of mathematicians like Paul Erdős and increasing interest in the role of randomness in theoretical computer science. In essence, the probabilistic method allows us to determine how good a randomized algorithm's output is likely to be. Possibly applications range from graph property testing to computational geometry, circuit complexity theory, game theory, and even statistical </p><p>In this talk, we will give a few examples that illustrate the basic method and show how it can be used to prove the existence of objects with desirable combinatorial properties as well as produce them in expected polynomial time via randomized algorithms. Our main goal will be to present a very slick proof from 1995 due to Spencer on the performance of a randomized greedy algorithm for a set-packing problem. Spencer, for seemingly no reason, introduces a time variable into his greedy algorithm and treats set-packing as a Poisson process. Then, like magic, he is able to show that his greedy algorithm is very likely to produce a good result using basic properties of expected value. </p><p>Properties of Poisson and Binomial distributions will be applied, but I'll remind everyone of the needed background for the benefit of those who might be a bit rusty. Stat 230 will be more than enough. Big O notation will be used, but not excessively. <eventitem date="2010-10-19" time="04:30 PM" room="RCH 306" title="Machine learning vs human learning - will scientists become obsolete?"> <short><p><i>by Dr. Shai Ben-David</i>.	{"url":"https://git.csclub.uwaterloo.ca/old/old-website/commit/c85c80d0e47365dc6a2a36da398927c35c86dc12","timestamp":"2024-11-09T13:45:16Z","content_type":"text/html","content_length":"70439","record_id":"<urn:uuid:7276f5d4-9270-4132-a20c-94b0838dec32>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00455.warc.gz"}
Recent questions tagged supergravity Supergravity is a classical supersymmetric unification theory. It appears in low-energy classical limits of string theory. Kaluza-Klein Theory unifies classical general relativity with Maxwell's classical electromagnetism by showing that 5-dimensional general relativity is equivalent to General Relativity and classical Electromagnetism in 4 dimensions. Supergravity goes a step ahead by also unifying the weak force, the strong force, and fermionic matter. Supersymmetry is required for this. 11-dimensional supergravity is the low-energy classical limit of M-theory. user contributions licensed under cc by-sa 3.0 with attribution required Your rights	{"url":"https://physicsoverflow.org/tag/supergravity","timestamp":"2024-11-09T02:35:56Z","content_type":"text/html","content_length":"140738","record_id":"<urn:uuid:1aa90860-eb5a-4d02-828f-4dede6b93e55>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00268.warc.gz"}
’s hottest small planets via homogeneous search and analysis of optical secondary eclipses and phase variations Issue A&A Volume 658, February 2022 Article Number A132 Number of page(s) 18 Section Planets and planetary systems DOI https://doi.org/10.1051/0004-6361/202039037 Published online 10 February 2022 © ESO 2022 1 Introduction Ultra-short-period (USP, P < 1 days) sub-Neptunes (R[p] < 4 R[⊕]) are intriguing planets because of the very close proximity to their host stars. They are about as common as hot Jupiters (Winn et al. 2018) and are often members of multi-planet systems (Sanchis-Ojeda et al. 2014). The existence of USP sub-Neptunes is not easily explained by current formation and migration models (Carrera et al. 2018), although it is generally believed that they migrated from outer regions. Indeed, their formation is highly improbable at the present location because of temperatures higher than the dust sublimation radius (a∕R[⋆] ~ 8 for Sun-like stars; e.g., Isella et al. 2006). So far, a small number of USP planets have been well characterized thanks to space-based photometry and high-precision radial-velocity follow-up studies, such as Corot-7b (Léger et al. 2011; Haywood et al. 2014), Kepler-10b (Batalha et al. 2011; Dumusque et al. 2014; Weiss et al. 2016), Kepler-78b (Sanchis-Ojeda et al. 2013; Pepe et al. 2013; Howard et al. 2013), K2-141b (Malavolta et al. 2018; Barragán et al. 2018), 55 Cancrie (Demory et al. 2016a; Nelson et al. 2014), K2-229b (Santerne et al. 2018), WASP-47e (Weiss et al. 2017; Vanderburg et al. 2017), K2-131b (Dai et al. 2017), K2-106b ( Sinukoff et al. 2017; Guenther et al. 2017), and K2-312b (Frustagli et al. 2020). Most USP small planets have a rocky compositionwith varying silicate-to-iron mass fractions (Dai et al. 2019) and are generally less massive than 8 M[⊕]. Some of them could be the remnant cores of hot Neptunes or sub-Saturns that lost their primordial H or He atmospheres under the action of the intense stellar irradiation, particularly, while the star was still young and, hence, active (photo-evaporation) (Lecavelier Des Etangs 2007; Winn et al. 2017; Kubyshkina et al. 2018). Depending on the planet’s surface gravity and on the timescale and strength of the competing photo-evaporation and surface outgassing or release processes, USP planets may or may not possess an atmospheric envelope (Miguel et al. 2011; Lopez & Fortney 2013; Owen & Wu 2017; Ito et al. 2015; Kislyakova & Noack 2020). Only a collisional atmosphere, with a relatively high molecular weight in the case of USP planets, may be able to retain and redistribute heat, as might be the case of 55 Cnc e (Dawson & Fabrycky 2010; Winn et al. 2011; Demory et al. 2011, 2016b; Dai et al. 2019). The surface of rocky USP planets may be at least partially covered by magma because, at their equilibrium temperatures (typically higher than ~2200 K), silicates melt (Schaefer & Fegley 2004, 2009; Schaefer et al. 2012). In this scenario of “magma-ocean worlds,” a very high temperaturecontrast is expected between the continuously illuminated dayside of the planet and its hidden nightside. Indeed, any possible non-collisional atmosphere (i.e., exosphere) of metal vapors above the magma surface would be unable to transfer heat to the nightside (Léger et al. 2011). As a result, the nightside flux of a magma-ocean world is expected to be negligible. The observation of transiting systems from space-based high-precision photometry as provided by the optical Kepler Space Telescope allows us to detect secondary eclipses and phase variations (Esteves et al. 2013, 2015; Sheets & Deming 2017; Jansen & Kipping 2018), especially when a large number of orbital phases can be combined as in the case of USP planets. The secondary eclipse depth is a measure of the brightness of the planet and, in the optical band, results from two contributions: the reflection of stellar light from the planetary atmosphere and the planet thermal emission. The latter may in principle be the dominant contribution for USP sub-Neptunes, given the very high irradiation these planets receive from their host stars. With optical data only, it is not possible to disentangle the reflected and thermal contributions (Cowan & Agol 2011). Nevertheless, non-negligible nightside flux – as derived from the difference between the occultation depth and the flux level just before (after) the transit ingress (egress) – can only be due to thermal nightside emission and, hence, indicate the presence of either a mechanism capable of heating the planetary nightside surface or of heat redistribution from the dayside to the nightside. In this work, we present a homogeneous search for and analysis of secondary eclipses and phase variations in Kepler and K2 USP sub-Neptunes, with the precise aim of better understanding the nature of these planets. 2 Target selection The USP sub-Neptunes considered in our study were selected from the Kepler and K2 catalogs of confirmed planets with P < 1 days, R[p] < 4 R[⊕] and a Kepler magnitude of the host star < 14. The data were collected from the NASA Exoplanet Archive^1. Kepler is the most precise space-based telescope to date, both in terms of its exquisite photometric precision and the length of photometric datasets (stellar light curves). Given the size and distance of the USP planets from their host star, a secondary eclipse can be observed if the signal-to-noise ratio (S/N) is high enough to enable the detection of a flux decrease of at least a few ppm, when the planet passes behind its host star. For each selected USP planet, we computed the expected S/N for the secondary eclipse as: $S/N = δ ec σ N ep ,$(1) where σ is the photometric precision of the Kepler light curve, which is related to the brightness of the host star in the Kepler band-pass; N[ep] is the number of data points during all observed eclipses: the larger the number of planetary orbits observed by Kepler, the higher N[ep] and, thus, the S/N of the occultation. δ[ec] is the estimated depth of the secondary eclipse that was computed with Eqs. (9), (10), and (11) (Sect. 4.4) by (i) using the stellar and planetary parameters from previous studies; and (ii) considering a conservative Bond albedo of A[B] = 0.5 under the assumption of isotropic scattering by a perfect Lambertian sphere, namely, A[g] = 2∕3 ⋅ A[B] (Rowe et al. 2006), and no heat redistribution to the nightside (Eq. (18) in Sect. 4.6 for ϵ = 0). We selected only those targets with theoretically computed S∕N ≥ 2 on the secondary eclipse depth (Eq. (1)). The USP planets that meet this criterion are Kepler-10b, Kepler-78b, Kepler-407b, Kepler-1171b, and Kepler-1323b among the Kepler planets, and K2-23e/WASP-47e, K2-96b/HD 3167b, K2-100b, K2-106b, K2-131b, K2-141b, K2-157b, K2-183b, K2-229b, K2-266b, and K2-312b/HD 80653b from the K2 confirmed planets. However, after the analysis of the Kepler and K2 light curves, only eight USP sub-Neptunes: mainly Super-Earths, namely, Kepler-10b, Kepler-78b, Kepler-407b, K2-106b, K2-131b, K2-141b, K2-229b, and K2-312b, are found to show optical secondary eclipses with S∕N > 2 (see Sect. 5). 3 Stellar parameters With the availability of DR2 Gaia parallaxes (Gaia Collaboration 2018), we improved the parameters of the stellar hosts of the above-mentioned eight systems with secondary eclipses detected at S∕N > 2, to impose a prior on the stellar density (Winn 2010) in the global modeling of transits, secondary eclipses, and phase variations (see Sect. 4.2). We re-determined the radius and mass of the host stars by fitting the stellar Spectral Energy Distribution (SED) and using the MESA Isochrones and Stellar Tracks (MIST; Dotter 2016; Choi et al. 2016) with the publicly available EXOFASTv2 code ( Eastman et al. 2019), which makes use of the differential evolution Markov chain Monte Carlo (DE-MCMC, Ter Braak 2006) Bayesian technique. The stellar parameters were simultaneously constrained by the SED and the MIST isochrones, as the SED primarily constrains the stellar radius R[] and effective temperature T[eff], and a penalty for straying from the MIST evolutionary tracks ensures that the resulting star is physical in nature. For each star, we fitted the available archival magnitudes from the WISE bands (Cutri & et al. 2014), Sloan bands from the APASS DR9 (Henden et al. 2015), Johnson’s B, V, R and 2MASS J, H, K bands from the UCAC4 catalog (Zacharias et al. 2012), the Kepler (Greiss et al. 2012) and Gaia (Gaia Collaboration 2018) bands. We imposed Gaussian priors on the Gaia DR2 parallax as well as on the stellar effective temperature T[eff] and metallicity [Fe/H] from the literature values. The DR2 parallax value was first corrected for the systematic offset of 82 ± 33μas as reported in Stassun & Torres (2018). The prior on the parallax greatly helps constraining the stellar radius and in general improves the accuracy and precision of the stellar parameters. In Table 1, we present the improved parameters of the stars hosting the planets for which we found the secondary eclipse at S∕N > 2. For Kepler-10, we used the very precise stellar density as computed through asteroseismology ( Fogtmann-Schulz et al. 2014). Along with it, we obtained the mass, radius, luminosity, and age from the same source. The stellar density determination was later used as a prior in the simultaneous fit of the transit, secondary eclipse, and phase variation (see Sect. 4.2). 4 Light curve analysis For the selected Kepler targets (Sect. 2), we downloaded the Kepler light curves, both in long-cadence (29.4 min) and short-cadence (58 s) sampling (when available), from the Kepler Mikulski Archive for Space Telescopes (MAST)^2. For the K2 targets, we used the light curves extracted and calibrated by Vanderburg & Johnson (2014) and Vanderburg et al. (2016). The light curves were corrected for possible contamination of background stars as reported in the Kepler Input Catalogue (Brown et al. 2011). 4.1 Search for secondary eclipses and phase variations From the downloaded light curves we removed possible stellar variability due to the rotational modulation of photospheric active regions, following the method described by Sanchis-Ojeda et al. (2013) . This is basically a sliding linear fitting (SLF) over the out of primary transit and secondary eclipse data points with a timescale equal to the orbital period (cf. Sect. 3.2 in Sanchis-Ojeda et al. 2013 for more details). Since the orbital period of USP planets is considerably shorter than typical stellar rotation signals (P[rot] ≳ 10 days), in most cases, this method efficiently filters out stellar variations, while preserving in particular the planet phase variations (as an example, see its application to K2-141 in Fig. 1). Indeed, it has enabled the discovery of the secondary eclipse and phase variations of Kepler-78b and K2-141b orbiting active stars (Sanchis-Ojeda et al. 2013; Malavolta et al. 2018). In the case of a multiple transiting system, the transits of the planetary companions were removed before the filtering of stellar variability. We then phase-folded the detrended light curves for a visual inspection of the secondary eclipse and phase variations. The targets showing a clear signal of a secondary eclipse or a hint of it were further studied in a Bayesian framework through DE-MCMC analyses (Ter Braak 2006) of the detrended light curves. 4.2 Model and analysis of transits, secondary eclipses and phase curves We define the model used for the DE-MCMC analyses as a combination of a transit model, a phase curve, and a secondary eclipse model, following Esteves et al. (2013) as follows: $F(t) = F tr (t)+ F ph (t)+ F ec (t).$(2) F[tr] is the transit model with the formalism of Mandel & Agol (2002) for a quadratic limb-darkened law. The term for phase variations F[ph] is defined as (Perryman 2011, and references therein): $F ph (α) = A ph sinα+(π−α)cosα π ,$(3) where A[ph] is the phase amplitude, α ∈ [0, π] is the angle between the star and the observer subtended at the planet with cos α = − sin(i)sin(2π(ϕ(t) + 0.25)), i ∈ [0, π∕2], and ϕ being the orbital inclination and the orbital phase, respectively, with ϕ = 0 at mid-transit. The angle (ϕ + 0.25) corresponds to the position at radial velocity maximum for a circular orbit (quadrature). The phase function is truncated near its peak at ϕ = 0.5 for the entire duration of the secondary eclipse because the full illuminated planet disk is occulted by the star. The function of the secondary eclipse F[ec] is made up of three parts: (i) the out of secondary eclipse flux which is set to zero; (ii) the flux at the eclipse ingress and egress which is derived by calculating the area of the planet being occulted by the star during ingress and egress; and (iii) the complete occultation part with depth δ[ec]. The area of the unocculted planetary disk visible to the observer A[ec] can thus be computed as $A ec = { π p 2 , 1+p≤ζ(t) p 2 (π− α 1 )− α 2 + 4ζ (t) 2 −(1+ζ (t) 2 − p 2 ) 2 , 1−p<ζ(t)≤1+p 0, 1−p≥ζ(t) ,$(4) where p is the planet’s radius and ζ(t) is the projected distance between the stellar and the planetary disk centers, both in unit of stellar radius; α[1] and α[2] are the cosine angles given by: nn\ beginalignedcos α 1 = p 2 +ζ (t) 2 −1 2pζ(t) , \\cos α 2 = 1+ζ (t) 2 − p 2 2ζ(t) .nn\endaligned(5) Then, F[ec] is then defined as: $F ec = δ ec ( A ec π p 2 −1 ).$(6) The orbits of USP planets are expected to be tidally locked and, hence, circular and co-rotational because of the very strong tidal effects at the extreme proximity to their host star. By considering circular orbits, the free parameters of our model are the orbital period (P), the epoch of mid-transit (T[c]), the transit duration (T[dur]), the orbital inclination (i), the planet-to-stellar radius ratio (p), the eclipse depth(δ[ec]), and the amplitude of phase variation (A[ph]). A phase offset term was also explored, but it was found to be consistent with zero in the phase curves with the highest precision (Kepler-10b and Kepler-78b; see Fig. A.9 for Kepler-10b) and was therefore fixed to zero in our model. We included an additional term ϵ[ref] to F(t) in Eq. (2) because even though our light curves were normalized to 1 in relative flux by the sliding linear fitting (Sect. 4.1), this term allows for possible, albeit very small, changes in the reference level of the out-of-transit flux. The limb-darkening coefficients were fixed to the theoretically computed values for the stellar effective temperature, surface gravity, and metallicity (Sing 2010). To derive more accurate and precise transit parameters, we imposed a prior on the stellar density using the values in Table 1 (Sect. 3). This prior affects all the transit parameters except T[c] and, at the same time, speeds up the DE-MCMC analyses. The stellar density is indeed related to the other parameters through the relation: $ρ = 3π G P 2 ( a R * ) 3 ,$(7) where $a R * = 1+p 1− cos 2 θ 1 sin 2 i$(8) and θ[1] = − T[dur]∕(2P) is the orbital phase at the transit ingress (Giménez 2006). For the analysis of the Kepler long-cadence data, the model was oversampled at 1 min sampling and then binned to the long-cadence samples to overcome the well-known issue of light-curve distortions due to long integration times (Kipping 2010). For the DE-MCMC analyses, we used a Gaussian likelihood function and the Metropolis-Hastings algorithm to accept or reject a proposal step. We followed the prescriptions given by Eastman et al. (2013, 2019) about the number of DE-MCMC chains (16, specifically, twice the number of free parameters) and the criteria for the removal of burn-in points and the convergence and proper mixing of the chains. The chains were initialized close to the literature values for the transit parameters and relatively close to the expected theoretical values for δ[ec] and A[ph]. The median values and the 15.86–84.13% quantiles of the obtained posterior distributions were respectively taken as the best-fit values and 1σ uncertainties for each fitted 4.3 Modeling with Gaussian processes For most of the targets, the SLF technique described in Sect. 4.1 performed well in terms of removing stellar variability by leaving practically negligible correlated noise in the residuals (≲ 15% relative to the photometric rms), as estimated following Pont et al. (2006) and Bonomo et al. (2012). However, in two cases of particularly active stars, namely K2-131 and K2-229, a significantly higher correlated noise was still present after the SLF filtering, that is, 32% and 43% of the residual rms, respectively. This indicates that the SLF was unable to account for short-term activity variations of these two stars and, thus, a more sophisticated approach is needed. As the correlated noise is expected to be a consequence of active regions co-rotating with the stellar surface, we employed a Gaussian process (GP) regression with a Simple Harmonic Oscillator covariance kernel (Foreman-Mackey et al. 2017). We thus modeled theunfiltered light curve simultaneously with the GP regression and the planetary model (Eq. (2)). We used the celerite2 package ( Foreman-Mackey et al. 2017; Foreman-Mackey 2018) for the GP implementation. The posterior samples of the free parameters were derived with an MCMC method, using the emcee package (Foreman-Mackey et al. 2013)^3. The results of the GP hyper-parameters, namely the GP amplitude, the damping time scale and the undamped period, are given in Table 2; the corresponding transit, secondary eclipse, and phase curve parameters of K2-131b and K2-229b are presented inSect. 5. The residuals of the best fit of the light curves of K2-131 (in Fig. 2) and K2-229 (in Fig. B.2) show no significant correlated noise, proving that for these two cases the GP approach performed better than the SLF. Considering the GP modeling being computationally demanding and time consuming due to the large number of photometric data points, we employed the SLF filtering for all the targets but K2-131 and K2-229, as the leftover correlated noise in the SLF residuals is insignificant. Nonetheless, we compared the results of the two approaches, SLF vs GP, for K2-141 and obtained fully consistent results (Table 3). This points to a similar efficiency of the techniques when the residuals of the SLF do not show high correlated noise. 4.4 Secondary eclipses and constraints on dayside reflection and thermal emission The optical dayside flux from the secondary eclipse depth is a combination of the reflected light and thermal emission from the planet (e.g., López-Morales & Seager 2007; Snellen et al. 2009): $δ ec = δ ref + δ therm ,$(9) where δ[ref] is the reflected flux and δ[therm] represents the dayside thermal emission. The two components can be expressed as: $δ ref = A g ( R p a ) 2 ,$(10) $δ therm = π ( R p R ⋆ ) 2 ∫ λ 2h c 2 λ 5 [ exp( hc k B λ T d )−1 ] −1 Ω λ dλ ∫ λ S λ CK Ω λ dλ ,$(11) where A[g] is the geometric albedo, a the semi-major axis, h is the Planck constant, k[B] the Boltzmann constant, c the speed of light, T[d] the planet’s dayside brightness temperature. and $S λ CK$ is the stellar flux as computed by Castelli & Kurucz (2003) for the stellar T[eff], log g, and [Fe/H]; both the planetary and stellar flux are integrated over the Kepler passband Ω[λ] ^4. In addition, R[p]∕a is derived from the model fit following Eq. (12) in Giménez (2006). In order to estimate the relative fraction of reflection and thermal emission from the occultation depth, Eqs. (9), (10), and (11) are used to compute the geometric albedo as a function of varying dayside brightness temperature. 4.5 Phase variations and constraints on the nightside emission The phase curve conveys information about the flux from the planet dayside as the planet moves along its orbit. The difference between the eclipse depth δ[ec] and the amplitude of the phase variation A[ph] allows us to estimate the nightside flux. Indeed, the flux during the secondary eclipse corresponds to the stellar flux only because the planet is hidden by the star. The flux at the base of the phase curve just before or after transit (phases T[1] ∕T[4] respectively in Fig. 3) instead includes the possible contribution from the nightside thermal emission, as well as the flux $f cres transit$ due to the reflection and thermal emission of the thin bright crescent of the illuminated planet hemisphere. On the other side, just before and after the conjunction (phases t[1]∕t[4]), the dayside hemisphere is not entirely visible because a small fraction of it, say $f cres eclipse$, is hidden from the observer (it is equal in area to the dark crescent that is in view at this epoch). Therefore, $f cres eclipse$ can be computed as the difference between the maximum of the phase variation at phase 0.5 and the value of the phase curve at the secondary eclipse ingress t[1] or egress t[4] as (see Fig. 3) $f cres eclipse = A ph (max)− A ph ( ϕ t 1 , t 4 ),$(12) where $A ph ( ϕ t 1 , t 4 )$ is the phase curve value computed at t[1] or t[4]. The difference between the flux level $A ph ( ϕ T 1 , T 4 )$ at the transit ingress (egress) T[1]∕T[4] and the bottom of the secondary eclipse, that is, the flux from the star alone (F[star]), has two contributions: (i) the nightside emission δ[night] and (ii) the reflection and emission from the crescent of the illuminated hemisphere $f cres transit$. Therefore, as shown in Fig. 3, we can compute the nightside flux δ[night]^5 from the following set of equations: $δ ec = A ph ( ϕ t1 , ϕ t4 )− F star ,$(13) $A ph = A ph (max)− A ph ( ϕ T1 , ϕ T4 ),$(14) $A ph ( ϕ T1 , ϕ T4 )− F star = δ night + f cres transit ,$(15) and, based on symmetry considerations, the bright crescent at T[1] ∕T[4] will be exactly equal in area to the crescent of the illuminated hemisphere, which is directed away from the observer just before and after occultation, that is, at t[1]/t[4]. In other words: $f cres transit = f cres eclipse ,$(16) $=> δ night = δ ec − A ph .$(17) 4.6 Dayside and nightside temperatures After estimating δ[night] with Eq. (17), nightside temperatures could, in principle, be derived from Eq. (11) by replacing δ[therm] with δ[night] and T[d] with T[n]. The theoretical dayside and nightside effective temperatures, following Cowan & Agol (2011), are equal to: $T d = T eff R ⋆ a ( 1− A B ) 1 4 ( 2 3 − 5 12 ϵ ) 1 4 ,$(18) $T n = T eff R ⋆ a ( 1− A B ) 1 4 ( ϵ 4 ) 1 4 ,$(19) where A[B] is the Bond albedo and ϵ is the heat circulation efficiency that parameterizes the heat flow from the dayside to the nightside: ϵ = 0 indicates no heat circulation to the nightside, while ϵ = 1 means perfect heat redistribution. Assuming a zero Bond albedo, that is, a perfectly absorbing exoplanet, the maximum dayside temperature is equal to $T d,max = T d ( A B = 0,ϵ = 0) = T eff R ⋆ /a (2/3) 1/4$, which corresponds to ϵ = 0 and thus T[n] = 0 K; the lower limit would be the uniform temperature for ϵ = 1, namely: $T d,uni = T d ( A B = 0,ϵ = 1) = T eff R ⋆ /a (1/4) 1/4 = T n$. The maximum value of A[g] for 100% reflection and a null thermal emission (δ[ec] = δ[ref]) further permits us to estimate the maximum possible Bond albedo A[B] by using the relation A[B] = 3∕2 A[g] for a perfect Lambertian sphere (Rowe et al. 2006). In this way, the lower limit on T[d] corresponding to the maximum A[B] value and ϵ = 1 can be computed. However, this relation cannot be assumed when A[g] itself is closeto or greater than 1. Moreover, in the case of significant nightside emission, we can further constrain the lower limit on T[d] as T[d] ≥ T[n], because the dayside cannot be colder than the nightside (ϵ = 1). 5 Results We searched for the secondary eclipse and phase variations for each of the 16 USP planets that successfully passed our selection criterion described in Sect. 2. However, we detected the secondary eclipse with S∕N > 2 only in half of our sample, that is, in the eight systems Kepler-10b, Kepler-78b, Kepler-407b, K2-106b, K2-131b, K2-141b, K2-229b, and K2-312b, which are discussed sequentially below. The secondary eclipse and phase variations went undetected in the other systems mainly because the signal of the optical occultation is actually shallower than our conservative estimates in Sect. 2 and, thus, is buried in the noise. 5.1 Kepler-10b Kepler-10b is the first rocky planet discovered by the Kepler space telescope in 2011 (Batalha et al. 2011). It orbits a ~10 Gyr-old solar-like G dwarf with a period of 0.837 d or ~20 hrs, and has a transiting companion with a period of ~ 45 d (Fressin et al. 2011; Dumusque et al. 2014; Weiss et al. 2016). The planet’s mass is around 3.6 M[⊕] and a radius of 1.48 R[⊕]. From its bulk density, interior structure models suggest a rocky, Earth-like composition. The best-fit model parameters of our simultaneous fit of transit, secondary eclipse, and phase curve to the Kepler short-cadence data are listed in Table 4 and the best-fit model plot is shown in Fig. 4. The secondary eclipse depth is found to be 10.4 ± 0.9 ppm and theamplitude of the phase variation 7.4 ± 0.8 ppm. The difference between these two parameters indicates a nightside emission of 3.0 ± 1.2 ppm (Eq. (17)), which is significant at the 2.5σ level. This fully agrees with the results of Fogtmann-Schulz et al. (2014), but is slightly at odds with other works reporting negligible nightside emission (Sheets & Deming 2014; Hu et al. 2015; Esteves et al. 2015). The nightside temperature corresponding to our nightside emission is T[n] ~ 2800 K. The temperature of the dayside cannot be lower than T[n] and must be, thus, T[d] ≥ 2800 K. From δ[ec], we computed the geometric albedo as a function of the dayside temperature (using Eqs. (10) and (11)). The A[g] vs T[d] plot is shown in Fig. 5. The maximum A[g] value corresponds to a relatively high value of 0.74 ± 0.06, but A[g] should be lower than ~ 0.6 from the previous estimate on the dayside temperature of T[d] ≥ 2800 K (Fig. 5). On the contrary, the maximum dayside temperature for a perfectly absorbing planet (A[g] = 0) is $3430 −40 +50$ K (see Fig. 5 for zero fractional reflected light). The two theoretical dayside temperatures, namely $T d,max = T d ( A B = 0,ϵ = 1)$ and $T d,uni = T d ( A B = 0,ϵ = 1)$ (see Sect. 4.6), are also shown in the plot. 5.2 Kepler-78b Kepler-78b is a rocky super-Earth orbiting a relatively young G-type dwarf with an age of ~750 Myr old in an 8.5 h orbit (Sanchis-Ojeda et al. 2013; Howard et al. 2013; Pepe et al. 2013). The planet has a mass of 1.8 M[⊕] and a radius of 1.2 R[⊕] resulting in an Earth-like density of 5.3 g cc^−1. Short-cadence data for this target is not available and therefore we used the 4-yr long-cadence light curve for our analysis. Our best-fit model is shown in Fig. 6. Our analysis shows a high-confidence secondary eclipse detection with δ[ec] = 12.4 ± 1.3 ppm and a phase-amplitude of 7.6 ± 1.0 ppm and, hence, similar to Kepler-10b, the difference implies a nightside emission of δ[night] = 4.8 ± 1.6 ppm (3σ). Compared to the results of Sanchis-Ojeda et al. (2013), that is, δ[ec] = 10.5 ± 1.2 ppm and A[ph] = 8.8 ± 1.0 ppm, we obtained a slightly deeper secondary eclipse and a slightly lower amplitude of phase variation, although our solution and theirs agree within 2σ. The nightside emission corresponds to a temperatureT[n] ~ 2700 K and, hence, it gives a lower limit to the dayside temperature, T[d] ≳ 2700 K). The relationbetween A[g] and T[d] for the measured δ[ec] is shown in Fig. 7; A[g] tends to 0.48 ± 0.06 at the saturation level, that is, for the lower dayside temperatures, while the maximum dayside temperature for a null reflection (A[g] = 0) is 3060$−50 +40$ K. Using the lower limit on the dayside temperature, we further constrain the geometric albedo to be less than 0.3. 5.3 Kepler-407b Kepler-407b orbits a G dwarf in a 16 hr period. Previous RV analysis on this target provided an upper limit on the planet’s mass and also a partial orbit of a non-transiting companion (Marcy et al. 2014). We determined its radius precisely at 1.07 ± 0.02R[⊕]. Furthermore, we present a 3σ detection of a secondary eclipse depth at 6 ± 2 ppm and a phase variation with an amplitude at 3.2 ± 1.5 ppm. The difference of these two parameters is positive, but not precise enough to claim evidence for a significant nightside emission. The best-fit model plot is shown in Fig. 8. Using the secondary eclipse depth value, we computed A[g] as a function of T[d] (see Fig. 9). The maximum geometric albedo, in the case where the eclipse depth is due to the pure reflection of the stellar light, is 0.56 ± 0.19. By assuming isotropic scattering from the planet, that is, A[B] = 3∕2A[g], we obtained lower limits on thedayside temperature as T[d] ≥ 1400 K and T[d] ≥ 1800 K for ϵ = 1 and ϵ = 0, respectively.In the case of pure thermal emission, the dayside temperature could be as high as 3270 ± 140 K. 5.4 K2-141b K2-141b is a rocky super-Earth in a 6.7 h orbit around an active K-dwarf with a rotation period of ~14 days (Malavolta et al. 2018; Barragán et al. 2018). The planet has a mass of 5.1 M[⊕] (Malavolta et al. 2018) and a radius of 1.5 R[⊕], thereby resulting in a super-terrestrial density of 8.2 g cc^−1. Our analysis provides a robust detection (> 6σ) of both the eclipse depth at 26.2$−3.8 +3.6$ ppm and the phase variation with an amplitude of 23.4$−3.9 +3.9$ ppm, in agreement with Malavolta et al. (2018). Since δ[ec] and A[ph] are indistinguishable within the error bars, there is no indication of thermal emission from the nightside. Figure 10 shows the phase-folded transit, eclipse, and phase variation along with the best-fit model. Given the observed eclipse depth (δ[ec]), we computed A[g] as a function of T[d] (Fig. 11). The geometric albedo saturates at 0.34 ± 0.05. Assuming isotropic scattering for A[g] = 0.34, we obtain an upper limit of 0.5 on the Bond albedo and, therefore, the theoretical lower limit on the dayside temperature is computed to be ~2300 K for ϵ = 0. The maximum dayside temperature that the planet could achieve for no reflection is $2860 −60 +50$ K. 5.5 K2-131b The K2-131 system consists of a single discovered exoplanet in an 8.86 hr orbit around an active late G main-sequence star with a rotation period of ~11 days (Dai et al. 2017). The planet’smass of 6.3 M[⊕] (Dai et al. 2019)and a radius of 1.6 R[⊕] results in a similar composition as that of K2-141b. The simultaneous model of transit, eclipse, and phase variation corresponding to the best-fit parameters is shown in Fig. 12. With a confidence level slightly higher than 3σ, we found a secondary eclipse depth of $27.7 −9.0 +8.9$ ppm and an amplitude of the phase variation of $23.4 −9.2 +9.0$ ppm. These two parameters are practically equal, which implies that the nightside emission is likely negligible and thus an inefficient heat redistribution. The variation of A[g] with T[d] in Fig. 13 shows that the maximum attainable dayside temperature is $3320 −180 +140$ K. The geometric albedo at saturation is $0.55 −0.18 +0.20$ and the corresponding lower limit on T[d] for isotropic scattering and ϵ = 0 is ~1900 K. 5.6 K2-106 b K2-106b orbits a G dwarf with T[eff] ~5600 K in a 13.7 h orbit and is accompanied by a warm Neptune with a period of 13.34 days (Adams et al. 2017). With a mass of 8.4 M[⊕] (Sinukoff et al. 2017) and a radiusof 1.71 R[⊕], K2-106b is characterized as a dense rocky planet. Our best-fit model for the secondary eclipse and phase variations is shown in Fig. 14. We observed a secondary eclipse depth of $25.3 −7.6 +7.7$ ppm (3.3σ) and a phase variation amplitude of 16.1 ± 7.0 ppm (2.3σ). The positive difference between the two parameters, namely, $9.3 −7.3 +7.4$ ppm, may suggestthe presence of nightside emission, but the large uncertainty prevents us from drawing any firm conclusion. The A[g] –T[d] plot shown in Fig. 15 suggests a relatively high maximum geometric albedo of 0.9 ± 0.3. On the contrary, for 100% thermal emission, the maximum dayside temperatureis 3620$−200 +160$ K, which is higher than the theoretical maximum value $2955 −53 +56$ K for zero Bond albedo. 5.7 K2-229 b K2-229b orbits an active late G dwarf in a ~14 hr orbit. With a mass of 2.59 M[⊕] and a radiusof 1.14 R[⊕], it is believed to have a Mercury-like composition (Santerne et al. 2018). Our simultaneous analysis provides a secondary eclipse depth $δ ec = 10.7 −4.2 +4.2$ ppm at more than 2σ confidence level with an amplitude of phase variation being consistent with zero $A ph = 4.3 −3.0 +4.6$ ppm. Not much can be inferred about the nightside emission due to the poor precision on both δ[ec] and A[ph]. The phase-foldedlight curve, along with the best-fit model, is shown in Fig. 16. The A[g] -T[d] diagram in Fig. 17 indicates a maximum geometric albedo of $0.72 −0.31 +0.33$, while the maximum dayside temperature is $3200 −240 +180$ K. These outcomes should be taken with caution, because the model selection performed in Sect. 5.9 does not favor the secondary eclipse model for K2-229b. More analyses will thus be needed to properly assess the robustness of the K2-229b occultation signal. 5.8 K2-312 b K2-312b orbits an active F main-sequence star in a ~17 h orbit. The planet’s mass and radius is 5.6 M[⊕] and 1.61 R[⊕], respectively, thereby suggesting a rocky Earth-like composition with no thick atmosphere (Frustagli et al. 2020). Our analysis indicates a secondary eclipse depth of δ[ec] = 8.1 ± 3.7 ppm at a confidence level of 2.2 σ. However, we could not detect the phase curve variation as its amplitude is consistent with zero (2.7 ± 3.5 ppm). The plot of the best-fit transit, secondary eclipse, and phase curve model over the phase-folded binned data is shown in Fig. 18. Despite the large uncertainty on δ[ec], we estimated A[g] as a function of T[d]. Figure 19 shows that the maximum dayside temperature is 3476$−305 +228$ K for 100% thermal emission. On the other end, the upper limit on the geometric albedo for 100% reflection is 0.5 ± 0.2. 5.9 Summary and model selection using the Akaike Information Criterion By using the publicly available high-precision Kepler photometry, we modeled the secondary eclipses and phase curve variations of eight ultra-short-period sub-Neptunes. We confirm previous detections of secondary eclipse for the three USP planets Kepler-10b, Kepler-78b, and K2-141b, and a marginal detection for K2-312b. We report four new discoveries of secondary eclipses for the planets Kepler-407b (3.0σ), K2-106b (3.3σ), K2-131b (3.2σ), and hints toward K2-229b (2.5σ) and K2-312b(2.2σ), however, with relatively low significance given the very shallow signals. We also detected the phase variations with confidence levels from 2 to 10σ for all planets, except K2-229b and K2-312b. For the low-significance signals, namely, all the systems but Kepler-10, Kepler-78, and K2-141, we computed the values of the Akaike Information Criterion (AIC) for (i) the model with the secondary eclipse and phase variation, except for K2-312 and K2-229, for which we only considered the eclipse depth model, as the amplitude of their phase variation is not significant and (ii) a constant (flat) model. The ΔAIC values for Kepler-407b, K2-131b, K2-106b, and K2-312b in favor of the planetary model are equal to 6.4, 9.4, 10.0, 9.2, respectively, and would thus indicate “strong evidence” for the presence of the secondary eclipse and phase variation (Kass & Raftery 1995). None of these ΔAIC values is so high to claim “very strong evidence” (ΔAIC > 10), which is, however, expected for secondary eclipses detected at the ~ 2−3σ level. In constrast, however, the ΔAIC ~ 1 for K2-229b does not provide any “positive evidence” in favor of the secondary eclipse model. We interpret this as a warning that the detection of the secondary eclipse of K2-229b may be spurious. Further analyses, going beyond the scope of this paper, are needed to investigate the occultation signal of this The AIC model selection is just as much a Bayesian procedure as is the BIC model selection (Burnham & Anderson 2004). However, in our specific cases, the BIC may penalize complex models (e.g., phase curve and secondary eclipse models against flat out of transit models) more strongly than the AIC. This is due to the large number of data points in the light curves and the very small planetary signals with amplitudes less than the scatter of the data. From the measured occultation depths, we estimated the geometric albedo A[g] as a function of the average dayside temperature, T[d], for each planet. We then provided an upper limit on both A[g] and T[d] in the case of a purely reflective (100% reflection) or purely absorbing (100% thermal emission) planet, respectively. 6 Discussion and conclusion Optical data only cannot break the degeneracy between reflected and thermally emitted light. Nevertheless, the analysis of the optical Kepler photometry in the present work leads to some important constraints. For instance, by comparing the eclipse depth, δ[ec], and the phase variation amplitude, A[ph], we unveiled nightside emission for Kepler-10b and Kepler-78b, and possibly Kepler-407b and K2-106b. Assuming that the dayside cannot be colder than the nightside, we obtained a lower limit on the dayside temperature, namely, T[d] ≳ T[n], which is comparable to the maximum theoretical value T[d,max] from thermal equilibrium considerations (see Sects. 5.1 and 5.2, along with related Figs. 5 and 7). This implies planetary temperatures hotter than T[d,max], possibly due to heat retention through greenhouse effects of a high molecular weight collisional atmosphere. The bulk density measured for the planets considered in this work indicates that they do not host a primary, hydrogen-dominated atmosphere, which was likely lost swiftly given the action of the intense stellar irradiation (e.g., Lopez 2017; Kubyshkina et al. 2018). Therefore, these planets may have quickly developed a secondary, possibly CO[2]-dominated, atmosphere as a result of volcanic activity or magma ocean outgassing (e.g., Elkins-Tanton 2008). Furthermore, if the secondary atmosphere formed while the star was still active, mass-loss driven by the intense stellar irradiation would have led to a complete escape also of this secondary atmosphere (e.g., Kulikov et al. 2006; Tian 2009), leaving behind a magma ocean on the dayside that would release heavy metals into an exosphere through non-thermal processes such as sputtering (e.g., Pfleger et al. 2015; Vidotto et al. 2018). However, an exosphere would not be able to redistribute heat because of its non-collisional nature and the magma ocean would be present only on the dayside, leaving a cold nightside (Léger et al. 2011). Kepler-78b orbits a young star (Sanchis-Ojeda et al. 2013). It is therefore possible that this planet still hosts part of the CO[2]-dominated atmosphere released following the loss of the primary hydrogen-dominated atmosphere. Indeed, the youth of the host star and the small orbital distance pose the condition for the presence of induction heating in the planetary interior (Kislyakova et al. 2017), which would significantly strengthen surface volcanism and outgassing counteracting escape (Kislyakova et al. 2018; Kislyakova & Noack 2020). If this secondary atmosphere is dense enough to be collisional, then heat could be carried from the day to the nightside. Furthermore, Kepler-78b orbits inside the Alfvén radius of the star (Strugarek et al. 2019), thus powering magnetic star-planet interactions that may provide further energy heating up the planetary atmosphere. In contrast, Kepler-10b orbits an old star (Fogtmann-Schulz et al. 2014) suggesting that the secondary atmosphere that was built up initially may have already been lost and that neither induction heating nor magnetic star-planet interaction would be present to support a collisional atmosphere against escape. However, a further secondary, CO[2]-dominated atmosphere may have built up over time from outgassing of the magma ocean present on the dayside as a result of the decreasing strength of mass loss with time due to the decreasing amount of high-energy radiation emitted by late-type stars with increasing age. The scenario of magma-ocean planets with no heat redistribution as described by Léger et al. (2011) may apply to both K2-141b and K2-131b. Indeed, for both planets we found fully consistent values of δ[ec] and A[ph], hence, there is no evidence for a significant nightside emission. For Kepler-407b, K2-106b, K2-229b, and K2-312b, the great uncertainties on δ[ec] and A[ph] prevent us from deriving useful constraints on possible nightside emissions. As mentioned before, solely optical photometry does not allow us to precisely measure geometric albedos. It is still debated whether the albedos of USP small planets should be prevalently high or low. High albedos (A[g] > 0.2) would require clouds of substantial reflective molecules in the secondary atmosphere (Demory 2014). Alternatively, for near-airless planets, high albedos could be the result of specular reflections from the moderately wavy lava surfaces made of metallic species such as iron oxides (Modirrousta-Galian et al. 2021). However, low albedos (A[g] ≲ 0.1) have also been predicted for lava-ocean planets under a different theoretical framework by Essack et al. (2020). In this case, the occultation signal would be mostly due to high thermal emission (e.g. Figs. 11, 13 ). Moreover, we found that the corresponding brightness temperatures for very low albedos and no heat redistribution (Sheets & Deming 2017) might be even higher than the maximum theoretical estimates. This would imply additional heat sources such as internal tidal or magnetic heating (e.g. Lanza 2021). Constraining the Bond albedo (A[B]) and the circulation efficiency (ϵ) requires a couple of assumptions, specifically: a scattering relationship between the Bond albedo and the geometric albedo, which we assumed isotropic in this work, and thermal equilibrium between the received stellar irradiation and the planet emission. Future infrared observations, for instance, with the forthcoming James Webb Space Telescope (Deming et al. 2009), should permit the degeneracy between the reflected and thermally emitted light to be broken and could also provide more precise A[g], T[d], and ϵ estimates. These values, in turn, will yield valuable information on the surface or atmospheric properties of USP small planets and will greatly help in understanding the nature of these extreme We acknowledge the computing centre of INAF – Osservatorio Astrofisico di Catania, under the coordination of the CHIPP project, for the availability of computing resources and support. We acknowledge financial contribution from the agreement ASI-INAF no. 2018-16-HH. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has also made use of publicly available Kepler data via the MAST archive and therefore, we acknowledge the funding for the Kepler space mission provided by the NASA Science Mission Directorate and Space Telescope Science Institute (STScI) for maintaining the archive. Appendix A Posterior distribution of the model parameters Appendix B Detrending of stellar variability using Gaussian processes All Tables All Figures Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform. 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IMO Maths Olympiad Class 7 - 2016 Previous Year Question Paper IMO Previous Year Question Papers 2016 for Class 7 - Free PDF Download The Science Olympiad Foundation conducts the International Mathematics Olympiad (IMO) from class 1 to 12. All the students are advised to participate in this examination so that they can improve in their skills. Students looking for the IMO 2016 Previous Year Question Papers for Class 7 can download them along with solutions in a PDF format. IMO 2016 previous year question papers are solved in a detailed manner by subject matter experts at Vedantu. These 2016 IMO Previous Year Question Papers for Class 7 are extremely useful for the students in Class 7 who are preparing for the IMO Exam 2024-25. Solving Previous Year Papers help in developing a solid foundation in the Maths subject. Prepare for the IMO exam 2024 by downloading this in a PDF format. Read the entire article to learn more about IMO Exam 2024-25, IMO Overview, and IMO Previous Year Question Papers for Class 7. FAQs on International Mathematics Olympiad (IMO) Previous Year Question Papers 2016 for Class 7 1. On what dates is the IMO 2024-25 Exam scheduled for Class 7? IMO Exam for the year 2024-25 will be conducted on the following dates: • 22nd October 2024 • 19th November 2024 • 12th December 2024 Download the IMO Previous Year Question Paper 2016 with solutions for Class 7 and start preparing today as the exam is just one month away. 2. Can I get the IMO 2016 previous year question paper for Class 7 for free? Yes, the IMO Previous Year Question Paper for Class 7 is absolutely free for all students. Any student can access the IMO Previous Year Question Paper for Class 7 with solutions in a PDF format from anywhere in the world. 3. How will IMO Previous Year Question Paper 2016 for Class 7 be helpful for students? The easiest way to become familiar with the IMO Exam pattern is to study IMO previous year question papers. They help you in learning the types of questions you should practise to be more prepared for the IMO Exam 2024–25 and also boost your confidence. 4. Does Vedantu offer any courses for Class 7 students to prepare for IMO? Yes, Vedantu offers many courses for Class 7 students to prepare for many Olympiads. You can check Vedantu’s crash course olympiad page to know the details of the IMO Exam courses. 5. Why should I solve the IMO previous year question paper 2016 for Class 7 the IMO exam? Students can benefit from solving PYQPs in a variety of ways. Solving these questions will help students in clarifying the concepts in their mind, which is very important for gaining a strong understanding of the subject. So, solving the 2016 IMO Previous Year Question Paper will help students to score better in their exams. 6. How can I score well in the IMO exam? 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This ensures that students will be able to access the previous year's question papers at any point in time. With this PDF, students will even be able to practice these questions at any time. 8. What is the Marking Scheme for the Class 7 2024 IMO exam? There are a total of 50 questions that will be asked during the 2024 IMO exams for Class 7. The Logical reasoning section of the paper has a total of 15 questions with each question consisting of one mark. Whereas, the mathematical reasoning section of the paper consists of 20 marks with one mark provided for each question. The everyday mathematics sections consist of 10 marks with 1 mark for each question. Finally, the achiever's section consists of 5 questions with each question weighing 3 marks. 9. Where can I find Study Material for IMO exam class 7? Vedantu provides students with the best study materials to prepare for the IMO exam for class 7. It provides all the important questions and answers to prepare for the exam. It contains all types of questions. Vedantu provides students with sample papers that consist of various questions that will help students to study and prepare well in their IMO exams. These sample papers are extremely important for all students attempting to write the exam. Also, students can download the PDF for free on the Vedantu mobile app.	{"url":"https://www.vedantu.com/olympiad/imo-maths-olympiad-previous-year-question-paper-class-7-2016","timestamp":"2024-11-05T07:13:51Z","content_type":"text/html","content_length":"334016","record_id":"<urn:uuid:d5bc28df-6fa9-4f04-8f98-fdeb90e81fc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00734.warc.gz"}
Welcome to the Sage Constructions documentation! Welcome to the Sage Constructions documentation!¶ This document collects the answers to some questions along the line “How do I construct … in Sage?” Though much of this material can be found in the manual or tutorial or string documentation of the Python code, it is hoped that this will provide the casual user with some basic examples on how to start using Sage in a useful way. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License. Please send suggestions, additions, corrections to the sage-devel Google group! The Sage wiki http://wiki.sagemath.org/ contains a wealth of information. Moreover, all these can be tested in the Sage interface http://sagecell.sagemath.org/ on the web.	{"url":"https://doc-gitlab.sagemath.org/html/en/constructions/index.html","timestamp":"2024-11-14T12:14:46Z","content_type":"text/html","content_length":"21138","record_id":"<urn:uuid:cccbb7f2-3468-4ad8-9479-a896f8a870aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00269.warc.gz"}
Schwarzpulver für Vorderladerkugel About points... We associate a certain number of points with each exercise. When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default". But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number That being said... How many "default points" should you associate with an exercise upon creation? As with difficulty, there is no straight forward and generally accepted way. But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise. Again, very vague... But the number should kind of represent the "work" required. About difficulty... We associate a certain difficulty with each exercise. When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default". But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here. Why we use chess pieces? Well... we like chess, we like playing around with $\LaTeX$-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want. That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc. In physics exercises, we try to follow this pattern: Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise Level 5 - Level 6 - No explanation / solution video to this exercise has yet been created. Visit our to see solutions to other exercises. Don't forget to subscribe to our channel, like the videos and leave comments! Schwarzpulver setzt bei der Explosion etwa HO Energie Explosionswärme frei. Wie viel Schwarzpulver ist mindestens nötig um einer Vorderladerkugel .g eine Schnelligkeit von vO zu verleihen? Man benötigt mindestens E frac sscmK v^ frac m qtyv^ Ek Energie. Dafür müssen mindestens m fracEH fracEkH mS Schwarzpulver explodieren. In Wirklichkeit wird sehr viel mehr Schwarzpulver benötigt denn der Wirkungsgrad eines Vorderlader-Gewehres ist sehr schlecht. Schwarzpulver setzt bei der Explosion etwa HO Energie Explosionswärme frei. Wie viel Schwarzpulver ist mindestens nötig um einer Vorderladerkugel .g eine Schnelligkeit von vO zu verleihen? Man benötigt mindestens E frac sscmK v^ frac m qtyv^ Ek Energie. Dafür müssen mindestens m fracEH fracEkH mS Schwarzpulver explodieren. In Wirklichkeit wird sehr viel mehr Schwarzpulver benötigt denn der Wirkungsgrad eines Vorderlader-Gewehres ist sehr schlecht. Contained in these collections:	{"url":"https://texercises.com/exercise/schwarzpulver-fur-vorderladerkugel/?col=energie","timestamp":"2024-11-07T15:28:26Z","content_type":"text/html","content_length":"83394","record_id":"<urn:uuid:d4add1e6-f999-4e38-b9bd-5e3ff2ca2400>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00249.warc.gz"}
seminars - Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X. A natural question arises when studying the configuration of those curves : If a point of X is contained in « many » exceptional curves, it is torsion on its fiber on E? In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if « many » equals 4 or more, then yes. With Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if « many » equals 9 or more, then yes. Additionnally we find counterexamples where a torsion point lies at the intersection lies at the intersection of 7 exceptional curves. Zoom 910 1345 711 Passcode 331303	{"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&l=en&sort_index=room&order_type=asc&page=89&document_srl=1007982","timestamp":"2024-11-11T09:51:29Z","content_type":"text/html","content_length":"50273","record_id":"<urn:uuid:eea5b7c0-158a-4f27-ac97-c279f6bc6bc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00448.warc.gz"}
In the circuit shown, if R = 0, then the phase angle between v(t) and i(t) is<br/> \| EXAMIANS The hexadecimal number system is also called base-16, a numeration system in which all numbers are represented using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F only. The system is base of 16. The hexadecimal numbers are used to represent binary numbers because of ease of conversion and compactness.	{"url":"https://examians.com/in-the-circuit-shown-if-r-0-then-the-phase-angle-between-vt-and-it-isbr","timestamp":"2024-11-05T22:30:16Z","content_type":"text/html","content_length":"57152","record_id":"<urn:uuid:ff62c76c-abc6-4ec7-b648-7cef13237cb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00772.warc.gz"}
DFS - Statement of Cash flows I'm busy revising DFS - Limited Company Accounts for my CBA in 3 weeks time. I've come stuck on statement of cash flows, right at the end - calculating the net increase/(decrease) in cash and cash Looking at the Osborne workbook, one example has cash & c/e at beginning of year as 135, cash & c/e at end of year as (486), net increase/(decrease) in cash and cash equivalents as (621). How is the (621) calculated, how do I interpret this as cash inflows/outflows to help me understand this? Also, I've come stuck on the treatment of gains/losses on disposal in the profit from operations section of the cash flow statement. Why are profits on disposal taken away from cash flows, and losses on disposal added in to cash flows? • You started with a positive balance of 135. That got reduced to 486 negative. 135 plus ? is (486) So ? is 621, which is the decrease in your cash flow. A decrease in your cash flow means you have more cash tied up in the items and therefore you got less cash to spent. Disposal issue: You have the gains on disposal added to the profit from operations, but this is not directly related to your operations, but instead to your investments part. So you take it out of the cash flows from operations. The losses get added to the cash flow from operations, because you deducted it earlier from the profit on operations. I hope this helps, • Some excellent advice there from Rinske. Gains/losses on disposal of non-current assets ALWAYS feature on the DFS/FNST paper so it's well worth practising lots and lots of these. Remember losses on disposal are the same as depreciation - they are 'paper' losses, not cash based so need to be added back. Gains are the opposite and need to be deducted from operating profit. Students often get confused with the add/subtract issue relating to the statement of cash flows, but the best way is to do lots of question practise to make it second nature. You may also find my article on IAS 7 helps explain this topic. Best regards • Hi Steve & Rinske, Thanks for you advice, it has helped a lot. However to quote 'The losses get added to the cash flow from operations, because you deducted it earlier from the profit on operations' - I'm not too sure what this means, can you explain? Many thanks, Hi Steve & Rinske, Thanks for you advice, it has helped a lot. However to quote 'The losses get added to the cash flow from operations, because you deducted it earlier from the profit on operations' - I'm not too sure what this means, can you explain? Many thanks, When you prepare an income statement or statement of comprehensive income, you deduct the losses from your investments from the revenue and all the other bits to get to the profit from operations, even though no actual cash went out. So when you start with the profit from operations in your statement of cash flow, you already deducted the investment loss from your opening figure and will now need to add it back to it, to get to the cash flows from operations. Hope that helps,	{"url":"https://forums.aat.org.uk/Forum/discussion/31398/dfs-statement-of-cash-flows","timestamp":"2024-11-04T20:38:03Z","content_type":"text/html","content_length":"296814","record_id":"<urn:uuid:58e3d0ef-27ef-4e18-b592-c288e8667082>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00204.warc.gz"}
[ANSWERED] The voltage across a capacitor with the value C 3 F is given - Kunduz The voltage across a capacitor with the value C 3 F is given Last updated: 1/21/2024 The voltage across a capacitor with the value C 3 F is given by the waveform in Fig P6 10 Find the waveform for the current in the capacitor for the time interval t3 t t5 given that t3 45s t5 86s Note The given values for C t1 t2 t3 and or t5 may be different than what you have used in previous questions v c t V 12 Frange t1 t2 t3 t4 t5 12 Figure P6 10 Notes on entering solution Enter your solution in Amps Pay attention to the interval given Enter your solution to 2 decimal points Do not include units in your answer ex 5W is entered as 5 00 t s	{"url":"https://kunduz.com/questions-and-answers/the-voltage-across-a-capacitor-with-the-value-c-3-f-is-given-by-the-waveform-in-fig-p6-10-find-the-waveform-for-the-current-in-the-capacitor-for-the-time-interval-t3-t-t5-given-that-t3-45s-t5-86s-note-330381/","timestamp":"2024-11-13T05:14:47Z","content_type":"text/html","content_length":"210101","record_id":"<urn:uuid:9f352a13-61c9-4691-88d5-eb6c6b8431ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00475.warc.gz"}
How do you write 1.5 million in numbers? How do you write 1.5 million in numbers? 1.5 million in numbers is 1,500,000. The number 1 million in numbers is 1,000,000. Writing Millions Sometimes in mathematics, we are given a specific number of millions in words. When this is the case, we need to know how to convert it to number form. Thankfully, we can do this in a straightforward manner using a little logic and multiplication. Leave a Comment	{"url":"https://thestudyish.com/how-do-you-write-1-5-million-in-numbers/","timestamp":"2024-11-13T04:20:32Z","content_type":"text/html","content_length":"49948","record_id":"<urn:uuid:16df15cf-0ed3-4e39-a38f-78c650ddd010>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00133.warc.gz"}
iff is used to create equivalence constraints between logic cases iff is used to define logic equivalence. As an example, the following code will ensure that a variable x satisfies a set of inequalities if and only if a binary variable d is true (value 1), and vice d = binvar(1); F = iff(d,Ax <= b); iff is mainly intended for (BINARY <-> BINARY), although it can be used also for more general constructions as above (bearing in mind that these models typically are numerically sensitive and may require a lot of binary variables to be modeled). The following code constrains a variable y if and only if a set of constraints on x hold. F = iff(Ax <= b, Ay <=y); The following code is equivalent d = binvar(1,1); F = iff(Ax <= b, d); F = iff(Ay <= b, d); iff is overloaded as == on constraints, hence the following code gives the same model. F = [(Ax <= b) == (Ay <= b)]; and so does the following model binvar d F = [d == (Ax <= b), d == (A*y <= b)]; Since iff is implemented using a big-M approach, it is crucial that all involved variables have explicit bound constraints. iff is very sensitive to modeling choices (is $10^{-5}$ really a positive number in practice?), so it should be avoided as much as possible. If you can rewrite your model to simple (BINARY-> Linear) using implies, do so! See for instance the logic programming example	{"url":"https://yalmip.github.io/command/iff/","timestamp":"2024-11-09T09:02:01Z","content_type":"text/html","content_length":"32533","record_id":"<urn:uuid:96d3ec61-3209-41d3-8ac9-e716da6ee872>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00506.warc.gz"}
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What Is X Modulus? Modulo arithmetic is one of the most important and widely used types of arithmetic. For those who are not familiar with it, here is a quick description. First, let’s define a number n, say “A”. If we have a finite group G, where each element I chooses an element in the set i, so that if I am chosen by an element in the set i then g will be chosen by an element in the set i. Go to the website for more info. Everything You Wanted To Know About X Mod And Were Too Embarrassed To Ask We then take the subset of all subsets of G called “G”, where we perform a modulo operation on the set I to get the value of the subset called “A”. We can also use this same method for any other finite group A. For example, we can use this method to find the value of x modulo n if we have n different numbers to work with, say “n+1” and “n-1”. This is a finite logic that can be performed on an n-dimensional graph, since the graphs can be generated from any shape. So now that we know what a mod n is, we can actually give a concrete definition. The definition is as follows: for any finite n, if there is a divisor k such that I is either the sum of k(I+1) or the product of k(i-1), then: for every I, if x mod n is k, then the square root of x mod n must be k. For some numbers k such as 0, we just need to find their log-norm which is defined as the difference between the normal value of x and the log-normal value of k. Then we can conclude that the value of x mod n is equal to the log-normal value of k. The proof of this is not as important as the fact that it can be done, and that was the purpose of the article. In other words, we want to show that the meaning of “x mod n” is the meaning of “k” when multiplied by k. In other words, we want to show that the formula for x and k is a finite logic that can be used to solve practical problems such as those concerning the square root of numbers. Indeed, all numbers, even “0”, are modular and so can be placed into this category.	{"url":"https://culture-multimedia.org/what-is-x-modulus/","timestamp":"2024-11-14T23:56:47Z","content_type":"text/html","content_length":"71816","record_id":"<urn:uuid:3047665f-d058-4fb6-8cfb-2b377c896afb>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00414.warc.gz"}
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Bella wants to conduct a survey to determine how many of the residents of Forks believe in supernatural creatures. Which of the following survey methods will allow Bella to make a valid conclusion about the percentage of residents of Forks that believe in supernatural creatures? Choose 1 answer: A: Ask a randomly selected sample of residents to answer the question anonymously. B: Ask a randomly selected sample of high school students to answer the question anonymously. Find an answer to your question 👍 “Bella wants to conduct a survey to determine how many of the residents of Forks believe in supernatural creatures. Which of the following ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions. Search for Other Answers	{"url":"https://cpep.org/mathematics/1541564-bella-wants-to-conduct-a-survey-to-determine-how-many-of-the-residents.html","timestamp":"2024-11-07T20:26:06Z","content_type":"text/html","content_length":"24102","record_id":"<urn:uuid:6550135b-bfd9-4556-b095-5fa5130ae555>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00297.warc.gz"}
Compute the difference between the median absolute deviations (MAD) for two response variables. The median absolute deviation (MAD) is: $ \mbox{MAD} = \mbox{median} \|x - \tilde{x}\| $ where $ \tilde{x} $ is the median of the variable. This statistic is sometimes used as an alternative to the standard deviation. The scaled MAD is defined as For normally distributed data, the scaled MAD is approximately equal to the standard deviation. For the differeence of median absolute deviations, the median absolute deviation is computed for each of two samples then their difference is taken. Syntax 1: LET <par> = DIFFERENCE OF MAD <y1> <y2> <SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the first response variable; <par> is a parameter where the computed difference of the median absolute deviations is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Syntax 2: LET <par> = DIFFERENCE OF SCALED MAD <y1> <y2> <SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the first response variable; <par> is a parameter where the computed difference of the scaled median absolute deviations is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. LET A = DIFFERENCE OF MAD Y1 Y2 LET A = DIFFERENCE OF MAD Y1 Y2 SUBSET X > 1 LET A = DIFFERENCE OF SCALED MAD Y1 Y2 Dataplot statistics can be used in a number of commands. For details, enter MEDIAN ABSOLUTE DEVIATION is a synonym for MAD MADN is a synonym for SCALED MAD NORMALIZED is a synonym for SCALED Related Commands: MAD = Compute the median absolute deviation. AAD = Compute the average absolute deviation. STANDARD DEVIATION = Compute the standard deviation. DIFFERENCE OF AAD = Compute the difference of average absolute deviations. DIFFERENCE OF SD = Compute the difference of standard deviations. DIFFERENCE OF BIWEIGHT SCALE = Compute the difference of biweight scales. DIFFERENCE OF BIWEIGHT MIDVARIANCE = Compute the difference of biweight midvariances. STATISTICS PLOT = Generate a statistic versus subset plot. BOOTSTRAP PLOT = Generate a bootstrap plot. TABULATE = Perform a tabulation for a specified statistic. Robust Estimation of Scale Implementation Date: 2016/02: Added DIFFERENCE OF SCALED MAD SKIP 25 READ IRIS.DAT Y1 TO Y4 X LET A = DIFFERENCE OF MAD Y1 Y2 TABULATE DIFFERENCE OF MAD Y1 Y2 X XTIC OFFSET 0.2 0.2 DIFFERENCE OF MAD PLOT Y1 Y2 X BOOTSTRAP DIFFERENCE OF MAD PLOT Y1 Y2 X Dataplot generated the following output. LET A = DIFFERENCE OF MAD Y1 Y2 THE COMPUTED VALUE OF THE CONSTANT A = 0.39999986E+00 TABULATE DIFFERENCE OF MAD Y1 Y2 X * Y1 AND Y2 X * DIFFERENCE OF MEDIAN ABSOLUTE 1.00000 * -0.500002E-01 2.00000 * 0.150000 3.00000 * 0.200000 GROUP-ID AND STATISTIC WRITTEN TO FILE DPST1F.DAT Privacy Policy/Security Notice Disclaimer \| FOIA NIST is an agency of the U.S. Commerce Department. Date created: 03/27/2003 Last updated: 11/09/2015 Please email comments on this WWW page to alan.heckert@nist.gov.	{"url":"https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/diffmad.htm","timestamp":"2024-11-10T01:13:42Z","content_type":"text/html","content_length":"12026","record_id":"<urn:uuid:d7bc58a0-c6c8-4c1b-9cea-7a20a09632b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00875.warc.gz"}
Type II Error in R » Data Science Tutorials Type II Error in R, we will discuss the concept of Errors in Hypothesis Testing, explore the different types of errors that may arise during this process, and learn how to calculate them. Hypothesis testing involves forming assumptions about a model or data set, which serve as the foundation for further analysis. The term ‘Error’ in Hypothesis Testing refers to the estimation of accepting or rejecting a specific hypothesis. Two primary types of errors can occur in this context: 1. Type I Error (Alpha Error): This error happens when we reject the Null Hypothesis, even though it is indeed true. It is also referred to as a ‘false positive,’ as it leads to incorrectly rejecting a correct hypothesis. 2. Type II Error (Beta Error): This error occurs when we fail to reject the Null Hypothesis despite it being incorrect, and the Alternative Hypothesis is correct. It is known as a ‘false negative,’ as it results in retaining an incorrect hypothesis. Understanding and managing these errors is crucial for conducting accurate hypothesis testing and drawing valid conclusions from the data. P(X) is the probability of the event X happening. Ho = NULL Hypothesis Ha = Alternative Hypothesis In this example, we will examine a jury/court decision-making process for a case, where the jury has two primary options: determining the convict is guilty or not guilty. These two decisions correspond to the two hypotheses considered in hypothesis testing. For every decision made by the jury, the reality can be either: Difference between sort and order in R » 1. The convict is genuinely guilty in real life, and the jury’s decision is that they are guilty. 2. The convict is not guilty in real life, and the jury’s decision is that they are not guilty. Now, let’s explore the two types of errors that can arise in this context: 1. Type I Error (False Conviction): This error occurs when the jury decides the convict is guilty, but in reality, they are not guilty. It results in an incorrect conviction and can have severe consequences for the individual’s life and future. 2. Type II Error (Missed Conviction): This error happens when the jury decides the convict is not guilty, even though they are genuinely guilty in reality. It leads to an incorrect acquittal, potentially allowing a criminal to go free and continue committing crimes. Minimizing these errors is crucial for maintaining justice and ensuring fair verdicts in court proceedings. Ho = Not Guilty Ha = Guilty #In this example, Type I Error will be: Innocent in Jail Type II Error will be: Guilty Set Free To Calculate the Type II Error in R Programming: Type II Error, also known as Beta Error, can be determined using a specific formula. However, in this article, we will demonstrate how to calculate Type II Error using the R programming language for enhanced efficiency and accuracy. P(Probability of failing to remove Ho / Probability of Ho being false ) = P(Accept Ho \| Ho False) # Parameters effect_size <-0.5 # The difference between null and alternative hypotheses sample_size <- 30 # The number of observations in each group sd <- 4 # The standard deviation alpha <- 0.05 # The significance level # Calculate Type II Error pwr_result <-pwr.t.test(n = sample_size,d = effect_size / sd,sig.level = alpha,type = "two.sample",alternative = "two.sided") type_II_error <- 1 - pwr_result$power # Print Type II Error How to choose optimal number of epochs in R » Data Science Tutorials	{"url":"https://datasciencetut.com/type-ii-error-in-r/","timestamp":"2024-11-06T20:58:08Z","content_type":"text/html","content_length":"109837","record_id":"<urn:uuid:1d22352e-5870-4017-b0d1-2900499f781d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00443.warc.gz"}
7.5 Sectors of Circles \| Education Auditorium top of page Chapter 7 Area and Volume 7.5 Sectors of Circles A semicircle is a plane figure that is formed by dividing a circle into exactly two parts, while a quarter circle is a quarter of a circle, formed by splitting a circle into 4 equal parts or a semicircle into 2 equal parts. In this section, we learn about area of a circle sector. If you're a student in a school/college that's based in England, you might be entitled for full access to our eLearning platform where you can access to all our tutorial videos and practice Please check with the head of Maths or deputy headteacher at your school/college to request access if they've already registered to our free pilot subscription. They can always reach us at the below Note: Due to our safeguard and chilled protection policy; we wouldn't be able to respond to students enquiries directly. AÂ semicircleÂ is a plane figure that is formed by dividing aÂ circleÂ into exactly two parts, while aÂ quarterÂ circleÂ is aÂ quarterÂ of aÂ circle, formed by splitting aÂ circleÂ into 4 equal parts or a semicircle into 2 equal parts. In this section, we learn about area of a circle sector. Calculating the area and perimeter of semicircles and quarter circles. Sectors of Circles - Area of a circle sector bottom of page	{"url":"https://www.education-auditorium.co.uk/gcse-maths-higher-ch-7-area-and-volume-7-7/7.5-sectors-of-circles","timestamp":"2024-11-04T15:39:22Z","content_type":"text/html","content_length":"1050500","record_id":"<urn:uuid:9a4c8e0a-4ca9-469d-bef4-b3946ff4a4aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00698.warc.gz"}
Go to Surface Area or Volume. A prism is a solid object with: • identical ends • flat faces • and the same cross section all along its length ! A cross section is the shape made by cutting straight across an object. The cross section of this object is a triangle ... .. it has the same cross section all along its length ... ... so it's a triangular prism. Try drawing a shape on a piece of paper (using straight lines) Then imagine it extending up from the sheet of paper ... ... that's a prism ! No Curves! A prism is a polyhedron, which means all faces are flat! For example, a cylinder is not a prism, because it has curved sides. The ends of a prism are parallel and each one is called a base. The side faces of a prism are parallelograms (4-sided shapes with opposite sides parallel) These are all Prisms: Square Prism: Cross-Section: Cube: Cross-Section: (yes, a cube is a prism, because it is a square all along its length) (Also see Rectangular Prisms ) Triangular Prism: Cross-Section: Pentagonal Prism: Cross-Section: and more! Example: This hexagonal ice crystal. It looks like a hexagon, but because it has some thickness it is actually a hexagonal prism! Photograph by NASA / Alexey Kljatov. Regular vs Irregular Prisms All the previous examples are Regular Prisms, because the cross section is regular (in other words it is a shape with equal edge lengths, and equal angles.) Here is an example of an Irregular Prism: Irregular Pentagonal Prism: It is "irregular" because the cross-section is not "regular" in shape. Right vs Oblique Prism When the two ends are perfectly aligned it is a Right Prism otherwise it is an Oblique Prism: Surface Area of a Prism Surface Area = 2 × Base Area + Base Perimeter × Length Example: What is the surface area of a prism where the base area is 25 m^2, the base perimeter is 24 m, and the length is 12 m: Surface Area = 2 × Base Area + Base Perimeter × Length = 2 × 25 m^2 + 24 m × 12 m = 50 m^2 + 288 m^2 = 338 m^2 (Note: we have an Area Calculation Tool) Volume of a Prism The Volume of a prism is the area of one end times the length of the prism. Volume = Base Area × Length Example: What is the volume of a prism where the base area is 25 m^2 and which is 12 m long: Volume = Area × Length = 25 m^2 × 12 m = 300 m^3 Play with it here. The formula also works when it "leans over" (oblique) but remember that the height is at right angles to the base: And this is why: The stack can lean over, but still has the same volume More About The Side Faces The side faces of a prism are parallelograms (4-sided shape with opposites sides parallel) A prism can lean to one side, making it an oblique prism, but the two ends are still parallel, and the side faces are still parallelograms! But if the two ends are not parallel it is not a prism. 639,640,863, 1826, 1827,864, 3379, 3377, 3378, 7649	{"url":"http://wegotthenumbers.org/prisms.html","timestamp":"2024-11-08T12:44:20Z","content_type":"text/html","content_length":"12117","record_id":"<urn:uuid:a5836dd7-4dac-4f7a-ad2b-bc8bd79d8aa7>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00004.warc.gz"}
How to calculate your ideal weight on growth and other indicators carefully for their body weight monitored more often women who dream of a slim figure, or professional athletes.But, nevertheless, find out how to calculate the ideal weight, it will be useful to everyone.Body weight above the norm can cause a variety of serious diseases.Too thin people are also at risk - with a lack of weight violated the metabolic processes in the body and the functioning of many systems.Monitor the ratio of height and weight just need to make timely adjustments in diet and lifestyle in general. How to determine the ideal weight for growth The simplest formula for calculating body mass increase compliance involves subtracting 110 from the rise in the see - women and 100 - men.This result can be considered arbitrary, since such a system calculation does not account for the many individual features.On the weight of a person affects not only his height and gender, but also the structure of the skeleton and physique.Many well-known formula for calculating the body mass index.The calculations are also used only data of height and weight, but insufficient for determining or excessive weight, this scheme is suitable.Necessary height in meters multiplied by 2, the result is divided by weight in kg.If you are seriously concerned about how to calculate your ideal weight, you can use the table with the detailed values.Conventionally, it is possible to assume that the value obtained is less than 18 indicates a lack of weight gain, and more than 30 - about obesity.All values between 18 and 25 are indicative of ideal weight, get results if the 25-30 - it makes sense to think about getting rid of a few kilograms, you are at risk. How to calculate your ideal weight: other ways correctly calculate the recommended weight is impossible without taking into account body type.Identify fine-boned, big-boned and normokostny views.At home, set his belonging to one of them is a snap.For this using conventional tape measures need to measure the circumference of your wrist.For women - less than 16 cm is asthenic type (fine-boned), more than 18 cm - hypersthenic or boned.We already know that men ideal weight is calculated differently than for a woman.Accordingly, shirokostnye the stronger sex have the wrist more than 20 cm in girth, and fine-boned - no more than 18. For representatives of relevant normalnokostnogo type formula minus 100-105 kg of growth.Very high (greater than 176 cm) is recommended to subtract 110 cm. If the type asthenic addition, reduce the result by 10 percent.Hypersthenic allowed to form, on the contrary, increased by 10 percent finite. How to calculate your ideal weight based on age and lifestyle?There are also other formulas to calculate the recommended body mass.For example, according Negleru must first calculate the difference between the available height and 152.4 cm, then divide it by 2.45, and multiplying by 900 grams.To add the result of 45 kg, since this weight, according to the method is the best when grown in 152.4 cm. The formula assumes Pine calculations using chest circumference and other indicators.Yet it is not necessary every day to recheck your weight ideal.It is enough to maintain optimal mobile lifestyle and watch your diet, then your weight will always be OK, but health problems will bypass.	{"url":"https://tipings.com/en/pages/134005","timestamp":"2024-11-05T10:13:27Z","content_type":"text/html","content_length":"17381","record_id":"<urn:uuid:e3def6b5-7be3-47ac-827d-a95d4704f43a>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00796.warc.gz"}
Oriented matroids with few mutations This paper defines a "connected sum" operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement of n projective planes in ℝP^ d-1 contains at least n simplicial regions and every plane is adjacent to at least d simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4 n pseudoplanes in ℝP^3 with only 3 n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the "strong-map conjecture" of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane. Dive into the research topics of 'Oriented matroids with few mutations'. Together they form a unique fingerprint.	{"url":"https://portal.fis.tum.de/en/publications/oriented-matroids-with-few-mutations","timestamp":"2024-11-04T14:35:49Z","content_type":"text/html","content_length":"48814","record_id":"<urn:uuid:aa6c414b-fbec-4daa-ab1c-eca4163cf05b>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00605.warc.gz"}
Amazing Short Story Behind Invention of Zero - Jokes To Text The great mathematician, Aryabhatta, once asked his wife: “Will you let me go out alone & enjoy drinks with my friends over every weekend, every month?” Wife: What is the Probability of me saying Yes as per your calculation ? That’s when Aryabhatta discovered Zero	{"url":"https://jokestotext.com/story-behind-invention-of-zero/","timestamp":"2024-11-06T11:29:16Z","content_type":"text/html","content_length":"52902","record_id":"<urn:uuid:09ac99a3-342d-4778-a6bd-1983da44f529>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00487.warc.gz"}
Studies of symmetries that give special quantum states the "right to exist" University of Waterloo In this thesis we study symmetric structures in Hilbert spaces known as symmetric informationally complete positive operator-valued measures (SIC-POVMs), mutually unbiased bases (MUBs), and MUB-balanced states. Our tools include symmetries such as the Weyl-Heisenberg (WH) group symmetry, Clifford unitaries, Zauner symmetry, and Galois-unitaries (g-unitaries). In the study of SIC-POVMs, we found their geometric significance as the ``most orthogonal'' bases on the cone of non-negative operators. While investigating SICs, we discovered a linear dependency property of the orbit of an arbitrary vector with the Zauner symmetry under the WH group. In dimension d = 3, the linear dependency structures arising from certain special SIC states are identified with the Hesse configuration known from the study of elliptic curves in mathematics. We provide an analytical explanation for linear dependencies in every dimension, and a numerical analysis based on exhaustive numerical searches in dimensions d = 4 to 9. We also study the relations among normal vectors of the hyperplanes spanned by the linearly dependent sets, and found 2-dimensional SICs embedded in the Hilbert space of dimension d = 6, and 3-dimensional SICs for d = 9. A full explanation is given for the case d = 6. Another study in the thesis focuses on the roles of g-unitaries in the theory of mutually unbiased bases. G-unitaries are, in general, non-linear operators defined to generalize the notion of anti-unitaries. Due to Wigner's theorem, their action has to be restricted to a smaller region of the Hilbert space, which consists of vectors whose components belong to a specific number field. G-unitaries are relevant to MUBs when this number field is the cyclotomic field. In this case, we found that g-unitaries simply permuted the bases in the standard set of MUBs in odd prime-power dimensions. With their action further restricted only to MUB vectors, g-unitaries can be represented by rotations in the Bloch space, just as ordinary unitary operators can. We identify g-unitaries that cycle through all d + 1 bases in prime power dimensions d = p^n where n is odd (the problem in even prime power dimensions has been solved using ordinary unitaries). Each of these MUB-cycling g-unitaries always leaves one state in the Hilbert space invariant. We provide a method for calculating these eigenvectors. Furthermore, we prove that when d = 3 mod 4, they are MUB-balanced states in the sense of Wootters & Sussman and Amburg et al. quantum physics, quantum information, symmetry, SIC-POVM, MUB, Galois unitary, g-unitary, Weyl-Heisenberg group, Clifford group, MUB-balanced, MUB-cycling	{"url":"https://uwspace.uwaterloo.ca/items/9ff5507d-af4e-4bfa-842f-7d34565ba9f3","timestamp":"2024-11-14T01:08:05Z","content_type":"text/html","content_length":"431130","record_id":"<urn:uuid:bce241bf-ab4f-44ae-8daa-7e9185a69fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00110.warc.gz"}
One Year Blogversary Event: Tommee Tippee Explora Section Plates Review and Giveaway! - CLOSED! I just love all of the feeding accessories over at Tommee Tippee ! You may remember my review for explora truly spill proof sippy cups. These are still Sam's favorite cups! (and mine since they don't leak!) I received another product from Tommee Tippee that I am excited to tell you about: explora section plates ! I may have mentioned a few times... that my sister-in-law is expecting her first baby! (I'm pretty excited for them!) Well since we don't know the gender of the baby yet, I picked out these plates in orange. These section plates are great for the baby since they are BPA free, dishwasher and microwave safe, and have three sections so that baby's food doesn't touch. (I still don't like for my food to touch!) You can also use them with Tommee Tippee's easi-mat to hold the section plate securely in place! That is a must for self feeder who likes to throw his or her food! BUY IT: You can purchase Tommee Tippee products at Babies R Us WIN IT: Thanks to Tommee Tippee , one lucky Adventures of a Thrifty Mommy reader will win a 2 pack of explora section plates like the ones I received! ( US Residents Only You MUST be a follower of Adventures of a Thrifty Mommy via Google Friend Connect (located at the right) for your entry to count. And you MUST include your email address with each entry. Mandatory Entry: Follow Adventures of a Thrifty Mommy via Google Friend Connect. Tommee Tippee and comment on another item you'd like to try. (both combined = 1 entry) That's all you have to do! But, if you'd like to receive extra entries you can: • Like Tommee Tippee on Facebook here. • Follow Tommee Tippee on Twitter here. • Like Adventures of a Thrifty Mommy on Facebook here. (1 entry) • Follow Adventures of a Thrifty Mommy on Networked Blogs. (1 entry) • Follow Adventures of a Thrifty Mommy on Twitter here. (1 entry) • Tweet this giveaway. (1 entry daily, include the link) • Enter another Adventures of a Thrifty Mommy giveaway. (1 entry for each giveaway entered) • Post the Adventures of a Thrifty Mommy button (located at the right) on your blog. (5 entries, leave me a link) • Post the One Year Blogversary button (located at the right) on your blog. (5 entries, leave me a link) Please include your email address with EACH entry. If you don't have your email attached to your comment, I won't be able to contact you if you win! Post a separate comment for each entry (if it says 3 entries, post 3 separate comments.) This giveaway ends October 24, 2011 at 11:59 pm. Good luck! * Lucky winner or winners will be selected through True Random Number Generator.* If your profile page does not show your email address, please include your email address in your comment. For example: adventuresofathriftymommy@gmail.com -- so that I may get in touch with you if you are selected as the giveaway winner. * Each giveaway winner has 48 hours to respond to my email about getting the awesome giveaway prize to him/her. If the winner does not reply to my email within 48 hours, I will choose another winner using the True Random Number Generator. * I do contact each winner via email Disclosure: I received the products mentioned above for this review. No monetary compensation was received by me. This is my completely honest opinion above and may differ from yours. Because I do not directly ship most giveaways from my home, I cannot be held liable for lost or not received products. 158 comments: I follow you on gfc and I like the spill proof trainer cup I like tommee tippee on fb I like you on fb I entered fire wire I entered scooby doo I entered babar I entered me bands I entered marcal I entered big bully I entered paper flavor I entered parenting infants I entered power capes I entered little one books I entered hair coverings I entered plasma car I entered rock n learn I entered costume squad I entered light affection I entered kidorable I entered sendafrizbee I entered doodlemark I entered kidtoons I entered exergen I entered five finger tees I entered thomas I entered flowerz in her hair I entered enjoy life foods entered novica I entered furniture fix I entered stinky kids I entered tiger balm I entered angelina ballerina I follow you on GFC. 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Electric Charge & Coulomb's Law UY1: Electric Charge & Coulomb’s Law There are exactly two kinds of electric charge: negative and positive. Electric charge is quantized: The magnitude e of charge of the electron or proton is a natural unit of charge. Every observable amount of electric charge Q is always an integer multiple n of this basic unit e: $$Q = ne$$ Principle of conservation of charge: The algebraic sum of all the electric charges in any closed system is constant. Opposite charges attract, and like charges repel A positive charge and a negative charge attract each other. Two positive charges or two negative charges repel each other. The magnitude of the electric force F between two point charges, q[1] and q[2], is directly proportional to the product of the charges and inversely proportional to the square of the distance r between them: $$F = k \frac{\|q_{1} q_{2} \|}{r^{2}}$$ The directions of the forces the two charges exert on each other are always along the line joining them. $$\vec{F}_{2 \, \text{on} \, 1} = \, – \vec{F}_{1 \, \text{on} \, 2}$$ The value of the proportionality constant k depends on the system of units used. In SI units: $$k = \frac{1}{4 \pi \epsilon_{0}} \approx 8.988 \times 10^{9} \, Nm^{2} C^{-2}$$ , where the permittivity of free space: $$\epsilon_{0} \approx 8.854 \times 10^{-12} \, N^{-1}m^{-2} C^{2}$$ The SI unit of electric charge is called one coulomb: $$1 C = 1 \, As$$ , where ampere A is a unit of electric current, equal to one coulomb per second. Principle of superposition of forces: When two charges exert forces simultaneously on a third charge, the total force acting on that charge is the vector sum of the forces that the two charges would exert individually. Next: The Electric Field As A Web Leave a Comment This site uses Akismet to reduce spam. Learn how your comment data is processed. Back To University Year 1 Physics Notes	{"url":"https://www.miniphysics.com/uy1-coulombs-law.html","timestamp":"2024-11-13T11:18:34Z","content_type":"text/html","content_length":"75786","record_id":"<urn:uuid:75580722-2e9d-40ff-bdc4-67f0a006fe8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00843.warc.gz"}
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Crash course in Gaussian integers Yesterday, I attended the UKMT mentoring conference in Murray Edwards College (affectionately abbreviated to ‘Medwards’), which mainly consisted of an excellent dinner in the Fellows’ dining room and informal discussion about various topics. Speaking of mentoring, I recently prepared for one of my mentees a crash course in the application of Gaussian integers to solving certain olympiad-style Diophantine equations. What follows is an almost verbatim regurgitation of that e-mail. Brief introduction So, basically a Gaussian integer is a complex number of the form a + bi, where a and b are both integers. When plotted in the complex plane, they form a square lattice, as shown in the left-hand diagram below. The hexagonal lattice of blue points is the ring of Eisenstein integers, which shares many of the same interesting properties as the Gaussian integers. Now, the most important fact about Gaussian integers is this: Gaussian integers are a Unique Factorisation Domain. The ordinary integers are also a Unique Factorisation Domain, as I’ll explain below. Unique factorisation domains Basically, the elements of $\mathbb{Z}$ (the ring of integers) can be categorised as follows: • Zero: 0 • Units: +1 and -1 • Irreducibles: +2, -2, +3, -3, +5, -5, +7, -7, +11, -11, … • Composite numbers: +4, -4, +6, -6, +8, -8, +9, -9, … Units are defined as elements whose reciprocals also belong to the ring. In particular, every non-zero element in a field is a unit. Irreducibles are defined to be elements that cannot be expressed as a product of two non-units; everything else is composite. Now, we define a prime to be a number p such that if p\|ab, then p\|a or p\|b. It’s straightforward to show logically that every prime must be an irreducible. It’s very non-trivial to show that the converse holds (that all irreducibles are primes), and there are rings in which this is not the case, such as $\mathbb{Z}[\sqrt{-3}]$. But for the integers, this is true, because they form a Principal Ideal Domain. The integers are also a Unique Factorisation Domain (this follows from the property that all irreducibles are primes), which essentially means that the Fundamental Theorem of Arithmetic holds. Every integer can be uniquely expressed as a product of irreducibles, up to reordering and multiplication by units. A proof of this from the prime property is given in the bottom-right corner of the second page of this GRM revision sheet I prepared earlier. Anyway, now that I’ve described these concepts in terms of the ordinary integers, I’ll now explain the Gaussian integers. The Gaussian integers The Gaussian integers, written Z[sqrt(-1)], also form a PID (hence a UFD), so we have the nice property that all irreducibles are primes. Firstly, we need to identify which elements are units; in this case, they are +1, -1, +i and -i. Note that the number 2 is not a prime in the Gaussian integers, since (1 + i)(1 – i) = 2, and neither 1 + i nor 1 – i is a unit. 1 + i and 1 – i are irreducibles (and called `associates’ since they differ only by multiplication by a unit, analogous to the irreducibles +3 and -3 in the ordinary integers). In fact, we can show the following: A positive integer is a Gaussian prime if and only if it is prime (as an ordinary integer) of the form p = 4k + 3. One direction is easy to prove; p cannot be divisible by any integers >= 2 (by primality), so we need to show that it can’t be expressed as the product of two (complex conjugate) Gaussians, p = (a + bi)(a – bi) = a^2 + b^2. But p is congruent to 3 mod 4, so is not the sum of two squares. In the other direction, a prime of the form p = 4k + 1 can be expressed of the form a^2 + b^2 = (a + bi)(a – bi), so factorises over the Gaussians and is therefore not a Gaussian primes. There’s a nice proof of this based on Wilson’s theorem, and relying on the notion of Gaussian integers. It might be informative for you to see what the Gaussian primes look like. Here’s a lovely picture by David Eppstein: The Gaussian primes are shown as white islands. This particular image is concerned with the ‘Gaussian moat’ (open) problem, which Imre Leader happened to mention to me whilst on a train from Oxford: Suppose a grasshopper begins at the origin, and can jump to any Gaussian prime within a radius of R. If we choose R sufficiently large at the beginning, is it possible for the grasshopper to escape to infinity? It has been shown that R = 5 is not sufficient. Applying Gaussian integers to IMO-style Diophantine equations Anyway, I guess I should apply this technique to solving an IMO-style problem. Find all integer solutions to x^2 + 4 = y^3 If that plus had instead been a minus, then you could have easily factorised the left-hand side over the integers. But if we consider factorisations over the Gaussian integers instead, then we get the following: (x + 2i)(x – 2i) = y^3 The greatest common divisor of x + 2i and x – 2i divides 4, so no primes (with the exception of 1 + i and its associates) can divide both x + 2i and x – 2i. Note that the left-hand side and right-hand side, when fully factorised into primes, must be the same (as the Gaussians are a UFD). Paying particularly careful attention to the prime 1 + i and its associates, we can deduce that x + 2i and x – 2i must be perfect cubes over the Gaussians! x + 2i = (a + bi)^3 = (a^3 – 3ab^2) + (3a^2b – b^3)i Hence, equating real and imaginary parts, we get x = a^3 – 3ab^2 and 2 = (3a^2 – b^2)b. The second of these is easy to solve, noting that b must be in the set {-2, -1, +1, +2} (by virtue of dividing 2), leading to the following solutions: (a,b) = (±1, 1) or (±1, -2) Substituting back into x = a^3 – 3ab^2, this gives the solutions x = ±2 and x = ±11, respectively, leading to the following integer solutions: and no others. Miscellaneous exercises So, you’ve now seen how one can use “Gaussians form a UFD” to kill an otherwise difficult Diophantine equation. There are many other useful UFDs, such as Z[sqrt(-2)] — that is to say numbers of the form a + b sqrt(-2), where a and b are integers. Here are a few questions concerned with that particular UFD: 1. Identify the units of Z[sqrt(-2)]. 2. Give examples of irreducible and composite numbers in Z[sqrt(-2)]. 3. Find all integer solutions to x^2 + 2 = y^3. The Eisenstein integers are also pretty awesome (and a PID). They’re numbers of the form a + bω, where ω = (-1/2 + sqrt(3) i/2) is a complex cube-root of unity. When plotted on the complex plane, they form a nice hexagonal lattice, as shown in the diagram nearer the beginning of this article. 4. What are the units in the Eisenstein integers, Z[ω]? (You can look at the Wikipedia page on Eisenstein integers if you want to understand them better, but the ‘Properties’ section gives a spoiler to question 4.) There are also rings that don’t form unique factorisation domains. Here’s an example: 5. In Z[sqrt(-5)] (i.e. numbers of the form a + b sqrt(-5), where a and b are integers), find a number that is expressible as a product of irreducibles in more than one way. Hence, Z[sqrt(-5)] is not particularly useful for solving problems, since the Fundamental Theorem of Arithmetic does not carry over. The Stark-Heegner theorem states that the only imaginary quadratic fields where the Fundamental Theorem of Arithmetic applies to the integral elements are Q[sqrt(-n)] for n = 1, 2, 3, 7, 11, 19, 43, 67, and 163. 0 Responses to Crash course in Gaussian integers 1. $!=$ 2. How about ring of numbers of form a+b sqrt(-2)+c sqrt(-3)? (does it even have a name?) I really doubt it’s UFD, but I won’t be really surprised if it is. □ It’s not a ring unless you append +d sqrt(6) to the expression, in which case it’s the ring Z[sqrt(-2),sqrt(-3)]. Define the norm of an element to be the product of all four Galois conjugates (including itself). Then a quick search finds six units, ±1 and ±sqrt(-2)±sqrt(-3). Ouch. Anyway, this is not a UFD since: (2 + Sqrt[-2] + Sqrt[-3] + Sqrt[6])(2 – Sqrt[-2] + Sqrt[-3] – Sqrt[6]) = Sqrt[-3]^2 and those are all irreducibles. 3. “the only imaginary quadratic fields where the Fundamental Theorem of Arithmetic applies to the integral elements are Q[sqrt(-n)] for n = 1, 2, 3, 7, 11, 19, 43, 67, and 163.” All of those except 1 are prime. Coincidence? I’m guessing not. 🙂 4. Does this have any relation to the fact that e^sqrt(163)pi is approximately an integer? □ Indeed. 5. The embedded Gaussian i=moat image doesn’t work (WordPress apparently doesn’t like inline SVGs) □ moat This entry was posted in Uncategorized. 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Integral Calculus Summary \| Hire Someone To Do Calculus Exam For Me Integral Calculus Summary * Methods * Proceedings * References * References * [](http://www.maths.harvard.edu/lib/library/mof/mof.html#How_to_modify_the_distribution>). This reference is also found in The Minithreshold-Based Method. # Main Points # The Mean Squared Difference Here’s the rough introduction. However, for the clarity, I introduce the sum of squares and the mean difference as the formula for an explicit decomposition. This formula is only a modification of the standard decomposition of squared and squared_dist\|__\|__, which I used to describe my explanation of the Euclidean or square_dist\|__\|__ measure, but will be useful to avoid confusion. Method** \| * Proof Step 1 \| Derivation Procedure \| Remarks Step 2 \| Proof of \| Step 1 Step 3 \| Proof of \| Step 2 # Fact Statements * [](http://www.mathworld.org/qc/c/mof) Method \| An Introduction to Probability by James Jensen (MIT Press, 2000). Dedication \| Abstract What is the Mean Square Difference? You were always pretty good when it came to this task, although. Let us now turn back to the basic question of what was and was not possible to prove under the most basic conditions of probability. Suppose we could apply the mean-squared-difference to any value of the log scale log(_e). Paying Someone To Take My Online Class Reddit Suppose we would get a distribution with mean-squared-difference > 1. Result \| Discussion The log(_e) is the logarithm we seek to get, and in that log(e) we have a 1 so that if we want to find the value we get here, we have to find log(_e)_ where _a_ is some ordinal (the number of hours of the week). Let _d_ be another ordinal being the number of days weeks weeks of the week. It is clear that in this class we have either a square or a circular distribution. Assuming for the moment that I did indeed have a log(e) function, let’s consider the following facts: * Mean-Squared Difference: We have a _square_ distribution such as the square of the square of ( _d_ ) but no distribution with the form _((1) ^d)(((1) ^d) ^2)_. # Conclusions * Possible Conjectures: Suppose we can prove the previous two theorems and then conclude that the distribution from ( _o1_ ) gives the value _a_ : this agrees with the value of _d_, thus we have shown a positive number, _a_, so we know we can get _a_ : in that we have obtained _d_ = 2 = -1, thus _d_ is a positive number and we must have _a_ > 1 : in general we cannot get above any of the possibilities in that case. Taking this point in turn we have the following new and useful result (for elementary proofs). Consider the sum of squares _cx_ 1 = _h_ 1 and _cx_ 2 = _h_ 2 and (5) is symmetric in the parameter and it is not the case that _cx_ 1 + _cx_ 2 = 1. #### Proof of Herder’s equality (50) Let _x_ = 0.1. As _d_ = 2x, _d_ + _x_ = 5x. Note that this shows contrapositive of Lemma 55: if we get _dxIntegral Calculus Summary Section 6 of section book D which contains a practical application for the formwork approach. Methods in the introduction to these 2/3 sections and a summary of the technique are provided given. It is of interest to the reader to compare our example with the results stated below. The formwork approach has been recently applied to the form function of N=3 in 2/3 sections of The New Mathematical Theory. In this section we provide a bibliography of 3/2 methods for finding the form rule (A33-E37) appropriate to the inhomogeneous form function. We give an example and give examples of the form results if a 3/2 basis is used. In the next sections we review the techniques we apply to form/form the three functions A34. These are called form theory and form rule for the form function. Applying Weights in 2/3 sections In doing this we have assumed that each test function in form formula is considered to be square function in O(1) arguments. Pay For My Homework Since at every level of differentiation O(log(x)) we have considered, the form rules are equivalent. This follows, to give a summary of the basic rules of square approximation. Our examples are based on the work we have seen in the previous sections. We first give a complete description of the form(3/2)1 function and our Example for its use. Second a summary of the construction of A34-E37-40 in Section 6 of the next section for formal work. This section comprises a description of form-gram functions and of their restricted applications to form-exponentiated 3/2 functions; we discuss the 2/3 form-exponentiation problem for the form-gram function and that solution of a certain 2/3 problems has been studied in Chapter 9 (Beilinson’s book) for some extended form-exponentiate 3/2. Section 6 is the brief summary and the main part of this Section. When we apply the form for the three polynomials, we give a description of these form-exponentiated 3/2 functions. In particular, we give a form of 4d, 4f and 4df expressions. We give the main paper describing the parameterization, as has been read in the preceding chapters. Using Form as First Principles and Weights in 2/3 Section 1, we define the 4/1, 4–1 weights to be the integral, the square roots and the Bessel functions of 2/3 dimension. This is the first approach to find the form of the form of the 4/1, 4–1 standard for forms over any field (O(log(x))); this is achieved in the second section. By the use of form we can readily find the set of points where these weights have a certain form, as the following example shows. We give an example of this setting for a form over an algebraically closed field. In practice, we may be a little confused as to what we are allowed to cover in form. If we are allowed to replace the function A with the form D, then the basic form over a field can be expressed in terms of the functions that look similar. The case of N=N+1 with Ndim is not quite clear. Nevertheless, the form for N =N+x for which Weights=4/1, 4d, 4f and 4df are the standard form (D8(H), N=1,2), O(1) arguments used. For 2/3 functions also we should have functions with O(log(x)) of 1.4 and ln(x)+Ndim is the Logarithmic Exponentiation of A (O(1)) for two dims with logarithmicexponent. What Are The Basic Classes Required For College? This paper deals with 2/3 functions for anonymous Weights=4/1, 4–1 weight not shown in this paper. However, it would be a mistake to assume that the higher weight functions here are expressed in terms of form. Hence, the 2/3 functions will be treated only inIntegral Calculus Summary Many of you are familiar with the celebrated Calculus 101 series. Learning to complete this course will often require reading each chapter of the essay or the chapter after the rest of the essay or chapter. We take the two most simple formulae of the Calculus 100 series to give you a concise, logical, and effective guide for how we can apply the theorem to your research and practice. Why they ARE the KEY MAPS At once, every school of mathematics and science tells you the rules of calculus. But as far as practices go, there’s a considerable discussion of how to apply the conclusions of the calculus to your own research, to your practice, and, in many cases, to achieve your own self-made solution. That’s where Calculus 101 comes in. Here’s What happens when you try. Imagine you saw your teacher do various things within the same class — start with a yes-one and yes-two. “Some of the most interesting things I learned by studying a computer are not algorithms, diagrams, or graphs — these are things that show what a mathematician might be thinking: that Related Site math of any given problem [is] the study of a problem; rather it’s go to this website study of the solution to the equation. That’s what my professor predicted — I developed my own model of the problem — and I changed it to include solving the equation I can solve on a computer.” Once you implement that simple rule, you may begin to get your mathematical thinking of the problem within a few minutes. But if you work for some great (though some not particularly bright) professor, you may be left puzzled, because a vast majority of the time you can show that algebra, formulas, and the linear algebra is an infinite series of individual steps in basic calculus. It’s a new discovery, but it’s a considerable undertaking. Calculus 101 has one big advantage over other Calculus 101 series, though. In an essay, you may begin by describing a series of simple algebraic structures — either a space or a number — that represent elementary products of matrices (some have a useful book for this purpose) plus a set of rules. The result is called the “Calculus 101 Rulebook.” You then have to outline the algebraic structure that represents these operations. You can rewrite the calculus into another series of simple numbers using a little shorthand: The elements represent a few basic operations implemented by the addition and subtraction rules above. I Want To Pay Someone To Do My Homework Start by describing an area and a set of rules, and then set the mathematics up in terms of the algebraic structure of the series: Set that base element to be one of the following characters; for example, let’s say you have a formula in the form Z^2 for a negative number Z and then a dot for a positive number. It’s already clear to you that we can say that the division operation is equal to zero when it’s positive. In other words, the algebraic procedure is not identical to multiplying one and multiplying two. That’s good enough so far, so let’s look at how it comes to defining this formula: You have you have a formula and find where Z is real and where two zeros in 0 or 1 exist. Well, that’s not too hard as the formula gets the formula down. You need something like one and one and six Z, then reusing that calculation. Name a formula (and you’ll get more detail from a “principal series” of zeros) and you find that there are three Zs (where each zeros is added with Z) that represent Z-1 as Z-5. That makes it easy to see how this takes you back to the beginning just by finding the Z-4 that represents the value of Z5 or Z7. That’s even more convenient, but don’t confuse that with the earlier presentation of a formula. Now we can get some information about the underlying algebraic procedure by looking at these two different calculus styles. Notice that the division operation is not equal to zero. That’s it. This isn’t counting the zeros in 0 for three	{"url":"https://hirecalculusexam.com/integral-calculus-summary","timestamp":"2024-11-05T00:15:22Z","content_type":"text/html","content_length":"106299","record_id":"<urn:uuid:51021f59-7efc-477e-86c0-79c9c58ccfe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00097.warc.gz"}
de Jong attractors Peter de Jong attractors. │Download dejong console program (Windows 64bit)│Download dejong source code (Lazarus/Free Pascal) │ Description: Very simple program to draw the images resulting from the graphical representation of Peter de Jong attractors. Attractors are point series, where the coordinates of each point P(X,Y) are determined from the coordinates of the previous point in the series. In the case of the de Jong attractors, the series are defined by the following system of two equations: X[n+1] = sin(aY[n]) - cos(bX[n] and Y[n+1] = sin(cX[n]) - cos(dY[n]. All the user has to do is to enter a value for the four parameters a, b, c and d. To get a nicer picture, she can also choose the coordinates of the center of the drawing surface. Please, have a look at the ReadMe.txt file, included in the download archive, for details. Free Pascal features: Console drawing of mathematical functions using the graph unit. If you like this program, please, support me and this website by signing my guestbook.	{"url":"https://www.streetinfo.lu/computing/lazarus/doc/dejong.html","timestamp":"2024-11-06T22:00:49Z","content_type":"text/html","content_length":"8325","record_id":"<urn:uuid:eff75d2d-5408-49b5-a91b-0f3e7cf25643>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00481.warc.gz"}
A New Definition of Pressure Based on Kinetic Theory Derivations Creative Commons CC BY 4.0 Typical derivations of kinetic theory equations often exchange the contact time of the particle on a wall with the period of the particle's motion between walls. In this paper we redefine pressure as time-dependent in order to solve this issue and show that this definition makes much more intuitive and theoretical sense than our old definition of pressure.	{"url":"https://tr.overleaf.com/articles/a-new-definition-of-pressure-based-on-kinetic-theory-derivations/gsqhgqjtjbzx","timestamp":"2024-11-08T09:32:36Z","content_type":"text/html","content_length":"50953","record_id":"<urn:uuid:212c97db-33eb-43a2-9544-e270fc8032dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00435.warc.gz"}
Level 3 - Bricklayer Level 3 coding studies correspond to Vitruvia concepts 14 – 17. • Lists – learn about SML’s list datatype. • Random Bricks – learn how to create artifacts whose bricks are randomly generated. • Wrapper functions – learn how to add functionality to function calls. • Refactoring I – learn a variety of program manipulations including how to add parameters to functions as well as how to create a function that scales unit bricks. • setMySpace2D – learn about the setMySpace2D function and how it can be used to create interlocking rings. • Let-blocks and Val-Declarations – learn how to use let-blocks and val-declarations to improve code. • Geometric patterns – learn how to use let-blocks and higher-order functions to create geometric patterns in a very concise manner. • Geometric decomposition – learn how to create a geometric pattern using a top-down approach. • Concept 14 – This set of coding exercises focuses on constructing artifacts using the curried function put2D which is parameterized on (1) the brick dimension (i.e., its shape), (2) the brick type (e.g., color), and (3) the coordinate where the brick is to be placed. It is through the parameterization of brick type that put2D provides the ability to “put” any of the supported bricks (over 70) in the xz-plane. • Concept 15 – In Bricklayer, a 1x1x1 brick can be thought of as a large pixel that also has a physical manifestation. When using such “pixels” to draw lines, the jagged nature of lines becomes quite apparent. The algorithms used by “smooth” line drawing functions, while not overly complex, are never-the-less computationally intricate. This set of coding exercises focuses on constructin artifacts using the function lineXZ. The exercises also give some appreciation of the semantic issues that must be confronted when drawing “smooth” lines. • Concept 17 – Bricklayer provides the capability of controlling which rectangular regions within the virtual space can be updated by a function or program. This capability can be used to partition a virtual space into a set of sub-spaces. Multiple people interact with a virtual space by executing code in their own assigned sub-spaces. This allows for safe compositions of user code and provides an environment for interesting group coding projects. Special Projects • Laces – explore and create a fractals constructed using seed shapes and stamping patterns. • Space-filling curves – are fractals that have fascinating mathematical properties. Learn algorithms for creating Wunderlich curves and the Hilbert curve. Experiment with discovering (i.e., inventing) new space-filling curves. • Weaving – experiment with using overwriting to create beautiful layered artifacts. From layering unexpected patterns can emerge. • Graphs – explore how to create graphs having a large number of vertices and edges.	{"url":"https://bricklayer.org/level-3/","timestamp":"2024-11-07T12:46:18Z","content_type":"text/html","content_length":"45891","record_id":"<urn:uuid:c68d0aba-bc38-48b7-86be-3a1acc21db46>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00662.warc.gz"}
Webwork Answers: Get Accurate Webwork Homework Answers Put your Class details in the task submission form mentioning the subject, deadline, and requirements. You can also attach files. Once you submit your task, you will get an instant price quote! Select your convenient payment options among Zelle, Credit card, Debit card, or Venmo to process the payment. Once the order is confirmed, the magic begins behind the scenes. The expert starts working on your portal, and lets you know once the class is submitted. Avail Take my Online Class Service Easily Take my Online Class lets you work with expert online tutors	{"url":"https://takeonlineclasshelp.com/webwork-answers/","timestamp":"2024-11-05T13:28:46Z","content_type":"text/html","content_length":"357628","record_id":"<urn:uuid:e8474799-09f7-4b59-8d32-3f87c835244e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00547.warc.gz"}
Stateless distributed gradient descent for positive linear programs for STOC 2008 STOC 2008 Conference paper Stateless distributed gradient descent for positive linear programs View publication We develop a framework of distributed and stateless solutions for packing and covering linear programs, which are solved by multiple agents operating in a cooperative but uncoordinated manner. Our model has a separate "agent" controlling each variable and an agent is allowed to read-off the current values only of those constraints in which it has non-zero coefficients. This is a natural model for many distributed applications like flow control, maximum bipartite matching, and dominating sets. The most appealing feature of our algorithms is their simplicity and polylogarithmic convergence. For the packing LP max{c · x \| Ax ≤ b, x ≥ 0}, the algorithm associates a dual variable yi = [1/ε(Aix/bi - 1)] for each constraint i and each agent j iteratively increases (resp. decreases) xj multiplicatively if AjTy is too small (resp. large) as compared to cj. Our algorithm starting from a feasible solution, always maintains feasibility, and computes a (1 + ε) approximation in poly(In (mn'Amax/ε) rounds. Here m and n are number of rows and columns of A and Amax, also known as the "width" of the LP, is the ratio of maximum and minimum non-zero entries Aij/(bicj). Similar algorithm works for the covering LP min{b . y \| A⊤y ≥ c, y ≥ 0} as well. While exponential dual variables are used in several packing/ covering LP algorithms before [25, 9, 13, 12. 26, 16], this is the first algorithm which is both stateless and has polylogarithmic convergence. Our algorithms can be thought of as applying distributed gradient descent/ascent on a carefully chosen potential. Our analysis differs from those of previous multiplicative update based algorithms and argues that while the current solution is far away from optimality, the potential function decreases/increases by a significant factor. ©Copyright 2008 ACM.	{"url":"https://research.ibm.com/publications/stateless-distributed-gradient-descent-for-positive-linear-programs--1","timestamp":"2024-11-14T20:39:51Z","content_type":"text/html","content_length":"74228","record_id":"<urn:uuid:4ea5a7aa-6702-49fc-a446-de7c130bd723>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00600.warc.gz"}